LMFDB - Embedded newform 686.2.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [686,2,Mod(1,686)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(686, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("686.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 686 = 2 \cdot 7^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	686.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$5.47773757866$
Analytic rank:	$1$
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			1.2
Root			$-1.24698$ of defining polynomial
Character	$\chi$	$=$	686.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} -1.55496 q^{3} +1.00000 q^{4} -1.44504 q^{5} -1.55496 q^{6} +1.00000 q^{8} -0.582105 q^{9} -1.44504 q^{10} -0.307979 q^{11} -1.55496 q^{12} +0.603875 q^{13} +2.24698 q^{15} +1.00000 q^{16} -5.54288 q^{17} -0.582105 q^{18} -3.89008 q^{19} -1.44504 q^{20} -0.307979 q^{22} -2.91185 q^{23} -1.55496 q^{24} -2.91185 q^{25} +0.603875 q^{26} +5.57002 q^{27} -0.594187 q^{29} +2.24698 q^{30} -8.07069 q^{31} +1.00000 q^{32} +0.478894 q^{33} -5.54288 q^{34} -0.582105 q^{36} -7.04892 q^{37} -3.89008 q^{38} -0.939001 q^{39} -1.44504 q^{40} +8.87800 q^{41} -10.9487 q^{43} -0.307979 q^{44} +0.841166 q^{45} -2.91185 q^{46} +7.36658 q^{47} -1.55496 q^{48} -2.91185 q^{50} +8.61894 q^{51} +0.603875 q^{52} +10.9487 q^{53} +5.57002 q^{54} +0.445042 q^{55} +6.04892 q^{57} -0.594187 q^{58} -11.6407 q^{59} +2.24698 q^{60} -1.52781 q^{61} -8.07069 q^{62} +1.00000 q^{64} -0.872625 q^{65} +0.478894 q^{66} +1.65279 q^{67} -5.54288 q^{68} +4.52781 q^{69} -9.44265 q^{71} -0.582105 q^{72} +13.1685 q^{73} -7.04892 q^{74} +4.52781 q^{75} -3.89008 q^{76} -0.939001 q^{78} +10.0368 q^{79} -1.44504 q^{80} -6.91484 q^{81} +8.87800 q^{82} +2.44504 q^{83} +8.00969 q^{85} -10.9487 q^{86} +0.923936 q^{87} -0.307979 q^{88} +9.35690 q^{89} +0.841166 q^{90} -2.91185 q^{92} +12.5496 q^{93} +7.36658 q^{94} +5.62133 q^{95} -1.55496 q^{96} -13.1347 q^{97} +0.179276 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} - 5 q^{3} + 3 q^{4} - 4 q^{5} - 5 q^{6} + 3 q^{8} + 4 q^{9} - 4 q^{10} - 6 q^{11} - 5 q^{12} - 7 q^{13} + 2 q^{15} + 3 q^{16} + 2 q^{17} + 4 q^{18} - 11 q^{19} - 4 q^{20} - 6 q^{22} - 5 q^{23}+ \cdots - 36 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 - 5 * q^3 + 3 * q^4 - 4 * q^5 - 5 * q^6 + 3 * q^8 + 4 * q^9 - 4 * q^10 - 6 * q^11 - 5 * q^12 - 7 * q^13 + 2 * q^15 + 3 * q^16 + 2 * q^17 + 4 * q^18 - 11 * q^19 - 4 * q^20 - 6 * q^22 - 5 * q^23 - 5 * q^24 - 5 * q^25 - 7 * q^26 - 8 * q^27 - 15 * q^29 + 2 * q^30 - 12 * q^31 + 3 * q^32 + 17 * q^33 + 2 * q^34 + 4 * q^36 - 12 * q^37 - 11 * q^38 + 7 * q^39 - 4 * q^40 + 7 * q^41 - q^43 - 6 * q^44 + 11 * q^45 - 5 * q^46 - 4 * q^47 - 5 * q^48 - 5 * q^50 - 8 * q^51 - 7 * q^52 + q^53 - 8 * q^54 + q^55 + 9 * q^57 - 15 * q^58 + 2 * q^59 + 2 * q^60 - 11 * q^61 - 12 * q^62 + 3 * q^64 + 14 * q^65 + 17 * q^66 - 13 * q^67 + 2 * q^68 + 20 * q^69 + 13 * q^71 + 4 * q^72 + 9 * q^73 - 12 * q^74 + 20 * q^75 - 11 * q^76 + 7 * q^78 + 2 * q^79 - 4 * q^80 + 27 * q^81 + 7 * q^82 + 7 * q^83 + 2 * q^85 - q^86 + 18 * q^87 - 6 * q^88 + 24 * q^89 + 11 * q^90 - 5 * q^92 + 41 * q^93 - 4 * q^94 + 24 * q^95 - 5 * q^96 + 7 * q^97 - 36 * 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
$3$	−1.55496	−0.897755	−0.448878	−	0.893593i	$-0.648176\pi$
$3$	−1.55496	−0.897755	−0.448878	+	0.893593i	$0.648176\pi$
$4$	1.00000	0.500000
$5$	−1.44504	−0.646242	−0.323121	−	0.946358i	$-0.604732\pi$
$5$	−1.44504	−0.646242	−0.323121	+	0.946358i	$0.604732\pi$
$6$	−1.55496	−0.634809
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	−0.582105	−0.194035
$10$	−1.44504	−0.456962
$11$	−0.307979	−0.0928590	−0.0464295	−	0.998922i	$-0.514784\pi$
$11$	−0.307979	−0.0928590	−0.0464295	+	0.998922i	$0.514784\pi$
$12$	−1.55496	−0.448878
$13$	0.603875	0.167485	0.0837425	−	0.996487i	$-0.473313\pi$
$13$	0.603875	0.167485	0.0837425	+	0.996487i	$0.473313\pi$
$14$	0	0
$15$	2.24698	0.580168
$16$	1.00000	0.250000
$17$	−5.54288	−1.34435	−0.672173	−	0.740395i	$-0.734639\pi$
$17$	−5.54288	−1.34435	−0.672173	+	0.740395i	$0.734639\pi$
$18$	−0.582105	−0.137204
$19$	−3.89008	−0.892446	−0.446223	−	0.894922i	$-0.647231\pi$
$19$	−3.89008	−0.892446	−0.446223	+	0.894922i	$0.647231\pi$
$20$	−1.44504	−0.323121
$21$	0	0
$22$	−0.307979	−0.0656612
$23$	−2.91185	−0.607164	−0.303582	−	0.952805i	$-0.598183\pi$
$23$	−2.91185	−0.607164	−0.303582	+	0.952805i	$0.598183\pi$
$24$	−1.55496	−0.317404
$25$	−2.91185	−0.582371
$26$	0.603875	0.118430
$27$	5.57002	1.07195
$28$	0	0
$29$	−0.594187	−0.110338	−0.0551689	−	0.998477i	$-0.517570\pi$
$29$	−0.594187	−0.110338	−0.0551689	+	0.998477i	$0.517570\pi$
$30$	2.24698	0.410240
$31$	−8.07069	−1.44954	−0.724769	−	0.688992i	$-0.758054\pi$
$31$	−8.07069	−1.44954	−0.724769	+	0.688992i	$0.758054\pi$
$32$	1.00000	0.176777
$33$	0.478894	0.0833647
$34$	−5.54288	−0.950595
$35$	0	0
$36$	−0.582105	−0.0970175
$37$	−7.04892	−1.15883	−0.579417	−	0.815031i	$-0.696720\pi$
$37$	−7.04892	−1.15883	−0.579417	+	0.815031i	$0.696720\pi$
$38$	−3.89008	−0.631055
$39$	−0.939001	−0.150361
$40$	−1.44504	−0.228481
$41$	8.87800	1.38651	0.693255	−	0.720692i	$-0.256176\pi$
$41$	8.87800	1.38651	0.693255	+	0.720692i	$0.256176\pi$
$42$	0	0
$43$	−10.9487	−1.66966	−0.834830	−	0.550508i	$-0.814434\pi$
$43$	−10.9487	−1.66966	−0.834830	+	0.550508i	$0.814434\pi$
$44$	−0.307979	−0.0464295
$45$	0.841166	0.125394
$46$	−2.91185	−0.429329
$47$	7.36658	1.07453	0.537263	−	0.843415i	$-0.319458\pi$
$47$	7.36658	1.07453	0.537263	+	0.843415i	$0.319458\pi$
$48$	−1.55496	−0.224439
$49$	0	0
$50$	−2.91185	−0.411798
$51$	8.61894	1.20689
$52$	0.603875	0.0837425
