LMFDB - Embedded newform 8016.2.a.be.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8016,2,Mod(1,8016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8016, 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("8016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8016 = 2^{4} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8016.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:	$64.0080822603$
Analytic rank:	$0$
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} - x^{10} - 33 x^{9} + 22 x^{8} + 417 x^{7} - 151 x^{6} - 2470 x^{5} + 272 x^{4} + 6584 x^{3} + \cdots + 242 $ x^11 - x^10 - 33x^9 + 22x^8 + 417x^7 - 151x^6 - 2470x^5 + 272x^4 + 6584x^3 + 292x^2 - 5687*x + 242
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 4008)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$-3.01685$ of defining polynomial
Character	$\chi$	$=$	8016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +4.01685 q^{5} +4.41557 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +4.01685 q^{5} +4.41557 q^{7} +1.00000 q^{9} -3.91880 q^{11} +1.33293 q^{13} +4.01685 q^{15} +5.36767 q^{17} -4.32150 q^{19} +4.41557 q^{21} -2.96332 q^{23} +11.1351 q^{25} +1.00000 q^{27} +0.641815 q^{29} +4.50410 q^{31} -3.91880 q^{33} +17.7367 q^{35} -3.58361 q^{37} +1.33293 q^{39} -8.78387 q^{41} -6.83087 q^{43} +4.01685 q^{45} +11.1340 q^{47} +12.4973 q^{49} +5.36767 q^{51} -1.90890 q^{53} -15.7412 q^{55} -4.32150 q^{57} +6.60930 q^{59} +0.00823330 q^{61} +4.41557 q^{63} +5.35420 q^{65} -4.03013 q^{67} -2.96332 q^{69} +2.34031 q^{71} +4.99602 q^{73} +11.1351 q^{75} -17.3037 q^{77} -1.33451 q^{79} +1.00000 q^{81} -14.1756 q^{83} +21.5611 q^{85} +0.641815 q^{87} +12.3702 q^{89} +5.88567 q^{91} +4.50410 q^{93} -17.3588 q^{95} +13.3417 q^{97} -3.91880 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 11 q^{3} + 10 q^{5} + q^{7} + 11 q^{9}+O(q^{10}) $ 11 * q + 11 * q^3 + 10 * q^5 + q^7 + 11 * q^9	$ 11 q + 11 q^{3} + 10 q^{5} + q^{7} + 11 q^{9} + q^{11} + 10 q^{13} + 10 q^{15} + 17 q^{17} - 2 q^{19} + q^{21} + 3 q^{23} + 21 q^{25} + 11 q^{27} + 17 q^{29} + 15 q^{31} + q^{33} - 11 q^{35} + 4 q^{37} + 10 q^{39} + 16 q^{41} - 10 q^{43} + 10 q^{45} + 16 q^{47} + 22 q^{49} + 17 q^{51} + 42 q^{53} + 5 q^{55} - 2 q^{57} + 2 q^{59} + 12 q^{61} + q^{63} + 10 q^{65} + q^{67} + 3 q^{69} + 9 q^{71} + 24 q^{73} + 21 q^{75} + 22 q^{77} + 30 q^{79} + 11 q^{81} - 16 q^{83} + 25 q^{85} + 17 q^{87} + 37 q^{89} - q^{91} + 15 q^{93} - 5 q^{95} + 4 q^{97} + q^{99}+O(q^{100}) $ 11 * q + 11 * q^3 + 10 * q^5 + q^7 + 11 * q^9 + q^11 + 10 * q^13 + 10 * q^15 + 17 * q^17 - 2 * q^19 + q^21 + 3 * q^23 + 21 * q^25 + 11 * q^27 + 17 * q^29 + 15 * q^31 + q^33 - 11 * q^35 + 4 * q^37 + 10 * q^39 + 16 * q^41 - 10 * q^43 + 10 * q^45 + 16 * q^47 + 22 * q^49 + 17 * q^51 + 42 * q^53 + 5 * q^55 - 2 * q^57 + 2 * q^59 + 12 * q^61 + q^63 + 10 * q^65 + q^67 + 3 * q^69 + 9 * q^71 + 24 * q^73 + 21 * q^75 + 22 * q^77 + 30 * q^79 + 11 * q^81 - 16 * q^83 + 25 * q^85 + 17 * q^87 + 37 * q^89 - q^91 + 15 * q^93 - 5 * q^95 + 4 * q^97 + q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	4.01685	1.79639	0.898195	−	0.439597i	$-0.144879\pi$
$5$	4.01685	1.79639	0.898195	+	0.439597i	$0.144879\pi$
$6$	0	0
$7$	4.41557	1.66893	0.834465	−	0.551062i	$-0.185777\pi$
$7$	4.41557	1.66893	0.834465	+	0.551062i	$0.185777\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.91880	−1.18156	−0.590781	−	0.806832i	$-0.701180\pi$
$11$	−3.91880	−1.18156	−0.590781	+	0.806832i	$0.701180\pi$
$12$	0	0
$13$	1.33293	0.369690	0.184845	−	0.982768i	$-0.440822\pi$
$13$	1.33293	0.369690	0.184845	+	0.982768i	$0.440822\pi$
$14$	0	0
$15$	4.01685	1.03715
$16$	0	0
$17$	5.36767	1.30185	0.650926	−	0.759141i	$-0.274381\pi$
$17$	5.36767	1.30185	0.650926	+	0.759141i	$0.274381\pi$
$18$	0	0
$19$	−4.32150	−0.991420	−0.495710	−	0.868488i	$-0.665092\pi$
$19$	−4.32150	−0.991420	−0.495710	+	0.868488i	$0.665092\pi$
$20$	0	0
$21$	4.41557	0.963557
$22$	0	0
$23$	−2.96332	−0.617894	−0.308947	−	0.951079i	$-0.599977\pi$
$23$	−2.96332	−0.617894	−0.308947	+	0.951079i	$0.599977\pi$
$24$	0	0
$25$	11.1351	2.22702
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	0.641815	0.119182	0.0595910	−	0.998223i	$-0.481020\pi$
$29$	0.641815	0.119182	0.0595910	+	0.998223i	$0.481020\pi$
$30$	0	0
$31$	4.50410	0.808960	0.404480	−	0.914547i	$-0.367453\pi$
$31$	4.50410	0.808960	0.404480	+	0.914547i	$0.367453\pi$
$32$	0	0
$33$	−3.91880	−0.682175
$34$	0	0
$35$	17.7367	2.99805
$36$	0	0
$37$	−3.58361	−0.589141	−0.294571	−	0.955630i	$-0.595177\pi$
$37$	−3.58361	−0.589141	−0.294571	+	0.955630i	$0.595177\pi$
$38$	0	0
$39$	1.33293	0.213440
$40$	0	0
$41$	−8.78387	−1.37181	−0.685905	−	0.727691i	$-0.740594\pi$
$41$	−8.78387	−1.37181	−0.685905	+	0.727691i	$0.740594\pi$
$42$	0	0
$43$	−6.83087	−1.04170	−0.520849	−	0.853649i	$-0.674385\pi$
