LMFDB - Embedded newform 2366.2.a.bd.1.3

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)

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")

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);

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 = [3,3,2,3,0] 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:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.24698$ 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)

$f(q)$	$=$	\(q+1.00000 q^{2} +2.24698 q^{3} +1.00000 q^{4} +1.69202 q^{5} +2.24698 q^{6} -1.00000 q^{7} +1.00000 q^{8} +2.04892 q^{9} +1.69202 q^{10} +0.445042 q^{11} +2.24698 q^{12} -1.00000 q^{14} +3.80194 q^{15} +1.00000 q^{16} -2.15883 q^{17} +2.04892 q^{18} +6.35690 q^{19} +1.69202 q^{20} -2.24698 q^{21} +0.445042 q^{22} -0.911854 q^{23} +2.24698 q^{24} -2.13706 q^{25} -2.13706 q^{27} -1.00000 q^{28} +3.58211 q^{29} +3.80194 q^{30} +8.89977 q^{31} +1.00000 q^{32} +1.00000 q^{33} -2.15883 q^{34} -1.69202 q^{35} +2.04892 q^{36} +10.8019 q^{37} +6.35690 q^{38} +1.69202 q^{40} +2.41789 q^{41} -2.24698 q^{42} +4.63102 q^{43} +0.445042 q^{44} +3.46681 q^{45} -0.911854 q^{46} -9.75063 q^{47} +2.24698 q^{48} +1.00000 q^{49} -2.13706 q^{50} -4.85086 q^{51} -8.74094 q^{53} -2.13706 q^{54} +0.753020 q^{55} -1.00000 q^{56} +14.2838 q^{57} +3.58211 q^{58} -10.1468 q^{59} +3.80194 q^{60} +1.37867 q^{61} +8.89977 q^{62} -2.04892 q^{63} +1.00000 q^{64} +1.00000 q^{66} +6.23490 q^{67} -2.15883 q^{68} -2.04892 q^{69} -1.69202 q^{70} -5.76271 q^{71} +2.04892 q^{72} -9.93661 q^{73} +10.8019 q^{74} -4.80194 q^{75} +6.35690 q^{76} -0.445042 q^{77} -6.30127 q^{79} +1.69202 q^{80} -10.9487 q^{81} +2.41789 q^{82} -2.91185 q^{83} -2.24698 q^{84} -3.65279 q^{85} +4.63102 q^{86} +8.04892 q^{87} +0.445042 q^{88} +18.1075 q^{89} +3.46681 q^{90} -0.911854 q^{92} +19.9976 q^{93} -9.75063 q^{94} +10.7560 q^{95} +2.24698 q^{96} -3.30798 q^{97} +1.00000 q^{98} +0.911854 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 2 q^{3} + 3 q^{4} + 2 q^{6} - 3 q^{7} + 3 q^{8} - 3 q^{9} + q^{11} + 2 q^{12} - 3 q^{14} + 7 q^{15} + 3 q^{16} + 2 q^{17} - 3 q^{18} + 15 q^{19} - 2 q^{21} + q^{22} + q^{23} + 2 q^{24}+ \cdots - q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 + 2 * q^3 + 3 * q^4 + 2 * q^6 - 3 * q^7 + 3 * q^8 - 3 * q^9 + q^11 + 2 * q^12 - 3 * q^14 + 7 * q^15 + 3 * q^16 + 2 * q^17 - 3 * q^18 + 15 * q^19 - 2 * q^21 + q^22 + q^23 + 2 * q^24 - q^25 - q^27 - 3 * q^28 + 5 * q^29 + 7 * q^30 + 4 * q^31 + 3 * q^32 + 3 * q^33 + 2 * q^34 - 3 * q^36 + 28 * q^37 + 15 * q^38 + 13 * q^41 - 2 * q^42 - q^43 + q^44 + 7 * q^45 + q^46 + 7 * q^47 + 2 * q^48 + 3 * q^49 - q^50 - q^51 - 12 * q^53 - q^54 + 7 * q^55 - 3 * q^56 + 10 * q^57 + 5 * q^58 - 3 * q^59 + 7 * q^60 - 3 * q^61 + 4 * q^62 + 3 * q^63 + 3 * q^64 + 3 * q^66 - 5 * q^67 + 2 * q^68 + 3 * q^69 - 3 * q^72 + 21 * q^73 + 28 * q^74 - 10 * q^75 + 15 * q^76 - q^77 - 2 * q^79 - q^81 + 13 * q^82 - 5 * q^83 - 2 * q^84 + 7 * q^85 - q^86 + 15 * q^87 + q^88 + 14 * q^89 + 7 * q^90 + q^92 + 19 * q^93 + 7 * q^94 - 7 * q^95 + 2 * q^96 - 15 * q^97 + 3 * q^98 - 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
$3$	2.24698	1.29729	0.648647	−	0.761089i	$-0.275335\pi$
$3$	2.24698	1.29729	0.648647	+	0.761089i	$0.275335\pi$
$4$	1.00000	0.500000
$5$	1.69202	0.756695	0.378348	−	0.925664i	$-0.376492\pi$
$5$	1.69202	0.756695	0.378348	+	0.925664i	$0.376492\pi$
$6$	2.24698	0.917326
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	2.04892	0.682972
$10$	1.69202	0.535064
$11$	0.445042	0.134185	0.0670926	−	0.997747i	$-0.478628\pi$
$11$	0.445042	0.134185	0.0670926	+	0.997747i	$0.478628\pi$
$12$	2.24698	0.648647
$13$	0	0
$13$	0	0
$14$	−1.00000	−0.267261
$15$	3.80194	0.981656
$16$	1.00000	0.250000
$17$	−2.15883	−0.523594	−0.261797	−	0.965123i	$-0.584315\pi$
$17$	−2.15883	−0.523594	−0.261797	+	0.965123i	$0.584315\pi$
$18$	2.04892	0.482934
$19$	6.35690	1.45837	0.729186	−	0.684316i	$-0.239899\pi$
$19$	6.35690	1.45837	0.729186	+	0.684316i	$0.239899\pi$
$20$	1.69202	0.378348
$21$	−2.24698	−0.490331
$22$	0.445042	0.0948832
$23$	−0.911854	−0.190135	−0.0950674	−	0.995471i	$-0.530307\pi$
$23$	−0.911854	−0.190135	−0.0950674	+	0.995471i	$0.530307\pi$
$24$	2.24698	0.458663
$25$	−2.13706	−0.427413
$26$	0	0
$27$	−2.13706	−0.411278
$28$	−1.00000	−0.188982
$29$	3.58211	0.665180	0.332590	−	0.943071i	$-0.392077\pi$
$29$	3.58211	0.665180	0.332590	+	0.943071i	$0.392077\pi$
$30$	3.80194	0.694136
$31$	8.89977	1.59845	0.799223	−	0.601034i	$-0.205245\pi$
$31$	8.89977	1.59845	0.799223	+	0.601034i	$0.205245\pi$
$32$	1.00000	0.176777
$33$	1.00000	0.174078
$34$	−2.15883	−0.370237
$35$	−1.69202	−0.286004
$36$	2.04892	0.341486
$37$	10.8019	1.77583	0.887914	−	0.460010i	$-0.152154\pi$
$37$	10.8019	1.77583	0.887914	+	0.460010i	$0.152154\pi$
$38$	6.35690	1.03122
$39$	0	0
$40$	1.69202	0.267532
$41$	2.41789	0.377612	0.188806	−	0.982014i	$-0.439538\pi$
$41$	2.41789	0.377612	0.188806	+	0.982014i	$0.439538\pi$
$42$	−2.24698	−0.346716
$43$	4.63102	0.706224	0.353112	−	0.935581i	$-0.385123\pi$
$43$	4.63102	0.706224	0.353112	+	0.935581i	$0.385123\pi$
$44$	0.445042	0.0670926
$45$	3.46681	0.516802
$46$	−0.911854	−0.134446
$47$	−9.75063	−1.42228	−0.711138	−	0.703053i	$-0.751820\pi$
$47$	−9.75063	−1.42228	−0.711138	+	0.703053i	$0.751820\pi$
$48$	2.24698	0.324324
$49$	1.00000	0.142857
$50$	−2.13706	−0.302226
$51$	−4.85086	−0.679256
$52$	0	0
$53$	−8.74094	−1.20066	−0.600330	−	0.799752i	$-0.704964\pi$
$53$	−8.74094	−1.20066	−0.600330	+	0.799752i	$0.704964\pi$
$54$	−2.13706	−0.290817
$55$	0.753020	0.101537
$56$	−1.00000	−0.133631
$57$	14.2838	1.89194
$58$	3.58211	0.470353
$59$	−10.1468	−1.32099	−0.660497	−	0.750828i	$-0.729655\pi$