$53$	10.9487	1.50392	0.751959	−	0.659210i	$-0.229109\pi$
$53$	10.9487	1.50392	0.751959	+	0.659210i	$0.229109\pi$
$54$	5.57002	0.757984
$55$	0.445042	0.0600094
$56$	0	0
$57$	6.04892	0.801199
$58$	−0.594187	−0.0780205
$59$	−11.6407	−1.51549	−0.757746	−	0.652550i	$-0.773699\pi$
$59$	−11.6407	−1.51549	−0.757746	+	0.652550i	$0.773699\pi$
$60$	2.24698	0.290084
$61$	−1.52781	−0.195616	−0.0978081	−	0.995205i	$-0.531183\pi$
$61$	−1.52781	−0.195616	−0.0978081	+	0.995205i	$0.531183\pi$
$62$	−8.07069	−1.02498
$63$	0	0
$64$	1.00000	0.125000
$65$	−0.872625	−0.108236
$66$	0.478894	0.0589477
$67$	1.65279	0.201921	0.100960	−	0.994890i	$-0.467808\pi$
$67$	1.65279	0.201921	0.100960	+	0.994890i	$0.467808\pi$
$68$	−5.54288	−0.672173
$69$	4.52781	0.545084
$70$	0	0
$71$	−9.44265	−1.12064	−0.560318	−	0.828277i	$-0.689321\pi$
$71$	−9.44265	−1.12064	−0.560318	+	0.828277i	$0.689321\pi$
$72$	−0.582105	−0.0686018
$73$	13.1685	1.54126	0.770629	−	0.637283i	$-0.219942\pi$
$73$	13.1685	1.54126	0.770629	+	0.637283i	$0.219942\pi$
$74$	−7.04892	−0.819420
$75$	4.52781	0.522827
$76$	−3.89008	−0.446223
$77$	0	0
$78$	−0.939001	−0.106321
$79$	10.0368	1.12923	0.564616	−	0.825354i	$-0.309024\pi$
$79$	10.0368	1.12923	0.564616	+	0.825354i	$0.309024\pi$
$80$	−1.44504	−0.161561
$81$	−6.91484	−0.768315
$82$	8.87800	0.980411
$83$	2.44504	0.268378	0.134189	−	0.990956i	$-0.457157\pi$
$83$	2.44504	0.268378	0.134189	+	0.990956i	$0.457157\pi$
$84$	0	0
$85$	8.00969	0.868773
$86$	−10.9487	−1.18063
$87$	0.923936	0.0990563
$88$	−0.307979	−0.0328306
$89$	9.35690	0.991829	0.495914	−	0.868371i	$-0.334833\pi$
$89$	9.35690	0.991829	0.495914	+	0.868371i	$0.334833\pi$
$90$	0.841166	0.0886667
$91$	0	0
$92$	−2.91185	−0.303582
$93$	12.5496	1.30133
$94$	7.36658	0.759805
$95$	5.62133	0.576737
$96$	−1.55496	−0.158702
$97$	−13.1347	−1.33362	−0.666812	−	0.745226i	$-0.732342\pi$
$97$	−13.1347	−1.33362	−0.666812	+	0.745226i	$0.732342\pi$
$98$	0	0
$99$	0.179276	0.0180179
$100$	−2.91185	−0.291185
$101$	8.74632	0.870291	0.435145	−	0.900360i	$-0.356697\pi$
$101$	8.74632	0.870291	0.435145	+	0.900360i	$0.356697\pi$
$102$	8.61894	0.853402
$103$	−4.76271	−0.469284	−0.234642	−	0.972082i	$-0.575392\pi$
$103$	−4.76271	−0.469284	−0.234642	+	0.972082i	$0.575392\pi$
$104$	0.603875	0.0592149
$105$	0	0
$106$	10.9487	1.06343
$107$	−7.71917	−0.746240	−0.373120	−	0.927783i	$-0.621712\pi$
$107$	−7.71917	−0.746240	−0.373120	+	0.927783i	$0.621712\pi$
$108$	5.57002	0.535976
$109$	−8.58211	−0.822017	−0.411008	−	0.911632i	$-0.634823\pi$
$109$	−8.58211	−0.822017	−0.411008	+	0.911632i	$0.634823\pi$
$110$	0.445042	0.0424331
$111$	10.9608	1.04035
$112$	0	0
$113$	18.5646	1.74642	0.873208	−	0.487349i	$-0.162036\pi$
$113$	18.5646	1.74642	0.873208	+	0.487349i	$0.162036\pi$
$114$	6.04892	0.566533
$115$	4.20775	0.392375
$116$	−0.594187	−0.0551689
$117$	−0.351519	−0.0324979
$118$	−11.6407	−1.07161
$119$	0	0
$120$	2.24698	0.205120
$121$	−10.9051	−0.991377
$122$	−1.52781	−0.138322
$123$	−13.8049	−1.24475
$124$	−8.07069	−0.724769
$125$	11.4330	1.02260
$126$	0	0
$127$	1.59850	0.141844	0.0709219	−	0.997482i	$-0.477406\pi$
$127$	1.59850	0.141844	0.0709219	+	0.997482i	$0.477406\pi$
$128$	1.00000	0.0883883
$129$	17.0248	1.49895
$130$	−0.872625	−0.0765343
$131$	−1.78017	−0.155534	−0.0777670	−	0.996972i	$-0.524779\pi$
$131$	−1.78017	−0.155534	−0.0777670	+	0.996972i	$0.524779\pi$
$132$	0.478894	0.0416823
$133$	0	0
$134$	1.65279	0.142780
$135$	−8.04892	−0.692741
$136$	−5.54288	−0.475298
$137$	0.362273	0.0309510	0.0154755	−	0.999880i	$-0.495074\pi$
$137$	0.362273	0.0309510	0.0154755	+	0.999880i	$0.495074\pi$
$138$	4.52781	0.385433
$139$	−9.59419	−0.813768	−0.406884	−	0.913480i	$-0.633385\pi$
$139$	−9.59419	−0.813768	−0.406884	+	0.913480i	$0.633385\pi$
$140$	0	0
$141$	−11.4547	−0.964662
$142$	−9.44265	−0.792409
$143$	−0.185981	−0.0155525
$144$	−0.582105	−0.0485088
$145$	0.858625	0.0713049
$146$	13.1685	1.08983
$147$	0	0
$148$	−7.04892	−0.579417
$149$	6.80194	0.557236	0.278618	−	0.960402i	$-0.410124\pi$
$149$	6.80194	0.557236	0.278618	+	0.960402i	$0.410124\pi$
$150$	4.52781	0.369694
$151$	23.4131	1.90533	0.952666	−	0.304019i	$-0.0983287\pi$
$151$	23.4131	1.90533	0.952666	+	0.304019i	$0.0983287\pi$
$152$	−3.89008	−0.315527
$153$	3.22654	0.260850
$154$	0	0
$155$	11.6625	0.936753
$156$	−0.939001	−0.0751803
$157$	19.2664	1.53762	0.768811	−	0.639476i	$-0.220849\pi$
$157$	19.2664	1.53762	0.768811	+	0.639476i	$0.220849\pi$
$158$	10.0368	0.798488
$159$	−17.0248	−1.35015
$160$	−1.44504	−0.114241
$161$	0	0
$162$	−6.91484	−0.543281
$163$	0.0174584	0.00136745	0.000683724	−	1.00000i	$-0.499782\pi$
$163$	0.0174584	0.00136745	0.000683724		1.00000i	$0.499782\pi$
$164$	8.87800	0.693255
$165$	−0.692021	−0.0538738
$166$	2.44504	0.189772
$167$	1.87800	0.145324	0.0726621	−	0.997357i	$-0.476851\pi$
$167$	1.87800	0.145324	0.0726621	+	0.997357i	$0.476851\pi$
$168$	0	0
$169$	−12.6353	−0.971949
$170$	8.00969	0.614315
$171$	2.26444	0.173166
$172$	−10.9487	−0.834830
$173$	−12.0683	−0.917535	−0.458768	−	0.888556i	$-0.651709\pi$
$173$	−12.0683	−0.917535	−0.458768	+	0.888556i	$0.651709\pi$
$174$	0.923936	0.0700434
$175$	0	0
$176$	−0.307979	−0.0232148
$177$	18.1008	1.36054
$178$	9.35690	0.701329
$179$	−24.4306	−1.82603	−0.913013	−	0.407930i	$-0.866251\pi$
$179$	−24.4306	−1.82603	−0.913013	+	0.407930i	$0.866251\pi$
$180$	0.841166	0.0626968
$181$	−7.85623	−0.583949	−0.291975	−	0.956426i	$-0.594312\pi$
$181$	−7.85623	−0.583949	−0.291975	+	0.956426i	$0.594312\pi$
$182$	0	0
$183$	2.37568	0.175615
$184$	−2.91185	−0.214665
$185$	10.1860	0.748888
$186$	12.5496	0.920180
$187$	1.70709	0.124835
$188$	7.36658	0.537263
$189$	0	0
$190$	5.62133	0.407814
$191$	5.75063	0.416101	0.208050	−	0.978118i	$-0.433288\pi$
