LMFDB - Embedded newform 725.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [725,2,Mod(1,725)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(725, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("725.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 725 = 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	725.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:	$5.78915414654$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 145)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	725.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.414214 q^{2} +2.00000 q^{3} -1.82843 q^{4} -0.828427 q^{6} +4.82843 q^{7} +1.58579 q^{8} +1.00000 q^{9} +0.828427 q^{11} -3.65685 q^{12} +2.00000 q^{13} -2.00000 q^{14} +3.00000 q^{16} -2.82843 q^{17} -0.414214 q^{18} -4.82843 q^{19} +9.65685 q^{21} -0.343146 q^{22} +3.17157 q^{23} +3.17157 q^{24} -0.828427 q^{26} -4.00000 q^{27} -8.82843 q^{28} +1.00000 q^{29} +6.48528 q^{31} -4.41421 q^{32} +1.65685 q^{33} +1.17157 q^{34} -1.82843 q^{36} +8.48528 q^{37} +2.00000 q^{38} +4.00000 q^{39} -6.00000 q^{41} -4.00000 q^{42} +6.00000 q^{43} -1.51472 q^{44} -1.31371 q^{46} +11.6569 q^{47} +6.00000 q^{48} +16.3137 q^{49} -5.65685 q^{51} -3.65685 q^{52} +3.65685 q^{53} +1.65685 q^{54} +7.65685 q^{56} -9.65685 q^{57} -0.414214 q^{58} -3.65685 q^{61} -2.68629 q^{62} +4.82843 q^{63} -4.17157 q^{64} -0.686292 q^{66} -6.48528 q^{67} +5.17157 q^{68} +6.34315 q^{69} -15.3137 q^{71} +1.58579 q^{72} -8.48528 q^{73} -3.51472 q^{74} +8.82843 q^{76} +4.00000 q^{77} -1.65685 q^{78} -2.48528 q^{79} -11.0000 q^{81} +2.48528 q^{82} -7.17157 q^{83} -17.6569 q^{84} -2.48528 q^{86} +2.00000 q^{87} +1.31371 q^{88} -7.65685 q^{89} +9.65685 q^{91} -5.79899 q^{92} +12.9706 q^{93} -4.82843 q^{94} -8.82843 q^{96} +12.4853 q^{97} -6.75736 q^{98} +0.828427 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 4 q^{6} + 4 q^{7} + 6 q^{8} + 2 q^{9} - 4 q^{11} + 4 q^{12} + 4 q^{13} - 4 q^{14} + 6 q^{16} + 2 q^{18} - 4 q^{19} + 8 q^{21} - 12 q^{22} + 12 q^{23} + 12 q^{24} + 4 q^{26}+ \cdots - 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 4 * q^3 + 2 * q^4 + 4 * q^6 + 4 * q^7 + 6 * q^8 + 2 * q^9 - 4 * q^11 + 4 * q^12 + 4 * q^13 - 4 * q^14 + 6 * q^16 + 2 * q^18 - 4 * q^19 + 8 * q^21 - 12 * q^22 + 12 * q^23 + 12 * q^24 + 4 * q^26 - 8 * q^27 - 12 * q^28 + 2 * q^29 - 4 * q^31 - 6 * q^32 - 8 * q^33 + 8 * q^34 + 2 * q^36 + 4 * q^38 + 8 * q^39 - 12 * q^41 - 8 * q^42 + 12 * q^43 - 20 * q^44 + 20 * q^46 + 12 * q^47 + 12 * q^48 + 10 * q^49 + 4 * q^52 - 4 * q^53 - 8 * q^54 + 4 * q^56 - 8 * q^57 + 2 * q^58 + 4 * q^61 - 28 * q^62 + 4 * q^63 - 14 * q^64 - 24 * q^66 + 4 * q^67 + 16 * q^68 + 24 * q^69 - 8 * q^71 + 6 * q^72 - 24 * q^74 + 12 * q^76 + 8 * q^77 + 8 * q^78 + 12 * q^79 - 22 * q^81 - 12 * q^82 - 20 * q^83 - 24 * q^84 + 12 * q^86 + 4 * q^87 - 20 * q^88 - 4 * q^89 + 8 * q^91 + 28 * q^92 - 8 * q^93 - 4 * q^94 - 12 * q^96 + 8 * q^97 - 22 * q^98 - 4 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−0.414214	−0.292893	−0.146447	−	0.989219i	$-0.546784\pi$
$2$	−0.414214	−0.292893	−0.146447	+	0.989219i	$0.546784\pi$
$3$	2.00000	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$3$	2.00000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	−1.82843	−0.914214
$5$	0	0
$5$	0	0
$6$	−0.828427	−0.338204
$7$	4.82843	1.82497	0.912487	−	0.409106i	$-0.134159\pi$
$7$	4.82843	1.82497	0.912487	+	0.409106i	$0.134159\pi$
$8$	1.58579	0.560660
$9$	1.00000	0.333333
$10$	0	0
$11$	0.828427	0.249780	0.124890	−	0.992171i	$-0.460142\pi$
$11$	0.828427	0.249780	0.124890	+	0.992171i	$0.460142\pi$
$12$	−3.65685	−1.05564
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	3.00000	0.750000
$17$	−2.82843	−0.685994	−0.342997	−	0.939336i	$-0.611442\pi$
$17$	−2.82843	−0.685994	−0.342997	+	0.939336i	$0.611442\pi$
$18$	−0.414214	−0.0976311
$19$	−4.82843	−1.10772	−0.553859	−	0.832611i	$-0.686845\pi$
$19$	−4.82843	−1.10772	−0.553859	+	0.832611i	$0.686845\pi$
$20$	0	0
$21$	9.65685	2.10730
$22$	−0.343146	−0.0731589
$23$	3.17157	0.661319	0.330659	−	0.943750i	$-0.392729\pi$
$23$	3.17157	0.661319	0.330659	+	0.943750i	$0.392729\pi$
$24$	3.17157	0.647395
$25$	0	0
$26$	−0.828427	−0.162468
$27$	−4.00000	−0.769800
$28$	−8.82843	−1.66842
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	6.48528	1.16479	0.582395	−	0.812906i	$-0.302116\pi$
$31$	6.48528	1.16479	0.582395	+	0.812906i	$0.302116\pi$
$32$	−4.41421	−0.780330
$33$	1.65685	0.288421
$34$	1.17157	0.200923
$35$	0	0
$36$	−1.82843	−0.304738
$37$	8.48528	1.39497	0.697486	−	0.716599i	$-0.254302\pi$
$37$	8.48528	1.39497	0.697486	+	0.716599i	$0.254302\pi$
$38$	2.00000	0.324443
$39$	4.00000	0.640513
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	−4.00000	−0.617213
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	−1.51472	−0.228352
$45$	0	0
$46$	−1.31371	−0.193696
$47$	11.6569	1.70033	0.850163	−	0.526519i	$-0.176503\pi$
$47$	11.6569	1.70033	0.850163	+	0.526519i	$0.176503\pi$
$48$	6.00000	0.866025
$49$	16.3137	2.33053
$50$	0	0
$51$	−5.65685	−0.792118
$52$	−3.65685	−0.507114
$53$	3.65685	0.502308	0.251154	−	0.967947i	$-0.419190\pi$
$53$	3.65685	0.502308	0.251154	+	0.967947i	$0.419190\pi$
$54$	1.65685	0.225469
$55$	0	0
$56$	7.65685	1.02319
$57$	−9.65685	−1.27908
$58$	−0.414214	−0.0543889