$59$	−10.1468	−1.32099	−0.660497	+	0.750828i	$0.729655\pi$
$60$	3.80194	0.490828
$61$	1.37867	0.176520	0.0882601	−	0.996097i	$-0.471869\pi$
$61$	1.37867	0.176520	0.0882601	+	0.996097i	$0.471869\pi$
$62$	8.89977	1.13027
$63$	−2.04892	−0.258139
$64$	1.00000	0.125000
$65$	0	0
$66$	1.00000	0.123091
$67$	6.23490	0.761714	0.380857	−	0.924634i	$-0.375629\pi$
$67$	6.23490	0.761714	0.380857	+	0.924634i	$0.375629\pi$
$68$	−2.15883	−0.261797
$69$	−2.04892	−0.246661
$70$	−1.69202	−0.202235
$71$	−5.76271	−0.683908	−0.341954	−	0.939717i	$-0.611089\pi$
$71$	−5.76271	−0.683908	−0.341954	+	0.939717i	$0.611089\pi$
$72$	2.04892	0.241467
$73$	−9.93661	−1.16299	−0.581496	−	0.813549i	$-0.697532\pi$
$73$	−9.93661	−1.16299	−0.581496	+	0.813549i	$0.697532\pi$
$74$	10.8019	1.25570
$75$	−4.80194	−0.554480
$76$	6.35690	0.729186
$77$	−0.445042	−0.0507172
$78$	0	0
$79$	−6.30127	−0.708949	−0.354474	−	0.935066i	$-0.615340\pi$
$79$	−6.30127	−0.708949	−0.354474	+	0.935066i	$0.615340\pi$
$80$	1.69202	0.189174
$81$	−10.9487	−1.21652
$82$	2.41789	0.267012
$83$	−2.91185	−0.319617	−0.159809	−	0.987148i	$-0.551088\pi$
$83$	−2.91185	−0.319617	−0.159809	+	0.987148i	$0.551088\pi$
$84$	−2.24698	−0.245166
$85$	−3.65279	−0.396201
$86$	4.63102	0.499376
$87$	8.04892	0.862935
$88$	0.445042	0.0474416
$89$	18.1075	1.91939	0.959697	−	0.281037i	$-0.0906785\pi$
$89$	18.1075	1.91939	0.959697	+	0.281037i	$0.0906785\pi$
$90$	3.46681	0.365434
$91$	0	0
$92$	−0.911854	−0.0950674
$93$	19.9976	2.07366
$94$	−9.75063	−1.00570
$95$	10.7560	1.10354
$96$	2.24698	0.229331
$97$	−3.30798	−0.335874	−0.167937	−	0.985798i	$-0.553711\pi$
$97$	−3.30798	−0.335874	−0.167937	+	0.985798i	$0.553711\pi$
$98$	1.00000	0.101015
$99$	0.911854	0.0916448
$100$	−2.13706	−0.213706
$101$	2.08815	0.207778	0.103889	−	0.994589i	$-0.466871\pi$
$101$	2.08815	0.207778	0.103889	+	0.994589i	$0.466871\pi$
$102$	−4.85086	−0.480306
$103$	−11.3013	−1.11355	−0.556774	−	0.830664i	$-0.687961\pi$
$103$	−11.3013	−1.11355	−0.556774	+	0.830664i	$0.687961\pi$
$104$	0	0
$105$	−3.80194	−0.371031
$106$	−8.74094	−0.848995
$107$	−7.13467	−0.689735	−0.344867	−	0.938651i	$-0.612076\pi$
$107$	−7.13467	−0.689735	−0.344867	+	0.938651i	$0.612076\pi$
$108$	−2.13706	−0.205639
$109$	−6.57002	−0.629294	−0.314647	−	0.949209i	$-0.601886\pi$
$109$	−6.57002	−0.629294	−0.314647	+	0.949209i	$0.601886\pi$
$110$	0.753020	0.0717977
$111$	24.2717	2.30377
$112$	−1.00000	−0.0944911
$113$	10.6015	0.997304	0.498652	−	0.866802i	$-0.333829\pi$
$113$	10.6015	0.997304	0.498652	+	0.866802i	$0.333829\pi$
$114$	14.2838	1.33780
$115$	−1.54288	−0.143874
$116$	3.58211	0.332590
$117$	0	0
$118$	−10.1468	−0.934084
$119$	2.15883	0.197900
$120$	3.80194	0.347068
$121$	−10.8019	−0.981994
$122$	1.37867	0.124819
$123$	5.43296	0.489874
$124$	8.89977	0.799223
$125$	−12.0761	−1.08012
$126$	−2.04892	−0.182532
$127$	−21.5646	−1.91355	−0.956776	−	0.290824i	$-0.906070\pi$
$127$	−21.5646	−1.91355	−0.956776	+	0.290824i	$0.906070\pi$
$128$	1.00000	0.0883883
$129$	10.4058	0.916181
$130$	0	0
$131$	3.14675	0.274933	0.137467	−	0.990506i	$-0.456104\pi$
$131$	3.14675	0.274933	0.137467	+	0.990506i	$0.456104\pi$
$132$	1.00000	0.0870388
$133$	−6.35690	−0.551213
$134$	6.23490	0.538613
$135$	−3.61596	−0.311212
$136$	−2.15883	−0.185118
$137$	−3.93362	−0.336072	−0.168036	−	0.985781i	$-0.553743\pi$
$137$	−3.93362	−0.336072	−0.168036	+	0.985781i	$0.553743\pi$
$138$	−2.04892	−0.174415
$139$	−11.2078	−0.950629	−0.475315	−	0.879816i	$-0.657666\pi$
$139$	−11.2078	−0.950629	−0.475315	+	0.879816i	$0.657666\pi$
$140$	−1.69202	−0.143002
$141$	−21.9095	−1.84511
$142$	−5.76271	−0.483596
$143$	0	0
$144$	2.04892	0.170743
$145$	6.06100	0.503339
$146$	−9.93661	−0.822360
$147$	2.24698	0.185328
$148$	10.8019	0.887914
$149$	−21.8726	−1.79188	−0.895938	−	0.444180i	$-0.853495\pi$
$149$	−21.8726	−1.79188	−0.895938	+	0.444180i	$0.853495\pi$
$150$	−4.80194	−0.392077
$151$	18.3002	1.48925	0.744625	−	0.667483i	$-0.232628\pi$
$151$	18.3002	1.48925	0.744625	+	0.667483i	$0.232628\pi$
$152$	6.35690	0.515612
$153$	−4.42327	−0.357600
$154$	−0.445042	−0.0358625
$155$	15.0586	1.20954
$156$	0	0
$157$	−10.8442	−0.865457	−0.432729	−	0.901524i	$-0.642449\pi$
$157$	−10.8442	−0.865457	−0.432729	+	0.901524i	$0.642449\pi$
$158$	−6.30127	−0.501302
$159$	−19.6407	−1.55761
$160$	1.69202	0.133766
$161$	0.911854	0.0718642
$162$	−10.9487	−0.860210
$163$	14.4058	1.12835	0.564175	−	0.825655i	$-0.309194\pi$
$163$	14.4058	1.12835	0.564175	+	0.825655i	$0.309194\pi$
$164$	2.41789	0.188806
$165$	1.69202	0.131724
$166$	−2.91185	−0.226004
$167$	−13.6963	−1.05985	−0.529927	−	0.848043i	$-0.677781\pi$
$167$	−13.6963	−1.05985	−0.529927	+	0.848043i	$0.677781\pi$
$168$	−2.24698	−0.173358
$169$	0	0
$170$	−3.65279	−0.280156
$171$	13.0248	0.996028
$172$	4.63102	0.353112
$173$	−4.47650	−0.340342	−0.170171	−	0.985415i	$-0.554432\pi$
$173$	−4.47650	−0.340342	−0.170171	+	0.985415i	$0.554432\pi$
$174$	8.04892	0.610187
$175$	2.13706	0.161547
$176$	0.445042	0.0335463
$177$	−22.7995	−1.71372
$178$	18.1075	1.35722
$179$	−23.0737	−1.72461	−0.862304	−	0.506392i	$-0.830979\pi$
$179$	−23.0737	−1.72461	−0.862304	+	0.506392i	$0.830979\pi$
$180$	3.46681	0.258401
$181$	8.46980	0.629555	0.314777	−	0.949165i	$-0.398070\pi$
$181$	8.46980	0.629555	0.314777	+	0.949165i	$0.398070\pi$
$182$	0	0
$183$	3.09783	0.228999
$184$	−0.911854	−0.0672228
$185$	18.2771	1.34376
$186$	19.9976	1.46630
$187$	−0.960771	−0.0702586
$188$	−9.75063	−0.711138
$189$	2.13706	0.155448
$190$	10.7560	0.780323
$191$	19.7235	1.42714	0.713570	−	0.700583i	$-0.247077\pi$
$191$	19.7235	1.42714	0.713570	+	0.700583i	$0.247077\pi$
$192$	2.24698	0.162162