$43$	−6.83087	−1.04170	−0.520849	+	0.853649i	$0.674385\pi$
$44$	0	0
$45$	4.01685	0.598797
$46$	0	0
$47$	11.1340	1.62406	0.812031	−	0.583615i	$-0.198362\pi$
$47$	11.1340	1.62406	0.812031	+	0.583615i	$0.198362\pi$
$48$	0	0
$49$	12.4973	1.78532
$50$	0	0
$51$	5.36767	0.751625
$52$	0	0
$53$	−1.90890	−0.262208	−0.131104	−	0.991369i	$-0.541852\pi$
$53$	−1.90890	−0.262208	−0.131104	+	0.991369i	$0.541852\pi$
$54$	0	0
$55$	−15.7412	−2.12255
$56$	0	0
$57$	−4.32150	−0.572397
$58$	0	0
$59$	6.60930	0.860458	0.430229	−	0.902720i	$-0.358433\pi$
$59$	6.60930	0.860458	0.430229	+	0.902720i	$0.358433\pi$
$60$	0	0
$61$	0.00823330	0.00105417	0.000527083	−	1.00000i	$-0.499832\pi$
$61$	0.00823330	0.00105417	0.000527083		1.00000i	$0.499832\pi$
$62$	0	0
$63$	4.41557	0.556310
$64$	0	0
$65$	5.35420	0.664107
$66$	0	0
$67$	−4.03013	−0.492359	−0.246179	−	0.969224i	$-0.579175\pi$
$67$	−4.03013	−0.492359	−0.246179	+	0.969224i	$0.579175\pi$
$68$	0	0
$69$	−2.96332	−0.356741
$70$	0	0
$71$	2.34031	0.277744	0.138872	−	0.990310i	$-0.455652\pi$
$71$	2.34031	0.277744	0.138872	+	0.990310i	$0.455652\pi$
$72$	0	0
$73$	4.99602	0.584740	0.292370	−	0.956305i	$-0.405556\pi$
$73$	4.99602	0.584740	0.292370	+	0.956305i	$0.405556\pi$
$74$	0	0
$75$	11.1351	1.28577
$76$	0	0
$77$	−17.3037	−1.97194
$78$	0	0
$79$	−1.33451	−0.150144	−0.0750718	−	0.997178i	$-0.523919\pi$
$79$	−1.33451	−0.150144	−0.0750718	+	0.997178i	$0.523919\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−14.1756	−1.55597	−0.777986	−	0.628281i	$-0.783759\pi$
$83$	−14.1756	−1.55597	−0.777986	+	0.628281i	$0.783759\pi$
$84$	0	0
$85$	21.5611	2.33863
$86$	0	0
$87$	0.641815	0.0688098
$88$	0	0
$89$	12.3702	1.31124	0.655621	−	0.755090i	$-0.272407\pi$
$89$	12.3702	1.31124	0.655621	+	0.755090i	$0.272407\pi$
$90$	0	0
$91$	5.88567	0.616986
$92$	0	0
$93$	4.50410	0.467053
$94$	0	0
$95$	−17.3588	−1.78098
$96$	0	0
$97$	13.3417	1.35464	0.677321	−	0.735688i	$-0.263141\pi$
$97$	13.3417	1.35464	0.677321	+	0.735688i	$0.263141\pi$
$98$	0	0
$99$	−3.91880	−0.393854
$100$	0	0
$101$	0.0616138	0.00613081	0.00306540	−	0.999995i	$-0.499024\pi$
$101$	0.0616138	0.00613081	0.00306540	+	0.999995i	$0.499024\pi$
$102$	0	0
$103$	1.58232	0.155910	0.0779552	−	0.996957i	$-0.475161\pi$
$103$	1.58232	0.155910	0.0779552	+	0.996957i	$0.475161\pi$
$104$	0	0
$105$	17.7367	1.73092
$106$	0	0
$107$	15.1136	1.46109	0.730544	−	0.682865i	$-0.239266\pi$
$107$	15.1136	1.46109	0.730544	+	0.682865i	$0.239266\pi$
$108$	0	0
$109$	−13.1743	−1.26187	−0.630934	−	0.775836i	$-0.717328\pi$
$109$	−13.1743	−1.26187	−0.630934	+	0.775836i	$0.717328\pi$
$110$	0	0
$111$	−3.58361	−0.340141
$112$	0	0
$113$	20.1380	1.89443	0.947213	−	0.320604i	$-0.103886\pi$
$113$	20.1380	1.89443	0.947213	+	0.320604i	$0.103886\pi$
$114$	0	0
$115$	−11.9032	−1.10998
$116$	0	0
$117$	1.33293	0.123230
$118$	0	0
$119$	23.7013	2.17270
$120$	0	0
$121$	4.35696	0.396088
$122$	0	0
$123$	−8.78387	−0.792015
$124$	0	0
$125$	24.6438	2.20421
$126$	0	0
$127$	−18.1668	−1.61204	−0.806019	−	0.591889i	$-0.798382\pi$
$127$	−18.1668	−1.61204	−0.806019	+	0.591889i	$0.798382\pi$
$128$	0	0
$129$	−6.83087	−0.601425
$130$	0	0
$131$	−11.0441	−0.964932	−0.482466	−	0.875915i	$-0.660259\pi$
$131$	−11.0441	−0.964932	−0.482466	+	0.875915i	$0.660259\pi$
$132$	0	0
$133$	−19.0819	−1.65461
$134$	0	0
$135$	4.01685	0.345716
$136$	0	0
$137$	14.8528	1.26896	0.634480	−	0.772939i	$-0.281214\pi$
$137$	14.8528	1.26896	0.634480	+	0.772939i	$0.281214\pi$
$138$	0	0
$139$	12.3986	1.05164	0.525818	−	0.850597i	$-0.323759\pi$
$139$	12.3986	1.05164	0.525818	+	0.850597i	$0.323759\pi$
$140$	0	0
$141$	11.1340	0.937652
$142$	0	0
$143$	−5.22350	−0.436811
$144$	0	0
$145$	2.57808	0.214098
$146$	0	0
$147$	12.4973	1.03076
$148$	0	0
$149$	−17.3220	−1.41908	−0.709538	−	0.704667i	$-0.751096\pi$
$149$	−17.3220	−1.41908	−0.709538	+	0.704667i	$0.751096\pi$
$150$	0	0
$151$	12.7672	1.03898	0.519490	−	0.854477i	$-0.326122\pi$
$151$	12.7672	1.03898	0.519490	+	0.854477i	$0.326122\pi$
$152$	0	0
$153$	5.36767	0.433951
$154$	0	0
$155$	18.0923	1.45321
$156$	0	0
$157$	1.00398	0.0801265	0.0400632	−	0.999197i	$-0.487244\pi$
$157$	1.00398	0.0801265	0.0400632	+	0.999197i	$0.487244\pi$
$158$	0	0
$159$	−1.90890	−0.151386
$160$	0	0
$161$	−13.0847	−1.03122
$162$	0	0
$163$	−9.70804	−0.760392	−0.380196	−	0.924906i	$-0.624143\pi$
$163$	−9.70804	−0.760392	−0.380196	+	0.924906i	$0.624143\pi$
$164$	0	0
$165$	−15.7412	−1.22545
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−11.2233	−0.863330
$170$	0	0
$171$	−4.32150	−0.330473
$172$	0	0
$173$	23.0933	1.75575	0.877876	−	0.478888i	$-0.158960\pi$
$173$	23.0933	1.75575	0.877876	+	0.478888i	$0.158960\pi$
$174$	0	0
$175$	49.1678	3.71674
$176$	0	0
$177$	6.60930	0.496786
$178$	0	0
$179$	18.0863	1.35183	0.675917	−	0.736978i	$-0.263748\pi$
$179$	18.0863	1.35183	0.675917	+	0.736978i	$0.263748\pi$
$180$	0	0
