LMFDB - Embedded newform 4008.2.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4008,2,Mod(1,4008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("4008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4008 = 2^{3} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4008.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:	$32.0040411301$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	4008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.46081 q^{5} +2.34017 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.46081 q^{5} +2.34017 q^{7} +1.00000 q^{9} -2.63090 q^{11} +2.00000 q^{13} -1.46081 q^{15} +1.61757 q^{17} -3.41855 q^{19} -2.34017 q^{21} -8.00000 q^{23} -2.86603 q^{25} -1.00000 q^{27} -7.26180 q^{29} +3.41855 q^{31} +2.63090 q^{33} +3.41855 q^{35} +0.340173 q^{37} -2.00000 q^{39} -8.63809 q^{41} -7.95774 q^{43} +1.46081 q^{45} -3.89269 q^{47} -1.52359 q^{49} -1.61757 q^{51} +6.29791 q^{53} -3.84324 q^{55} +3.41855 q^{57} -1.26180 q^{59} +12.5464 q^{61} +2.34017 q^{63} +2.92162 q^{65} -12.8794 q^{67} +8.00000 q^{69} -15.9155 q^{71} +1.50307 q^{73} +2.86603 q^{75} -6.15676 q^{77} +2.35577 q^{79} +1.00000 q^{81} +16.5958 q^{83} +2.36296 q^{85} +7.26180 q^{87} -11.9421 q^{89} +4.68035 q^{91} -3.41855 q^{93} -4.99386 q^{95} -1.23513 q^{97} -2.63090 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 6 q^{5} - 4 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 6 * q^5 - 4 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} + 6 q^{5} - 4 q^{7} + 3 q^{9} - 4 q^{11} + 6 q^{13} - 6 q^{15} + 4 q^{19} + 4 q^{21} - 24 q^{23} + 5 q^{25} - 3 q^{27} - 14 q^{29} - 4 q^{31} + 4 q^{33} - 4 q^{35} - 10 q^{37} - 6 q^{39} + 12 q^{41} - 8 q^{43} + 6 q^{45} + 11 q^{49} - 8 q^{53} - 18 q^{55} - 4 q^{57} + 4 q^{59} + 2 q^{61} - 4 q^{63} + 12 q^{65} - 26 q^{67} + 24 q^{69} - 16 q^{71} + 22 q^{73} - 5 q^{75} - 12 q^{77} + 10 q^{79} + 3 q^{81} - 4 q^{83} - 24 q^{85} + 14 q^{87} - 6 q^{89} - 8 q^{91} + 4 q^{93} + 20 q^{95} + 6 q^{97} - 4 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 6 * q^5 - 4 * q^7 + 3 * q^9 - 4 * q^11 + 6 * q^13 - 6 * q^15 + 4 * q^19 + 4 * q^21 - 24 * q^23 + 5 * q^25 - 3 * q^27 - 14 * q^29 - 4 * q^31 + 4 * q^33 - 4 * q^35 - 10 * q^37 - 6 * q^39 + 12 * q^41 - 8 * q^43 + 6 * q^45 + 11 * q^49 - 8 * q^53 - 18 * q^55 - 4 * q^57 + 4 * q^59 + 2 * q^61 - 4 * q^63 + 12 * q^65 - 26 * q^67 + 24 * q^69 - 16 * q^71 + 22 * q^73 - 5 * q^75 - 12 * q^77 + 10 * q^79 + 3 * q^81 - 4 * q^83 - 24 * q^85 + 14 * q^87 - 6 * q^89 - 8 * q^91 + 4 * q^93 + 20 * q^95 + 6 * q^97 - 4 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.46081	0.653295	0.326647	−	0.945146i	$-0.394081\pi$
$5$	1.46081	0.653295	0.326647	+	0.945146i	$0.394081\pi$
$6$	0	0
$7$	2.34017	0.884502	0.442251	−	0.896891i	$-0.354180\pi$
$7$	2.34017	0.884502	0.442251	+	0.896891i	$0.354180\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.63090	−0.793245	−0.396623	−	0.917982i	$-0.629818\pi$
$11$	−2.63090	−0.793245	−0.396623	+	0.917982i	$0.629818\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	−1.46081	−0.377180
$16$	0	0
$17$	1.61757	0.392318	0.196159	−	0.980572i	$-0.437153\pi$
$17$	1.61757	0.392318	0.196159	+	0.980572i	$0.437153\pi$
$18$	0	0
$19$	−3.41855	−0.784269	−0.392135	−	0.919908i	$-0.628263\pi$
$19$	−3.41855	−0.784269	−0.392135	+	0.919908i	$0.628263\pi$
$20$	0	0
$21$	−2.34017	−0.510668
$22$	0	0
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	0	0
$25$	−2.86603	−0.573206
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−7.26180	−1.34848	−0.674241	−	0.738512i	$-0.735529\pi$
$29$	−7.26180	−1.34848	−0.674241	+	0.738512i	$0.735529\pi$
$30$	0	0
$31$	3.41855	0.613990	0.306995	−	0.951711i	$-0.400677\pi$
$31$	3.41855	0.613990	0.306995	+	0.951711i	$0.400677\pi$
$32$	0	0
$33$	2.63090	0.457980
$34$	0	0
$35$	3.41855	0.577841
$36$	0	0
$37$	0.340173	0.0559241	0.0279620	−	0.999609i	$-0.491098\pi$
$37$	0.340173	0.0559241	0.0279620	+	0.999609i	$0.491098\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	−8.63809	−1.34904	−0.674521	−	0.738256i	$-0.735650\pi$
$41$	−8.63809	−1.34904	−0.674521	+	0.738256i	$0.735650\pi$
$42$	0	0
$43$	−7.95774	−1.21354	−0.606772	−	0.794876i	$-0.707536\pi$
$43$	−7.95774	−1.21354	−0.606772	+	0.794876i	$0.707536\pi$
$44$	0	0
$45$	1.46081	0.217765
$46$	0	0
$47$	−3.89269	−0.567808	−0.283904	−	0.958853i	$-0.591630\pi$
$47$	−3.89269	−0.567808	−0.283904	+	0.958853i	$0.591630\pi$
$48$	0	0
$49$	−1.52359	−0.217656
$50$	0	0
$51$	−1.61757	−0.226505
$52$	0	0
$53$	6.29791	0.865085	0.432542	−	0.901614i	$-0.357617\pi$
$53$	6.29791	0.865085	0.432542	+	0.901614i	$0.357617\pi$
$54$	0	0
$55$	−3.84324	−0.518223
$56$	0	0
$57$	3.41855	0.452798
$58$	0	0
$59$	−1.26180	−0.164272	−0.0821359	−	0.996621i	$-0.526174\pi$