$193$	−9.15883	−0.659267	−0.329634	−	0.944109i	$-0.606925\pi$
$193$	−9.15883	−0.659267	−0.329634	+	0.944109i	$0.606925\pi$
$194$	−3.30798	−0.237499
$195$	0	0
$196$	1.00000	0.0714286
$197$	3.73019	0.265765	0.132882	−	0.991132i	$-0.457577\pi$
$197$	3.73019	0.265765	0.132882	+	0.991132i	$0.457577\pi$
$198$	0.911854	0.0648026
$199$	19.5918	1.38883	0.694413	−	0.719577i	$-0.255664\pi$
$199$	19.5918	1.38883	0.694413	+	0.719577i	$0.255664\pi$
$200$	−2.13706	−0.151113
$201$	14.0097	0.988167
$202$	2.08815	0.146921
$203$	−3.58211	−0.251414
$204$	−4.85086	−0.339628
$205$	4.09113	0.285737
$206$	−11.3013	−0.787397
$207$	−1.86831	−0.129857
$208$	0	0
$209$	2.82908	0.195692
$210$	−3.80194	−0.262359
$211$	23.3860	1.60996	0.804978	−	0.593305i	$-0.202177\pi$
$211$	23.3860	1.60996	0.804978	+	0.593305i	$0.202177\pi$
$212$	−8.74094	−0.600330
$213$	−12.9487	−0.887230
$214$	−7.13467	−0.487716
$215$	7.83579	0.534396
$216$	−2.13706	−0.145409
$217$	−8.89977	−0.604156
$218$	−6.57002	−0.444978
$219$	−22.3274	−1.50874
$220$	0.753020	0.0507686
$221$	0	0
$222$	24.2717	1.62901
$223$	−21.7482	−1.45637	−0.728185	−	0.685381i	$-0.759636\pi$
$223$	−21.7482	−1.45637	−0.728185	+	0.685381i	$0.759636\pi$
$224$	−1.00000	−0.0668153
$225$	−4.37867	−0.291911
$226$	10.6015	0.705200
$227$	6.55927	0.435354	0.217677	−	0.976021i	$-0.430152\pi$
$227$	6.55927	0.435354	0.217677	+	0.976021i	$0.430152\pi$
$228$	14.2838	0.945969
$229$	−0.948690	−0.0626912	−0.0313456	−	0.999509i	$-0.509979\pi$
$229$	−0.948690	−0.0626912	−0.0313456	+	0.999509i	$0.509979\pi$
$230$	−1.54288	−0.101734
$231$	−1.00000	−0.0657952
$232$	3.58211	0.235177
$233$	−15.4765	−1.01390	−0.506950	−	0.861976i	$-0.669227\pi$
$233$	−15.4765	−1.01390	−0.506950	+	0.861976i	$0.669227\pi$
$234$	0	0
$235$	−16.4983	−1.07623
$236$	−10.1468	−0.660497
$237$	−14.1588	−0.919715
$238$	2.15883	0.139936
$239$	25.3448	1.63942	0.819710	−	0.572779i	$-0.194135\pi$
$239$	25.3448	1.63942	0.819710	+	0.572779i	$0.194135\pi$
$240$	3.80194	0.245414
$241$	−28.2664	−1.82080	−0.910398	−	0.413733i	$-0.864225\pi$
$241$	−28.2664	−1.82080	−0.910398	+	0.413733i	$0.864225\pi$
$242$	−10.8019	−0.694375
$243$	−18.1903	−1.16691
$244$	1.37867	0.0882601
$245$	1.69202	0.108099
$246$	5.43296	0.346393
$247$	0	0
$248$	8.89977	0.565136
$249$	−6.54288	−0.414638
$250$	−12.0761	−0.763757
$251$	−13.7235	−0.866218	−0.433109	−	0.901341i	$-0.642584\pi$
$251$	−13.7235	−0.866218	−0.433109	+	0.901341i	$0.642584\pi$
$252$	−2.04892	−0.129070
$253$	−0.405813	−0.0255133
$254$	−21.5646	−1.35309
$255$	−8.20775	−0.513989
$256$	1.00000	0.0625000
$257$	−13.3502	−0.832762	−0.416381	−	0.909190i	$-0.636702\pi$
$257$	−13.3502	−0.832762	−0.416381	+	0.909190i	$0.636702\pi$
$258$	10.4058	0.647838
$259$	−10.8019	−0.671200
$260$	0	0
$261$	7.33944	0.454300
$262$	3.14675	0.194407
$263$	19.3153	1.19103	0.595515	−	0.803344i	$-0.296948\pi$
$263$	19.3153	1.19103	0.595515	+	0.803344i	$0.296948\pi$
$264$	1.00000	0.0615457
$265$	−14.7899	−0.908534
$266$	−6.35690	−0.389766
$267$	40.6872	2.49002
$268$	6.23490	0.380857
$269$	24.2935	1.48120	0.740601	−	0.671946i	$-0.234541\pi$
$269$	24.2935	1.48120	0.740601	+	0.671946i	$0.234541\pi$
$270$	−3.61596	−0.220060
$271$	29.0368	1.76386	0.881931	−	0.471378i	$-0.156243\pi$
$271$	29.0368	1.76386	0.881931	+	0.471378i	$0.156243\pi$
$272$	−2.15883	−0.130899
$273$	0	0
$274$	−3.93362	−0.237639
$275$	−0.951083	−0.0573524
$276$	−2.04892	−0.123330
$277$	1.19029	0.0715177	0.0357589	−	0.999360i	$-0.488615\pi$
$277$	1.19029	0.0715177	0.0357589	+	0.999360i	$0.488615\pi$
$278$	−11.2078	−0.672196
$279$	18.2349	1.09169
$280$	−1.69202	−0.101118
$281$	21.2325	1.26663	0.633313	−	0.773896i	$-0.281695\pi$
$281$	21.2325	1.26663	0.633313	+	0.773896i	$0.281695\pi$
$282$	−21.9095	−1.30469
$283$	11.6756	0.694044	0.347022	−	0.937857i	$-0.387193\pi$
$283$	11.6756	0.694044	0.347022	+	0.937857i	$0.387193\pi$
$284$	−5.76271	−0.341954
$285$	24.1685	1.43162
$286$	0	0
$287$	−2.41789	−0.142724
$288$	2.04892	0.120734
$289$	−12.3394	−0.725849
$290$	6.06100	0.355914
$291$	−7.43296	−0.435728
$292$	−9.93661	−0.581496
$293$	14.4101	0.841848	0.420924	−	0.907096i	$-0.361706\pi$
$293$	14.4101	0.841848	0.420924	+	0.907096i	$0.361706\pi$
$294$	2.24698	0.131047
$295$	−17.1685	−0.999590
$296$	10.8019	0.627850
$297$	−0.951083	−0.0551874
$298$	−21.8726	−1.26705
$299$	0	0
$300$	−4.80194	−0.277240
$301$	−4.63102	−0.266928
$302$	18.3002	1.05306
$303$	4.69202	0.269550
$304$	6.35690	0.364593
$305$	2.33273	0.133572
$306$	−4.42327	−0.252862
$307$	2.55257	0.145683	0.0728413	−	0.997344i	$-0.476793\pi$
$307$	2.55257	0.145683	0.0728413	+	0.997344i	$0.476793\pi$
$308$	−0.445042	−0.0253586
$309$	−25.3937	−1.44460
$310$	15.0586	0.855271
$311$	−1.63533	−0.0927313	−0.0463657	−	0.998925i	$-0.514764\pi$
$311$	−1.63533	−0.0927313	−0.0463657	+	0.998925i	$0.514764\pi$
$312$	0	0
$313$	−12.7724	−0.721939	−0.360969	−	0.932578i	$-0.617554\pi$
$313$	−12.7724	−0.721939	−0.360969	+	0.932578i	$0.617554\pi$
$314$	−10.8442	−0.611971
$315$	−3.46681	−0.195333
$316$	−6.30127	−0.354474
$317$	−10.9041	−0.612434	−0.306217	−	0.951962i	$-0.599063\pi$
$317$	−10.9041	−0.612434	−0.306217	+	0.951962i	$0.599063\pi$
$318$	−19.6407	−1.10140
$319$	1.59419	0.0892573
$320$	1.69202	0.0945869
$321$	−16.0315	−0.894789
$322$	0.911854	0.0508156
$323$	−13.7235	−0.763595
$324$	−10.9487	−0.608261
$325$	0	0
$326$	14.4058	0.797864
$327$	−14.7627	−0.816380
$328$	2.41789	0.133506
$329$	9.75063	0.537569
$330$	1.69202	0.0931427
$331$	9.94869	0.546829	0.273415	−	0.961896i	$-0.411847\pi$
$331$	9.94869	0.546829	0.273415	+	0.961896i	$0.411847\pi$
$332$	−2.91185	−0.159809
$333$	22.1323	1.21284
$334$	−13.6963	−0.749430
$335$	10.5496	0.576385
$336$	−2.24698	−0.122583
