LMFDB - Embedded newform 4016.2.a.f.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4016,2,Mod(1,4016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4016, 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("4016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4016 = 2^{4} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4016.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$32.0679214517$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.60853001.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 8x^{4} + 15x^{3} + 20x^{2} - 12x - 5 $ x^6 - 3x^5 - 8x^4 + 15x^3 + 20x^2 - 12*x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 502)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.57676$ of defining polynomial
Character	$\chi$	$=$	4016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.87121 q^{3} -3.46767 q^{5} -0.332919 q^{7} +5.24384 q^{9} +O(q^{10})$	$q+2.87121 q^{3} -3.46767 q^{5} -0.332919 q^{7} +5.24384 q^{9} -2.63967 q^{11} -1.68584 q^{13} -9.95640 q^{15} +7.27596 q^{17} -0.928472 q^{19} -0.955880 q^{21} -6.01336 q^{23} +7.02472 q^{25} +6.44253 q^{27} -6.02227 q^{29} +1.65844 q^{31} -7.57905 q^{33} +1.15445 q^{35} +2.35858 q^{37} -4.84041 q^{39} +3.70781 q^{41} -11.4727 q^{43} -18.1839 q^{45} -2.14124 q^{47} -6.88917 q^{49} +20.8908 q^{51} +4.74332 q^{53} +9.15351 q^{55} -2.66584 q^{57} -8.11822 q^{59} -12.6397 q^{61} -1.74577 q^{63} +5.84595 q^{65} +4.45533 q^{67} -17.2656 q^{69} +8.96205 q^{71} -8.01468 q^{73} +20.1694 q^{75} +0.878797 q^{77} -16.0788 q^{79} +2.76632 q^{81} -15.6018 q^{83} -25.2306 q^{85} -17.2912 q^{87} +14.3458 q^{89} +0.561250 q^{91} +4.76172 q^{93} +3.21963 q^{95} -4.15255 q^{97} -13.8420 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{3} - q^{5} - 6 q^{7} + 15 q^{9}+O(q^{10}) $ 6 * q - q^3 - q^5 - 6 * q^7 + 15 * q^9	$ 6 q - q^{3} - q^{5} - 6 q^{7} + 15 q^{9} - q^{11} - 5 q^{13} - 2 q^{15} + 8 q^{17} - 3 q^{19} - 14 q^{21} - 18 q^{23} - q^{25} - 16 q^{27} + q^{29} - 6 q^{31} - 16 q^{33} - 6 q^{35} - 13 q^{37} + 6 q^{39} + 4 q^{41} + 5 q^{43} - 23 q^{45} - 8 q^{47} + 8 q^{49} + 16 q^{51} - 3 q^{53} + 30 q^{55} - 24 q^{57} + 5 q^{59} - 61 q^{61} + 27 q^{63} - q^{65} + 13 q^{67} - 21 q^{69} - 22 q^{71} + 6 q^{73} + 30 q^{75} + 4 q^{77} - 28 q^{79} + 2 q^{81} - 14 q^{83} - 16 q^{85} + 24 q^{87} + 18 q^{89} + 16 q^{91} + 27 q^{93} - 20 