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−3.65685	−0.468212	−0.234106	−	0.972211i	$-0.575216\pi$
$61$	−3.65685	−0.468212	−0.234106	+	0.972211i	$0.575216\pi$
$62$	−2.68629	−0.341159
$63$	4.82843	0.608325
$64$	−4.17157	−0.521447
$65$	0	0
$66$	−0.686292	−0.0844766
$67$	−6.48528	−0.792303	−0.396152	−	0.918185i	$-0.629655\pi$
$67$	−6.48528	−0.792303	−0.396152	+	0.918185i	$0.629655\pi$
$68$	5.17157	0.627145
$69$	6.34315	0.763625
$70$	0	0
$71$	−15.3137	−1.81740	−0.908701	−	0.417447i	$-0.862925\pi$
$71$	−15.3137	−1.81740	−0.908701	+	0.417447i	$0.862925\pi$
$72$	1.58579	0.186887
$73$	−8.48528	−0.993127	−0.496564	−	0.868000i	$-0.665405\pi$
$73$	−8.48528	−0.993127	−0.496564	+	0.868000i	$0.665405\pi$
$74$	−3.51472	−0.408578
$75$	0	0
$76$	8.82843	1.01269
$77$	4.00000	0.455842
$78$	−1.65685	−0.187602
$79$	−2.48528	−0.279616	−0.139808	−	0.990179i	$-0.544649\pi$
$79$	−2.48528	−0.279616	−0.139808	+	0.990179i	$0.544649\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	2.48528	0.274453
$83$	−7.17157	−0.787182	−0.393591	−	0.919286i	$-0.628767\pi$
$83$	−7.17157	−0.787182	−0.393591	+	0.919286i	$0.628767\pi$
$84$	−17.6569	−1.92652
$85$	0	0
$86$	−2.48528	−0.267995
$87$	2.00000	0.214423
$88$	1.31371	0.140042
$89$	−7.65685	−0.811625	−0.405812	−	0.913956i	$-0.633011\pi$
$89$	−7.65685	−0.811625	−0.405812	+	0.913956i	$0.633011\pi$
$90$	0	0
$91$	9.65685	1.01231
$92$	−5.79899	−0.604586
$93$	12.9706	1.34498
$94$	−4.82843	−0.498014
$95$	0	0
$96$	−8.82843	−0.901048
$97$	12.4853	1.26769	0.633844	−	0.773461i	$-0.281476\pi$
$97$	12.4853	1.26769	0.633844	+	0.773461i	$0.281476\pi$
$98$	−6.75736	−0.682596
$99$	0.828427	0.0832601
$100$	0	0
$101$	15.6569	1.55792	0.778958	−	0.627077i	$-0.215749\pi$
$101$	15.6569	1.55792	0.778958	+	0.627077i	$0.215749\pi$
$102$	2.34315	0.232006
$103$	−16.1421	−1.59053	−0.795266	−	0.606261i	$-0.792669\pi$
$103$	−16.1421	−1.59053	−0.795266	+	0.606261i	$0.792669\pi$
$104$	3.17157	0.310998
$105$	0	0
$106$	−1.51472	−0.147122
$107$	−20.1421	−1.94721	−0.973607	−	0.228232i	$-0.926706\pi$
$107$	−20.1421	−1.94721	−0.973607	+	0.228232i	$0.926706\pi$
$108$	7.31371	0.703762
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	16.9706	1.61077
$112$	14.4853	1.36873
$113$	2.82843	0.266076	0.133038	−	0.991111i	$-0.457527\pi$
$113$	2.82843	0.266076	0.133038	+	0.991111i	$0.457527\pi$
$114$	4.00000	0.374634
$115$	0	0
$116$	−1.82843	−0.169765
$117$	2.00000	0.184900
$118$	0	0
$119$	−13.6569	−1.25192
$120$	0	0
$121$	−10.3137	−0.937610
$122$	1.51472	0.137136
$123$	−12.0000	−1.08200
$124$	−11.8579	−1.06487
$125$	0	0
$126$	−2.00000	−0.178174
$127$	−6.00000	−0.532414	−0.266207	−	0.963916i	$-0.585770\pi$
$127$	−6.00000	−0.532414	−0.266207	+	0.963916i	$0.585770\pi$
$128$	10.5563	0.933058
$129$	12.0000	1.05654
$130$	0	0
$131$	−12.1421	−1.06086	−0.530432	−	0.847728i	$-0.677970\pi$
$131$	−12.1421	−1.06086	−0.530432	+	0.847728i	$0.677970\pi$
$132$	−3.02944	−0.263679
$133$	−23.3137	−2.02155
$134$	2.68629	0.232060
$135$	0	0
$136$	−4.48528	−0.384610
$137$	5.17157	0.441837	0.220919	−	0.975292i	$-0.429094\pi$
$137$	5.17157	0.441837	0.220919	+	0.975292i	$0.429094\pi$
$138$	−2.62742	−0.223661
$139$	21.6569	1.83691	0.918455	−	0.395525i	$-0.129437\pi$
$139$	21.6569	1.83691	0.918455	+	0.395525i	$0.129437\pi$
$140$	0	0
$141$	23.3137	1.96337
$142$	6.34315	0.532305
$143$	1.65685	0.138553
$144$	3.00000	0.250000
$145$	0	0
$146$	3.51472	0.290880
$147$	32.6274	2.69106
$148$	−15.5147	−1.27530
$149$	9.31371	0.763009	0.381504	−	0.924367i	$-0.375406\pi$
$149$	9.31371	0.763009	0.381504	+	0.924367i	$0.375406\pi$
$150$	0	0
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	−7.65685	−0.621053
$153$	−2.82843	−0.228665
$154$	−1.65685	−0.133513
$155$	0	0
$156$	−7.31371	−0.585565
$157$	−0.485281	−0.0387297	−0.0193648	−	0.999812i	$-0.506164\pi$
$157$	−0.485281	−0.0387297	−0.0193648	+	0.999812i	$0.506164\pi$
$158$	1.02944	0.0818976
$159$	7.31371	0.580015
$160$	0	0
$161$	15.3137	1.20689
$162$	4.55635	0.357981
$163$	8.34315	0.653486	0.326743	−	0.945113i	$-0.394049\pi$
$163$	8.34315	0.653486	0.326743	+	0.945113i	$0.394049\pi$
$164$	10.9706	0.856657
$165$	0	0
$166$	2.97056	0.230560
$167$	2.48528	0.192317	0.0961584	−	0.995366i	$-0.469344\pi$
$167$	2.48528	0.192317	0.0961584	+	0.995366i	$0.469344\pi$
$168$	15.3137	1.18148
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−4.82843	−0.369239
$172$	−10.9706	−0.836498
$173$	−17.3137	−1.31634	−0.658168	−	0.752871i	$-0.728669\pi$
$173$	−17.3137	−1.31634	−0.658168	+	0.752871i	$0.728669\pi$
$174$	−0.828427	−0.0628029
$175$	0	0
$176$	2.48528	0.187335
$177$	0	0
$178$	3.17157	0.237719
$179$	−23.3137	−1.74255	−0.871274	−	0.490797i	$-0.836706\pi$
$179$	−23.3137	−1.74255	−0.871274	+	0.490797i	$0.836706\pi$
$180$	0	0
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	−4.00000	−0.296500
$183$	−7.31371	−0.540645
$184$	5.02944	0.370775
$185$	0	0
$186$	−5.37258	−0.393937
$187$	−2.34315	−0.171348
$188$	−21.3137	−1.55446
$189$	−19.3137	−1.40487
$190$	0	0
$191$	−20.8284	−1.50709	−0.753546	−	0.657395i	$-0.771658\pi$
$191$	−20.8284	−1.50709	−0.753546	+	0.657395i	$0.771658\pi$
$192$	−8.34315	−0.602115
$193$	−4.48528	−0.322858	−0.161429	−	0.986884i	$-0.551610\pi$
$193$	−4.48528	−0.322858	−0.161429	+	0.986884i	$0.551610\pi$
$194$	−5.17157	−0.371297
$195$	0	0
$196$	−29.8284	−2.13060
$197$	19.6569	1.40049	0.700246	−	0.713901i	$-0.253073\pi$