$337$	8.06829	0.439508	0.219754	−	0.975555i	$-0.429475\pi$
$337$	8.06829	0.439508	0.219754	+	0.975555i	$0.429475\pi$
$338$	0	0
$339$	23.8213	1.29380
$340$	−3.65279	−0.198101
$341$	3.96077	0.214488
$342$	13.0248	0.704298
$343$	−1.00000	−0.0539949
$344$	4.63102	0.249688
$345$	−3.46681	−0.186647
$346$	−4.47650	−0.240658
$347$	−33.1769	−1.78103	−0.890514	−	0.454955i	$-0.849655\pi$
$347$	−33.1769	−1.78103	−0.890514	+	0.454955i	$0.849655\pi$
$348$	8.04892	0.431467
$349$	24.6872	1.32148	0.660739	−	0.750616i	$-0.270243\pi$
$349$	24.6872	1.32148	0.660739	+	0.750616i	$0.270243\pi$
$350$	2.13706	0.114231
$351$	0	0
$352$	0.445042	0.0237208
$353$	22.5754	1.20157	0.600784	−	0.799412i	$-0.294855\pi$
$353$	22.5754	1.20157	0.600784	+	0.799412i	$0.294855\pi$
$354$	−22.7995	−1.21178
$355$	−9.75063	−0.517510
$356$	18.1075	0.959697
$357$	4.85086	0.256734
$358$	−23.0737	−1.21948
$359$	2.30798	0.121810	0.0609052	−	0.998144i	$-0.480601\pi$
$359$	2.30798	0.121810	0.0609052	+	0.998144i	$0.480601\pi$
$360$	3.46681	0.182717
$361$	21.4101	1.12685
$362$	8.46980	0.445163
$363$	−24.2717	−1.27394
$364$	0	0
$365$	−16.8130	−0.880030
$366$	3.09783	0.161926
$367$	25.1685	1.31379	0.656893	−	0.753984i	$-0.271870\pi$
$367$	25.1685	1.31379	0.656893	+	0.753984i	$0.271870\pi$
$368$	−0.911854	−0.0475337
$369$	4.95407	0.257898
$370$	18.2771	0.950182
$371$	8.74094	0.453807
$372$	19.9976	1.03683
$373$	11.6407	0.602733	0.301367	−	0.953508i	$-0.402557\pi$
$373$	11.6407	0.602733	0.301367	+	0.953508i	$0.402557\pi$
$374$	−0.960771	−0.0496803
$375$	−27.1347	−1.40123
$376$	−9.75063	−0.502850
$377$	0	0
$378$	2.13706	0.109919
$379$	−16.5623	−0.850746	−0.425373	−	0.905018i	$-0.639857\pi$
$379$	−16.5623	−0.850746	−0.425373	+	0.905018i	$0.639857\pi$
$380$	10.7560	0.551771
$381$	−48.4553	−2.48244
$382$	19.7235	1.00914
$383$	−7.39506	−0.377870	−0.188935	−	0.981990i	$-0.560504\pi$
$383$	−7.39506	−0.377870	−0.188935	+	0.981990i	$0.560504\pi$
$384$	2.24698	0.114666
$385$	−0.753020	−0.0383775
$386$	−9.15883	−0.466172
$387$	9.48858	0.482332
$388$	−3.30798	−0.167937
$389$	23.4674	1.18984	0.594922	−	0.803783i	$-0.297183\pi$
$389$	23.4674	1.18984	0.594922	+	0.803783i	$0.297183\pi$
$390$	0	0
$391$	1.96854	0.0995534
$392$	1.00000	0.0505076
$393$	7.07069	0.356669
$394$	3.73019	0.187924
$395$	−10.6619	−0.536458
$396$	0.911854	0.0458224
$397$	−0.0499823	−0.00250854	−0.00125427	−	0.999999i	$-0.500399\pi$
$397$	−0.0499823	−0.00250854	−0.00125427	+	0.999999i	$0.500399\pi$
$398$	19.5918	0.982048
$399$	−14.2838	−0.715085
$400$	−2.13706	−0.106853
$401$	4.84010	0.241703	0.120852	−	0.992671i	$-0.461437\pi$
$401$	4.84010	0.241703	0.120852	+	0.992671i	$0.461437\pi$
$402$	14.0097	0.698740
$403$	0	0
$404$	2.08815	0.103889
$405$	−18.5254	−0.920535
$406$	−3.58211	−0.177777
$407$	4.80731	0.238290
$408$	−4.85086	−0.240153
$409$	29.8866	1.47780	0.738899	−	0.673816i	$-0.235346\pi$
$409$	29.8866	1.47780	0.738899	+	0.673816i	$0.235346\pi$
$410$	4.09113	0.202047
$411$	−8.83877	−0.435985
$412$	−11.3013	−0.556774
$413$	10.1468	0.499289
$414$	−1.86831	−0.0918226
$415$	−4.92692	−0.241853
$416$	0	0
$417$	−25.1836	−1.23325
$418$	2.82908	0.138375
$419$	8.13898	0.397615	0.198808	−	0.980039i	$-0.436293\pi$
$419$	8.13898	0.397615	0.198808	+	0.980039i	$0.436293\pi$
$420$	−3.80194	−0.185516
$421$	−17.0398	−0.830470	−0.415235	−	0.909714i	$-0.636301\pi$
$421$	−17.0398	−0.830470	−0.415235	+	0.909714i	$0.636301\pi$
$422$	23.3860	1.13841
$423$	−19.9782	−0.971375
$424$	−8.74094	−0.424498
$425$	4.61356	0.223791
$426$	−12.9487	−0.627366
$427$	−1.37867	−0.0667183
$428$	−7.13467	−0.344867
$429$	0	0
$430$	7.83579	0.377875
$431$	−6.95779	−0.335145	−0.167572	−	0.985860i	$-0.553593\pi$
$431$	−6.95779	−0.335145	−0.167572	+	0.985860i	$0.553593\pi$
$432$	−2.13706	−0.102820
$433$	−6.80731	−0.327139	−0.163569	−	0.986532i	$-0.552301\pi$
$433$	−6.80731	−0.327139	−0.163569	+	0.986532i	$0.552301\pi$
$434$	−8.89977	−0.427203
$435$	13.6189	0.652978
$436$	−6.57002	−0.314647
$437$	−5.79656	−0.277287
$438$	−22.3274	−1.06684
$439$	26.3351	1.25691	0.628453	−	0.777847i	$-0.283688\pi$
$439$	26.3351	1.25691	0.628453	+	0.777847i	$0.283688\pi$
$440$	0.753020	0.0358988
$441$	2.04892	0.0975675
$442$	0	0
$443$	7.32437	0.347991	0.173996	−	0.984746i	$-0.444332\pi$
$443$	7.32437	0.347991	0.173996	+	0.984746i	$0.444332\pi$
$444$	24.2717	1.15189
$445$	30.6383	1.45240
$446$	−21.7482	−1.02981
$447$	−49.1473	−2.32459
$448$	−1.00000	−0.0472456
$449$	−41.4228	−1.95486	−0.977431	−	0.211253i	$-0.932245\pi$
$449$	−41.4228	−1.95486	−0.977431	+	0.211253i	$0.932245\pi$
$450$	−4.37867	−0.206412
$451$	1.07606	0.0506699
$452$	10.6015	0.498652
$453$	41.1202	1.93200
$454$	6.55927	0.307842
$455$	0	0
$456$	14.2838	0.668901
$457$	11.7125	0.547886	0.273943	−	0.961746i	$-0.411672\pi$
$457$	11.7125	0.547886	0.273943	+	0.961746i	$0.411672\pi$
$458$	−0.948690	−0.0443294
$459$	4.61356	0.215343
$460$	−1.54288	−0.0719370
$461$	−20.6461	−0.961584	−0.480792	−	0.876835i	$-0.659651\pi$
$461$	−20.6461	−0.961584	−0.480792	+	0.876835i	$0.659651\pi$
$462$	−1.00000	−0.0465242
$463$	10.9312	0.508017	0.254009	−	0.967202i	$-0.418251\pi$
$463$	10.9312	0.508017	0.254009	+	0.967202i	$0.418251\pi$
$464$	3.58211	0.166295
$465$	33.8364	1.56912
$466$	−15.4765	−0.716935
$467$	15.4450	0.714711	0.357356	−	0.933968i	$-0.383678\pi$
$467$	15.4450	0.714711	0.357356	+	0.933968i	$0.383678\pi$
$468$	0	0
$469$	−6.23490	−0.287901
$470$	−16.4983	−0.761008
$471$	−24.3666	−1.12275
$472$	−10.1468	−0.467042
$473$	2.06100	0.0947648
$474$	−14.1588	−0.650337
$475$	−13.5851	−0.623327
$476$	2.15883	0.0989500
$477$	−17.9095	−0.820018
$478$	25.3448	1.15924
$479$	25.2828	1.15520	0.577599	−	0.816321i	$-0.303990\pi$