q^{95} + 16 q^{97} - 5 q^{99}+O(q^{100}) $ 6 * q - q^3 - q^5 - 6 * q^7 + 15 * q^9 - q^11 - 5 * q^13 - 2 * q^15 + 8 * q^17 - 3 * q^19 - 14 * q^21 - 18 * q^23 - q^25 - 16 * q^27 + q^29 - 6 * q^31 - 16 * q^33 - 6 * q^35 - 13 * q^37 + 6 * q^39 + 4 * q^41 + 5 * q^43 - 23 * q^45 - 8 * q^47 + 8 * q^49 + 16 * q^51 - 3 * q^53 + 30 * q^55 - 24 * q^57 + 5 * q^59 - 61 * q^61 + 27 * q^63 - q^65 + 13 * q^67 - 21 * q^69 - 22 * q^71 + 6 * q^73 + 30 * q^75 + 4 * q^77 - 28 * q^79 + 2 * q^81 - 14 * q^83 - 16 * q^85 + 24 * q^87 + 18 * q^89 + 16 * q^91 + 27 * q^93 - 20 * q^95 + 16 * q^97 - 5 * 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
$3$	2.87121	1.65769	0.828846	−	0.559476i	$-0.188998\pi$
$3$	2.87121	1.65769	0.828846	+	0.559476i	$0.188998\pi$
$4$	0	0
$5$	−3.46767	−1.55079	−0.775394	−	0.631478i	$-0.782449\pi$
$5$	−3.46767	−1.55079	−0.775394	+	0.631478i	$0.782449\pi$
$6$	0	0
$7$	−0.332919	−0.125832	−0.0629158	−	0.998019i	$-0.520040\pi$
$7$	−0.332919	−0.125832	−0.0629158	+	0.998019i	$0.520040\pi$
$8$	0	0
$9$	5.24384	1.74795
$10$	0	0
$11$	−2.63967	−0.795892	−0.397946	−	0.917409i	$-0.630277\pi$
$11$	−2.63967	−0.795892	−0.397946	+	0.917409i	$0.630277\pi$
$12$	0	0
$13$	−1.68584	−0.467569	−0.233785	−	0.972288i	$-0.575111\pi$
$13$	−1.68584	−0.467569	−0.233785	+	0.972288i	$0.575111\pi$
$14$	0	0
$15$	−9.95640	−2.57073
$16$	0	0
$17$	7.27596	1.76468	0.882340	−	0.470613i	$-0.155967\pi$
$17$	7.27596	1.76468	0.882340	+	0.470613i	$0.155967\pi$
$18$	0	0
$19$	−0.928472	−0.213006	−0.106503	−	0.994312i	$-0.533965\pi$
$19$	−0.928472	−0.213006	−0.106503	+	0.994312i	$0.533965\pi$
$20$	0	0
$21$	−0.955880	−0.208590
$22$	0	0
$23$	−6.01336	−1.25387	−0.626936	−	0.779071i	$-0.715691\pi$
$23$	−6.01336	−1.25387	−0.626936	+	0.779071i	$0.715691\pi$
$24$	0	0
$25$	7.02472	1.40494
$26$	0	0
$27$	6.44253	1.23986
$28$	0	0
$29$	−6.02227	−1.11831	−0.559154	−	0.829064i	$-0.688874\pi$
$29$	−6.02227	−1.11831	−0.559154	+	0.829064i	$0.688874\pi$
$30$	0	0
$31$	1.65844	0.297864	0.148932	−	0.988847i	$-0.452416\pi$
$31$	1.65844	0.297864	0.148932	+	0.988847i	$0.452416\pi$
$32$	0	0
$33$	−7.57905	−1.31934
$34$	0	0
$35$	1.15445	0.195138
$36$	0	0
$37$	2.35858	0.387748	0.193874	−	0.981026i	$-0.437895\pi$
$37$	2.35858	0.387748	0.193874	+	0.981026i	$0.437895\pi$
$38$	0	0
$39$	−4.84041	−0.775086
$40$	0	0
$41$	3.70781	0.579063	0.289532	−	0.957168i	$-0.406500\pi$