$59$	−1.26180	−0.164272	−0.0821359	+	0.996621i	$0.526174\pi$
$60$	0	0
$61$	12.5464	1.60640	0.803199	−	0.595710i	$-0.203129\pi$
$61$	12.5464	1.60640	0.803199	+	0.595710i	$0.203129\pi$
$62$	0	0
$63$	2.34017	0.294834
$64$	0	0
$65$	2.92162	0.362383
$66$	0	0
$67$	−12.8794	−1.57346	−0.786732	−	0.617294i	$-0.788229\pi$
$67$	−12.8794	−1.57346	−0.786732	+	0.617294i	$0.788229\pi$
$68$	0	0
$69$	8.00000	0.963087
$70$	0	0
$71$	−15.9155	−1.88882	−0.944410	−	0.328770i	$-0.893366\pi$
$71$	−15.9155	−1.88882	−0.944410	+	0.328770i	$0.893366\pi$
$72$	0	0
$73$	1.50307	0.175921	0.0879606	−	0.996124i	$-0.471965\pi$
$73$	1.50307	0.175921	0.0879606	+	0.996124i	$0.471965\pi$
$74$	0	0
$75$	2.86603	0.330941
$76$	0	0
$77$	−6.15676	−0.701627
$78$	0	0
$79$	2.35577	0.265045	0.132522	−	0.991180i	$-0.457692\pi$
$79$	2.35577	0.265045	0.132522	+	0.991180i	$0.457692\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	16.5958	1.82163	0.910814	−	0.412816i	$-0.135455\pi$
$83$	16.5958	1.82163	0.910814	+	0.412816i	$0.135455\pi$
$84$	0	0
$85$	2.36296	0.256299
$86$	0	0
$87$	7.26180	0.778546
$88$	0	0
$89$	−11.9421	−1.26586	−0.632932	−	0.774207i	$-0.718149\pi$
$89$	−11.9421	−1.26586	−0.632932	+	0.774207i	$0.718149\pi$
$90$	0	0
$91$	4.68035	0.490634
$92$	0	0
$93$	−3.41855	−0.354487
$94$	0	0
$95$	−4.99386	−0.512359
$96$	0	0
$97$	−1.23513	−0.125409	−0.0627044	−	0.998032i	$-0.519973\pi$
$97$	−1.23513	−0.125409	−0.0627044	+	0.998032i	$0.519973\pi$
$98$	0	0
$99$	−2.63090	−0.264415
$100$	0	0
$101$	4.87936	0.485515	0.242757	−	0.970087i	$-0.421948\pi$
$101$	4.87936	0.485515	0.242757	+	0.970087i	$0.421948\pi$
$102$	0	0
$103$	−3.12064	−0.307486	−0.153743	−	0.988111i	$-0.549133\pi$
$103$	−3.12064	−0.307486	−0.153743	+	0.988111i	$0.549133\pi$
$104$	0	0
$105$	−3.41855	−0.333616
$106$	0	0
$107$	6.04945	0.584822	0.292411	−	0.956293i	$-0.405542\pi$
$107$	6.04945	0.584822	0.292411	+	0.956293i	$0.405542\pi$
$108$	0	0
$109$	7.94214	0.760719	0.380360	−	0.924839i	$-0.375800\pi$
$109$	7.94214	0.760719	0.380360	+	0.924839i	$0.375800\pi$
$110$	0	0
$111$	−0.340173	−0.0322878
$112$	0	0
$113$	7.21953	0.679157	0.339578	−	0.940578i	$-0.389716\pi$
$113$	7.21953	0.679157	0.339578	+	0.940578i	$0.389716\pi$
$114$	0	0
$115$	−11.6865	−1.08977
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	3.78539	0.347006
$120$	0	0
$121$	−4.07838	−0.370762
$122$	0	0
$123$	8.63809	0.778870
$124$	0	0
$125$	−11.4908	−1.02777
$126$	0	0
$127$	3.60197	0.319623	0.159811	−	0.987148i	$-0.448911\pi$
$127$	3.60197	0.319623	0.159811	+	0.987148i	$0.448911\pi$
$128$	0	0
$129$	7.95774	0.700640
$130$	0	0
$131$	0.183417	0.0160253	0.00801263	−	0.999968i	$-0.497449\pi$
$131$	0.183417	0.0160253	0.00801263	+	0.999968i	$0.497449\pi$
$132$	0	0
$133$	−8.00000	−0.693688
$134$	0	0
$135$	−1.46081	−0.125727
$136$	0	0
$137$	−1.81658	−0.155201	−0.0776006	−	0.996985i	$-0.524726\pi$
$137$	−1.81658	−0.155201	−0.0776006	+	0.996985i	$0.524726\pi$
$138$	0	0
$139$	11.4608	0.972093	0.486047	−	0.873933i	$-0.338439\pi$
$139$	11.4608	0.972093	0.486047	+	0.873933i	$0.338439\pi$
$140$	0	0
$141$	3.89269	0.327824
$142$	0	0
$143$	−5.26180	−0.440013
$144$	0	0
$145$	−10.6081	−0.880956
$146$	0	0
$147$	1.52359	0.125664
$148$	0	0
$149$	−11.1929	−0.916956	−0.458478	−	0.888706i	$-0.651605\pi$
$149$	−11.1929	−0.916956	−0.458478	+	0.888706i	$0.651605\pi$
$150$	0	0
$151$	7.06278	0.574761	0.287380	−	0.957817i	$-0.407216\pi$
$151$	7.06278	0.574761	0.287380	+	0.957817i	$0.407216\pi$
$152$	0	0
$153$	1.61757	0.130773
$154$	0	0
$155$	4.99386	0.401116
$156$	0	0
$157$	1.60197	0.127851	0.0639255	−	0.997955i	$-0.479638\pi$
$157$	1.60197	0.127851	0.0639255	+	0.997955i	$0.479638\pi$
$158$	0	0
$159$	−6.29791	−0.499457
$160$	0	0
$161$	−18.7214	−1.47545
$162$	0	0
$163$	−11.4608	−0.897680	−0.448840	−	0.893612i	$-0.648163\pi$
$163$	−11.4608	−0.897680	−0.448840	+	0.893612i	$0.648163\pi$
$164$	0	0
$165$	3.84324	0.299196
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−3.41855	−0.261423
$172$	0	0
$173$	20.8371	1.58422	0.792108	−	0.610381i	$-0.208984\pi$
$173$	20.8371	1.58422	0.792108	+	0.610381i	$0.208984\pi$
$174$	0	0
$175$	−6.70701	−0.507002
$176$	0	0
$177$	1.26180	0.0948423
$178$	0	0
$179$	−14.1568	−1.05813	−0.529063	−	0.848583i	$-0.677456\pi$
$179$	−14.1568	−1.05813	−0.529063	+	0.848583i	$0.677456\pi$
$180$	0	0
$181$	8.81432	0.655163	0.327581	−	0.944823i	$-0.393766\pi$
$181$	8.81432	0.655163	0.327581	+	0.944823i	$0.393766\pi$
$182$	0	0
$183$	−12.5464	−0.927455
$184$	0	0
$185$	0.496928	0.0365349
$186$	0	0
$187$	−4.25565	−0.311204
$188$	0	0
$189$	−2.34017	−0.170223
$190$	0	0
$191$	−16.7877	−1.21471	−0.607356	−	0.794430i	$-0.707770\pi$
$191$	−16.7877	−1.21471	−0.607356	+	0.794430i	$0.707770\pi$
$192$	0	0
$193$	7.26180	0.522715	0.261358	−	0.965242i	$-0.415830\pi$