$191$	5.75063	0.416101	0.208050	+	0.978118i	$0.433288\pi$
$192$	−1.55496	−0.112219
$193$	21.3991	1.54034	0.770171	−	0.637838i	$-0.220171\pi$
$193$	21.3991	1.54034	0.770171	+	0.637838i	$0.220171\pi$
$194$	−13.1347	−0.943014
$195$	1.35690	0.0971693
$196$	0	0
$197$	−13.9312	−0.992559	−0.496280	−	0.868163i	$-0.665301\pi$
$197$	−13.9312	−0.992559	−0.496280	+	0.868163i	$0.665301\pi$
$198$	0.179276	0.0127406
$199$	−5.05131	−0.358078	−0.179039	−	0.983842i	$-0.557299\pi$
$199$	−5.05131	−0.358078	−0.179039	+	0.983842i	$0.557299\pi$
$200$	−2.91185	−0.205899
$201$	−2.57002	−0.181275
$202$	8.74632	0.615389
$203$	0	0
$204$	8.61894	0.603447
$205$	−12.8291	−0.896022
$206$	−4.76271	−0.331834
$207$	1.69501	0.117811
$208$	0.603875	0.0418712
$209$	1.19806	0.0828717
$210$	0	0
$211$	−1.61596	−0.111247	−0.0556235	−	0.998452i	$-0.517715\pi$
$211$	−1.61596	−0.111247	−0.0556235	+	0.998452i	$0.517715\pi$
$212$	10.9487	0.751959
$213$	14.6829	1.00606
$214$	−7.71917	−0.527671
$215$	15.8213	1.07900
$216$	5.57002	0.378992
$217$	0	0
$218$	−8.58211	−0.581254
$219$	−20.4765	−1.38367
$220$	0.445042	0.0300047
$221$	−3.34721	−0.225158
$222$	10.9608	0.735639
$223$	−27.8418	−1.86442	−0.932211	−	0.361915i	$-0.882123\pi$
$223$	−27.8418	−1.86442	−0.932211	+	0.361915i	$0.882123\pi$
$224$	0	0
$225$	1.69501	0.113000
$226$	18.5646	1.23490
$227$	−18.2892	−1.21390	−0.606948	−	0.794741i	$-0.707606\pi$
$227$	−18.2892	−1.21390	−0.606948	+	0.794741i	$0.707606\pi$
$228$	6.04892	0.400599
$229$	−20.5375	−1.35716	−0.678578	−	0.734528i	$-0.737403\pi$
$229$	−20.5375	−1.35716	−0.678578	+	0.734528i	$0.737403\pi$
$230$	4.20775	0.277451
$231$	0	0
$232$	−0.594187	−0.0390103
$233$	−6.44504	−0.422229	−0.211114	−	0.977461i	$-0.567709\pi$
$233$	−6.44504	−0.422229	−0.211114	+	0.977461i	$0.567709\pi$
$234$	−0.351519	−0.0229795
$235$	−10.6450	−0.694405
$236$	−11.6407	−0.757746
$237$	−15.6069	−1.01377
$238$	0	0
$239$	−14.4155	−0.932461	−0.466231	−	0.884663i	$-0.654388\pi$
$239$	−14.4155	−0.932461	−0.466231	+	0.884663i	$0.654388\pi$
$240$	2.24698	0.145042
$241$	0.339437	0.0218651	0.0109325	−	0.999940i	$-0.496520\pi$
$241$	0.339437	0.0218651	0.0109325	+	0.999940i	$0.496520\pi$
$242$	−10.9051	−0.701010
$243$	−5.95779	−0.382192
$244$	−1.52781	−0.0978081
$245$	0	0
$246$	−13.8049	−0.880170
$247$	−2.34913	−0.149471
$248$	−8.07069	−0.512489
$249$	−3.80194	−0.240938
$250$	11.4330	0.723084
$251$	3.78554	0.238941	0.119471	−	0.992838i	$-0.461880\pi$
$251$	3.78554	0.238941	0.119471	+	0.992838i	$0.461880\pi$
$252$	0	0
$253$	0.896789	0.0563806
$254$	1.59850	0.100299
$255$	−12.4547	−0.779945
$256$	1.00000	0.0625000
$257$	31.4523	1.96194	0.980971	−	0.194152i	$-0.0621955\pi$
$257$	31.4523	1.96194	0.980971	+	0.194152i	$0.0621955\pi$
$258$	17.0248	1.05991
$259$	0	0
$260$	−0.872625	−0.0541179
$261$	0.345879	0.0214094
$262$	−1.78017	−0.109979
$263$	14.8334	0.914666	0.457333	−	0.889295i	$-0.348805\pi$
$263$	14.8334	0.914666	0.457333	+	0.889295i	$0.348805\pi$
$264$	0.478894	0.0294739
$265$	−15.8213	−0.971896
$266$	0	0
$267$	−14.5496	−0.890420
$268$	1.65279	0.100960
$269$	−12.4969	−0.761952	−0.380976	−	0.924585i	$-0.624412\pi$
$269$	−12.4969	−0.761952	−0.380976	+	0.924585i	$0.624412\pi$
$270$	−8.04892	−0.489842
$271$	4.35450	0.264517	0.132259	−	0.991215i	$-0.457777\pi$
$271$	4.35450	0.264517	0.132259	+	0.991215i	$0.457777\pi$
$272$	−5.54288	−0.336086
$273$	0	0
$274$	0.362273	0.0218857
$275$	0.896789	0.0540784
$276$	4.52781	0.272542
$277$	−13.2403	−0.795531	−0.397766	−	0.917487i	$-0.630214\pi$
$277$	−13.2403	−0.795531	−0.397766	+	0.917487i	$0.630214\pi$
$278$	−9.59419	−0.575421
$279$	4.69799	0.281261
$280$	0	0
$281$	12.2513	0.730851	0.365425	−	0.930841i	$-0.380924\pi$
$281$	12.2513	0.730851	0.365425	+	0.930841i	$0.380924\pi$
$282$	−11.4547	−0.682119
$283$	−5.84117	−0.347221	−0.173611	−	0.984814i	$-0.555543\pi$
$283$	−5.84117	−0.347221	−0.173611	+	0.984814i	$0.555543\pi$
$284$	−9.44265	−0.560318
$285$	−8.74094	−0.517769
$286$	−0.185981	−0.0109973
$287$	0	0
$288$	−0.582105	−0.0343009
$289$	13.7235	0.807264
$290$	0.858625	0.0504202
$291$	20.4239	1.19727
$292$	13.1685	0.770629
$293$	4.10752	0.239964	0.119982	−	0.992776i	$-0.461716\pi$
$293$	4.10752	0.239964	0.119982	+	0.992776i	$0.461716\pi$
$294$	0	0
$295$	16.8213	0.979375
$296$	−7.04892	−0.409710
$297$	−1.71545	−0.0995404
$298$	6.80194	0.394026
$299$	−1.75840	−0.101691
$300$	4.52781	0.261413
$301$	0	0
$302$	23.4131	1.34727
$303$	−13.6002	−0.781308
$304$	−3.89008	−0.223112
$305$	2.20775	0.126415
$306$	3.22654	0.184449
$307$	−29.4010	−1.67801	−0.839003	−	0.544127i	$-0.816861\pi$
$307$	−29.4010	−1.67801	−0.839003	+	0.544127i	$0.816861\pi$
$308$	0	0
$309$	7.40581	0.421302
$310$	11.6625	0.662384
$311$	−23.0465	−1.30685	−0.653424	−	0.756992i	$-0.726668\pi$
$311$	−23.0465	−1.30685	−0.653424	+	0.756992i	$0.726668\pi$
$312$	−0.939001	−0.0531605
$313$	15.8931	0.898329	0.449165	−	0.893449i	$-0.351722\pi$
$313$	15.8931	0.898329	0.449165	+	0.893449i	$0.351722\pi$
$314$	19.2664	1.08726
$315$	0	0
$316$	10.0368	0.564616
$317$	−20.8998	−1.17385	−0.586924	−	0.809642i	$-0.699661\pi$
$317$	−20.8998	−1.17385	−0.586924	+	0.809642i	$0.699661\pi$
$318$	−17.0248	−0.954701
$319$	0.182997	0.0102459
$320$	−1.44504	−0.0807803
$321$	12.0030	0.669941
$322$	0	0
$323$	21.5623	1.19976
$324$	−6.91484	−0.384158
$325$	−1.75840	−0.0975383
$326$	0.0174584	0.000966931 0
$327$	13.3448	0.737970
$328$	8.87800	0.490206
$329$	0	0
$330$	−0.692021	−0.0380945
$331$	12.3123	0.676745	0.338372	−	0.941012i	$-0.390124\pi$
$331$	12.3123	0.676745	0.338372	+	0.941012i	$0.390124\pi$
$332$	2.44504	0.134189
$333$	4.10321	0.224855
$334$	1.87800	0.102760
$335$	−2.38835	−0.130490
$336$	0	0
