LMFDB - Embedded newform 6016.2.a.m.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6016,2,Mod(1,6016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6016, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6016 = 2^{7} \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6016.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:	$48.0380018560$
Analytic rank:	$1$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 4 x^{12} - 22 x^{11} + 96 x^{10} + 163 x^{9} - 840 x^{8} - 438 x^{7} + 3358 x^{6} + 5 x^{5} + \cdots - 832 $ x^13 - 4x^12 - 22x^11 + 96x^10 + 163x^9 - 840x^8 - 438x^7 + 3358x^6 + 5x^5 - 6230x^4 + 1488x^3 + 4720x^2 - 1440x - 832
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.73608$ of defining polynomial
Character	$\chi$	$=$	6016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73608 q^{3} +3.07388 q^{5} -4.60716 q^{7} +0.0139658 q^{9} +O(q^{10})$	$q-1.73608 q^{3} +3.07388 q^{5} -4.60716 q^{7} +0.0139658 q^{9} +3.25201 q^{11} +3.13174 q^{13} -5.33650 q^{15} +1.98403 q^{17} -3.15860 q^{19} +7.99840 q^{21} -8.21953 q^{23} +4.44876 q^{25} +5.18399 q^{27} +1.32121 q^{29} +0.155177 q^{31} -5.64575 q^{33} -14.1619 q^{35} +0.122812 q^{37} -5.43694 q^{39} -6.73192 q^{41} +10.2888 q^{43} +0.0429292 q^{45} +1.00000 q^{47} +14.2260 q^{49} -3.44443 q^{51} -9.44846 q^{53} +9.99631 q^{55} +5.48358 q^{57} -14.5361 q^{59} +0.165968 q^{61} -0.0643427 q^{63} +9.62660 q^{65} -0.709327 q^{67} +14.2697 q^{69} +6.43089 q^{71} -13.5046 q^{73} -7.72340 q^{75} -14.9826 q^{77} +2.71846 q^{79} -9.04170 q^{81} +15.2501 q^{83} +6.09867 q^{85} -2.29372 q^{87} +11.6474 q^{89} -14.4284 q^{91} -0.269400 q^{93} -9.70918 q^{95} +2.82193 q^{97} +0.0454170 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 4 q^{3} - 6 q^{5} + 2 q^{7} + 21 q^{9}+O(q^{10}) $ 13 * q - 4 * q^3 - 6 * q^5 + 2 * q^7 + 21 * q^9	$ 13 q - 4 q^{3} - 6 q^{5} + 2 q^{7} + 21 q^{9} - 10 q^{11} - 4 q^{13} - 14 q^{15} + 10 q^{17} - 8 q^{19} - 10 q^{21} - 18 q^{23} + 23 q^{25} - 16 q^{27} - 14 q^{29} - 4 q^{31} + 14 q^{33} - 14 q^{35} - 16 q^{37} - 12 q^{39} + 10 q^{41} - 12 q^{43} - 10 q^{45} + 13 q^{47} + 9 q^{49} - 22 q^{51} - 26 q^{53} + 2 q^{55} + 20 q^{57} - 30 q^{59} - 18 q^{61} - 12 q^{63} - 4 q^{65} - 4 q^{67} - 2 q^{69} - 36 q^{71} + 10 q^{73} - 38 q^{75} - 42 q^{77} + 21 q^{81} - 12 q^{83} - 4 q^{85} - 6 q^{87} + 50 q^{89} + 4 q^{91} - 52 q^{93} - 8 q^{95} - 10 q^{97} - 34 q^{99}+O(q^{100}) $ 13 * q - 4 * q^3 - 6 * q^5 + 2 * q^7 + 21 * q^9 - 10 * q^11 - 4 * q^13 - 14 * q^15 + 10 * q^17 - 8 * q^19 - 10 * q^21 - 18 * q^23 + 23 * q^25 - 16 * q^27 - 14 * q^29 - 4 * q^31 + 14 * q^33 - 14 * q^35 - 16 * q^37 - 12 * q^39 + 10 * q^41 - 12 * q^43 - 10 * q^45 + 13 * q^47 + 9 * q^49 - 22 * q^51 - 26 * q^53 + 2 * q^55 + 20 * q^57 - 30 * q^59 - 18 * q^61 - 12 * q^63 - 4 * q^65 - 4 * q^67 - 2 * q^69 - 36 * q^71 + 10 * q^73 - 38 * q^75 - 42 * q^77 + 21 * q^81 - 12 * q^83 - 4 * q^85 - 6 * q^87 + 50 * q^89 + 4 * q^91 - 52 * q^93 - 8 * q^95 - 10 * q^97 - 34 * 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
$3$	−1.73608	−1.00232	−0.501162	−	0.865353i	$-0.667094\pi$
$3$	−1.73608	−1.00232	−0.501162	+	0.865353i	$0.667094\pi$
$4$	0	0
$5$	3.07388	1.37468	0.687341	−	0.726335i	$-0.258778\pi$
$5$	3.07388	1.37468	0.687341	+	0.726335i	$0.258778\pi$
$6$	0	0
$7$	−4.60716	−1.74134	−0.870672	−	0.491864i	$-0.836316\pi$
$7$	−4.60716	−1.74134	−0.870672	+	0.491864i	$0.836316\pi$
$8$	0	0
$9$	0.0139658	0.00465526
$10$	0	0
$11$	3.25201	0.980519	0.490260	−	0.871576i	$-0.336902\pi$
$11$	3.25201	0.980519	0.490260	+	0.871576i	$0.336902\pi$
$12$	0	0
$13$	3.13174	0.868588	0.434294	−	0.900771i	$-0.356998\pi$
$13$	3.13174	0.868588	0.434294	+	0.900771i	$0.356998\pi$
$14$	0	0
$15$	−5.33650	−1.37788
$16$	0	0
$17$	1.98403	0.481197	0.240599	−	0.970625i	$-0.422656\pi$
$17$	1.98403	0.481197	0.240599	+	0.970625i	$0.422656\pi$
$18$	0	0
$19$	−3.15860	−0.724634	−0.362317	−	0.932055i	$-0.618014\pi$
$19$	−3.15860	−0.724634	−0.362317	+	0.932055i	$0.618014\pi$
$20$	0	0
$21$	7.99840	1.74539
$22$	0	0
$23$	−8.21953	−1.71389	−0.856946	−	0.515407i	$-0.827641\pi$
$23$	−8.21953	−1.71389	−0.856946	+	0.515407i	$0.827641\pi$
$24$	0	0
$25$	4.44876	0.889753
$26$	0	0
$27$	5.18399	0.997659
$28$	0	0
$29$	1.32121	0.245343	0.122671	−	0.992447i	$-0.460854\pi$
$29$	1.32121	0.245343	0.122671	+	0.992447i	$0.460854\pi$
$30$	0	0
$31$	0.155177	0.0278706	0.0139353	−	0.999903i	$-0.495564\pi$
$31$	0.155177	0.0278706	0.0139353	+	0.999903i	$0.495564\pi$
$32$	0	0
$33$	−5.64575	−0.982799
$34$	0	0
$35$	−14.1619	−2.39380
$36$	0	0
$37$	0.122812	0.0201902	0.0100951	−	0.999949i	$-0.496787\pi$
$37$	0.122812	0.0201902	0.0100951	+	0.999949i	$0.496787\pi$
$38$	0	0
$39$	−5.43694	−0.870607
$40$	0	0
$41$	−6.73192	−1.05135	−0.525675	−	0.850686i	$-0.676187\pi$
$41$	−6.73192	−1.05135	−0.525675	+	0.850686i	$0.676187\pi$
$42$	0	0
$43$	10.2888	1.56903	0.784514	−	0.620111i	$-0.212912\pi$
