LMFDB - Embedded newform 3016.2.a.i.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3016,2,Mod(1,3016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3016, 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("3016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3016 = 2^{3} \cdot 13 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3016.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:	$24.0828812496$
Analytic rank:	$1$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 5x^{10} - 9x^{9} + 65x^{8} + 19x^{7} - 298x^{6} + 17x^{5} + 541x^{4} - 60x^{3} - 287x^{2} + 16x + 16 $ x^11 - 5x^10 - 9x^9 + 65x^8 + 19x^7 - 298x^6 + 17x^5 + 541x^4 - 60x^3 - 287x^2 + 16x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.820253$ of defining polynomial
Character	$\chi$	$=$	3016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.82025 q^{3} +1.95827 q^{5} -3.06441 q^{7} +0.313322 q^{9} +O(q^{10})$	$q-1.82025 q^{3} +1.95827 q^{5} -3.06441 q^{7} +0.313322 q^{9} -1.72679 q^{11} +1.00000 q^{13} -3.56454 q^{15} +4.39466 q^{17} +5.69706 q^{19} +5.57801 q^{21} -7.84294 q^{23} -1.16520 q^{25} +4.89043 q^{27} -1.00000 q^{29} +6.22509 q^{31} +3.14320 q^{33} -6.00094 q^{35} +6.19789 q^{37} -1.82025 q^{39} -1.68643 q^{41} +0.296799 q^{43} +0.613569 q^{45} -9.08016 q^{47} +2.39063 q^{49} -7.99940 q^{51} +10.8622 q^{53} -3.38152 q^{55} -10.3701 q^{57} -13.6326 q^{59} -10.9698 q^{61} -0.960150 q^{63} +1.95827 q^{65} -4.19784 q^{67} +14.2761 q^{69} +4.40460 q^{71} +4.21587 q^{73} +2.12095 q^{75} +5.29161 q^{77} -5.34401 q^{79} -9.84180 q^{81} -8.57508 q^{83} +8.60592 q^{85} +1.82025 q^{87} -15.1463 q^{89} -3.06441 q^{91} -11.3312 q^{93} +11.1564 q^{95} -12.0902 q^{97} -0.541043 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q - 6 q^{3} - 2 q^{5} - 3 q^{7} + 11 q^{9}+O(q^{10}) $ 11 * q - 6 * q^3 - 2 * q^5 - 3 * q^7 + 11 * q^9	$ 11 q - 6 q^{3} - 2 q^{5} - 3 q^{7} + 11 q^{9} - 13 q^{11} + 11 q^{13} - 8 q^{15} - 6 q^{17} - 12 q^{19} + q^{21} - 13 q^{23} + 11 q^{25} - 24 q^{27} - 11 q^{29} - 11 q^{31} + 17 q^{33} - 4 q^{35} + 11 q^{37} - 6 q^{39} - 9 q^{41} - 30 q^{43} - 16 q^{45} - q^{47} - 4 q^{49} - 13 q^{51} - 9 q^{53} - q^{55} + 2 q^{57} - 9 q^{59} - 5 q^{61} - 6 q^{63} - 2 q^{65} - 25 q^{67} - 26 q^{71} + 10 q^{73} - 41 q^{75} - 8 q^{77} - 14 q^{79} + 3 q^{81} - 6 q^{83} + 19 q^{85} + 6 q^{87} - 11 q^{89} - 3 q^{91} - 3 q^{93} - 31 q^{95} + 12 q^{97} - 35 q^{99}+O(q^{100}) $ 11 * q - 6 * q^3 - 2 * q^5 - 3 * q^7 + 11 * q^9 - 13 * q^11 + 11 * q^13 - 8 * q^15 - 6 * q^17 - 12 * q^19 + q^21 - 13 * q^23 + 11 * q^25 - 24 * q^27 - 11 * q^29 - 11 * q^31 + 17 * q^33 - 4 * q^35 + 11 * q^37 - 6 * q^39 - 9 * q^41 - 30 * q^43 - 16 * q^45 - q^47 - 4 * q^49 - 13 * q^51 - 9 * q^53 - q^55 + 2 * q^57 - 9 * q^59 - 5 * q^61 - 6 * q^63 - 2 * q^65 - 25 * q^67 - 26 * q^71 + 10 * q^73 - 41 * q^75 - 8 * q^77 - 14 * q^79 + 3 * q^81 - 6 * q^83 + 19 * q^85 + 6 * q^87 - 11 * q^89 - 3 * q^91 - 3 * q^93 - 31 * q^95 + 12 * q^97 - 35 * 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
$3$	−1.82025	−1.05092	−0.525462	−	0.850817i	$-0.676107\pi$
$3$	−1.82025	−1.05092	−0.525462	+	0.850817i	$0.676107\pi$
$4$	0	0
$5$	1.95827	0.875763	0.437881	−	0.899033i	$-0.355729\pi$
$5$	1.95827	0.875763	0.437881	+	0.899033i	$0.355729\pi$
$6$	0	0
$7$	−3.06441	−1.15824	−0.579120	−	0.815242i	$-0.696604\pi$
$7$	−3.06441	−1.15824	−0.579120	+	0.815242i	$0.696604\pi$
$8$	0	0
$9$	0.313322	0.104441
$10$	0	0
$11$	−1.72679	−0.520647	−0.260324	−	0.965521i	$-0.583829\pi$
$11$	−1.72679	−0.520647	−0.260324	+	0.965521i	$0.583829\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−3.56454	−0.920360
$16$	0	0
$17$	4.39466	1.06586	0.532931	−	0.846159i	$-0.321090\pi$
$17$	4.39466	1.06586	0.532931	+	0.846159i	$0.321090\pi$
$18$	0	0
$19$	5.69706	1.30699	0.653497	−	0.756929i	$-0.273301\pi$
$19$	5.69706	1.30699	0.653497	+	0.756929i	$0.273301\pi$
$20$	0	0
$21$	5.57801	1.21722
$22$	0	0
$23$	−7.84294	−1.63537	−0.817683	−	0.575668i	$-0.804742\pi$
$23$	−7.84294	−1.63537	−0.817683	+	0.575668i	$0.804742\pi$
$24$	0	0
$25$	−1.16520	−0.233039
$26$	0	0
$27$	4.89043	0.941164
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	6.22509	1.11806	0.559029	−	0.829148i	$-0.311174\pi$
$31$	6.22509	1.11806	0.559029	+	0.829148i	$0.311174\pi$
$32$	0	0
$33$	3.14320	0.547161
$34$	0	0
$35$	−6.00094	−1.01434
$36$	0	0
$37$	6.19789	1.01893	0.509464	−	0.860492i	$-0.329844\pi$
$37$	6.19789	1.01893	0.509464	+	0.860492i	$0.329844\pi$
$38$	0	0
$39$	−1.82025	−0.291474
$40$	0	0
$41$	−1.68643	−0.263376	−0.131688	−	0.991291i	$-0.542040\pi$
$41$	−1.68643	−0.263376	−0.131688	+	0.991291i	$0.542040\pi$
$42$	0	0
$43$	0.296799	0.0452615	0.0226307	−	0.999744i	$-0.492796\pi$
$43$	0.296799	0.0452615	0.0226307	+	0.999744i	$0.492796\pi$
$44$	0	0
$45$	0.613569	0.0914654
$46$	0	0