$197$	19.6569	1.40049	0.700246	+	0.713901i	$0.253073\pi$
$198$	−0.343146	−0.0243863
$199$	12.0000	0.850657	0.425329	−	0.905039i	$-0.360158\pi$
$199$	12.0000	0.850657	0.425329	+	0.905039i	$0.360158\pi$
$200$	0	0
$201$	−12.9706	−0.914873
$202$	−6.48528	−0.456303
$203$	4.82843	0.338889
$204$	10.3431	0.724165
$205$	0	0
$206$	6.68629	0.465856
$207$	3.17157	0.220440
$208$	6.00000	0.416025
$209$	−4.00000	−0.276686
$210$	0	0
$211$	−0.828427	−0.0570313	−0.0285156	−	0.999593i	$-0.509078\pi$
$211$	−0.828427	−0.0570313	−0.0285156	+	0.999593i	$0.509078\pi$
$212$	−6.68629	−0.459216
$213$	−30.6274	−2.09856
$214$	8.34315	0.570326
$215$	0	0
$216$	−6.34315	−0.431596
$217$	31.3137	2.12571
$218$	−0.828427	−0.0561082
$219$	−16.9706	−1.14676
$220$	0	0
$221$	−5.65685	−0.380521
$222$	−7.02944	−0.471785
$223$	17.7990	1.19191	0.595954	−	0.803018i	$-0.296774\pi$
$223$	17.7990	1.19191	0.595954	+	0.803018i	$0.296774\pi$
$224$	−21.3137	−1.42408
$225$	0	0
$226$	−1.17157	−0.0779319
$227$	−20.1421	−1.33688	−0.668440	−	0.743766i	$-0.733038\pi$
$227$	−20.1421	−1.33688	−0.668440	+	0.743766i	$0.733038\pi$
$228$	17.6569	1.16935
$229$	−2.00000	−0.132164	−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000	−0.132164	−0.0660819	+	0.997814i	$0.521050\pi$
$230$	0	0
$231$	8.00000	0.526361
$232$	1.58579	0.104112
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	−0.828427	−0.0541560
$235$	0	0
$236$	0	0
$237$	−4.97056	−0.322873
$238$	5.65685	0.366679
$239$	−0.686292	−0.0443925	−0.0221963	−	0.999754i	$-0.507066\pi$
$239$	−0.686292	−0.0443925	−0.0221963	+	0.999754i	$0.507066\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	4.27208	0.274620
$243$	−10.0000	−0.641500
$244$	6.68629	0.428046
$245$	0	0
$246$	4.97056	0.316912
$247$	−9.65685	−0.614451
$248$	10.2843	0.653052
$249$	−14.3431	−0.908960
$250$	0	0
$251$	8.82843	0.557245	0.278623	−	0.960401i	$-0.410122\pi$
$251$	8.82843	0.557245	0.278623	+	0.960401i	$0.410122\pi$
$252$	−8.82843	−0.556139
$253$	2.62742	0.165184
$254$	2.48528	0.155940
$255$	0	0
$256$	3.97056	0.248160
$257$	−6.68629	−0.417079	−0.208540	−	0.978014i	$-0.566871\pi$
$257$	−6.68629	−0.417079	−0.208540	+	0.978014i	$0.566871\pi$
$258$	−4.97056	−0.309454
$259$	40.9706	2.54579
$260$	0	0
$261$	1.00000	0.0618984
$262$	5.02944	0.310720
$263$	19.6569	1.21209	0.606047	−	0.795429i	$-0.292754\pi$
$263$	19.6569	1.21209	0.606047	+	0.795429i	$0.292754\pi$
$264$	2.62742	0.161706
$265$	0	0
$266$	9.65685	0.592100
$267$	−15.3137	−0.937184
$268$	11.8579	0.724334
$269$	−21.3137	−1.29952	−0.649760	−	0.760140i	$-0.725131\pi$
$269$	−21.3137	−1.29952	−0.649760	+	0.760140i	$0.725131\pi$
$270$	0	0
$271$	−9.79899	−0.595246	−0.297623	−	0.954683i	$-0.596194\pi$
$271$	−9.79899	−0.595246	−0.297623	+	0.954683i	$0.596194\pi$
$272$	−8.48528	−0.514496
$273$	19.3137	1.16892
$274$	−2.14214	−0.129411
$275$	0	0
$276$	−11.5980	−0.698116
$277$	3.65685	0.219719	0.109860	−	0.993947i	$-0.464960\pi$
$277$	3.65685	0.219719	0.109860	+	0.993947i	$0.464960\pi$
$278$	−8.97056	−0.538019
$279$	6.48528	0.388264
$280$	0	0
$281$	−29.3137	−1.74871	−0.874355	−	0.485288i	$-0.838715\pi$
$281$	−29.3137	−1.74871	−0.874355	+	0.485288i	$0.838715\pi$
$282$	−9.65685	−0.575057
$283$	−4.82843	−0.287020	−0.143510	−	0.989649i	$-0.545839\pi$
$283$	−4.82843	−0.287020	−0.143510	+	0.989649i	$0.545839\pi$
$284$	28.0000	1.66149
$285$	0	0
$286$	−0.686292	−0.0405813
$287$	−28.9706	−1.71008
$288$	−4.41421	−0.260110
$289$	−9.00000	−0.529412
$290$	0	0
$291$	24.9706	1.46380
$292$	15.5147	0.907930
$293$	−8.48528	−0.495715	−0.247858	−	0.968796i	$-0.579727\pi$
$293$	−8.48528	−0.495715	−0.247858	+	0.968796i	$0.579727\pi$
$294$	−13.5147	−0.788194
$295$	0	0
$296$	13.4558	0.782105
$297$	−3.31371	−0.192281
$298$	−3.85786	−0.223480
$299$	6.34315	0.366834
$300$	0	0
$301$	28.9706	1.66984
$302$	4.97056	0.286024
$303$	31.3137	1.79893
$304$	−14.4853	−0.830788
$305$	0	0
$306$	1.17157	0.0669744
$307$	22.9706	1.31100	0.655500	−	0.755195i	$-0.272458\pi$
$307$	22.9706	1.31100	0.655500	+	0.755195i	$0.272458\pi$
$308$	−7.31371	−0.416737
$309$	−32.2843	−1.83659
$310$	0	0
$311$	14.4853	0.821385	0.410692	−	0.911774i	$-0.365287\pi$
$311$	14.4853	0.821385	0.410692	+	0.911774i	$0.365287\pi$
$312$	6.34315	0.359110
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	0.201010	0.0113437
$315$	0	0
$316$	4.54416	0.255629
$317$	−2.82843	−0.158860	−0.0794301	−	0.996840i	$-0.525310\pi$
$317$	−2.82843	−0.158860	−0.0794301	+	0.996840i	$0.525310\pi$
$318$	−3.02944	−0.169882
$319$	0.828427	0.0463830
$320$	0	0
$321$	−40.2843	−2.24845
$322$	−6.34315	−0.353490
$323$	13.6569	0.759888
$324$	20.1127	1.11737
$325$	0	0
$326$	−3.45584	−0.191402
$327$	4.00000	0.221201
$328$	−9.51472	−0.525362
$329$	56.2843	3.10305
$330$	0	0
$331$	21.7990	1.19818	0.599090	−	0.800681i	$-0.295529\pi$
$331$	21.7990	1.19818	0.599090	+	0.800681i	$0.295529\pi$
$332$	13.1127	0.719653
$333$	8.48528	0.464991
$334$	−1.02944	−0.0563283
$335$	0	0
$336$	28.9706	1.58047
$337$	1.17157	0.0638196	0.0319098	−	0.999491i	$-0.489841\pi$
$337$	1.17157	0.0638196	0.0319098	+	0.999491i	$0.489841\pi$
$338$	3.72792	0.202772
$339$	5.65685	0.307238
$340$	0	0
$341$	5.37258	0.290942
$342$	2.00000	0.108148
$343$	44.9706	2.42818
$344$	9.51472	0.512999
$345$	0	0
$346$	7.17157	0.385546
$347$	8.14214	0.437093	0.218546	−	0.975827i	$-0.429869\pi$