$43$	10.2888	1.56903	0.784514	+	0.620111i	$0.212912\pi$
$44$	0	0
$45$	0.0429292	0.00639951
$46$	0	0
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	0	0
$49$	14.2260	2.03228
$50$	0	0
$51$	−3.44443	−0.482316
$52$	0	0
$53$	−9.44846	−1.29785	−0.648923	−	0.760854i	$-0.724780\pi$
$53$	−9.44846	−1.29785	−0.648923	+	0.760854i	$0.724780\pi$
$54$	0	0
$55$	9.99631	1.34790
$56$	0	0
$57$	5.48358	0.726318
$58$	0	0
$59$	−14.5361	−1.89244	−0.946222	−	0.323517i	$-0.895135\pi$
$59$	−14.5361	−1.89244	−0.946222	+	0.323517i	$0.895135\pi$
$60$	0	0
$61$	0.165968	0.0212500	0.0106250	−	0.999944i	$-0.496618\pi$
$61$	0.165968	0.0212500	0.0106250	+	0.999944i	$0.496618\pi$
$62$	0	0
$63$	−0.0643427	−0.00810642
$64$	0	0
$65$	9.62660	1.19403
$66$	0	0
$67$	−0.709327	−0.0866581	−0.0433291	−	0.999061i	$-0.513796\pi$
$67$	−0.709327	−0.0866581	−0.0433291	+	0.999061i	$0.513796\pi$
$68$	0	0
$69$	14.2697	1.71788
$70$	0	0
$71$	6.43089	0.763206	0.381603	−	0.924326i	$-0.375372\pi$
$71$	6.43089	0.763206	0.381603	+	0.924326i	$0.375372\pi$
$72$	0	0
$73$	−13.5046	−1.58060	−0.790299	−	0.612722i	$-0.790075\pi$
$73$	−13.5046	−1.58060	−0.790299	+	0.612722i	$0.790075\pi$
$74$	0	0
$75$	−7.72340	−0.891821
$76$	0	0
$77$	−14.9826	−1.70742
$78$	0	0
$79$	2.71846	0.305851	0.152925	−	0.988238i	$-0.451131\pi$
$79$	2.71846	0.305851	0.152925	+	0.988238i	$0.451131\pi$
$80$	0	0
$81$	−9.04170	−1.00463
$82$	0	0
$83$	15.2501	1.67391	0.836957	−	0.547268i	$-0.184332\pi$
$83$	15.2501	1.67391	0.836957	+	0.547268i	$0.184332\pi$
$84$	0	0
$85$	6.09867	0.661494
$86$	0	0
$87$	−2.29372	−0.245913
$88$	0	0
$89$	11.6474	1.23462	0.617311	−	0.786719i	$-0.288222\pi$
$89$	11.6474	1.23462	0.617311	+	0.786719i	$0.288222\pi$
$90$	0	0
$91$	−14.4284	−1.51251
$92$	0	0
$93$	−0.269400	−0.0279354
$94$	0	0
$95$	−9.70918	−0.996141
$96$	0	0
$97$	2.82193	0.286524	0.143262	−	0.989685i	$-0.454241\pi$
$97$	2.82193	0.286524	0.143262	+	0.989685i	$0.454241\pi$
$98$	0	0
$99$	0.0454170	0.00456458
$100$	0	0
$101$	8.97861	0.893405	0.446703	−	0.894683i	$-0.352598\pi$
$101$	8.97861	0.893405	0.446703	+	0.894683i	$0.352598\pi$
$102$	0	0
$103$	10.0164	0.986947	0.493474	−	0.869761i	$-0.335727\pi$
$103$	10.0164	0.986947	0.493474	+	0.869761i	$0.335727\pi$
$104$	0	0
$105$	24.5861	2.39936
$106$	0	0
$107$	9.24291	0.893546	0.446773	−	0.894647i	$-0.352573\pi$
$107$	9.24291	0.893546	0.446773	+	0.894647i	$0.352573\pi$
$108$	0	0
$109$	−3.46112	−0.331515	−0.165758	−	0.986167i	$-0.553007\pi$
$109$	−3.46112	−0.331515	−0.165758	+	0.986167i	$0.553007\pi$
$110$	0	0
$111$	−0.213211	−0.0202371
$112$	0	0
$113$	−9.51981	−0.895548	−0.447774	−	0.894147i	$-0.647783\pi$
$113$	−9.51981	−0.895548	−0.447774	+	0.894147i	$0.647783\pi$
$114$	0	0
$115$	−25.2659	−2.35606
$116$	0	0
$117$	0.0437372	0.00404351
$118$	0	0
$119$	−9.14074	−0.837930
$120$	0	0
$121$	−0.424405	−0.0385823
$122$	0	0
$123$	11.6871	1.05379
$124$	0	0
$125$	−1.69444	−0.151555
$126$	0	0
$127$	4.15758	0.368926	0.184463	−	0.982839i	$-0.440945\pi$
$127$	4.15758	0.368926	0.184463	+	0.982839i	$0.440945\pi$
$128$	0	0
$129$	−17.8622	−1.57268
$130$	0	0
$131$	−19.3061	−1.68678	−0.843391	−	0.537300i	$-0.819444\pi$
$131$	−19.3061	−1.68678	−0.843391	+	0.537300i	$0.819444\pi$
$132$	0	0
$133$	14.5522	1.26184
$134$	0	0
$135$	15.9350	1.37146
$136$	0	0
$137$	7.76257	0.663201	0.331601	−	0.943420i	$-0.392411\pi$
$137$	7.76257	0.663201	0.331601	+	0.943420i	$0.392411\pi$
$138$	0	0
$139$	−13.1398	−1.11450	−0.557251	−	0.830344i	$-0.688144\pi$
$139$	−13.1398	−1.11450	−0.557251	+	0.830344i	$0.688144\pi$
$140$	0	0
$141$	−1.73608	−0.146204
$142$	0	0
$143$	10.1845	0.851667
$144$	0	0
$145$	4.06125	0.337268
$146$	0	0
$147$	−24.6974	−2.03701
$148$	0	0
$149$	−16.9653	−1.38985	−0.694924	−	0.719083i	$-0.744562\pi$
$149$	−16.9653	−1.38985	−0.694924	+	0.719083i	$0.744562\pi$
$150$	0	0
$151$	−24.2794	−1.97583	−0.987914	−	0.155006i	$-0.950460\pi$
$151$	−24.2794	−1.97583	−0.987914	+	0.155006i	$0.950460\pi$
$152$	0	0
$153$	0.0277085	0.00224010
$154$	0	0
$155$	0.476996	0.0383133
$156$	0	0
$157$	−10.4430	−0.833439	−0.416720	−	0.909035i	$-0.636820\pi$
$157$	−10.4430	−0.833439	−0.416720	+	0.909035i	$0.636820\pi$
$158$	0	0
$159$	16.4033	1.30086
$160$	0	0
$161$	37.8687	2.98447
$162$	0	0
$163$	3.46132	0.271112	0.135556	−	0.990770i	$-0.456718\pi$
$163$	3.46132	0.271112	0.135556	+	0.990770i	$0.456718\pi$
$164$	0	0
$165$	−17.3544	−1.35104
$166$	0	0
$167$	−22.3535	−1.72977	−0.864883	−	0.501973i	$-0.832608\pi$
$167$	−22.3535	−1.72977	−0.864883	+	0.501973i	$0.832608\pi$
$168$	0	0
$169$	−3.19222	−0.245555
$170$	0	0
$171$	−0.0441124	−0.00337336
$172$	0	0
$173$	7.68171	0.584030	0.292015	−	0.956414i	$-0.405674\pi$
$173$	7.68171	0.584030	0.292015	+	0.956414i	$0.405674\pi$
$174$	0	0
$175$	−20.4962	−1.54937
$176$	0	0
$177$	25.2359	1.89684
$178$	0	0
$179$	−17.1009	−1.27818	−0.639091	−	0.769131i	$-0.720689\pi$