$47$	−9.08016	−1.32448	−0.662239	−	0.749293i	$-0.730394\pi$
$47$	−9.08016	−1.32448	−0.662239	+	0.749293i	$0.730394\pi$
$48$	0	0
$49$	2.39063	0.341519
$50$	0	0
$51$	−7.99940	−1.12014
$52$	0	0
$53$	10.8622	1.49204	0.746021	−	0.665922i	$-0.231962\pi$
$53$	10.8622	1.49204	0.746021	+	0.665922i	$0.231962\pi$
$54$	0	0
$55$	−3.38152	−0.455964
$56$	0	0
$57$	−10.3701	−1.37355
$58$	0	0
$59$	−13.6326	−1.77482	−0.887410	−	0.460982i	$-0.847497\pi$
$59$	−13.6326	−1.77482	−0.887410	+	0.460982i	$0.847497\pi$
$60$	0	0
$61$	−10.9698	−1.40454	−0.702272	−	0.711908i	$-0.747831\pi$
$61$	−10.9698	−1.40454	−0.702272	+	0.711908i	$0.747831\pi$
$62$	0	0
$63$	−0.960150	−0.120968
$64$	0	0
$65$	1.95827	0.242893
$66$	0	0
$67$	−4.19784	−0.512848	−0.256424	−	0.966564i	$-0.582544\pi$
$67$	−4.19784	−0.512848	−0.256424	+	0.966564i	$0.582544\pi$
$68$	0	0
$69$	14.2761	1.71865
$70$	0	0
$71$	4.40460	0.522730	0.261365	−	0.965240i	$-0.415827\pi$
$71$	4.40460	0.522730	0.261365	+	0.965240i	$0.415827\pi$
$72$	0	0
$73$	4.21587	0.493430	0.246715	−	0.969088i	$-0.420649\pi$
$73$	4.21587	0.493430	0.246715	+	0.969088i	$0.420649\pi$
$74$	0	0
$75$	2.12095	0.244906
$76$	0	0
$77$	5.29161	0.603034
$78$	0	0
$79$	−5.34401	−0.601248	−0.300624	−	0.953743i	$-0.597195\pi$
$79$	−5.34401	−0.601248	−0.300624	+	0.953743i	$0.597195\pi$
$80$	0	0
$81$	−9.84180	−1.09353
$82$	0	0
$83$	−8.57508	−0.941238	−0.470619	−	0.882337i	$-0.655969\pi$
$83$	−8.57508	−0.941238	−0.470619	+	0.882337i	$0.655969\pi$
$84$	0	0
$85$	8.60592	0.933443
$86$	0	0
$87$	1.82025	0.195152
$88$	0	0
$89$	−15.1463	−1.60550	−0.802751	−	0.596315i	$-0.796631\pi$
$89$	−15.1463	−1.60550	−0.802751	+	0.596315i	$0.796631\pi$
$90$	0	0
$91$	−3.06441	−0.321238
$92$	0	0
$93$	−11.3312	−1.17499
$94$	0	0
$95$	11.1564	1.14462
$96$	0	0
$97$	−12.0902	−1.22758	−0.613789	−	0.789470i	$-0.710356\pi$
$97$	−12.0902	−1.22758	−0.613789	+	0.789470i	$0.710356\pi$
$98$	0	0
$99$	−0.541043	−0.0543768
$100$	0	0
$101$	4.86258	0.483845	0.241922	−	0.970296i	$-0.422222\pi$
$101$	4.86258	0.483845	0.241922	+	0.970296i	$0.422222\pi$
$102$	0	0
$103$	10.8900	1.07302	0.536511	−	0.843893i	$-0.319742\pi$
$103$	10.8900	1.07302	0.536511	+	0.843893i	$0.319742\pi$
$104$	0	0
$105$	10.9232	1.06600
$106$	0	0
$107$	2.75430	0.266268	0.133134	−	0.991098i	$-0.457496\pi$
$107$	2.75430	0.266268	0.133134	+	0.991098i	$0.457496\pi$
$108$	0	0
$109$	1.75758	0.168345	0.0841727	−	0.996451i	$-0.473175\pi$
$109$	1.75758	0.168345	0.0841727	+	0.996451i	$0.473175\pi$
$110$	0	0
$111$	−11.2817	−1.07081
$112$	0	0
$113$	4.87693	0.458783	0.229392	−	0.973334i	$-0.426326\pi$
$113$	4.87693	0.458783	0.229392	+	0.973334i	$0.426326\pi$
$114$	0	0
$115$	−15.3586	−1.43219
$116$	0	0
$117$	0.313322	0.0289667
$118$	0	0
$119$	−13.4671	−1.23452
$120$	0	0
$121$	−8.01819	−0.728926
$122$	0	0
$123$	3.06973	0.276788
$124$	0	0
$125$	−12.0731	−1.07985
$126$	0	0
$127$	−6.34584	−0.563102	−0.281551	−	0.959546i	$-0.590849\pi$
$127$	−6.34584	−0.563102	−0.281551	+	0.959546i	$0.590849\pi$
$128$	0	0
$129$	−0.540250	−0.0475664
$130$	0	0
$131$	4.85101	0.423835	0.211917	−	0.977288i	$-0.432029\pi$
$131$	4.85101	0.423835	0.211917	+	0.977288i	$0.432029\pi$
$132$	0	0
$133$	−17.4581	−1.51381
$134$	0	0
$135$	9.57677	0.824237
$136$	0	0
$137$	13.8850	1.18627	0.593136	−	0.805102i	$-0.297890\pi$
$137$	13.8850	1.18627	0.593136	+	0.805102i	$0.297890\pi$
$138$	0	0
$139$	0.470397	0.0398986	0.0199493	−	0.999801i	$-0.493650\pi$
$139$	0.470397	0.0398986	0.0199493	+	0.999801i	$0.493650\pi$
$140$	0	0
$141$	16.5282	1.39192
$142$	0	0
$143$	−1.72679	−0.144402
$144$	0	0
$145$	−1.95827	−0.162625
$146$	0	0
$147$	−4.35156	−0.358911
$148$	0	0
$149$	2.65017	0.217111	0.108555	−	0.994090i	$-0.465378\pi$
$149$	2.65017	0.217111	0.108555	+	0.994090i	$0.465378\pi$
$150$	0	0
$151$	−19.9726	−1.62535	−0.812673	−	0.582720i	$-0.801988\pi$
$151$	−19.9726	−1.62535	−0.812673	+	0.582720i	$0.801988\pi$
$152$	0	0
$153$	1.37695	0.111320
$154$	0	0
$155$	12.1904	0.979154
$156$	0	0
$157$	−15.2668	−1.21843	−0.609214	−	0.793006i	$-0.708515\pi$
$157$	−15.2668	−1.21843	−0.609214	+	0.793006i	$0.708515\pi$
$158$	0	0
$159$	−19.7720	−1.56802
$160$	0	0
$161$	24.0340	1.89415
$162$	0	0
$163$	23.4389	1.83587	0.917937	−	0.396727i	$-0.129854\pi$
$163$	23.4389	1.83587	0.917937	+	0.396727i	$0.129854\pi$
$164$	0	0
$165$	6.15522	0.479183
$166$	0	0
$167$	−4.36530	−0.337797	−0.168898	−	0.985633i	$-0.554021\pi$
$167$	−4.36530	−0.337797	−0.168898	+	0.985633i	$0.554021\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	1.78502	0.136504
$172$	0	0
$173$	−16.0363	−1.21922	−0.609610	−	0.792702i	$-0.708674\pi$
$173$	−16.0363	−1.21922	−0.609610	+	0.792702i	$0.708674\pi$
$174$	0	0
$175$	3.57064	0.269915
$176$	0	0
$177$	24.8149	1.86520
$178$	0	0
$179$	−19.6691	−1.47014	−0.735070	−	0.677991i	$-0.762851\pi$
$179$	−19.6691	−1.47014	−0.735070	+	0.677991i	$0.762851\pi$
$180$	0	0