$479$	25.2828	1.15520	0.577599	+	0.816321i	$0.303990\pi$
$480$	3.80194	0.173534
$481$	0	0
$482$	−28.2664	−1.28750
$483$	2.04892	0.0932290
$484$	−10.8019	−0.490997
$485$	−5.59717	−0.254154
$486$	−18.1903	−0.825128
$487$	1.01938	0.0461924	0.0230962	−	0.999733i	$-0.492648\pi$
$487$	1.01938	0.0461924	0.0230962	+	0.999733i	$0.492648\pi$
$488$	1.37867	0.0624093
$489$	32.3696	1.46380
$490$	1.69202	0.0764377
$491$	−36.7198	−1.65714	−0.828570	−	0.559886i	$-0.810845\pi$
$491$	−36.7198	−1.65714	−0.828570	+	0.559886i	$0.810845\pi$
$492$	5.43296	0.244937
$493$	−7.73317	−0.348284
$494$	0	0
$495$	1.54288	0.0693471
$496$	8.89977	0.399612
$497$	5.76271	0.258493
$498$	−6.54288	−0.293193
$499$	−41.4685	−1.85638	−0.928192	−	0.372102i	$-0.878637\pi$
$499$	−41.4685	−1.85638	−0.928192	+	0.372102i	$0.878637\pi$
$500$	−12.0761	−0.540058
$501$	−30.7754	−1.37494
$502$	−13.7235	−0.612509
$503$	−2.15346	−0.0960179	−0.0480089	−	0.998847i	$-0.515288\pi$
$503$	−2.15346	−0.0960179	−0.0480089	+	0.998847i	$0.515288\pi$
$504$	−2.04892	−0.0912660
$505$	3.53319	0.157225
$506$	−0.405813	−0.0180406
$507$	0	0
$508$	−21.5646	−0.956776
$509$	−7.62565	−0.338001	−0.169000	−	0.985616i	$-0.554054\pi$
$509$	−7.62565	−0.338001	−0.169000	+	0.985616i	$0.554054\pi$
$510$	−8.20775	−0.363445
$511$	9.93661	0.439570
$512$	1.00000	0.0441942
$513$	−13.5851	−0.599796
$514$	−13.3502	−0.588852
$515$	−19.1220	−0.842616
$516$	10.4058	0.458090
$517$	−4.33944	−0.190848
$518$	−10.8019	−0.474610
$519$	−10.0586	−0.441524
$520$	0	0
$521$	20.8377	0.912917	0.456458	−	0.889745i	$-0.349118\pi$
$521$	20.8377	0.912917	0.456458	+	0.889745i	$0.349118\pi$
$522$	7.33944	0.321238
$523$	38.3491	1.67689	0.838445	−	0.544986i	$-0.183465\pi$
$523$	38.3491	1.67689	0.838445	+	0.544986i	$0.183465\pi$
$524$	3.14675	0.137467
$525$	4.80194	0.209574
$526$	19.3153	0.842186
$527$	−19.2131	−0.836937
$528$	1.00000	0.0435194
$529$	−22.1685	−0.963849
$530$	−14.7899	−0.642430
$531$	−20.7899	−0.902203
$532$	−6.35690	−0.275606
$533$	0	0
$534$	40.6872	1.76071
$535$	−12.0720	−0.521919
$536$	6.23490	0.269307
$537$	−51.8461	−2.23732
$538$	24.2935	1.04737
$539$	0.445042	0.0191693
$540$	−3.61596	−0.155606
$541$	−5.46740	−0.235062	−0.117531	−	0.993069i	$-0.537498\pi$
$541$	−5.46740	−0.235062	−0.117531	+	0.993069i	$0.537498\pi$
$542$	29.0368	1.24724
$543$	19.0315	0.816718
$544$	−2.15883	−0.0925592
$545$	−11.1166	−0.476184
$546$	0	0
$547$	−6.26636	−0.267930	−0.133965	−	0.990986i	$-0.542771\pi$
$547$	−6.26636	−0.267930	−0.133965	+	0.990986i	$0.542771\pi$
$548$	−3.93362	−0.168036
$549$	2.82477	0.120558
$550$	−0.951083	−0.0405543
$551$	22.7711	0.970080
$552$	−2.04892	−0.0872077
$553$	6.30127	0.267957
$554$	1.19029	0.0505707
$555$	41.0683	1.74325
$556$	−11.2078	−0.475315
$557$	−23.1371	−0.980349	−0.490174	−	0.871624i	$-0.663067\pi$
$557$	−23.1371	−0.980349	−0.490174	+	0.871624i	$0.663067\pi$
$558$	18.2349	0.771945
$559$	0	0
$560$	−1.69202	−0.0715010
$561$	−2.15883	−0.0911460
$562$	21.2325	0.895639
$563$	1.21685	0.0512841	0.0256420	−	0.999671i	$-0.491837\pi$
$563$	1.21685	0.0512841	0.0256420	+	0.999671i	$0.491837\pi$
$564$	−21.9095	−0.922555
$565$	17.9379	0.754655
$566$	11.6756	0.490763
$567$	10.9487	0.459802
$568$	−5.76271	−0.241798
$569$	−28.5816	−1.19820	−0.599102	−	0.800673i	$-0.704476\pi$
$569$	−28.5816	−1.19820	−0.599102	+	0.800673i	$0.704476\pi$
$570$	24.1685	1.01231
$571$	−32.1825	−1.34680	−0.673398	−	0.739280i	$-0.735166\pi$
$571$	−32.1825	−1.34680	−0.673398	+	0.739280i	$0.735166\pi$
$572$	0	0
$573$	44.3183	1.85142
$574$	−2.41789	−0.100921
$575$	1.94869	0.0812660
$576$	2.04892	0.0853716
$577$	−2.94810	−0.122731	−0.0613655	−	0.998115i	$-0.519546\pi$
$577$	−2.94810	−0.122731	−0.0613655	+	0.998115i	$0.519546\pi$
$578$	−12.3394	−0.513253
$579$	−20.5797	−0.855264
$580$	6.06100	0.251669
$581$	2.91185	0.120804
$582$	−7.43296	−0.308106
$583$	−3.89008	−0.161111
$584$	−9.93661	−0.411180
$585$	0	0
$586$	14.4101	0.595277
$587$	2.77538	0.114552	0.0572761	−	0.998358i	$-0.481758\pi$
$587$	2.77538	0.114552	0.0572761	+	0.998358i	$0.481758\pi$
$588$	2.24698	0.0926639
$589$	56.5749	2.33113
$590$	−17.1685	−0.706817
$591$	8.38165	0.344775
$592$	10.8019	0.443957
$593$	21.5241	0.883888	0.441944	−	0.897043i	$-0.354289\pi$
$593$	21.5241	0.883888	0.441944	+	0.897043i	$0.354289\pi$
$594$	−0.951083	−0.0390234
$595$	3.65279	0.149750
$596$	−21.8726	−0.895938
$597$	44.0224	1.80172
$598$	0	0
$599$	23.3230	0.952954	0.476477	−	0.879187i	$-0.341914\pi$
$599$	23.3230	0.952954	0.476477	+	0.879187i	$0.341914\pi$
$600$	−4.80194	−0.196038
$601$	15.7429	0.642165	0.321082	−	0.947051i	$-0.395953\pi$
$601$	15.7429	0.642165	0.321082	+	0.947051i	$0.395953\pi$
$602$	−4.63102	−0.188746
$603$	12.7748	0.520230
$604$	18.3002	0.744625
$605$	−18.2771	−0.743070
$606$	4.69202	0.190600
$607$	−19.7845	−0.803027	−0.401514	−	0.915853i	$-0.631516\pi$
$607$	−19.7845	−0.803027	−0.401514	+	0.915853i	$0.631516\pi$
$608$	6.35690	0.257806
$609$	−8.04892	−0.326159
$610$	2.33273	0.0944496
$611$	0	0
$612$	−4.42327	−0.178800
$613$	−7.24996	−0.292823	−0.146412	−	0.989224i	$-0.546772\pi$
$613$	−7.24996	−0.292823	−0.146412	+	0.989224i	$0.546772\pi$
$614$	2.55257	0.103013
$615$	9.19269	0.370685
$616$	−0.445042	−0.0179312
$617$	−36.7415	−1.47916	−0.739579	−	0.673070i	$-0.764975\pi$
$617$	−36.7415	−1.47916	−0.739579	+	0.673070i	$0.764975\pi$
$618$	−25.3937	−1.02149
$619$	−17.4959	−0.703219	−0.351609	−	0.936147i	$-0.614365\pi$
$619$	−17.4959	−0.703219	−0.351609	+	0.936147i	$0.614365\pi$
$620$	15.0586	0.604768
$621$	1.94869	0.0781982
$622$	−1.63533	−0.0655709
$623$	−18.1075	−0.725463
$624$	0	0
$625$	−9.74764	−0.389906
$626$	−12.7724	−0.510488
$627$	6.35690	0.253870
$628$	−10.8442	−0.432729