$41$	3.70781	0.579063	0.289532	+	0.957168i	$0.406500\pi$
$42$	0	0
$43$	−11.4727	−1.74957	−0.874784	−	0.484513i	$-0.838997\pi$
$43$	−11.4727	−1.74957	−0.874784	+	0.484513i	$0.838997\pi$
$44$	0	0
$45$	−18.1839	−2.71069
$46$	0	0
$47$	−2.14124	−0.312332	−0.156166	−	0.987731i	$-0.549914\pi$
$47$	−2.14124	−0.312332	−0.156166	+	0.987731i	$0.549914\pi$
$48$	0	0
$49$	−6.88917	−0.984166
$50$	0	0
$51$	20.8908	2.92530
$52$	0	0
$53$	4.74332	0.651546	0.325773	−	0.945448i	$-0.394376\pi$
$53$	4.74332	0.651546	0.325773	+	0.945448i	$0.394376\pi$
$54$	0	0
$55$	9.15351	1.23426
$56$	0	0
$57$	−2.66584	−0.353099
$58$	0	0
$59$	−8.11822	−1.05690	−0.528451	−	0.848964i	$-0.677227\pi$
$59$	−8.11822	−1.05690	−0.528451	+	0.848964i	$0.677227\pi$
$60$	0	0
$61$	−12.6397	−1.61834	−0.809172	−	0.587572i	$-0.800084\pi$
$61$	−12.6397	−1.61834	−0.809172	+	0.587572i	$0.800084\pi$
$62$	0	0
$63$	−1.74577	−0.219947
$64$	0	0
$65$	5.84595	0.725101
$66$	0	0
$67$	4.45533	0.544305	0.272153	−	0.962254i	$-0.412264\pi$
$67$	4.45533	0.544305	0.272153	+	0.962254i	$0.412264\pi$
$68$	0	0
$69$	−17.2656	−2.07853
$70$	0	0
$71$	8.96205	1.06360	0.531800	−	0.846870i	$-0.321516\pi$
$71$	8.96205	1.06360	0.531800	+	0.846870i	$0.321516\pi$
$72$	0	0
$73$	−8.01468	−0.938048	−0.469024	−	0.883186i	$-0.655394\pi$
$73$	−8.01468	−0.938048	−0.469024	+	0.883186i	$0.655394\pi$
$74$	0	0
$75$	20.1694	2.32897
$76$	0	0
$77$	0.878797	0.100148
$78$	0	0
$79$	−16.0788	−1.80901	−0.904506	−	0.426462i	$-0.859760\pi$
$79$	−16.0788	−1.80901	−0.904506	+	0.426462i	$0.859760\pi$
$80$	0	0
$81$	2.76632	0.307369
$82$	0	0
$83$	−15.6018	−1.71252	−0.856259	−	0.516547i	$-0.827217\pi$
$83$	−15.6018	−1.71252	−0.856259	+	0.516547i	$0.827217\pi$
$84$	0	0
$85$	−25.2306	−2.73664
$86$	0	0
$87$	−17.2912	−1.85381
$88$	0	0
$89$	14.3458	1.52065	0.760324	−	0.649543i	$-0.225040\pi$
$89$	14.3458	1.52065	0.760324	+	0.649543i	$0.225040\pi$
$90$	0	0
$91$	0.561250	0.0588350
$92$	0	0
$93$	4.76172	0.493767
$94$	0	0
$95$	3.21963	0.330327
$96$	0	0
$97$	−4.15255	−0.421628	−0.210814	−	0.977526i	$-0.567611\pi$
$97$	−4.15255	−0.421628	−0.210814	+	0.977526i	$0.567611\pi$
$98$	0	0
$99$	−13.8420	−1.39118
$100$	0	0
$101$	−5.83675	−0.580779	−0.290389	−	0.956909i	$-0.593785\pi$
$101$	−5.83675	−0.580779	−0.290389	+	0.956909i	$0.593785\pi$
$102$	0	0
$103$	−8.18219	−0.806215	−0.403107	−	0.915153i	$-0.632070\pi$
$103$	−8.18219	−0.806215	−0.403107	+	0.915153i	$0.632070\pi$