$337$	−2.92692	−0.159439	−0.0797197	−	0.996817i	$-0.525403\pi$
$337$	−2.92692	−0.159439	−0.0797197	+	0.996817i	$0.525403\pi$
$338$	−12.6353	−0.687272
$339$	−28.8672	−1.56785
$340$	8.00969	0.434386
$341$	2.48560	0.134603
$342$	2.26444	0.122447
$343$	0	0
$344$	−10.9487	−0.590314
$345$	−6.54288	−0.352257
$346$	−12.0683	−0.648796
$347$	−0.262045	−0.0140673	−0.00703366	−	0.999975i	$-0.502239\pi$
$347$	−0.262045	−0.0140673	−0.00703366	+	0.999975i	$0.502239\pi$
$348$	0.923936	0.0495281
$349$	−30.6310	−1.63964	−0.819821	−	0.572621i	$-0.805927\pi$
$349$	−30.6310	−1.63964	−0.819821	+	0.572621i	$0.805927\pi$
$350$	0	0
$351$	3.36360	0.179536
$352$	−0.307979	−0.0164153
$353$	18.5429	0.986938	0.493469	−	0.869763i	$-0.335729\pi$
$353$	18.5429	0.986938	0.493469	+	0.869763i	$0.335729\pi$
$354$	18.1008	0.962048
$355$	13.6450	0.724203
$356$	9.35690	0.495914
$357$	0	0
$358$	−24.4306	−1.29120
$359$	9.18060	0.484534	0.242267	−	0.970210i	$-0.422109\pi$
$359$	9.18060	0.484534	0.242267	+	0.970210i	$0.422109\pi$
$360$	0.841166	0.0443334
$361$	−3.86725	−0.203539
$362$	−7.85623	−0.412914
$363$	16.9571	0.890014
$364$	0	0
$365$	−19.0291	−0.996027
$366$	2.37568	0.124179
$367$	−24.1642	−1.26136	−0.630681	−	0.776042i	$-0.717224\pi$
$367$	−24.1642	−1.26136	−0.630681	+	0.776042i	$0.717224\pi$
$368$	−2.91185	−0.151791
$369$	−5.16793	−0.269032
$370$	10.1860	0.529544
$371$	0	0
$372$	12.5496	0.650665
$373$	25.9148	1.34182	0.670910	−	0.741539i	$-0.265904\pi$
$373$	25.9148	1.34182	0.670910	+	0.741539i	$0.265904\pi$
$374$	1.70709	0.0882714
$375$	−17.7778	−0.918040
$376$	7.36658	0.379903
$377$	−0.358815	−0.0184799
$378$	0	0
$379$	32.0844	1.64807	0.824033	−	0.566542i	$-0.191719\pi$
$379$	32.0844	1.64807	0.824033	+	0.566542i	$0.191719\pi$
$380$	5.62133	0.288368
$381$	−2.48560	−0.127341
$382$	5.75063	0.294228
$383$	−12.6280	−0.645263	−0.322631	−	0.946525i	$-0.604567\pi$
$383$	−12.6280	−0.645263	−0.322631	+	0.946525i	$0.604567\pi$
$384$	−1.55496	−0.0793511
$385$	0	0
$386$	21.3991	1.08919
$387$	6.37329	0.323973
$388$	−13.1347	−0.666812
$389$	0.341830	0.0173315	0.00866574	−	0.999962i	$-0.497242\pi$
$389$	0.341830	0.0173315	0.00866574	+	0.999962i	$0.497242\pi$
$390$	1.35690	0.0687091
$391$	16.1400	0.816237
$392$	0	0
$393$	2.76809	0.139631
$394$	−13.9312	−0.701845
$395$	−14.5036	−0.729758
$396$	0.179276	0.00900895
$397$	8.78986	0.441150	0.220575	−	0.975370i	$-0.429207\pi$
$397$	8.78986	0.441150	0.220575	+	0.975370i	$0.429207\pi$
$398$	−5.05131	−0.253199
$399$	0	0
$400$	−2.91185	−0.145593
$401$	9.49934	0.474374	0.237187	−	0.971464i	$-0.423775\pi$
$401$	9.49934	0.474374	0.237187	+	0.971464i	$0.423775\pi$
$402$	−2.57002	−0.128181
$403$	−4.87369	−0.242776
$404$	8.74632	0.435145
$405$	9.99223	0.496518
$406$	0	0
$407$	2.17092	0.107608
$408$	8.61894	0.426701
$409$	−17.5526	−0.867918	−0.433959	−	0.900933i	$-0.642884\pi$
$409$	−17.5526	−0.867918	−0.433959	+	0.900933i	$0.642884\pi$
$410$	−12.8291	−0.633583
$411$	−0.563319	−0.0277865
$412$	−4.76271	−0.234642
$413$	0	0
$414$	1.69501	0.0833050
$415$	−3.53319	−0.173437
$416$	0.603875	0.0296074
$417$	14.9186	0.730565
$418$	1.19806	0.0585991
$419$	−19.0965	−0.932925	−0.466463	−	0.884541i	$-0.654472\pi$
$419$	−19.0965	−0.932925	−0.466463	+	0.884541i	$0.654472\pi$
$420$	0	0
$421$	−19.9095	−0.970328	−0.485164	−	0.874423i	$-0.661240\pi$
$421$	−19.9095	−0.970328	−0.485164	+	0.874423i	$0.661240\pi$
$422$	−1.61596	−0.0786636
$423$	−4.28813	−0.208496
$424$	10.9487	0.531715
$425$	16.1400	0.782907
$426$	14.6829	0.711390
$427$	0	0
$428$	−7.71917	−0.373120
$429$	0.289192	0.0139623
$430$	15.8213	0.762972
$431$	−6.21073	−0.299161	−0.149580	−	0.988750i	$-0.547792\pi$
$431$	−6.21073	−0.299161	−0.149580	+	0.988750i	$0.547792\pi$
$432$	5.57002	0.267988
$433$	−8.81163	−0.423460	−0.211730	−	0.977328i	$-0.567910\pi$
$433$	−8.81163	−0.423460	−0.211730	+	0.977328i	$0.567910\pi$
$434$	0	0
$435$	−1.33513	−0.0640144
$436$	−8.58211	−0.411008
$437$	11.3274	0.541861
$438$	−20.4765	−0.978405
$439$	−36.1672	−1.72617	−0.863083	−	0.505062i	$-0.831470\pi$
$439$	−36.1672	−1.72617	−0.863083	+	0.505062i	$0.831470\pi$
$440$	0.445042	0.0212165
$441$	0	0
$442$	−3.34721	−0.159210
$443$	20.2881	0.963918	0.481959	−	0.876194i	$-0.339925\pi$
$443$	20.2881	0.963918	0.481959	+	0.876194i	$0.339925\pi$
$444$	10.9608	0.520175
$445$	−13.5211	−0.640962
$446$	−27.8418	−1.31835
$447$	−10.5767	−0.500262
$448$	0	0
$449$	−31.3448	−1.47925	−0.739627	−	0.673017i	$-0.764998\pi$
$449$	−31.3448	−1.47925	−0.739627	+	0.673017i	$0.764998\pi$
$450$	1.69501	0.0799033
$451$	−2.73423	−0.128750
$452$	18.5646	0.873208
$453$	−36.4064	−1.71052
$454$	−18.2892	−0.858354
$455$	0	0
$456$	6.04892	0.283267
$457$	8.78315	0.410858	0.205429	−	0.978672i	$-0.434141\pi$
$457$	8.78315	0.410858	0.205429	+	0.978672i	$0.434141\pi$
$458$	−20.5375	−0.959654
$459$	−30.8740	−1.44107
$460$	4.20775	0.196187
$461$	16.0968	0.749701	0.374851	−	0.927085i	$-0.377694\pi$
$461$	16.0968	0.749701	0.374851	+	0.927085i	$0.377694\pi$
$462$	0	0
$463$	−1.68963	−0.0785237	−0.0392618	−	0.999229i	$-0.512501\pi$
$463$	−1.68963	−0.0785237	−0.0392618	+	0.999229i	$0.512501\pi$
$464$	−0.594187	−0.0275844
$465$	−18.1347	−0.840975
$466$	−6.44504	−0.298561
$467$	33.5623	1.55308	0.776538	−	0.630070i	$-0.216974\pi$
$467$	33.5623	1.55308	0.776538	+	0.630070i	$0.216974\pi$
$468$	−0.351519	−0.0162490
$469$	0	0
$470$	−10.6450	−0.491018
$471$	−29.9584	−1.38041
$472$	−11.6407	−0.535807
$473$	3.37196	0.155043
$474$	−15.6069	−0.716847
$475$	11.3274	0.519735
$476$	0	0
$477$	−6.37329	−0.291813
$478$	−14.4155	−0.659350
$479$	7.94331	0.362939	0.181470	−	0.983397i	$-0.441915\pi$
$479$	7.94331	0.362939	0.181470	+	0.983397i	$0.441915\pi$