$181$	10.0862	0.749700	0.374850	−	0.927085i	$-0.377694\pi$
$181$	10.0862	0.749700	0.374850	+	0.927085i	$0.377694\pi$
$182$	0	0
$183$	19.9679	1.47607
$184$	0	0
$185$	12.1371	0.892339
$186$	0	0
$187$	−7.58867	−0.554938
$188$	0	0
$189$	−14.9863	−1.09009
$190$	0	0
$191$	−14.7710	−1.06879	−0.534396	−	0.845234i	$-0.679461\pi$
$191$	−14.7710	−1.06879	−0.534396	+	0.845234i	$0.679461\pi$
$192$	0	0
$193$	−18.5099	−1.33237	−0.666186	−	0.745786i	$-0.732074\pi$
$193$	−18.5099	−1.33237	−0.666186	+	0.745786i	$0.732074\pi$
$194$	0	0
$195$	−3.56454	−0.255262
$196$	0	0
$197$	−4.63352	−0.330125	−0.165062	−	0.986283i	$-0.552783\pi$
$197$	−4.63352	−0.330125	−0.165062	+	0.986283i	$0.552783\pi$
$198$	0	0
$199$	5.10731	0.362048	0.181024	−	0.983479i	$-0.442059\pi$
$199$	5.10731	0.362048	0.181024	+	0.983479i	$0.442059\pi$
$200$	0	0
$201$	7.64114	0.538964
$202$	0	0
$203$	3.06441	0.215080
$204$	0	0
$205$	−3.30248	−0.230655
$206$	0	0
$207$	−2.45737	−0.170799
$208$	0	0
$209$	−9.83763	−0.680483
$210$	0	0
$211$	−28.2932	−1.94778	−0.973892	−	0.227011i	$-0.927105\pi$
$211$	−28.2932	−1.94778	−0.973892	+	0.227011i	$0.927105\pi$
$212$	0	0
$213$	−8.01749	−0.549349
$214$	0	0
$215$	0.581212	0.0396383
$216$	0	0
$217$	−19.0762	−1.29498
$218$	0	0
$219$	−7.67396	−0.518558
$220$	0	0
$221$	4.39466	0.295617
$222$	0	0
$223$	2.31412	0.154965	0.0774823	−	0.996994i	$-0.475312\pi$
$223$	2.31412	0.154965	0.0774823	+	0.996994i	$0.475312\pi$
$224$	0	0
$225$	−0.365082	−0.0243388
$226$	0	0
$227$	11.3833	0.755539	0.377770	−	0.925900i	$-0.376691\pi$
$227$	11.3833	0.755539	0.377770	+	0.925900i	$0.376691\pi$
$228$	0	0
$229$	−14.5462	−0.961238	−0.480619	−	0.876930i	$-0.659588\pi$
$229$	−14.5462	−0.961238	−0.480619	+	0.876930i	$0.659588\pi$
$230$	0	0
$231$	−9.63206	−0.633743
$232$	0	0
$233$	0.360181	0.0235962	0.0117981	−	0.999930i	$-0.496244\pi$
$233$	0.360181	0.0235962	0.0117981	+	0.999930i	$0.496244\pi$
$234$	0	0
$235$	−17.7814	−1.15993
$236$	0	0
$237$	9.72745	0.631866
$238$	0	0
$239$	1.34134	0.0867642	0.0433821	−	0.999059i	$-0.486187\pi$
$239$	1.34134	0.0867642	0.0433821	+	0.999059i	$0.486187\pi$
$240$	0	0
$241$	−8.76508	−0.564609	−0.282304	−	0.959325i	$-0.591099\pi$
$241$	−8.76508	−0.564609	−0.282304	+	0.959325i	$0.591099\pi$
$242$	0	0
$243$	3.24326	0.208055
$244$	0	0
$245$	4.68150	0.299090
$246$	0	0
$247$	5.69706	0.362495
$248$	0	0
$249$	15.6088	0.989169
$250$	0	0
$251$	−5.71619	−0.360802	−0.180401	−	0.983593i	$-0.557740\pi$
$251$	−5.71619	−0.360802	−0.180401	+	0.983593i	$0.557740\pi$
$252$	0	0
$253$	13.5431	0.851449
$254$	0	0
$255$	−15.6649	−0.980977
$256$	0	0
$257$	−6.58324	−0.410651	−0.205326	−	0.978694i	$-0.565825\pi$
$257$	−6.58324	−0.410651	−0.205326	+	0.978694i	$0.565825\pi$
$258$	0	0
$259$	−18.9929	−1.18016
$260$	0	0
$261$	−0.313322	−0.0193942
$262$	0	0
$263$	12.3569	0.761959	0.380980	−	0.924583i	$-0.375587\pi$
$263$	12.3569	0.761959	0.380980	+	0.924583i	$0.375587\pi$
$264$	0	0
$265$	21.2711	1.30668
$266$	0	0
$267$	27.5701	1.68726
$268$	0	0
$269$	−19.6983	−1.20102	−0.600512	−	0.799616i	$-0.705037\pi$
$269$	−19.6983	−1.20102	−0.600512	+	0.799616i	$0.705037\pi$
$270$	0	0
$271$	5.85191	0.355478	0.177739	−	0.984078i	$-0.443122\pi$
$271$	5.85191	0.355478	0.177739	+	0.984078i	$0.443122\pi$
$272$	0	0
$273$	5.57801	0.337597
$274$	0	0
$275$	2.01205	0.121331
$276$	0	0
$277$	6.27322	0.376922	0.188461	−	0.982081i	$-0.439650\pi$
$277$	6.27322	0.376922	0.188461	+	0.982081i	$0.439650\pi$
$278$	0	0
$279$	1.95046	0.116771
$280$	0	0
$281$	−14.2933	−0.852669	−0.426335	−	0.904565i	$-0.640195\pi$
$281$	−14.2933	−0.852669	−0.426335	+	0.904565i	$0.640195\pi$
$282$	0	0
$283$	−30.1084	−1.78976	−0.894881	−	0.446305i	$-0.852740\pi$
$283$	−30.1084	−1.78976	−0.894881	+	0.446305i	$0.852740\pi$
$284$	0	0
$285$	−20.3074	−1.20291
$286$	0	0
$287$	5.16792	0.305053
$288$	0	0
$289$	2.31306	0.136062
$290$	0	0
$291$	22.0073	1.29009
$292$	0	0
$293$	−24.7219	−1.44427	−0.722136	−	0.691751i	$-0.756839\pi$
$293$	−24.7219	−1.44427	−0.722136	+	0.691751i	$0.756839\pi$
$294$	0	0
$295$	−26.6963	−1.55432
$296$	0	0
$297$	−8.44476	−0.490015
$298$	0	0
$299$	−7.84294	−0.453569
$300$	0	0
$301$	−0.909517	−0.0524237
$302$	0	0
$303$	−8.85113	−0.508484
$304$	0	0
$305$	−21.4819	−1.23005
$306$	0	0
$307$	−25.9924	−1.48347	−0.741733	−	0.670695i	$-0.765996\pi$
$307$	−25.9924	−1.48347	−0.741733	+	0.670695i	$0.765996\pi$
$308$	0	0
$309$	−19.8225	−1.12766
$310$	0	0
$311$	23.8285	1.35119	0.675596	−	0.737272i	$-0.263886\pi$
$311$	23.8285	1.35119	0.675596	+	0.737272i	$0.263886\pi$
$312$	0	0
$313$	−2.83541	−0.160267	−0.0801334	−	0.996784i	$-0.525535\pi$
$313$	−2.83541	−0.160267	−0.0801334	+	0.996784i	$0.525535\pi$
$314$	0	0
$315$	−1.88023	−0.105939
$316$	0	0
$317$	−18.7421	−1.05266	−0.526330	−	0.850280i	$-0.676432\pi$
$317$	−18.7421	−1.05266	−0.526330	+	0.850280i	$0.676432\pi$
$318$	0	0
$319$	1.72679	0.0966818
$320$	0	0