$181$	−18.8159	−1.39858	−0.699289	−	0.714839i	$-0.746500\pi$
$181$	−18.8159	−1.39858	−0.699289	+	0.714839i	$0.746500\pi$
$182$	0	0
$183$	0.00823330	0.000608623 0
$184$	0	0
$185$	−14.3948	−1.05833
$186$	0	0
$187$	−21.0348	−1.53822
$188$	0	0
$189$	4.41557	0.321186
$190$	0	0
$191$	−5.24773	−0.379712	−0.189856	−	0.981812i	$-0.560802\pi$
$191$	−5.24773	−0.379712	−0.189856	+	0.981812i	$0.560802\pi$
$192$	0	0
$193$	15.4295	1.11064	0.555321	−	0.831636i	$-0.312595\pi$
$193$	15.4295	1.11064	0.555321	+	0.831636i	$0.312595\pi$
$194$	0	0
$195$	5.35420	0.383422
$196$	0	0
$197$	−4.27605	−0.304656	−0.152328	−	0.988330i	$-0.548677\pi$
$197$	−4.27605	−0.304656	−0.152328	+	0.988330i	$0.548677\pi$
$198$	0	0
$199$	−8.93953	−0.633707	−0.316853	−	0.948475i	$-0.602626\pi$
$199$	−8.93953	−0.633707	−0.316853	+	0.948475i	$0.602626\pi$
$200$	0	0
$201$	−4.03013	−0.284263
$202$	0	0
$203$	2.83398	0.198906
$204$	0	0
$205$	−35.2835	−2.46431
$206$	0	0
$207$	−2.96332	−0.205965
$208$	0	0
$209$	16.9351	1.17142
$210$	0	0
$211$	−20.7250	−1.42677	−0.713383	−	0.700774i	$-0.752838\pi$
$211$	−20.7250	−1.42677	−0.713383	+	0.700774i	$0.752838\pi$
$212$	0	0
$213$	2.34031	0.160356
$214$	0	0
$215$	−27.4386	−1.87130
$216$	0	0
$217$	19.8882	1.35010
$218$	0	0
$219$	4.99602	0.337600
$220$	0	0
$221$	7.15476	0.481281
$222$	0	0
$223$	−10.2434	−0.685948	−0.342974	−	0.939345i	$-0.611434\pi$
$223$	−10.2434	−0.685948	−0.342974	+	0.939345i	$0.611434\pi$
$224$	0	0
$225$	11.1351	0.742340
$226$	0	0
$227$	−17.0924	−1.13446	−0.567230	−	0.823559i	$-0.691985\pi$
$227$	−17.0924	−1.13446	−0.567230	+	0.823559i	$0.691985\pi$
$228$	0	0
$229$	24.4530	1.61590	0.807949	−	0.589252i	$-0.200578\pi$
$229$	24.4530	1.61590	0.807949	+	0.589252i	$0.200578\pi$
$230$	0	0
$231$	−17.3037	−1.13850
$232$	0	0
$233$	−19.7612	−1.29460	−0.647300	−	0.762236i	$-0.724102\pi$
$233$	−19.7612	−1.29460	−0.647300	+	0.762236i	$0.724102\pi$
$234$	0	0
$235$	44.7236	2.91745
$236$	0	0
$237$	−1.33451	−0.0866854
$238$	0	0
$239$	−13.5708	−0.877819	−0.438910	−	0.898531i	$-0.644635\pi$
$239$	−13.5708	−0.877819	−0.438910	+	0.898531i	$0.644635\pi$
$240$	0	0
$241$	19.3143	1.24414	0.622072	−	0.782960i	$-0.286291\pi$
$241$	19.3143	1.24414	0.622072	+	0.782960i	$0.286291\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	50.1997	3.20714
$246$	0	0
$247$	−5.76028	−0.366518
$248$	0	0
$249$	−14.1756	−0.898341
$250$	0	0
$251$	10.8135	0.682540	0.341270	−	0.939965i	$-0.389143\pi$
$251$	10.8135	0.682540	0.341270	+	0.939965i	$0.389143\pi$
$252$	0	0
$253$	11.6126	0.730080
$254$	0	0
$255$	21.5611	1.35021
$256$	0	0
$257$	−0.473302	−0.0295238	−0.0147619	−	0.999891i	$-0.504699\pi$
$257$	−0.473302	−0.0295238	−0.0147619	+	0.999891i	$0.504699\pi$
$258$	0	0
$259$	−15.8237	−0.983235
$260$	0	0
$261$	0.641815	0.0397274
$262$	0	0
$263$	−5.76686	−0.355600	−0.177800	−	0.984067i	$-0.556898\pi$
$263$	−5.76686	−0.355600	−0.177800	+	0.984067i	$0.556898\pi$
$264$	0	0
$265$	−7.66777	−0.471028
$266$	0	0
$267$	12.3702	0.757046
$268$	0	0
$269$	−2.62828	−0.160249	−0.0801247	−	0.996785i	$-0.525532\pi$
$269$	−2.62828	−0.160249	−0.0801247	+	0.996785i	$0.525532\pi$
$270$	0	0
$271$	−1.50936	−0.0916873	−0.0458437	−	0.998949i	$-0.514598\pi$
$271$	−1.50936	−0.0916873	−0.0458437	+	0.998949i	$0.514598\pi$
$272$	0	0
$273$	5.88567	0.356217
$274$	0	0
$275$	−43.6362	−2.63136
$276$	0	0
$277$	−11.3632	−0.682751	−0.341375	−	0.939927i	$-0.610893\pi$
$277$	−11.3632	−0.682751	−0.341375	+	0.939927i	$0.610893\pi$
$278$	0	0
$279$	4.50410	0.269653
$280$	0	0
$281$	−26.3821	−1.57382	−0.786911	−	0.617066i	$-0.788321\pi$
$281$	−26.3821	−1.57382	−0.786911	+	0.617066i	$0.788321\pi$
$282$	0	0
$283$	−19.9376	−1.18517	−0.592584	−	0.805509i	$-0.701892\pi$
$283$	−19.9376	−1.18517	−0.592584	+	0.805509i	$0.701892\pi$
$284$	0	0
$285$	−17.3588	−1.02825
$286$	0	0
$287$	−38.7858	−2.28945
$288$	0	0
$289$	11.8119	0.694819
$290$	0	0
$291$	13.3417	0.782103
$292$	0	0
$293$	−30.6368	−1.78982	−0.894911	−	0.446244i	$-0.852761\pi$
$293$	−30.6368	−1.78982	−0.894911	+	0.446244i	$0.852761\pi$
$294$	0	0
$295$	26.5486	1.54572
$296$	0	0
$297$	−3.91880	−0.227392
$298$	0	0
$299$	−3.94991	−0.228429
$300$	0	0
$301$	−30.1622	−1.73852
$302$	0	0
$303$	0.0616138	0.00353962
$304$	0	0
$305$	0.0330719	0.00189369
$306$	0	0
$307$	0.699590	0.0399277	0.0199638	−	0.999801i	$-0.493645\pi$
$307$	0.699590	0.0399277	0.0199638	+	0.999801i	$0.493645\pi$
$308$	0	0
$309$	1.58232	0.0900149
$310$	0	0
$311$	−15.0625	−0.854118	−0.427059	−	0.904224i	$-0.640450\pi$
$311$	−15.0625	−0.854118	−0.427059	+	0.904224i	$0.640450\pi$
$312$	0	0
$313$	−7.76381	−0.438836	−0.219418	−	0.975631i	$-0.570416\pi$
$313$	−7.76381	−0.438836	−0.219418	+	0.975631i	$0.570416\pi$
$314$	0	0
$315$	17.7367	0.999350
$316$	0	0
$317$	12.0203	0.675129	0.337565	−	0.941302i	$-0.390397\pi$
$317$	12.0203	0.675129	0.337565	+	0.941302i	$0.390397\pi$
$318$	0	0
