LMFDB - Embedded newform 4018.2.a.ba.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4018,2,Mod(1,4018)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4018, 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("4018.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4018 = 2 \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4018.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:	$32.0838915322$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 82)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	4018.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.41421 q^{3} +1.00000 q^{4} +2.82843 q^{5} -1.41421 q^{6} +1.00000 q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.41421 q^{3} +1.00000 q^{4} +2.82843 q^{5} -1.41421 q^{6} +1.00000 q^{8} -1.00000 q^{9} +2.82843 q^{10} +4.24264 q^{11} -1.41421 q^{12} -4.00000 q^{15} +1.00000 q^{16} -7.65685 q^{17} -1.00000 q^{18} +5.41421 q^{19} +2.82843 q^{20} +4.24264 q^{22} +1.17157 q^{23} -1.41421 q^{24} +3.00000 q^{25} +5.65685 q^{27} -1.65685 q^{29} -4.00000 q^{30} +1.17157 q^{31} +1.00000 q^{32} -6.00000 q^{33} -7.65685 q^{34} -1.00000 q^{36} +8.48528 q^{37} +5.41421 q^{38} +2.82843 q^{40} +1.00000 q^{41} -1.65685 q^{43} +4.24264 q^{44} -2.82843 q^{45} +1.17157 q^{46} +9.07107 q^{47} -1.41421 q^{48} +3.00000 q^{50} +10.8284 q^{51} +12.0000 q^{53} +5.65685 q^{54} +12.0000 q^{55} -7.65685 q^{57} -1.65685 q^{58} +1.17157 q^{59} -4.00000 q^{60} -6.00000 q^{61} +1.17157 q^{62} +1.00000 q^{64} -6.00000 q^{66} -8.24264 q^{67} -7.65685 q^{68} -1.65685 q^{69} -0.585786 q^{71} -1.00000 q^{72} +13.6569 q^{73} +8.48528 q^{74} -4.24264 q^{75} +5.41421 q^{76} -10.2426 q^{79} +2.82843 q^{80} -5.00000 q^{81} +1.00000 q^{82} -17.6569 q^{83} -21.6569 q^{85} -1.65685 q^{86} +2.34315 q^{87} +4.24264 q^{88} +11.6569 q^{89} -2.82843 q^{90} +1.17157 q^{92} -1.65685 q^{93} +9.07107 q^{94} +15.3137 q^{95} -1.41421 q^{96} -3.65685 q^{97} -4.24264 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9} - 8 q^{15} + 2 q^{16} - 4 q^{17} - 2 q^{18} + 8 q^{19} + 8 q^{23} + 6 q^{25} + 8 q^{29} - 8 q^{30} + 8 q^{31} + 2 q^{32} - 12 q^{33} - 4 q^{34} - 2 q^{36} + 8 q^{38} + 2 q^{41} + 8 q^{43} + 8 q^{46} + 4 q^{47} + 6 q^{50} + 16 q^{51} + 24 q^{53} + 24 q^{55} - 4 q^{57} + 8 q^{58} + 8 q^{59} - 8 q^{60} - 12 q^{61} + 8 q^{62} + 2 q^{64} - 12 q^{66} - 8 q^{67} - 4 q^{68} + 8 q^{69} - 4 q^{71} - 2 q^{72} + 16 q^{73} + 8 q^{76} - 12 q^{79} - 10 q^{81} + 2 q^{82} - 24 q^{83} - 32 q^{85} + 8 q^{86} + 16 q^{87} + 12 q^{89} + 8 q^{92} + 8 q^{93} + 4 q^{94} + 8 q^{95} + 4 q^{97}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9 - 8 * q^15 + 2 * q^16 - 4 * q^17 - 2 * q^18 + 8 * q^19 + 8 * q^23 + 6 * q^25 + 8 * q^29 - 8 * q^30 + 8 * q^31 + 2 * q^32 - 12 * q^33 - 4 * q^34 - 2 * q^36 + 8 * q^38 + 2 * q^41 + 8 * q^43 + 8 * q^46 + 4 * q^47 + 6 * q^50 + 16 * q^51 + 24 * q^53 + 24 * q^55 - 4 * q^57 + 8 * q^58 + 8 * q^59 - 8 * q^60 - 12 * q^61 + 8 * q^62 + 2 * q^64 - 12 * q^66 - 8 * q^67 - 4 * q^68 + 8 * q^69 - 4 * q^71 - 2 * q^72 + 16 * q^73 + 8 * q^76 - 12 * q^79 - 10 * q^81 + 2 * q^82 - 24 * q^83 - 32 * q^85 + 8 * q^86 + 16 * q^87 + 12 * q^89 + 8 * q^92 + 8 * q^93 + 4 * q^94 + 8 * q^95 + 4 * q^97

Display 100 coefficients 10 coefficients

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
$3$	−1.41421	−0.816497	−0.408248	−	0.912871i	$-0.633860\pi$
$3$	−1.41421	−0.816497	−0.408248	+	0.912871i	$0.633860\pi$
$4$	1.00000	0.500000
$5$	2.82843	1.26491	0.632456	−	0.774597i	$-0.282047\pi$
$5$	2.82843	1.26491	0.632456	+	0.774597i	$0.282047\pi$
$6$	−1.41421	−0.577350
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	−1.00000	−0.333333
$10$	2.82843	0.894427
$11$	4.24264	1.27920	0.639602	−	0.768706i	$-0.279099\pi$
$11$	4.24264	1.27920	0.639602	+	0.768706i	$0.279099\pi$
$12$	−1.41421	−0.408248
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	−4.00000	−1.03280
$16$	1.00000	0.250000
$17$	−7.65685	−1.85706	−0.928530	−	0.371257i	$-0.878927\pi$
$17$	−7.65685	−1.85706	−0.928530	+	0.371257i	$0.878927\pi$
$18$	−1.00000	−0.235702
$19$	5.41421	1.24211	0.621053	−	0.783769i	$-0.286705\pi$
$19$	5.41421	1.24211	0.621053	+	0.783769i	$0.286705\pi$
$20$	2.82843	0.632456
$21$	0	0
$22$	4.24264	0.904534
$23$	1.17157	0.244290	0.122145	−	0.992512i	$-0.461023\pi$
$23$	1.17157	0.244290	0.122145	+	0.992512i	$0.461023\pi$
$24$	−1.41421	−0.288675
$25$	3.00000	0.600000
$26$	0	0
$27$	5.65685	1.08866
$28$	0	0
$29$	−1.65685	−0.307670	−0.153835	−	0.988097i	$-0.549162\pi$
$29$	−1.65685	−0.307670	−0.153835	+	0.988097i	$0.549162\pi$
$30$	−4.00000	−0.730297
$31$	1.17157	0.210421	0.105210	−	0.994450i	$-0.466448\pi$
$31$	1.17157	0.210421	0.105210	+	0.994450i	$0.466448\pi$
$32$	1.00000	0.176777
$33$	−6.00000	−1.04447
$34$	−7.65685	−1.31314
$35$	0	0
$36$	−1.00000	−0.166667
$37$	8.48528	1.39497	0.697486	−	0.716599i	$-0.254302\pi$
$37$	8.48528	1.39497	0.697486	+	0.716599i	$0.254302\pi$
$38$	5.41421	0.878301
$39$	0	0
$40$	2.82843	0.447214
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−1.65685	−0.252668	−0.126334	−	0.991988i	$-0.540321\pi$
$43$	−1.65685	−0.252668	−0.126334	+	0.991988i	$0.540321\pi$
$44$	4.24264	0.639602
$45$	−2.82843	−0.421637
$46$	1.17157	0.172739
$47$	9.07107	1.32315	0.661576	−	0.749878i	$-0.269888\pi$
$47$	9.07107	1.32315	0.661576	+	0.749878i	$0.269888\pi$
$48$	−1.41421	−0.204124
$49$	0	0
$50$	3.00000	0.424264
$51$	10.8284	1.51628
$52$	0	0
