LMFDB - Embedded newform 4014.2.a.x.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4014,2,Mod(1,4014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4014, 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("4014.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4014 = 2 \cdot 3^{2} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4014.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:	$32.0519513713$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 14x^{6} + 28x^{5} + 43x^{4} - 90x^{3} - 23x^{2} + 82x - 28 $ x^8 - 2x^7 - 14x^6 + 28x^5 + 43x^4 - 90x^3 - 23x^2 + 82*x - 28
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.769489$ of defining polynomial
Character	$\chi$	$=$	4014.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +2.40529 q^{5} +2.01389 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +2.40529 q^{5} +2.01389 q^{7} -1.00000 q^{8} -2.40529 q^{10} +4.74119 q^{11} -1.18892 q^{13} -2.01389 q^{14} +1.00000 q^{16} +2.84640 q^{17} +6.39576 q^{19} +2.40529 q^{20} -4.74119 q^{22} -3.82143 q^{23} +0.785402 q^{25} +1.18892 q^{26} +2.01389 q^{28} +1.12694 q^{29} +4.42777 q^{31} -1.00000 q^{32} -2.84640 q^{34} +4.84397 q^{35} -4.49968 q^{37} -6.39576 q^{38} -2.40529 q^{40} +12.3787 q^{41} -2.24510 q^{43} +4.74119 q^{44} +3.82143 q^{46} -3.85599 q^{47} -2.94426 q^{49} -0.785402 q^{50} -1.18892 q^{52} +8.02407 q^{53} +11.4039 q^{55} -2.01389 q^{56} -1.12694 q^{58} +8.06772 q^{59} -0.752213 q^{61} -4.42777 q^{62} +1.00000 q^{64} -2.85970 q^{65} -5.18406 q^{67} +2.84640 q^{68} -4.84397 q^{70} -10.8588 q^{71} -13.8923 q^{73} +4.49968 q^{74} +6.39576 q^{76} +9.54821 q^{77} -1.50068 q^{79} +2.40529 q^{80} -12.3787 q^{82} +2.56673 q^{83} +6.84640 q^{85} +2.24510 q^{86} -4.74119 q^{88} +8.08800 q^{89} -2.39435 q^{91} -3.82143 q^{92} +3.85599 q^{94} +15.3836 q^{95} +2.17296 q^{97} +2.94426 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + 8 q^{4} + 6 q^{5} - 6 q^{7} - 8 q^{8}+O(q^{10}) $ 8 * q - 8 * q^2 + 8 * q^4 + 6 * q^5 - 6 * q^7 - 8 * q^8	$ 8 q - 8 q^{2} + 8 q^{4} + 6 q^{5} - 6 q^{7} - 8 q^{8} - 6 q^{10} + 11 q^{11} - q^{13} + 6 q^{14} + 8 q^{16} + 16 q^{17} - q^{19} + 6 q^{20} - 11 q^{22} + 14 q^{23} + 10 q^{25} + q^{26} - 6 q^{28} + 21 q^{29} - 6 q^{31} - 8 q^{32} - 16 q^{34} + 8 q^{35} - 14 q^{37} + q^{38} - 6 q^{40} + 16 q^{41} - 29 q^{43} + 11 q^{44} - 14 q^{46} + 9 q^{47} - 2 q^{49} - 10 q^{50} - q^{52} + 11 q^{53} - 22 q^{55} + 6 q^{56} - 21 q^{58} + 21 q^{59} + 3 q^{61} + 6 q^{62} + 8 q^{64} + 24 q^{65} - 20 q^{67} + 16 q^{68} - 8 q^{70} + 32 q^{71} + 13 q^{73} + 14 q^{74} - q^{76} + 4 q^{77} + 21 q^{79} + 6 q^{80} - 16 q^{82} + 28 q^{83} - 14 q^{85} + 29 q^{86} - 11 q^{88} + 54 q^{89} - 36 q^{91} + 14 q^{92} - 9 q^{94} + 30 q^{95} + 10 q^{97} + 2 q^{98}+O(q^{100}) $ 8 * q - 8 * q^2 + 8 * q^4 + 6 * q^5 - 6 * q^7 - 8 * q^8 - 6 * q^10 + 11 * q^11 - q^13 + 6 * q^14 + 8 * q^16 + 16 * q^17 - q^19 + 6 * q^20 - 11 * q^22 + 14 * q^23 + 10 * q^25 + q^26 - 6 * q^28 + 21 * q^29 - 6 * q^31 - 8 * q^32 - 16 * q^34 + 8 * q^35 - 14 * q^37 + q^38 - 6 * q^40 + 16 * q^41 - 29 * q^43 + 11 * q^44 - 14 * q^46 + 9 * q^47 - 2 * q^49 - 10 * q^50 - q^52 + 11 * q^53 - 22 * q^55 + 6 * q^56 - 21 * q^58 + 21 * q^59 + 3 * q^61 + 6 * q^62 + 8 * q^64 + 24 * q^65 - 20 * q^67 + 16 * q^68 - 8 * q^70 + 32 * q^71 + 13 * q^73 + 14 * q^74 - q^76 + 4 * q^77 + 21 * q^79 + 6 * q^80 - 16 * q^82 + 28 * q^83 - 14 * q^85 + 29 * q^86 - 11 * q^88 + 54 * q^89 - 36 * q^91 + 14 * q^92 - 9 * q^94 + 30 * q^95 + 10 * q^97 + 2 * q^98

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
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	2.40529	1.07568	0.537838	−	0.843048i	$-0.319241\pi$
$5$	2.40529	1.07568	0.537838	+	0.843048i	$0.319241\pi$
$6$	0	0
$7$	2.01389	0.761177	0.380589	−	0.924744i	$-0.375721\pi$
$7$	2.01389	0.761177	0.380589	+	0.924744i	$0.375721\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−2.40529	−0.760618
$11$	4.74119	1.42952	0.714761	−	0.699369i	$-0.246536\pi$
$11$	4.74119	1.42952	0.714761	+	0.699369i	$0.246536\pi$
$12$	0	0
$13$	−1.18892	−0.329748	−0.164874	−	0.986315i	$-0.552722\pi$
$13$	−1.18892	−0.329748	−0.164874	+	0.986315i	$0.552722\pi$
$14$	−2.01389	−0.538234
$15$	0	0
$16$	1.00000	0.250000
$17$	2.84640	0.690353	0.345177	−	0.938538i	$-0.387819\pi$
$17$	2.84640	0.690353	0.345177	+	0.938538i	$0.387819\pi$
$18$	0	0
$19$	6.39576	1.46729	0.733643	−	0.679535i	$-0.237818\pi$
$19$	6.39576	1.46729	0.733643	+	0.679535i	$0.237818\pi$
$20$	2.40529	0.537838
$21$	0	0
$22$	−4.74119	−1.01082
$23$	−3.82143	−0.796824	−0.398412	−	0.917207i	$-0.630439\pi$
$23$	−3.82143	−0.796824	−0.398412	+	0.917207i	$0.630439\pi$
$24$	0	0
$25$	0.785402	0.157080
$26$	1.18892	0.233167
$27$	0	0
$28$	2.01389	0.380589
$29$	1.12694	0.209267	0.104634	−	0.994511i	$-0.466633\pi$
$29$	1.12694	0.209267	0.104634	+	0.994511i	$0.466633\pi$
$30$	0	0
$31$	4.42777	0.795252	0.397626	−	0.917548i	$-0.369834\pi$
$31$	4.42777	0.795252	0.397626	+	0.917548i	$0.369834\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−2.84640	−0.488153
$35$	4.84397	0.818781
$36$	0	0
$37$	−4.49968	−0.739743	−0.369872	−	0.929083i	$-0.620598\pi$
$37$	−4.49968	−0.739743	−0.369872	+	0.929083i	$0.620598\pi$
$38$	−6.39576	−1.03753
$39$	0	0
$40$	−2.40529	−0.380309