$319$	−2.51514	−0.140821
$320$	0	0
$321$	15.1136	0.843560
$322$	0	0
$323$	−23.1964	−1.29068
$324$	0	0
$325$	14.8424	0.823306
$326$	0	0
$327$	−13.1743	−0.728540
$328$	0	0
$329$	49.1630	2.71044
$330$	0	0
$331$	4.62791	0.254373	0.127186	−	0.991879i	$-0.459405\pi$
$331$	4.62791	0.254373	0.127186	+	0.991879i	$0.459405\pi$
$332$	0	0
$333$	−3.58361	−0.196380
$334$	0	0
$335$	−16.1884	−0.884469
$336$	0	0
$337$	−4.04164	−0.220162	−0.110081	−	0.993923i	$-0.535111\pi$
$337$	−4.04164	−0.220162	−0.110081	+	0.993923i	$0.535111\pi$
$338$	0	0
$339$	20.1380	1.09375
$340$	0	0
$341$	−17.6506	−0.955836
$342$	0	0
$343$	24.2736	1.31065
$344$	0	0
$345$	−11.9032	−0.640847
$346$	0	0
$347$	37.0090	1.98675	0.993373	−	0.114933i	$-0.0366653\pi$
$347$	37.0090	1.98675	0.993373	+	0.114933i	$0.0366653\pi$
$348$	0	0
$349$	−5.18184	−0.277378	−0.138689	−	0.990336i	$-0.544289\pi$
$349$	−5.18184	−0.277378	−0.138689	+	0.990336i	$0.544289\pi$
$350$	0	0
$351$	1.33293	0.0711468
$352$	0	0
$353$	−28.9215	−1.53934	−0.769668	−	0.638444i	$-0.779578\pi$
$353$	−28.9215	−1.53934	−0.769668	+	0.638444i	$0.779578\pi$
$354$	0	0
$355$	9.40068	0.498937
$356$	0	0
$357$	23.7013	1.25441
$358$	0	0
$359$	−32.0737	−1.69279	−0.846393	−	0.532559i	$-0.821230\pi$
$359$	−32.0737	−1.69279	−0.846393	+	0.532559i	$0.821230\pi$
$360$	0	0
$361$	−0.324636	−0.0170861
$362$	0	0
$363$	4.35696	0.228681
$364$	0	0
$365$	20.0683	1.05042
$366$	0	0
$367$	2.98571	0.155853	0.0779264	−	0.996959i	$-0.475170\pi$
$367$	2.98571	0.155853	0.0779264	+	0.996959i	$0.475170\pi$
$368$	0	0
$369$	−8.78387	−0.457270
$370$	0	0
$371$	−8.42889	−0.437606
$372$	0	0
$373$	−26.5306	−1.37370	−0.686852	−	0.726797i	$-0.741008\pi$
$373$	−26.5306	−1.37370	−0.686852	+	0.726797i	$0.741008\pi$
$374$	0	0
$375$	24.6438	1.27260
$376$	0	0
$377$	0.855498	0.0440604
$378$	0	0
$379$	−11.6976	−0.600864	−0.300432	−	0.953803i	$-0.597131\pi$
$379$	−11.6976	−0.600864	−0.300432	+	0.953803i	$0.597131\pi$
$380$	0	0
$381$	−18.1668	−0.930711
$382$	0	0
$383$	14.4782	0.739800	0.369900	−	0.929072i	$-0.379392\pi$
$383$	14.4782	0.739800	0.369900	+	0.929072i	$0.379392\pi$
$384$	0	0
$385$	−69.5065	−3.54238
$386$	0	0
$387$	−6.83087	−0.347233
$388$	0	0
$389$	33.9739	1.72255	0.861273	−	0.508143i	$-0.169668\pi$
$389$	33.9739	1.72255	0.861273	+	0.508143i	$0.169668\pi$
$390$	0	0
$391$	−15.9061	−0.804406
$392$	0	0
$393$	−11.0441	−0.557104
$394$	0	0
$395$	−5.36051	−0.269716
$396$	0	0
$397$	21.9115	1.09971	0.549853	−	0.835262i	$-0.314684\pi$
$397$	21.9115	1.09971	0.549853	+	0.835262i	$0.314684\pi$
$398$	0	0
$399$	−19.0819	−0.955290
$400$	0	0
$401$	−15.2013	−0.759115	−0.379557	−	0.925168i	$-0.623924\pi$
$401$	−15.2013	−0.759115	−0.379557	+	0.925168i	$0.623924\pi$
$402$	0	0
$403$	6.00367	0.299064
$404$	0	0
$405$	4.01685	0.199599
$406$	0	0
$407$	14.0434	0.696106
$408$	0	0
$409$	−22.8815	−1.13141	−0.565707	−	0.824606i	$-0.691397\pi$
$409$	−22.8815	−1.13141	−0.565707	+	0.824606i	$0.691397\pi$
$410$	0	0
$411$	14.8528	0.732635
$412$	0	0
$413$	29.1838	1.43604
$414$	0	0
$415$	−56.9412	−2.79513
$416$	0	0
$417$	12.3986	0.607162
$418$	0	0
$419$	7.44289	0.363609	0.181804	−	0.983335i	$-0.441806\pi$
$419$	7.44289	0.363609	0.181804	+	0.983335i	$0.441806\pi$
$420$	0	0
$421$	−12.2244	−0.595780	−0.297890	−	0.954600i	$-0.596283\pi$
$421$	−12.2244	−0.595780	−0.297890	+	0.954600i	$0.596283\pi$
$422$	0	0
$423$	11.1340	0.541354
$424$	0	0
$425$	59.7696	2.89925
$426$	0	0
$427$	0.0363547	0.00175933
$428$	0	0
$429$	−5.22350	−0.252193
$430$	0	0
$431$	−7.14112	−0.343976	−0.171988	−	0.985099i	$-0.555019\pi$
$431$	−7.14112	−0.343976	−0.171988	+	0.985099i	$0.555019\pi$
$432$	0	0
$433$	−6.96710	−0.334818	−0.167409	−	0.985888i	$-0.553540\pi$
$433$	−6.96710	−0.334818	−0.167409	+	0.985888i	$0.553540\pi$
$434$	0	0
$435$	2.57808	0.123609
$436$	0	0
$437$	12.8060	0.612593
$438$	0	0
$439$	9.01824	0.430417	0.215209	−	0.976568i	$-0.430957\pi$
$439$	9.01824	0.430417	0.215209	+	0.976568i	$0.430957\pi$
$440$	0	0
$441$	12.4973	0.595108
$442$	0	0
$443$	−24.8441	−1.18038	−0.590189	−	0.807265i	$-0.700947\pi$
$443$	−24.8441	−1.18038	−0.590189	+	0.807265i	$0.700947\pi$
$444$	0	0
$445$	49.6894	2.35550
$446$	0	0
$447$	−17.3220	−0.819304
$448$	0	0
$449$	−6.68633	−0.315548	−0.157774	−	0.987475i	$-0.550432\pi$
$449$	−6.68633	−0.315548	−0.157774	+	0.987475i	$0.550432\pi$
$450$	0	0
$451$	34.4222	1.62088
$452$	0	0
$453$	12.7672	0.599855
$454$	0	0
$455$	23.6419	1.10835
$456$	0	0
$457$	−10.5537	−0.493683	−0.246842	−	0.969056i	$-0.579393\pi$
$457$	−10.5537	−0.493683	−0.246842	+	0.969056i	$0.579393\pi$
$458$	0	0
$459$	5.36767	0.250542
$460$	0	0
$461$	−17.5758	−0.818587	−0.409293	−	0.912403i	$-0.634225\pi$
$461$	−17.5758	−0.818587	−0.409293	+	0.912403i	$0.634225\pi$
$462$	0	0
$463$	0.242028	0.0112480	0.00562400	−	0.999984i	$-0.498210\pi$
$463$	0.242028	0.0112480	0.00562400	+	0.999984i	$0.498210\pi$