$53$	12.0000	1.64833	0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000	1.64833	0.824163	+	0.566352i	$0.191646\pi$
$54$	5.65685	0.769800
$55$	12.0000	1.61808
$56$	0	0
$57$	−7.65685	−1.01418
$58$	−1.65685	−0.217556
$59$	1.17157	0.152526	0.0762629	−	0.997088i	$-0.475701\pi$
$59$	1.17157	0.152526	0.0762629	+	0.997088i	$0.475701\pi$
$60$	−4.00000	−0.516398
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	1.17157	0.148790
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	−6.00000	−0.738549
$67$	−8.24264	−1.00700	−0.503499	−	0.863996i	$-0.667954\pi$
$67$	−8.24264	−1.00700	−0.503499	+	0.863996i	$0.667954\pi$
$68$	−7.65685	−0.928530
$69$	−1.65685	−0.199462
$70$	0	0
$71$	−0.585786	−0.0695201	−0.0347600	−	0.999396i	$-0.511067\pi$
$71$	−0.585786	−0.0695201	−0.0347600	+	0.999396i	$0.511067\pi$
$72$	−1.00000	−0.117851
$73$	13.6569	1.59841	0.799207	−	0.601056i	$-0.205253\pi$
$73$	13.6569	1.59841	0.799207	+	0.601056i	$0.205253\pi$
$74$	8.48528	0.986394
$75$	−4.24264	−0.489898
$76$	5.41421	0.621053
$77$	0	0
$78$	0	0
$79$	−10.2426	−1.15239	−0.576194	−	0.817313i	$-0.695463\pi$
$79$	−10.2426	−1.15239	−0.576194	+	0.817313i	$0.695463\pi$
$80$	2.82843	0.316228
$81$	−5.00000	−0.555556
$82$	1.00000	0.110432
$83$	−17.6569	−1.93809	−0.969046	−	0.246881i	$-0.920594\pi$
$83$	−17.6569	−1.93809	−0.969046	+	0.246881i	$0.920594\pi$
$84$	0	0
$85$	−21.6569	−2.34902
$86$	−1.65685	−0.178663
$87$	2.34315	0.251212
$88$	4.24264	0.452267
$89$	11.6569	1.23562	0.617812	−	0.786326i	$-0.288019\pi$
$89$	11.6569	1.23562	0.617812	+	0.786326i	$0.288019\pi$
$90$	−2.82843	−0.298142
$91$	0	0
$92$	1.17157	0.122145
$93$	−1.65685	−0.171808
$94$	9.07107	0.935609
$95$	15.3137	1.57115
$96$	−1.41421	−0.144338
$97$	−3.65685	−0.371297	−0.185649	−	0.982616i	$-0.559439\pi$
$97$	−3.65685	−0.371297	−0.185649	+	0.982616i	$0.559439\pi$
$98$	0	0
$99$	−4.24264	−0.426401
$100$	3.00000	0.300000
$101$	6.34315	0.631167	0.315583	−	0.948898i	$-0.397800\pi$
$101$	6.34315	0.631167	0.315583	+	0.948898i	$0.397800\pi$
$102$	10.8284	1.07217
$103$	−4.48528	−0.441948	−0.220974	−	0.975280i	$-0.570924\pi$
$103$	−4.48528	−0.441948	−0.220974	+	0.975280i	$0.570924\pi$
$104$	0	0
$105$	0	0
$106$	12.0000	1.16554
$107$	6.34315	0.613215	0.306608	−	0.951836i	$-0.400806\pi$
$107$	6.34315	0.613215	0.306608	+	0.951836i	$0.400806\pi$
$108$	5.65685	0.544331
$109$	12.0000	1.14939	0.574696	−	0.818367i	$-0.305120\pi$
$109$	12.0000	1.14939	0.574696	+	0.818367i	$0.305120\pi$
$110$	12.0000	1.14416
$111$	−12.0000	−1.13899
$112$	0	0
$113$	−1.65685	−0.155864	−0.0779319	−	0.996959i	$-0.524832\pi$
$113$	−1.65685	−0.155864	−0.0779319	+	0.996959i	$0.524832\pi$
$114$	−7.65685	−0.717130
$115$	3.31371	0.309005
$116$	−1.65685	−0.153835
$117$	0	0
$118$	1.17157	0.107852
$119$	0	0
$120$	−4.00000	−0.365148
$121$	7.00000	0.636364
$122$	−6.00000	−0.543214
$123$	−1.41421	−0.127515
$124$	1.17157	0.105210
$125$	−5.65685	−0.505964
$126$	0	0
$127$	18.1421	1.60985	0.804927	−	0.593374i	$-0.202204\pi$
$127$	18.1421	1.60985	0.804927	+	0.593374i	$0.202204\pi$
$128$	1.00000	0.0883883
$129$	2.34315	0.206302
$130$	0	0
$131$	10.1421	0.886123	0.443061	−	0.896491i	$-0.353892\pi$
$131$	10.1421	0.886123	0.443061	+	0.896491i	$0.353892\pi$
$132$	−6.00000	−0.522233
$133$	0	0
$134$	−8.24264	−0.712056
$135$	16.0000	1.37706
$136$	−7.65685	−0.656570
$137$	5.31371	0.453981	0.226990	−	0.973897i	$-0.427111\pi$
$137$	5.31371	0.453981	0.226990	+	0.973897i	$0.427111\pi$
$138$	−1.65685	−0.141041
$139$	−12.4853	−1.05899	−0.529494	−	0.848314i	$-0.677618\pi$
$139$	−12.4853	−1.05899	−0.529494	+	0.848314i	$0.677618\pi$
$140$	0	0
$141$	−12.8284	−1.08035
$142$	−0.585786	−0.0491581
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	−4.68629	−0.389175
$146$	13.6569	1.13025
$147$	0	0
$148$	8.48528	0.697486
$149$	3.31371	0.271470	0.135735	−	0.990745i	$-0.456660\pi$
$149$	3.31371	0.271470	0.135735	+	0.990745i	$0.456660\pi$
$150$	−4.24264	−0.346410
$151$	−11.8995	−0.968367	−0.484184	−	0.874966i	$-0.660883\pi$
$151$	−11.8995	−0.968367	−0.484184	+	0.874966i	$0.660883\pi$
$152$	5.41421	0.439151
$153$	7.65685	0.619020
$154$	0	0
$155$	3.31371	0.266163
$156$	0	0
$157$	20.9706	1.67363	0.836817	−	0.547483i	$-0.184414\pi$
$157$	20.9706	1.67363	0.836817	+	0.547483i	$0.184414\pi$
$158$	−10.2426	−0.814861
$159$	−16.9706	−1.34585
$160$	2.82843	0.223607
$161$	0	0
$162$	−5.00000	−0.392837
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	1.00000	0.0780869
$165$	−16.9706	−1.32116
$166$	−17.6569	−1.37044
$167$	18.7279	1.44921	0.724605	−	0.689164i	$-0.242022\pi$
$167$	18.7279	1.44921	0.724605	+	0.689164i	$0.242022\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	−21.6569	−1.66100
$171$	−5.41421	−0.414035
$172$	−1.65685	−0.126334
$173$	−13.3137	−1.01222	−0.506111	−	0.862468i	$-0.668917\pi$
$173$	−13.3137	−1.01222	−0.506111	+	0.862468i	$0.668917\pi$
$174$	2.34315	0.177633
$175$	0	0
$176$	4.24264	0.319801
$177$	−1.65685	−0.124537
$178$	11.6569	0.873718
$179$	−15.5563	−1.16274	−0.581368	−	0.813641i	$-0.697482\pi$
$179$	−15.5563	−1.16274	−0.581368	+	0.813641i	$0.697482\pi$
$180$	−2.82843	−0.210819
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	8.48528	0.627250
$184$	1.17157	0.0863695
$185$	24.0000	1.76452
$186$	−1.65685	−0.121486
$187$	−32.4853	−2.37556
$188$	9.07107	0.661576
$189$	0	0
$190$	15.3137	1.11097
$191$	−2.92893	−0.211930	−0.105965	−	0.994370i	$-0.533793\pi$
$191$	−2.92893	−0.211930	−0.105965	+	0.994370i	$0.533793\pi$
$192$	−1.41421	−0.102062