$347$	8.14214	0.437093	0.218546	+	0.975827i	$0.429869\pi$
$348$	−3.65685	−0.196028
$349$	20.6274	1.10416	0.552080	−	0.833791i	$-0.313834\pi$
$349$	20.6274	1.10416	0.552080	+	0.833791i	$0.313834\pi$
$350$	0	0
$351$	−8.00000	−0.427008
$352$	−3.65685	−0.194911
$353$	4.34315	0.231162	0.115581	−	0.993298i	$-0.463127\pi$
$353$	4.34315	0.231162	0.115581	+	0.993298i	$0.463127\pi$
$354$	0	0
$355$	0	0
$356$	14.0000	0.741999
$357$	−27.3137	−1.44559
$358$	9.65685	0.510381
$359$	−3.85786	−0.203610	−0.101805	−	0.994804i	$-0.532462\pi$
$359$	−3.85786	−0.203610	−0.101805	+	0.994804i	$0.532462\pi$
$360$	0	0
$361$	4.31371	0.227037
$362$	2.48528	0.130623
$363$	−20.6274	−1.08266
$364$	−17.6569	−0.925471
$365$	0	0
$366$	3.02944	0.158351
$367$	−18.0000	−0.939592	−0.469796	−	0.882775i	$-0.655673\pi$
$367$	−18.0000	−0.939592	−0.469796	+	0.882775i	$0.655673\pi$
$368$	9.51472	0.495989
$369$	−6.00000	−0.312348
$370$	0	0
$371$	17.6569	0.916698
$372$	−23.7157	−1.22960
$373$	6.97056	0.360922	0.180461	−	0.983582i	$-0.442241\pi$
$373$	6.97056	0.360922	0.180461	+	0.983582i	$0.442241\pi$
$374$	0.970563	0.0501866
$375$	0	0
$376$	18.4853	0.953306
$377$	2.00000	0.103005
$378$	8.00000	0.411476
$379$	22.4853	1.15499	0.577496	−	0.816394i	$-0.304030\pi$
$379$	22.4853	1.15499	0.577496	+	0.816394i	$0.304030\pi$
$380$	0	0
$381$	−12.0000	−0.614779
$382$	8.62742	0.441417
$383$	−2.48528	−0.126992	−0.0634960	−	0.997982i	$-0.520225\pi$
$383$	−2.48528	−0.126992	−0.0634960	+	0.997982i	$0.520225\pi$
$384$	21.1127	1.07740
$385$	0	0
$386$	1.85786	0.0945628
$387$	6.00000	0.304997
$388$	−22.8284	−1.15894
$389$	−29.3137	−1.48626	−0.743132	−	0.669145i	$-0.766661\pi$
$389$	−29.3137	−1.48626	−0.743132	+	0.669145i	$0.766661\pi$
$390$	0	0
$391$	−8.97056	−0.453661
$392$	25.8701	1.30664
$393$	−24.2843	−1.22498
$394$	−8.14214	−0.410195
$395$	0	0
$396$	−1.51472	−0.0761175
$397$	19.6569	0.986549	0.493275	−	0.869874i	$-0.335800\pi$
$397$	19.6569	0.986549	0.493275	+	0.869874i	$0.335800\pi$
$398$	−4.97056	−0.249152
$399$	−46.6274	−2.33429
$400$	0	0
$401$	−6.68629	−0.333897	−0.166949	−	0.985966i	$-0.553391\pi$
$401$	−6.68629	−0.333897	−0.166949	+	0.985966i	$0.553391\pi$
$402$	5.37258	0.267960
$403$	12.9706	0.646110
$404$	−28.6274	−1.42427
$405$	0	0
$406$	−2.00000	−0.0992583
$407$	7.02944	0.348436
$408$	−8.97056	−0.444109
$409$	−2.97056	−0.146885	−0.0734424	−	0.997299i	$-0.523399\pi$
$409$	−2.97056	−0.146885	−0.0734424	+	0.997299i	$0.523399\pi$
$410$	0	0
$411$	10.3431	0.510190
$412$	29.5147	1.45409
$413$	0	0
$414$	−1.31371	−0.0645653
$415$	0	0
$416$	−8.82843	−0.432849
$417$	43.3137	2.12108
$418$	1.65685	0.0810394
$419$	28.9706	1.41530	0.707652	−	0.706561i	$-0.249754\pi$
$419$	28.9706	1.41530	0.707652	+	0.706561i	$0.249754\pi$
$420$	0	0
$421$	18.9706	0.924569	0.462284	−	0.886732i	$-0.347030\pi$
$421$	18.9706	0.924569	0.462284	+	0.886732i	$0.347030\pi$
$422$	0.343146	0.0167041
$423$	11.6569	0.566776
$424$	5.79899	0.281624
$425$	0	0
$426$	12.6863	0.614653
$427$	−17.6569	−0.854475
$428$	36.8284	1.78017
$429$	3.31371	0.159987
$430$	0	0
$431$	3.31371	0.159616	0.0798079	−	0.996810i	$-0.474569\pi$
$431$	3.31371	0.159616	0.0798079	+	0.996810i	$0.474569\pi$
$432$	−12.0000	−0.577350
$433$	29.1716	1.40190	0.700948	−	0.713212i	$-0.252760\pi$
$433$	29.1716	1.40190	0.700948	+	0.713212i	$0.252760\pi$
$434$	−12.9706	−0.622607
$435$	0	0
$436$	−3.65685	−0.175132
$437$	−15.3137	−0.732554
$438$	7.02944	0.335880
$439$	−10.3431	−0.493651	−0.246826	−	0.969060i	$-0.579388\pi$
$439$	−10.3431	−0.493651	−0.246826	+	0.969060i	$0.579388\pi$
$440$	0	0
$441$	16.3137	0.776843
$442$	2.34315	0.111452
$443$	7.65685	0.363788	0.181894	−	0.983318i	$-0.441777\pi$
$443$	7.65685	0.363788	0.181894	+	0.983318i	$0.441777\pi$
$444$	−31.0294	−1.47259
$445$	0	0
$446$	−7.37258	−0.349102
$447$	18.6274	0.881047
$448$	−20.1421	−0.951626
$449$	11.6569	0.550121	0.275060	−	0.961427i	$-0.411302\pi$
$449$	11.6569	0.550121	0.275060	+	0.961427i	$0.411302\pi$
$450$	0	0
$451$	−4.97056	−0.234055
$452$	−5.17157	−0.243250
$453$	−24.0000	−1.12762
$454$	8.34315	0.391563
$455$	0	0
$456$	−15.3137	−0.717130
$457$	−19.6569	−0.919509	−0.459754	−	0.888046i	$-0.652063\pi$
$457$	−19.6569	−0.919509	−0.459754	+	0.888046i	$0.652063\pi$
$458$	0.828427	0.0387099
$459$	11.3137	0.528079
$460$	0	0
$461$	−35.6569	−1.66071	−0.830353	−	0.557238i	$-0.811861\pi$
$461$	−35.6569	−1.66071	−0.830353	+	0.557238i	$0.811861\pi$
$462$	−3.31371	−0.154168
$463$	−21.7990	−1.01308	−0.506542	−	0.862215i	$-0.669077\pi$
$463$	−21.7990	−1.01308	−0.506542	+	0.862215i	$0.669077\pi$
$464$	3.00000	0.139272
$465$	0	0
$466$	7.45584	0.345385
$467$	−10.9706	−0.507657	−0.253829	−	0.967249i	$-0.581690\pi$
$467$	−10.9706	−0.507657	−0.253829	+	0.967249i	$0.581690\pi$
$468$	−3.65685	−0.169038
$469$	−31.3137	−1.44593
$470$	0	0
$471$	−0.970563	−0.0447212
$472$	0	0
$473$	4.97056	0.228547
$474$	2.05887	0.0945672
$475$	0	0
$476$	24.9706	1.14452
$477$	3.65685	0.167436
$478$	0.284271	0.0130023
$479$	7.17157	0.327678	0.163839	−	0.986487i	$-0.447612\pi$
$479$	7.17157	0.327678	0.163839	+	0.986487i	$0.447612\pi$
$480$	0	0
$481$	16.9706	0.773791
$482$	−4.14214	−0.188669
$483$	30.6274	1.39360
$484$	18.8579	0.857176
$485$	0	0
$486$	4.14214	0.187891
$487$	−9.79899	−0.444035	−0.222017	−	0.975043i	$-0.571264\pi$
$487$	−9.79899	−0.444035	−0.222017	+	0.975043i	$0.571264\pi$
$488$	−5.79899	−0.262508