$464$	0	0
$465$	18.0923	0.839010
$466$	0	0
$467$	−32.1216	−1.48641	−0.743205	−	0.669064i	$-0.766695\pi$
$467$	−32.1216	−1.48641	−0.743205	+	0.669064i	$0.766695\pi$
$468$	0	0
$469$	−17.7953	−0.821712
$470$	0	0
$471$	1.00398	0.0462611
$472$	0	0
$473$	26.7688	1.23083
$474$	0	0
$475$	−48.1203	−2.20791
$476$	0	0
$477$	−1.90890	−0.0874026
$478$	0	0
$479$	−15.7450	−0.719405	−0.359703	−	0.933067i	$-0.617122\pi$
$479$	−15.7450	−0.719405	−0.359703	+	0.933067i	$0.617122\pi$
$480$	0	0
$481$	−4.77671	−0.217799
$482$	0	0
$483$	−13.0847	−0.595376
$484$	0	0
$485$	53.5915	2.43347
$486$	0	0
$487$	−22.7962	−1.03299	−0.516496	−	0.856289i	$-0.672764\pi$
$487$	−22.7962	−1.03299	−0.516496	+	0.856289i	$0.672764\pi$
$488$	0	0
$489$	−9.70804	−0.439013
$490$	0	0
$491$	2.40062	0.108338	0.0541692	−	0.998532i	$-0.482749\pi$
$491$	2.40062	0.108338	0.0541692	+	0.998532i	$0.482749\pi$
$492$	0	0
$493$	3.44505	0.155157
$494$	0	0
$495$	−15.7412	−0.707515
$496$	0	0
$497$	10.3338	0.463535
$498$	0	0
$499$	15.1018	0.676049	0.338024	−	0.941137i	$-0.390241\pi$
$499$	15.1018	0.676049	0.338024	+	0.941137i	$0.390241\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	35.3038	1.57412	0.787059	−	0.616878i	$-0.211603\pi$
$503$	35.3038	1.57412	0.787059	+	0.616878i	$0.211603\pi$
$504$	0	0
$505$	0.247494	0.0110133
$506$	0	0
$507$	−11.2233	−0.498444
$508$	0	0
$509$	−6.51112	−0.288601	−0.144300	−	0.989534i	$-0.546093\pi$
$509$	−6.51112	−0.288601	−0.144300	+	0.989534i	$0.546093\pi$
$510$	0	0
$511$	22.0603	0.975890
$512$	0	0
$513$	−4.32150	−0.190799
$514$	0	0
$515$	6.35594	0.280076
$516$	0	0
$517$	−43.6319	−1.91893
$518$	0	0
$519$	23.0933	1.01368
$520$	0	0
$521$	−16.3745	−0.717380	−0.358690	−	0.933457i	$-0.616776\pi$
$521$	−16.3745	−0.717380	−0.358690	+	0.933457i	$0.616776\pi$
$522$	0	0
$523$	−33.9619	−1.48505	−0.742526	−	0.669817i	$-0.766372\pi$
$523$	−33.9619	−1.48505	−0.742526	+	0.669817i	$0.766372\pi$
$524$	0	0
$525$	49.1678	2.14586
$526$	0	0
$527$	24.1765	1.05315
$528$	0	0
$529$	−14.2188	−0.618207
$530$	0	0
$531$	6.60930	0.286819
$532$	0	0
$533$	−11.7083	−0.507144
$534$	0	0
$535$	60.7092	2.62469
$536$	0	0
$537$	18.0863	0.780482
$538$	0	0
$539$	−48.9743	−2.10947
$540$	0	0
$541$	8.99706	0.386814	0.193407	−	0.981119i	$-0.438046\pi$
$541$	8.99706	0.386814	0.193407	+	0.981119i	$0.438046\pi$
$542$	0	0
$543$	−18.8159	−0.807470
$544$	0	0
$545$	−52.9192	−2.26681
$546$	0	0
$547$	20.4728	0.875355	0.437677	−	0.899132i	$-0.355801\pi$
$547$	20.4728	0.875355	0.437677	+	0.899132i	$0.355801\pi$
$548$	0	0
$549$	0.00823330	0.000351389 0
$550$	0	0
$551$	−2.77360	−0.118160
$552$	0	0
$553$	−5.89260	−0.250579
$554$	0	0
$555$	−14.3948	−0.611026
$556$	0	0
$557$	20.5319	0.869965	0.434983	−	0.900439i	$-0.356754\pi$
$557$	20.5319	0.869965	0.434983	+	0.900439i	$0.356754\pi$
$558$	0	0
$559$	−9.10511	−0.385105
$560$	0	0
$561$	−21.0348	−0.888091
$562$	0	0
$563$	19.3977	0.817514	0.408757	−	0.912643i	$-0.365962\pi$
$563$	19.3977	0.817514	0.408757	+	0.912643i	$0.365962\pi$
$564$	0	0
$565$	80.8915	3.40313
$566$	0	0
$567$	4.41557	0.185437
$568$	0	0
$569$	32.4646	1.36098	0.680492	−	0.732755i	$-0.261766\pi$
$569$	32.4646	1.36098	0.680492	+	0.732755i	$0.261766\pi$
$570$	0	0
$571$	−26.5493	−1.11105	−0.555527	−	0.831499i	$-0.687483\pi$
$571$	−26.5493	−1.11105	−0.555527	+	0.831499i	$0.687483\pi$
$572$	0	0
$573$	−5.24773	−0.219227
$574$	0	0
$575$	−32.9968	−1.37606
$576$	0	0
$577$	−2.82004	−0.117400	−0.0586999	−	0.998276i	$-0.518696\pi$
$577$	−2.82004	−0.117400	−0.0586999	+	0.998276i	$0.518696\pi$
$578$	0	0
$579$	15.4295	0.641230
$580$	0	0
$581$	−62.5933	−2.59681
$582$	0	0
$583$	7.48059	0.309815
$584$	0	0
$585$	5.35420	0.221369
$586$	0	0
$587$	26.9907	1.11403	0.557013	−	0.830504i	$-0.311948\pi$
$587$	26.9907	1.11403	0.557013	+	0.830504i	$0.311948\pi$
$588$	0	0
$589$	−19.4645	−0.802019
$590$	0	0
$591$	−4.27605	−0.175893
$592$	0	0
$593$	37.1868	1.52708	0.763538	−	0.645762i	$-0.223460\pi$
$593$	37.1868	1.52708	0.763538	+	0.645762i	$0.223460\pi$
$594$	0	0
$595$	95.2048	3.90302
$596$	0	0
$597$	−8.93953	−0.365871
$598$	0	0
$599$	−28.6209	−1.16942	−0.584709	−	0.811243i	$-0.698791\pi$
$599$	−28.6209	−1.16942	−0.584709	+	0.811243i	$0.698791\pi$
$600$	0	0
$601$	−16.6774	−0.680285	−0.340142	−	0.940374i	$-0.610475\pi$
$601$	−16.6774	−0.680285	−0.340142	+	0.940374i	$0.610475\pi$
$602$	0	0
$603$	−4.03013	−0.164120
$604$	0	0
$605$	17.5013	0.711528
$606$	0	0
$607$	23.8248	0.967020	0.483510	−	0.875339i	$-0.339362\pi$
$607$	23.8248	0.967020	0.483510	+	0.875339i	$0.339362\pi$
$608$	0	0
$609$	2.83398	0.114839
$610$	0	0
$611$	14.8409	0.600399
$612$	0	0
$613$	33.8091	1.36554	0.682769	−	0.730635i	$-0.260776\pi$
$613$	33.8091	1.36554	0.682769	+	0.730635i	$0.260776\pi$
$614$	0	0
$615$	−35.2835	−1.42277
$616$	0	0
$617$	1.75349	0.0705928	0.0352964	−	0.999377i	$-0.488762\pi$