$480$	2.24698	0.102560
$481$	−4.25667	−0.194087
$482$	0.339437	0.0154610
$483$	0	0
$484$	−10.9051	−0.495689
$485$	18.9801	0.861844
$486$	−5.95779	−0.270251
$487$	35.8998	1.62677	0.813387	−	0.581723i	$-0.197621\pi$
$487$	35.8998	1.62677	0.813387	+	0.581723i	$0.197621\pi$
$488$	−1.52781	−0.0691608
$489$	−0.0271471	−0.00122763
$490$	0	0
$491$	−6.17092	−0.278490	−0.139245	−	0.990258i	$-0.544467\pi$
$491$	−6.17092	−0.278490	−0.139245	+	0.990258i	$0.544467\pi$
$492$	−13.8049	−0.622374
$493$	3.29350	0.148332
$494$	−2.34913	−0.105692
$495$	−0.259061	−0.0116439
$496$	−8.07069	−0.362385
$497$	0	0
$498$	−3.80194	−0.170369
$499$	5.19673	0.232638	0.116319	−	0.993212i	$-0.462891\pi$
$499$	5.19673	0.232638	0.116319	+	0.993212i	$0.462891\pi$
$500$	11.4330	0.511298
$501$	−2.92021	−0.130466
$502$	3.78554	0.168957
$503$	−11.4058	−0.508560	−0.254280	−	0.967131i	$-0.581838\pi$
$503$	−11.4058	−0.508560	−0.254280	+	0.967131i	$0.581838\pi$
$504$	0	0
$505$	−12.6388	−0.562419
$506$	0.896789	0.0398671
$507$	19.6474	0.872572
$508$	1.59850	0.0709219
$509$	−42.1323	−1.86748	−0.933740	−	0.357951i	$-0.883475\pi$
$509$	−42.1323	−1.86748	−0.933740	+	0.357951i	$0.883475\pi$
$510$	−12.4547	−0.551505
$511$	0	0
$512$	1.00000	0.0441942
$513$	−21.6679	−0.956659
$514$	31.4523	1.38730
$515$	6.88231	0.303271
$516$	17.0248	0.749473
$517$	−2.26875	−0.0997795
$518$	0	0
$519$	18.7657	0.823722
$520$	−0.872625	−0.0382672
$521$	21.4722	0.940714	0.470357	−	0.882476i	$-0.344125\pi$
$521$	21.4722	0.940714	0.470357	+	0.882476i	$0.344125\pi$
$522$	0.345879	0.0151387
$523$	−12.0479	−0.526816	−0.263408	−	0.964685i	$-0.584847\pi$
$523$	−12.0479	−0.526816	−0.263408	+	0.964685i	$0.584847\pi$
$524$	−1.78017	−0.0777670
$525$	0	0
$526$	14.8334	0.646767
$527$	44.7348	1.94868
$528$	0.478894	0.0208412
$529$	−14.5211	−0.631352
$530$	−15.8213	−0.687234
$531$	6.77612	0.294059
$532$	0	0
$533$	5.36121	0.232220
$534$	−14.5496	−0.629622
$535$	11.1545	0.482252
$536$	1.65279	0.0713898
$537$	37.9885	1.63933
$538$	−12.4969	−0.538781
$539$	0	0
$540$	−8.04892	−0.346370
$541$	−25.5375	−1.09794	−0.548971	−	0.835841i	$-0.684980\pi$
$541$	−25.5375	−1.09794	−0.548971	+	0.835841i	$0.684980\pi$
$542$	4.35450	0.187042
$543$	12.2161	0.524244
$544$	−5.54288	−0.237649
$545$	12.4015	0.531222
$546$	0	0
$547$	−24.1497	−1.03257	−0.516284	−	0.856417i	$-0.672685\pi$
$547$	−24.1497	−1.03257	−0.516284	+	0.856417i	$0.672685\pi$
$548$	0.362273	0.0154755
$549$	0.889347	0.0379564
$550$	0.896789	0.0382392
$551$	2.31144	0.0984705
$552$	4.52781	0.192716
$553$	0	0
$554$	−13.2403	−0.562525
$555$	−15.8388	−0.672318
$556$	−9.59419	−0.406884
$557$	8.36765	0.354549	0.177274	−	0.984161i	$-0.443272\pi$
$557$	8.36765	0.354549	0.177274	+	0.984161i	$0.443272\pi$
$558$	4.69799	0.198882
$559$	−6.61165	−0.279643
$560$	0	0
$561$	−2.65445	−0.112071
$562$	12.2513	0.516790
$563$	36.8568	1.55333	0.776665	−	0.629914i	$-0.216910\pi$
$563$	36.8568	1.55333	0.776665	+	0.629914i	$0.216910\pi$
$564$	−11.4547	−0.482331
$565$	−26.8267	−1.12861
$566$	−5.84117	−0.245523
$567$	0	0
$568$	−9.44265	−0.396205
$569$	−39.8135	−1.66907	−0.834535	−	0.550954i	$-0.814264\pi$
$569$	−39.8135	−1.66907	−0.834535	+	0.550954i	$0.814264\pi$
$570$	−8.74094	−0.366118
$571$	−12.1062	−0.506629	−0.253314	−	0.967384i	$-0.581521\pi$
$571$	−12.1062	−0.506629	−0.253314	+	0.967384i	$0.581521\pi$
$572$	−0.185981	−0.00777624
$573$	−8.94198	−0.373557
$574$	0	0
$575$	8.47889	0.353594
$576$	−0.582105	−0.0242544
$577$	8.17928	0.340508	0.170254	−	0.985400i	$-0.445541\pi$
$577$	8.17928	0.340508	0.170254	+	0.985400i	$0.445541\pi$
$578$	13.7235	0.570822
$579$	−33.2747	−1.38285
$580$	0.858625	0.0356525
$581$	0	0
$582$	20.4239	0.846596
$583$	−3.37196	−0.139652
$584$	13.1685	0.544917
$585$	0.507960	0.0210016
$586$	4.10752	0.169680
$587$	11.5386	0.476248	0.238124	−	0.971235i	$-0.423468\pi$
$587$	11.5386	0.476248	0.238124	+	0.971235i	$0.423468\pi$
$588$	0	0
$589$	31.3957	1.29364
$590$	16.8213	0.692523
$591$	21.6625	0.891075
$592$	−7.04892	−0.289709
$593$	38.6679	1.58790	0.793949	−	0.607984i	$-0.208022\pi$
$593$	38.6679	1.58790	0.793949	+	0.607984i	$0.208022\pi$
$594$	−1.71545	−0.0703857
$595$	0	0
$596$	6.80194	0.278618
$597$	7.85458	0.321466
$598$	−1.75840	−0.0719062
$599$	25.5894	1.04555	0.522777	−	0.852469i	$-0.324896\pi$
$599$	25.5894	1.04555	0.522777	+	0.852469i	$0.324896\pi$
$600$	4.52781	0.184847
$601$	−0.881723	−0.0359662	−0.0179831	−	0.999838i	$-0.505725\pi$
$601$	−0.881723	−0.0359662	−0.0179831	+	0.999838i	$0.505725\pi$
$602$	0	0
$603$	−0.962099	−0.0391797
$604$	23.4131	0.952666
$605$	15.7584	0.640670
$606$	−13.6002	−0.552469
$607$	7.62804	0.309613	0.154806	−	0.987945i	$-0.450525\pi$
$607$	7.62804	0.309613	0.154806	+	0.987945i	$0.450525\pi$
$608$	−3.89008	−0.157764
$609$	0	0
$610$	2.20775	0.0893892
$611$	4.44850	0.179967
$612$	3.22654	0.130425
$613$	30.1575	1.21805	0.609025	−	0.793151i	$-0.291561\pi$
$613$	30.1575	1.21805	0.609025	+	0.793151i	$0.291561\pi$
$614$	−29.4010	−1.18653
$615$	19.9487	0.804409
$616$	0	0
$617$	−1.54958	−0.0623838	−0.0311919	−	0.999513i	$-0.509930\pi$
$617$	−1.54958	−0.0623838	−0.0311919	+	0.999513i	$0.509930\pi$
$618$	7.40581	0.297905
$619$	−6.86294	−0.275845	−0.137922	−	0.990443i	$-0.544042\pi$
$619$	−6.86294	−0.275845	−0.137922	+	0.990443i	$0.544042\pi$
$620$	11.6625	0.468377
$621$	−16.2191	−0.650850
$622$	−23.0465	−0.924081
$623$	0	0
$624$	−0.939001	−0.0375901
$625$	−1.96184	−0.0784735
$626$	15.8931	0.635215
$627$	−1.86294	−0.0743985
$628$	19.2664	0.768811
$629$	39.0713	1.55787
$630$	0	0
$631$	−9.08708	−0.361751	−0.180875	−	0.983506i	$-0.557893\pi$
$631$	−9.08708	−0.361751	−0.180875	+	0.983506i	$0.557893\pi$
$632$	10.0368	0.399244