$41$	12.3787	1.93323	0.966617	−	0.256226i	$-0.0824790\pi$
$41$	12.3787	1.93323	0.966617	+	0.256226i	$0.0824790\pi$
$42$	0	0
$43$	−2.24510	−0.342375	−0.171187	−	0.985238i	$-0.554760\pi$
$43$	−2.24510	−0.342375	−0.171187	+	0.985238i	$0.554760\pi$
$44$	4.74119	0.714761
$45$	0	0
$46$	3.82143	0.563440
$47$	−3.85599	−0.562454	−0.281227	−	0.959641i	$-0.590741\pi$
$47$	−3.85599	−0.562454	−0.281227	+	0.959641i	$0.590741\pi$
$48$	0	0
$49$	−2.94426	−0.420609
$50$	−0.785402	−0.111073
$51$	0	0
$52$	−1.18892	−0.164874
$53$	8.02407	1.10219	0.551095	−	0.834442i	$-0.314210\pi$
$53$	8.02407	1.10219	0.551095	+	0.834442i	$0.314210\pi$
$54$	0	0
$55$	11.4039	1.53770
$56$	−2.01389	−0.269117
$57$	0	0
$58$	−1.12694	−0.147974
$59$	8.06772	1.05033	0.525164	−	0.851001i	$-0.324004\pi$
$59$	8.06772	1.05033	0.525164	+	0.851001i	$0.324004\pi$
$60$	0	0
$61$	−0.752213	−0.0963111	−0.0481555	−	0.998840i	$-0.515334\pi$
$61$	−0.752213	−0.0963111	−0.0481555	+	0.998840i	$0.515334\pi$
$62$	−4.42777	−0.562328
$63$	0	0
$64$	1.00000	0.125000
$65$	−2.85970	−0.354702
$66$	0	0
$67$	−5.18406	−0.633334	−0.316667	−	0.948537i	$-0.602564\pi$
$67$	−5.18406	−0.633334	−0.316667	+	0.948537i	$0.602564\pi$
$68$	2.84640	0.345177
$69$	0	0
$70$	−4.84397	−0.578965
$71$	−10.8588	−1.28870	−0.644348	−	0.764732i	$-0.722871\pi$
$71$	−10.8588	−1.28870	−0.644348	+	0.764732i	$0.722871\pi$
$72$	0	0
$73$	−13.8923	−1.62597	−0.812983	−	0.582288i	$-0.802158\pi$
$73$	−13.8923	−1.62597	−0.812983	+	0.582288i	$0.802158\pi$
$74$	4.49968	0.523077
$75$	0	0
$76$	6.39576	0.733643
$77$	9.54821	1.08812
$78$	0	0
$79$	−1.50068	−0.168839	−0.0844197	−	0.996430i	$-0.526904\pi$
$79$	−1.50068	−0.168839	−0.0844197	+	0.996430i	$0.526904\pi$
$80$	2.40529	0.268919
$81$	0	0
$82$	−12.3787	−1.36700
$83$	2.56673	0.281735	0.140867	−	0.990028i	$-0.455011\pi$
$83$	2.56673	0.281735	0.140867	+	0.990028i	$0.455011\pi$
$84$	0	0
$85$	6.84640	0.742597
$86$	2.24510	0.242096
$87$	0	0
$88$	−4.74119	−0.505412
$89$	8.08800	0.857326	0.428663	−	0.903465i	$-0.358985\pi$
$89$	8.08800	0.857326	0.428663	+	0.903465i	$0.358985\pi$
$90$	0	0
$91$	−2.39435	−0.250996
$92$	−3.82143	−0.398412
$93$	0	0
$94$	3.85599	0.397715
$95$	15.3836	1.57833
$96$	0	0
$97$	2.17296	0.220631	0.110315	−	0.993897i	$-0.464814\pi$
$97$	2.17296	0.220631	0.110315	+	0.993897i	$0.464814\pi$
$98$	2.94426	0.297415
$99$	0	0
$100$	0.785402	0.0785402
$101$	10.1071	1.00570	0.502850	−	0.864374i	$-0.332285\pi$
$101$	10.1071	1.00570	0.502850	+	0.864374i	$0.332285\pi$
$102$	0	0
$103$	−6.65688	−0.655922	−0.327961	−	0.944691i	$-0.606361\pi$
$103$	−6.65688	−0.655922	−0.327961	+	0.944691i	$0.606361\pi$
$104$	1.18892	0.116583
$105$	0	0
$106$	−8.02407	−0.779366
$107$	13.0227	1.25896	0.629478	−	0.777018i	$-0.283269\pi$
$107$	13.0227	1.25896	0.629478	+	0.777018i	$0.283269\pi$
$108$	0	0
$109$	−16.5118	−1.58154	−0.790771	−	0.612112i	$-0.790320\pi$
$109$	−16.5118	−1.58154	−0.790771	+	0.612112i	$0.790320\pi$
$110$	−11.4039	−1.08732
$111$	0	0
$112$	2.01389	0.190294
$113$	−13.6166	−1.28094	−0.640469	−	0.767984i	$-0.721260\pi$
$113$	−13.6166	−1.28094	−0.640469	+	0.767984i	$0.721260\pi$
$114$	0	0
$115$	−9.19164	−0.857125
$116$	1.12694	0.104634
$117$	0	0
$118$	−8.06772	−0.742694
$119$	5.73232	0.525481
$120$	0	0
$121$	11.4788	1.04353
$122$	0.752213	0.0681022
$123$	0	0
$124$	4.42777	0.397626
$125$	−10.1373	−0.906709
$126$	0	0
$127$	8.40728	0.746025	0.373013	−	0.927826i	$-0.378325\pi$
$127$	8.40728	0.746025	0.373013	+	0.927826i	$0.378325\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	2.85970	0.250812
$131$	12.4493	1.08770	0.543852	−	0.839181i	$-0.316965\pi$
$131$	12.4493	1.08770	0.543852	+	0.839181i	$0.316965\pi$
$132$	0	0
$133$	12.8803	1.11687
$134$	5.18406	0.447835
$135$	0	0
$136$	−2.84640	−0.244077
$137$	−4.23462	−0.361788	−0.180894	−	0.983503i	$-0.557899\pi$
$137$	−4.23462	−0.361788	−0.180894	+	0.983503i	$0.557899\pi$
$138$	0	0
$139$	17.8592	1.51480	0.757400	−	0.652952i	$-0.226470\pi$
$139$	17.8592	1.51480	0.757400	+	0.652952i	$0.226470\pi$
$140$	4.84397	0.409390
$141$	0	0
$142$	10.8588	0.911246
$143$	−5.63690	−0.471381
$144$	0	0
$145$	2.71061	0.225104
$146$	13.8923	1.14973
$147$	0	0
$148$	−4.49968	−0.369872
$149$	−3.20765	−0.262781	−0.131391	−	0.991331i	$-0.541944\pi$
$149$	−3.20765	−0.262781	−0.131391	+	0.991331i	$0.541944\pi$
$150$	0	0
$151$	−10.7045	−0.871117	−0.435559	−	0.900160i	$-0.643449\pi$
$151$	−10.7045	−0.871117	−0.435559	+	0.900160i	$0.643449\pi$
$152$	−6.39576	−0.518764
$153$	0	0
$154$	−9.54821	−0.769416
$155$	10.6501	0.855434
$156$	0	0
$157$	−15.9572	−1.27352	−0.636760	−	0.771062i	$-0.719726\pi$
$157$	−15.9572	−1.27352	−0.636760	+	0.771062i	$0.719726\pi$
$158$	1.50068	0.119387
$159$	0	0
$160$	−2.40529	−0.190155
$161$	−7.69593	−0.606524
$162$	0	0
$163$	−10.4348	−0.817314	−0.408657	−	0.912688i	$-0.634003\pi$
$163$	−10.4348	−0.817314	−0.408657	+	0.912688i	$0.634003\pi$
$164$	12.3787	0.966617
$165$	0	0
$166$	−2.56673	−0.199217
$167$	−20.8806	−1.61579	−0.807893	−	0.589329i	$-0.799392\pi$
$167$	−20.8806	−1.61579	−0.807893	+	0.589329i	$0.799392\pi$
$168$	0	0
$169$	−11.5865	−0.891266
$170$	−6.84640	−0.525095
$171$	0	0
$172$	−2.24510	−0.171187
$173$	−16.1164	−1.22531	−0.612653	−	0.790352i	$-0.709898\pi$
$173$	−16.1164	−1.22531	−0.612653	+	0.790352i	$0.709898\pi$
$174$	0	0