$179$	−17.1009	−1.27818	−0.639091	+	0.769131i	$0.720689\pi$
$180$	0	0
$181$	−7.53867	−0.560345	−0.280173	−	0.959950i	$-0.590392\pi$
$181$	−7.53867	−0.560345	−0.280173	+	0.959950i	$0.590392\pi$
$182$	0	0
$183$	−0.288133	−0.0212994
$184$	0	0
$185$	0.377510	0.0277551
$186$	0	0
$187$	6.45208	0.471823
$188$	0	0
$189$	−23.8835	−1.73727
$190$	0	0
$191$	−6.47173	−0.468278	−0.234139	−	0.972203i	$-0.575227\pi$
$191$	−6.47173	−0.468278	−0.234139	+	0.972203i	$0.575227\pi$
$192$	0	0
$193$	−14.0571	−1.01185	−0.505927	−	0.862576i	$-0.668849\pi$
$193$	−14.0571	−1.01185	−0.505927	+	0.862576i	$0.668849\pi$
$194$	0	0
$195$	−16.7125	−1.19681
$196$	0	0
$197$	4.06861	0.289876	0.144938	−	0.989441i	$-0.453702\pi$
$197$	4.06861	0.289876	0.144938	+	0.989441i	$0.453702\pi$
$198$	0	0
$199$	−0.747244	−0.0529707	−0.0264854	−	0.999649i	$-0.508432\pi$
$199$	−0.747244	−0.0529707	−0.0264854	+	0.999649i	$0.508432\pi$
$200$	0	0
$201$	1.23145	0.0868596
$202$	0	0
$203$	−6.08703	−0.427226
$204$	0	0
$205$	−20.6931	−1.44527
$206$	0	0
$207$	−0.114792	−0.00797862
$208$	0	0
$209$	−10.2718	−0.710517
$210$	0	0
$211$	13.4493	0.925890	0.462945	−	0.886387i	$-0.346793\pi$
$211$	13.4493	0.925890	0.462945	+	0.886387i	$0.346793\pi$
$212$	0	0
$213$	−11.1645	−0.764980
$214$	0	0
$215$	31.6266	2.15692
$216$	0	0
$217$	−0.714927	−0.0485324
$218$	0	0
$219$	23.4451	1.58427
$220$	0	0
$221$	6.21345	0.417962
$222$	0	0
$223$	4.82629	0.323193	0.161596	−	0.986857i	$-0.448336\pi$
$223$	4.82629	0.323193	0.161596	+	0.986857i	$0.448336\pi$
$224$	0	0
$225$	0.0621305	0.00414203
$226$	0	0
$227$	20.9131	1.38805	0.694026	−	0.719950i	$-0.255835\pi$
$227$	20.9131	1.38805	0.694026	+	0.719950i	$0.255835\pi$
$228$	0	0
$229$	10.5871	0.699615	0.349808	−	0.936822i	$-0.386247\pi$
$229$	10.5871	0.699615	0.349808	+	0.936822i	$0.386247\pi$
$230$	0	0
$231$	26.0109	1.71139
$232$	0	0
$233$	10.8038	0.707781	0.353891	−	0.935287i	$-0.384859\pi$
$233$	10.8038	0.707781	0.353891	+	0.935287i	$0.384859\pi$
$234$	0	0
$235$	3.07388	0.200518
$236$	0	0
$237$	−4.71946	−0.306562
$238$	0	0
$239$	−3.23907	−0.209518	−0.104759	−	0.994498i	$-0.533407\pi$
$239$	−3.23907	−0.209518	−0.104759	+	0.994498i	$0.533407\pi$
$240$	0	0
$241$	−13.1864	−0.849411	−0.424706	−	0.905332i	$-0.639622\pi$
$241$	−13.1864	−0.849411	−0.424706	+	0.905332i	$0.639622\pi$
$242$	0	0
$243$	0.145136	0.00931045
$244$	0	0
$245$	43.7290	2.79374
$246$	0	0
$247$	−9.89192	−0.629408
$248$	0	0
$249$	−26.4753	−1.67781
$250$	0	0
$251$	−6.22110	−0.392672	−0.196336	−	0.980537i	$-0.562904\pi$
$251$	−6.22110	−0.392672	−0.196336	+	0.980537i	$0.562904\pi$
$252$	0	0
$253$	−26.7300	−1.68050
$254$	0	0
$255$	−10.5878	−0.663031
$256$	0	0
$257$	20.8976	1.30356	0.651778	−	0.758410i	$-0.274023\pi$
$257$	20.8976	1.30356	0.651778	+	0.758410i	$0.274023\pi$
$258$	0	0
$259$	−0.565815	−0.0351580
$260$	0	0
$261$	0.0184517	0.00114213
$262$	0	0
$263$	−6.26030	−0.386026	−0.193013	−	0.981196i	$-0.561826\pi$
$263$	−6.26030	−0.386026	−0.193013	+	0.981196i	$0.561826\pi$
$264$	0	0
$265$	−29.0435	−1.78413
$266$	0	0
$267$	−20.2208	−1.23749
$268$	0	0
$269$	−14.9502	−0.911527	−0.455764	−	0.890101i	$-0.650634\pi$
$269$	−14.9502	−0.911527	−0.455764	+	0.890101i	$0.650634\pi$
$270$	0	0
$271$	8.82152	0.535869	0.267935	−	0.963437i	$-0.413659\pi$
$271$	8.82152	0.535869	0.267935	+	0.963437i	$0.413659\pi$
$272$	0	0
$273$	25.0489	1.51603
$274$	0	0
$275$	14.4674	0.872419
$276$	0	0
$277$	−27.1518	−1.63139	−0.815697	−	0.578479i	$-0.803647\pi$
$277$	−27.1518	−1.63139	−0.815697	+	0.578479i	$0.803647\pi$
$278$	0	0
$279$	0.00216717	0.000129745 0
$280$	0	0
$281$	30.7428	1.83396	0.916981	−	0.398930i	$-0.130618\pi$
$281$	30.7428	1.83396	0.916981	+	0.398930i	$0.130618\pi$
$282$	0	0
$283$	17.7217	1.05345	0.526723	−	0.850037i	$-0.323421\pi$
$283$	17.7217	1.05345	0.526723	+	0.850037i	$0.323421\pi$
$284$	0	0
$285$	16.8559	0.998457
$286$	0	0
$287$	31.0151	1.83076
$288$	0	0
$289$	−13.0636	−0.768449
$290$	0	0
$291$	−4.89909	−0.287190
$292$	0	0
$293$	−33.4734	−1.95554	−0.977768	−	0.209690i	$-0.932755\pi$
$293$	−33.4734	−1.95554	−0.977768	+	0.209690i	$0.932755\pi$
$294$	0	0
$295$	−44.6824	−2.60151
$296$	0	0
$297$	16.8584	0.978224
$298$	0	0
$299$	−25.7414	−1.48866
$300$	0	0
$301$	−47.4022	−2.73222
$302$	0	0
$303$	−15.5876	−0.895482
$304$	0	0
$305$	0.510166	0.0292120
$306$	0	0
$307$	1.04732	0.0597735	0.0298867	−	0.999553i	$-0.490485\pi$
$307$	1.04732	0.0597735	0.0298867	+	0.999553i	$0.490485\pi$
$308$	0	0
$309$	−17.3893	−0.989242
$310$	0	0
$311$	12.4806	0.707712	0.353856	−	0.935300i	$-0.384870\pi$
$311$	12.4806	0.707712	0.353856	+	0.935300i	$0.384870\pi$
$312$	0	0
$313$	19.1869	1.08451	0.542255	−	0.840214i	$-0.317571\pi$
$313$	19.1869	1.08451	0.542255	+	0.840214i	$0.317571\pi$
$314$	0	0
$315$	−0.197782	−0.0111438
$316$	0	0
$317$	18.2453	1.02476	0.512379	−	0.858760i	$-0.328764\pi$
$317$	18.2453	1.02476	0.512379	+	0.858760i	$0.328764\pi$
$318$	0	0
$319$	4.29659	0.240563