$321$	−5.01352	−0.279827
$322$	0	0
$323$	25.0366	1.39308
$324$	0	0
$325$	−1.16520	−0.0646335
$326$	0	0
$327$	−3.19924	−0.176918
$328$	0	0
$329$	27.8254	1.53406
$330$	0	0
$331$	15.9894	0.878859	0.439430	−	0.898277i	$-0.355181\pi$
$331$	15.9894	0.878859	0.439430	+	0.898277i	$0.355181\pi$
$332$	0	0
$333$	1.94194	0.106418
$334$	0	0
$335$	−8.22049	−0.449134
$336$	0	0
$337$	12.9335	0.704533	0.352267	−	0.935900i	$-0.385411\pi$
$337$	12.9335	0.704533	0.352267	+	0.935900i	$0.385411\pi$
$338$	0	0
$339$	−8.87725	−0.482146
$340$	0	0
$341$	−10.7494	−0.582114
$342$	0	0
$343$	14.1250	0.762679
$344$	0	0
$345$	27.9565	1.50513
$346$	0	0
$347$	31.9198	1.71355	0.856773	−	0.515693i	$-0.172466\pi$
$347$	31.9198	1.71355	0.856773	+	0.515693i	$0.172466\pi$
$348$	0	0
$349$	−17.5444	−0.939129	−0.469565	−	0.882898i	$-0.655589\pi$
$349$	−17.5444	−0.939129	−0.469565	+	0.882898i	$0.655589\pi$
$350$	0	0
$351$	4.89043	0.261032
$352$	0	0
$353$	7.34871	0.391132	0.195566	−	0.980691i	$-0.437346\pi$
$353$	7.34871	0.391132	0.195566	+	0.980691i	$0.437346\pi$
$354$	0	0
$355$	8.62537	0.457787
$356$	0	0
$357$	24.5135	1.29739
$358$	0	0
$359$	−3.32747	−0.175617	−0.0878086	−	0.996137i	$-0.527986\pi$
$359$	−3.32747	−0.175617	−0.0878086	+	0.996137i	$0.527986\pi$
$360$	0	0
$361$	13.4565	0.708235
$362$	0	0
$363$	14.5951	0.766046
$364$	0	0
$365$	8.25580	0.432128
$366$	0	0
$367$	19.2767	1.00623	0.503117	−	0.864218i	$-0.332186\pi$
$367$	19.2767	1.00623	0.503117	+	0.864218i	$0.332186\pi$
$368$	0	0
$369$	−0.528397	−0.0275072
$370$	0	0
$371$	−33.2864	−1.72814
$372$	0	0
$373$	29.7999	1.54298	0.771491	−	0.636241i	$-0.219511\pi$
$373$	29.7999	1.54298	0.771491	+	0.636241i	$0.219511\pi$
$374$	0	0
$375$	21.9761	1.13484
$376$	0	0
$377$	−1.00000	−0.0515026
$378$	0	0
$379$	−1.10217	−0.0566145	−0.0283073	−	0.999599i	$-0.509012\pi$
$379$	−1.10217	−0.0566145	−0.0283073	+	0.999599i	$0.509012\pi$
$380$	0	0
$381$	11.5510	0.591778
$382$	0	0
$383$	−7.67519	−0.392184	−0.196092	−	0.980586i	$-0.562825\pi$
$383$	−7.67519	−0.392184	−0.196092	+	0.980586i	$0.562825\pi$
$384$	0	0
$385$	10.3624	0.528115
$386$	0	0
$387$	0.0929939	0.00472715
$388$	0	0
$389$	−16.1649	−0.819593	−0.409796	−	0.912177i	$-0.634400\pi$
$389$	−16.1649	−0.819593	−0.409796	+	0.912177i	$0.634400\pi$
$390$	0	0
$391$	−34.4671	−1.74308
$392$	0	0
$393$	−8.83007	−0.445418
$394$	0	0
$395$	−10.4650	−0.526551
$396$	0	0
$397$	−16.0384	−0.804942	−0.402471	−	0.915433i	$-0.631849\pi$
$397$	−16.0384	−0.804942	−0.402471	+	0.915433i	$0.631849\pi$
$398$	0	0
$399$	31.7782	1.59090
$400$	0	0
$401$	31.5935	1.57770	0.788851	−	0.614584i	$-0.210676\pi$
$401$	31.5935	1.57770	0.788851	+	0.614584i	$0.210676\pi$
$402$	0	0
$403$	6.22509	0.310094
$404$	0	0
$405$	−19.2729	−0.957676
$406$	0	0
$407$	−10.7025	−0.530502
$408$	0	0
$409$	8.42460	0.416570	0.208285	−	0.978068i	$-0.433212\pi$
$409$	8.42460	0.416570	0.208285	+	0.978068i	$0.433212\pi$
$410$	0	0
$411$	−25.2741	−1.24668
$412$	0	0
$413$	41.7761	2.05567
$414$	0	0
$415$	−16.7923	−0.824301
$416$	0	0
$417$	−0.856242	−0.0419303
$418$	0	0
$419$	−38.3193	−1.87202	−0.936010	−	0.351974i	$-0.885510\pi$
$419$	−38.3193	−1.87202	−0.936010	+	0.351974i	$0.885510\pi$
$420$	0	0
$421$	9.73460	0.474436	0.237218	−	0.971456i	$-0.423764\pi$
$421$	9.73460	0.474436	0.237218	+	0.971456i	$0.423764\pi$
$422$	0	0
$423$	−2.84502	−0.138330
$424$	0	0
$425$	−5.12064	−0.248388
$426$	0	0
$427$	33.6162	1.62680
$428$	0	0
$429$	3.14320	0.151755
$430$	0	0
$431$	−20.9399	−1.00864	−0.504320	−	0.863517i	$-0.668257\pi$
$431$	−20.9399	−1.00864	−0.504320	+	0.863517i	$0.668257\pi$
$432$	0	0
$433$	27.8506	1.33842	0.669208	−	0.743075i	$-0.266633\pi$
$433$	27.8506	1.33842	0.669208	+	0.743075i	$0.266633\pi$
$434$	0	0
$435$	3.56454	0.170907
$436$	0	0
$437$	−44.6817	−2.13742
$438$	0	0
$439$	36.0013	1.71825	0.859124	−	0.511767i	$-0.171009\pi$
$439$	36.0013	1.71825	0.859124	+	0.511767i	$0.171009\pi$
$440$	0	0
$441$	0.749040	0.0356685
$442$	0	0
$443$	−31.2951	−1.48687	−0.743437	−	0.668806i	$-0.766806\pi$
$443$	−31.2951	−1.48687	−0.743437	+	0.668806i	$0.766806\pi$
$444$	0	0
$445$	−29.6604	−1.40604
$446$	0	0
$447$	−4.82398	−0.228167
$448$	0	0
$449$	−34.4143	−1.62411	−0.812055	−	0.583581i	$-0.801651\pi$
$449$	−34.4143	−1.62411	−0.812055	+	0.583581i	$0.801651\pi$
$450$	0	0
$451$	2.91211	0.137126
$452$	0	0
$453$	36.3552	1.70811
$454$	0	0
$455$	−6.00094	−0.281328
$456$	0	0
$457$	9.50403	0.444580	0.222290	−	0.974981i	$-0.428647\pi$
$457$	9.50403	0.444580	0.222290	+	0.974981i	$0.428647\pi$
$458$	0	0
$459$	21.4918	1.00315
$460$	0	0
$461$	29.5355	1.37561	0.687803	−	0.725898i	$-0.258575\pi$
$461$	29.5355	1.37561	0.687803	+	0.725898i	$0.258575\pi$
$462$	0	0
$463$	−0.572458	−0.0266044	−0.0133022	−	0.999912i	$-0.504234\pi$
$463$	−0.572458	−0.0266044	−0.0133022	+	0.999912i	$0.504234\pi$
$464$	0	0
$465$	−22.1896	−1.02902
$466$	0	0