$629$	−23.3196	−0.929813
$630$	−3.46681	−0.138121
$631$	31.4644	1.25258	0.626289	−	0.779591i	$-0.284573\pi$
$631$	31.4644	1.25258	0.626289	+	0.779591i	$0.284573\pi$
$632$	−6.30127	−0.250651
$633$	52.5478	2.08859
$634$	−10.9041	−0.433057
$635$	−36.4878	−1.44798
$636$	−19.6407	−0.778805
$637$	0	0
$638$	1.59419	0.0631145
$639$	−11.8073	−0.467090
$640$	1.69202	0.0668830
$641$	−24.6799	−0.974799	−0.487400	−	0.873179i	$-0.662054\pi$
$641$	−24.6799	−0.974799	−0.487400	+	0.873179i	$0.662054\pi$
$642$	−16.0315	−0.632711
$643$	−11.1588	−0.440061	−0.220031	−	0.975493i	$-0.570616\pi$
$643$	−11.1588	−0.440061	−0.220031	+	0.975493i	$0.570616\pi$
$644$	0.911854	0.0359321
$645$	17.6069	0.693269
$646$	−13.7235	−0.539943
$647$	14.8791	0.584956	0.292478	−	0.956272i	$-0.405520\pi$
$647$	14.8791	0.584956	0.292478	+	0.956272i	$0.405520\pi$
$648$	−10.9487	−0.430105
$649$	−4.51573	−0.177258
$650$	0	0
$651$	−19.9976	−0.783768
$652$	14.4058	0.564175
$653$	−5.13600	−0.200987	−0.100494	−	0.994938i	$-0.532042\pi$
$653$	−5.13600	−0.200987	−0.100494	+	0.994938i	$0.532042\pi$
$654$	−14.7627	−0.577268
$655$	5.32437	0.208040
$656$	2.41789	0.0944029
$657$	−20.3593	−0.794292
$658$	9.75063	0.380119
$659$	8.42998	0.328385	0.164193	−	0.986428i	$-0.447498\pi$
$659$	8.42998	0.328385	0.164193	+	0.986428i	$0.447498\pi$
$660$	1.69202	0.0658618
$661$	20.3793	0.792661	0.396331	−	0.918108i	$-0.370283\pi$
$661$	20.3793	0.792661	0.396331	+	0.918108i	$0.370283\pi$
$662$	9.94869	0.386667
$663$	0	0
$664$	−2.91185	−0.113002
$665$	−10.7560	−0.417100
$666$	22.1323	0.857608
$667$	−3.26636	−0.126474
$668$	−13.6963	−0.529927
$669$	−48.8678	−1.88934
$670$	10.5496	0.407566
$671$	0.613564	0.0236864
$672$	−2.24698	−0.0866791
$673$	21.6383	0.834096	0.417048	−	0.908884i	$-0.363065\pi$
$673$	21.6383	0.834096	0.417048	+	0.908884i	$0.363065\pi$
$674$	8.06829	0.310779
$675$	4.56704	0.175785
$676$	0	0
$677$	40.5351	1.55789	0.778945	−	0.627092i	$-0.215755\pi$
$677$	40.5351	1.55789	0.778945	+	0.627092i	$0.215755\pi$
$678$	23.8213	0.914852
$679$	3.30798	0.126949
$680$	−3.65279	−0.140078
$681$	14.7385	0.564782
$682$	3.96077	0.151666
$683$	21.6969	0.830210	0.415105	−	0.909774i	$-0.363745\pi$
$683$	21.6969	0.830210	0.415105	+	0.909774i	$0.363745\pi$
$684$	13.0248	0.498014
$685$	−6.65578	−0.254304
$686$	−1.00000	−0.0381802
$687$	−2.13169	−0.0813289
$688$	4.63102	0.176556
$689$	0	0
$690$	−3.46681	−0.131979
$691$	28.0610	1.06749	0.533745	−	0.845645i	$-0.320784\pi$
$691$	28.0610	1.06749	0.533745	+	0.845645i	$0.320784\pi$
$692$	−4.47650	−0.170171
$693$	−0.911854	−0.0346385
$694$	−33.1769	−1.25938
$695$	−18.9638	−0.719336
$696$	8.04892	0.305093
$697$	−5.21983	−0.197715
$698$	24.6872	0.934426
$699$	−34.7754	−1.31533
$700$	2.13706	0.0807734
$701$	−13.7380	−0.518875	−0.259438	−	0.965760i	$-0.583537\pi$
$701$	−13.7380	−0.518875	−0.259438	+	0.965760i	$0.583537\pi$
$702$	0	0
$703$	68.6668	2.58982
$704$	0.445042	0.0167731
$705$	−37.0713	−1.39619
$706$	22.5754	0.849636
$707$	−2.08815	−0.0785328
$708$	−22.7995	−0.856859
$709$	−23.0586	−0.865984	−0.432992	−	0.901398i	$-0.642542\pi$
$709$	−23.0586	−0.865984	−0.432992	+	0.901398i	$0.642542\pi$
$710$	−9.75063	−0.365935
$711$	−12.9108	−0.484192
$712$	18.1075	0.678608
$713$	−8.11529	−0.303920
$714$	4.85086	0.181539
$715$	0	0
$716$	−23.0737	−0.862304
$717$	56.9493	2.12681
$718$	2.30798	0.0861330
$719$	22.8750	0.853094	0.426547	−	0.904465i	$-0.359730\pi$
$719$	22.8750	0.853094	0.426547	+	0.904465i	$0.359730\pi$
$720$	3.46681	0.129200
$721$	11.3013	0.420881
$722$	21.4101	0.796802
$723$	−63.5139	−2.36211
$724$	8.46980	0.314777
$725$	−7.65519	−0.284306
$726$	−24.2717	−0.900809
$727$	9.76377	0.362118	0.181059	−	0.983472i	$-0.442047\pi$
$727$	9.76377	0.362118	0.181059	+	0.983472i	$0.442047\pi$
$728$	0	0
$729$	−8.02715	−0.297302
$730$	−16.8130	−0.622275
$731$	−9.99761	−0.369775
$732$	3.09783	0.114499
$733$	2.65519	0.0980715	0.0490358	−	0.998797i	$-0.484385\pi$
$733$	2.65519	0.0980715	0.0490358	+	0.998797i	$0.484385\pi$
$734$	25.1685	0.928987
$735$	3.80194	0.140237
$736$	−0.911854	−0.0336114
$737$	2.77479	0.102211
$738$	4.95407	0.182362
$739$	0.135144	0.00497137	0.00248568	−	0.999997i	$-0.499209\pi$
$739$	0.135144	0.00497137	0.00248568	+	0.999997i	$0.499209\pi$
$740$	18.2771	0.671880
$741$	0	0
$742$	8.74094	0.320890
$743$	−38.1943	−1.40121	−0.700607	−	0.713547i	$-0.747087\pi$
$743$	−38.1943	−1.40121	−0.700607	+	0.713547i	$0.747087\pi$
$744$	19.9976	0.733148
$745$	−37.0090	−1.35590
$746$	11.6407	0.426197
$747$	−5.96615	−0.218290
$748$	−0.960771	−0.0351293
$749$	7.13467	0.260695
$750$	−27.1347	−0.990818
$751$	31.9614	1.16629	0.583143	−	0.812369i	$-0.301823\pi$
$751$	31.9614	1.16629	0.583143	+	0.812369i	$0.301823\pi$
$752$	−9.75063	−0.355569
$753$	−30.8364	−1.12374
$754$	0	0
$755$	30.9643	1.12691
$756$	2.13706	0.0777242
$757$	−51.4258	−1.86910	−0.934551	−	0.355829i	$-0.884198\pi$
$757$	−51.4258	−1.86910	−0.934551	+	0.355829i	$0.884198\pi$
$758$	−16.5623	−0.601568
$759$	−0.911854	−0.0330982
$760$	10.7560	0.390161
$761$	29.4782	1.06858	0.534291	−	0.845301i	$-0.320579\pi$
$761$	29.4782	1.06858	0.534291	+	0.845301i	$0.320579\pi$
$762$	−48.4553	−1.75535
$763$	6.57002	0.237851
$764$	19.7235	0.713570
$765$	−7.48427	−0.270594
$766$	−7.39506	−0.267194
$767$	0	0
$768$	2.24698	0.0810809
$769$	−24.7114	−0.891116	−0.445558	−	0.895253i	$-0.646995\pi$
$769$	−24.7114	−0.891116	−0.445558	+	0.895253i	$0.646995\pi$
$770$	−0.753020	−0.0271370
$771$	−29.9976	−1.08034
$772$	−9.15883	−0.329634
$773$	−22.3032	−0.802190	−0.401095	−	0.916036i	$-0.631370\pi$
$773$	−22.3032	−0.802190	−0.401095	+	0.916036i	$0.631370\pi$
$774$	9.48858	0.341060
$775$	−19.0194	−0.683196