$320$	0	0
$321$	−16.0464	−0.895623
$322$	0	0
$323$	−6.26676	−0.348692
$324$	0	0
$325$	13.9324	0.772828
$326$	0	0
$327$	6.00877	0.332286
$328$	0	0
$329$	−4.60716	−0.254001
$330$	0	0
$331$	−21.3888	−1.17563	−0.587817	−	0.808994i	$-0.700013\pi$
$331$	−21.3888	−1.17563	−0.587817	+	0.808994i	$0.700013\pi$
$332$	0	0
$333$	0.00171517	9.39905e−5 0
$334$	0	0
$335$	−2.18039	−0.119127
$336$	0	0
$337$	−5.73773	−0.312554	−0.156277	−	0.987713i	$-0.549949\pi$
$337$	−5.73773	−0.312554	−0.156277	+	0.987713i	$0.549949\pi$
$338$	0	0
$339$	16.5271	0.897630
$340$	0	0
$341$	0.504638	0.0273277
$342$	0	0
$343$	−33.2912	−1.79756
$344$	0	0
$345$	43.8635	2.36153
$346$	0	0
$347$	22.0471	1.18355	0.591776	−	0.806103i	$-0.298427\pi$
$347$	22.0471	1.18355	0.591776	+	0.806103i	$0.298427\pi$
$348$	0	0
$349$	−32.7353	−1.75228	−0.876139	−	0.482059i	$-0.839889\pi$
$349$	−32.7353	−1.75228	−0.876139	+	0.482059i	$0.839889\pi$
$350$	0	0
$351$	16.2349	0.866554
$352$	0	0
$353$	−19.5157	−1.03872	−0.519359	−	0.854556i	$-0.673829\pi$
$353$	−19.5157	−1.03872	−0.519359	+	0.854556i	$0.673829\pi$
$354$	0	0
$355$	19.7678	1.04917
$356$	0	0
$357$	15.8690	0.839878
$358$	0	0
$359$	−35.1527	−1.85529	−0.927644	−	0.373465i	$-0.878170\pi$
$359$	−35.1527	−1.85529	−0.927644	+	0.373465i	$0.878170\pi$
$360$	0	0
$361$	−9.02322	−0.474906
$362$	0	0
$363$	0.736800	0.0386720
$364$	0	0
$365$	−41.5117	−2.17282
$366$	0	0
$367$	−29.3033	−1.52962	−0.764809	−	0.644257i	$-0.777167\pi$
$367$	−29.3033	−1.52962	−0.764809	+	0.644257i	$0.777167\pi$
$368$	0	0
$369$	−0.0940166	−0.00489431
$370$	0	0
$371$	43.5306	2.26000
$372$	0	0
$373$	−16.7829	−0.868986	−0.434493	−	0.900675i	$-0.643072\pi$
$373$	−16.7829	−0.868986	−0.434493	+	0.900675i	$0.643072\pi$
$374$	0	0
$375$	2.94168	0.151908
$376$	0	0
$377$	4.13768	0.213102
$378$	0	0
$379$	2.81712	0.144706	0.0723529	−	0.997379i	$-0.476949\pi$
$379$	2.81712	0.144706	0.0723529	+	0.997379i	$0.476949\pi$
$380$	0	0
$381$	−7.21788	−0.369783
$382$	0	0
$383$	−24.5324	−1.25355	−0.626774	−	0.779201i	$-0.715625\pi$
$383$	−24.5324	−1.25355	−0.626774	+	0.779201i	$0.715625\pi$
$384$	0	0
$385$	−46.0547	−2.34716
$386$	0	0
$387$	0.143691	0.00730424
$388$	0	0
$389$	−18.7162	−0.948948	−0.474474	−	0.880269i	$-0.657362\pi$
$389$	−18.7162	−0.948948	−0.474474	+	0.880269i	$0.657362\pi$
$390$	0	0
$391$	−16.3078	−0.824720
$392$	0	0
$393$	33.5169	1.69070
$394$	0	0
$395$	8.35624	0.420448
$396$	0	0
$397$	−2.38286	−0.119592	−0.0597962	−	0.998211i	$-0.519045\pi$
$397$	−2.38286	−0.119592	−0.0597962	+	0.998211i	$0.519045\pi$
$398$	0	0
$399$	−25.2638	−1.26477
$400$	0	0
$401$	−19.4749	−0.972531	−0.486265	−	0.873811i	$-0.661641\pi$
$401$	−19.4749	−0.972531	−0.486265	+	0.873811i	$0.661641\pi$
$402$	0	0
$403$	0.485974	0.0242081
$404$	0	0
$405$	−27.7931	−1.38105
$406$	0	0
$407$	0.399386	0.0197968
$408$	0	0
$409$	8.04740	0.397919	0.198959	−	0.980008i	$-0.436244\pi$
$409$	8.04740	0.397919	0.198959	+	0.980008i	$0.436244\pi$
$410$	0	0
$411$	−13.4764	−0.664743
$412$	0	0
$413$	66.9704	3.29540
$414$	0	0
$415$	46.8770	2.30110
$416$	0	0
$417$	22.8117	1.11709
$418$	0	0
$419$	29.2752	1.43019	0.715094	−	0.699028i	$-0.246384\pi$
$419$	29.2752	1.43019	0.715094	+	0.699028i	$0.246384\pi$
$420$	0	0
$421$	14.6616	0.714564	0.357282	−	0.933997i	$-0.383704\pi$
$421$	14.6616	0.714564	0.357282	+	0.933997i	$0.383704\pi$
$422$	0	0
$423$	0.0139658	0.000679040 0
$424$	0	0
$425$	8.82647	0.428147
$426$	0	0
$427$	−0.764642	−0.0370036
$428$	0	0
$429$	−17.6810	−0.853647
$430$	0	0
$431$	−25.3220	−1.21972	−0.609859	−	0.792510i	$-0.708774\pi$
$431$	−25.3220	−1.21972	−0.609859	+	0.792510i	$0.708774\pi$
$432$	0	0
$433$	−9.52001	−0.457502	−0.228751	−	0.973485i	$-0.573464\pi$
$433$	−9.52001	−0.457502	−0.228751	+	0.973485i	$0.573464\pi$
$434$	0	0
$435$	−7.05064	−0.338052
$436$	0	0
$437$	25.9623	1.24194
$438$	0	0
$439$	−38.7419	−1.84905	−0.924524	−	0.381123i	$-0.875537\pi$
$439$	−38.7419	−1.84905	−0.924524	+	0.381123i	$0.875537\pi$
$440$	0	0
$441$	0.198677	0.00946081
$442$	0	0
$443$	−22.1440	−1.05209	−0.526046	−	0.850456i	$-0.676326\pi$
$443$	−22.1440	−1.05209	−0.526046	+	0.850456i	$0.676326\pi$
$444$	0	0
$445$	35.8028	1.69721
$446$	0	0
$447$	29.4530	1.39308
$448$	0	0
$449$	7.16829	0.338293	0.169146	−	0.985591i	$-0.445899\pi$
$449$	7.16829	0.338293	0.169146	+	0.985591i	$0.445899\pi$
$450$	0	0
$451$	−21.8923	−1.03087
$452$	0	0
$453$	42.1509	1.98042
$454$	0	0
$455$	−44.3513	−2.07922
$456$	0	0
$457$	−9.81999	−0.459360	−0.229680	−	0.973266i	$-0.573768\pi$
$457$	−9.81999	−0.459360	−0.229680	+	0.973266i	$0.573768\pi$
$458$	0	0
$459$	10.2852	0.480071
$460$	0	0
$461$	−22.4375	−1.04502	−0.522510	−	0.852633i	$-0.675004\pi$
$461$	−22.4375	−1.04502	−0.522510	+	0.852633i	$0.675004\pi$
$462$	0	0
$463$	−12.6027	−0.585698	−0.292849	−	0.956159i	$-0.594603\pi$
$463$	−12.6027	−0.585698	−0.292849	+	0.956159i	$0.594603\pi$
$464$	0	0
$465$	−0.828103	−0.0384024
$466$	0	0