$633$	2.51275	0.0998727
$634$	−20.8998	−0.830036
$635$	−2.30990	−0.0916655
$636$	−17.0248	−0.675075
$637$	0	0
$638$	0.182997	0.00724491
$639$	5.49662	0.217443
$640$	−1.44504	−0.0571203
$641$	−24.0954	−0.951713	−0.475856	−	0.879523i	$-0.657862\pi$
$641$	−24.0954	−0.951713	−0.475856	+	0.879523i	$0.657862\pi$
$642$	12.0030	0.473720
$643$	44.6915	1.76246	0.881231	−	0.472685i	$-0.156715\pi$
$643$	44.6915	1.76246	0.881231	+	0.472685i	$0.156715\pi$
$644$	0	0
$645$	−24.6015	−0.968682
$646$	21.5623	0.848356
$647$	12.1558	0.477896	0.238948	−	0.971032i	$-0.423197\pi$
$647$	12.1558	0.477896	0.238948	+	0.971032i	$0.423197\pi$
$648$	−6.91484	−0.271640
$649$	3.58509	0.140727
$650$	−1.75840	−0.0689700
$651$	0	0
$652$	0.0174584	0.000683724 0
$653$	−27.2868	−1.06781	−0.533907	−	0.845543i	$-0.679277\pi$
$653$	−27.2868	−1.06781	−0.533907	+	0.845543i	$0.679277\pi$
$654$	13.3448	0.521824
$655$	2.57242	0.100513
$656$	8.87800	0.346628
$657$	−7.66547	−0.299058
$658$	0	0
$659$	−0.126310	−0.00492033	−0.00246016	−	0.999997i	$-0.500783\pi$
$659$	−0.126310	−0.00492033	−0.00246016	+	0.999997i	$0.500783\pi$
$660$	−0.692021	−0.0269369
$661$	15.5603	0.605227	0.302613	−	0.953113i	$-0.402141\pi$
$661$	15.5603	0.605227	0.302613	+	0.953113i	$0.402141\pi$
$662$	12.3123	0.478531
$663$	5.20477	0.202136
$664$	2.44504	0.0948860
$665$	0	0
$666$	4.10321	0.158996
$667$	1.73019	0.0669930
$668$	1.87800	0.0726621
$669$	43.2928	1.67380
$670$	−2.38835	−0.0922702
$671$	0.470533	0.0181647
$672$	0	0
$673$	20.5295	0.791353	0.395676	−	0.918390i	$-0.370510\pi$
$673$	20.5295	0.791353	0.395676	+	0.918390i	$0.370510\pi$
$674$	−2.92692	−0.112741
$675$	−16.2191	−0.624273
$676$	−12.6353	−0.485974
$677$	38.8998	1.49504	0.747520	−	0.664239i	$-0.231244\pi$
$677$	38.8998	1.49504	0.747520	+	0.664239i	$0.231244\pi$
$678$	−28.8672	−1.10864
$679$	0	0
$680$	8.00969	0.307158
$681$	28.4389	1.08978
$682$	2.48560	0.0951785
$683$	−32.0344	−1.22576	−0.612882	−	0.790174i	$-0.709990\pi$
$683$	−32.0344	−1.22576	−0.612882	+	0.790174i	$0.709990\pi$
$684$	2.26444	0.0865830
$685$	−0.523499	−0.0200019
$686$	0	0
$687$	31.9350	1.21839
$688$	−10.9487	−0.417415
$689$	6.61165	0.251884
$690$	−6.54288	−0.249083
$691$	−3.86964	−0.147208	−0.0736040	−	0.997288i	$-0.523450\pi$
$691$	−3.86964	−0.147208	−0.0736040	+	0.997288i	$0.523450\pi$
$692$	−12.0683	−0.458768
$693$	0	0
$694$	−0.262045	−0.00994710
$695$	13.8640	0.525892
$696$	0.923936	0.0350217
$697$	−49.2097	−1.86395
$698$	−30.6310	−1.15940
$699$	10.0218	0.379058
$700$	0	0
$701$	38.5411	1.45568	0.727838	−	0.685749i	$-0.240525\pi$
$701$	38.5411	1.45568	0.727838	+	0.685749i	$0.240525\pi$
$702$	3.36360	0.126951
$703$	27.4209	1.03420
$704$	−0.307979	−0.0116074
$705$	16.5526	0.623406
$706$	18.5429	0.697870
$707$	0	0
$708$	18.1008	0.680270
$709$	−2.56033	−0.0961554	−0.0480777	−	0.998844i	$-0.515310\pi$
$709$	−2.56033	−0.0961554	−0.0480777	+	0.998844i	$0.515310\pi$
$710$	13.6450	0.512089
$711$	−5.84249	−0.219111
$712$	9.35690	0.350664
$713$	23.5007	0.880107
$714$	0	0
$715$	0.268750	0.0100507
$716$	−24.4306	−0.913013
$717$	22.4155	0.837122
$718$	9.18060	0.342617
$719$	38.6752	1.44234	0.721170	−	0.692758i	$-0.243605\pi$
$719$	38.6752	1.44234	0.721170	+	0.692758i	$0.243605\pi$
$720$	0.841166	0.0313484
$721$	0	0
$722$	−3.86725	−0.143924
$723$	−0.527811	−0.0196295
$724$	−7.85623	−0.291975
$725$	1.73019	0.0642575
$726$	16.9571	0.629335
$727$	19.0597	0.706884	0.353442	−	0.935456i	$-0.385011\pi$
$727$	19.0597	0.706884	0.353442	+	0.935456i	$0.385011\pi$
$728$	0	0
$729$	30.0086	1.11143
$730$	−19.0291	−0.704297
$731$	60.6872	2.24460
$732$	2.37568	0.0878077
$733$	−30.2626	−1.11778	−0.558888	−	0.829243i	$-0.688772\pi$
$733$	−30.2626	−1.11778	−0.558888	+	0.829243i	$0.688772\pi$
$734$	−24.1642	−0.891917
$735$	0	0
$736$	−2.91185	−0.107332
$737$	−0.509025	−0.0187502
$738$	−5.16793	−0.190234
$739$	−6.89738	−0.253724	−0.126862	−	0.991920i	$-0.540491\pi$
$739$	−6.89738	−0.253724	−0.126862	+	0.991920i	$0.540491\pi$
$740$	10.1860	0.374444
$741$	3.65279	0.134189
$742$	0	0
$743$	0.440730	0.0161688	0.00808441	−	0.999967i	$-0.497427\pi$
$743$	0.440730	0.0161688	0.00808441	+	0.999967i	$0.497427\pi$
$744$	12.5496	0.460090
$745$	−9.82908	−0.360110
$746$	25.9148	0.948810
$747$	−1.42327	−0.0520748
$748$	1.70709	0.0624173
$749$	0	0
$750$	−17.7778	−0.649153
$751$	−18.0664	−0.659251	−0.329626	−	0.944112i	$-0.606923\pi$
$751$	−18.0664	−0.659251	−0.329626	+	0.944112i	$0.606923\pi$
$752$	7.36658	0.268632
$753$	−5.88636	−0.214511
$754$	−0.358815	−0.0130673
$755$	−33.8329	−1.23131
$756$	0	0
$757$	36.8491	1.33930	0.669651	−	0.742676i	$-0.266444\pi$
$757$	36.8491	1.33930	0.669651	+	0.742676i	$0.266444\pi$
$758$	32.0844	1.16536
$759$	−1.39447	−0.0506160
$760$	5.62133	0.203907
$761$	−3.29350	−0.119389	−0.0596947	−	0.998217i	$-0.519013\pi$
$761$	−3.29350	−0.119389	−0.0596947	+	0.998217i	$0.519013\pi$
$762$	−2.48560	−0.0900437
$763$	0	0
$764$	5.75063	0.208050
$765$	−4.66248	−0.168572
$766$	−12.6280	−0.456270
$767$	−7.02954	−0.253822
$768$	−1.55496	−0.0561097
$769$	19.1752	0.691476	0.345738	−	0.938331i	$-0.387629\pi$
$769$	19.1752	0.691476	0.345738	+	0.938331i	$0.387629\pi$
$770$	0	0
$771$	−48.9071	−1.76135
$772$	21.3991	0.770171
$773$	−35.8297	−1.28870	−0.644352	−	0.764729i	$-0.722873\pi$
$773$	−35.8297	−1.28870	−0.644352	+	0.764729i	$0.722873\pi$
$774$	6.37329	0.229083
$775$	23.5007	0.844169
$776$	−13.1347	−0.471507
$777$	0	0
$778$	0.341830	0.0122552
$779$	−34.5362	−1.23739
$780$	1.35690	0.0485847
$781$	2.90813	0.104061
$782$	16.1400	0.577167
$783$	−3.30963	−0.118277
$784$	0	0
$785$	−27.8407	−0.993677
$786$	2.76809	0.0987344
$787$	2.63209	0.0938238	0.0469119	−	0.998899i	$-0.485062\pi$