$467$	35.3155	1.63420	0.817102	−	0.576493i	$-0.195579\pi$
$467$	35.3155	1.63420	0.817102	+	0.576493i	$0.195579\pi$
$468$	0	0
$469$	12.8639	0.594001
$470$	0	0
$471$	27.7895	1.28047
$472$	0	0
$473$	−0.512511	−0.0235653
$474$	0	0
$475$	−6.63819	−0.304581
$476$	0	0
$477$	3.40338	0.155830
$478$	0	0
$479$	19.5729	0.894309	0.447155	−	0.894457i	$-0.352437\pi$
$479$	19.5729	0.894309	0.447155	+	0.894457i	$0.352437\pi$
$480$	0	0
$481$	6.19789	0.282600
$482$	0	0
$483$	−43.7480	−1.99060
$484$	0	0
$485$	−23.6759	−1.07507
$486$	0	0
$487$	−24.2809	−1.10027	−0.550136	−	0.835075i	$-0.685424\pi$
$487$	−24.2809	−1.10027	−0.550136	+	0.835075i	$0.685424\pi$
$488$	0	0
$489$	−42.6647	−1.92936
$490$	0	0
$491$	−15.2609	−0.688715	−0.344358	−	0.938839i	$-0.611903\pi$
$491$	−15.2609	−0.688715	−0.344358	+	0.938839i	$0.611903\pi$
$492$	0	0
$493$	−4.39466	−0.197926
$494$	0	0
$495$	−1.05951	−0.0476212
$496$	0	0
$497$	−13.4975	−0.605446
$498$	0	0
$499$	38.3805	1.71815	0.859075	−	0.511850i	$-0.171040\pi$
$499$	38.3805	1.71815	0.859075	+	0.511850i	$0.171040\pi$
$500$	0	0
$501$	7.94595	0.354999
$502$	0	0
$503$	−0.583558	−0.0260196	−0.0130098	−	0.999915i	$-0.504141\pi$
$503$	−0.583558	−0.0260196	−0.0130098	+	0.999915i	$0.504141\pi$
$504$	0	0
$505$	9.52222	0.423733
$506$	0	0
$507$	−1.82025	−0.0808403
$508$	0	0
$509$	16.1431	0.715530	0.357765	−	0.933812i	$-0.383539\pi$
$509$	16.1431	0.715530	0.357765	+	0.933812i	$0.383539\pi$
$510$	0	0
$511$	−12.9192	−0.571511
$512$	0	0
$513$	27.8611	1.23010
$514$	0	0
$515$	21.3255	0.939713
$516$	0	0
$517$	15.6795	0.689586
$518$	0	0
$519$	29.1902	1.28131
$520$	0	0
$521$	−27.0119	−1.18341	−0.591706	−	0.806154i	$-0.701546\pi$
$521$	−27.0119	−1.18341	−0.591706	+	0.806154i	$0.701546\pi$
$522$	0	0
$523$	33.2266	1.45290	0.726448	−	0.687221i	$-0.241170\pi$
$523$	33.2266	1.45290	0.726448	+	0.687221i	$0.241170\pi$
$524$	0	0
$525$	−6.49948	−0.283660
$526$	0	0
$527$	27.3572	1.19170
$528$	0	0
$529$	38.5117	1.67442
$530$	0	0
$531$	−4.27141	−0.185364
$532$	0	0
$533$	−1.68643	−0.0730474
$534$	0	0
$535$	5.39365	0.233188
$536$	0	0
$537$	35.8028	1.54501
$538$	0	0
$539$	−4.12813	−0.177811
$540$	0	0
$541$	34.6484	1.48965	0.744825	−	0.667260i	$-0.232533\pi$
$541$	34.6484	1.48965	0.744825	+	0.667260i	$0.232533\pi$
$542$	0	0
$543$	−18.3594	−0.787878
$544$	0	0
$545$	3.44180	0.147431
$546$	0	0
$547$	−25.7388	−1.10051	−0.550255	−	0.834997i	$-0.685469\pi$
$547$	−25.7388	−1.10051	−0.550255	+	0.834997i	$0.685469\pi$
$548$	0	0
$549$	−3.43710	−0.146692
$550$	0	0
$551$	−5.69706	−0.242703
$552$	0	0
$553$	16.3763	0.696389
$554$	0	0
$555$	−22.0926	−0.937780
$556$	0	0
$557$	8.85411	0.375161	0.187580	−	0.982249i	$-0.439935\pi$
$557$	8.85411	0.375161	0.187580	+	0.982249i	$0.439935\pi$
$558$	0	0
$559$	0.296799	0.0125533
$560$	0	0
$561$	13.8133	0.583198
$562$	0	0
$563$	6.46667	0.272537	0.136269	−	0.990672i	$-0.456489\pi$
$563$	6.46667	0.272537	0.136269	+	0.990672i	$0.456489\pi$
$564$	0	0
$565$	9.55033	0.401785
$566$	0	0
$567$	30.1593	1.26657
$568$	0	0
$569$	11.3259	0.474805	0.237403	−	0.971411i	$-0.423704\pi$
$569$	11.3259	0.474805	0.237403	+	0.971411i	$0.423704\pi$
$570$	0	0
$571$	−7.53325	−0.315257	−0.157628	−	0.987499i	$-0.550385\pi$
$571$	−7.53325	−0.315257	−0.157628	+	0.987499i	$0.550385\pi$
$572$	0	0
$573$	26.8870	1.12322
$574$	0	0
$575$	9.13857	0.381105
$576$	0	0
$577$	−7.97561	−0.332029	−0.166014	−	0.986123i	$-0.553090\pi$
$577$	−7.97561	−0.332029	−0.166014	+	0.986123i	$0.553090\pi$
$578$	0	0
$579$	33.6927	1.40022
$580$	0	0
$581$	26.2776	1.09018
$582$	0	0
$583$	−18.7568	−0.776828
$584$	0	0
$585$	0.613569	0.0253679
$586$	0	0
$587$	0.750536	0.0309780	0.0154890	−	0.999880i	$-0.495070\pi$
$587$	0.750536	0.0309780	0.0154890	+	0.999880i	$0.495070\pi$
$588$	0	0
$589$	35.4647	1.46130
$590$	0	0
$591$	8.43419	0.346936
$592$	0	0
$593$	18.0575	0.741532	0.370766	−	0.928726i	$-0.379095\pi$
$593$	18.0575	0.741532	0.370766	+	0.928726i	$0.379095\pi$
$594$	0	0
$595$	−26.3721	−1.08115
$596$	0	0
$597$	−9.29660	−0.380484
$598$	0	0
$599$	16.1989	0.661871	0.330935	−	0.943653i	$-0.392636\pi$
$599$	16.1989	0.661871	0.330935	+	0.943653i	$0.392636\pi$
$600$	0	0
$601$	−39.5825	−1.61460	−0.807301	−	0.590139i	$-0.799073\pi$
$601$	−39.5825	−1.61460	−0.807301	+	0.590139i	$0.799073\pi$
$602$	0	0
$603$	−1.31528	−0.0535623
$604$	0	0
$605$	−15.7017	−0.638367
$606$	0	0
$607$	34.4816	1.39957	0.699783	−	0.714355i	$-0.253280\pi$
$607$	34.4816	1.39957	0.699783	+	0.714355i	$0.253280\pi$
$608$	0	0
$609$	−5.57801	−0.226032
$610$	0	0
$611$	−9.08016	−0.367344
$612$	0	0
$613$	−19.7005	−0.795697	−0.397848	−	0.917451i	$-0.630243\pi$
$613$	−19.7005	−0.795697	−0.397848	+	0.917451i	$0.630243\pi$
$614$	0	0
$615$	6.01135	0.242401
$616$	0	0
$617$	43.4104	1.74764	0.873819	−	0.486252i	$-0.161636\pi$
$617$	43.4104	1.74764	0.873819	+	0.486252i	$0.161636\pi$
$618$	0	0