$175$	1.58171	0.119566
$176$	4.74119	0.357380
$177$	0	0
$178$	−8.08800	−0.606221
$179$	−9.04126	−0.675776	−0.337888	−	0.941186i	$-0.609712\pi$
$179$	−9.04126	−0.675776	−0.337888	+	0.941186i	$0.609712\pi$
$180$	0	0
$181$	14.2580	1.05979	0.529894	−	0.848064i	$-0.322232\pi$
$181$	14.2580	1.05979	0.529894	+	0.848064i	$0.322232\pi$
$182$	2.39435	0.177481
$183$	0	0
$184$	3.82143	0.281720
$185$	−10.8230	−0.795724
$186$	0	0
$187$	13.4953	0.986874
$188$	−3.85599	−0.281227
$189$	0	0
$190$	−15.3836	−1.11605
$191$	−3.78584	−0.273934	−0.136967	−	0.990576i	$-0.543735\pi$
$191$	−3.78584	−0.273934	−0.136967	+	0.990576i	$0.543735\pi$
$192$	0	0
$193$	1.43542	0.103323	0.0516617	−	0.998665i	$-0.483548\pi$
$193$	1.43542	0.103323	0.0516617	+	0.998665i	$0.483548\pi$
$194$	−2.17296	−0.156009
$195$	0	0
$196$	−2.94426	−0.210305
$197$	9.87791	0.703772	0.351886	−	0.936043i	$-0.385540\pi$
$197$	9.87791	0.703772	0.351886	+	0.936043i	$0.385540\pi$
$198$	0	0
$199$	3.75481	0.266171	0.133086	−	0.991105i	$-0.457511\pi$
$199$	3.75481	0.266171	0.133086	+	0.991105i	$0.457511\pi$
$200$	−0.785402	−0.0555363
$201$	0	0
$202$	−10.1071	−0.711137
$203$	2.26953	0.159290
$204$	0	0
$205$	29.7744	2.07953
$206$	6.65688	0.463807
$207$	0	0
$208$	−1.18892	−0.0824369
$209$	30.3235	2.09752
$210$	0	0
$211$	−17.6476	−1.21491	−0.607454	−	0.794355i	$-0.707809\pi$
$211$	−17.6476	−1.21491	−0.607454	+	0.794355i	$0.707809\pi$
$212$	8.02407	0.551095
$213$	0	0
$214$	−13.0227	−0.890217
$215$	−5.40012	−0.368285
$216$	0	0
$217$	8.91703	0.605328
$218$	16.5118	1.11832
$219$	0	0
$220$	11.4039	0.768851
$221$	−3.38415	−0.227642
$222$	0	0
$223$	1.00000	0.0669650
$223$	1.00000	0.0669650
$224$	−2.01389	−0.134558
$225$	0	0
$226$	13.6166	0.905760
$227$	5.69750	0.378156	0.189078	−	0.981962i	$-0.439450\pi$
$227$	5.69750	0.378156	0.189078	+	0.981962i	$0.439450\pi$
$228$	0	0
$229$	−26.0839	−1.72367	−0.861836	−	0.507187i	$-0.830685\pi$
$229$	−26.0839	−1.72367	−0.861836	+	0.507187i	$0.830685\pi$
$230$	9.19164	0.606079
$231$	0	0
$232$	−1.12694	−0.0739872
$233$	25.2515	1.65428	0.827141	−	0.561994i	$-0.189966\pi$
$233$	25.2515	1.65428	0.827141	+	0.561994i	$0.189966\pi$
$234$	0	0
$235$	−9.27477	−0.605019
$236$	8.06772	0.525164
$237$	0	0
$238$	−5.73232	−0.371571
$239$	7.89413	0.510629	0.255314	−	0.966858i	$-0.417821\pi$
$239$	7.89413	0.510629	0.255314	+	0.966858i	$0.417821\pi$
$240$	0	0
$241$	27.7248	1.78591	0.892955	−	0.450146i	$-0.148628\pi$
$241$	27.7248	1.78591	0.892955	+	0.450146i	$0.148628\pi$
$242$	−11.4788	−0.737888
$243$	0	0
$244$	−0.752213	−0.0481555
$245$	−7.08180	−0.452439
$246$	0	0
$247$	−7.60406	−0.483834
$248$	−4.42777	−0.281164
$249$	0	0
$250$	10.1373	0.641140
$251$	−16.6848	−1.05314	−0.526569	−	0.850132i	$-0.676522\pi$
$251$	−16.6848	−1.05314	−0.526569	+	0.850132i	$0.676522\pi$
$252$	0	0
$253$	−18.1181	−1.13908
$254$	−8.40728	−0.527519
$255$	0	0
$256$	1.00000	0.0625000
$257$	9.32829	0.581883	0.290941	−	0.956741i	$-0.406032\pi$
$257$	9.32829	0.581883	0.290941	+	0.956741i	$0.406032\pi$
$258$	0	0
$259$	−9.06185	−0.563076
$260$	−2.85970	−0.177351
$261$	0	0
$262$	−12.4493	−0.769122
$263$	−23.0617	−1.42205	−0.711023	−	0.703169i	$-0.751768\pi$
$263$	−23.0617	−1.42205	−0.711023	+	0.703169i	$0.751768\pi$
$264$	0	0
$265$	19.3002	1.18560
$266$	−12.8803	−0.789743
$267$	0	0
$268$	−5.18406	−0.316667
$269$	1.30960	0.0798476	0.0399238	−	0.999203i	$-0.487288\pi$
$269$	1.30960	0.0798476	0.0399238	+	0.999203i	$0.487288\pi$
$270$	0	0
$271$	7.50256	0.455748	0.227874	−	0.973691i	$-0.426823\pi$
$271$	7.50256	0.455748	0.227874	+	0.973691i	$0.426823\pi$
$272$	2.84640	0.172588
$273$	0	0
$274$	4.23462	0.255823
$275$	3.72373	0.224550
$276$	0	0
$277$	4.02078	0.241585	0.120793	−	0.992678i	$-0.461456\pi$
$277$	4.02078	0.241585	0.120793	+	0.992678i	$0.461456\pi$
$278$	−17.8592	−1.07112
$279$	0	0
$280$	−4.84397	−0.289483
$281$	−7.37207	−0.439781	−0.219890	−	0.975525i	$-0.570570\pi$
$281$	−7.37207	−0.439781	−0.219890	+	0.975525i	$0.570570\pi$
$282$	0	0
$283$	2.96020	0.175966	0.0879829	−	0.996122i	$-0.471958\pi$
$283$	2.96020	0.175966	0.0879829	+	0.996122i	$0.471958\pi$
$284$	−10.8588	−0.644348
$285$	0	0
$286$	5.63690	0.333317
$287$	24.9294	1.47153
$288$	0	0
$289$	−8.89801	−0.523413
$290$	−2.71061	−0.159173
$291$	0	0
$292$	−13.8923	−0.812983
$293$	22.6359	1.32240	0.661202	−	0.750208i	$-0.270047\pi$
$293$	22.6359	1.32240	0.661202	+	0.750208i	$0.270047\pi$
$294$	0	0
$295$	19.4052	1.12981
$296$	4.49968	0.261539
$297$	0	0
$298$	3.20765	0.185814
$299$	4.54339	0.262751
$300$	0	0
$301$	−4.52138	−0.260608
$302$	10.7045	0.615973
$303$	0	0
$304$	6.39576	0.366822
$305$	−1.80929	−0.103600
$306$	0	0
$307$	30.8339	1.75978	0.879891	−	0.475175i	$-0.157615\pi$
$307$	30.8339	1.75978	0.879891	+	0.475175i	$0.157615\pi$
$308$	9.54821	0.544060
$309$	0	0
$310$	−10.6501	−0.604883
$311$	10.2265	0.579891	0.289945	−	0.957043i	$-0.406363\pi$
$311$	10.2265	0.579891	0.289945	+	0.957043i	$0.406363\pi$
$312$	0	0
$313$	−8.10343	−0.458033	−0.229017	−	0.973423i	$-0.573551\pi$
$313$	−8.10343	−0.458033	−0.229017	+	0.973423i	$0.573551\pi$
$314$	15.9572	0.900514
$315$	0	0
$316$	−1.50068	−0.0844197
$317$	−6.55901	−0.368391	−0.184195	−	0.982890i	$-0.558968\pi$
$317$	−6.55901	−0.368391	−0.184195	+	0.982890i	$0.558968\pi$
$318$	0	0
$319$	5.34303	0.299152