$467$	6.03412	0.279226	0.139613	−	0.990206i	$-0.455414\pi$
$467$	6.03412	0.279226	0.139613	+	0.990206i	$0.455414\pi$
$468$	0	0
$469$	3.26799	0.150902
$470$	0	0
$471$	18.1298	0.835377
$472$	0	0
$473$	33.4593	1.53846
$474$	0	0
$475$	−14.0519	−0.644745
$476$	0	0
$477$	−0.131955	−0.00604181
$478$	0	0
$479$	−15.6740	−0.716163	−0.358082	−	0.933690i	$-0.616569\pi$
$479$	−15.6740	−0.716163	−0.358082	+	0.933690i	$0.616569\pi$
$480$	0	0
$481$	0.384615	0.0175369
$482$	0	0
$483$	−65.7431	−2.99141
$484$	0	0
$485$	8.67428	0.393879
$486$	0	0
$487$	10.6721	0.483600	0.241800	−	0.970326i	$-0.422262\pi$
$487$	10.6721	0.483600	0.241800	+	0.970326i	$0.422262\pi$
$488$	0	0
$489$	−6.00913	−0.271742
$490$	0	0
$491$	6.25802	0.282421	0.141210	−	0.989980i	$-0.454901\pi$
$491$	6.25802	0.282421	0.141210	+	0.989980i	$0.454901\pi$
$492$	0	0
$493$	2.62132	0.118058
$494$	0	0
$495$	0.139606	0.00627484
$496$	0	0
$497$	−29.6282	−1.32900
$498$	0	0
$499$	9.25077	0.414121	0.207061	−	0.978328i	$-0.433610\pi$
$499$	9.25077	0.414121	0.207061	+	0.978328i	$0.433610\pi$
$500$	0	0
$501$	38.8074	1.73379
$502$	0	0
$503$	−30.5842	−1.36368	−0.681840	−	0.731501i	$-0.738820\pi$
$503$	−30.5842	−1.36368	−0.681840	+	0.731501i	$0.738820\pi$
$504$	0	0
$505$	27.5992	1.22815
$506$	0	0
$507$	5.54194	0.246126
$508$	0	0
$509$	−8.64623	−0.383237	−0.191619	−	0.981469i	$-0.561374\pi$
$509$	−8.64623	−0.383237	−0.191619	+	0.981469i	$0.561374\pi$
$510$	0	0
$511$	62.2180	2.75236
$512$	0	0
$513$	−16.3742	−0.722937
$514$	0	0
$515$	30.7893	1.35674
$516$	0	0
$517$	3.25201	0.143023
$518$	0	0
$519$	−13.3360	−0.585387
$520$	0	0
$521$	29.6556	1.29924	0.649618	−	0.760261i	$-0.274929\pi$
$521$	29.6556	1.29924	0.649618	+	0.760261i	$0.274929\pi$
$522$	0	0
$523$	−31.8295	−1.39181	−0.695903	−	0.718135i	$-0.744996\pi$
$523$	−31.8295	−1.39181	−0.695903	+	0.718135i	$0.744996\pi$
$524$	0	0
$525$	35.5830	1.55297
$526$	0	0
$527$	0.307876	0.0134113
$528$	0	0
$529$	44.5607	1.93742
$530$	0	0
$531$	−0.203009	−0.00880983
$532$	0	0
$533$	−21.0826	−0.913189
$534$	0	0
$535$	28.4116	1.22834
$536$	0	0
$537$	29.6885	1.28115
$538$	0	0
$539$	46.2630	1.99269
$540$	0	0
$541$	23.6460	1.01662	0.508311	−	0.861174i	$-0.330270\pi$
$541$	23.6460	1.01662	0.508311	+	0.861174i	$0.330270\pi$
$542$	0	0
$543$	13.0877	0.561648
$544$	0	0
$545$	−10.6391	−0.455728
$546$	0	0
$547$	24.2872	1.03844	0.519222	−	0.854639i	$-0.326222\pi$
$547$	24.2872	1.03844	0.519222	+	0.854639i	$0.326222\pi$
$548$	0	0
$549$	0.00231787	9.89244e−5 0
$550$	0	0
$551$	−4.17318	−0.177783
$552$	0	0
$553$	−12.5244	−0.532592
$554$	0	0
$555$	−0.655386	−0.0278196
$556$	0	0
$557$	9.78198	0.414476	0.207238	−	0.978291i	$-0.433553\pi$
$557$	9.78198	0.414476	0.207238	+	0.978291i	$0.433553\pi$
$558$	0	0
$559$	32.2218	1.36284
$560$	0	0
$561$	−11.2013	−0.472920
$562$	0	0
$563$	−41.7247	−1.75849	−0.879243	−	0.476374i	$-0.841951\pi$
$563$	−41.7247	−1.75849	−0.879243	+	0.476374i	$0.841951\pi$
$564$	0	0
$565$	−29.2628	−1.23109
$566$	0	0
$567$	41.6566	1.74941
$568$	0	0
$569$	24.0759	1.00931	0.504656	−	0.863320i	$-0.331619\pi$
$569$	24.0759	1.00931	0.504656	+	0.863320i	$0.331619\pi$
$570$	0	0
$571$	−34.2010	−1.43127	−0.715634	−	0.698475i	$-0.753862\pi$
$571$	−34.2010	−1.43127	−0.715634	+	0.698475i	$0.753862\pi$
$572$	0	0
$573$	11.2354	0.469366
$574$	0	0
$575$	−36.5668	−1.52494
$576$	0	0
$577$	−24.3536	−1.01385	−0.506927	−	0.861989i	$-0.669219\pi$
$577$	−24.3536	−1.01385	−0.506927	+	0.861989i	$0.669219\pi$
$578$	0	0
$579$	24.4042	1.01421
$580$	0	0
$581$	−70.2597	−2.91486
$582$	0	0
$583$	−30.7265	−1.27256
$584$	0	0
$585$	0.134443	0.00555854
$586$	0	0
$587$	−9.03895	−0.373078	−0.186539	−	0.982448i	$-0.559727\pi$
$587$	−9.03895	−0.373078	−0.186539	+	0.982448i	$0.559727\pi$
$588$	0	0
$589$	−0.490143	−0.0201960
$590$	0	0
$591$	−7.06342	−0.290550
$592$	0	0
$593$	−33.7598	−1.38635	−0.693175	−	0.720769i	$-0.743789\pi$
$593$	−33.7598	−1.38635	−0.693175	+	0.720769i	$0.743789\pi$
$594$	0	0
$595$	−28.0976	−1.15189
$596$	0	0
$597$	1.29727	0.0530939
$598$	0	0
$599$	−4.43195	−0.181085	−0.0905423	−	0.995893i	$-0.528860\pi$
$599$	−4.43195	−0.181085	−0.0905423	+	0.995893i	$0.528860\pi$
$600$	0	0
$601$	−29.6734	−1.21040	−0.605202	−	0.796072i	$-0.706908\pi$
$601$	−29.6734	−1.21040	−0.605202	+	0.796072i	$0.706908\pi$
$602$	0	0
$603$	−0.00990632	−0.000403417 0
$604$	0	0
$605$	−1.30457	−0.0530384
$606$	0	0
$607$	−10.6283	−0.431390	−0.215695	−	0.976461i	$-0.569202\pi$
$607$	−10.6283	−0.431390	−0.215695	+	0.976461i	$0.569202\pi$
$608$	0	0
$609$	10.5676	0.428219
$610$	0	0
$611$	3.13174	0.126697
$612$	0	0
$613$	22.3184	0.901432	0.450716	−	0.892667i	$-0.351169\pi$
$613$	22.3184	0.901432	0.450716	+	0.892667i	$0.351169\pi$
$614$	0	0
$615$	35.9249	1.44863
$616$	0	0
$617$	9.57779	0.385587	0.192794	−	0.981239i	$-0.438245\pi$
$617$	9.57779	0.385587	0.192794	+	0.981239i	$0.438245\pi$
$618$	0	0
$619$	−28.3889	−1.14105	−0.570523	−	0.821282i	$-0.693259\pi$