$193$	15.6569	1.12701	0.563503	−	0.826114i	$-0.309454\pi$
$193$	15.6569	1.12701	0.563503	+	0.826114i	$0.309454\pi$
$194$	−3.65685	−0.262547
$195$	0	0
$196$	0	0
$197$	9.31371	0.663574	0.331787	−	0.943354i	$-0.392348\pi$
$197$	9.31371	0.663574	0.331787	+	0.943354i	$0.392348\pi$
$198$	−4.24264	−0.301511
$199$	2.24264	0.158977	0.0794883	−	0.996836i	$-0.474671\pi$
$199$	2.24264	0.158977	0.0794883	+	0.996836i	$0.474671\pi$
$200$	3.00000	0.212132
$201$	11.6569	0.822211
$202$	6.34315	0.446302
$203$	0	0
$204$	10.8284	0.758142
$205$	2.82843	0.197546
$206$	−4.48528	−0.312504
$207$	−1.17157	−0.0814299
$208$	0	0
$209$	22.9706	1.58891
$210$	0	0
$211$	3.07107	0.211421	0.105711	−	0.994397i	$-0.466288\pi$
$211$	3.07107	0.211421	0.105711	+	0.994397i	$0.466288\pi$
$212$	12.0000	0.824163
$213$	0.828427	0.0567629
$214$	6.34315	0.433609
$215$	−4.68629	−0.319602
$216$	5.65685	0.384900
$217$	0	0
$218$	12.0000	0.812743
$219$	−19.3137	−1.30510
$220$	12.0000	0.809040
$221$	0	0
$222$	−12.0000	−0.805387
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	−3.00000	−0.200000
$226$	−1.65685	−0.110212
$227$	−13.8995	−0.922542	−0.461271	−	0.887259i	$-0.652606\pi$
$227$	−13.8995	−0.922542	−0.461271	+	0.887259i	$0.652606\pi$
$228$	−7.65685	−0.507088
$229$	8.00000	0.528655	0.264327	−	0.964433i	$-0.414850\pi$
$229$	8.00000	0.528655	0.264327	+	0.964433i	$0.414850\pi$
$230$	3.31371	0.218499
$231$	0	0
$232$	−1.65685	−0.108778
$233$	27.6569	1.81186	0.905930	−	0.423427i	$-0.139173\pi$
$233$	27.6569	1.81186	0.905930	+	0.423427i	$0.139173\pi$
$234$	0	0
$235$	25.6569	1.67367
$236$	1.17157	0.0762629
$237$	14.4853	0.940920
$238$	0	0
$239$	−0.585786	−0.0378914	−0.0189457	−	0.999821i	$-0.506031\pi$
$239$	−0.585786	−0.0378914	−0.0189457	+	0.999821i	$0.506031\pi$
$240$	−4.00000	−0.258199
$241$	−18.0000	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$242$	7.00000	0.449977
$243$	−9.89949	−0.635053
$244$	−6.00000	−0.384111
$245$	0	0
$246$	−1.41421	−0.0901670
$247$	0	0
$248$	1.17157	0.0743950
$249$	24.9706	1.58245
$250$	−5.65685	−0.357771
$251$	−2.14214	−0.135210	−0.0676052	−	0.997712i	$-0.521536\pi$
$251$	−2.14214	−0.135210	−0.0676052	+	0.997712i	$0.521536\pi$
$252$	0	0
$253$	4.97056	0.312497
$254$	18.1421	1.13834
$255$	30.6274	1.91796
$256$	1.00000	0.0625000
$257$	−13.3137	−0.830486	−0.415243	−	0.909710i	$-0.636304\pi$
$257$	−13.3137	−0.830486	−0.415243	+	0.909710i	$0.636304\pi$
$258$	2.34315	0.145878
$259$	0	0
$260$	0	0
$261$	1.65685	0.102557
$262$	10.1421	0.626583
$263$	−6.72792	−0.414861	−0.207431	−	0.978250i	$-0.566510\pi$
$263$	−6.72792	−0.414861	−0.207431	+	0.978250i	$0.566510\pi$
$264$	−6.00000	−0.369274
$265$	33.9411	2.08499
$266$	0	0
$267$	−16.4853	−1.00888
$268$	−8.24264	−0.503499
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	16.0000	0.973729
$271$	20.4853	1.24439	0.622196	−	0.782861i	$-0.286241\pi$
$271$	20.4853	1.24439	0.622196	+	0.782861i	$0.286241\pi$
$272$	−7.65685	−0.464265
$273$	0	0
$274$	5.31371	0.321013
$275$	12.7279	0.767523
$276$	−1.65685	−0.0997309
$277$	24.4853	1.47118	0.735589	−	0.677428i	$-0.236906\pi$
$277$	24.4853	1.47118	0.735589	+	0.677428i	$0.236906\pi$
$278$	−12.4853	−0.748817
$279$	−1.17157	−0.0701402
$280$	0	0
$281$	−22.9706	−1.37031	−0.685154	−	0.728398i	$-0.740265\pi$
$281$	−22.9706	−1.37031	−0.685154	+	0.728398i	$0.740265\pi$
$282$	−12.8284	−0.763922
$283$	11.5147	0.684479	0.342239	−	0.939613i	$-0.388815\pi$
$283$	11.5147	0.684479	0.342239	+	0.939613i	$0.388815\pi$
$284$	−0.585786	−0.0347600
$285$	−21.6569	−1.28284
$286$	0	0
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	41.6274	2.44867
$290$	−4.68629	−0.275189
$291$	5.17157	0.303163
$292$	13.6569	0.799207
$293$	17.6569	1.03152	0.515762	−	0.856732i	$-0.327509\pi$
$293$	17.6569	1.03152	0.515762	+	0.856732i	$0.327509\pi$
$294$	0	0
$295$	3.31371	0.192932
$296$	8.48528	0.493197
$297$	24.0000	1.39262
$298$	3.31371	0.191958
$299$	0	0
$300$	−4.24264	−0.244949
$301$	0	0
$302$	−11.8995	−0.684739
$303$	−8.97056	−0.515345
$304$	5.41421	0.310526
$305$	−16.9706	−0.971732
$306$	7.65685	0.437713
$307$	−12.4853	−0.712573	−0.356286	−	0.934377i	$-0.615957\pi$
$307$	−12.4853	−0.712573	−0.356286	+	0.934377i	$0.615957\pi$
$308$	0	0
$309$	6.34315	0.360849
$310$	3.31371	0.188206
$311$	−5.07107	−0.287554	−0.143777	−	0.989610i	$-0.545925\pi$
$311$	−5.07107	−0.287554	−0.143777	+	0.989610i	$0.545925\pi$
$312$	0	0
$313$	1.31371	0.0742552	0.0371276	−	0.999311i	$-0.488179\pi$
$313$	1.31371	0.0742552	0.0371276	+	0.999311i	$0.488179\pi$
$314$	20.9706	1.18344
$315$	0	0
$316$	−10.2426	−0.576194
$317$	−11.3137	−0.635441	−0.317721	−	0.948184i	$-0.602917\pi$
$317$	−11.3137	−0.635441	−0.317721	+	0.948184i	$0.602917\pi$
$318$	−16.9706	−0.951662
$319$	−7.02944	−0.393573
$320$	2.82843	0.158114
$321$	−8.97056	−0.500688
$322$	0	0
$323$	−41.4558	−2.30666
$324$	−5.00000	−0.277778
$325$	0	0
$326$	−4.00000	−0.221540
$327$	−16.9706	−0.938474
$328$	1.00000	0.0552158
$329$	0	0
$330$	−16.9706	−0.934199
$331$	5.89949	0.324266	0.162133	−	0.986769i	$-0.448163\pi$
$331$	5.89949	0.324266	0.162133	+	0.986769i	$0.448163\pi$
$332$	−17.6569	−0.969046
$333$	−8.48528	−0.464991
$334$	18.7279	1.02475
$335$	−23.3137	−1.27376
$336$	0	0
$337$	−12.0000	−0.653682	−0.326841	−	0.945079i	$-0.605984\pi$
$337$	−12.0000	−0.653682	−0.326841	+	0.945079i	$0.605984\pi$
$338$	−13.0000	−0.707107
$339$	2.34315	0.127262
$340$	−21.6569	−1.17451
$341$	4.97056	0.269171
$342$	−5.41421	−0.292767
$343$	0	0
$344$	−1.65685	−0.0893316