$489$	16.6863	0.754580
$490$	0	0
$491$	−7.45584	−0.336478	−0.168239	−	0.985746i	$-0.553808\pi$
$491$	−7.45584	−0.336478	−0.168239	+	0.985746i	$0.553808\pi$
$492$	21.9411	0.989182
$493$	−2.82843	−0.127386
$494$	4.00000	0.179969
$495$	0	0
$496$	19.4558	0.873593
$497$	−73.9411	−3.31671
$498$	5.94113	0.266228
$499$	36.0000	1.61158	0.805791	−	0.592200i	$-0.201741\pi$
$499$	36.0000	1.61158	0.805791	+	0.592200i	$0.201741\pi$
$500$	0	0
$501$	4.97056	0.222068
$502$	−3.65685	−0.163213
$503$	30.0000	1.33763	0.668817	−	0.743427i	$-0.266801\pi$
$503$	30.0000	1.33763	0.668817	+	0.743427i	$0.266801\pi$
$504$	7.65685	0.341063
$505$	0	0
$506$	−1.08831	−0.0483814
$507$	−18.0000	−0.799408
$508$	10.9706	0.486740
$509$	0.627417	0.0278098	0.0139049	−	0.999903i	$-0.495574\pi$
$509$	0.627417	0.0278098	0.0139049	+	0.999903i	$0.495574\pi$
$510$	0	0
$511$	−40.9706	−1.81243
$512$	−22.7574	−1.00574
$513$	19.3137	0.852721
$514$	2.76955	0.122160
$515$	0	0
$516$	−21.9411	−0.965904
$517$	9.65685	0.424708
$518$	−16.9706	−0.745644
$519$	−34.6274	−1.51997
$520$	0	0
$521$	−21.3137	−0.933771	−0.466885	−	0.884318i	$-0.654624\pi$
$521$	−21.3137	−0.933771	−0.466885	+	0.884318i	$0.654624\pi$
$522$	−0.414214	−0.0181296
$523$	−2.48528	−0.108674	−0.0543369	−	0.998523i	$-0.517304\pi$
$523$	−2.48528	−0.108674	−0.0543369	+	0.998523i	$0.517304\pi$
$524$	22.2010	0.969856
$525$	0	0
$526$	−8.14214	−0.355014
$527$	−18.3431	−0.799040
$528$	4.97056	0.216316
$529$	−12.9411	−0.562658
$530$	0	0
$531$	0	0
$532$	42.6274	1.84813
$533$	−12.0000	−0.519778
$534$	6.34315	0.274495
$535$	0	0
$536$	−10.2843	−0.444213
$537$	−46.6274	−2.01212
$538$	8.82843	0.380621
$539$	13.5147	0.582120
$540$	0	0
$541$	−5.02944	−0.216232	−0.108116	−	0.994138i	$-0.534482\pi$
$541$	−5.02944	−0.216232	−0.108116	+	0.994138i	$0.534482\pi$
$542$	4.05887	0.174344
$543$	−12.0000	−0.514969
$544$	12.4853	0.535302
$545$	0	0
$546$	−8.00000	−0.342368
$547$	2.48528	0.106263	0.0531315	−	0.998588i	$-0.483080\pi$
$547$	2.48528	0.106263	0.0531315	+	0.998588i	$0.483080\pi$
$548$	−9.45584	−0.403934
$549$	−3.65685	−0.156071
$550$	0	0
$551$	−4.82843	−0.205698
$552$	10.0589	0.428134
$553$	−12.0000	−0.510292
$554$	−1.51472	−0.0643542
$555$	0	0
$556$	−39.5980	−1.67933
$557$	27.9411	1.18390	0.591952	−	0.805973i	$-0.298358\pi$
$557$	27.9411	1.18390	0.591952	+	0.805973i	$0.298358\pi$
$558$	−2.68629	−0.113720
$559$	12.0000	0.507546
$560$	0	0
$561$	−4.68629	−0.197855
$562$	12.1421	0.512185
$563$	−7.65685	−0.322698	−0.161349	−	0.986897i	$-0.551584\pi$
$563$	−7.65685	−0.322698	−0.161349	+	0.986897i	$0.551584\pi$
$564$	−42.6274	−1.79494
$565$	0	0
$566$	2.00000	0.0840663
$567$	−53.1127	−2.23052
$568$	−24.2843	−1.01895
$569$	27.6569	1.15944	0.579718	−	0.814817i	$-0.303163\pi$
$569$	27.6569	1.15944	0.579718	+	0.814817i	$0.303163\pi$
$570$	0	0
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	−3.02944	−0.126667
$573$	−41.6569	−1.74024
$574$	12.0000	0.500870
$575$	0	0
$576$	−4.17157	−0.173816
$577$	−23.7990	−0.990765	−0.495382	−	0.868675i	$-0.664972\pi$
$577$	−23.7990	−0.990765	−0.495382	+	0.868675i	$0.664972\pi$
$578$	3.72792	0.155061
$579$	−8.97056	−0.372804
$580$	0	0
$581$	−34.6274	−1.43659
$582$	−10.3431	−0.428737
$583$	3.02944	0.125466
$584$	−13.4558	−0.556807
$585$	0	0
$586$	3.51472	0.145192
$587$	29.7990	1.22994	0.614968	−	0.788552i	$-0.289169\pi$
$587$	29.7990	1.22994	0.614968	+	0.788552i	$0.289169\pi$
$588$	−59.6569	−2.46021
$589$	−31.3137	−1.29026
$590$	0	0
$591$	39.3137	1.61715
$592$	25.4558	1.04623
$593$	7.65685	0.314429	0.157215	−	0.987564i	$-0.449749\pi$
$593$	7.65685	0.314429	0.157215	+	0.987564i	$0.449749\pi$
$594$	1.37258	0.0563178
$595$	0	0
$596$	−17.0294	−0.697553
$597$	24.0000	0.982255
$598$	−2.62742	−0.107443
$599$	−37.7990	−1.54442	−0.772212	−	0.635364i	$-0.780850\pi$
$599$	−37.7990	−1.54442	−0.772212	+	0.635364i	$0.780850\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	−12.0000	−0.489083
$603$	−6.48528	−0.264101
$604$	21.9411	0.892772
$605$	0	0
$606$	−12.9706	−0.526893
$607$	−9.02944	−0.366494	−0.183247	−	0.983067i	$-0.558661\pi$
$607$	−9.02944	−0.366494	−0.183247	+	0.983067i	$0.558661\pi$
$608$	21.3137	0.864385
$609$	9.65685	0.391315
$610$	0	0
$611$	23.3137	0.943172
$612$	5.17157	0.209048
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	−9.51472	−0.383983
$615$	0	0
$616$	6.34315	0.255573
$617$	9.17157	0.369234	0.184617	−	0.982811i	$-0.440896\pi$
$617$	9.17157	0.369234	0.184617	+	0.982811i	$0.440896\pi$
$618$	13.3726	0.537924
$619$	−9.79899	−0.393855	−0.196927	−	0.980418i	$-0.563096\pi$
$619$	−9.79899	−0.393855	−0.196927	+	0.980418i	$0.563096\pi$
$620$	0	0
$621$	−12.6863	−0.509083
$622$	−6.00000	−0.240578
$623$	−36.9706	−1.48119
$624$	12.0000	0.480384
$625$	0	0
$626$	−2.48528	−0.0993318
$627$	−8.00000	−0.319489
$628$	0.887302	0.0354072
$629$	−24.0000	−0.956943
$630$	0	0
$631$	36.9706	1.47177	0.735887	−	0.677104i	$-0.236765\pi$
$631$	36.9706	1.47177	0.735887	+	0.677104i	$0.236765\pi$
$632$	−3.94113	−0.156770
$633$	−1.65685	−0.0658540
$634$	1.17157	0.0465291
$635$	0	0
$636$	−13.3726	−0.530257
$637$	32.6274	1.29275
$638$	−0.343146	−0.0135853
$639$	−15.3137	−0.605801
$640$	0	0
$641$	0.627417	0.0247815	0.0123907	−	0.999923i	$-0.496056\pi$
$641$	0.627417	0.0247815	0.0123907	+	0.999923i	$0.496056\pi$
$642$	16.6863	0.658555
$643$	−19.4558	−0.767264	−0.383632	−	0.923486i	$-0.625327\pi$