$619$	−28.3889	−1.14105	−0.570523	+	0.821282i	$0.693259\pi$
$620$	0	0
$621$	−42.6100	−1.70988
$622$	0	0
$623$	−53.6615	−2.14990
$624$	0	0
$625$	−27.4523	−1.09809
$626$	0	0
$627$	17.8327	0.712169
$628$	0	0
$629$	0.243662	0.00971545
$630$	0	0
$631$	−6.87974	−0.273878	−0.136939	−	0.990579i	$-0.543726\pi$
$631$	−6.87974	−0.273878	−0.136939	+	0.990579i	$0.543726\pi$
$632$	0	0
$633$	−23.3491	−0.928043
$634$	0	0
$635$	12.7799	0.507156
$636$	0	0
$637$	44.5520	1.76521
$638$	0	0
$639$	0.0898125	0.00355293
$640$	0	0
$641$	−24.4052	−0.963946	−0.481973	−	0.876186i	$-0.660080\pi$
$641$	−24.4052	−0.963946	−0.481973	+	0.876186i	$0.660080\pi$
$642$	0	0
$643$	16.5483	0.652601	0.326301	−	0.945266i	$-0.394198\pi$
$643$	16.5483	0.652601	0.326301	+	0.945266i	$0.394198\pi$
$644$	0	0
$645$	−54.9062	−2.16193
$646$	0	0
$647$	−8.07755	−0.317561	−0.158781	−	0.987314i	$-0.550756\pi$
$647$	−8.07755	−0.317561	−0.158781	+	0.987314i	$0.550756\pi$
$648$	0	0
$649$	−47.2717	−1.85558
$650$	0	0
$651$	1.24117	0.0486452
$652$	0	0
$653$	−1.66715	−0.0652405	−0.0326202	−	0.999468i	$-0.510385\pi$
$653$	−1.66715	−0.0652405	−0.0326202	+	0.999468i	$0.510385\pi$
$654$	0	0
$655$	−59.3447	−2.31879
$656$	0	0
$657$	−0.188603	−0.00735810
$658$	0	0
$659$	0.748190	0.0291453	0.0145727	−	0.999894i	$-0.495361\pi$
$659$	0.748190	0.0291453	0.0145727	+	0.999894i	$0.495361\pi$
$660$	0	0
$661$	31.3464	1.21923	0.609616	−	0.792697i	$-0.291324\pi$
$661$	31.3464	1.21923	0.609616	+	0.792697i	$0.291324\pi$
$662$	0	0
$663$	−10.7870	−0.418934
$664$	0	0
$665$	44.7318	1.73463
$666$	0	0
$667$	−10.8597	−0.420490
$668$	0	0
$669$	−8.37882	−0.323944
$670$	0	0
$671$	0.539730	0.0208360
$672$	0	0
$673$	39.4025	1.51886	0.759428	−	0.650591i	$-0.225479\pi$
$673$	39.4025	1.51886	0.759428	+	0.650591i	$0.225479\pi$
$674$	0	0
$675$	23.0623	0.887670
$676$	0	0
$677$	−12.0347	−0.462532	−0.231266	−	0.972891i	$-0.574287\pi$
$677$	−12.0347	−0.462532	−0.231266	+	0.972891i	$0.574287\pi$
$678$	0	0
$679$	−13.0011	−0.498936
$680$	0	0
$681$	−36.3068	−1.39128
$682$	0	0
$683$	6.61906	0.253271	0.126636	−	0.991949i	$-0.459582\pi$
$683$	6.61906	0.253271	0.126636	+	0.991949i	$0.459582\pi$
$684$	0	0
$685$	23.8612	0.911691
$686$	0	0
$687$	−18.3800	−0.701242
$688$	0	0
$689$	−29.5901	−1.12729
$690$	0	0
$691$	20.6843	0.786868	0.393434	−	0.919353i	$-0.371287\pi$
$691$	20.6843	0.786868	0.393434	+	0.919353i	$0.371287\pi$
$692$	0	0
$693$	−0.209243	−0.00794850
$694$	0	0
$695$	−40.3902	−1.53209
$696$	0	0
$697$	−13.3563	−0.505906
$698$	0	0
$699$	−18.7563	−0.709427
$700$	0	0
$701$	−2.10692	−0.0795773	−0.0397887	−	0.999208i	$-0.512668\pi$
$701$	−2.10692	−0.0795773	−0.0397887	+	0.999208i	$0.512668\pi$
$702$	0	0
$703$	−0.387914	−0.0146305
$704$	0	0
$705$	−5.33650	−0.200984
$706$	0	0
$707$	−41.3659	−1.55573
$708$	0	0
$709$	18.2299	0.684637	0.342318	−	0.939584i	$-0.388788\pi$
$709$	18.2299	0.684637	0.342318	+	0.939584i	$0.388788\pi$
$710$	0	0
$711$	0.0379655	0.00142382
$712$	0	0
$713$	−1.27548	−0.0477672
$714$	0	0
$715$	31.3058	1.17077
$716$	0	0
$717$	5.62328	0.210005
$718$	0	0
$719$	3.52201	0.131349	0.0656744	−	0.997841i	$-0.479080\pi$
$719$	3.52201	0.131349	0.0656744	+	0.997841i	$0.479080\pi$
$720$	0	0
$721$	−46.1473	−1.71862
$722$	0	0
$723$	22.8926	0.851386
$724$	0	0
$725$	5.87775	0.218294
$726$	0	0
$727$	45.9002	1.70234	0.851172	−	0.524886i	$-0.175892\pi$
$727$	45.9002	1.70234	0.851172	+	0.524886i	$0.175892\pi$
$728$	0	0
$729$	26.8731	0.995301
$730$	0	0
$731$	20.4133	0.755012
$732$	0	0
$733$	−8.94933	−0.330551	−0.165275	−	0.986247i	$-0.552851\pi$
$733$	−8.94933	−0.330551	−0.165275	+	0.986247i	$0.552851\pi$
$734$	0	0
$735$	−75.9169	−2.80024
$736$	0	0
$737$	−2.30674	−0.0849700
$738$	0	0
$739$	−0.905173	−0.0332973	−0.0166487	−	0.999861i	$-0.505300\pi$
$739$	−0.905173	−0.0332973	−0.0166487	+	0.999861i	$0.505300\pi$
$740$	0	0
$741$	17.1731	0.630871
$742$	0	0
$743$	46.2792	1.69782	0.848909	−	0.528539i	$-0.177260\pi$
$743$	46.2792	1.69782	0.848909	+	0.528539i	$0.177260\pi$
$744$	0	0
$745$	−52.1493	−1.91060
$746$	0	0
$747$	0.212980	0.00779252
$748$	0	0
$749$	−42.5836	−1.55597
$750$	0	0
$751$	25.5173	0.931140	0.465570	−	0.885011i	$-0.345849\pi$
$751$	25.5173	0.931140	0.465570	+	0.885011i	$0.345849\pi$
$752$	0	0
$753$	10.8003	0.393585
$754$	0	0
$755$	−74.6320	−2.71614
$756$	0	0
$757$	24.1051	0.876115	0.438058	−	0.898947i	$-0.355667\pi$
$757$	24.1051	0.876115	0.438058	+	0.898947i	$0.355667\pi$
$758$	0	0
$759$	46.4054	1.68441
$760$	0	0
$761$	45.5094	1.64972	0.824858	−	0.565341i	$-0.191255\pi$
$761$	45.5094	1.64972	0.824858	+	0.565341i	$0.191255\pi$
$762$	0	0
$763$	15.9460	0.577282
$764$	0	0
$765$	0.0851727	0.00307943
$766$	0	0
$767$	−45.5234	−1.64375
$768$	0	0
$769$	−1.88574	−0.0680017	−0.0340008	−	0.999422i	$-0.510825\pi$
$769$	−1.88574	−0.0680017	−0.0340008	+	0.999422i	$0.510825\pi$
$770$	0	0
$771$	−36.2799	−1.30659
$772$	0	0