$345$	−4.68629	−0.252301
$346$	−13.3137	−0.715749
$347$	21.8995	1.17563	0.587813	−	0.808997i	$-0.299989\pi$
$347$	21.8995	1.17563	0.587813	+	0.808997i	$0.299989\pi$
$348$	2.34315	0.125606
$349$	−15.5147	−0.830484	−0.415242	−	0.909711i	$-0.636303\pi$
$349$	−15.5147	−0.830484	−0.415242	+	0.909711i	$0.636303\pi$
$350$	0	0
$351$	0	0
$352$	4.24264	0.226134
$353$	3.31371	0.176371	0.0881855	−	0.996104i	$-0.471893\pi$
$353$	3.31371	0.176371	0.0881855	+	0.996104i	$0.471893\pi$
$354$	−1.65685	−0.0880608
$355$	−1.65685	−0.0879367
$356$	11.6569	0.617812
$357$	0	0
$358$	−15.5563	−0.822179
$359$	13.6569	0.720781	0.360391	−	0.932801i	$-0.382643\pi$
$359$	13.6569	0.720781	0.360391	+	0.932801i	$0.382643\pi$
$360$	−2.82843	−0.149071
$361$	10.3137	0.542827
$362$	0	0
$363$	−9.89949	−0.519589
$364$	0	0
$365$	38.6274	2.02185
$366$	8.48528	0.443533
$367$	13.6569	0.712882	0.356441	−	0.934318i	$-0.383990\pi$
$367$	13.6569	0.712882	0.356441	+	0.934318i	$0.383990\pi$
$368$	1.17157	0.0610725
$369$	−1.00000	−0.0520579
$370$	24.0000	1.24770
$371$	0	0
$372$	−1.65685	−0.0859039
$373$	−32.6274	−1.68938	−0.844692	−	0.535253i	$-0.820216\pi$
$373$	−32.6274	−1.68938	−0.844692	+	0.535253i	$0.820216\pi$
$374$	−32.4853	−1.67977
$375$	8.00000	0.413118
$376$	9.07107	0.467805
$377$	0	0
$378$	0	0
$379$	−21.4558	−1.10211	−0.551056	−	0.834468i	$-0.685775\pi$
$379$	−21.4558	−1.10211	−0.551056	+	0.834468i	$0.685775\pi$
$380$	15.3137	0.785577
$381$	−25.6569	−1.31444
$382$	−2.92893	−0.149857
$383$	3.89949	0.199255	0.0996274	−	0.995025i	$-0.468235\pi$
$383$	3.89949	0.199255	0.0996274	+	0.995025i	$0.468235\pi$
$384$	−1.41421	−0.0721688
$385$	0	0
$386$	15.6569	0.796913
$387$	1.65685	0.0842226
$388$	−3.65685	−0.185649
$389$	−9.31371	−0.472224	−0.236112	−	0.971726i	$-0.575873\pi$
$389$	−9.31371	−0.472224	−0.236112	+	0.971726i	$0.575873\pi$
$390$	0	0
$391$	−8.97056	−0.453661
$392$	0	0
$393$	−14.3431	−0.723516
$394$	9.31371	0.469218
$395$	−28.9706	−1.45767
$396$	−4.24264	−0.213201
$397$	4.97056	0.249465	0.124733	−	0.992190i	$-0.460193\pi$
$397$	4.97056	0.249465	0.124733	+	0.992190i	$0.460193\pi$
$398$	2.24264	0.112413
$399$	0	0
$400$	3.00000	0.150000
$401$	−11.3137	−0.564980	−0.282490	−	0.959270i	$-0.591160\pi$
$401$	−11.3137	−0.564980	−0.282490	+	0.959270i	$0.591160\pi$
$402$	11.6569	0.581391
$403$	0	0
$404$	6.34315	0.315583
$405$	−14.1421	−0.702728
$406$	0	0
$407$	36.0000	1.78445
$408$	10.8284	0.536087
$409$	13.6569	0.675288	0.337644	−	0.941274i	$-0.390370\pi$
$409$	13.6569	0.675288	0.337644	+	0.941274i	$0.390370\pi$
$410$	2.82843	0.139686
$411$	−7.51472	−0.370674
$412$	−4.48528	−0.220974
$413$	0	0
$414$	−1.17157	−0.0575797
$415$	−49.9411	−2.45151
$416$	0	0
$417$	17.6569	0.864660
$418$	22.9706	1.12353
$419$	−17.6569	−0.862594	−0.431297	−	0.902210i	$-0.641944\pi$
$419$	−17.6569	−0.862594	−0.431297	+	0.902210i	$0.641944\pi$
$420$	0	0
$421$	−8.97056	−0.437198	−0.218599	−	0.975815i	$-0.570149\pi$
$421$	−8.97056	−0.437198	−0.218599	+	0.975815i	$0.570149\pi$
$422$	3.07107	0.149497
$423$	−9.07107	−0.441050
$424$	12.0000	0.582772
$425$	−22.9706	−1.11424
$426$	0.828427	0.0401374
$427$	0	0
$428$	6.34315	0.306608
$429$	0	0
$430$	−4.68629	−0.225993
$431$	−5.65685	−0.272481	−0.136241	−	0.990676i	$-0.543502\pi$
$431$	−5.65685	−0.272481	−0.136241	+	0.990676i	$0.543502\pi$
$432$	5.65685	0.272166
$433$	35.9411	1.72722	0.863610	−	0.504160i	$-0.168198\pi$
$433$	35.9411	1.72722	0.863610	+	0.504160i	$0.168198\pi$
$434$	0	0
$435$	6.62742	0.317760
$436$	12.0000	0.574696
$437$	6.34315	0.303434
$438$	−19.3137	−0.922845
$439$	−8.58579	−0.409777	−0.204889	−	0.978785i	$-0.565683\pi$
$439$	−8.58579	−0.409777	−0.204889	+	0.978785i	$0.565683\pi$
$440$	12.0000	0.572078
$441$	0	0
$442$	0	0
$443$	−9.17157	−0.435755	−0.217877	−	0.975976i	$-0.569913\pi$
$443$	−9.17157	−0.435755	−0.217877	+	0.975976i	$0.569913\pi$
$444$	−12.0000	−0.569495
$445$	32.9706	1.56295
$446$	0	0
$447$	−4.68629	−0.221654
$448$	0	0
$449$	−5.31371	−0.250769	−0.125385	−	0.992108i	$-0.540017\pi$
$449$	−5.31371	−0.250769	−0.125385	+	0.992108i	$0.540017\pi$
$450$	−3.00000	−0.141421
$451$	4.24264	0.199778
$452$	−1.65685	−0.0779319
$453$	16.8284	0.790668
$454$	−13.8995	−0.652336
$455$	0	0
$456$	−7.65685	−0.358565
$457$	24.6274	1.15202	0.576011	−	0.817442i	$-0.304609\pi$
$457$	24.6274	1.15202	0.576011	+	0.817442i	$0.304609\pi$
$458$	8.00000	0.373815
$459$	−43.3137	−2.02171
$460$	3.31371	0.154502
$461$	−19.7990	−0.922131	−0.461065	−	0.887366i	$-0.652533\pi$
$461$	−19.7990	−0.922131	−0.461065	+	0.887366i	$0.652533\pi$
$462$	0	0
$463$	−3.61522	−0.168014	−0.0840068	−	0.996465i	$-0.526772\pi$
$463$	−3.61522	−0.168014	−0.0840068	+	0.996465i	$0.526772\pi$
$464$	−1.65685	−0.0769175
$465$	−4.68629	−0.217322
$466$	27.6569	1.28118
$467$	5.85786	0.271070	0.135535	−	0.990773i	$-0.456725\pi$
$467$	5.85786	0.271070	0.135535	+	0.990773i	$0.456725\pi$
$468$	0	0
$469$	0	0
$470$	25.6569	1.18346
$471$	−29.6569	−1.36652
$472$	1.17157	0.0539260
$473$	−7.02944	−0.323214
$474$	14.4853	0.665331
$475$	16.2426	0.745263
$476$	0	0
$477$	−12.0000	−0.549442
$478$	−0.585786	−0.0267932
$479$	−20.5858	−0.940589	−0.470294	−	0.882510i	$-0.655852\pi$
$479$	−20.5858	−0.940589	−0.470294	+	0.882510i	$0.655852\pi$
$480$	−4.00000	−0.182574
$481$	0	0
$482$	−18.0000	−0.819878
$483$	0	0
$484$	7.00000	0.318182
$485$	−10.3431	−0.469658
$486$	−9.89949	−0.449050
$487$	−3.51472	−0.159267	−0.0796336	−	0.996824i	$-0.525375\pi$
$487$	−3.51472	−0.159267	−0.0796336	+	0.996824i	$0.525375\pi$