$320$	2.40529	0.134460
$321$	0	0
$322$	7.69593	0.428877
$323$	18.2049	1.01295
$324$	0	0
$325$	−0.933781	−0.0517969
$326$	10.4348	0.577928
$327$	0	0
$328$	−12.3787	−0.683501
$329$	−7.76553	−0.428128
$330$	0	0
$331$	12.0429	0.661939	0.330970	−	0.943641i	$-0.392624\pi$
$331$	12.0429	0.661939	0.330970	+	0.943641i	$0.392624\pi$
$332$	2.56673	0.140867
$333$	0	0
$334$	20.8806	1.14253
$335$	−12.4692	−0.681263
$336$	0	0
$337$	−9.69914	−0.528346	−0.264173	−	0.964475i	$-0.585099\pi$
$337$	−9.69914	−0.528346	−0.264173	+	0.964475i	$0.585099\pi$
$338$	11.5865	0.630221
$339$	0	0
$340$	6.84640	0.371298
$341$	20.9929	1.13683
$342$	0	0
$343$	−20.0266	−1.08134
$344$	2.24510	0.121048
$345$	0	0
$346$	16.1164	0.866422
$347$	−2.34351	−0.125806	−0.0629030	−	0.998020i	$-0.520036\pi$
$347$	−2.34351	−0.125806	−0.0629030	+	0.998020i	$0.520036\pi$
$348$	0	0
$349$	13.5512	0.725379	0.362690	−	0.931910i	$-0.381859\pi$
$349$	13.5512	0.725379	0.362690	+	0.931910i	$0.381859\pi$
$350$	−1.58171	−0.0845459
$351$	0	0
$352$	−4.74119	−0.252706
$353$	28.5501	1.51957	0.759784	−	0.650175i	$-0.225304\pi$
$353$	28.5501	1.51957	0.759784	+	0.650175i	$0.225304\pi$
$354$	0	0
$355$	−26.1184	−1.38622
$356$	8.08800	0.428663
$357$	0	0
$358$	9.04126	0.477846
$359$	1.51560	0.0799903	0.0399951	−	0.999200i	$-0.487266\pi$
$359$	1.51560	0.0799903	0.0399951	+	0.999200i	$0.487266\pi$
$360$	0	0
$361$	21.9057	1.15293
$362$	−14.2580	−0.749383
$363$	0	0
$364$	−2.39435	−0.125498
$365$	−33.4149	−1.74901
$366$	0	0
$367$	−3.26419	−0.170389	−0.0851945	−	0.996364i	$-0.527151\pi$
$367$	−3.26419	−0.170389	−0.0851945	+	0.996364i	$0.527151\pi$
$368$	−3.82143	−0.199206
$369$	0	0
$370$	10.8230	0.562662
$371$	16.1596	0.838962
$372$	0	0
$373$	−19.3552	−1.00217	−0.501086	−	0.865397i	$-0.667066\pi$
$373$	−19.3552	−1.00217	−0.501086	+	0.865397i	$0.667066\pi$
$374$	−13.4953	−0.697826
$375$	0	0
$376$	3.85599	0.198858
$377$	−1.33984	−0.0690054
$378$	0	0
$379$	12.7890	0.656925	0.328462	−	0.944517i	$-0.393470\pi$
$379$	12.7890	0.656925	0.328462	+	0.944517i	$0.393470\pi$
$380$	15.3836	0.789163
$381$	0	0
$382$	3.78584	0.193700
$383$	17.4215	0.890197	0.445099	−	0.895481i	$-0.353169\pi$
$383$	17.4215	0.890197	0.445099	+	0.895481i	$0.353169\pi$
$384$	0	0
$385$	22.9662	1.17046
$386$	−1.43542	−0.0730607
$387$	0	0
$388$	2.17296	0.110315
$389$	20.2383	1.02612	0.513060	−	0.858353i	$-0.328512\pi$
$389$	20.2383	1.02612	0.513060	+	0.858353i	$0.328512\pi$
$390$	0	0
$391$	−10.8773	−0.550090
$392$	2.94426	0.148708
$393$	0	0
$394$	−9.87791	−0.497642
$395$	−3.60956	−0.181617
$396$	0	0
$397$	35.6524	1.78935	0.894673	−	0.446723i	$-0.147409\pi$
$397$	35.6524	1.78935	0.894673	+	0.446723i	$0.147409\pi$
$398$	−3.75481	−0.188211
$399$	0	0
$400$	0.785402	0.0392701
$401$	27.5517	1.37587	0.687933	−	0.725774i	$-0.258518\pi$
$401$	27.5517	1.37587	0.687933	+	0.725774i	$0.258518\pi$
$402$	0	0
$403$	−5.26428	−0.262232
$404$	10.1071	0.502850
$405$	0	0
$406$	−2.26953	−0.112635
$407$	−21.3338	−1.05748
$408$	0	0
$409$	3.65303	0.180631	0.0903154	−	0.995913i	$-0.471212\pi$
$409$	3.65303	0.180631	0.0903154	+	0.995913i	$0.471212\pi$
$410$	−29.7744	−1.47045
$411$	0	0
$412$	−6.65688	−0.327961
$413$	16.2475	0.799486
$414$	0	0
$415$	6.17371	0.303056
$416$	1.18892	0.0582917
$417$	0	0
$418$	−30.3235	−1.48317
$419$	28.4301	1.38890	0.694451	−	0.719540i	$-0.255647\pi$
$419$	28.4301	1.38890	0.694451	+	0.719540i	$0.255647\pi$
$420$	0	0
$421$	22.7754	1.11001	0.555004	−	0.831848i	$-0.312717\pi$
$421$	22.7754	1.11001	0.555004	+	0.831848i	$0.312717\pi$
$422$	17.6476	0.859070
$423$	0	0
$424$	−8.02407	−0.389683
$425$	2.23557	0.108441
$426$	0	0
$427$	−1.51487	−0.0733098
$428$	13.0227	0.629478
$429$	0	0
$430$	5.40012	0.260417
$431$	−21.2326	−1.02274	−0.511370	−	0.859361i	$-0.670862\pi$
$431$	−21.2326	−1.02274	−0.511370	+	0.859361i	$0.670862\pi$
$432$	0	0
$433$	30.5653	1.46887	0.734437	−	0.678677i	$-0.237446\pi$
$433$	30.5653	1.46887	0.734437	+	0.678677i	$0.237446\pi$
$434$	−8.91703	−0.428031
$435$	0	0
$436$	−16.5118	−0.790771
$437$	−24.4410	−1.16917
$438$	0	0
$439$	−18.8453	−0.899437	−0.449718	−	0.893170i	$-0.648476\pi$
$439$	−18.8453	−0.899437	−0.449718	+	0.893170i	$0.648476\pi$
$440$	−11.4039	−0.543660
$441$	0	0
$442$	3.38415	0.160967
$443$	−6.37575	−0.302921	−0.151460	−	0.988463i	$-0.548398\pi$
$443$	−6.37575	−0.302921	−0.151460	+	0.988463i	$0.548398\pi$
$444$	0	0
$445$	19.4539	0.922205
$446$	−1.00000	−0.0473514
$447$	0	0
$448$	2.01389	0.0951472
$449$	−23.2926	−1.09925	−0.549624	−	0.835412i	$-0.685229\pi$
$449$	−23.2926	−1.09925	−0.549624	+	0.835412i	$0.685229\pi$
$450$	0	0
$451$	58.6899	2.76360
$452$	−13.6166	−0.640469
$453$	0	0
$454$	−5.69750	−0.267397
$455$	−5.75911	−0.269991
$456$	0	0
$457$	26.7034	1.24913	0.624567	−	0.780971i	$-0.285276\pi$
$457$	26.7034	1.24913	0.624567	+	0.780971i	$0.285276\pi$
$458$	26.0839	1.21882
$459$	0	0
$460$	−9.19164	−0.428562
$461$	1.37642	0.0641063	0.0320532	−	0.999486i	$-0.489795\pi$
$461$	1.37642	0.0641063	0.0320532	+	0.999486i	$0.489795\pi$
$462$	0	0
$463$	−9.04789	−0.420491	−0.210246	−	0.977649i	$-0.567426\pi$
$463$	−9.04789	−0.420491	−0.210246	+	0.977649i	$0.567426\pi$
$464$	1.12694	0.0523168
$465$	0	0
$466$	−25.2515	−1.16975
$467$	10.7756	0.498637	0.249319	−	0.968421i	$-0.419793\pi$
$467$	10.7756	0.498637	0.249319	+	0.968421i	$0.419793\pi$