$776$	−3.30798	−0.118750
$777$	−24.2717	−0.870744
$778$	23.4674	0.841347
$779$	15.3703	0.550698
$780$	0	0
$781$	−2.56465	−0.0917703
$782$	1.96854	0.0703949
$783$	−7.65519	−0.273574
$784$	1.00000	0.0357143
$785$	−18.3485	−0.654887
$786$	7.07069	0.252203
$787$	−16.4896	−0.587792	−0.293896	−	0.955837i	$-0.594952\pi$
$787$	−16.4896	−0.587792	−0.293896	+	0.955837i	$0.594952\pi$
$788$	3.73019	0.132882
$789$	43.4010	1.54512
$790$	−10.6619	−0.379333
$791$	−10.6015	−0.376945
$792$	0.911854	0.0324013
$793$	0	0
$794$	−0.0499823	−0.00177380
$795$	−33.2325	−1.17864
$796$	19.5918	0.694413
$797$	−1.00106	−0.0354595	−0.0177298	−	0.999843i	$-0.505644\pi$
$797$	−1.00106	−0.0354595	−0.0177298	+	0.999843i	$0.505644\pi$
$798$	−14.2838	−0.505642
$799$	21.0500	0.744695
$800$	−2.13706	−0.0755566
$801$	37.1008	1.31089
$802$	4.84010	0.170910
$803$	−4.42221	−0.156056
$804$	14.0097	0.494084
$805$	1.54288	0.0543793
$806$	0	0
$807$	54.5870	1.92155
$808$	2.08815	0.0734607
$809$	−49.9172	−1.75500	−0.877498	−	0.479580i	$-0.840789\pi$
$809$	−49.9172	−1.75500	−0.877498	+	0.479580i	$0.840789\pi$
$810$	−18.5254	−0.650917
$811$	32.9554	1.15722	0.578610	−	0.815604i	$-0.303595\pi$
$811$	32.9554	1.15722	0.578610	+	0.815604i	$0.303595\pi$
$812$	−3.58211	−0.125707
$813$	65.2452	2.28825
$814$	4.80731	0.168496
$815$	24.3749	0.853817
$816$	−4.85086	−0.169814
$817$	29.4389	1.02994
$818$	29.8866	1.04496
$819$	0	0
$820$	4.09113	0.142868
$821$	29.3129	1.02303	0.511513	−	0.859275i	$-0.329085\pi$
$821$	29.3129	1.02303	0.511513	+	0.859275i	$0.329085\pi$
$822$	−8.83877	−0.308288
$823$	−11.3013	−0.393938	−0.196969	−	0.980410i	$-0.563110\pi$
$823$	−11.3013	−0.393938	−0.196969	+	0.980410i	$0.563110\pi$
$824$	−11.3013	−0.393699
$825$	−2.13706	−0.0744030
$826$	10.1468	0.353051
$827$	−32.8568	−1.14254	−0.571272	−	0.820761i	$-0.693550\pi$
$827$	−32.8568	−1.14254	−0.571272	+	0.820761i	$0.693550\pi$
$828$	−1.86831	−0.0649284
$829$	36.6853	1.27413	0.637067	−	0.770809i	$-0.280148\pi$
$829$	36.6853	1.27413	0.637067	+	0.770809i	$0.280148\pi$
$830$	−4.92692	−0.171016
$831$	2.67456	0.0927796
$832$	0	0
$833$	−2.15883	−0.0747992
$834$	−25.1836	−0.872036
$835$	−23.1745	−0.801986
$836$	2.82908	0.0978459
$837$	−19.0194	−0.657406
$838$	8.13898	0.281156
$839$	−23.7687	−0.820586	−0.410293	−	0.911954i	$-0.634574\pi$
$839$	−23.7687	−0.820586	−0.410293	+	0.911954i	$0.634574\pi$
$840$	−3.80194	−0.131179
$841$	−16.1685	−0.557535
$842$	−17.0398	−0.587231
$843$	47.7090	1.64319
$844$	23.3860	0.804978
$845$	0	0
$846$	−19.9782	−0.686866
$847$	10.8019	0.371159
$848$	−8.74094	−0.300165
$849$	26.2349	0.900379
$850$	4.61356	0.158244
$851$	−9.84979	−0.337646
$852$	−12.9487	−0.443615
$853$	33.6437	1.15194	0.575969	−	0.817471i	$-0.304625\pi$
$853$	33.6437	1.15194	0.575969	+	0.817471i	$0.304625\pi$
$854$	−1.37867	−0.0471770
$855$	22.0382	0.753689
$856$	−7.13467	−0.243858
$857$	−35.2646	−1.20461	−0.602307	−	0.798264i	$-0.705752\pi$
$857$	−35.2646	−1.20461	−0.602307	+	0.798264i	$0.705752\pi$
$858$	0	0
$859$	9.80433	0.334519	0.167260	−	0.985913i	$-0.446508\pi$
$859$	9.80433	0.334519	0.167260	+	0.985913i	$0.446508\pi$
$860$	7.83579	0.267198
$861$	−5.43296	−0.185155
$862$	−6.95779	−0.236983
$863$	34.2586	1.16618	0.583088	−	0.812409i	$-0.301844\pi$
$863$	34.2586	1.16618	0.583088	+	0.812409i	$0.301844\pi$
$864$	−2.13706	−0.0727044
$865$	−7.57434	−0.257535
$866$	−6.80731	−0.231322
$867$	−27.7265	−0.941640
$868$	−8.89977	−0.302078
$869$	−2.80433	−0.0951304
$870$	13.6189	0.461725
$871$	0	0
$872$	−6.57002	−0.222489
$873$	−6.77777	−0.229393
$874$	−5.79656	−0.196072
$875$	12.0761	0.408245
$876$	−22.3274	−0.754371
$877$	−28.5174	−0.962964	−0.481482	−	0.876456i	$-0.659901\pi$
$877$	−28.5174	−0.962964	−0.481482	+	0.876456i	$0.659901\pi$
$878$	26.3351	0.888767
$879$	32.3793	1.09213
$880$	0.753020	0.0253843
$881$	−55.3193	−1.86376	−0.931878	−	0.362773i	$-0.881830\pi$
$881$	−55.3193	−1.86376	−0.931878	+	0.362773i	$0.881830\pi$
$882$	2.04892	0.0689906
$883$	0.815938	0.0274585	0.0137293	−	0.999906i	$-0.495630\pi$
$883$	0.815938	0.0274585	0.0137293	+	0.999906i	$0.495630\pi$
$884$	0	0
$885$	−38.5773	−1.29676
$886$	7.32437	0.246067
$887$	39.8713	1.33875	0.669374	−	0.742926i	$-0.266563\pi$
$887$	39.8713	1.33875	0.669374	+	0.742926i	$0.266563\pi$
$888$	24.2717	0.814506
$889$	21.5646	0.723255
$890$	30.6383	1.02700
$891$	−4.87263	−0.163239
$892$	−21.7482	−0.728185
$893$	−61.9837	−2.07421
$894$	−49.1473	−1.64373
$895$	−39.0411	−1.30500
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−41.4228	−1.38230
$899$	31.8799	1.06325
$900$	−4.37867	−0.145956
$901$	18.8702	0.628659
$902$	1.07606	0.0358290
$903$	−10.4058	−0.346284
$904$	10.6015	0.352600
$905$	14.3311	0.476381
$906$	41.1202	1.36613
$907$	5.54719	0.184191	0.0920957	−	0.995750i	$-0.470643\pi$
$907$	5.54719	0.184191	0.0920957	+	0.995750i	$0.470643\pi$
$908$	6.55927	0.217677
$909$	4.27844	0.141907
$910$	0	0
$911$	12.3357	0.408701	0.204350	−	0.978898i	$-0.434492\pi$
$911$	12.3357	0.408701	0.204350	+	0.978898i	$0.434492\pi$
$912$	14.2838	0.472984
$913$	−1.29590	−0.0428879
$914$	11.7125	0.387414
$915$	5.24160	0.173282
$916$	−0.948690	−0.0313456
$917$	−3.14675	−0.103915
$918$	4.61356	0.152270
$919$	5.22223	0.172265	0.0861327	−	0.996284i	$-0.472549\pi$
$919$	5.22223	0.172265	0.0861327	+	0.996284i	$0.472549\pi$
$920$	−1.54288	−0.0508671
$921$	5.73556	0.188993
$922$	−20.6461	−0.679943
$923$	0	0
$924$	−1.00000	−0.0328976
$925$	−23.0844	−0.759011
$926$	10.9312	0.359223
$927$	−23.1554	−0.760522
$928$	3.58211	0.117588
$929$	10.9957	0.360757	0.180378	−	0.983597i	$-0.442268\pi$
$929$	10.9957	0.360757	0.180378	+	0.983597i	$0.442268\pi$
$930$	33.8364	1.10954
$931$	6.35690	0.208339