$193$	7.26180	0.522715	0.261358	+	0.965242i	$0.415830\pi$
$194$	0	0
$195$	−2.92162	−0.209222
$196$	0	0
$197$	0.114495	0.00815744	0.00407872	−	0.999992i	$-0.498702\pi$
$197$	0.114495	0.00815744	0.00407872	+	0.999992i	$0.498702\pi$
$198$	0	0
$199$	−11.9155	−0.844666	−0.422333	−	0.906441i	$-0.638789\pi$
$199$	−11.9155	−0.844666	−0.422333	+	0.906441i	$0.638789\pi$
$200$	0	0
$201$	12.8794	0.908440
$202$	0	0
$203$	−16.9939	−1.19273
$204$	0	0
$205$	−12.6186	−0.881322
$206$	0	0
$207$	−8.00000	−0.556038
$208$	0	0
$209$	8.99386	0.622118
$210$	0	0
$211$	−11.2039	−0.771311	−0.385655	−	0.922643i	$-0.626025\pi$
$211$	−11.2039	−0.771311	−0.385655	+	0.922643i	$0.626025\pi$
$212$	0	0
$213$	15.9155	1.09051
$214$	0	0
$215$	−11.6248	−0.792802
$216$	0	0
$217$	8.00000	0.543075
$218$	0	0
$219$	−1.50307	−0.101568
$220$	0	0
$221$	3.23513	0.217619
$222$	0	0
$223$	2.34017	0.156710	0.0783548	−	0.996926i	$-0.475033\pi$
$223$	2.34017	0.156710	0.0783548	+	0.996926i	$0.475033\pi$
$224$	0	0
$225$	−2.86603	−0.191069
$226$	0	0
$227$	−19.8310	−1.31623	−0.658113	−	0.752919i	$-0.728645\pi$
$227$	−19.8310	−1.31623	−0.658113	+	0.752919i	$0.728645\pi$
$228$	0	0
$229$	18.7031	1.23594	0.617969	−	0.786203i	$-0.287956\pi$
$229$	18.7031	1.23594	0.617969	+	0.786203i	$0.287956\pi$
$230$	0	0
$231$	6.15676	0.405085
$232$	0	0
$233$	−22.8638	−1.49785	−0.748927	−	0.662652i	$-0.769431\pi$
$233$	−22.8638	−1.49785	−0.748927	+	0.662652i	$0.769431\pi$
$234$	0	0
$235$	−5.68649	−0.370946
$236$	0	0
$237$	−2.35577	−0.153024
$238$	0	0
$239$	17.2534	1.11603	0.558014	−	0.829831i	$-0.311563\pi$
$239$	17.2534	1.11603	0.558014	+	0.829831i	$0.311563\pi$
$240$	0	0
$241$	11.8576	0.763816	0.381908	−	0.924200i	$-0.375267\pi$
$241$	11.8576	0.763816	0.381908	+	0.924200i	$0.375267\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−2.22568	−0.142193
$246$	0	0
$247$	−6.83710	−0.435034
$248$	0	0
$249$	−16.5958	−1.05172
$250$	0	0
$251$	−17.4908	−1.10401	−0.552004	−	0.833841i	$-0.686137\pi$
$251$	−17.4908	−1.10401	−0.552004	+	0.833841i	$0.686137\pi$
$252$	0	0
$253$	21.0472	1.32322
$254$	0	0
$255$	−2.36296	−0.147974
$256$	0	0
$257$	4.97826	0.310535	0.155268	−	0.987872i	$-0.450376\pi$
$257$	4.97826	0.310535	0.155268	+	0.987872i	$0.450376\pi$
$258$	0	0
$259$	0.796064	0.0494650
$260$	0	0
$261$	−7.26180	−0.449494
$262$	0	0
$263$	26.1711	1.61378	0.806891	−	0.590701i	$-0.201149\pi$
$263$	26.1711	1.61378	0.806891	+	0.590701i	$0.201149\pi$
$264$	0	0
$265$	9.20006	0.565155
$266$	0	0
$267$	11.9421	0.730847
$268$	0	0
$269$	3.37629	0.205856	0.102928	−	0.994689i	$-0.467179\pi$
$269$	3.37629	0.205856	0.102928	+	0.994689i	$0.467179\pi$
$270$	0	0
$271$	8.29791	0.504062	0.252031	−	0.967719i	$-0.418901\pi$
$271$	8.29791	0.504062	0.252031	+	0.967719i	$0.418901\pi$
$272$	0	0
$273$	−4.68035	−0.283267
$274$	0	0
$275$	7.54023	0.454693
$276$	0	0
$277$	−16.3090	−0.979911	−0.489956	−	0.871747i	$-0.662987\pi$
$277$	−16.3090	−0.979911	−0.489956	+	0.871747i	$0.662987\pi$
$278$	0	0
$279$	3.41855	0.204663
$280$	0	0
$281$	−10.8638	−0.648078	−0.324039	−	0.946044i	$-0.605041\pi$
$281$	−10.8638	−0.648078	−0.324039	+	0.946044i	$0.605041\pi$
$282$	0	0
$283$	16.8950	1.00430	0.502151	−	0.864780i	$-0.332542\pi$
$283$	16.8950	1.00430	0.502151	+	0.864780i	$0.332542\pi$
$284$	0	0
$285$	4.99386	0.295811
$286$	0	0
$287$	−20.2146	−1.19323
$288$	0	0
$289$	−14.3835	−0.846087
$290$	0	0
$291$	1.23513	0.0724048
$292$	0	0
$293$	33.0349	1.92992	0.964960	−	0.262399i	$-0.0845135\pi$
$293$	33.0349	1.92992	0.964960	+	0.262399i	$0.0845135\pi$
$294$	0	0
$295$	−1.84324	−0.107318
$296$	0	0
$297$	2.63090	0.152660
$298$	0	0
$299$	−16.0000	−0.925304
$300$	0	0
$301$	−18.6225	−1.07338
$302$	0	0
$303$	−4.87936	−0.280312
$304$	0	0
$305$	18.3279	1.04945
$306$	0	0
$307$	−9.21953	−0.526187	−0.263093	−	0.964770i	$-0.584743\pi$
$307$	−9.21953	−0.526187	−0.263093	+	0.964770i	$0.584743\pi$
$308$	0	0
$309$	3.12064	0.177527
$310$	0	0
$311$	−13.8576	−0.785794	−0.392897	−	0.919583i	$-0.628527\pi$
$311$	−13.8576	−0.785794	−0.392897	+	0.919583i	$0.628527\pi$
$312$	0	0
$313$	−12.8371	−0.725596	−0.362798	−	0.931868i	$-0.618178\pi$
$313$	−12.8371	−0.725596	−0.362798	+	0.931868i	$0.618178\pi$
$314$	0	0
$315$	3.41855	0.192614
$316$	0	0
$317$	−6.09890	−0.342548	−0.171274	−	0.985223i	$-0.554788\pi$
$317$	−6.09890	−0.342548	−0.171274	+	0.985223i	$0.554788\pi$
$318$	0	0
$319$	19.1050	1.06968
$320$	0	0
$321$	−6.04945	−0.337647
$322$	0	0
$323$	−5.52973	−0.307683
$324$	0	0
$325$	−5.73206	−0.317958
$326$	0	0
$327$	−7.94214	−0.439201
$328$	0	0
$329$	−9.10957	−0.502227
$330$	0	0
$331$	−12.8482	−0.706199	−0.353100	−	0.935586i	$-0.614872\pi$
$331$	−12.8482	−0.706199	−0.353100	+	0.935586i	$0.614872\pi$
$332$	0	0
$333$	0.340173	0.0186414
$334$	0	0
$335$	−18.8143	−1.02794
$336$	0	0