$468$	0	0
$469$	−10.4401	−0.482080
$470$	9.27477	0.427813
$471$	0	0
$472$	−8.06772	−0.371347
$473$	−10.6444	−0.489432
$474$	0	0
$475$	5.02324	0.230482
$476$	5.73232	0.262741
$477$	0	0
$478$	−7.89413	−0.361069
$479$	−30.3795	−1.38808	−0.694038	−	0.719939i	$-0.744170\pi$
$479$	−30.3795	−1.38808	−0.694038	+	0.719939i	$0.744170\pi$
$480$	0	0
$481$	5.34977	0.243929
$482$	−27.7248	−1.26283
$483$	0	0
$484$	11.4788	0.521765
$485$	5.22659	0.237327
$486$	0	0
$487$	2.38443	0.108049	0.0540244	−	0.998540i	$-0.482795\pi$
$487$	2.38443	0.108049	0.0540244	+	0.998540i	$0.482795\pi$
$488$	0.752213	0.0340511
$489$	0	0
$490$	7.08180	0.319923
$491$	−31.8464	−1.43721	−0.718603	−	0.695420i	$-0.755218\pi$
$491$	−31.8464	−1.43721	−0.718603	+	0.695420i	$0.755218\pi$
$492$	0	0
$493$	3.20772	0.144468
$494$	7.60406	0.342123
$495$	0	0
$496$	4.42777	0.198813
$497$	−21.8683	−0.980927
$498$	0	0
$499$	−14.0800	−0.630309	−0.315154	−	0.949040i	$-0.602056\pi$
$499$	−14.0800	−0.630309	−0.315154	+	0.949040i	$0.602056\pi$
$500$	−10.1373	−0.453355
$501$	0	0
$502$	16.6848	0.744681
$503$	−32.4556	−1.44713	−0.723563	−	0.690259i	$-0.757497\pi$
$503$	−32.4556	−1.44713	−0.723563	+	0.690259i	$0.757497\pi$
$504$	0	0
$505$	24.3106	1.08181
$506$	18.1181	0.805449
$507$	0	0
$508$	8.40728	0.373013
$509$	21.5874	0.956843	0.478422	−	0.878130i	$-0.341209\pi$
$509$	21.5874	0.956843	0.478422	+	0.878130i	$0.341209\pi$
$510$	0	0
$511$	−27.9774	−1.23765
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−9.32829	−0.411453
$515$	−16.0117	−0.705560
$516$	0	0
$517$	−18.2820	−0.804040
$518$	9.06185	0.398155
$519$	0	0
$520$	2.85970	0.125406
$521$	−9.26775	−0.406027	−0.203014	−	0.979176i	$-0.565074\pi$
$521$	−9.26775	−0.406027	−0.203014	+	0.979176i	$0.565074\pi$
$522$	0	0
$523$	29.3899	1.28513	0.642565	−	0.766231i	$-0.277870\pi$
$523$	29.3899	1.28513	0.642565	+	0.766231i	$0.277870\pi$
$524$	12.4493	0.543852
$525$	0	0
$526$	23.0617	1.00554
$527$	12.6032	0.549004
$528$	0	0
$529$	−8.39665	−0.365072
$530$	−19.3002	−0.838346
$531$	0	0
$532$	12.8803	0.558433
$533$	−14.7174	−0.637479
$534$	0	0
$535$	31.3234	1.35423
$536$	5.18406	0.223917
$537$	0	0
$538$	−1.30960	−0.0564608
$539$	−13.9593	−0.601270
$540$	0	0
$541$	33.8384	1.45483	0.727413	−	0.686200i	$-0.240722\pi$
$541$	33.8384	1.45483	0.727413	+	0.686200i	$0.240722\pi$
$542$	−7.50256	−0.322263
$543$	0	0
$544$	−2.84640	−0.122038
$545$	−39.7156	−1.70123
$546$	0	0
$547$	−3.35049	−0.143257	−0.0716284	−	0.997431i	$-0.522820\pi$
$547$	−3.35049	−0.143257	−0.0716284	+	0.997431i	$0.522820\pi$
$548$	−4.23462	−0.180894
$549$	0	0
$550$	−3.72373	−0.158781
$551$	7.20763	0.307055
$552$	0	0
$553$	−3.02219	−0.128517
$554$	−4.02078	−0.170826
$555$	0	0
$556$	17.8592	0.757400
$557$	19.3954	0.821811	0.410905	−	0.911678i	$-0.365213\pi$
$557$	19.3954	0.821811	0.410905	+	0.911678i	$0.365213\pi$
$558$	0	0
$559$	2.66925	0.112897
$560$	4.84397	0.204695
$561$	0	0
$562$	7.37207	0.310972
$563$	−17.1711	−0.723674	−0.361837	−	0.932241i	$-0.617850\pi$
$563$	−17.1711	−0.723674	−0.361837	+	0.932241i	$0.617850\pi$
$564$	0	0
$565$	−32.7517	−1.37788
$566$	−2.96020	−0.124427
$567$	0	0
$568$	10.8588	0.455623
$569$	−9.86548	−0.413583	−0.206791	−	0.978385i	$-0.566302\pi$
$569$	−9.86548	−0.413583	−0.206791	+	0.978385i	$0.566302\pi$
$570$	0	0
$571$	−42.2688	−1.76889	−0.884446	−	0.466642i	$-0.845464\pi$
$571$	−42.2688	−1.76889	−0.884446	+	0.466642i	$0.845464\pi$
$572$	−5.63690	−0.235691
$573$	0	0
$574$	−24.9294	−1.04053
$575$	−3.00136	−0.125165
$576$	0	0
$577$	36.0725	1.50172	0.750858	−	0.660463i	$-0.229640\pi$
$577$	36.0725	1.50172	0.750858	+	0.660463i	$0.229640\pi$
$578$	8.89801	0.370109
$579$	0	0
$580$	2.71061	0.112552
$581$	5.16910	0.214450
$582$	0	0
$583$	38.0436	1.57560
$584$	13.8923	0.574866
$585$	0	0
$586$	−22.6359	−0.935080
$587$	−20.5128	−0.846654	−0.423327	−	0.905977i	$-0.639138\pi$
$587$	−20.5128	−0.846654	−0.423327	+	0.905977i	$0.639138\pi$
$588$	0	0
$589$	28.3190	1.16686
$590$	−19.4052	−0.798899
$591$	0	0
$592$	−4.49968	−0.184936
$593$	10.3527	0.425135	0.212568	−	0.977146i	$-0.431817\pi$
$593$	10.3527	0.425135	0.212568	+	0.977146i	$0.431817\pi$
$594$	0	0
$595$	13.7879	0.565248
$596$	−3.20765	−0.131391
$597$	0	0
$598$	−4.54339	−0.185793
$599$	−30.3548	−1.24026	−0.620131	−	0.784498i	$-0.712921\pi$
$599$	−30.3548	−1.24026	−0.620131	+	0.784498i	$0.712921\pi$
$600$	0	0
$601$	−1.02586	−0.0418456	−0.0209228	−	0.999781i	$-0.506660\pi$
$601$	−1.02586	−0.0418456	−0.0209228	+	0.999781i	$0.506660\pi$
$602$	4.52138	0.184278
$603$	0	0
$604$	−10.7045	−0.435559
$605$	27.6099	1.12250
$606$	0	0
$607$	−43.7954	−1.77760	−0.888799	−	0.458297i	$-0.848460\pi$
$607$	−43.7954	−1.77760	−0.888799	+	0.458297i	$0.848460\pi$
$608$	−6.39576	−0.259382
$609$	0	0
$610$	1.80929	0.0732560
$611$	4.58448	0.185468
$612$	0	0
$613$	28.5306	1.15234	0.576169	−	0.817330i	$-0.304547\pi$
$613$	28.5306	1.15234	0.576169	+	0.817330i	$0.304547\pi$
$614$	−30.8339	−1.24435
$615$	0	0
$616$	−9.54821	−0.384708
$617$	9.85911	0.396913	0.198456	−	0.980110i	$-0.436407\pi$
$617$	9.85911	0.396913	0.198456	+	0.980110i	$0.436407\pi$
$618$	0	0
$619$	−1.03126	−0.0414498	−0.0207249	−	0.999785i	$-0.506597\pi$
$619$	−1.03126	−0.0414498	−0.0207249	+	0.999785i	$0.506597\pi$
$620$	10.6501	0.427717
$621$	0	0