$488$	−6.00000	−0.271607
$489$	5.65685	0.255812
$490$	0	0
$491$	−31.7990	−1.43507	−0.717534	−	0.696523i	$-0.754729\pi$
$491$	−31.7990	−1.43507	−0.717534	+	0.696523i	$0.754729\pi$
$492$	−1.41421	−0.0637577
$493$	12.6863	0.571362
$494$	0	0
$495$	−12.0000	−0.539360
$496$	1.17157	0.0526052
$497$	0	0
$498$	24.9706	1.11896
$499$	−28.5269	−1.27704	−0.638520	−	0.769605i	$-0.720453\pi$
$499$	−28.5269	−1.27704	−0.638520	+	0.769605i	$0.720453\pi$
$500$	−5.65685	−0.252982
$501$	−26.4853	−1.18328
$502$	−2.14214	−0.0956082
$503$	−1.07107	−0.0477566	−0.0238783	−	0.999715i	$-0.507601\pi$
$503$	−1.07107	−0.0477566	−0.0238783	+	0.999715i	$0.507601\pi$
$504$	0	0
$505$	17.9411	0.798370
$506$	4.97056	0.220968
$507$	18.3848	0.816497
$508$	18.1421	0.804927
$509$	0.686292	0.0304193	0.0152097	−	0.999884i	$-0.495158\pi$
$509$	0.686292	0.0304193	0.0152097	+	0.999884i	$0.495158\pi$
$510$	30.6274	1.35620
$511$	0	0
$512$	1.00000	0.0441942
$513$	30.6274	1.35223
$514$	−13.3137	−0.587243
$515$	−12.6863	−0.559025
$516$	2.34315	0.103151
$517$	38.4853	1.69258
$518$	0	0
$519$	18.8284	0.826476
$520$	0	0
$521$	2.68629	0.117689	0.0588443	−	0.998267i	$-0.481258\pi$
$521$	2.68629	0.117689	0.0588443	+	0.998267i	$0.481258\pi$
$522$	1.65685	0.0725185
$523$	15.3137	0.669622	0.334811	−	0.942285i	$-0.391328\pi$
$523$	15.3137	0.669622	0.334811	+	0.942285i	$0.391328\pi$
$524$	10.1421	0.443061
$525$	0	0
$526$	−6.72792	−0.293351
$527$	−8.97056	−0.390764
$528$	−6.00000	−0.261116
$529$	−21.6274	−0.940322
$530$	33.9411	1.47431
$531$	−1.17157	−0.0508419
$532$	0	0
$533$	0	0
$534$	−16.4853	−0.713388
$535$	17.9411	0.775662
$536$	−8.24264	−0.356028
$537$	22.0000	0.949370
$538$	−18.0000	−0.776035
$539$	0	0
$540$	16.0000	0.688530
$541$	8.48528	0.364811	0.182405	−	0.983223i	$-0.441612\pi$
$541$	8.48528	0.364811	0.182405	+	0.983223i	$0.441612\pi$
$542$	20.4853	0.879918
$543$	0	0
$544$	−7.65685	−0.328285
$545$	33.9411	1.45388
$546$	0	0
$547$	−9.41421	−0.402523	−0.201261	−	0.979538i	$-0.564504\pi$
$547$	−9.41421	−0.402523	−0.201261	+	0.979538i	$0.564504\pi$
$548$	5.31371	0.226990
$549$	6.00000	0.256074
$550$	12.7279	0.542720
$551$	−8.97056	−0.382159
$552$	−1.65685	−0.0705204
$553$	0	0
$554$	24.4853	1.04028
$555$	−33.9411	−1.44072
$556$	−12.4853	−0.529494
$557$	−5.65685	−0.239689	−0.119844	−	0.992793i	$-0.538240\pi$
$557$	−5.65685	−0.239689	−0.119844	+	0.992793i	$0.538240\pi$
$558$	−1.17157	−0.0495966
$559$	0	0
$560$	0	0
$561$	45.9411	1.93964
$562$	−22.9706	−0.968955
$563$	0.727922	0.0306783	0.0153391	−	0.999882i	$-0.495117\pi$
$563$	0.727922	0.0306783	0.0153391	+	0.999882i	$0.495117\pi$
$564$	−12.8284	−0.540174
$565$	−4.68629	−0.197154
$566$	11.5147	0.484000
$567$	0	0
$568$	−0.585786	−0.0245791
$569$	24.0000	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$570$	−21.6569	−0.907106
$571$	32.2426	1.34931	0.674656	−	0.738132i	$-0.264292\pi$
$571$	32.2426	1.34931	0.674656	+	0.738132i	$0.264292\pi$
$572$	0	0
$573$	4.14214	0.173040
$574$	0	0
$575$	3.51472	0.146574
$576$	−1.00000	−0.0416667
$577$	10.0000	0.416305	0.208153	−	0.978096i	$-0.433255\pi$
$577$	10.0000	0.416305	0.208153	+	0.978096i	$0.433255\pi$
$578$	41.6274	1.73147
$579$	−22.1421	−0.920196
$580$	−4.68629	−0.194588
$581$	0	0
$582$	5.17157	0.214369
$583$	50.9117	2.10855
$584$	13.6569	0.565125
$585$	0	0
$586$	17.6569	0.729398
$587$	−9.89949	−0.408596	−0.204298	−	0.978909i	$-0.565491\pi$
$587$	−9.89949	−0.408596	−0.204298	+	0.978909i	$0.565491\pi$
$588$	0	0
$589$	6.34315	0.261365
$590$	3.31371	0.136423
$591$	−13.1716	−0.541806
$592$	8.48528	0.348743
$593$	−13.3137	−0.546728	−0.273364	−	0.961911i	$-0.588136\pi$
$593$	−13.3137	−0.546728	−0.273364	+	0.961911i	$0.588136\pi$
$594$	24.0000	0.984732
$595$	0	0
$596$	3.31371	0.135735
$597$	−3.17157	−0.129804
$598$	0	0
$599$	−3.51472	−0.143608	−0.0718038	−	0.997419i	$-0.522876\pi$
$599$	−3.51472	−0.143608	−0.0718038	+	0.997419i	$0.522876\pi$
$600$	−4.24264	−0.173205
$601$	10.9706	0.447499	0.223749	−	0.974647i	$-0.428170\pi$
$601$	10.9706	0.447499	0.223749	+	0.974647i	$0.428170\pi$
$602$	0	0
$603$	8.24264	0.335666
$604$	−11.8995	−0.484184
$605$	19.7990	0.804943
$606$	−8.97056	−0.364404
$607$	−27.3137	−1.10863	−0.554315	−	0.832307i	$-0.687020\pi$
$607$	−27.3137	−1.10863	−0.554315	+	0.832307i	$0.687020\pi$
$608$	5.41421	0.219575
$609$	0	0
$610$	−16.9706	−0.687118
$611$	0	0
$612$	7.65685	0.309510
$613$	−10.8284	−0.437356	−0.218678	−	0.975797i	$-0.570174\pi$
$613$	−10.8284	−0.437356	−0.218678	+	0.975797i	$0.570174\pi$
$614$	−12.4853	−0.503865
$615$	−4.00000	−0.161296
$616$	0	0
$617$	−33.3137	−1.34116	−0.670580	−	0.741837i	$-0.733955\pi$
$617$	−33.3137	−1.34116	−0.670580	+	0.741837i	$0.733955\pi$
$618$	6.34315	0.255159
$619$	−38.4264	−1.54449	−0.772244	−	0.635326i	$-0.780866\pi$
$619$	−38.4264	−1.54449	−0.772244	+	0.635326i	$0.780866\pi$
$620$	3.31371	0.133082
$621$	6.62742	0.265949
$622$	−5.07107	−0.203331
$623$	0	0
$624$	0	0
$625$	−31.0000	−1.24000
$626$	1.31371	0.0525064
$627$	−32.4853	−1.29734
$628$	20.9706	0.836817
$629$	−64.9706	−2.59055
$630$	0	0
$631$	−33.9411	−1.35117	−0.675587	−	0.737280i	$-0.736110\pi$
$631$	−33.9411	−1.35117	−0.675587	+	0.737280i	$0.736110\pi$
$632$	−10.2426	−0.407430
$633$	−4.34315	−0.172625
$634$	−11.3137	−0.449325
$635$	51.3137	2.03632
$636$	−16.9706	−0.672927
$637$	0	0
$638$	−7.02944	−0.278298
$639$	0.585786	0.0231734
$640$	2.82843	0.111803
$641$	1.02944	0.0406603	0.0203302	−	0.999793i	$-0.493528\pi$