$617$	1.75349	0.0705928	0.0352964	+	0.999377i	$0.488762\pi$
$618$	0	0
$619$	43.9500	1.76650	0.883251	−	0.468901i	$-0.155350\pi$
$619$	43.9500	1.76650	0.883251	+	0.468901i	$0.155350\pi$
$620$	0	0
$621$	−2.96332	−0.118914
$622$	0	0
$623$	54.6217	2.18837
$624$	0	0
$625$	43.3149	1.73260
$626$	0	0
$627$	16.9351	0.676322
$628$	0	0
$629$	−19.2356	−0.766975
$630$	0	0
$631$	−23.5677	−0.938215	−0.469108	−	0.883141i	$-0.655424\pi$
$631$	−23.5677	−0.938215	−0.469108	+	0.883141i	$0.655424\pi$
$632$	0	0
$633$	−20.7250	−0.823744
$634$	0	0
$635$	−72.9731	−2.89585
$636$	0	0
$637$	16.6581	0.660016
$638$	0	0
$639$	2.34031	0.0925813
$640$	0	0
$641$	−45.5487	−1.79907	−0.899533	−	0.436854i	$-0.856093\pi$
$641$	−45.5487	−1.79907	−0.899533	+	0.436854i	$0.856093\pi$
$642$	0	0
$643$	−2.60193	−0.102610	−0.0513050	−	0.998683i	$-0.516338\pi$
$643$	−2.60193	−0.102610	−0.0513050	+	0.998683i	$0.516338\pi$
$644$	0	0
$645$	−27.4386	−1.08039
$646$	0	0
$647$	0.0801218	0.00314991	0.00157496	−	0.999999i	$-0.499499\pi$
$647$	0.0801218	0.00314991	0.00157496	+	0.999999i	$0.499499\pi$
$648$	0	0
$649$	−25.9005	−1.01668
$650$	0	0
$651$	19.8882	0.779479
$652$	0	0
$653$	40.6905	1.59234	0.796171	−	0.605072i	$-0.206856\pi$
$653$	40.6905	1.59234	0.796171	+	0.605072i	$0.206856\pi$
$654$	0	0
$655$	−44.3627	−1.73339
$656$	0	0
$657$	4.99602	0.194913
$658$	0	0
$659$	−40.4684	−1.57642	−0.788212	−	0.615404i	$-0.788993\pi$
$659$	−40.4684	−1.57642	−0.788212	+	0.615404i	$0.788993\pi$
$660$	0	0
$661$	−18.0618	−0.702523	−0.351262	−	0.936277i	$-0.614247\pi$
$661$	−18.0618	−0.702523	−0.351262	+	0.936277i	$0.614247\pi$
$662$	0	0
$663$	7.15476	0.277868
$664$	0	0
$665$	−76.6491	−2.97233
$666$	0	0
$667$	−1.90190	−0.0736419
$668$	0	0
$669$	−10.2434	−0.396032
$670$	0	0
$671$	−0.0322646	−0.00124556
$672$	0	0
$673$	33.2906	1.28326	0.641630	−	0.767015i	$-0.278259\pi$
$673$	33.2906	1.28326	0.641630	+	0.767015i	$0.278259\pi$
$674$	0	0
$675$	11.1351	0.428590
$676$	0	0
$677$	41.0968	1.57948	0.789738	−	0.613444i	$-0.210216\pi$
$677$	41.0968	1.57948	0.789738	+	0.613444i	$0.210216\pi$
$678$	0	0
$679$	58.9111	2.26080
$680$	0	0
$681$	−17.0924	−0.654981
$682$	0	0
$683$	30.2871	1.15890	0.579451	−	0.815007i	$-0.303267\pi$
$683$	30.2871	1.15890	0.579451	+	0.815007i	$0.303267\pi$
$684$	0	0
$685$	59.6615	2.27955
$686$	0	0
$687$	24.4530	0.932939
$688$	0	0
$689$	−2.54444	−0.0969355
$690$	0	0
$691$	35.2114	1.33951	0.669753	−	0.742584i	$-0.266400\pi$
$691$	35.2114	1.33951	0.669753	+	0.742584i	$0.266400\pi$
$692$	0	0
$693$	−17.3037	−0.657314
$694$	0	0
$695$	49.8033	1.88915
$696$	0	0
$697$	−47.1490	−1.78589
$698$	0	0
$699$	−19.7612	−0.747437
$700$	0	0
$701$	13.6585	0.515876	0.257938	−	0.966162i	$-0.416957\pi$
$701$	13.6585	0.515876	0.257938	+	0.966162i	$0.416957\pi$
$702$	0	0
$703$	15.4866	0.584086
$704$	0	0
$705$	44.7236	1.68439
$706$	0	0
$707$	0.272060	0.0102319
$708$	0	0
$709$	−28.6575	−1.07626	−0.538128	−	0.842863i	$-0.680868\pi$
$709$	−28.6575	−1.07626	−0.538128	+	0.842863i	$0.680868\pi$
$710$	0	0
$711$	−1.33451	−0.0500479
$712$	0	0
$713$	−13.3471	−0.499851
$714$	0	0
$715$	−20.9820	−0.784683
$716$	0	0
$717$	−13.5708	−0.506809
$718$	0	0
$719$	49.4611	1.84459	0.922295	−	0.386487i	$-0.126312\pi$
$719$	49.4611	1.84459	0.922295	+	0.386487i	$0.126312\pi$
$720$	0	0
$721$	6.98684	0.260204
$722$	0	0
$723$	19.3143	0.718307
$724$	0	0
$725$	7.14667	0.265421
$726$	0	0
$727$	−14.4539	−0.536067	−0.268034	−	0.963410i	$-0.586374\pi$
$727$	−14.4539	−0.536067	−0.268034	+	0.963410i	$0.586374\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−36.6659	−1.35614
$732$	0	0
$733$	−23.5450	−0.869653	−0.434827	−	0.900514i	$-0.643190\pi$
$733$	−23.5450	−0.869653	−0.434827	+	0.900514i	$0.643190\pi$
$734$	0	0
$735$	50.1997	1.85164
$736$	0	0
$737$	15.7933	0.581752
$738$	0	0
$739$	12.1706	0.447704	0.223852	−	0.974623i	$-0.428137\pi$
$739$	12.1706	0.447704	0.223852	+	0.974623i	$0.428137\pi$
$740$	0	0
$741$	−5.76028	−0.211609
$742$	0	0
$743$	22.7882	0.836017	0.418008	−	0.908443i	$-0.362728\pi$
$743$	22.7882	0.836017	0.418008	+	0.908443i	$0.362728\pi$
$744$	0	0
$745$	−69.5800	−2.54921
$746$	0	0
$747$	−14.1756	−0.518658
$748$	0	0
$749$	66.7353	2.43845
$750$	0	0
$751$	35.0391	1.27859	0.639297	−	0.768960i	$-0.279225\pi$
$751$	35.0391	1.27859	0.639297	+	0.768960i	$0.279225\pi$
$752$	0	0
$753$	10.8135	0.394065
$754$	0	0
$755$	51.2839	1.86641
$756$	0	0
$757$	0.874303	0.0317771	0.0158885	−	0.999874i	$-0.494942\pi$
$757$	0.874303	0.0317771	0.0158885	+	0.999874i	$0.494942\pi$
$758$	0	0
$759$	11.6126	0.421512
$760$	0	0
$761$	−21.2232	−0.769340	−0.384670	−	0.923054i	$-0.625685\pi$
$761$	−21.2232	−0.769340	−0.384670	+	0.923054i	$0.625685\pi$
$762$	0	0
$763$	−58.1720	−2.10597
$764$	0	0
$765$	21.5611	0.779545
$766$	0	0
$767$	8.80977	0.318102
$768$	0	0
$769$	−53.3473	−1.92375	−0.961876	−	0.273487i	$-0.911823\pi$
$769$	−53.3473	−1.92375	−0.961876	+	0.273487i	$0.911823\pi$