$622$	−10.2265	−0.410045
$623$	16.2883	0.652577
$624$	0	0
$625$	−28.3102	−1.13241
$626$	8.10343	0.323878
$627$	0	0
$628$	−15.9572	−0.636760
$629$	−12.8079	−0.510684
$630$	0	0
$631$	−37.2675	−1.48360	−0.741798	−	0.670624i	$-0.766026\pi$
$631$	−37.2675	−1.48360	−0.741798	+	0.670624i	$0.766026\pi$
$632$	1.50068	0.0596937
$633$	0	0
$634$	6.55901	0.260491
$635$	20.2219	0.802482
$636$	0	0
$637$	3.50050	0.138695
$638$	−5.34303	−0.211533
$639$	0	0
$640$	−2.40529	−0.0950773
$641$	7.84927	0.310028	0.155014	−	0.987912i	$-0.450458\pi$
$641$	7.84927	0.310028	0.155014	+	0.987912i	$0.450458\pi$
$642$	0	0
$643$	15.5032	0.611386	0.305693	−	0.952130i	$-0.401112\pi$
$643$	15.5032	0.611386	0.305693	+	0.952130i	$0.401112\pi$
$644$	−7.69593	−0.303262
$645$	0	0
$646$	−18.2049	−0.716261
$647$	4.52462	0.177881	0.0889405	−	0.996037i	$-0.471652\pi$
$647$	4.52462	0.177881	0.0889405	+	0.996037i	$0.471652\pi$
$648$	0	0
$649$	38.2506	1.50147
$650$	0.933781	0.0366259
$651$	0	0
$652$	−10.4348	−0.408657
$653$	−19.0951	−0.747250	−0.373625	−	0.927580i	$-0.621885\pi$
$653$	−19.0951	−0.747250	−0.373625	+	0.927580i	$0.621885\pi$
$654$	0	0
$655$	29.9442	1.17002
$656$	12.3787	0.483308
$657$	0	0
$658$	7.76553	0.302732
$659$	−38.2638	−1.49055	−0.745273	−	0.666759i	$-0.767681\pi$
$659$	−38.2638	−1.49055	−0.745273	+	0.666759i	$0.767681\pi$
$660$	0	0
$661$	−21.6666	−0.842733	−0.421367	−	0.906890i	$-0.638449\pi$
$661$	−21.6666	−0.842733	−0.421367	+	0.906890i	$0.638449\pi$
$662$	−12.0429	−0.468062
$663$	0	0
$664$	−2.56673	−0.0996083
$665$	30.9809	1.20139
$666$	0	0
$667$	−4.30652	−0.166749
$668$	−20.8806	−0.807893
$669$	0	0
$670$	12.4692	0.481726
$671$	−3.56638	−0.137679
$672$	0	0
$673$	12.3952	0.477800	0.238900	−	0.971044i	$-0.423213\pi$
$673$	12.3952	0.477800	0.238900	+	0.971044i	$0.423213\pi$
$674$	9.69914	0.373597
$675$	0	0
$676$	−11.5865	−0.445633
$677$	30.1779	1.15983	0.579916	−	0.814677i	$-0.303085\pi$
$677$	30.1779	1.15983	0.579916	+	0.814677i	$0.303085\pi$
$678$	0	0
$679$	4.37609	0.167939
$680$	−6.84640	−0.262548
$681$	0	0
$682$	−20.9929	−0.803860
$683$	−26.7463	−1.02342	−0.511710	−	0.859158i	$-0.670988\pi$
$683$	−26.7463	−1.02342	−0.511710	+	0.859158i	$0.670988\pi$
$684$	0	0
$685$	−10.1855	−0.389167
$686$	20.0266	0.764620
$687$	0	0
$688$	−2.24510	−0.0855937
$689$	−9.53999	−0.363445
$690$	0	0
$691$	30.9967	1.17917	0.589584	−	0.807707i	$-0.299292\pi$
$691$	30.9967	1.17917	0.589584	+	0.807707i	$0.299292\pi$
$692$	−16.1164	−0.612653
$693$	0	0
$694$	2.34351	0.0889582
$695$	42.9565	1.62943
$696$	0	0
$697$	35.2348	1.33461
$698$	−13.5512	−0.512920
$699$	0	0
$700$	1.58171	0.0597830
$701$	24.0991	0.910210	0.455105	−	0.890438i	$-0.349602\pi$
$701$	24.0991	0.910210	0.455105	+	0.890438i	$0.349602\pi$
$702$	0	0
$703$	−28.7789	−1.08542
$704$	4.74119	0.178690
$705$	0	0
$706$	−28.5501	−1.07450
$707$	20.3546	0.765515
$708$	0	0
$709$	25.3365	0.951530	0.475765	−	0.879572i	$-0.342171\pi$
$709$	25.3365	0.951530	0.475765	+	0.879572i	$0.342171\pi$
$710$	26.1184	0.980206
$711$	0	0
$712$	−8.08800	−0.303110
$713$	−16.9204	−0.633675
$714$	0	0
$715$	−13.5584	−0.507054
$716$	−9.04126	−0.337888
$717$	0	0
$718$	−1.51560	−0.0565617
$719$	−12.4378	−0.463853	−0.231926	−	0.972733i	$-0.574503\pi$
$719$	−12.4378	−0.463853	−0.231926	+	0.972733i	$0.574503\pi$
$720$	0	0
$721$	−13.4062	−0.499273
$722$	−21.9057	−0.815245
$723$	0	0
$724$	14.2580	0.529894
$725$	0.885100	0.0328718
$726$	0	0
$727$	29.4778	1.09327	0.546636	−	0.837370i	$-0.315908\pi$
$727$	29.4778	1.09327	0.546636	+	0.837370i	$0.315908\pi$
$728$	2.39435	0.0887407
$729$	0	0
$730$	33.4149	1.23674
$731$	−6.39046	−0.236360
$732$	0	0
$733$	16.0193	0.591686	0.295843	−	0.955237i	$-0.404400\pi$
$733$	16.0193	0.591686	0.295843	+	0.955237i	$0.404400\pi$
$734$	3.26419	0.120483
$735$	0	0
$736$	3.82143	0.140860
$737$	−24.5786	−0.905365
$738$	0	0
$739$	−50.0856	−1.84243	−0.921214	−	0.389056i	$-0.872801\pi$
$739$	−50.0856	−1.84243	−0.921214	+	0.389056i	$0.872801\pi$
$740$	−10.8230	−0.397862
$741$	0	0
$742$	−16.1596	−0.593236
$743$	34.5932	1.26910	0.634550	−	0.772882i	$-0.281186\pi$
$743$	34.5932	1.26910	0.634550	+	0.772882i	$0.281186\pi$
$744$	0	0
$745$	−7.71532	−0.282667
$746$	19.3552	0.708643
$747$	0	0
$748$	13.4953	0.493437
$749$	26.2263	0.958289
$750$	0	0
$751$	4.49127	0.163889	0.0819443	−	0.996637i	$-0.473887\pi$
$751$	4.49127	0.163889	0.0819443	+	0.996637i	$0.473887\pi$
$752$	−3.85599	−0.140614
$753$	0	0
$754$	1.33984	0.0487942
$755$	−25.7473	−0.937041
$756$	0	0
$757$	−42.4682	−1.54353	−0.771767	−	0.635906i	$-0.780627\pi$
$757$	−42.4682	−1.54353	−0.771767	+	0.635906i	$0.780627\pi$
$758$	−12.7890	−0.464516
$759$	0	0
$760$	−15.3836	−0.558023
$761$	−31.2837	−1.13403	−0.567017	−	0.823706i	$-0.691902\pi$
$761$	−31.2837	−1.13403	−0.567017	+	0.823706i	$0.691902\pi$
$762$	0	0
$763$	−33.2529	−1.20383
$764$	−3.78584	−0.136967
$765$	0	0
$766$	−17.4215	−0.629465
$767$	−9.59189	−0.346343
$768$	0	0
$769$	6.55761	0.236473	0.118237	−	0.992985i	$-0.462276\pi$
$769$	6.55761	0.236473	0.118237	+	0.992985i	$0.462276\pi$
$770$	−22.9662	−0.827643
$771$	0	0
$772$	1.43542	0.0516617
$773$	52.5611	1.89049	0.945246	−	0.326360i	$-0.105822\pi$
$773$	52.5611	1.89049	0.945246	+	0.326360i	$0.105822\pi$
$774$	0	0
$775$	3.47758	0.124918
$776$	−2.17296	−0.0780047
$777$	0	0