$773$	−30.9069	−1.11164	−0.555821	−	0.831302i	$-0.687596\pi$
$773$	−30.9069	−1.11164	−0.555821	+	0.831302i	$0.687596\pi$
$774$	0	0
$775$	0.690346	0.0247980
$776$	0	0
$777$	0.982299	0.0352398
$778$	0	0
$779$	21.2635	0.761843
$780$	0	0
$781$	20.9133	0.748338
$782$	0	0
$783$	6.84914	0.244768
$784$	0	0
$785$	−32.1005	−1.14571
$786$	0	0
$787$	32.9261	1.17369	0.586844	−	0.809700i	$-0.300370\pi$
$787$	32.9261	1.17369	0.586844	+	0.809700i	$0.300370\pi$
$788$	0	0
$789$	10.8684	0.386924
$790$	0	0
$791$	43.8593	1.55946
$792$	0	0
$793$	0.519768	0.0184575
$794$	0	0
$795$	50.4217	1.78827
$796$	0	0
$797$	−41.1351	−1.45708	−0.728540	−	0.685004i	$-0.759801\pi$
$797$	−41.1351	−1.45708	−0.728540	+	0.685004i	$0.759801\pi$
$798$	0	0
$799$	1.98403	0.0701898
$800$	0	0
$801$	0.162665	0.00574749
$802$	0	0
$803$	−43.9172	−1.54981
$804$	0	0
$805$	116.404	4.10271
$806$	0	0
$807$	25.9546	0.913646
$808$	0	0
$809$	−1.47110	−0.0517210	−0.0258605	−	0.999666i	$-0.508233\pi$
$809$	−1.47110	−0.0517210	−0.0258605	+	0.999666i	$0.508233\pi$
$810$	0	0
$811$	−42.7445	−1.50096	−0.750480	−	0.660893i	$-0.770178\pi$
$811$	−42.7445	−1.50096	−0.750480	+	0.660893i	$0.770178\pi$
$812$	0	0
$813$	−15.3148	−0.537115
$814$	0	0
$815$	10.6397	0.372693
$816$	0	0
$817$	−32.4983	−1.13697
$818$	0	0
$819$	−0.201504	−0.00704114
$820$	0	0
$821$	−3.12827	−0.109177	−0.0545887	−	0.998509i	$-0.517385\pi$
$821$	−3.12827	−0.109177	−0.0545887	+	0.998509i	$0.517385\pi$
$822$	0	0
$823$	53.9811	1.88166	0.940832	−	0.338873i	$-0.110046\pi$
$823$	53.9811	1.88166	0.940832	+	0.338873i	$0.110046\pi$
$824$	0	0
$825$	−25.1166	−0.874448
$826$	0	0
$827$	23.6763	0.823306	0.411653	−	0.911341i	$-0.364952\pi$
$827$	23.6763	0.823306	0.411653	+	0.911341i	$0.364952\pi$
$828$	0	0
$829$	4.39129	0.152516	0.0762580	−	0.997088i	$-0.475703\pi$
$829$	4.39129	0.152516	0.0762580	+	0.997088i	$0.475703\pi$
$830$	0	0
$831$	47.1377	1.63519
$832$	0	0
$833$	28.2247	0.977928
$834$	0	0
$835$	−68.7121	−2.37788
$836$	0	0
$837$	0.804436	0.0278054
$838$	0	0
$839$	−10.4724	−0.361549	−0.180774	−	0.983525i	$-0.557860\pi$
$839$	−10.4724	−0.361549	−0.180774	+	0.983525i	$0.557860\pi$
$840$	0	0
$841$	−27.2544	−0.939807
$842$	0	0
$843$	−53.3719	−1.83823
$844$	0	0
$845$	−9.81251	−0.337561
$846$	0	0
$847$	1.95530	0.0671850
$848$	0	0
$849$	−30.7662	−1.05589
$850$	0	0
$851$	−1.00946	−0.0346037
$852$	0	0
$853$	12.5901	0.431076	0.215538	−	0.976495i	$-0.430850\pi$
$853$	12.5901	0.431076	0.215538	+	0.976495i	$0.430850\pi$
$854$	0	0
$855$	−0.135596	−0.00463730
$856$	0	0
$857$	−9.23699	−0.315529	−0.157765	−	0.987477i	$-0.550429\pi$
$857$	−9.23699	−0.315529	−0.157765	+	0.987477i	$0.550429\pi$
$858$	0	0
$859$	9.28932	0.316948	0.158474	−	0.987363i	$-0.449343\pi$
$859$	9.28932	0.316948	0.158474	+	0.987363i	$0.449343\pi$
$860$	0	0
$861$	−53.8446	−1.83502
$862$	0	0
$863$	25.5775	0.870667	0.435333	−	0.900269i	$-0.356631\pi$
$863$	25.5775	0.870667	0.435333	+	0.900269i	$0.356631\pi$
$864$	0	0
$865$	23.6127	0.802855
$866$	0	0
$867$	22.6795	0.770236
$868$	0	0
$869$	8.84048	0.299893
$870$	0	0
$871$	−2.22143	−0.0752702
$872$	0	0
$873$	0.0394105	0.00133384
$874$	0	0
$875$	7.80656	0.263910
$876$	0	0
$877$	−21.9782	−0.742151	−0.371076	−	0.928603i	$-0.621011\pi$
$877$	−21.9782	−0.742151	−0.371076	+	0.928603i	$0.621011\pi$
$878$	0	0
$879$	58.1124	1.96008
$880$	0	0
$881$	36.8279	1.24076	0.620381	−	0.784301i	$-0.286978\pi$
$881$	36.8279	1.24076	0.620381	+	0.784301i	$0.286978\pi$
$882$	0	0
$883$	−4.67060	−0.157178	−0.0785890	−	0.996907i	$-0.525041\pi$
$883$	−4.67060	−0.157178	−0.0785890	+	0.996907i	$0.525041\pi$
$884$	0	0
$885$	77.5721	2.60756
$886$	0	0
$887$	46.3977	1.55788	0.778941	−	0.627097i	$-0.215757\pi$
$887$	46.3977	1.55788	0.778941	+	0.627097i	$0.215757\pi$
$888$	0	0
$889$	−19.1547	−0.642427
$890$	0	0
$891$	−29.4037	−0.985062
$892$	0	0
$893$	−3.15860	−0.105699
$894$	0	0
$895$	−52.5662	−1.75709
$896$	0	0
$897$	44.6891	1.49213
$898$	0	0
$899$	0.205022	0.00683785
$900$	0	0
$901$	−18.7460	−0.624520
$902$	0	0
$903$	82.2939	2.73857
$904$	0	0
$905$	−23.1730	−0.770297
$906$	0	0
$907$	17.5707	0.583426	0.291713	−	0.956506i	$-0.405775\pi$
$907$	17.5707	0.583426	0.291713	+	0.956506i	$0.405775\pi$
$908$	0	0
$909$	0.125393	0.00415904
$910$	0	0
$911$	−20.6650	−0.684660	−0.342330	−	0.939580i	$-0.611216\pi$
$911$	−20.6650	−0.684660	−0.342330	+	0.939580i	$0.611216\pi$
$912$	0	0
$913$	49.5935	1.64131
$914$	0	0
$915$	−0.885688	−0.0292799
$916$	0	0
$917$	88.9464	2.93727
$918$	0	0
$919$	−7.64800	−0.252284	−0.126142	−	0.992012i	$-0.540260\pi$
$919$	−7.64800	−0.252284	−0.126142	+	0.992012i	$0.540260\pi$
$920$	0	0
$921$	−1.81822	−0.0599125
$922$	0	0
$923$	20.1399	0.662911
$924$	0	0
$925$	0.546361	0.0179643
$926$	0	0
$927$	0.139887	0.00459450
$928$	0	0
$929$	19.5411	0.641121	0.320561	−	0.947228i	$-0.396129\pi$
$929$	19.5411	0.641121	0.320561	+	0.947228i	$0.396129\pi$
$930$	0	0
$931$	−44.9342	−1.47266