$619$	−0.0177227	−0.000712337 0	−0.000356169	−	1.00000i	$-0.500113\pi$
$619$	−0.0177227	−0.000712337 0	−0.000356169		1.00000i	$0.500113\pi$
$620$	0	0
$621$	−38.3554	−1.53915
$622$	0	0
$623$	46.4145	1.85956
$624$	0	0
$625$	−17.8163	−0.712653
$626$	0	0
$627$	17.9070	0.715136
$628$	0	0
$629$	27.2376	1.08604
$630$	0	0
$631$	1.68409	0.0670424	0.0335212	−	0.999438i	$-0.489328\pi$
$631$	1.68409	0.0670424	0.0335212	+	0.999438i	$0.489328\pi$
$632$	0	0
$633$	51.5008	2.04697
$634$	0	0
$635$	−12.4268	−0.493144
$636$	0	0
$637$	2.39063	0.0947204
$638$	0	0
$639$	1.38006	0.0545943
$640$	0	0
$641$	11.4863	0.453680	0.226840	−	0.973932i	$-0.427160\pi$
$641$	11.4863	0.453680	0.226840	+	0.973932i	$0.427160\pi$
$642$	0	0
$643$	16.8357	0.663935	0.331967	−	0.943291i	$-0.392288\pi$
$643$	16.8357	0.663935	0.331967	+	0.943291i	$0.392288\pi$
$644$	0	0
$645$	−1.05795	−0.0416569
$646$	0	0
$647$	25.0219	0.983713	0.491856	−	0.870676i	$-0.336318\pi$
$647$	25.0219	0.983713	0.491856	+	0.870676i	$0.336318\pi$
$648$	0	0
$649$	23.5407	0.924055
$650$	0	0
$651$	34.7236	1.36092
$652$	0	0
$653$	−43.1461	−1.68844	−0.844219	−	0.535998i	$-0.819935\pi$
$653$	−43.1461	−1.68844	−0.844219	+	0.535998i	$0.819935\pi$
$654$	0	0
$655$	9.49957	0.371179
$656$	0	0
$657$	1.32093	0.0515343
$658$	0	0
$659$	19.1675	0.746660	0.373330	−	0.927699i	$-0.378216\pi$
$659$	19.1675	0.746660	0.373330	+	0.927699i	$0.378216\pi$
$660$	0	0
$661$	48.5296	1.88758	0.943791	−	0.330544i	$-0.107232\pi$
$661$	48.5296	1.88758	0.943791	+	0.330544i	$0.107232\pi$
$662$	0	0
$663$	−7.99940	−0.310671
$664$	0	0
$665$	−34.1877	−1.32574
$666$	0	0
$667$	7.84294	0.303680
$668$	0	0
$669$	−4.21228	−0.162856
$670$	0	0
$671$	18.9426	0.731273
$672$	0	0
$673$	1.02944	0.0396820	0.0198410	−	0.999803i	$-0.493684\pi$
$673$	1.02944	0.0396820	0.0198410	+	0.999803i	$0.493684\pi$
$674$	0	0
$675$	−5.69831	−0.219328
$676$	0	0
$677$	4.98344	0.191529	0.0957645	−	0.995404i	$-0.469470\pi$
$677$	4.98344	0.191529	0.0957645	+	0.995404i	$0.469470\pi$
$678$	0	0
$679$	37.0495	1.42183
$680$	0	0
$681$	−20.7206	−0.794014
$682$	0	0
$683$	−17.7732	−0.680073	−0.340036	−	0.940412i	$-0.610439\pi$
$683$	−17.7732	−0.680073	−0.340036	+	0.940412i	$0.610439\pi$
$684$	0	0
$685$	27.1904	1.03889
$686$	0	0
$687$	26.4777	1.01019
$688$	0	0
$689$	10.8622	0.413818
$690$	0	0
$691$	6.69117	0.254544	0.127272	−	0.991868i	$-0.459378\pi$
$691$	6.69117	0.254544	0.127272	+	0.991868i	$0.459378\pi$
$692$	0	0
$693$	1.65798	0.0629814
$694$	0	0
$695$	0.921162	0.0349417
$696$	0	0
$697$	−7.41129	−0.280723
$698$	0	0
$699$	−0.655620	−0.0247978
$700$	0	0
$701$	24.7275	0.933945	0.466973	−	0.884272i	$-0.345345\pi$
$701$	24.7275	0.933945	0.466973	+	0.884272i	$0.345345\pi$
$702$	0	0
$703$	35.3097	1.33173
$704$	0	0
$705$	32.3666	1.21900
$706$	0	0
$707$	−14.9010	−0.560408
$708$	0	0
$709$	−0.378330	−0.0142085	−0.00710425	−	0.999975i	$-0.502261\pi$
$709$	−0.378330	−0.0142085	−0.00710425	+	0.999975i	$0.502261\pi$
$710$	0	0
$711$	−1.67440	−0.0627948
$712$	0	0
$713$	−48.8230	−1.82844
$714$	0	0
$715$	−3.38152	−0.126462
$716$	0	0
$717$	−2.44158	−0.0911825
$718$	0	0
$719$	−9.38570	−0.350027	−0.175014	−	0.984566i	$-0.555997\pi$
$719$	−9.38570	−0.350027	−0.175014	+	0.984566i	$0.555997\pi$
$720$	0	0
$721$	−33.3714	−1.24282
$722$	0	0
$723$	15.9547	0.593361
$724$	0	0
$725$	1.16520	0.0432743
$726$	0	0
$727$	18.5717	0.688787	0.344394	−	0.938825i	$-0.388085\pi$
$727$	18.5717	0.688787	0.344394	+	0.938825i	$0.388085\pi$
$728$	0	0
$729$	23.6218	0.874883
$730$	0	0
$731$	1.30433	0.0482425
$732$	0	0
$733$	9.00990	0.332788	0.166394	−	0.986059i	$-0.446788\pi$
$733$	9.00990	0.332788	0.166394	+	0.986059i	$0.446788\pi$
$734$	0	0
$735$	−8.52151	−0.314321
$736$	0	0
$737$	7.24880	0.267013
$738$	0	0
$739$	−33.6523	−1.23792	−0.618960	−	0.785423i	$-0.712446\pi$
$739$	−33.6523	−1.23792	−0.618960	+	0.785423i	$0.712446\pi$
$740$	0	0
$741$	−10.3701	−0.380955
$742$	0	0
$743$	−30.4672	−1.11773	−0.558866	−	0.829258i	$-0.688764\pi$
$743$	−30.4672	−1.11773	−0.558866	+	0.829258i	$0.688764\pi$
$744$	0	0
$745$	5.18974	0.190137
$746$	0	0
$747$	−2.68677	−0.0983036
$748$	0	0
$749$	−8.44031	−0.308402
$750$	0	0
$751$	−38.7191	−1.41288	−0.706440	−	0.707773i	$-0.749700\pi$
$751$	−38.7191	−1.41288	−0.706440	+	0.707773i	$0.749700\pi$
$752$	0	0
$753$	10.4049	0.379176
$754$	0	0
$755$	−39.1116	−1.42342
$756$	0	0
$757$	21.5925	0.784795	0.392397	−	0.919796i	$-0.371646\pi$
$757$	21.5925	0.784795	0.392397	+	0.919796i	$0.371646\pi$
$758$	0	0
$759$	−24.6519	−0.894808
$760$	0	0
$761$	−41.8800	−1.51815	−0.759074	−	0.651004i	$-0.774348\pi$
$761$	−41.8800	−1.51815	−0.759074	+	0.651004i	$0.774348\pi$
$762$	0	0
$763$	−5.38595	−0.194984
$764$	0	0
$765$	2.69643	0.0974895
$766$	0	0
$767$	−13.6326	−0.492246
$768$	0	0
$769$	−3.00667	−0.108423	−0.0542116	−	0.998529i	$-0.517265\pi$
$769$	−3.00667	−0.108423	−0.0542116	+	0.998529i	$0.517265\pi$