$337$	−26.5958	−1.44877	−0.724383	−	0.689397i	$-0.757875\pi$
$337$	−26.5958	−1.44877	−0.724383	+	0.689397i	$0.757875\pi$
$338$	0	0
$339$	−7.21953	−0.392111
$340$	0	0
$341$	−8.99386	−0.487045
$342$	0	0
$343$	−19.9467	−1.07702
$344$	0	0
$345$	11.6865	0.629179
$346$	0	0
$347$	−35.3028	−1.89516	−0.947578	−	0.319525i	$-0.896477\pi$
$347$	−35.3028	−1.89516	−0.947578	+	0.319525i	$0.896477\pi$
$348$	0	0
$349$	−9.81658	−0.525470	−0.262735	−	0.964868i	$-0.584624\pi$
$349$	−9.81658	−0.525470	−0.262735	+	0.964868i	$0.584624\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	0	0
$353$	14.1834	0.754907	0.377454	−	0.926029i	$-0.376800\pi$
$353$	14.1834	0.754907	0.377454	+	0.926029i	$0.376800\pi$
$354$	0	0
$355$	−23.2495	−1.23396
$356$	0	0
$357$	−3.78539	−0.200344
$358$	0	0
$359$	7.20394	0.380209	0.190105	−	0.981764i	$-0.439117\pi$
$359$	7.20394	0.380209	0.190105	+	0.981764i	$0.439117\pi$
$360$	0	0
$361$	−7.31351	−0.384922
$362$	0	0
$363$	4.07838	0.214059
$364$	0	0
$365$	2.19570	0.114928
$366$	0	0
$367$	−29.4596	−1.53778	−0.768889	−	0.639382i	$-0.779190\pi$
$367$	−29.4596	−1.53778	−0.768889	+	0.639382i	$0.779190\pi$
$368$	0	0
$369$	−8.63809	−0.449681
$370$	0	0
$371$	14.7382	0.765169
$372$	0	0
$373$	28.3857	1.46976	0.734879	−	0.678198i	$-0.237239\pi$
$373$	28.3857	1.46976	0.734879	+	0.678198i	$0.237239\pi$
$374$	0	0
$375$	11.4908	0.593382
$376$	0	0
$377$	−14.5236	−0.748003
$378$	0	0
$379$	−14.4280	−0.741117	−0.370558	−	0.928809i	$-0.620834\pi$
$379$	−14.4280	−0.741117	−0.370558	+	0.928809i	$0.620834\pi$
$380$	0	0
$381$	−3.60197	−0.184534
$382$	0	0
$383$	−14.7298	−0.752657	−0.376329	−	0.926486i	$-0.622814\pi$
$383$	−14.7298	−0.752657	−0.376329	+	0.926486i	$0.622814\pi$
$384$	0	0
$385$	−8.99386	−0.458369
$386$	0	0
$387$	−7.95774	−0.404515
$388$	0	0
$389$	−28.1145	−1.42546	−0.712731	−	0.701438i	$-0.752542\pi$
$389$	−28.1145	−1.42546	−0.712731	+	0.701438i	$0.752542\pi$
$390$	0	0
$391$	−12.9405	−0.654431
$392$	0	0
$393$	−0.183417	−0.00925219
$394$	0	0
$395$	3.44134	0.173152
$396$	0	0
$397$	33.3256	1.67256	0.836282	−	0.548299i	$-0.184724\pi$
$397$	33.3256	1.67256	0.836282	+	0.548299i	$0.184724\pi$
$398$	0	0
$399$	8.00000	0.400501
$400$	0	0
$401$	−16.8794	−0.842915	−0.421458	−	0.906848i	$-0.638481\pi$
$401$	−16.8794	−0.842915	−0.421458	+	0.906848i	$0.638481\pi$
$402$	0	0
$403$	6.83710	0.340580
$404$	0	0
$405$	1.46081	0.0725883
$406$	0	0
$407$	−0.894960	−0.0443615
$408$	0	0
$409$	13.3958	0.662378	0.331189	−	0.943564i	$-0.392550\pi$
$409$	13.3958	0.662378	0.331189	+	0.943564i	$0.392550\pi$
$410$	0	0
$411$	1.81658	0.0896054
$412$	0	0
$413$	−2.95282	−0.145299
$414$	0	0
$415$	24.2434	1.19006
$416$	0	0
$417$	−11.4608	−0.561238
$418$	0	0
$419$	13.8348	0.675876	0.337938	−	0.941168i	$-0.390271\pi$
$419$	13.8348	0.675876	0.337938	+	0.941168i	$0.390271\pi$
$420$	0	0
$421$	8.07223	0.393417	0.196708	−	0.980462i	$-0.436975\pi$
$421$	8.07223	0.393417	0.196708	+	0.980462i	$0.436975\pi$
$422$	0	0
$423$	−3.89269	−0.189269
$424$	0	0
$425$	−4.63600	−0.224879
$426$	0	0
$427$	29.3607	1.42086
$428$	0	0
$429$	5.26180	0.254042
$430$	0	0
$431$	−26.7070	−1.28643	−0.643216	−	0.765685i	$-0.722400\pi$
$431$	−26.7070	−1.28643	−0.643216	+	0.765685i	$0.722400\pi$
$432$	0	0
$433$	−17.5936	−0.845492	−0.422746	−	0.906248i	$-0.638934\pi$
$433$	−17.5936	−0.845492	−0.422746	+	0.906248i	$0.638934\pi$
$434$	0	0
$435$	10.6081	0.508620
$436$	0	0
$437$	27.3484	1.30825
$438$	0	0
$439$	35.5741	1.69786	0.848929	−	0.528507i	$-0.177248\pi$
$439$	35.5741	1.69786	0.848929	+	0.528507i	$0.177248\pi$
$440$	0	0
$441$	−1.52359	−0.0725519
$442$	0	0
$443$	−0.948284	−0.0450543	−0.0225272	−	0.999746i	$-0.507171\pi$
$443$	−0.948284	−0.0450543	−0.0225272	+	0.999746i	$0.507171\pi$
$444$	0	0
$445$	−17.4452	−0.826982
$446$	0	0
$447$	11.1929	0.529405
$448$	0	0
$449$	19.9565	0.941806	0.470903	−	0.882185i	$-0.343928\pi$
$449$	19.9565	0.941806	0.470903	+	0.882185i	$0.343928\pi$
$450$	0	0
$451$	22.7259	1.07012
$452$	0	0
$453$	−7.06278	−0.331838
$454$	0	0
$455$	6.83710	0.320528
$456$	0	0
$457$	13.9155	0.650939	0.325469	−	0.945553i	$-0.394478\pi$
$457$	13.9155	0.650939	0.325469	+	0.945553i	$0.394478\pi$
$458$	0	0
$459$	−1.61757	−0.0755015
$460$	0	0
$461$	28.9216	1.34701	0.673507	−	0.739181i	$-0.264787\pi$
$461$	28.9216	1.34701	0.673507	+	0.739181i	$0.264787\pi$
$462$	0	0
$463$	−0.396809	−0.0184413	−0.00922064	−	0.999957i	$-0.502935\pi$
$463$	−0.396809	−0.0184413	−0.00922064	+	0.999957i	$0.502935\pi$
$464$	0	0
$465$	−4.99386	−0.231585
$466$	0	0
$467$	19.5708	0.905627	0.452814	−	0.891605i	$-0.350420\pi$
$467$	19.5708	0.905627	0.452814	+	0.891605i	$0.350420\pi$
$468$	0	0
$469$	−30.1399	−1.39173
$470$	0	0
$471$	−1.60197	−0.0738148
$472$	0	0
$473$	20.9360	0.962638
$474$	0	0
$475$	9.79767	0.449548
$476$	0	0
$477$	6.29791	0.288362