$641$	1.02944	0.0406603	0.0203302	+	0.999793i	$0.493528\pi$
$642$	−8.97056	−0.354040
$643$	−6.87006	−0.270929	−0.135464	−	0.990782i	$-0.543253\pi$
$643$	−6.87006	−0.270929	−0.135464	+	0.990782i	$0.543253\pi$
$644$	0	0
$645$	6.62742	0.260954
$646$	−41.4558	−1.63106
$647$	−44.4853	−1.74890	−0.874448	−	0.485118i	$-0.838776\pi$
$647$	−44.4853	−1.74890	−0.874448	+	0.485118i	$0.838776\pi$
$648$	−5.00000	−0.196419
$649$	4.97056	0.195112
$650$	0	0
$651$	0	0
$652$	−4.00000	−0.156652
$653$	−11.3137	−0.442740	−0.221370	−	0.975190i	$-0.571053\pi$
$653$	−11.3137	−0.442740	−0.221370	+	0.975190i	$0.571053\pi$
$654$	−16.9706	−0.663602
$655$	28.6863	1.12087
$656$	1.00000	0.0390434
$657$	−13.6569	−0.532805
$658$	0	0
$659$	13.4142	0.522544	0.261272	−	0.965265i	$-0.415858\pi$
$659$	13.4142	0.522544	0.261272	+	0.965265i	$0.415858\pi$
$660$	−16.9706	−0.660578
$661$	−41.4558	−1.61245	−0.806223	−	0.591612i	$-0.798492\pi$
$661$	−41.4558	−1.61245	−0.806223	+	0.591612i	$0.798492\pi$
$662$	5.89949	0.229290
$663$	0	0
$664$	−17.6569	−0.685219
$665$	0	0
$666$	−8.48528	−0.328798
$667$	−1.94113	−0.0751607
$668$	18.7279	0.724605
$669$	0	0
$670$	−23.3137	−0.900687
$671$	−25.4558	−0.982712
$672$	0	0
$673$	−50.9706	−1.96477	−0.982385	−	0.186866i	$-0.940167\pi$
$673$	−50.9706	−1.96477	−0.982385	+	0.186866i	$0.940167\pi$
$674$	−12.0000	−0.462223
$675$	16.9706	0.653197
$676$	−13.0000	−0.500000
$677$	38.1421	1.46592	0.732961	−	0.680271i	$-0.238138\pi$
$677$	38.1421	1.46592	0.732961	+	0.680271i	$0.238138\pi$
$678$	2.34315	0.0899880
$679$	0	0
$680$	−21.6569	−0.830502
$681$	19.6569	0.753252
$682$	4.97056	0.190333
$683$	−41.4142	−1.58467	−0.792335	−	0.610086i	$-0.791135\pi$
$683$	−41.4142	−1.58467	−0.792335	+	0.610086i	$0.791135\pi$
$684$	−5.41421	−0.207018
$685$	15.0294	0.574245
$686$	0	0
$687$	−11.3137	−0.431645
$688$	−1.65685	−0.0631670
$689$	0	0
$690$	−4.68629	−0.178404
$691$	−12.2426	−0.465732	−0.232866	−	0.972509i	$-0.574810\pi$
$691$	−12.2426	−0.465732	−0.232866	+	0.972509i	$0.574810\pi$
$692$	−13.3137	−0.506111
$693$	0	0
$694$	21.8995	0.831293
$695$	−35.3137	−1.33953
$696$	2.34315	0.0888167
$697$	−7.65685	−0.290024
$698$	−15.5147	−0.587241
$699$	−39.1127	−1.47938
$700$	0	0
$701$	25.4558	0.961454	0.480727	−	0.876870i	$-0.340373\pi$
$701$	25.4558	0.961454	0.480727	+	0.876870i	$0.340373\pi$
$702$	0	0
$703$	45.9411	1.73270
$704$	4.24264	0.159901
$705$	−36.2843	−1.36654
$706$	3.31371	0.124713
$707$	0	0
$708$	−1.65685	−0.0622684
$709$	−21.6569	−0.813340	−0.406670	−	0.913575i	$-0.633310\pi$
$709$	−21.6569	−0.813340	−0.406670	+	0.913575i	$0.633310\pi$
$710$	−1.65685	−0.0621806
$711$	10.2426	0.384129
$712$	11.6569	0.436859
$713$	1.37258	0.0514036
$714$	0	0
$715$	0	0
$716$	−15.5563	−0.581368
$717$	0.828427	0.0309382
$718$	13.6569	0.509669
$719$	12.1005	0.451273	0.225636	−	0.974212i	$-0.427554\pi$
$719$	12.1005	0.451273	0.225636	+	0.974212i	$0.427554\pi$
$720$	−2.82843	−0.105409
$721$	0	0
$722$	10.3137	0.383836
$723$	25.4558	0.946713
$724$	0	0
$725$	−4.97056	−0.184602
$726$	−9.89949	−0.367405
$727$	−29.0711	−1.07819	−0.539093	−	0.842246i	$-0.681233\pi$
$727$	−29.0711	−1.07819	−0.539093	+	0.842246i	$0.681233\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	38.6274	1.42966
$731$	12.6863	0.469219
$732$	8.48528	0.313625
$733$	−26.6863	−0.985681	−0.492840	−	0.870120i	$-0.664041\pi$
$733$	−26.6863	−0.985681	−0.492840	+	0.870120i	$0.664041\pi$
$734$	13.6569	0.504084
$735$	0	0
$736$	1.17157	0.0431847
$737$	−34.9706	−1.28816
$738$	−1.00000	−0.0368105
$739$	−19.0294	−0.700009	−0.350005	−	0.936748i	$-0.613820\pi$
$739$	−19.0294	−0.700009	−0.350005	+	0.936748i	$0.613820\pi$
$740$	24.0000	0.882258
$741$	0	0
$742$	0	0
$743$	3.51472	0.128943	0.0644713	−	0.997920i	$-0.479464\pi$
$743$	3.51472	0.128943	0.0644713	+	0.997920i	$0.479464\pi$
$744$	−1.65685	−0.0607432
$745$	9.37258	0.343385
$746$	−32.6274	−1.19457
$747$	17.6569	0.646031
$748$	−32.4853	−1.18778
$749$	0	0
$750$	8.00000	0.292119
$751$	−50.7279	−1.85109	−0.925544	−	0.378640i	$-0.876392\pi$
$751$	−50.7279	−1.85109	−0.925544	+	0.378640i	$0.876392\pi$
$752$	9.07107	0.330788
$753$	3.02944	0.110399
$754$	0	0
$755$	−33.6569	−1.22490
$756$	0	0
$757$	30.6274	1.11317	0.556586	−	0.830790i	$-0.312111\pi$
$757$	30.6274	1.11317	0.556586	+	0.830790i	$0.312111\pi$
$758$	−21.4558	−0.779311
$759$	−7.02944	−0.255152
$760$	15.3137	0.555487
$761$	36.0000	1.30500	0.652499	−	0.757789i	$-0.273720\pi$
$761$	36.0000	1.30500	0.652499	+	0.757789i	$0.273720\pi$
$762$	−25.6569	−0.929450
$763$	0	0
$764$	−2.92893	−0.105965
$765$	21.6569	0.783005
$766$	3.89949	0.140894
$767$	0	0
$768$	−1.41421	−0.0510310
$769$	−0.627417	−0.0226252	−0.0113126	−	0.999936i	$-0.503601\pi$
$769$	−0.627417	−0.0226252	−0.0113126	+	0.999936i	$0.503601\pi$
$770$	0	0
$771$	18.8284	0.678089
$772$	15.6569	0.563503
$773$	−11.3137	−0.406926	−0.203463	−	0.979083i	$-0.565220\pi$
$773$	−11.3137	−0.406926	−0.203463	+	0.979083i	$0.565220\pi$
$774$	1.65685	0.0595544
$775$	3.51472	0.126252
$776$	−3.65685	−0.131273
$777$	0	0
$778$	−9.31371	−0.333913
$779$	5.41421	0.193984
$780$	0	0
$781$	−2.48528	−0.0889304
$782$	−8.97056	−0.320787
$783$	−9.37258	−0.334949
$784$	0	0
$785$	59.3137	2.11700
$786$	−14.3431	−0.511603
$787$	−19.5147	−0.695625	−0.347812	−	0.937564i	$-0.613075\pi$
$787$	−19.5147	−0.695625	−0.347812	+	0.937564i	$0.613075\pi$
$788$	9.31371	0.331787
$789$	9.51472	0.338733
$790$	−28.9706	−1.03073
$791$	0	0
$792$	−4.24264	−0.150756
$793$	0	0
$794$	4.97056	0.176399