$770$	0	0
$771$	11.9832	0.431563
$772$	0	0
$773$	−6.98189	−0.251121	−0.125561	−	0.992086i	$-0.540073\pi$
$773$	−6.98189	−0.251121	−0.125561	+	0.992086i	$0.540073\pi$
$774$	0	0
$775$	−7.25345	−0.260551
$776$	0	0
$777$	34.5719	1.24026
$778$	0	0
$779$	−9.60769	−0.344231
$780$	0	0
$781$	−7.60583	−0.272158
$782$	0	0
$783$	−4.89043	−0.174770
$784$	0	0
$785$	−29.8965	−1.06705
$786$	0	0
$787$	9.47159	0.337626	0.168813	−	0.985648i	$-0.446007\pi$
$787$	9.47159	0.337626	0.168813	+	0.985648i	$0.446007\pi$
$788$	0	0
$789$	−22.4927	−0.800761
$790$	0	0
$791$	−14.9449	−0.531381
$792$	0	0
$793$	−10.9698	−0.389551
$794$	0	0
$795$	−38.7189	−1.37322
$796$	0	0
$797$	−43.9874	−1.55811	−0.779057	−	0.626953i	$-0.784302\pi$
$797$	−43.9874	−1.55811	−0.779057	+	0.626953i	$0.784302\pi$
$798$	0	0
$799$	−39.9042	−1.41171
$800$	0	0
$801$	−4.74567	−0.167680
$802$	0	0
$803$	−7.27993	−0.256903
$804$	0	0
$805$	47.0650	1.65882
$806$	0	0
$807$	35.8558	1.26219
$808$	0	0
$809$	−45.7748	−1.60936	−0.804678	−	0.593712i	$-0.797662\pi$
$809$	−45.7748	−1.60936	−0.804678	+	0.593712i	$0.797662\pi$
$810$	0	0
$811$	−17.0097	−0.597293	−0.298646	−	0.954364i	$-0.596535\pi$
$811$	−17.0097	−0.597293	−0.298646	+	0.954364i	$0.596535\pi$
$812$	0	0
$813$	−10.6520	−0.373580
$814$	0	0
$815$	45.8995	1.60779
$816$	0	0
$817$	1.69088	0.0591565
$818$	0	0
$819$	−0.960150	−0.0335503
$820$	0	0
$821$	−7.21904	−0.251946	−0.125973	−	0.992034i	$-0.540205\pi$
$821$	−7.21904	−0.251946	−0.125973	+	0.992034i	$0.540205\pi$
$822$	0	0
$823$	−18.2508	−0.636182	−0.318091	−	0.948060i	$-0.603042\pi$
$823$	−18.2508	−0.636182	−0.318091	+	0.948060i	$0.603042\pi$
$824$	0	0
$825$	−3.66244	−0.127510
$826$	0	0
$827$	−9.09319	−0.316201	−0.158101	−	0.987423i	$-0.550537\pi$
$827$	−9.09319	−0.316201	−0.158101	+	0.987423i	$0.550537\pi$
$828$	0	0
$829$	−32.5549	−1.13068	−0.565339	−	0.824858i	$-0.691255\pi$
$829$	−32.5549	−1.13068	−0.565339	+	0.824858i	$0.691255\pi$
$830$	0	0
$831$	−11.4189	−0.396116
$832$	0	0
$833$	10.5060	0.364012
$834$	0	0
$835$	−8.54841	−0.295830
$836$	0	0
$837$	30.4434	1.05228
$838$	0	0
$839$	−40.5100	−1.39856	−0.699280	−	0.714848i	$-0.746496\pi$
$839$	−40.5100	−1.39856	−0.699280	+	0.714848i	$0.746496\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	26.0175	0.896091
$844$	0	0
$845$	1.95827	0.0673664
$846$	0	0
$847$	24.5711	0.844271
$848$	0	0
$849$	54.8050	1.88090
$850$	0	0
$851$	−48.6097	−1.66632
$852$	0	0
$853$	−41.0212	−1.40454	−0.702269	−	0.711911i	$-0.747830\pi$
$853$	−41.0212	−1.40454	−0.702269	+	0.711911i	$0.747830\pi$
$854$	0	0
$855$	3.49554	0.119545
$856$	0	0
$857$	−14.0856	−0.481155	−0.240578	−	0.970630i	$-0.577337\pi$
$857$	−14.0856	−0.481155	−0.240578	+	0.970630i	$0.577337\pi$
$858$	0	0
$859$	18.6597	0.636662	0.318331	−	0.947980i	$-0.396878\pi$
$859$	18.6597	0.636662	0.318331	+	0.947980i	$0.396878\pi$
$860$	0	0
$861$	−9.40693	−0.320587
$862$	0	0
$863$	52.1495	1.77519	0.887595	−	0.460624i	$-0.152374\pi$
$863$	52.1495	1.77519	0.887595	+	0.460624i	$0.152374\pi$
$864$	0	0
$865$	−31.4034	−1.06775
$866$	0	0
$867$	−4.21036	−0.142991
$868$	0	0
$869$	9.22799	0.313038
$870$	0	0
$871$	−4.19784	−0.142239
$872$	0	0
$873$	−3.78815	−0.128209
$874$	0	0
$875$	36.9970	1.25073
$876$	0	0
$877$	−4.00005	−0.135072	−0.0675361	−	0.997717i	$-0.521514\pi$
$877$	−4.00005	−0.135072	−0.0675361	+	0.997717i	$0.521514\pi$
$878$	0	0
$879$	45.0002	1.51782
$880$	0	0
$881$	−32.0930	−1.08124	−0.540620	−	0.841267i	$-0.681810\pi$
$881$	−32.0930	−1.08124	−0.540620	+	0.841267i	$0.681810\pi$
$882$	0	0
$883$	−27.1061	−0.912193	−0.456096	−	0.889930i	$-0.650753\pi$
$883$	−27.1061	−0.912193	−0.456096	+	0.889930i	$0.650753\pi$
$884$	0	0
$885$	48.5941	1.63347
$886$	0	0
$887$	−13.0378	−0.437766	−0.218883	−	0.975751i	$-0.570241\pi$
$887$	−13.0378	−0.437766	−0.218883	+	0.975751i	$0.570241\pi$
$888$	0	0
$889$	19.4463	0.652208
$890$	0	0
$891$	16.9947	0.569345
$892$	0	0
$893$	−51.7302	−1.73108
$894$	0	0
$895$	−38.5174	−1.28749
$896$	0	0
$897$	14.2761	0.476666
$898$	0	0
$899$	−6.22509	−0.207618
$900$	0	0
$901$	47.7358	1.59031
$902$	0	0
$903$	1.65555	0.0550933
$904$	0	0
$905$	19.7514	0.656560
$906$	0	0
$907$	−21.0512	−0.698994	−0.349497	−	0.936937i	$-0.613648\pi$
$907$	−21.0512	−0.698994	−0.349497	+	0.936937i	$0.613648\pi$
$908$	0	0
$909$	1.52356	0.0505331
$910$	0	0
$911$	−17.2881	−0.572779	−0.286390	−	0.958113i	$-0.592455\pi$
$911$	−17.2881	−0.572779	−0.286390	+	0.958113i	$0.592455\pi$
$912$	0	0
$913$	14.8074	0.490053
$914$	0	0
$915$	39.1025	1.29269
$916$	0	0
$917$	−14.8655	−0.490902
$918$	0	0
$919$	14.8244	0.489011	0.244506	−	0.969648i	$-0.421374\pi$
$919$	14.8244	0.489011	0.244506	+	0.969648i	$0.421374\pi$
$920$	0	0
$921$	47.3128	1.55901
$922$	0	0
$923$	4.40460	0.144979
$924$	0	0
$925$	−7.22176	−0.237450
$926$	0	0
$927$	3.41208	0.112067
$928$	0	0
$929$	44.8011	1.46988	0.734939	−	0.678134i	$-0.237211\pi$