$478$	0	0
$479$	7.23513	0.330582	0.165291	−	0.986245i	$-0.447144\pi$
$479$	7.23513	0.330582	0.165291	+	0.986245i	$0.447144\pi$
$480$	0	0
$481$	0.680346	0.0310211
$482$	0	0
$483$	18.7214	0.851852
$484$	0	0
$485$	−1.80430	−0.0819289
$486$	0	0
$487$	−3.78661	−0.171588	−0.0857938	−	0.996313i	$-0.527343\pi$
$487$	−3.78661	−0.171588	−0.0857938	+	0.996313i	$0.527343\pi$
$488$	0	0
$489$	11.4608	0.518276
$490$	0	0
$491$	−33.5669	−1.51485	−0.757426	−	0.652921i	$-0.773544\pi$
$491$	−33.5669	−1.51485	−0.757426	+	0.652921i	$0.773544\pi$
$492$	0	0
$493$	−11.7464	−0.529033
$494$	0	0
$495$	−3.84324	−0.172741
$496$	0	0
$497$	−37.2450	−1.67067
$498$	0	0
$499$	24.7948	1.10997	0.554985	−	0.831861i	$-0.312724\pi$
$499$	24.7948	1.10997	0.554985	+	0.831861i	$0.312724\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	−31.6248	−1.41008	−0.705039	−	0.709168i	$-0.749071\pi$
$503$	−31.6248	−1.41008	−0.705039	+	0.709168i	$0.749071\pi$
$504$	0	0
$505$	7.12783	0.317184
$506$	0	0
$507$	9.00000	0.399704
$508$	0	0
$509$	−27.6430	−1.22525	−0.612627	−	0.790372i	$-0.709887\pi$
$509$	−27.6430	−1.22525	−0.612627	+	0.790372i	$0.709887\pi$
$510$	0	0
$511$	3.51745	0.155603
$512$	0	0
$513$	3.41855	0.150933
$514$	0	0
$515$	−4.55866	−0.200879
$516$	0	0
$517$	10.2413	0.450411
$518$	0	0
$519$	−20.8371	−0.914647
$520$	0	0
$521$	22.2667	0.975523	0.487761	−	0.872977i	$-0.337814\pi$
$521$	22.2667	0.975523	0.487761	+	0.872977i	$0.337814\pi$
$522$	0	0
$523$	12.5113	0.547081	0.273541	−	0.961860i	$-0.411805\pi$
$523$	12.5113	0.547081	0.273541	+	0.961860i	$0.411805\pi$
$524$	0	0
$525$	6.70701	0.292718
$526$	0	0
$527$	5.52973	0.240879
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	−1.26180	−0.0547572
$532$	0	0
$533$	−17.2762	−0.748314
$534$	0	0
$535$	8.83710	0.382061
$536$	0	0
$537$	14.1568	0.610909
$538$	0	0
$539$	4.00841	0.172654
$540$	0	0
$541$	10.6270	0.456891	0.228446	−	0.973557i	$-0.426636\pi$
$541$	10.6270	0.456891	0.228446	+	0.973557i	$0.426636\pi$
$542$	0	0
$543$	−8.81432	−0.378258
$544$	0	0
$545$	11.6020	0.496974
$546$	0	0
$547$	22.9060	0.979391	0.489695	−	0.871894i	$-0.337108\pi$
$547$	22.9060	0.979391	0.489695	+	0.871894i	$0.337108\pi$
$548$	0	0
$549$	12.5464	0.535466
$550$	0	0
$551$	24.8248	1.05757
$552$	0	0
$553$	5.51291	0.234433
$554$	0	0
$555$	−0.496928	−0.0210934
$556$	0	0
$557$	−21.3340	−0.903952	−0.451976	−	0.892030i	$-0.649281\pi$
$557$	−21.3340	−0.903952	−0.451976	+	0.892030i	$0.649281\pi$
$558$	0	0
$559$	−15.9155	−0.673153
$560$	0	0
$561$	4.25565	0.179674
$562$	0	0
$563$	29.5441	1.24514	0.622568	−	0.782566i	$-0.286089\pi$
$563$	29.5441	1.24514	0.622568	+	0.782566i	$0.286089\pi$
$564$	0	0
$565$	10.5464	0.443689
$566$	0	0
$567$	2.34017	0.0982780
$568$	0	0
$569$	−45.6730	−1.91471	−0.957355	−	0.288913i	$-0.906706\pi$
$569$	−45.6730	−1.91471	−0.957355	+	0.288913i	$0.906706\pi$
$570$	0	0
$571$	−29.4485	−1.23238	−0.616191	−	0.787597i	$-0.711325\pi$
$571$	−29.4485	−1.23238	−0.616191	+	0.787597i	$0.711325\pi$
$572$	0	0
$573$	16.7877	0.701314
$574$	0	0
$575$	22.9282	0.956174
$576$	0	0
$577$	38.3195	1.59526	0.797630	−	0.603147i	$-0.206087\pi$
$577$	38.3195	1.59526	0.797630	+	0.603147i	$0.206087\pi$
$578$	0	0
$579$	−7.26180	−0.301790
$580$	0	0
$581$	38.8371	1.61123
$582$	0	0
$583$	−16.5692	−0.686225
$584$	0	0
$585$	2.92162	0.120794
$586$	0	0
$587$	−37.9877	−1.56792	−0.783960	−	0.620811i	$-0.786803\pi$
$587$	−37.9877	−1.56792	−0.783960	+	0.620811i	$0.786803\pi$
$588$	0	0
$589$	−11.6865	−0.481533
$590$	0	0
$591$	−0.114495	−0.00470970
$592$	0	0
$593$	22.9516	0.942509	0.471255	−	0.881997i	$-0.343801\pi$
$593$	22.9516	0.942509	0.471255	+	0.881997i	$0.343801\pi$
$594$	0	0
$595$	5.52973	0.226697
$596$	0	0
$597$	11.9155	0.487668
$598$	0	0
$599$	11.8843	0.485579	0.242789	−	0.970079i	$-0.421938\pi$
$599$	11.8843	0.485579	0.242789	+	0.970079i	$0.421938\pi$
$600$	0	0
$601$	13.1506	0.536425	0.268212	−	0.963360i	$-0.413567\pi$
$601$	13.1506	0.536425	0.268212	+	0.963360i	$0.413567\pi$
$602$	0	0
$603$	−12.8794	−0.524488
$604$	0	0
$605$	−5.95774	−0.242217
$606$	0	0
$607$	28.7226	1.16581	0.582907	−	0.812539i	$-0.301915\pi$
$607$	28.7226	1.16581	0.582907	+	0.812539i	$0.301915\pi$
$608$	0	0
$609$	16.9939	0.688626
$610$	0	0
$611$	−7.78539	−0.314963
$612$	0	0
$613$	2.97107	0.120000	0.0600002	−	0.998198i	$-0.480890\pi$
$613$	2.97107	0.120000	0.0600002	+	0.998198i	$0.480890\pi$
$614$	0	0
$615$	12.6186	0.508832
$616$	0	0
$617$	−6.68035	−0.268941	−0.134470	−	0.990918i	$-0.542933\pi$
$617$	−6.68035	−0.268941	−0.134470	+	0.990918i	$0.542933\pi$
$618$	0	0
$619$	32.1990	1.29419	0.647094	−	0.762410i	$-0.275984\pi$
$619$	32.1990	1.29419	0.647094	+	0.762410i	$0.275984\pi$
$620$	0	0
$621$	8.00000	0.321029
$622$	0	0
$623$	−27.9467	−1.11966
$624$	0	0
$625$	−2.45571	−0.0982285