$787$	2.63209	0.0938238	0.0469119	+	0.998899i	$0.485062\pi$
$788$	−13.9312	−0.496280
$789$	−23.0653	−0.821147
$790$	−14.5036	−0.516017
$791$	0	0
$792$	0.179276	0.00637029
$793$	−0.922608	−0.0327628
$794$	8.78986	0.311940
$795$	24.6015	0.872525
$796$	−5.05131	−0.179039
$797$	20.2674	0.717909	0.358954	−	0.933355i	$-0.383133\pi$
$797$	20.2674	0.717909	0.358954	+	0.933355i	$0.383133\pi$
$798$	0	0
$799$	−40.8321	−1.44453
$800$	−2.91185	−0.102950
$801$	−5.44670	−0.192450
$802$	9.49934	0.335433
$803$	−4.05562	−0.143120
$804$	−2.57002	−0.0906377
$805$	0	0
$806$	−4.87369	−0.171668
$807$	19.4322	0.684047
$808$	8.74632	0.307694
$809$	40.7144	1.43144	0.715721	−	0.698387i	$-0.246098\pi$
$809$	40.7144	1.43144	0.715721	+	0.698387i	$0.246098\pi$
$810$	9.99223	0.351091
$811$	3.19806	0.112299	0.0561496	−	0.998422i	$-0.482118\pi$
$811$	3.19806	0.112299	0.0561496	+	0.998422i	$0.482118\pi$
$812$	0	0
$813$	−6.77107	−0.237472
$814$	2.17092	0.0760905
$815$	−0.0252281	−0.000883702 0
$816$	8.61894	0.301723
$817$	42.5913	1.49008
$818$	−17.5526	−0.613711
$819$	0	0
$820$	−12.8291	−0.448011
$821$	11.7616	0.410484	0.205242	−	0.978711i	$-0.434202\pi$
$821$	11.7616	0.410484	0.205242	+	0.978711i	$0.434202\pi$
$822$	−0.563319	−0.0196480
$823$	−23.7694	−0.828550	−0.414275	−	0.910152i	$-0.635965\pi$
$823$	−23.7694	−0.828550	−0.414275	+	0.910152i	$0.635965\pi$
$824$	−4.76271	−0.165917
$825$	−1.39447	−0.0485492
$826$	0	0
$827$	−14.2631	−0.495977	−0.247988	−	0.968763i	$-0.579770\pi$
$827$	−14.2631	−0.495977	−0.247988	+	0.968763i	$0.579770\pi$
$828$	1.69501	0.0589055
$829$	−35.7060	−1.24012	−0.620061	−	0.784554i	$-0.712892\pi$
$829$	−35.7060	−1.24012	−0.620061	+	0.784554i	$0.712892\pi$
$830$	−3.53319	−0.122639
$831$	20.5881	0.714192
$832$	0.603875	0.0209356
$833$	0	0
$834$	14.9186	0.516587
$835$	−2.71379	−0.0939146
$836$	1.19806	0.0414358
$837$	−44.9539	−1.55383
$838$	−19.0965	−0.659678
$839$	−4.55496	−0.157255	−0.0786273	−	0.996904i	$-0.525054\pi$
$839$	−4.55496	−0.157255	−0.0786273	+	0.996904i	$0.525054\pi$
$840$	0	0
$841$	−28.6469	−0.987826
$842$	−19.9095	−0.686125
$843$	−19.0502	−0.656125
$844$	−1.61596	−0.0556235
$845$	18.2586	0.628114
$846$	−4.28813	−0.147429
$847$	0	0
$848$	10.9487	0.375980
$849$	9.08277	0.311720
$850$	16.1400	0.553599
$851$	20.5254	0.703602
$852$	14.6829	0.503029
$853$	−38.7560	−1.32698	−0.663490	−	0.748185i	$-0.730926\pi$
$853$	−38.7560	−1.32698	−0.663490	+	0.748185i	$0.730926\pi$
$854$	0	0
$855$	−3.27221	−0.111907
$856$	−7.71917	−0.263836
$857$	−5.07905	−0.173497	−0.0867485	−	0.996230i	$-0.527648\pi$
$857$	−5.07905	−0.173497	−0.0867485	+	0.996230i	$0.527648\pi$
$858$	0.289192	0.00987286
$859$	−18.9396	−0.646211	−0.323105	−	0.946363i	$-0.604727\pi$
$859$	−18.9396	−0.646211	−0.323105	+	0.946363i	$0.604727\pi$
$860$	15.8213	0.539502
$861$	0	0
$862$	−6.21073	−0.211538
$863$	−47.6920	−1.62346	−0.811728	−	0.584036i	$-0.801473\pi$
$863$	−47.6920	−1.62346	−0.811728	+	0.584036i	$0.801473\pi$
$864$	5.57002	0.189496
$865$	17.4392	0.592950
$866$	−8.81163	−0.299431
$867$	−21.3394	−0.724725
$868$	0	0
$869$	−3.09113	−0.104859
$870$	−1.33513	−0.0452650
$871$	0.998081	0.0338187
$872$	−8.58211	−0.290627
$873$	7.64576	0.258770
$874$	11.3274	0.383154
$875$	0	0
$876$	−20.4765	−0.691837
$877$	−25.3752	−0.856860	−0.428430	−	0.903575i	$-0.640933\pi$
$877$	−25.3752	−0.856860	−0.428430	+	0.903575i	$0.640933\pi$
$878$	−36.1672	−1.22058
$879$	−6.38703	−0.215429
$880$	0.445042	0.0150024
$881$	4.35258	0.146642	0.0733211	−	0.997308i	$-0.476640\pi$
$881$	4.35258	0.146642	0.0733211	+	0.997308i	$0.476640\pi$
$882$	0	0
$883$	39.6088	1.33294	0.666471	−	0.745531i	$-0.267804\pi$
$883$	39.6088	1.33294	0.666471	+	0.745531i	$0.267804\pi$
$884$	−3.34721	−0.112579
$885$	−26.1564	−0.879239
$886$	20.2881	0.681593
$887$	−18.8619	−0.633320	−0.316660	−	0.948539i	$-0.602561\pi$
$887$	−18.8619	−0.633320	−0.316660	+	0.948539i	$0.602561\pi$
$888$	10.9608	0.367819
$889$	0	0
$890$	−13.5211	−0.453229
$891$	2.12962	0.0713450
$892$	−27.8418	−0.932211
$893$	−28.6566	−0.958958
$894$	−10.5767	−0.353739
$895$	35.3032	1.18006
$896$	0	0
$897$	2.73423	0.0912934
$898$	−31.3448	−1.04599
$899$	4.79550	0.159939
$900$	1.69501	0.0565002
$901$	−60.6872	−2.02178
$902$	−2.73423	−0.0910400
$903$	0	0
$904$	18.5646	0.617451
$905$	11.3526	0.377373
$906$	−36.4064	−1.20952
$907$	27.5502	0.914788	0.457394	−	0.889264i	$-0.348783\pi$
$907$	27.5502	0.914788	0.457394	+	0.889264i	$0.348783\pi$
$908$	−18.2892	−0.606948
$909$	−5.09128	−0.168867
$910$	0	0
$911$	−20.8576	−0.691042	−0.345521	−	0.938411i	$-0.612298\pi$
$911$	−20.8576	−0.691042	−0.345521	+	0.938411i	$0.612298\pi$
$912$	6.04892	0.200300
$913$	−0.753020	−0.0249213
$914$	8.78315	0.290521
$915$	−3.43296	−0.113490
$916$	−20.5375	−0.678578
$917$	0	0
$918$	−30.8740	−1.01899
$919$	38.1237	1.25758	0.628792	−	0.777574i	$-0.283550\pi$
$919$	38.1237	1.25758	0.628792	+	0.777574i	$0.283550\pi$
$920$	4.20775	0.138725
$921$	45.7174	1.50644
$922$	16.0968	0.530119
$923$	−5.70218	−0.187690
$924$	0	0
$925$	20.5254	0.674872
$926$	−1.68963	−0.0555246
$927$	2.77240	0.0910575
$928$	−0.594187	−0.0195051
$929$	−0.287273	−0.00942513	−0.00471256	−	0.999989i	$-0.501500\pi$
$929$	−0.287273	−0.00942513	−0.00471256	+	0.999989i	$0.501500\pi$
$930$	−18.1347	−0.594659
$931$	0	0
$932$	−6.44504	−0.211114
$933$	35.8364	1.17323
$934$	33.5623	1.09819
$935$	−2.46681	−0.0806734
$936$	−0.351519	−0.0114898
$937$	29.3773	0.959716	0.479858	−	0.877346i	$-0.340688\pi$
$937$	29.3773	0.959716	0.479858	+	0.877346i	$0.340688\pi$
$938$	0	0
$939$	−24.7131	−0.806480
$940$	−10.6450	−0.347202
$941$	−25.2392	−0.822775	−0.411387	−	0.911461i	$-0.634956\pi$
$941$	−25.2392	−0.822775	−0.411387	+	0.911461i	$0.634956\pi$