$778$	−20.2383	−0.725577
$779$	79.1714	2.83661
$780$	0	0
$781$	−51.4834	−1.84222
$782$	10.8773	0.388972
$783$	0	0
$784$	−2.94426	−0.105152
$785$	−38.3815	−1.36990
$786$	0	0
$787$	6.76198	0.241038	0.120519	−	0.992711i	$-0.461544\pi$
$787$	6.76198	0.241038	0.120519	+	0.992711i	$0.461544\pi$
$788$	9.87791	0.351886
$789$	0	0
$790$	3.60956	0.128422
$791$	−27.4222	−0.975021
$792$	0	0
$793$	0.894323	0.0317584
$794$	−35.6524	−1.26526
$795$	0	0
$796$	3.75481	0.133086
$797$	−21.7630	−0.770884	−0.385442	−	0.922732i	$-0.625951\pi$
$797$	−21.7630	−0.770884	−0.385442	+	0.922732i	$0.625951\pi$
$798$	0	0
$799$	−10.9757	−0.388292
$800$	−0.785402	−0.0277681
$801$	0	0
$802$	−27.5517	−0.972885
$803$	−65.8658	−2.32435
$804$	0	0
$805$	−18.5109	−0.652424
$806$	5.26428	0.185426
$807$	0	0
$808$	−10.1071	−0.355568
$809$	−42.8621	−1.50695	−0.753476	−	0.657476i	$-0.771624\pi$
$809$	−42.8621	−1.50695	−0.753476	+	0.657476i	$0.771624\pi$
$810$	0	0
$811$	−35.0823	−1.23191	−0.615954	−	0.787782i	$-0.711229\pi$
$811$	−35.0823	−1.23191	−0.615954	+	0.787782i	$0.711229\pi$
$812$	2.26953	0.0796448
$813$	0	0
$814$	21.3338	0.747750
$815$	−25.0986	−0.879165
$816$	0	0
$817$	−14.3591	−0.502362
$818$	−3.65303	−0.127725
$819$	0	0
$820$	29.7744	1.03977
$821$	−27.8203	−0.970935	−0.485467	−	0.874255i	$-0.661351\pi$
$821$	−27.8203	−0.970935	−0.485467	+	0.874255i	$0.661351\pi$
$822$	0	0
$823$	−54.8522	−1.91203	−0.956014	−	0.293321i	$-0.905239\pi$
$823$	−54.8522	−1.91203	−0.956014	+	0.293321i	$0.905239\pi$
$824$	6.65688	0.231903
$825$	0	0
$826$	−16.2475	−0.565322
$827$	−4.18301	−0.145457	−0.0727287	−	0.997352i	$-0.523171\pi$
$827$	−4.18301	−0.145457	−0.0727287	+	0.997352i	$0.523171\pi$
$828$	0	0
$829$	14.9795	0.520261	0.260130	−	0.965574i	$-0.416234\pi$
$829$	14.9795	0.520261	0.260130	+	0.965574i	$0.416234\pi$
$830$	−6.17371	−0.214293
$831$	0	0
$832$	−1.18892	−0.0412185
$833$	−8.38055	−0.290369
$834$	0	0
$835$	−50.2237	−1.73806
$836$	30.3235	1.04876
$837$	0	0
$838$	−28.4301	−0.982102
$839$	−24.9957	−0.862947	−0.431473	−	0.902126i	$-0.642006\pi$
$839$	−24.9957	−0.862947	−0.431473	+	0.902126i	$0.642006\pi$
$840$	0	0
$841$	−27.7300	−0.956207
$842$	−22.7754	−0.784894
$843$	0	0
$844$	−17.6476	−0.607454
$845$	−27.8688	−0.958715
$846$	0	0
$847$	23.1171	0.794312
$848$	8.02407	0.275548
$849$	0	0
$850$	−2.23557	−0.0766793
$851$	17.1952	0.589445
$852$	0	0
$853$	−47.3594	−1.62156	−0.810778	−	0.585354i	$-0.800956\pi$
$853$	−47.3594	−1.62156	−0.810778	+	0.585354i	$0.800956\pi$
$854$	1.51487	0.0518379
$855$	0	0
$856$	−13.0227	−0.445108
$857$	−5.11464	−0.174713	−0.0873564	−	0.996177i	$-0.527842\pi$
$857$	−5.11464	−0.174713	−0.0873564	+	0.996177i	$0.527842\pi$
$858$	0	0
$859$	−16.8719	−0.575661	−0.287831	−	0.957681i	$-0.592934\pi$
$859$	−16.8719	−0.575661	−0.287831	+	0.957681i	$0.592934\pi$
$860$	−5.40012	−0.184142
$861$	0	0
$862$	21.2326	0.723186
$863$	19.1439	0.651665	0.325833	−	0.945428i	$-0.394355\pi$
$863$	19.1439	0.651665	0.325833	+	0.945428i	$0.394355\pi$
$864$	0	0
$865$	−38.7645	−1.31803
$866$	−30.5653	−1.03865
$867$	0	0
$868$	8.91703	0.302664
$869$	−7.11499	−0.241359
$870$	0	0
$871$	6.16345	0.208840
$872$	16.5118	0.559160
$873$	0	0
$874$	24.4410	0.826728
$875$	−20.4154	−0.690166
$876$	0	0
$877$	8.33682	0.281514	0.140757	−	0.990044i	$-0.455046\pi$
$877$	8.33682	0.281514	0.140757	+	0.990044i	$0.455046\pi$
$878$	18.8453	0.635998
$879$	0	0
$880$	11.4039	0.384426
$881$	32.7135	1.10215	0.551074	−	0.834457i	$-0.314218\pi$
$881$	32.7135	1.10215	0.551074	+	0.834457i	$0.314218\pi$
$882$	0	0
$883$	−17.0448	−0.573604	−0.286802	−	0.957990i	$-0.592592\pi$
$883$	−17.0448	−0.573604	−0.286802	+	0.957990i	$0.592592\pi$
$884$	−3.38415	−0.113821
$885$	0	0
$886$	6.37575	0.214197
$887$	−14.5117	−0.487256	−0.243628	−	0.969869i	$-0.578338\pi$
$887$	−14.5117	−0.487256	−0.243628	+	0.969869i	$0.578338\pi$
$888$	0	0
$889$	16.9313	0.567857
$890$	−19.4539	−0.652098
$891$	0	0
$892$	1.00000	0.0334825
$893$	−24.6620	−0.825282
$894$	0	0
$895$	−21.7468	−0.726916
$896$	−2.01389	−0.0672792
$897$	0	0
$898$	23.2926	0.777285
$899$	4.98983	0.166420
$900$	0	0
$901$	22.8397	0.760900
$902$	−58.6899	−1.95416
$903$	0	0
$904$	13.6166	0.452880
$905$	34.2946	1.13999
$906$	0	0
$907$	−14.2526	−0.473251	−0.236625	−	0.971601i	$-0.576041\pi$
$907$	−14.2526	−0.473251	−0.236625	+	0.971601i	$0.576041\pi$
$908$	5.69750	0.189078
$909$	0	0
$910$	5.75911	0.190913
$911$	2.43816	0.0807800	0.0403900	−	0.999184i	$-0.487140\pi$
$911$	2.43816	0.0807800	0.0403900	+	0.999184i	$0.487140\pi$
$912$	0	0
$913$	12.1693	0.402746
$914$	−26.7034	−0.883271
$915$	0	0
$916$	−26.0839	−0.861836
$917$	25.0715	0.827935
$918$	0	0
$919$	12.4074	0.409284	0.204642	−	0.978837i	$-0.434397\pi$
$919$	12.4074	0.409284	0.204642	+	0.978837i	$0.434397\pi$
$920$	9.19164	0.303039
$921$	0	0
$922$	−1.37642	−0.0453300
$923$	12.9102	0.424945
$924$	0	0
$925$	−3.53406	−0.116199
$926$	9.04789	0.297332
$927$	0	0
$928$	−1.12694	−0.0369936
$929$	−36.3335	−1.19206	−0.596031	−	0.802961i	$-0.703256\pi$
$929$	−36.3335	−1.19206	−0.596031	+	0.802961i	$0.703256\pi$
$930$	0	0
$931$	−18.8308	−0.617154
$932$	25.2515	0.827141
$933$	0	0
$934$	−10.7756	−0.352590
$935$	32.4601	1.06156
$936$	0	0
$937$	−23.2332	−0.758997	−0.379498	−	0.925192i	$-0.623903\pi$
$937$	−23.2332	−0.758997	−0.379498	+	0.925192i	$0.623903\pi$