$104$	0	0
$105$	3.31467	0.323479
$106$	0	0
$107$	−7.88103	−0.761888	−0.380944	−	0.924598i	$-0.624401\pi$
$107$	−7.88103	−0.761888	−0.380944	+	0.924598i	$0.624401\pi$
$108$	0	0
$109$	0.784054	0.0750988	0.0375494	−	0.999295i	$-0.488045\pi$
$109$	0.784054	0.0750988	0.0375494	+	0.999295i	$0.488045\pi$
$110$	0	0
$111$	6.77197	0.642767
$112$	0	0
$113$	4.56909	0.429823	0.214912	−	0.976633i	$-0.431054\pi$
$113$	4.56909	0.429823	0.214912	+	0.976633i	$0.431054\pi$
$114$	0	0
$115$	20.8523	1.94449
$116$	0	0
$117$	−8.84030	−0.817286
$118$	0	0
$119$	−2.42230	−0.222052
$120$	0	0
$121$	−4.03212	−0.366556
$122$	0	0
$123$	10.6459	0.959909
$124$	0	0
$125$	−7.02106	−0.627983
$126$	0	0
$127$	9.50205	0.843171	0.421585	−	0.906789i	$-0.361474\pi$
$127$	9.50205	0.843171	0.421585	+	0.906789i	$0.361474\pi$
$128$	0	0
$129$	−32.9405	−2.90025
$130$	0	0
$131$	17.1953	1.50236	0.751180	−	0.660098i	$-0.229485\pi$
$131$	17.1953	1.50236	0.751180	+	0.660098i	$0.229485\pi$
$132$	0	0
$133$	0.309106	0.0268029
$134$	0	0
$135$	−22.3405	−1.92277
$136$	0	0
$137$	17.4605	1.49175	0.745877	−	0.666083i	$-0.232031\pi$
$137$	17.4605	1.49175	0.745877	+	0.666083i	$0.232031\pi$
$138$	0	0
$139$	−22.0794	−1.87275	−0.936376	−	0.350999i	$-0.885842\pi$
$139$	−22.0794	−1.87275	−0.936376	+	0.350999i	$0.885842\pi$
$140$	0	0
$141$	−6.14795	−0.517751
$142$	0	0
$143$	4.45008	0.372134
$144$	0	0
$145$	20.8832	1.73426
$146$	0	0
$147$	−19.7802	−1.63145
$148$	0	0
$149$	4.97305	0.407408	0.203704	−	0.979033i	$-0.434702\pi$
$149$	4.97305	0.407408	0.203704	+	0.979033i	$0.434702\pi$
$150$	0	0
$151$	17.2470	1.40354	0.701769	−	0.712405i	$-0.252394\pi$
$151$	17.2470	1.40354	0.701769	+	0.712405i	$0.252394\pi$
$152$	0	0
$153$	38.1539	3.08456
$154$	0	0
$155$	−5.75091	−0.461924
$156$	0	0
$157$	9.09247	0.725658	0.362829	−	0.931856i	$-0.381811\pi$
$157$	9.09247	0.725658	0.362829	+	0.931856i	$0.381811\pi$
$158$	0	0
$159$	13.6191	1.08006
$160$	0	0
$161$	2.00196	0.157777
$162$	0	0
$163$	−10.5804	−0.828718	−0.414359	−	0.910114i	$-0.635994\pi$
$163$	−10.5804	−0.828718	−0.414359	+	0.910114i	$0.635994\pi$
$164$	0	0
$165$	26.2816	2.04602
$166$	0	0
$167$	−4.18065	−0.323508	−0.161754	−	0.986831i	$-0.551715\pi$
$167$	−4.18065	−0.323508	−0.161754	+	0.986831i	$0.551715\pi$
$168$	0	0
$169$	−10.1579	−0.781379
$170$	0	0
$171$	−4.86876	−0.372323
$172$	0	0