$643$	−19.4558	−0.767264	−0.383632	+	0.923486i	$0.625327\pi$
$644$	−28.0000	−1.10335
$645$	0	0
$646$	−5.65685	−0.222566
$647$	41.1127	1.61631	0.808153	−	0.588972i	$-0.200467\pi$
$647$	41.1127	1.61631	0.808153	+	0.588972i	$0.200467\pi$
$648$	−17.4437	−0.685251
$649$	0	0
$650$	0	0
$651$	62.6274	2.45456
$652$	−15.2548	−0.597425
$653$	17.1716	0.671976	0.335988	−	0.941866i	$-0.390930\pi$
$653$	17.1716	0.671976	0.335988	+	0.941866i	$0.390930\pi$
$654$	−1.65685	−0.0647881
$655$	0	0
$656$	−18.0000	−0.702782
$657$	−8.48528	−0.331042
$658$	−23.3137	−0.908863
$659$	1.79899	0.0700787	0.0350393	−	0.999386i	$-0.488844\pi$
$659$	1.79899	0.0700787	0.0350393	+	0.999386i	$0.488844\pi$
$660$	0	0
$661$	26.0000	1.01128	0.505641	−	0.862744i	$-0.331256\pi$
$661$	26.0000	1.01128	0.505641	+	0.862744i	$0.331256\pi$
$662$	−9.02944	−0.350939
$663$	−11.3137	−0.439388
$664$	−11.3726	−0.441342
$665$	0	0
$666$	−3.51472	−0.136193
$667$	3.17157	0.122804
$668$	−4.54416	−0.175819
$669$	35.5980	1.37630
$670$	0	0
$671$	−3.02944	−0.116950
$672$	−42.6274	−1.64439
$673$	−22.9706	−0.885450	−0.442725	−	0.896657i	$-0.645988\pi$
$673$	−22.9706	−0.885450	−0.442725	+	0.896657i	$0.645988\pi$
$674$	−0.485281	−0.0186923
$675$	0	0
$676$	16.4558	0.632917
$677$	36.7696	1.41317	0.706584	−	0.707629i	$-0.250235\pi$
$677$	36.7696	1.41317	0.706584	+	0.707629i	$0.250235\pi$
$678$	−2.34315	−0.0899880
$679$	60.2843	2.31350
$680$	0	0
$681$	−40.2843	−1.54370
$682$	−2.22540	−0.0852148
$683$	−11.8579	−0.453729	−0.226864	−	0.973926i	$-0.572847\pi$
$683$	−11.8579	−0.453729	−0.226864	+	0.973926i	$0.572847\pi$
$684$	8.82843	0.337563
$685$	0	0
$686$	−18.6274	−0.711198
$687$	−4.00000	−0.152610
$688$	18.0000	0.686244
$689$	7.31371	0.278630
$690$	0	0
$691$	−44.9706	−1.71076	−0.855380	−	0.518000i	$-0.826677\pi$
$691$	−44.9706	−1.71076	−0.855380	+	0.518000i	$0.826677\pi$
$692$	31.6569	1.20341
$693$	4.00000	0.151947
$694$	−3.37258	−0.128022
$695$	0	0
$696$	3.17157	0.120218
$697$	16.9706	0.642806
$698$	−8.54416	−0.323401
$699$	−36.0000	−1.36165
$700$	0	0
$701$	6.68629	0.252538	0.126269	−	0.991996i	$-0.459700\pi$
$701$	6.68629	0.252538	0.126269	+	0.991996i	$0.459700\pi$
$702$	3.31371	0.125068
$703$	−40.9706	−1.54523
$704$	−3.45584	−0.130247
$705$	0	0
$706$	−1.79899	−0.0677059
$707$	75.5980	2.84315
$708$	0	0
$709$	−22.0000	−0.826227	−0.413114	−	0.910679i	$-0.635559\pi$
$709$	−22.0000	−0.826227	−0.413114	+	0.910679i	$0.635559\pi$
$710$	0	0
$711$	−2.48528	−0.0932053
$712$	−12.1421	−0.455046
$713$	20.5685	0.770298
$714$	11.3137	0.423405
$715$	0	0
$716$	42.6274	1.59306
$717$	−1.37258	−0.0512601
$718$	1.59798	0.0596361
$719$	−34.6274	−1.29138	−0.645692	−	0.763598i	$-0.723431\pi$
$719$	−34.6274	−1.29138	−0.645692	+	0.763598i	$0.723431\pi$
$720$	0	0
$721$	−77.9411	−2.90268
$722$	−1.78680	−0.0664977
$723$	20.0000	0.743808
$724$	10.9706	0.407718
$725$	0	0
$726$	8.54416	0.317103
$727$	23.9411	0.887927	0.443964	−	0.896045i	$-0.353572\pi$
$727$	23.9411	0.887927	0.443964	+	0.896045i	$0.353572\pi$
$728$	15.3137	0.567564
$729$	13.0000	0.481481
$730$	0	0
$731$	−16.9706	−0.627679
$732$	13.3726	0.494265
$733$	22.8284	0.843187	0.421594	−	0.906785i	$-0.361471\pi$
$733$	22.8284	0.843187	0.421594	+	0.906785i	$0.361471\pi$
$734$	7.45584	0.275200
$735$	0	0
$736$	−14.0000	−0.516047
$737$	−5.37258	−0.197902
$738$	2.48528	0.0914845
$739$	14.4853	0.532850	0.266425	−	0.963856i	$-0.414158\pi$
$739$	14.4853	0.532850	0.266425	+	0.963856i	$0.414158\pi$
$740$	0	0
$741$	−19.3137	−0.709507
$742$	−7.31371	−0.268495
$743$	52.6274	1.93071	0.965356	−	0.260935i	$-0.0840309\pi$
$743$	52.6274	1.93071	0.965356	+	0.260935i	$0.0840309\pi$
$744$	20.5685	0.754079
$745$	0	0
$746$	−2.88730	−0.105712
$747$	−7.17157	−0.262394
$748$	4.28427	0.156648
$749$	−97.2548	−3.55361
$750$	0	0
$751$	16.1421	0.589035	0.294517	−	0.955646i	$-0.404841\pi$
$751$	16.1421	0.589035	0.294517	+	0.955646i	$0.404841\pi$
$752$	34.9706	1.27525
$753$	17.6569	0.643452
$754$	−0.828427	−0.0301695
$755$	0	0
$756$	35.3137	1.28435
$757$	−19.5147	−0.709275	−0.354637	−	0.935004i	$-0.615396\pi$
$757$	−19.5147	−0.709275	−0.354637	+	0.935004i	$0.615396\pi$
$758$	−9.31371	−0.338289
$759$	5.25483	0.190738
$760$	0	0
$761$	8.62742	0.312744	0.156372	−	0.987698i	$-0.450020\pi$
$761$	8.62742	0.312744	0.156372	+	0.987698i	$0.450020\pi$
$762$	4.97056	0.180064
$763$	9.65685	0.349602
$764$	38.0833	1.37780
$765$	0	0
$766$	1.02944	0.0371951
$767$	0	0
$768$	7.94113	0.286551
$769$	−15.6569	−0.564601	−0.282300	−	0.959326i	$-0.591097\pi$
$769$	−15.6569	−0.564601	−0.282300	+	0.959326i	$0.591097\pi$
$770$	0	0
$771$	−13.3726	−0.481602
$772$	8.20101	0.295161
$773$	8.48528	0.305194	0.152597	−	0.988288i	$-0.451236\pi$
$773$	8.48528	0.305194	0.152597	+	0.988288i	$0.451236\pi$
$774$	−2.48528	−0.0893316
$775$	0	0
$776$	19.7990	0.710742
$777$	81.9411	2.93962
$778$	12.1421	0.435317
$779$	28.9706	1.03798
$780$	0	0
$781$	−12.6863	−0.453951
$782$	3.71573	0.132874
$783$	−4.00000	−0.142948
$784$	48.9411	1.74790
$785$	0	0
$786$	10.0589	0.358788
$787$	−17.7990	−0.634465	−0.317233	−	0.948348i	$-0.602754\pi$
$787$	−17.7990	−0.634465	−0.317233	+	0.948348i	$0.602754\pi$
$788$	−35.9411	−1.28035
$789$	39.3137	1.39961
$790$	0	0
$791$	13.6569	0.485582
$792$	1.31371	0.0466806
$793$	−7.31371	−0.259717
$794$	−8.14214	−0.288954
$795$	0	0
$796$	−21.9411	−0.777683