$626$	0	0
$627$	−8.99386	−0.359180
$628$	0	0
$629$	0.550252	0.0219400
$630$	0	0
$631$	32.9627	1.31222	0.656111	−	0.754664i	$-0.272200\pi$
$631$	32.9627	1.31222	0.656111	+	0.754664i	$0.272200\pi$
$632$	0	0
$633$	11.2039	0.445316
$634$	0	0
$635$	5.26180	0.208808
$636$	0	0
$637$	−3.04718	−0.120734
$638$	0	0
$639$	−15.9155	−0.629607
$640$	0	0
$641$	−30.4547	−1.20289	−0.601444	−	0.798915i	$-0.705408\pi$
$641$	−30.4547	−1.20289	−0.601444	+	0.798915i	$0.705408\pi$
$642$	0	0
$643$	36.5490	1.44135	0.720677	−	0.693271i	$-0.243831\pi$
$643$	36.5490	1.44135	0.720677	+	0.693271i	$0.243831\pi$
$644$	0	0
$645$	11.6248	0.457724
$646$	0	0
$647$	−45.1605	−1.77544	−0.887720	−	0.460383i	$-0.847712\pi$
$647$	−45.1605	−1.77544	−0.887720	+	0.460383i	$0.847712\pi$
$648$	0	0
$649$	3.31965	0.130308
$650$	0	0
$651$	−8.00000	−0.313545
$652$	0	0
$653$	−6.99386	−0.273691	−0.136845	−	0.990592i	$-0.543696\pi$
$653$	−6.99386	−0.273691	−0.136845	+	0.990592i	$0.543696\pi$
$654$	0	0
$655$	0.267938	0.0104692
$656$	0	0
$657$	1.50307	0.0586404
$658$	0	0
$659$	35.1338	1.36862	0.684309	−	0.729192i	$-0.260104\pi$
$659$	35.1338	1.36862	0.684309	+	0.729192i	$0.260104\pi$
$660$	0	0
$661$	−48.4846	−1.88583	−0.942917	−	0.333028i	$-0.891930\pi$
$661$	−48.4846	−1.88583	−0.942917	+	0.333028i	$0.891930\pi$
$662$	0	0
$663$	−3.23513	−0.125642
$664$	0	0
$665$	−11.6865	−0.453183
$666$	0	0
$667$	58.0944	2.24942
$668$	0	0
$669$	−2.34017	−0.0904763
$670$	0	0
$671$	−33.0082	−1.27427
$672$	0	0
$673$	16.2101	0.624853	0.312426	−	0.949942i	$-0.398858\pi$
$673$	16.2101	0.624853	0.312426	+	0.949942i	$0.398858\pi$
$674$	0	0
$675$	2.86603	0.110314
$676$	0	0
$677$	−37.7152	−1.44951	−0.724757	−	0.689004i	$-0.758048\pi$
$677$	−37.7152	−1.44951	−0.724757	+	0.689004i	$0.758048\pi$
$678$	0	0
$679$	−2.89043	−0.110924
$680$	0	0
$681$	19.8310	0.759924
$682$	0	0
$683$	18.4391	0.705551	0.352776	−	0.935708i	$-0.385238\pi$
$683$	18.4391	0.705551	0.352776	+	0.935708i	$0.385238\pi$
$684$	0	0
$685$	−2.65368	−0.101392
$686$	0	0
$687$	−18.7031	−0.713569
$688$	0	0
$689$	12.5958	0.479863
$690$	0	0
$691$	8.86499	0.337240	0.168620	−	0.985681i	$-0.446069\pi$
$691$	8.86499	0.337240	0.168620	+	0.985681i	$0.446069\pi$
$692$	0	0
$693$	−6.15676	−0.233876
$694$	0	0
$695$	16.7421	0.635063
$696$	0	0
$697$	−13.9727	−0.529253
$698$	0	0
$699$	22.8638	0.864787
$700$	0	0
$701$	36.8059	1.39014	0.695070	−	0.718942i	$-0.255373\pi$
$701$	36.8059	1.39014	0.695070	+	0.718942i	$0.255373\pi$
$702$	0	0
$703$	−1.16290	−0.0438595
$704$	0	0
$705$	5.68649	0.214166
$706$	0	0
$707$	11.4186	0.429439
$708$	0	0
$709$	22.1978	0.833656	0.416828	−	0.908985i	$-0.363142\pi$
$709$	22.1978	0.833656	0.416828	+	0.908985i	$0.363142\pi$
$710$	0	0
$711$	2.35577	0.0883483
$712$	0	0
$713$	−27.3484	−1.02421
$714$	0	0
$715$	−7.68649	−0.287458
$716$	0	0
$717$	−17.2534	−0.644339
$718$	0	0
$719$	21.3919	0.797783	0.398891	−	0.916998i	$-0.369395\pi$
$719$	21.3919	0.797783	0.398891	+	0.916998i	$0.369395\pi$
$720$	0	0
$721$	−7.30283	−0.271972
$722$	0	0
$723$	−11.8576	−0.440990
$724$	0	0
$725$	20.8125	0.772958
$726$	0	0
$727$	34.2979	1.27204	0.636020	−	0.771673i	$-0.280580\pi$
$727$	34.2979	1.27204	0.636020	+	0.771673i	$0.280580\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−12.8722	−0.476095
$732$	0	0
$733$	−26.5418	−0.980345	−0.490173	−	0.871625i	$-0.663066\pi$
$733$	−26.5418	−0.980345	−0.490173	+	0.871625i	$0.663066\pi$
$734$	0	0
$735$	2.22568	0.0820954
$736$	0	0
$737$	33.8843	1.24814
$738$	0	0
$739$	37.5741	1.38219	0.691093	−	0.722766i	$-0.257130\pi$
$739$	37.5741	1.38219	0.691093	+	0.722766i	$0.257130\pi$
$740$	0	0
$741$	6.83710	0.251167
$742$	0	0
$743$	−34.3545	−1.26035	−0.630173	−	0.776455i	$-0.717016\pi$
$743$	−34.3545	−1.26035	−0.630173	+	0.776455i	$0.717016\pi$
$744$	0	0
$745$	−16.3507	−0.599042
$746$	0	0
$747$	16.5958	0.607209
$748$	0	0
$749$	14.1568	0.517277
$750$	0	0
$751$	28.4403	1.03780	0.518901	−	0.854835i	$-0.326342\pi$
$751$	28.4403	1.03780	0.518901	+	0.854835i	$0.326342\pi$
$752$	0	0
$753$	17.4908	0.637400
$754$	0	0
$755$	10.3174	0.375488
$756$	0	0
$757$	14.7337	0.535504	0.267752	−	0.963488i	$-0.413719\pi$
$757$	14.7337	0.535504	0.267752	+	0.963488i	$0.413719\pi$
$758$	0	0
$759$	−21.0472	−0.763964
$760$	0	0
$761$	−11.6742	−0.423190	−0.211595	−	0.977357i	$-0.567866\pi$
$761$	−11.6742	−0.423190	−0.211595	+	0.977357i	$0.567866\pi$
$762$	0	0
$763$	18.5860	0.672858
$764$	0	0
$765$	2.36296	0.0854330
$766$	0	0
$767$	−2.52359	−0.0911216
$768$	0	0
$769$	−20.3545	−0.734004	−0.367002	−	0.930220i	$-0.619616\pi$
$769$	−20.3545	−0.734004	−0.367002	+	0.930220i	$0.619616\pi$
$770$	0	0
$771$	−4.97826	−0.179288
$772$	0	0
$773$	8.28354	0.297938	0.148969	−	0.988842i	$-0.452405\pi$
$773$	8.28354	0.297938	0.148969	+	0.988842i	$0.452405\pi$