$173$	8.42554	0.640582	0.320291	−	0.947319i	$-0.396219\pi$
$173$	8.42554	0.640582	0.320291	+	0.947319i	$0.396219\pi$
$174$	0	0
$175$	−2.33866	−0.176786
$176$	0	0
$177$	−23.3091	−1.75202
$178$	0	0
$179$	10.1032	0.755149	0.377574	−	0.925979i	$-0.376758\pi$
$179$	10.1032	0.755149	0.377574	+	0.925979i	$0.376758\pi$
$180$	0	0
$181$	−21.3453	−1.58658	−0.793290	−	0.608844i	$-0.791634\pi$
$181$	−21.3453	−1.58658	−0.793290	+	0.608844i	$0.791634\pi$
$182$	0	0
$183$	−36.2911	−2.68272
$184$	0	0
$185$	−8.17877	−0.601315
$186$	0	0
$187$	−19.2062	−1.40449
$188$	0	0
$189$	−2.14484	−0.156014
$190$	0	0
$191$	−16.5123	−1.19479	−0.597393	−	0.801948i	$-0.703797\pi$
$191$	−16.5123	−1.19479	−0.597393	+	0.801948i	$0.703797\pi$
$192$	0	0
$193$	−24.7051	−1.77831	−0.889155	−	0.457606i	$-0.848707\pi$
$193$	−24.7051	−1.77831	−0.889155	+	0.457606i	$0.848707\pi$
$194$	0	0
$195$	16.7849	1.20199
$196$	0	0
$197$	−11.4968	−0.819113	−0.409557	−	0.912285i	$-0.634317\pi$
$197$	−11.4968	−0.819113	−0.409557	+	0.912285i	$0.634317\pi$
$198$	0	0
$199$	3.67140	0.260258	0.130129	−	0.991497i	$-0.458461\pi$
$199$	3.67140	0.260258	0.130129	+	0.991497i	$0.458461\pi$
$200$	0	0
$201$	12.7922	0.902291
$202$	0	0
$203$	2.00493	0.140718
$204$	0	0
$205$	−12.8575	−0.898004
$206$	0	0
$207$	−31.5331	−2.19170
$208$	0	0
$209$	2.45086	0.169530
$210$	0	0
$211$	−7.54460	−0.519392	−0.259696	−	0.965690i	$-0.583622\pi$
$211$	−7.54460	−0.519392	−0.259696	+	0.965690i	$0.583622\pi$
$212$	0	0
$213$	25.7319	1.76312
$214$	0	0
$215$	39.7835	2.71321
$216$	0	0
$217$	−0.552125	−0.0374807
$218$	0	0
$219$	−23.0118	−1.55499
$220$	0	0
$221$	−12.2661	−0.825110
$222$	0	0
$223$	3.43600	0.230092	0.115046	−	0.993360i	$-0.463299\pi$
$223$	3.43600	0.230092	0.115046	+	0.993360i	$0.463299\pi$
$224$	0	0
$225$	36.8365	2.45577
$226$	0	0
$227$	0.625444	0.0415122	0.0207561	−	0.999785i	$-0.493393\pi$
$227$	0.625444	0.0415122	0.0207561	+	0.999785i	$0.493393\pi$
$228$	0	0
$229$	29.5883	1.95525	0.977625	−	0.210353i	$-0.0674613\pi$
$229$	29.5883	1.95525	0.977625	+	0.210353i	$0.0674613\pi$
$230$	0	0
$231$	2.52321	0.166015
$232$	0	0
$233$	−7.54515	−0.494299	−0.247150	−	0.968977i	$-0.579494\pi$
$233$	−7.54515	−0.494299	−0.247150	+	0.968977i	$0.579494\pi$
$234$	0	0
$235$	7.42512	0.484361
$236$	0	0
$237$	−46.1657	−2.99879
$238$	0	0
$239$	−5.03167	−0.325471	−0.162736	−	0.986670i	$-0.552032\pi$
$239$	−5.03167	−0.325471	−0.162736	+	0.986670i	$0.552032\pi$