$932$	0	0
$933$	−21.6674	−0.709358
$934$	0	0
$935$	19.8330	0.648607
$936$	0	0
$937$	37.1182	1.21260	0.606299	−	0.795236i	$-0.292653\pi$
$937$	37.1182	1.21260	0.606299	+	0.795236i	$0.292653\pi$
$938$	0	0
$939$	−33.3100	−1.08703
$940$	0	0
$941$	52.4125	1.70860	0.854300	−	0.519781i	$-0.173986\pi$
$941$	52.4125	1.70860	0.854300	+	0.519781i	$0.173986\pi$
$942$	0	0
$943$	55.3332	1.80190
$944$	0	0
$945$	−73.4151	−2.38819
$946$	0	0
$947$	28.2704	0.918665	0.459333	−	0.888264i	$-0.348089\pi$
$947$	28.2704	0.918665	0.459333	+	0.888264i	$0.348089\pi$
$948$	0	0
$949$	−42.2930	−1.37289
$950$	0	0
$951$	−31.6752	−1.02714
$952$	0	0
$953$	−15.9896	−0.517955	−0.258977	−	0.965883i	$-0.583385\pi$
$953$	−15.9896	−0.517955	−0.258977	+	0.965883i	$0.583385\pi$
$954$	0	0
$955$	−19.8933	−0.643733
$956$	0	0
$957$	−7.45922	−0.241122
$958$	0	0
$959$	−35.7634	−1.15486
$960$	0	0
$961$	−30.9759	−0.999223
$962$	0	0
$963$	0.129085	0.00415969
$964$	0	0
$965$	−43.2099	−1.39098
$966$	0	0
$967$	9.13852	0.293875	0.146938	−	0.989146i	$-0.453058\pi$
$967$	9.13852	0.293875	0.146938	+	0.989146i	$0.453058\pi$
$968$	0	0
$969$	10.8796	0.349502
$970$	0	0
$971$	−48.8938	−1.56908	−0.784539	−	0.620080i	$-0.787100\pi$
$971$	−48.8938	−1.56908	−0.784539	+	0.620080i	$0.787100\pi$
$972$	0	0
$973$	60.5372	1.94073
$974$	0	0
$975$	−24.1877	−0.774625
$976$	0	0
$977$	−14.5766	−0.466346	−0.233173	−	0.972435i	$-0.574911\pi$
$977$	−14.5766	−0.466346	−0.233173	+	0.972435i	$0.574911\pi$
$978$	0	0
$979$	37.8775	1.21057
$980$	0	0
$981$	−0.0483373	−0.00154329
$982$	0	0
$983$	23.4721	0.748645	0.374322	−	0.927299i	$-0.377875\pi$
$983$	23.4721	0.748645	0.374322	+	0.927299i	$0.377875\pi$
$984$	0	0
$985$	12.5064	0.398488
$986$	0	0
$987$	7.99840	0.254592
$988$	0	0
$989$	−84.5692	−2.68914
$990$	0	0
$991$	26.2885	0.835083	0.417541	−	0.908658i	$-0.362892\pi$
$991$	26.2885	0.835083	0.417541	+	0.908658i	$0.362892\pi$
$992$	0	0
$993$	37.1326	1.17837
$994$	0	0
$995$	−2.29694	−0.0728179
$996$	0	0
$997$	−19.4651	−0.616465	−0.308232	−	0.951311i	$-0.599737\pi$
$997$	−19.4651	−0.616465	−0.308232	+	0.951311i	$0.599737\pi$
$998$	0	0
$999$	0.636656	0.0201429

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6016.2.a.m.1.5	✓	13
4.3	odd	2		6016.2.a.o.1.9	yes	13
8.3	odd	2		6016.2.a.n.1.5	yes	13
8.5	even	2		6016.2.a.p.1.9	yes	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6016.2.a.m.1.5	✓	13	1.1	even	1	trivial
6016.2.a.n.1.5	yes	13	8.3	odd	2
6016.2.a.o.1.9	yes	13	4.3	odd	2
6016.2.a.p.1.9	yes	13	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6016 = 2^{7} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6016.a (trivial)

\(f(q)\)	\(=\)	\(q-1.73608 q^{3} +3.07388 q^{5} -4.60716 q^{7} +0.0139658 q^{9} +O(q^{10})\)	\(q-1.73608 q^{3} +3.07388 q^{5} -4.60716 q^{7} +0.0139658 q^{9} +3.25201 q^{11} +3.13174 q^{13} -5.33650 q^{15} +1.98403 q^{17} -3.15860 q^{19} +7.99840 q^{21} -8.21953 q^{23} +4.44876 q^{25} +5.18399 q^{27} +1.32121 q^{29} +0.155177 q^{31} -5.64575 q^{33} -14.1619 q^{35} +0.122812 q^{37} -5.43694 q^{39} -6.73192 q^{41} +10.2888 q^{43} +0.0429292 q^{45} +1.00000 q^{47} +14.2260 q^{49} -3.44443 q^{51} -9.44846 q^{53} +9.99631 q^{55} +5.48358 q^{57} -14.5361 q^{59} +0.165968 q^{61} -0.0643427 q^{63} +9.62660 q^{65} -0.709327 q^{67} +14.2697 q^{69} +6.43089 q^{71} -13.5046 q^{73} -7.72340 q^{75} -14.9826 q^{77} +2.71846 q^{79} -9.04170 q^{81} +15.2501 q^{83} +6.09867 q^{85} -2.29372 q^{87} +11.6474 q^{89} -14.4284 q^{91} -0.269400 q^{93} -9.70918 q^{95} +2.82193 q^{97} +0.0454170 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 4 q^{3} - 6 q^{5} + 2 q^{7} + 21 q^{9}+O(q^{10}) \) 13 * q - 4 * q^3 - 6 * q^5 + 2 * q^7 + 21 * q^9	\( 13 q - 4 q^{3} - 6 q^{5} + 2 q^{7} + 21 q^{9} - 10 q^{11} - 4 q^{13} - 14 q^{15} + 10 q^{17} - 8 q^{19} - 10 q^{21} - 18 q^{23} + 23 q^{25} - 16 q^{27} - 14 q^{29} - 4 q^{31} + 14 q^{33} - 14 q^{35} - 16 q^{37} - 12 q^{39} + 10 q^{41} - 12 q^{43} - 10 q^{45} + 13 q^{47} + 9 q^{49} - 22 q^{51} - 26 q^{53} + 2 q^{55} + 20 q^{57} - 30 q^{59} - 18 q^{61} - 12 q^{63} - 4 q^{65} - 4 q^{67} - 2 q^{69} - 36 q^{71} + 10 q^{73} - 38 q^{75} - 42 q^{77} + 21 q^{81} - 12 q^{83} - 4 q^{85} - 6 q^{87} + 50 q^{89} + 4 q^{91} - 52 q^{93} - 8 q^{95} - 10 q^{97} - 34 q^{99}+O(q^{100}) \) 13 * q - 4 * q^3 - 6 * q^5 + 2 * q^7 + 21 * q^9 - 10 * q^11 - 4 * q^13 - 14 * q^15 + 10 * q^17 - 8 * q^19 - 10 * q^21 - 18 * q^23 + 23 * q^25 - 16 * q^27 - 14 * q^29 - 4 * q^31 + 14 * q^33 - 14 * q^35 - 16 * q^37 - 12 * q^39 + 10 * q^41 - 12 * q^43 - 10 * q^45 + 13 * q^47 + 9 * q^49 - 22 * q^51 - 26 * q^53 + 2 * q^55 + 20 * q^57 - 30 * q^59 - 18 * q^61 - 12 * q^63 - 4 * q^65 - 4 * q^67 - 2 * q^69 - 36 * q^71 + 10 * q^73 - 38 * q^75 - 42 * q^77 + 21 * q^81 - 12 * q^83 - 4 * q^85 - 6 * q^87 + 50 * q^89 + 4 * q^91 - 52 * q^93 - 8 * q^95 - 10 * q^97 - 34 * q^99

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