$774$	0	0
$775$	−9.79767	−0.351943
$776$	0	0
$777$	−0.796064	−0.0285586
$778$	0	0
$779$	29.5297	1.05801
$780$	0	0
$781$	41.8720	1.49830
$782$	0	0
$783$	7.26180	0.259515
$784$	0	0
$785$	2.34017	0.0835244
$786$	0	0
$787$	8.14116	0.290201	0.145100	−	0.989417i	$-0.453649\pi$
$787$	8.14116	0.290201	0.145100	+	0.989417i	$0.453649\pi$
$788$	0	0
$789$	−26.1711	−0.931717
$790$	0	0
$791$	16.8950	0.600716
$792$	0	0
$793$	25.0928	0.891070
$794$	0	0
$795$	−9.20006	−0.326293
$796$	0	0
$797$	15.8588	0.561749	0.280875	−	0.959744i	$-0.409376\pi$
$797$	15.8588	0.561749	0.280875	+	0.959744i	$0.409376\pi$
$798$	0	0
$799$	−6.29669	−0.222761
$800$	0	0
$801$	−11.9421	−0.421955
$802$	0	0
$803$	−3.95443	−0.139549
$804$	0	0
$805$	−27.3484	−0.963905
$806$	0	0
$807$	−3.37629	−0.118851
$808$	0	0
$809$	31.8043	1.11818	0.559090	−	0.829107i	$-0.311151\pi$
$809$	31.8043	1.11818	0.559090	+	0.829107i	$0.311151\pi$
$810$	0	0
$811$	−4.55356	−0.159897	−0.0799486	−	0.996799i	$-0.525476\pi$
$811$	−4.55356	−0.159897	−0.0799486	+	0.996799i	$0.525476\pi$
$812$	0	0
$813$	−8.29791	−0.291020
$814$	0	0
$815$	−16.7421	−0.586449
$816$	0	0
$817$	27.2039	0.951745
$818$	0	0
$819$	4.68035	0.163545
$820$	0	0
$821$	16.4235	0.573183	0.286592	−	0.958053i	$-0.407478\pi$
$821$	16.4235	0.573183	0.286592	+	0.958053i	$0.407478\pi$
$822$	0	0
$823$	−1.46081	−0.0509207	−0.0254603	−	0.999676i	$-0.508105\pi$
$823$	−1.46081	−0.0509207	−0.0254603	+	0.999676i	$0.508105\pi$
$824$	0	0
$825$	−7.54023	−0.262517
$826$	0	0
$827$	−21.0472	−0.731882	−0.365941	−	0.930638i	$-0.619253\pi$
$827$	−21.0472	−0.731882	−0.365941	+	0.930638i	$0.619253\pi$
$828$	0	0
$829$	−31.9421	−1.10940	−0.554698	−	0.832052i	$-0.687166\pi$
$829$	−31.9421	−1.10940	−0.554698	+	0.832052i	$0.687166\pi$
$830$	0	0
$831$	16.3090	0.565752
$832$	0	0
$833$	−2.46451	−0.0853902
$834$	0	0
$835$	1.46081	0.0505535
$836$	0	0
$837$	−3.41855	−0.118162
$838$	0	0
$839$	−36.3584	−1.25523	−0.627616	−	0.778523i	$-0.715969\pi$
$839$	−36.3584	−1.25523	−0.627616	+	0.778523i	$0.715969\pi$
$840$	0	0
$841$	23.7337	0.818402
$842$	0	0
$843$	10.8638	0.374168
$844$	0	0
$845$	−13.1473	−0.452281
$846$	0	0
$847$	−9.54411	−0.327939
$848$	0	0
$849$	−16.8950	−0.579834
$850$	0	0
$851$	−2.72138	−0.0932878
$852$	0	0
$853$	−12.5464	−0.429580	−0.214790	−	0.976660i	$-0.568907\pi$
$853$	−12.5464	−0.429580	−0.214790	+	0.976660i	$0.568907\pi$
$854$	0	0
$855$	−4.99386	−0.170786
$856$	0	0
$857$	−0.523590	−0.0178855	−0.00894275	−	0.999960i	$-0.502847\pi$
$857$	−0.523590	−0.0178855	−0.00894275	+	0.999960i	$0.502847\pi$
$858$	0	0
$859$	5.75872	0.196485	0.0982426	−	0.995162i	$-0.468678\pi$
$859$	5.75872	0.196485	0.0982426	+	0.995162i	$0.468678\pi$
$860$	0	0
$861$	20.2146	0.688912
$862$	0	0
$863$	−22.2868	−0.758653	−0.379327	−	0.925263i	$-0.623844\pi$
$863$	−22.2868	−0.758653	−0.379327	+	0.925263i	$0.623844\pi$
$864$	0	0
$865$	30.4391	1.03496
$866$	0	0
$867$	14.3835	0.488489
$868$	0	0
$869$	−6.19779	−0.210246
$870$	0	0
$871$	−25.7587	−0.872801
$872$	0	0
$873$	−1.23513	−0.0418029
$874$	0	0
$875$	−26.8904	−0.909062
$876$	0	0
$877$	−34.4801	−1.16431	−0.582155	−	0.813078i	$-0.697790\pi$
$877$	−34.4801	−1.16431	−0.582155	+	0.813078i	$0.697790\pi$
$878$	0	0
$879$	−33.0349	−1.11424
$880$	0	0
$881$	9.64423	0.324922	0.162461	−	0.986715i	$-0.448057\pi$
$881$	9.64423	0.324922	0.162461	+	0.986715i	$0.448057\pi$
$882$	0	0
$883$	6.68488	0.224964	0.112482	−	0.993654i	$-0.464120\pi$
$883$	6.68488	0.224964	0.112482	+	0.993654i	$0.464120\pi$
$884$	0	0
$885$	1.84324	0.0619600
$886$	0	0
$887$	33.9109	1.13862	0.569309	−	0.822124i	$-0.307211\pi$
$887$	33.9109	1.13862	0.569309	+	0.822124i	$0.307211\pi$
$888$	0	0
$889$	8.42923	0.282707
$890$	0	0
$891$	−2.63090	−0.0881384
$892$	0	0
$893$	13.3074	0.445314
$894$	0	0
$895$	−20.6803	−0.691268
$896$	0	0
$897$	16.0000	0.534224
$898$	0	0
$899$	−24.8248	−0.827954
$900$	0	0
$901$	10.1873	0.339388
$902$	0	0
$903$	18.6225	0.619718
$904$	0	0
$905$	12.8760	0.428014
$906$	0	0
$907$	−2.72383	−0.0904433	−0.0452216	−	0.998977i	$-0.514399\pi$
$907$	−2.72383	−0.0904433	−0.0452216	+	0.998977i	$0.514399\pi$
$908$	0	0
$909$	4.87936	0.161838
$910$	0	0
$911$	28.9405	0.958843	0.479421	−	0.877585i	$-0.340847\pi$
$911$	28.9405	0.958843	0.479421	+	0.877585i	$0.340847\pi$
$912$	0	0
$913$	−43.6619	−1.44500
$914$	0	0
$915$	−18.3279	−0.605901
$916$	0	0
$917$	0.429229	0.0141744
$918$	0	0
$919$	33.4740	1.10420	0.552102	−	0.833777i	$-0.313826\pi$
$919$	33.4740	1.10420	0.552102	+	0.833777i	$0.313826\pi$
$920$	0	0
$921$	9.21953	0.303794
$922$	0	0
$923$	−31.8310	−1.04773
$924$	0	0
$925$	−0.974946	−0.0320560
$926$	0	0
$927$	−3.12064	−0.102495
$928$	0	0
$929$	−13.2663	−0.435254	−0.217627	−	0.976032i	$-0.569832\pi$
$929$	−13.2663	−0.435254	−0.217627	+	0.976032i	$0.569832\pi$
$930$	0	0