$240$	0	0
$241$	18.1185	1.16711	0.583557	−	0.812072i	$-0.301660\pi$
$241$	18.1185	1.16711	0.583557	+	0.812072i	$0.301660\pi$
$242$	0	0
$243$	−11.3849	−0.730341
$244$	0	0
$245$	23.8893	1.52623
$246$	0	0
$247$	1.56526	0.0995951
$248$	0	0
$249$	−44.7960	−2.83883
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	15.8733	0.997946
$254$	0	0
$255$	−72.4423	−4.53652
$256$	0	0
$257$	−23.6398	−1.47461	−0.737305	−	0.675560i	$-0.763902\pi$
$257$	−23.6398	−1.47461	−0.737305	+	0.675560i	$0.763902\pi$
$258$	0	0
$259$	−0.785216	−0.0487909
$260$	0	0
$261$	−31.5798	−1.95474
$262$	0	0
$263$	12.0307	0.741845	0.370923	−	0.928664i	$-0.379041\pi$
$263$	12.0307	0.741845	0.370923	+	0.928664i	$0.379041\pi$
$264$	0	0
$265$	−16.4483	−1.01041
$266$	0	0
$267$	41.1897	2.52077
$268$	0	0
$269$	−14.2435	−0.868442	−0.434221	−	0.900806i	$-0.642976\pi$
$269$	−14.2435	−0.868442	−0.434221	+	0.900806i	$0.642976\pi$
$270$	0	0
$271$	2.76475	0.167947	0.0839733	−	0.996468i	$-0.473239\pi$
$271$	2.76475	0.167947	0.0839733	+	0.996468i	$0.473239\pi$
$272$	0	0
$273$	1.61146	0.0975303
$274$	0	0
$275$	−18.5430	−1.11818
$276$	0	0
$277$	19.9499	1.19867	0.599336	−	0.800498i	$-0.295431\pi$
$277$	19.9499	1.19867	0.599336	+	0.800498i	$0.295431\pi$
$278$	0	0
$279$	8.69658	0.520651
$280$	0	0
$281$	17.9104	1.06844	0.534222	−	0.845344i	$-0.320605\pi$
$281$	17.9104	1.06844	0.534222	+	0.845344i	$0.320605\pi$
$282$	0	0
$283$	32.4275	1.92761	0.963807	−	0.266601i	$-0.0859005\pi$
$283$	32.4275	1.92761	0.963807	+	0.266601i	$0.0859005\pi$
$284$	0	0
$285$	9.24424	0.547582
$286$	0	0
$287$	−1.23440	−0.0728644
$288$	0	0
$289$	35.9396	2.11409
$290$	0	0
$291$	−11.9228	−0.698929
$292$	0	0
$293$	−13.7594	−0.803833	−0.401916	−	0.915676i	$-0.631656\pi$
$293$	−13.7594	−0.803833	−0.401916	+	0.915676i	$0.631656\pi$
$294$	0	0
$295$	28.1513	1.63903
$296$	0	0
$297$	−17.0062	−0.986798
$298$	0	0
$299$	10.1376	0.586272
$300$	0	0
$301$	3.81947	0.220151
$302$	0	0
$303$	−16.7585	−0.962753
$304$	0	0
$305$	43.8302	2.50971
$306$	0	0
$307$	5.16638	0.294861	0.147430	−	0.989072i	$-0.452900\pi$
$307$	5.16638	0.294861	0.147430	+	0.989072i	$0.452900\pi$
$308$	0	0
$309$	−23.4928	−1.33646
$310$	0	0
$311$	−4.19074	−0.237635	−0.118817	−	0.992916i	$-0.537910\pi$
$311$	−4.19074	−0.237635	−0.118817	+	0.992916i	$0.537910\pi$
$312$	0	0
$313$	12.7944	0.723182	0.361591	−	0.932337i	$-0.382234\pi$