$795$	−48.0000	−1.70238
$796$	2.24264	0.0794883
$797$	−21.3137	−0.754970	−0.377485	−	0.926016i	$-0.623211\pi$
$797$	−21.3137	−0.754970	−0.377485	+	0.926016i	$0.623211\pi$
$798$	0	0
$799$	−69.4558	−2.45717
$800$	3.00000	0.106066
$801$	−11.6569	−0.411875
$802$	−11.3137	−0.399501
$803$	57.9411	2.04470
$804$	11.6569	0.411106
$805$	0	0
$806$	0	0
$807$	25.4558	0.896088
$808$	6.34315	0.223151
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	−14.1421	−0.496904
$811$	−8.20101	−0.287976	−0.143988	−	0.989579i	$-0.545993\pi$
$811$	−8.20101	−0.287976	−0.143988	+	0.989579i	$0.545993\pi$
$812$	0	0
$813$	−28.9706	−1.01604
$814$	36.0000	1.26180
$815$	−11.3137	−0.396302
$816$	10.8284	0.379071
$817$	−8.97056	−0.313840
$818$	13.6569	0.477501
$819$	0	0
$820$	2.82843	0.0987730
$821$	−6.14214	−0.214362	−0.107181	−	0.994240i	$-0.534182\pi$
$821$	−6.14214	−0.214362	−0.107181	+	0.994240i	$0.534182\pi$
$822$	−7.51472	−0.262106
$823$	−47.2132	−1.64575	−0.822874	−	0.568223i	$-0.807631\pi$
$823$	−47.2132	−1.64575	−0.822874	+	0.568223i	$0.807631\pi$
$824$	−4.48528	−0.156252
$825$	−18.0000	−0.626680
$826$	0	0
$827$	28.9289	1.00596	0.502979	−	0.864299i	$-0.332237\pi$
$827$	28.9289	1.00596	0.502979	+	0.864299i	$0.332237\pi$
$828$	−1.17157	−0.0407150
$829$	18.8284	0.653938	0.326969	−	0.945035i	$-0.393973\pi$
$829$	18.8284	0.653938	0.326969	+	0.945035i	$0.393973\pi$
$830$	−49.9411	−1.73348
$831$	−34.6274	−1.20121
$832$	0	0
$833$	0	0
$834$	17.6569	0.611407
$835$	52.9706	1.83312
$836$	22.9706	0.794454
$837$	6.62742	0.229077
$838$	−17.6569	−0.609946
$839$	38.5269	1.33010	0.665048	−	0.746800i	$-0.268411\pi$
$839$	38.5269	1.33010	0.665048	+	0.746800i	$0.268411\pi$
$840$	0	0
$841$	−26.2548	−0.905339
$842$	−8.97056	−0.309146
$843$	32.4853	1.11885
$844$	3.07107	0.105711
$845$	−36.7696	−1.26491
$846$	−9.07107	−0.311870
$847$	0	0
$848$	12.0000	0.412082
$849$	−16.2843	−0.558875
$850$	−22.9706	−0.787884
$851$	9.94113	0.340777
$852$	0.828427	0.0283814
$853$	14.0000	0.479351	0.239675	−	0.970853i	$-0.422959\pi$
$853$	14.0000	0.479351	0.239675	+	0.970853i	$0.422959\pi$
$854$	0	0
$855$	−15.3137	−0.523718
$856$	6.34315	0.216804
$857$	−42.3431	−1.44641	−0.723207	−	0.690631i	$-0.757333\pi$
$857$	−42.3431	−1.44641	−0.723207	+	0.690631i	$0.757333\pi$
$858$	0	0
$859$	−29.4558	−1.00502	−0.502510	−	0.864571i	$-0.667590\pi$
$859$	−29.4558	−1.00502	−0.502510	+	0.864571i	$0.667590\pi$
$860$	−4.68629	−0.159801
$861$	0	0
$862$	−5.65685	−0.192673
$863$	53.2548	1.81282	0.906408	−	0.422404i	$-0.138814\pi$
$863$	53.2548	1.81282	0.906408	+	0.422404i	$0.138814\pi$
$864$	5.65685	0.192450
$865$	−37.6569	−1.28037
$866$	35.9411	1.22133
$867$	−58.8701	−1.99933
$868$	0	0
$869$	−43.4558	−1.47414
$870$	6.62742	0.224690
$871$	0	0
$872$	12.0000	0.406371
$873$	3.65685	0.123766
$874$	6.34315	0.214560
$875$	0	0
$876$	−19.3137	−0.652550
$877$	10.0000	0.337676	0.168838	−	0.985644i	$-0.445999\pi$
$877$	10.0000	0.337676	0.168838	+	0.985644i	$0.445999\pi$
$878$	−8.58579	−0.289756
$879$	−24.9706	−0.842236
$880$	12.0000	0.404520
$881$	−28.6274	−0.964482	−0.482241	−	0.876039i	$-0.660177\pi$
$881$	−28.6274	−0.964482	−0.482241	+	0.876039i	$0.660177\pi$
$882$	0	0
$883$	−34.3848	−1.15714	−0.578570	−	0.815633i	$-0.696389\pi$
$883$	−34.3848	−1.15714	−0.578570	+	0.815633i	$0.696389\pi$
$884$	0	0
$885$	−4.68629	−0.157528
$886$	−9.17157	−0.308125
$887$	21.5563	0.723791	0.361896	−	0.932219i	$-0.382130\pi$
$887$	21.5563	0.723791	0.361896	+	0.932219i	$0.382130\pi$
$888$	−12.0000	−0.402694
$889$	0	0
$890$	32.9706	1.10518
$891$	−21.2132	−0.710669
$892$	0	0
$893$	49.1127	1.64349
$894$	−4.68629	−0.156733
$895$	−44.0000	−1.47076
$896$	0	0
$897$	0	0
$898$	−5.31371	−0.177321
$899$	−1.94113	−0.0647402
$900$	−3.00000	−0.100000
$901$	−91.8823	−3.06104
$902$	4.24264	0.141264
$903$	0	0
$904$	−1.65685	−0.0551062
$905$	0	0
$906$	16.8284	0.559087
$907$	6.82843	0.226734	0.113367	−	0.993553i	$-0.463836\pi$
$907$	6.82843	0.226734	0.113367	+	0.993553i	$0.463836\pi$
$908$	−13.8995	−0.461271
$909$	−6.34315	−0.210389
$910$	0	0
$911$	51.3137	1.70010	0.850050	−	0.526703i	$-0.176572\pi$
$911$	51.3137	1.70010	0.850050	+	0.526703i	$0.176572\pi$
$912$	−7.65685	−0.253544
$913$	−74.9117	−2.47922
$914$	24.6274	0.814603
$915$	24.0000	0.793416
$916$	8.00000	0.264327
$917$	0	0
$918$	−43.3137	−1.42957
$919$	11.6985	0.385897	0.192949	−	0.981209i	$-0.438195\pi$
$919$	11.6985	0.385897	0.192949	+	0.981209i	$0.438195\pi$
$920$	3.31371	0.109250
$921$	17.6569	0.581813
$922$	−19.7990	−0.652045
$923$	0	0
$924$	0	0
$925$	25.4558	0.836983
$926$	−3.61522	−0.118804
$927$	4.48528	0.147316
$928$	−1.65685	−0.0543889
$929$	−51.6569	−1.69481	−0.847403	−	0.530950i	$-0.821835\pi$
$929$	−51.6569	−1.69481	−0.847403	+	0.530950i	$0.821835\pi$
$930$	−4.68629	−0.153670
$931$	0	0
$932$	27.6569	0.905930
$933$	7.17157	0.234787
$934$	5.85786	0.191675
$935$	−91.8823	−3.00487
$936$	0	0
$937$	10.9706	0.358393	0.179196	−	0.983813i	$-0.442650\pi$
$937$	10.9706	0.358393	0.179196	+	0.983813i	$0.442650\pi$
$938$	0	0
$939$	−1.85786	−0.0606291
$940$	25.6569	0.836834
$941$	−32.6274	−1.06362	−0.531812	−	0.846863i	$-0.678489\pi$
$941$	−32.6274	−1.06362	−0.531812	+	0.846863i	$0.678489\pi$
$942$	−29.6569	−0.966273
$943$	1.17157	0.0381517
$944$	1.17157	0.0381314
$945$	0	0
$946$	−7.02944	−0.228547
$947$	−13.4558	−0.437256	−0.218628	−	0.975808i	$-0.570158\pi$
$947$	−13.4558	−0.437256	−0.218628	+	0.975808i	$0.570158\pi$
$948$	14.4853	0.470460
$949$	0	0
$950$	16.2426	0.526981