$929$	44.8011	1.46988	0.734939	+	0.678134i	$0.237211\pi$
$930$	0	0
$931$	13.6196	0.446364
$932$	0	0
$933$	−43.3740	−1.42000
$934$	0	0
$935$	−14.8606	−0.485994
$936$	0	0
$937$	5.07973	0.165947	0.0829737	−	0.996552i	$-0.473558\pi$
$937$	5.07973	0.165947	0.0829737	+	0.996552i	$0.473558\pi$
$938$	0	0
$939$	5.16116	0.168428
$940$	0	0
$941$	−41.1077	−1.34007	−0.670036	−	0.742329i	$-0.733721\pi$
$941$	−41.1077	−1.34007	−0.670036	+	0.742329i	$0.733721\pi$
$942$	0	0
$943$	13.2266	0.430717
$944$	0	0
$945$	−29.3472	−0.954664
$946$	0	0
$947$	−28.0173	−0.910441	−0.455220	−	0.890379i	$-0.650439\pi$
$947$	−28.0173	−0.910441	−0.455220	+	0.890379i	$0.650439\pi$
$948$	0	0
$949$	4.21587	0.136853
$950$	0	0
$951$	34.1153	1.10627
$952$	0	0
$953$	34.9981	1.13370	0.566850	−	0.823821i	$-0.308162\pi$
$953$	34.9981	1.13370	0.566850	+	0.823821i	$0.308162\pi$
$954$	0	0
$955$	−28.9256	−0.936009
$956$	0	0
$957$	−3.14320	−0.101605
$958$	0	0
$959$	−42.5492	−1.37399
$960$	0	0
$961$	7.75169	0.250055
$962$	0	0
$963$	0.862983	0.0278092
$964$	0	0
$965$	−36.2473	−1.16684
$966$	0	0
$967$	−39.2791	−1.26313	−0.631565	−	0.775323i	$-0.717587\pi$
$967$	−39.2791	−1.26313	−0.631565	+	0.775323i	$0.717587\pi$
$968$	0	0
$969$	−45.5730	−1.46402
$970$	0	0
$971$	−48.6998	−1.56285	−0.781425	−	0.623999i	$-0.785507\pi$
$971$	−48.6998	−1.56285	−0.781425	+	0.623999i	$0.785507\pi$
$972$	0	0
$973$	−1.44149	−0.0462121
$974$	0	0
$975$	2.12095	0.0679248
$976$	0	0
$977$	−24.9920	−0.799565	−0.399782	−	0.916610i	$-0.630914\pi$
$977$	−24.9920	−0.799565	−0.399782	+	0.916610i	$0.630914\pi$
$978$	0	0
$979$	26.1545	0.835900
$980$	0	0
$981$	0.550689	0.0175821
$982$	0	0
$983$	52.9254	1.68806	0.844029	−	0.536298i	$-0.180178\pi$
$983$	52.9254	1.68806	0.844029	+	0.536298i	$0.180178\pi$
$984$	0	0
$985$	−9.07367	−0.289111
$986$	0	0
$987$	−50.6492	−1.61218
$988$	0	0
$989$	−2.32778	−0.0740191
$990$	0	0
$991$	56.5939	1.79777	0.898883	−	0.438189i	$-0.144380\pi$
$991$	56.5939	1.79777	0.898883	+	0.438189i	$0.144380\pi$
$992$	0	0
$993$	−29.1048	−0.923614
$994$	0	0
$995$	10.0015	0.317068
$996$	0	0
$997$	−30.5154	−0.966431	−0.483216	−	0.875501i	$-0.660531\pi$
$997$	−30.5154	−0.966431	−0.483216	+	0.875501i	$0.660531\pi$
$998$	0	0
$999$	30.3104	0.958978

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3016.2.a.i.1.4	✓	11
4.3	odd	2		6032.2.a.bc.1.8		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3016.2.a.i.1.4	✓	11	1.1	even	1	trivial
6032.2.a.bc.1.8		11	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:	\( N \)	\(=\)	\( 3016 = 2^{3} \cdot 13 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3016.a (trivial)

\(f(q)\)	\(=\)	\(q-1.82025 q^{3} +1.95827 q^{5} -3.06441 q^{7} +0.313322 q^{9} +O(q^{10})\)	\(q-1.82025 q^{3} +1.95827 q^{5} -3.06441 q^{7} +0.313322 q^{9} -1.72679 q^{11} +1.00000 q^{13} -3.56454 q^{15} +4.39466 q^{17} +5.69706 q^{19} +5.57801 q^{21} -7.84294 q^{23} -1.16520 q^{25} +4.89043 q^{27} -1.00000 q^{29} +6.22509 q^{31} +3.14320 q^{33} -6.00094 q^{35} +6.19789 q^{37} -1.82025 q^{39} -1.68643 q^{41} +0.296799 q^{43} +0.613569 q^{45} -9.08016 q^{47} +2.39063 q^{49} -7.99940 q^{51} +10.8622 q^{53} -3.38152 q^{55} -10.3701 q^{57} -13.6326 q^{59} -10.9698 q^{61} -0.960150 q^{63} +1.95827 q^{65} -4.19784 q^{67} +14.2761 q^{69} +4.40460 q^{71} +4.21587 q^{73} +2.12095 q^{75} +5.29161 q^{77} -5.34401 q^{79} -9.84180 q^{81} -8.57508 q^{83} +8.60592 q^{85} +1.82025 q^{87} -15.1463 q^{89} -3.06441 q^{91} -11.3312 q^{93} +11.1564 q^{95} -12.0902 q^{97} -0.541043 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 6 q^{3} - 2 q^{5} - 3 q^{7} + 11 q^{9}+O(q^{10}) \) 11 * q - 6 * q^3 - 2 * q^5 - 3 * q^7 + 11 * q^9	\( 11 q - 6 q^{3} - 2 q^{5} - 3 q^{7} + 11 q^{9} - 13 q^{11} + 11 q^{13} - 8 q^{15} - 6 q^{17} - 12 q^{19} + q^{21} - 13 q^{23} + 11 q^{25} - 24 q^{27} - 11 q^{29} - 11 q^{31} + 17 q^{33} - 4 q^{35} + 11 q^{37} - 6 q^{39} - 9 q^{41} - 30 q^{43} - 16 q^{45} - q^{47} - 4 q^{49} - 13 q^{51} - 9 q^{53} - q^{55} + 2 q^{57} - 9 q^{59} - 5 q^{61} - 6 q^{63} - 2 q^{65} - 25 q^{67} - 26 q^{71} + 10 q^{73} - 41 q^{75} - 8 q^{77} - 14 q^{79} + 3 q^{81} - 6 q^{83} + 19 q^{85} + 6 q^{87} - 11 q^{89} - 3 q^{91} - 3 q^{93} - 31 q^{95} + 12 q^{97} - 35 q^{99}+O(q^{100}) \) 11 * q - 6 * q^3 - 2 * q^5 - 3 * q^7 + 11 * q^9 - 13 * q^11 + 11 * q^13 - 8 * q^15 - 6 * q^17 - 12 * q^19 + q^21 - 13 * q^23 + 11 * q^25 - 24 * q^27 - 11 * q^29 - 11 * q^31 + 17 * q^33 - 4 * q^35 + 11 * q^37 - 6 * q^39 - 9 * q^41 - 30 * q^43 - 16 * q^45 - q^47 - 4 * q^49 - 13 * q^51 - 9 * q^53 - q^55 + 2 * q^57 - 9 * q^59 - 5 * q^61 - 6 * q^63 - 2 * q^65 - 25 * q^67 - 26 * q^71 + 10 * q^73 - 41 * q^75 - 8 * q^77 - 14 * q^79 + 3 * q^81 - 6 * q^83 + 19 * q^85 + 6 * q^87 - 11 * q^89 - 3 * q^91 - 3 * q^93 - 31 * q^95 + 12 * q^97 - 35 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more