$313$	12.7944	0.723182	0.361591	+	0.932337i	$0.382234\pi$
$314$	0	0
$315$	6.05376	0.341091
$316$	0	0
$317$	27.6798	1.55465	0.777325	−	0.629099i	$-0.216576\pi$
$317$	27.6798	1.55465	0.777325	+	0.629099i	$0.216576\pi$
$318$	0	0
$319$	15.8968	0.890052
$320$	0	0
$321$	−22.6281	−1.26298
$322$	0	0
$323$	−6.75553	−0.375888
$324$	0	0
$325$	−11.8426	−0.656909
$326$	0	0
$327$	2.25118	0.124491
$328$	0	0
$329$	0.712860	0.0393013
$330$	0	0
$331$	0.379612	0.0208654	0.0104327	−	0.999946i	$-0.496679\pi$
$331$	0.379612	0.0208654	0.0104327	+	0.999946i	$0.496679\pi$
$332$	0	0
$333$	12.3680	0.677763
$334$	0	0
$335$	−15.4496	−0.844102
$336$	0	0
$337$	−30.2405	−1.64730	−0.823652	−	0.567095i	$-0.808067\pi$
$337$	−30.2405	−1.64730	−0.823652	+	0.567095i	$0.808067\pi$
$338$	0	0
$339$	13.1188	0.712515
$340$	0	0
$341$	−4.37774	−0.237068
$342$	0	0
$343$	4.62397	0.249671
$344$	0	0
$345$	59.8714	3.22337
$346$	0	0
$347$	34.7166	1.86368	0.931841	−	0.362866i	$-0.118202\pi$
$347$	34.7166	1.86368	0.931841	+	0.362866i	$0.118202\pi$
$348$	0	0
$349$	3.54207	0.189603	0.0948013	−	0.995496i	$-0.469778\pi$
$349$	3.54207	0.189603	0.0948013	+	0.995496i	$0.469778\pi$
$350$	0	0
$351$	−10.8611	−0.579723
$352$	0	0
$353$	3.65146	0.194348	0.0971738	−	0.995267i	$-0.469020\pi$
$353$	3.65146	0.194348	0.0971738	+	0.995267i	$0.469020\pi$
$354$	0	0
$355$	−31.0774	−1.64942
$356$	0	0
$357$	−6.95494	−0.368094
$358$	0	0
$359$	−4.84760	−0.255846	−0.127923	−	0.991784i	$-0.540831\pi$
$359$	−4.84760	−0.255846	−0.127923	+	0.991784i	$0.540831\pi$
$360$	0	0
$361$	−18.1379	−0.954628
$362$	0	0
$363$	−11.5771	−0.607638
$364$	0	0
$365$	27.7923	1.45471
$366$	0	0
$367$	−3.48483	−0.181907	−0.0909534	−	0.995855i	$-0.528991\pi$
$367$	−3.48483	−0.181907	−0.0909534	+	0.995855i	$0.528991\pi$
$368$	0	0
$369$	19.4432	1.01217
$370$	0	0
$371$	−1.57914	−0.0819850
$372$	0	0
$373$	−17.9597	−0.929918	−0.464959	−	0.885332i	$-0.653931\pi$
$373$	−17.9597	−0.929918	−0.464959	+	0.885332i	$0.653931\pi$
$374$	0	0
$375$	−20.1589	−1.04100
$376$	0	0
$377$	10.1526	0.522886
$378$	0	0
$379$	30.6100	1.57233	0.786165	−	0.618016i	$-0.212063\pi$
$379$	30.6100	1.57233	0.786165	+	0.618016i	$0.212063\pi$
$380$	0	0
$381$	27.2824	1.39772
$382$	0	0
$383$	10.8403	0.553914	0.276957	−	0.960882i	$-0.410674\pi$
$383$	10.8403	0.553914	0.276957	+	0.960882i	$0.410674\pi$
$384$	0	0
$385$	−3.04738	−0.155309
$386$	0	0