$797$	5.85786	0.207496	0.103748	−	0.994604i	$-0.466916\pi$
$797$	5.85786	0.207496	0.103748	+	0.994604i	$0.466916\pi$
$798$	19.3137	0.683698
$799$	−32.9706	−1.16641
$800$	0	0
$801$	−7.65685	−0.270542
$802$	2.76955	0.0977963
$803$	−7.02944	−0.248063
$804$	23.7157	0.836389
$805$	0	0
$806$	−5.37258	−0.189241
$807$	−42.6274	−1.50056
$808$	24.8284	0.873461
$809$	42.2843	1.48664	0.743318	−	0.668938i	$-0.233251\pi$
$809$	42.2843	1.48664	0.743318	+	0.668938i	$0.233251\pi$
$810$	0	0
$811$	37.6569	1.32231	0.661155	−	0.750249i	$-0.270066\pi$
$811$	37.6569	1.32231	0.661155	+	0.750249i	$0.270066\pi$
$812$	−8.82843	−0.309817
$813$	−19.5980	−0.687331
$814$	−2.91169	−0.102055
$815$	0	0
$816$	−16.9706	−0.594089
$817$	−28.9706	−1.01355
$818$	1.23045	0.0430216
$819$	9.65685	0.337438
$820$	0	0
$821$	−22.6863	−0.791757	−0.395879	−	0.918303i	$-0.629560\pi$
$821$	−22.6863	−0.791757	−0.395879	+	0.918303i	$0.629560\pi$
$822$	−4.28427	−0.149431
$823$	30.9706	1.07957	0.539783	−	0.841804i	$-0.318506\pi$
$823$	30.9706	1.07957	0.539783	+	0.841804i	$0.318506\pi$
$824$	−25.5980	−0.891748
$825$	0	0
$826$	0	0
$827$	−17.3137	−0.602057	−0.301028	−	0.953615i	$-0.597330\pi$
$827$	−17.3137	−0.602057	−0.301028	+	0.953615i	$0.597330\pi$
$828$	−5.79899	−0.201529
$829$	20.6274	0.716420	0.358210	−	0.933641i	$-0.383387\pi$
$829$	20.6274	0.716420	0.358210	+	0.933641i	$0.383387\pi$
$830$	0	0
$831$	7.31371	0.253710
$832$	−8.34315	−0.289247
$833$	−46.1421	−1.59873
$834$	−17.9411	−0.621250
$835$	0	0
$836$	7.31371	0.252950
$837$	−25.9411	−0.896656
$838$	−12.0000	−0.414533
$839$	2.48528	0.0858014	0.0429007	−	0.999079i	$-0.486340\pi$
$839$	2.48528	0.0858014	0.0429007	+	0.999079i	$0.486340\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−7.85786	−0.270800
$843$	−58.6274	−2.01924
$844$	1.51472	0.0521388
$845$	0	0
$846$	−4.82843	−0.166005
$847$	−49.7990	−1.71111
$848$	10.9706	0.376731
$849$	−9.65685	−0.331422
$850$	0	0
$851$	26.9117	0.922521
$852$	56.0000	1.91853
$853$	−51.1127	−1.75007	−0.875033	−	0.484064i	$-0.839160\pi$
$853$	−51.1127	−1.75007	−0.875033	+	0.484064i	$0.839160\pi$
$854$	7.31371	0.250270
$855$	0	0
$856$	−31.9411	−1.09173
$857$	−3.37258	−0.115205	−0.0576026	−	0.998340i	$-0.518346\pi$
$857$	−3.37258	−0.115205	−0.0576026	+	0.998340i	$0.518346\pi$
$858$	−1.37258	−0.0468592
$859$	−56.4264	−1.92524	−0.962622	−	0.270848i	$-0.912696\pi$
$859$	−56.4264	−1.92524	−0.962622	+	0.270848i	$0.912696\pi$
$860$	0	0
$861$	−57.9411	−1.97463
$862$	−1.37258	−0.0467504
$863$	36.1421	1.23029	0.615146	−	0.788413i	$-0.289097\pi$
$863$	36.1421	1.23029	0.615146	+	0.788413i	$0.289097\pi$
$864$	17.6569	0.600698
$865$	0	0
$866$	−12.0833	−0.410606
$867$	−18.0000	−0.611312
$868$	−57.2548	−1.94336
$869$	−2.05887	−0.0698425
$870$	0	0
$871$	−12.9706	−0.439491
$872$	3.17157	0.107403
$873$	12.4853	0.422563
$874$	6.34315	0.214560
$875$	0	0
$876$	31.0294	1.04839
$877$	−38.2843	−1.29277	−0.646384	−	0.763012i	$-0.723720\pi$
$877$	−38.2843	−1.29277	−0.646384	+	0.763012i	$0.723720\pi$
$878$	4.28427	0.144587
$879$	−16.9706	−0.572403
$880$	0	0
$881$	29.3137	0.987604	0.493802	−	0.869574i	$-0.335607\pi$
$881$	29.3137	0.987604	0.493802	+	0.869574i	$0.335607\pi$
$882$	−6.75736	−0.227532
$883$	14.4853	0.487469	0.243734	−	0.969842i	$-0.421628\pi$
$883$	14.4853	0.487469	0.243734	+	0.969842i	$0.421628\pi$
$884$	10.3431	0.347878
$885$	0	0
$886$	−3.17157	−0.106551
$887$	6.68629	0.224504	0.112252	−	0.993680i	$-0.464194\pi$
$887$	6.68629	0.224504	0.112252	+	0.993680i	$0.464194\pi$
$888$	26.9117	0.903097
$889$	−28.9706	−0.971641
$890$	0	0
$891$	−9.11270	−0.305287
$892$	−32.5442	−1.08966
$893$	−56.2843	−1.88348
$894$	−7.71573	−0.258053
$895$	0	0
$896$	50.9706	1.70281
$897$	12.6863	0.423583
$898$	−4.82843	−0.161127
$899$	6.48528	0.216296
$900$	0	0
$901$	−10.3431	−0.344580
$902$	2.05887	0.0685530
$903$	57.9411	1.92816
$904$	4.48528	0.149178
$905$	0	0
$906$	9.94113	0.330272
$907$	10.0000	0.332045	0.166022	−	0.986122i	$-0.446908\pi$
$907$	10.0000	0.332045	0.166022	+	0.986122i	$0.446908\pi$
$908$	36.8284	1.22219
$909$	15.6569	0.519305
$910$	0	0
$911$	32.1421	1.06492	0.532458	−	0.846456i	$-0.321268\pi$
$911$	32.1421	1.06492	0.532458	+	0.846456i	$0.321268\pi$
$912$	−28.9706	−0.959311
$913$	−5.94113	−0.196623
$914$	8.14214	0.269318
$915$	0	0
$916$	3.65685	0.120826
$917$	−58.6274	−1.93605
$918$	−4.68629	−0.154671
$919$	−36.0000	−1.18753	−0.593765	−	0.804638i	$-0.702359\pi$
$919$	−36.0000	−1.18753	−0.593765	+	0.804638i	$0.702359\pi$
$920$	0	0
$921$	45.9411	1.51381
$922$	14.7696	0.486409
$923$	−30.6274	−1.00811
$924$	−14.6274	−0.481207
$925$	0	0
$926$	9.02944	0.296726
$927$	−16.1421	−0.530177
$928$	−4.41421	−0.144904
$929$	−4.62742	−0.151821	−0.0759103	−	0.997115i	$-0.524186\pi$
$929$	−4.62742	−0.151821	−0.0759103	+	0.997115i	$0.524186\pi$
$930$	0	0
$931$	−78.7696	−2.58157
$932$	32.9117	1.07806
$933$	28.9706	0.948454
$934$	4.54416	0.148689
$935$	0	0
$936$	3.17157	0.103666
$937$	−19.6569	−0.642161	−0.321081	−	0.947052i	$-0.604046\pi$
$937$	−19.6569	−0.642161	−0.321081	+	0.947052i	$0.604046\pi$
$938$	12.9706	0.423504
$939$	12.0000	0.391605
$940$	0	0
$941$	−27.9411	−0.910855	−0.455427	−	0.890273i	$-0.650514\pi$
$941$	−27.9411	−0.910855	−0.455427	+	0.890273i	$0.650514\pi$
$942$	0.402020	0.0130985
$943$	−19.0294	−0.619684
$944$	0	0
$945$	0	0
$946$	−2.05887	−0.0669398