$770$	0	0
$771$	−0.473302	−0.0170456
$772$	0	0
$773$	11.4161	0.410609	0.205305	−	0.978698i	$-0.434181\pi$
$773$	11.4161	0.410609	0.205305	+	0.978698i	$0.434181\pi$
$774$	0	0
$775$	50.1535	1.80157
$776$	0	0
$777$	−15.8237	−0.567671
$778$	0	0
$779$	37.9595	1.36004
$780$	0	0
$781$	−9.17120	−0.328172
$782$	0	0
$783$	0.641815	0.0229366
$784$	0	0
$785$	4.03285	0.143938
$786$	0	0
$787$	37.1510	1.32429	0.662145	−	0.749375i	$-0.269646\pi$
$787$	37.1510	1.32429	0.662145	+	0.749375i	$0.269646\pi$
$788$	0	0
$789$	−5.76686	−0.205306
$790$	0	0
$791$	88.9209	3.16166
$792$	0	0
$793$	0.0109745	0.000389714 0
$794$	0	0
$795$	−7.66777	−0.271948
$796$	0	0
$797$	12.4526	0.441093	0.220547	−	0.975376i	$-0.429216\pi$
$797$	12.4526	0.441093	0.220547	+	0.975376i	$0.429216\pi$
$798$	0	0
$799$	59.7637	2.11429
$800$	0	0
$801$	12.3702	0.437081
$802$	0	0
$803$	−19.5784	−0.690906
$804$	0	0
$805$	−52.5594	−1.85248
$806$	0	0
$807$	−2.62828	−0.0925200
$808$	0	0
$809$	50.0293	1.75894	0.879469	−	0.475957i	$-0.157898\pi$
$809$	50.0293	1.75894	0.879469	+	0.475957i	$0.157898\pi$
$810$	0	0
$811$	−40.8278	−1.43366	−0.716829	−	0.697249i	$-0.754407\pi$
$811$	−40.8278	−1.43366	−0.716829	+	0.697249i	$0.754407\pi$
$812$	0	0
$813$	−1.50936	−0.0529357
$814$	0	0
$815$	−38.9957	−1.36596
$816$	0	0
$817$	29.5196	1.03276
$818$	0	0
$819$	5.88567	0.205662
$820$	0	0
$821$	−27.7502	−0.968490	−0.484245	−	0.874932i	$-0.660906\pi$
$821$	−27.7502	−0.968490	−0.484245	+	0.874932i	$0.660906\pi$
$822$	0	0
$823$	44.3295	1.54523	0.772614	−	0.634876i	$-0.218949\pi$
$823$	44.3295	1.54523	0.772614	+	0.634876i	$0.218949\pi$
$824$	0	0
$825$	−43.6362	−1.51922
$826$	0	0
$827$	8.50957	0.295907	0.147953	−	0.988994i	$-0.452731\pi$
$827$	8.50957	0.295907	0.147953	+	0.988994i	$0.452731\pi$
$828$	0	0
$829$	−1.03665	−0.0360045	−0.0180022	−	0.999838i	$-0.505731\pi$
$829$	−1.03665	−0.0360045	−0.0180022	+	0.999838i	$0.505731\pi$
$830$	0	0
$831$	−11.3632	−0.394186
$832$	0	0
$833$	67.0813	2.32423
$834$	0	0
$835$	−4.01685	−0.139009
$836$	0	0
$837$	4.50410	0.155684
$838$	0	0
$839$	26.5492	0.916580	0.458290	−	0.888803i	$-0.348462\pi$
$839$	26.5492	0.916580	0.458290	+	0.888803i	$0.348462\pi$
$840$	0	0
$841$	−28.5881	−0.985796
$842$	0	0
$843$	−26.3821	−0.908647
$844$	0	0
$845$	−45.0823	−1.55088
$846$	0	0
$847$	19.2385	0.661042
$848$	0	0
$849$	−19.9376	−0.684257
$850$	0	0
$851$	10.6194	0.364027
$852$	0	0
$853$	−37.6529	−1.28921	−0.644606	−	0.764515i	$-0.722978\pi$
$853$	−37.6529	−1.28921	−0.644606	+	0.764515i	$0.722978\pi$
$854$	0	0
$855$	−17.3588	−0.593659
$856$	0	0
$857$	8.32877	0.284505	0.142253	−	0.989830i	$-0.454565\pi$
$857$	8.32877	0.284505	0.142253	+	0.989830i	$0.454565\pi$
$858$	0	0
$859$	−6.27936	−0.214249	−0.107125	−	0.994246i	$-0.534164\pi$
$859$	−6.27936	−0.214249	−0.107125	+	0.994246i	$0.534164\pi$
$860$	0	0
$861$	−38.7858	−1.32182
$862$	0	0
$863$	−0.750794	−0.0255573	−0.0127787	−	0.999918i	$-0.504068\pi$
$863$	−0.750794	−0.0255573	−0.0127787	+	0.999918i	$0.504068\pi$
$864$	0	0
$865$	92.7624	3.15402
$866$	0	0
$867$	11.8119	0.401154
$868$	0	0
$869$	5.22965	0.177404
$870$	0	0
$871$	−5.37190	−0.182020
$872$	0	0
$873$	13.3417	0.451547
$874$	0	0
$875$	108.816	3.67866
$876$	0	0
$877$	−2.08436	−0.0703839	−0.0351920	−	0.999381i	$-0.511204\pi$
$877$	−2.08436	−0.0703839	−0.0351920	+	0.999381i	$0.511204\pi$
$878$	0	0
$879$	−30.6368	−1.03335
$880$	0	0
$881$	−27.7826	−0.936018	−0.468009	−	0.883724i	$-0.655029\pi$
$881$	−27.7826	−0.936018	−0.468009	+	0.883724i	$0.655029\pi$
$882$	0	0
$883$	5.29815	0.178297	0.0891484	−	0.996018i	$-0.471585\pi$
$883$	5.29815	0.178297	0.0891484	+	0.996018i	$0.471585\pi$
$884$	0	0
$885$	26.5486	0.892421
$886$	0	0
$887$	48.2315	1.61946	0.809728	−	0.586805i	$-0.199615\pi$
$887$	48.2315	1.61946	0.809728	+	0.586805i	$0.199615\pi$
$888$	0	0
$889$	−80.2166	−2.69038
$890$	0	0
$891$	−3.91880	−0.131285
$892$	0	0
$893$	−48.1156	−1.61013
$894$	0	0
$895$	72.6500	2.42842
$896$	0	0
$897$	−3.94991	−0.131884
$898$	0	0
$899$	2.89080	0.0964135
$900$	0	0
$901$	−10.2464	−0.341356
$902$	0	0
$903$	−30.1622	−1.00374
$904$	0	0
$905$	−75.5809	−2.51239
$906$	0	0
$907$	23.6247	0.784445	0.392222	−	0.919870i	$-0.371706\pi$
$907$	23.6247	0.784445	0.392222	+	0.919870i	$0.371706\pi$
$908$	0	0
$909$	0.0616138	0.00204360
$910$	0	0
$911$	31.1298	1.03138	0.515688	−	0.856777i	$-0.327536\pi$
$911$	31.1298	1.03138	0.515688	+	0.856777i	$0.327536\pi$
$912$	0	0
$913$	55.5512	1.83848
$914$	0	0
$915$	0.0330719	0.00109332
$916$	0	0
$917$	−48.7662	−1.61040
$918$	0	0
$919$	−33.7448	−1.11314	−0.556569	−	0.830801i	$-0.687883\pi$
$919$	−33.7448	−1.11314	−0.556569	+	0.830801i	$0.687883\pi$
$920$	0	0
$921$	0.699590	0.0230523
$922$	0	0
$923$	3.11948	0.102679
$924$	0	0
$925$	−39.9038	−1.31203
$926$	0	0
$927$	1.58232	0.0519702
$928$	0	0
$929$	−21.5423	−0.706779	−0.353390	−	0.935476i	$-0.614971\pi$