$951$	16.0000	0.518836
$952$	0	0
$953$	26.3431	0.853338	0.426669	−	0.904408i	$-0.359687\pi$
$953$	26.3431	0.853338	0.426669	+	0.904408i	$0.359687\pi$
$954$	−12.0000	−0.388514
$955$	−8.28427	−0.268073
$956$	−0.585786	−0.0189457
$957$	9.94113	0.321351
$958$	−20.5858	−0.665097
$959$	0	0
$960$	−4.00000	−0.129099
$961$	−29.6274	−0.955723
$962$	0	0
$963$	−6.34315	−0.204405
$964$	−18.0000	−0.579741
$965$	44.2843	1.42556
$966$	0	0
$967$	56.6690	1.82235	0.911177	−	0.412015i	$-0.135175\pi$
$967$	56.6690	1.82235	0.911177	+	0.412015i	$0.135175\pi$
$968$	7.00000	0.224989
$969$	58.6274	1.88338
$970$	−10.3431	−0.332098
$971$	32.5269	1.04384	0.521919	−	0.852995i	$-0.325216\pi$
$971$	32.5269	1.04384	0.521919	+	0.852995i	$0.325216\pi$
$972$	−9.89949	−0.317526
$973$	0	0
$974$	−3.51472	−0.112619
$975$	0	0
$976$	−6.00000	−0.192055
$977$	8.34315	0.266921	0.133460	−	0.991054i	$-0.457391\pi$
$977$	8.34315	0.266921	0.133460	+	0.991054i	$0.457391\pi$
$978$	5.65685	0.180886
$979$	49.4558	1.58062
$980$	0	0
$981$	−12.0000	−0.383131
$982$	−31.7990	−1.01475
$983$	−23.7990	−0.759070	−0.379535	−	0.925177i	$-0.623916\pi$
$983$	−23.7990	−0.759070	−0.379535	+	0.925177i	$0.623916\pi$
$984$	−1.41421	−0.0450835
$985$	26.3431	0.839362
$986$	12.6863	0.404014
$987$	0	0
$988$	0	0
$989$	−1.94113	−0.0617242
$990$	−12.0000	−0.381385
$991$	2.72792	0.0866553	0.0433277	−	0.999061i	$-0.486204\pi$
$991$	2.72792	0.0866553	0.0433277	+	0.999061i	$0.486204\pi$
$992$	1.17157	0.0371975
$993$	−8.34315	−0.264762
$994$	0	0
$995$	6.34315	0.201091
$996$	24.9706	0.791223
$997$	45.9411	1.45497	0.727485	−	0.686124i	$-0.240689\pi$
$997$	45.9411	1.45497	0.727485	+	0.686124i	$0.240689\pi$
$998$	−28.5269	−0.903004
$999$	48.0000	1.51865

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4018.2.a.ba.1.1		2
7.6	odd	2		82.2.a.b.1.2	✓	2
21.20	even	2		738.2.a.k.1.2		2
28.27	even	2		656.2.a.e.1.1		2
35.13	even	4		2050.2.c.l.1149.2		4
35.27	even	4		2050.2.c.l.1149.3		4
35.34	odd	2		2050.2.a.h.1.1		2
56.13	odd	2		2624.2.a.j.1.1		2
56.27	even	2		2624.2.a.l.1.2		2
77.76	even	2		9922.2.a.i.1.2		2
84.83	odd	2		5904.2.a.z.1.2		2
287.286	odd	2		3362.2.a.m.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
82.2.a.b.1.2	✓	2	7.6	odd	2
656.2.a.e.1.1		2	28.27	even	2
738.2.a.k.1.2		2	21.20	even	2
2050.2.a.h.1.1		2	35.34	odd	2
2050.2.c.l.1149.2		4	35.13	even	4
2050.2.c.l.1149.3		4	35.27	even	4
2624.2.a.j.1.1		2	56.13	odd	2
2624.2.a.l.1.2		2	56.27	even	2
3362.2.a.m.1.1		2	287.286	odd	2
4018.2.a.ba.1.1		2	1.1	even	1	trivial
5904.2.a.z.1.2		2	84.83	odd	2
9922.2.a.i.1.2		2	77.76	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:	\( N \)	\(=\)	\( 4018 = 2 \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4018.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.41421 q^{3} +1.00000 q^{4} +2.82843 q^{5} -1.41421 q^{6} +1.00000 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.41421 q^{3} +1.00000 q^{4} +2.82843 q^{5} -1.41421 q^{6} +1.00000 q^{8} -1.00000 q^{9} +2.82843 q^{10} +4.24264 q^{11} -1.41421 q^{12} -4.00000 q^{15} +1.00000 q^{16} -7.65685 q^{17} -1.00000 q^{18} +5.41421 q^{19} +2.82843 q^{20} +4.24264 q^{22} +1.17157 q^{23} -1.41421 q^{24} +3.00000 q^{25} +5.65685 q^{27} -1.65685 q^{29} -4.00000 q^{30} +1.17157 q^{31} +1.00000 q^{32} -6.00000 q^{33} -7.65685 q^{34} -1.00000 q^{36} +8.48528 q^{37} +5.41421 q^{38} +2.82843 q^{40} +1.00000 q^{41} -1.65685 q^{43} +4.24264 q^{44} -2.82843 q^{45} +1.17157 q^{46} +9.07107 q^{47} -1.41421 q^{48} +3.00000 q^{50} +10.8284 q^{51} +12.0000 q^{53} +5.65685 q^{54} +12.0000 q^{55} -7.65685 q^{57} -1.65685 q^{58} +1.17157 q^{59} -4.00000 q^{60} -6.00000 q^{61} +1.17157 q^{62} +1.00000 q^{64} -6.00000 q^{66} -8.24264 q^{67} -7.65685 q^{68} -1.65685 q^{69} -0.585786 q^{71} -1.00000 q^{72} +13.6569 q^{73} +8.48528 q^{74} -4.24264 q^{75} +5.41421 q^{76} -10.2426 q^{79} +2.82843 q^{80} -5.00000 q^{81} +1.00000 q^{82} -17.6569 q^{83} -21.6569 q^{85} -1.65685 q^{86} +2.34315 q^{87} +4.24264 q^{88} +11.6569 q^{89} -2.82843 q^{90} +1.17157 q^{92} -1.65685 q^{93} +9.07107 q^{94} +15.3137 q^{95} -1.41421 q^{96} -3.65685 q^{97} -4.24264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9} - 8 q^{15} + 2 q^{16} - 4 q^{17} - 2 q^{18} + 8 q^{19} + 8 q^{23} + 6 q^{25} + 8 q^{29} - 8 q^{30} + 8 q^{31} + 2 q^{32} - 12 q^{33} - 4 q^{34} - 2 q^{36} + 8 q^{38} + 2 q^{41} + 8 q^{43} + 8 q^{46} + 4 q^{47} + 6 q^{50} + 16 q^{51} + 24 q^{53} + 24 q^{55} - 4 q^{57} + 8 q^{58} + 8 q^{59} - 8 q^{60} - 12 q^{61} + 8 q^{62} + 2 q^{64} - 12 q^{66} - 8 q^{67} - 4 q^{68} + 8 q^{69} - 4 q^{71} - 2 q^{72} + 16 q^{73} + 8 q^{76} - 12 q^{79} - 10 q^{81} + 2 q^{82} - 24 q^{83} - 32 q^{85} + 8 q^{86} + 16 q^{87} + 12 q^{89} + 8 q^{92} + 8 q^{93} + 4 q^{94} + 8 q^{95} + 4 q^{97}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9 - 8 * q^15 + 2 * q^16 - 4 * q^17 - 2 * q^18 + 8 * q^19 + 8 * q^23 + 6 * q^25 + 8 * q^29 - 8 * q^30 + 8 * q^31 + 2 * q^32 - 12 * q^33 - 4 * q^34 - 2 * q^36 + 8 * q^38 + 2 * q^41 + 8 * q^43 + 8 * q^46 + 4 * q^47 + 6 * q^50 + 16 * q^51 + 24 * q^53 + 24 * q^55 - 4 * q^57 + 8 * q^58 + 8 * q^59 - 8 * q^60 - 12 * q^61 + 8 * q^62 + 2 * q^64 - 12 * q^66 - 8 * q^67 - 4 * q^68 + 8 * q^69 - 4 * q^71 - 2 * q^72 + 16 * q^73 + 8 * q^76 - 12 * q^79 - 10 * q^81 + 2 * q^82 - 24 * q^83 - 32 * q^85 + 8 * q^86 + 16 * q^87 + 12 * q^89 + 8 * q^92 + 8 * q^93 + 4 * q^94 + 8 * q^95 + 4 * q^97

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