$938$	10.4401	0.340882
$939$	0	0
$940$	−9.27477	−0.302510
$941$	5.96353	0.194406	0.0972028	−	0.995265i	$-0.469010\pi$
$941$	5.96353	0.194406	0.0972028	+	0.995265i	$0.469010\pi$
$942$	0	0
$943$	−47.3045	−1.54045
$944$	8.06772	0.262582
$945$	0	0
$946$	10.6444	0.346081
$947$	−29.5857	−0.961407	−0.480703	−	0.876883i	$-0.659619\pi$
$947$	−29.5857	−0.961407	−0.480703	+	0.876883i	$0.659619\pi$
$948$	0	0
$949$	16.5168	0.536158
$950$	−5.02324	−0.162975
$951$	0	0
$952$	−5.73232	−0.185786
$953$	3.89630	0.126214	0.0631068	−	0.998007i	$-0.479899\pi$
$953$	3.89630	0.126214	0.0631068	+	0.998007i	$0.479899\pi$
$954$	0	0
$955$	−9.10603	−0.294664
$956$	7.89413	0.255314
$957$	0	0
$958$	30.3795	0.981518
$959$	−8.52805	−0.275385
$960$	0	0
$961$	−11.3948	−0.367575
$962$	−5.34977	−0.172484
$963$	0	0
$964$	27.7248	0.892955
$965$	3.45258	0.111143
$966$	0	0
$967$	−18.2682	−0.587466	−0.293733	−	0.955888i	$-0.594898\pi$
$967$	−18.2682	−0.587466	−0.293733	+	0.955888i	$0.594898\pi$
$968$	−11.4788	−0.368944
$969$	0	0
$970$	−5.22659	−0.167816
$971$	2.71632	0.0871709	0.0435854	−	0.999050i	$-0.486122\pi$
$971$	2.71632	0.0871709	0.0435854	+	0.999050i	$0.486122\pi$
$972$	0	0
$973$	35.9664	1.15303
$974$	−2.38443	−0.0764020
$975$	0	0
$976$	−0.752213	−0.0240778
$977$	20.2415	0.647582	0.323791	−	0.946129i	$-0.395043\pi$
$977$	20.2415	0.647582	0.323791	+	0.946129i	$0.395043\pi$
$978$	0	0
$979$	38.3467	1.22557
$980$	−7.08180	−0.226220
$981$	0	0
$982$	31.8464	1.01626
$983$	−56.9307	−1.81581	−0.907903	−	0.419180i	$-0.862318\pi$
$983$	−56.9307	−1.81581	−0.907903	+	0.419180i	$0.862318\pi$
$984$	0	0
$985$	23.7592	0.757031
$986$	−3.20772	−0.102155
$987$	0	0
$988$	−7.60406	−0.241917
$989$	8.57951	0.272813
$990$	0	0
$991$	1.60887	0.0511073	0.0255537	−	0.999673i	$-0.491865\pi$
$991$	1.60887	0.0511073	0.0255537	+	0.999673i	$0.491865\pi$
$992$	−4.42777	−0.140582
$993$	0	0
$994$	21.8683	0.693620
$995$	9.03138	0.286314
$996$	0	0
$997$	−15.9624	−0.505535	−0.252768	−	0.967527i	$-0.581341\pi$
$997$	−15.9624	−0.505535	−0.252768	+	0.967527i	$0.581341\pi$
$998$	14.0800	0.445695
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4014.2.a.x.1.6	✓	8
3.2	odd	2		4014.2.a.y.1.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4014.2.a.x.1.6	✓	8	1.1	even	1	trivial
4014.2.a.y.1.3	yes	8	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4014.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.40529 q^{5} +2.01389 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.40529 q^{5} +2.01389 q^{7} -1.00000 q^{8} -2.40529 q^{10} +4.74119 q^{11} -1.18892 q^{13} -2.01389 q^{14} +1.00000 q^{16} +2.84640 q^{17} +6.39576 q^{19} +2.40529 q^{20} -4.74119 q^{22} -3.82143 q^{23} +0.785402 q^{25} +1.18892 q^{26} +2.01389 q^{28} +1.12694 q^{29} +4.42777 q^{31} -1.00000 q^{32} -2.84640 q^{34} +4.84397 q^{35} -4.49968 q^{37} -6.39576 q^{38} -2.40529 q^{40} +12.3787 q^{41} -2.24510 q^{43} +4.74119 q^{44} +3.82143 q^{46} -3.85599 q^{47} -2.94426 q^{49} -0.785402 q^{50} -1.18892 q^{52} +8.02407 q^{53} +11.4039 q^{55} -2.01389 q^{56} -1.12694 q^{58} +8.06772 q^{59} -0.752213 q^{61} -4.42777 q^{62} +1.00000 q^{64} -2.85970 q^{65} -5.18406 q^{67} +2.84640 q^{68} -4.84397 q^{70} -10.8588 q^{71} -13.8923 q^{73} +4.49968 q^{74} +6.39576 q^{76} +9.54821 q^{77} -1.50068 q^{79} +2.40529 q^{80} -12.3787 q^{82} +2.56673 q^{83} +6.84640 q^{85} +2.24510 q^{86} -4.74119 q^{88} +8.08800 q^{89} -2.39435 q^{91} -3.82143 q^{92} +3.85599 q^{94} +15.3836 q^{95} +2.17296 q^{97} +2.94426 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 8 q^{4} + 6 q^{5} - 6 q^{7} - 8 q^{8}+O(q^{10}) \) 8 * q - 8 * q^2 + 8 * q^4 + 6 * q^5 - 6 * q^7 - 8 * q^8	\( 8 q - 8 q^{2} + 8 q^{4} + 6 q^{5} - 6 q^{7} - 8 q^{8} - 6 q^{10} + 11 q^{11} - q^{13} + 6 q^{14} + 8 q^{16} + 16 q^{17} - q^{19} + 6 q^{20} - 11 q^{22} + 14 q^{23} + 10 q^{25} + q^{26} - 6 q^{28} + 21 q^{29} - 6 q^{31} - 8 q^{32} - 16 q^{34} + 8 q^{35} - 14 q^{37} + q^{38} - 6 q^{40} + 16 q^{41} - 29 q^{43} + 11 q^{44} - 14 q^{46} + 9 q^{47} - 2 q^{49} - 10 q^{50} - q^{52} + 11 q^{53} - 22 q^{55} + 6 q^{56} - 21 q^{58} + 21 q^{59} + 3 q^{61} + 6 q^{62} + 8 q^{64} + 24 q^{65} - 20 q^{67} + 16 q^{68} - 8 q^{70} + 32 q^{71} + 13 q^{73} + 14 q^{74} - q^{76} + 4 q^{77} + 21 q^{79} + 6 q^{80} - 16 q^{82} + 28 q^{83} - 14 q^{85} + 29 q^{86} - 11 q^{88} + 54 q^{89} - 36 q^{91} + 14 q^{92} - 9 q^{94} + 30 q^{95} + 10 q^{97} + 2 q^{98}+O(q^{100}) \) 8 * q - 8 * q^2 + 8 * q^4 + 6 * q^5 - 6 * q^7 - 8 * q^8 - 6 * q^10 + 11 * q^11 - q^13 + 6 * q^14 + 8 * q^16 + 16 * q^17 - q^19 + 6 * q^20 - 11 * q^22 + 14 * q^23 + 10 * q^25 + q^26 - 6 * q^28 + 21 * q^29 - 6 * q^31 - 8 * q^32 - 16 * q^34 + 8 * q^35 - 14 * q^37 + q^38 - 6 * q^40 + 16 * q^41 - 29 * q^43 + 11 * q^44 - 14 * q^46 + 9 * q^47 - 2 * q^49 - 10 * q^50 - q^52 + 11 * q^53 - 22 * q^55 + 6 * q^56 - 21 * q^58 + 21 * q^59 + 3 * q^61 + 6 * q^62 + 8 * q^64 + 24 * q^65 - 20 * q^67 + 16 * q^68 - 8 * q^70 + 32 * q^71 + 13 * q^73 + 14 * q^74 - q^76 + 4 * q^77 + 21 * q^79 + 6 * q^80 - 16 * q^82 + 28 * q^83 - 14 * q^85 + 29 * q^86 - 11 * q^88 + 54 * q^89 - 36 * q^91 + 14 * q^92 - 9 * q^94 + 30 * q^95 + 10 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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