$931$	5.20847	0.170701
$932$	0	0
$933$	13.8576	0.453678
$934$	0	0
$935$	−6.21670	−0.203308
$936$	0	0
$937$	−10.2679	−0.335439	−0.167719	−	0.985835i	$-0.553640\pi$
$937$	−10.2679	−0.335439	−0.167719	+	0.985835i	$0.553640\pi$
$938$	0	0
$939$	12.8371	0.418923
$940$	0	0
$941$	34.1867	1.11446	0.557228	−	0.830360i	$-0.311865\pi$
$941$	34.1867	1.11446	0.557228	+	0.830360i	$0.311865\pi$
$942$	0	0
$943$	69.1047	2.25036
$944$	0	0
$945$	−3.41855	−0.111205
$946$	0	0
$947$	−10.2185	−0.332056	−0.166028	−	0.986121i	$-0.553094\pi$
$947$	−10.2185	−0.332056	−0.166028	+	0.986121i	$0.553094\pi$
$948$	0	0
$949$	3.00614	0.0975835
$950$	0	0
$951$	6.09890	0.197770
$952$	0	0
$953$	−18.2257	−0.590388	−0.295194	−	0.955437i	$-0.595384\pi$
$953$	−18.2257	−0.590388	−0.295194	+	0.955437i	$0.595384\pi$
$954$	0	0
$955$	−24.5236	−0.793565
$956$	0	0
$957$	−19.1050	−0.617578
$958$	0	0
$959$	−4.25112	−0.137276
$960$	0	0
$961$	−19.3135	−0.623016
$962$	0	0
$963$	6.04945	0.194941
$964$	0	0
$965$	10.6081	0.341487
$966$	0	0
$967$	34.3234	1.10376	0.551882	−	0.833922i	$-0.313910\pi$
$967$	34.3234	1.10376	0.551882	+	0.833922i	$0.313910\pi$
$968$	0	0
$969$	5.52973	0.177641
$970$	0	0
$971$	41.8043	1.34156	0.670782	−	0.741655i	$-0.265959\pi$
$971$	41.8043	1.34156	0.670782	+	0.741655i	$0.265959\pi$
$972$	0	0
$973$	26.8203	0.859819
$974$	0	0
$975$	5.73206	0.183573
$976$	0	0
$977$	−7.50638	−0.240150	−0.120075	−	0.992765i	$-0.538314\pi$
$977$	−7.50638	−0.240150	−0.120075	+	0.992765i	$0.538314\pi$
$978$	0	0
$979$	31.4186	1.00414
$980$	0	0
$981$	7.94214	0.253573
$982$	0	0
$983$	25.5441	0.814731	0.407365	−	0.913265i	$-0.366448\pi$
$983$	25.5441	0.814731	0.407365	+	0.913265i	$0.366448\pi$
$984$	0	0
$985$	0.167256	0.00532921
$986$	0	0
$987$	9.10957	0.289961
$988$	0	0
$989$	63.6619	2.02433
$990$	0	0
$991$	−24.9926	−0.793917	−0.396959	−	0.917837i	$-0.629934\pi$
$991$	−24.9926	−0.793917	−0.396959	+	0.917837i	$0.629934\pi$
$992$	0	0
$993$	12.8482	0.407724
$994$	0	0
$995$	−17.4063	−0.551816
$996$	0	0
$997$	−31.2846	−0.990793	−0.495396	−	0.868667i	$-0.664977\pi$
$997$	−31.2846	−0.990793	−0.495396	+	0.868667i	$0.664977\pi$
$998$	0	0
$999$	−0.340173	−0.0107626

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4008.2.a.e.1.2	✓	3
4.3	odd	2		8016.2.a.n.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.e.1.2	✓	3	1.1	even	1	trivial
8016.2.a.n.1.2		3	4.3	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:	\( N \)	\(=\)	\( 4008 = 2^{3} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4008.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.46081 q^{5} +2.34017 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.46081 q^{5} +2.34017 q^{7} +1.00000 q^{9} -2.63090 q^{11} +2.00000 q^{13} -1.46081 q^{15} +1.61757 q^{17} -3.41855 q^{19} -2.34017 q^{21} -8.00000 q^{23} -2.86603 q^{25} -1.00000 q^{27} -7.26180 q^{29} +3.41855 q^{31} +2.63090 q^{33} +3.41855 q^{35} +0.340173 q^{37} -2.00000 q^{39} -8.63809 q^{41} -7.95774 q^{43} +1.46081 q^{45} -3.89269 q^{47} -1.52359 q^{49} -1.61757 q^{51} +6.29791 q^{53} -3.84324 q^{55} +3.41855 q^{57} -1.26180 q^{59} +12.5464 q^{61} +2.34017 q^{63} +2.92162 q^{65} -12.8794 q^{67} +8.00000 q^{69} -15.9155 q^{71} +1.50307 q^{73} +2.86603 q^{75} -6.15676 q^{77} +2.35577 q^{79} +1.00000 q^{81} +16.5958 q^{83} +2.36296 q^{85} +7.26180 q^{87} -11.9421 q^{89} +4.68035 q^{91} -3.41855 q^{93} -4.99386 q^{95} -1.23513 q^{97} -2.63090 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 6 q^{5} - 4 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 6 * q^5 - 4 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} + 6 q^{5} - 4 q^{7} + 3 q^{9} - 4 q^{11} + 6 q^{13} - 6 q^{15} + 4 q^{19} + 4 q^{21} - 24 q^{23} + 5 q^{25} - 3 q^{27} - 14 q^{29} - 4 q^{31} + 4 q^{33} - 4 q^{35} - 10 q^{37} - 6 q^{39} + 12 q^{41} - 8 q^{43} + 6 q^{45} + 11 q^{49} - 8 q^{53} - 18 q^{55} - 4 q^{57} + 4 q^{59} + 2 q^{61} - 4 q^{63} + 12 q^{65} - 26 q^{67} + 24 q^{69} - 16 q^{71} + 22 q^{73} - 5 q^{75} - 12 q^{77} + 10 q^{79} + 3 q^{81} - 4 q^{83} - 24 q^{85} + 14 q^{87} - 6 q^{89} - 8 q^{91} + 4 q^{93} + 20 q^{95} + 6 q^{97} - 4 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 6 * q^5 - 4 * q^7 + 3 * q^9 - 4 * q^11 + 6 * q^13 - 6 * q^15 + 4 * q^19 + 4 * q^21 - 24 * q^23 + 5 * q^25 - 3 * q^27 - 14 * q^29 - 4 * q^31 + 4 * q^33 - 4 * q^35 - 10 * q^37 - 6 * q^39 + 12 * q^41 - 8 * q^43 + 6 * q^45 + 11 * q^49 - 8 * q^53 - 18 * q^55 - 4 * q^57 + 4 * q^59 + 2 * q^61 - 4 * q^63 + 12 * q^65 - 26 * q^67 + 24 * q^69 - 16 * q^71 + 22 * q^73 - 5 * q^75 - 12 * q^77 + 10 * q^79 + 3 * q^81 - 4 * q^83 - 24 * q^85 + 14 * q^87 - 6 * q^89 - 8 * q^91 + 4 * q^93 + 20 * q^95 + 6 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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