$947$	−44.9117	−1.45943	−0.729717	−	0.683749i	$-0.760348\pi$
$947$	−44.9117	−1.45943	−0.729717	+	0.683749i	$0.760348\pi$
$948$	9.08831	0.295175
$949$	−16.9706	−0.550888
$950$	0	0
$951$	−5.65685	−0.183436
$952$	−21.6569	−0.701903
$953$	−29.3137	−0.949564	−0.474782	−	0.880103i	$-0.657473\pi$
$953$	−29.3137	−0.949564	−0.474782	+	0.880103i	$0.657473\pi$
$954$	−1.51472	−0.0490408
$955$	0	0
$956$	1.25483	0.0405842
$957$	1.65685	0.0535585
$958$	−2.97056	−0.0959745
$959$	24.9706	0.806342
$960$	0	0
$961$	11.0589	0.356738
$962$	−7.02944	−0.226638
$963$	−20.1421	−0.649071
$964$	−18.2843	−0.588897
$965$	0	0
$966$	−12.6863	−0.408175
$967$	−14.9706	−0.481421	−0.240710	−	0.970597i	$-0.577380\pi$
$967$	−14.9706	−0.481421	−0.240710	+	0.970597i	$0.577380\pi$
$968$	−16.3553	−0.525681
$969$	27.3137	0.877443
$970$	0	0
$971$	28.1421	0.903124	0.451562	−	0.892240i	$-0.350867\pi$
$971$	28.1421	0.903124	0.451562	+	0.892240i	$0.350867\pi$
$972$	18.2843	0.586468
$973$	104.569	3.35231
$974$	4.05887	0.130055
$975$	0	0
$976$	−10.9706	−0.351159
$977$	2.68629	0.0859421	0.0429710	−	0.999076i	$-0.486318\pi$
$977$	2.68629	0.0859421	0.0429710	+	0.999076i	$0.486318\pi$
$978$	−6.91169	−0.221011
$979$	−6.34315	−0.202728
$980$	0	0
$981$	2.00000	0.0638551
$982$	3.08831	0.0985520
$983$	9.31371	0.297061	0.148531	−	0.988908i	$-0.452546\pi$
$983$	9.31371	0.297061	0.148531	+	0.988908i	$0.452546\pi$
$984$	−19.0294	−0.606636
$985$	0	0
$986$	1.17157	0.0373105
$987$	112.569	3.58310
$988$	17.6569	0.561739
$989$	19.0294	0.605101
$990$	0	0
$991$	52.0000	1.65183	0.825917	−	0.563791i	$-0.190658\pi$
$991$	52.0000	1.65183	0.825917	+	0.563791i	$0.190658\pi$
$992$	−28.6274	−0.908921
$993$	43.5980	1.38354
$994$	30.6274	0.971443
$995$	0	0
$996$	26.2254	0.830983
$997$	−6.82843	−0.216258	−0.108129	−	0.994137i	$-0.534486\pi$
$997$	−6.82843	−0.216258	−0.108129	+	0.994137i	$0.534486\pi$
$998$	−14.9117	−0.472021
$999$	−33.9411	−1.07385

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	725.2.a.c.1.1		2
3.2	odd	2		6525.2.a.p.1.2		2
5.2	odd	4		725.2.b.c.349.2		4
5.3	odd	4		725.2.b.c.349.3		4
5.4	even	2		145.2.a.b.1.2	✓	2
15.14	odd	2		1305.2.a.n.1.1		2
20.19	odd	2		2320.2.a.k.1.2		2
35.34	odd	2		7105.2.a.e.1.2		2
40.19	odd	2		9280.2.a.w.1.2		2
40.29	even	2		9280.2.a.be.1.1		2
145.144	even	2		4205.2.a.d.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.a.b.1.2	✓	2	5.4	even	2
725.2.a.c.1.1		2	1.1	even	1	trivial
725.2.b.c.349.2		4	5.2	odd	4
725.2.b.c.349.3		4	5.3	odd	4
1305.2.a.n.1.1		2	15.14	odd	2
2320.2.a.k.1.2		2	20.19	odd	2
4205.2.a.d.1.1		2	145.144	even	2
6525.2.a.p.1.2		2	3.2	odd	2
7105.2.a.e.1.2		2	35.34	odd	2
9280.2.a.w.1.2		2	40.19	odd	2
9280.2.a.be.1.1		2	40.29	even	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:	\( N \)	\(=\)	\( 725 = 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	725.a (trivial)

\(f(q)\)	\(=\)	\(q-0.414214 q^{2} +2.00000 q^{3} -1.82843 q^{4} -0.828427 q^{6} +4.82843 q^{7} +1.58579 q^{8} +1.00000 q^{9} +0.828427 q^{11} -3.65685 q^{12} +2.00000 q^{13} -2.00000 q^{14} +3.00000 q^{16} -2.82843 q^{17} -0.414214 q^{18} -4.82843 q^{19} +9.65685 q^{21} -0.343146 q^{22} +3.17157 q^{23} +3.17157 q^{24} -0.828427 q^{26} -4.00000 q^{27} -8.82843 q^{28} +1.00000 q^{29} +6.48528 q^{31} -4.41421 q^{32} +1.65685 q^{33} +1.17157 q^{34} -1.82843 q^{36} +8.48528 q^{37} +2.00000 q^{38} +4.00000 q^{39} -6.00000 q^{41} -4.00000 q^{42} +6.00000 q^{43} -1.51472 q^{44} -1.31371 q^{46} +11.6569 q^{47} +6.00000 q^{48} +16.3137 q^{49} -5.65685 q^{51} -3.65685 q^{52} +3.65685 q^{53} +1.65685 q^{54} +7.65685 q^{56} -9.65685 q^{57} -0.414214 q^{58} -3.65685 q^{61} -2.68629 q^{62} +4.82843 q^{63} -4.17157 q^{64} -0.686292 q^{66} -6.48528 q^{67} +5.17157 q^{68} +6.34315 q^{69} -15.3137 q^{71} +1.58579 q^{72} -8.48528 q^{73} -3.51472 q^{74} +8.82843 q^{76} +4.00000 q^{77} -1.65685 q^{78} -2.48528 q^{79} -11.0000 q^{81} +2.48528 q^{82} -7.17157 q^{83} -17.6569 q^{84} -2.48528 q^{86} +2.00000 q^{87} +1.31371 q^{88} -7.65685 q^{89} +9.65685 q^{91} -5.79899 q^{92} +12.9706 q^{93} -4.82843 q^{94} -8.82843 q^{96} +12.4853 q^{97} -6.75736 q^{98} +0.828427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 4 q^{6} + 4 q^{7} + 6 q^{8} + 2 q^{9} - 4 q^{11} + 4 q^{12} + 4 q^{13} - 4 q^{14} + 6 q^{16} + 2 q^{18} - 4 q^{19} + 8 q^{21} - 12 q^{22} + 12 q^{23} + 12 q^{24} + 4 q^{26}+ \cdots - 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 4 * q^3 + 2 * q^4 + 4 * q^6 + 4 * q^7 + 6 * q^8 + 2 * q^9 - 4 * q^11 + 4 * q^12 + 4 * q^13 - 4 * q^14 + 6 * q^16 + 2 * q^18 - 4 * q^19 + 8 * q^21 - 12 * q^22 + 12 * q^23 + 12 * q^24 + 4 * q^26 - 8 * q^27 - 12 * q^28 + 2 * q^29 - 4 * q^31 - 6 * q^32 - 8 * q^33 + 8 * q^34 + 2 * q^36 + 4 * q^38 + 8 * q^39 - 12 * q^41 - 8 * q^42 + 12 * q^43 - 20 * q^44 + 20 * q^46 + 12 * q^47 + 12 * q^48 + 10 * q^49 + 4 * q^52 - 4 * q^53 - 8 * q^54 + 4 * q^56 - 8 * q^57 + 2 * q^58 + 4 * q^61 - 28 * q^62 + 4 * q^63 - 14 * q^64 - 24 * q^66 + 4 * q^67 + 16 * q^68 + 24 * q^69 - 8 * q^71 + 6 * q^72 - 24 * q^74 + 12 * q^76 + 8 * q^77 + 8 * q^78 + 12 * q^79 - 22 * q^81 - 12 * q^82 - 20 * q^83 - 24 * q^84 + 12 * q^86 + 4 * q^87 - 20 * q^88 - 4 * q^89 + 8 * q^91 + 28 * q^92 - 8 * q^93 - 4 * q^94 - 12 * q^96 + 8 * q^97 - 22 * q^98 - 4 * q^99

Introduction

L-functions

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