$932$	−15.4765	−0.506950
$933$	−3.67456	−0.120300
$934$	15.4450	0.505377
$935$	−1.62565	−0.0531643
$936$	0	0
$937$	−21.5190	−0.702994	−0.351497	−	0.936189i	$-0.614327\pi$
$937$	−21.5190	−0.702994	−0.351497	+	0.936189i	$0.614327\pi$
$938$	−6.23490	−0.203577
$939$	−28.6993	−0.936567
$940$	−16.4983	−0.538114
$941$	−11.2784	−0.367667	−0.183833	−	0.982957i	$-0.558851\pi$
$941$	−11.2784	−0.367667	−0.183833	+	0.982957i	$0.558851\pi$
$942$	−24.3666	−0.793906
$943$	−2.20477	−0.0717971
$944$	−10.1468	−0.330249
$945$	3.61596	0.117627
$946$	2.06100	0.0670089
$947$	−1.92154	−0.0624417	−0.0312209	−	0.999513i	$-0.509940\pi$
$947$	−1.92154	−0.0624417	−0.0312209	+	0.999513i	$0.509940\pi$
$948$	−14.1588	−0.459858
$949$	0	0
$950$	−13.5851	−0.440758
$951$	−24.5013	−0.794508
$952$	2.15883	0.0699682
$953$	32.0538	1.03833	0.519163	−	0.854676i	$-0.326244\pi$
$953$	32.0538	1.03833	0.519163	+	0.854676i	$0.326244\pi$
$954$	−17.9095	−0.579840
$955$	33.3726	1.07991
$956$	25.3448	0.819710
$957$	3.58211	0.115793
$958$	25.2828	0.816849
$959$	3.93362	0.127023
$960$	3.80194	0.122707
$961$	48.2059	1.55503
$962$	0	0
$963$	−14.6183	−0.471070
$964$	−28.2664	−0.910398
$965$	−15.4969	−0.498864
$966$	2.04892	0.0659228
$967$	47.5080	1.52775	0.763876	−	0.645362i	$-0.223294\pi$
$967$	47.5080	1.52775	0.763876	+	0.645362i	$0.223294\pi$
$968$	−10.8019	−0.347187
$969$	−30.8364	−0.990607
$970$	−5.59717	−0.179714
$971$	36.0224	1.15601	0.578006	−	0.816032i	$-0.303831\pi$
$971$	36.0224	1.15601	0.578006	+	0.816032i	$0.303831\pi$
$972$	−18.1903	−0.583454
$973$	11.2078	0.359304
$974$	1.01938	0.0326630
$975$	0	0
$976$	1.37867	0.0441300
$977$	−28.2620	−0.904183	−0.452091	−	0.891972i	$-0.649322\pi$
$977$	−28.2620	−0.904183	−0.452091	+	0.891972i	$0.649322\pi$
$978$	32.3696	1.03506
$979$	8.05861	0.257554
$980$	1.69202	0.0540496
$981$	−13.4614	−0.429791
$982$	−36.7198	−1.17177
$983$	8.60196	0.274360	0.137180	−	0.990546i	$-0.456196\pi$
$983$	8.60196	0.274360	0.137180	+	0.990546i	$0.456196\pi$
$984$	5.43296	0.173196
$985$	6.31155	0.201103
$986$	−7.73317	−0.246274
$987$	21.9095	0.697386
$988$	0	0
$989$	−4.22282	−0.134278
$990$	1.54288	0.0490358
$991$	−17.8549	−0.567180	−0.283590	−	0.958946i	$-0.591525\pi$
$991$	−17.8549	−0.567180	−0.283590	+	0.958946i	$0.591525\pi$
$992$	8.89977	0.282568
$993$	22.3545	0.709399
$994$	5.76271	0.182782
$995$	33.1497	1.05092
$996$	−6.54288	−0.207319
$997$	−36.3247	−1.15041	−0.575207	−	0.818008i	$-0.695079\pi$
$997$	−36.3247	−1.15041	−0.575207	+	0.818008i	$0.695079\pi$
$998$	−41.4685	−1.31266
$999$	−23.0844	−0.730359

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2366.2.a.bd.1.3	yes	3
13.5	odd	4		2366.2.d.p.337.3		6
13.8	odd	4		2366.2.d.p.337.6		6
13.12	even	2		2366.2.a.y.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2366.2.a.y.1.3	✓	3	13.12	even	2
2366.2.a.bd.1.3	yes	3	1.1	even	1	trivial
2366.2.d.p.337.3		6	13.5	odd	4
2366.2.d.p.337.6		6	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.24698 q^{3} +1.00000 q^{4} +1.69202 q^{5} +2.24698 q^{6} -1.00000 q^{7} +1.00000 q^{8} +2.04892 q^{9} +1.69202 q^{10} +0.445042 q^{11} +2.24698 q^{12} -1.00000 q^{14} +3.80194 q^{15} +1.00000 q^{16} -2.15883 q^{17} +2.04892 q^{18} +6.35690 q^{19} +1.69202 q^{20} -2.24698 q^{21} +0.445042 q^{22} -0.911854 q^{23} +2.24698 q^{24} -2.13706 q^{25} -2.13706 q^{27} -1.00000 q^{28} +3.58211 q^{29} +3.80194 q^{30} +8.89977 q^{31} +1.00000 q^{32} +1.00000 q^{33} -2.15883 q^{34} -1.69202 q^{35} +2.04892 q^{36} +10.8019 q^{37} +6.35690 q^{38} +1.69202 q^{40} +2.41789 q^{41} -2.24698 q^{42} +4.63102 q^{43} +0.445042 q^{44} +3.46681 q^{45} -0.911854 q^{46} -9.75063 q^{47} +2.24698 q^{48} +1.00000 q^{49} -2.13706 q^{50} -4.85086 q^{51} -8.74094 q^{53} -2.13706 q^{54} +0.753020 q^{55} -1.00000 q^{56} +14.2838 q^{57} +3.58211 q^{58} -10.1468 q^{59} +3.80194 q^{60} +1.37867 q^{61} +8.89977 q^{62} -2.04892 q^{63} +1.00000 q^{64} +1.00000 q^{66} +6.23490 q^{67} -2.15883 q^{68} -2.04892 q^{69} -1.69202 q^{70} -5.76271 q^{71} +2.04892 q^{72} -9.93661 q^{73} +10.8019 q^{74} -4.80194 q^{75} +6.35690 q^{76} -0.445042 q^{77} -6.30127 q^{79} +1.69202 q^{80} -10.9487 q^{81} +2.41789 q^{82} -2.91185 q^{83} -2.24698 q^{84} -3.65279 q^{85} +4.63102 q^{86} +8.04892 q^{87} +0.445042 q^{88} +18.1075 q^{89} +3.46681 q^{90} -0.911854 q^{92} +19.9976 q^{93} -9.75063 q^{94} +10.7560 q^{95} +2.24698 q^{96} -3.30798 q^{97} +1.00000 q^{98} +0.911854 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 2 q^{3} + 3 q^{4} + 2 q^{6} - 3 q^{7} + 3 q^{8} - 3 q^{9} + q^{11} + 2 q^{12} - 3 q^{14} + 7 q^{15} + 3 q^{16} + 2 q^{17} - 3 q^{18} + 15 q^{19} - 2 q^{21} + q^{22} + q^{23} + 2 q^{24}+ \cdots - q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 + 2 * q^3 + 3 * q^4 + 2 * q^6 - 3 * q^7 + 3 * q^8 - 3 * q^9 + q^11 + 2 * q^12 - 3 * q^14 + 7 * q^15 + 3 * q^16 + 2 * q^17 - 3 * q^18 + 15 * q^19 - 2 * q^21 + q^22 + q^23 + 2 * q^24 - q^25 - q^27 - 3 * q^28 + 5 * q^29 + 7 * q^30 + 4 * q^31 + 3 * q^32 + 3 * q^33 + 2 * q^34 - 3 * q^36 + 28 * q^37 + 15 * q^38 + 13 * q^41 - 2 * q^42 - q^43 + q^44 + 7 * q^45 + q^46 + 7 * q^47 + 2 * q^48 + 3 * q^49 - q^50 - q^51 - 12 * q^53 - q^54 + 7 * q^55 - 3 * q^56 + 10 * q^57 + 5 * q^58 - 3 * q^59 + 7 * q^60 - 3 * q^61 + 4 * q^62 + 3 * q^63 + 3 * q^64 + 3 * q^66 - 5 * q^67 + 2 * q^68 + 3 * q^69 - 3 * q^72 + 21 * q^73 + 28 * q^74 - 10 * q^75 + 15 * q^76 - q^77 - 2 * q^79 - q^81 + 13 * q^82 - 5 * q^83 - 2 * q^84 + 7 * q^85 - q^86 + 15 * q^87 + q^88 + 14 * q^89 + 7 * q^90 + q^92 + 19 * q^93 + 7 * q^94 - 7 * q^95 + 2 * q^96 - 15 * q^97 + 3 * q^98 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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