$942$	−29.9584	−0.976097
$943$	−25.8514	−0.841839
$944$	−11.6407	−0.378873
$945$	0	0
$946$	3.37196	0.109632
$947$	−33.3773	−1.08462	−0.542309	−	0.840179i	$-0.682450\pi$
$947$	−33.3773	−1.08462	−0.542309	+	0.840179i	$0.682450\pi$
$948$	−15.6069	−0.506887
$949$	7.95215	0.258138
$950$	11.3274	0.367508
$951$	32.4983	1.05383
$952$	0	0
$953$	29.0683	0.941614	0.470807	−	0.882236i	$-0.343963\pi$
$953$	29.0683	0.941614	0.470807	+	0.882236i	$0.343963\pi$
$954$	−6.37329	−0.206343
$955$	−8.30990	−0.268902
$956$	−14.4155	−0.466231
$957$	−0.284552	−0.00919827
$958$	7.94331	0.256637
$959$	0	0
$960$	2.24698	0.0725210
$961$	34.1360	1.10116
$962$	−4.25667	−0.137240
$963$	4.49337	0.144797
$964$	0.339437	0.0109325
$965$	−30.9226	−0.995434
$966$	0	0
$967$	−45.7036	−1.46973	−0.734865	−	0.678214i	$-0.762754\pi$
$967$	−45.7036	−1.46973	−0.734865	+	0.678214i	$0.762754\pi$
$968$	−10.9051	−0.350505
$969$	−33.5284	−1.07709
$970$	18.9801	0.609416
$971$	30.9885	0.994469	0.497234	−	0.867616i	$-0.334349\pi$
$971$	30.9885	0.994469	0.497234	+	0.867616i	$0.334349\pi$
$972$	−5.95779	−0.191096
$973$	0	0
$974$	35.8998	1.15030
$975$	2.73423	0.0875656
$976$	−1.52781	−0.0489040
$977$	−58.6480	−1.87632	−0.938158	−	0.346207i	$-0.887469\pi$
$977$	−58.6480	−1.87632	−0.938158	+	0.346207i	$0.887469\pi$
$978$	−0.0271471	−0.000868068 0
$979$	−2.88172	−0.0921003
$980$	0	0
$981$	4.99569	0.159500
$982$	−6.17092	−0.196922
$983$	−1.33214	−0.0424887	−0.0212444	−	0.999774i	$-0.506763\pi$
$983$	−1.33214	−0.0424887	−0.0212444	+	0.999774i	$0.506763\pi$
$984$	−13.8049	−0.440085
$985$	20.1312	0.641434
$986$	3.29350	0.104887
$987$	0	0
$988$	−2.34913	−0.0747357
$989$	31.8810	1.01376
$990$	−0.259061	−0.00823351
$991$	−56.5126	−1.79518	−0.897591	−	0.440829i	$-0.854684\pi$
$991$	−56.5126	−1.79518	−0.897591	+	0.440829i	$0.854684\pi$
$992$	−8.07069	−0.256245
$993$	−19.1451	−0.607551
$994$	0	0
$995$	7.29935	0.231405
$996$	−3.80194	−0.120469
$997$	−55.3032	−1.75147	−0.875735	−	0.482792i	$-0.839623\pi$
$997$	−55.3032	−1.75147	−0.875735	+	0.482792i	$0.839623\pi$
$998$	5.19673	0.164500
$999$	−39.2626	−1.24221

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	686.2.a.c.1.2	✓	3
3.2	odd	2		6174.2.a.e.1.2		3
4.3	odd	2		5488.2.a.f.1.2		3
7.2	even	3		686.2.c.b.361.2		6
7.3	odd	6		686.2.c.a.667.2		6
7.4	even	3		686.2.c.b.667.2		6
7.5	odd	6		686.2.c.a.361.2		6
7.6	odd	2		686.2.a.d.1.2	yes	3
21.20	even	2		6174.2.a.c.1.2		3
28.27	even	2		5488.2.a.a.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
686.2.a.c.1.2	✓	3	1.1	even	1	trivial
686.2.a.d.1.2	yes	3	7.6	odd	2
686.2.c.a.361.2		6	7.5	odd	6
686.2.c.a.667.2		6	7.3	odd	6
686.2.c.b.361.2		6	7.2	even	3
686.2.c.b.667.2		6	7.4	even	3
5488.2.a.a.1.2		3	28.27	even	2
5488.2.a.f.1.2		3	4.3	odd	2
6174.2.a.c.1.2		3	21.20	even	2
6174.2.a.e.1.2		3	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 686 = 2 \cdot 7^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	686.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.55496 q^{3} +1.00000 q^{4} -1.44504 q^{5} -1.55496 q^{6} +1.00000 q^{8} -0.582105 q^{9} -1.44504 q^{10} -0.307979 q^{11} -1.55496 q^{12} +0.603875 q^{13} +2.24698 q^{15} +1.00000 q^{16} -5.54288 q^{17} -0.582105 q^{18} -3.89008 q^{19} -1.44504 q^{20} -0.307979 q^{22} -2.91185 q^{23} -1.55496 q^{24} -2.91185 q^{25} +0.603875 q^{26} +5.57002 q^{27} -0.594187 q^{29} +2.24698 q^{30} -8.07069 q^{31} +1.00000 q^{32} +0.478894 q^{33} -5.54288 q^{34} -0.582105 q^{36} -7.04892 q^{37} -3.89008 q^{38} -0.939001 q^{39} -1.44504 q^{40} +8.87800 q^{41} -10.9487 q^{43} -0.307979 q^{44} +0.841166 q^{45} -2.91185 q^{46} +7.36658 q^{47} -1.55496 q^{48} -2.91185 q^{50} +8.61894 q^{51} +0.603875 q^{52} +10.9487 q^{53} +5.57002 q^{54} +0.445042 q^{55} +6.04892 q^{57} -0.594187 q^{58} -11.6407 q^{59} +2.24698 q^{60} -1.52781 q^{61} -8.07069 q^{62} +1.00000 q^{64} -0.872625 q^{65} +0.478894 q^{66} +1.65279 q^{67} -5.54288 q^{68} +4.52781 q^{69} -9.44265 q^{71} -0.582105 q^{72} +13.1685 q^{73} -7.04892 q^{74} +4.52781 q^{75} -3.89008 q^{76} -0.939001 q^{78} +10.0368 q^{79} -1.44504 q^{80} -6.91484 q^{81} +8.87800 q^{82} +2.44504 q^{83} +8.00969 q^{85} -10.9487 q^{86} +0.923936 q^{87} -0.307979 q^{88} +9.35690 q^{89} +0.841166 q^{90} -2.91185 q^{92} +12.5496 q^{93} +7.36658 q^{94} +5.62133 q^{95} -1.55496 q^{96} -13.1347 q^{97} +0.179276 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - 5 q^{3} + 3 q^{4} - 4 q^{5} - 5 q^{6} + 3 q^{8} + 4 q^{9} - 4 q^{10} - 6 q^{11} - 5 q^{12} - 7 q^{13} + 2 q^{15} + 3 q^{16} + 2 q^{17} + 4 q^{18} - 11 q^{19} - 4 q^{20} - 6 q^{22} - 5 q^{23}+ \cdots - 36 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 - 5 * q^3 + 3 * q^4 - 4 * q^5 - 5 * q^6 + 3 * q^8 + 4 * q^9 - 4 * q^10 - 6 * q^11 - 5 * q^12 - 7 * q^13 + 2 * q^15 + 3 * q^16 + 2 * q^17 + 4 * q^18 - 11 * q^19 - 4 * q^20 - 6 * q^22 - 5 * q^23 - 5 * q^24 - 5 * q^25 - 7 * q^26 - 8 * q^27 - 15 * q^29 + 2 * q^30 - 12 * q^31 + 3 * q^32 + 17 * q^33 + 2 * q^34 + 4 * q^36 - 12 * q^37 - 11 * q^38 + 7 * q^39 - 4 * q^40 + 7 * q^41 - q^43 - 6 * q^44 + 11 * q^45 - 5 * q^46 - 4 * q^47 - 5 * q^48 - 5 * q^50 - 8 * q^51 - 7 * q^52 + q^53 - 8 * q^54 + q^55 + 9 * q^57 - 15 * q^58 + 2 * q^59 + 2 * q^60 - 11 * q^61 - 12 * q^62 + 3 * q^64 + 14 * q^65 + 17 * q^66 - 13 * q^67 + 2 * q^68 + 20 * q^69 + 13 * q^71 + 4 * q^72 + 9 * q^73 - 12 * q^74 + 20 * q^75 - 11 * q^76 + 7 * q^78 + 2 * q^79 - 4 * q^80 + 27 * q^81 + 7 * q^82 + 7 * q^83 + 2 * q^85 - q^86 + 18 * q^87 - 6 * q^88 + 24 * q^89 + 11 * q^90 - 5 * q^92 + 41 * q^93 - 4 * q^94 + 24 * q^95 - 5 * q^96 + 7 * q^97 - 36 * q^99

Introduction

L-functions

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