$929$	−21.5423	−0.706779	−0.353390	+	0.935476i	$0.614971\pi$
$930$	0	0
$931$	−54.0070	−1.77001
$932$	0	0
$933$	−15.0625	−0.493125
$934$	0	0
$935$	−84.4937	−2.76324
$936$	0	0
$937$	52.7027	1.72172	0.860861	−	0.508841i	$-0.169926\pi$
$937$	52.7027	1.72172	0.860861	+	0.508841i	$0.169926\pi$
$938$	0	0
$939$	−7.76381	−0.253362
$940$	0	0
$941$	55.8970	1.82219	0.911094	−	0.412198i	$-0.135239\pi$
$941$	55.8970	1.82219	0.911094	+	0.412198i	$0.135239\pi$
$942$	0	0
$943$	26.0294	0.847633
$944$	0	0
$945$	17.7367	0.576975
$946$	0	0
$947$	9.46512	0.307575	0.153788	−	0.988104i	$-0.450853\pi$
$947$	9.46512	0.307575	0.153788	+	0.988104i	$0.450853\pi$
$948$	0	0
$949$	6.65937	0.216172
$950$	0	0
$951$	12.0203	0.389786
$952$	0	0
$953$	−20.4719	−0.663151	−0.331575	−	0.943429i	$-0.607580\pi$
$953$	−20.4719	−0.663151	−0.331575	+	0.943429i	$0.607580\pi$
$954$	0	0
$955$	−21.0794	−0.682112
$956$	0	0
$957$	−2.51514	−0.0813030
$958$	0	0
$959$	65.5836	2.11781
$960$	0	0
$961$	−10.7131	−0.345584
$962$	0	0
$963$	15.1136	0.487030
$964$	0	0
$965$	61.9782	1.99515
$966$	0	0
$967$	−60.8243	−1.95598	−0.977988	−	0.208660i	$-0.933090\pi$
$967$	−60.8243	−1.95598	−0.977988	+	0.208660i	$0.933090\pi$
$968$	0	0
$969$	−23.1964	−0.745176
$970$	0	0
$971$	−45.6775	−1.46586	−0.732930	−	0.680304i	$-0.761848\pi$
$971$	−45.6775	−1.46586	−0.732930	+	0.680304i	$0.761848\pi$
$972$	0	0
$973$	54.7469	1.75511
$974$	0	0
$975$	14.8424	0.475336
$976$	0	0
$977$	−56.6492	−1.81237	−0.906183	−	0.422885i	$-0.861018\pi$
$977$	−56.6492	−1.81237	−0.906183	+	0.422885i	$0.861018\pi$
$978$	0	0
$979$	−48.4765	−1.54931
$980$	0	0
$981$	−13.1743	−0.420623
$982$	0	0
$983$	39.9879	1.27542	0.637708	−	0.770278i	$-0.279883\pi$
$983$	39.9879	1.27542	0.637708	+	0.770278i	$0.279883\pi$
$984$	0	0
$985$	−17.1762	−0.547281
$986$	0	0
$987$	49.1630	1.56488
$988$	0	0
$989$	20.2420	0.643659
$990$	0	0
$991$	−13.6527	−0.433692	−0.216846	−	0.976206i	$-0.569577\pi$
$991$	−13.6527	−0.433692	−0.216846	+	0.976206i	$0.569577\pi$
$992$	0	0
$993$	4.62791	0.146862
$994$	0	0
$995$	−35.9088	−1.13838
$996$	0	0
$997$	16.9689	0.537410	0.268705	−	0.963223i	$-0.413404\pi$
$997$	16.9689	0.537410	0.268705	+	0.963223i	$0.413404\pi$
$998$	0	0
$999$	−3.58361	−0.113380

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.be.1.11		11
4.3	odd	2		4008.2.a.k.1.11	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.k.1.11	✓	11	4.3	odd	2
8016.2.a.be.1.11		11	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8016.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +4.01685 q^{5} +4.41557 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +4.01685 q^{5} +4.41557 q^{7} +1.00000 q^{9} -3.91880 q^{11} +1.33293 q^{13} +4.01685 q^{15} +5.36767 q^{17} -4.32150 q^{19} +4.41557 q^{21} -2.96332 q^{23} +11.1351 q^{25} +1.00000 q^{27} +0.641815 q^{29} +4.50410 q^{31} -3.91880 q^{33} +17.7367 q^{35} -3.58361 q^{37} +1.33293 q^{39} -8.78387 q^{41} -6.83087 q^{43} +4.01685 q^{45} +11.1340 q^{47} +12.4973 q^{49} +5.36767 q^{51} -1.90890 q^{53} -15.7412 q^{55} -4.32150 q^{57} +6.60930 q^{59} +0.00823330 q^{61} +4.41557 q^{63} +5.35420 q^{65} -4.03013 q^{67} -2.96332 q^{69} +2.34031 q^{71} +4.99602 q^{73} +11.1351 q^{75} -17.3037 q^{77} -1.33451 q^{79} +1.00000 q^{81} -14.1756 q^{83} +21.5611 q^{85} +0.641815 q^{87} +12.3702 q^{89} +5.88567 q^{91} +4.50410 q^{93} -17.3588 q^{95} +13.3417 q^{97} -3.91880 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 11 q^{3} + 10 q^{5} + q^{7} + 11 q^{9}+O(q^{10}) \) 11 * q + 11 * q^3 + 10 * q^5 + q^7 + 11 * q^9	\( 11 q + 11 q^{3} + 10 q^{5} + q^{7} + 11 q^{9} + q^{11} + 10 q^{13} + 10 q^{15} + 17 q^{17} - 2 q^{19} + q^{21} + 3 q^{23} + 21 q^{25} + 11 q^{27} + 17 q^{29} + 15 q^{31} + q^{33} - 11 q^{35} + 4 q^{37} + 10 q^{39} + 16 q^{41} - 10 q^{43} + 10 q^{45} + 16 q^{47} + 22 q^{49} + 17 q^{51} + 42 q^{53} + 5 q^{55} - 2 q^{57} + 2 q^{59} + 12 q^{61} + q^{63} + 10 q^{65} + q^{67} + 3 q^{69} + 9 q^{71} + 24 q^{73} + 21 q^{75} + 22 q^{77} + 30 q^{79} + 11 q^{81} - 16 q^{83} + 25 q^{85} + 17 q^{87} + 37 q^{89} - q^{91} + 15 q^{93} - 5 q^{95} + 4 q^{97} + q^{99}+O(q^{100}) \) 11 * q + 11 * q^3 + 10 * q^5 + q^7 + 11 * q^9 + q^11 + 10 * q^13 + 10 * q^15 + 17 * q^17 - 2 * q^19 + q^21 + 3 * q^23 + 21 * q^25 + 11 * q^27 + 17 * q^29 + 15 * q^31 + q^33 - 11 * q^35 + 4 * q^37 + 10 * q^39 + 16 * q^41 - 10 * q^43 + 10 * q^45 + 16 * q^47 + 22 * q^49 + 17 * q^51 + 42 * q^53 + 5 * q^55 - 2 * q^57 + 2 * q^59 + 12 * q^61 + q^63 + 10 * q^65 + q^67 + 3 * q^69 + 9 * q^71 + 24 * q^73 + 21 * q^75 + 22 * q^77 + 30 * q^79 + 11 * q^81 - 16 * q^83 + 25 * q^85 + 17 * q^87 + 37 * q^89 - q^91 + 15 * q^93 - 5 * q^95 + 4 * q^97 + q^99

Introduction

L-functions

Modular forms

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Database

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