$387$	−60.1609	−3.05815
$388$	0	0
$389$	−6.32688	−0.320785	−0.160393	−	0.987053i	$-0.551276\pi$
$389$	−6.32688	−0.320785	−0.160393	+	0.987053i	$0.551276\pi$
$390$	0	0
$391$	−43.7529	−2.21268
$392$	0	0
$393$	49.3712	2.49045
$394$	0	0
$395$	55.7561	2.80539
$396$	0	0
$397$	1.30995	0.0657443	0.0328721	−	0.999460i	$-0.489535\pi$
$397$	1.30995	0.0657443	0.0328721	+	0.999460i	$0.489535\pi$
$398$	0	0
$399$	0.887508	0.0444310
$400$	0	0
$401$	−7.65847	−0.382446	−0.191223	−	0.981547i	$-0.561245\pi$
$401$	−7.65847	−0.382446	−0.191223	+	0.981547i	$0.561245\pi$
$402$	0	0
$403$	−2.79587	−0.139272
$404$	0	0
$405$	−9.59268	−0.476664
$406$	0	0
$407$	−6.22588	−0.308605
$408$	0	0
$409$	11.4433	0.565836	0.282918	−	0.959144i	$-0.408698\pi$
$409$	11.4433	0.565836	0.282918	+	0.959144i	$0.408698\pi$
$410$	0	0
$411$	50.1328	2.47287
$412$	0	0
$413$	2.70271	0.132992
$414$	0	0
$415$	54.1018	2.65575
$416$	0	0
$417$	−63.3946	−3.10445
$418$	0	0
$419$	23.8803	1.16663	0.583314	−	0.812247i	$-0.301756\pi$
$419$	23.8803	1.16663	0.583314	+	0.812247i	$0.301756\pi$
$420$	0	0
$421$	8.76894	0.427372	0.213686	−	0.976902i	$-0.431453\pi$
$421$	8.76894	0.427372	0.213686	+	0.976902i	$0.431453\pi$
$422$	0	0
$423$	−11.2283	−0.545940
$424$	0	0
$425$	51.1116	2.47928
$426$	0	0
$427$	4.20799	0.203639
$428$	0	0
$429$	12.7771	0.616885
$430$	0	0
$431$	−2.33291	−0.112372	−0.0561861	−	0.998420i	$-0.517894\pi$
$431$	−2.33291	−0.112372	−0.0561861	+	0.998420i	$0.517894\pi$
$432$	0	0
$433$	9.50733	0.456893	0.228447	−	0.973556i	$-0.426635\pi$
$433$	9.50733	0.456893	0.228447	+	0.973556i	$0.426635\pi$
$434$	0	0
$435$	59.9601	2.87487
$436$	0	0
$437$	5.58324	0.267082
$438$	0	0
$439$	13.6735	0.652601	0.326300	−	0.945266i	$-0.394198\pi$
$439$	13.6735	0.652601	0.326300	+	0.945266i	$0.394198\pi$
$440$	0	0
$441$	−36.1257	−1.72027
$442$	0	0
$443$	−31.6415	−1.50333	−0.751666	−	0.659544i	$-0.770750\pi$
$443$	−31.6415	−1.50333	−0.751666	+	0.659544i	$0.770750\pi$
$444$	0	0
$445$	−49.7464	−2.35820
$446$	0	0
$447$	14.2786	0.675357
$448$	0	0
$449$	14.7952	0.698230	0.349115	−	0.937080i	$-0.386482\pi$
$449$	14.7952	0.698230	0.349115	+	0.937080i	$0.386482\pi$
$450$	0	0
$451$	−9.78742	−0.460871
$452$	0	0
$453$	49.5196	2.32663
$454$	0	0
$455$	−1.94623	−0.0912406
$456$	0	0
$457$	−42.3253	−1.97989	−0.989947	−	0.141442i	$-0.954826\pi$
$457$	−42.3253	−1.97989	−0.989947	+	0.141442i	$0.954826\pi$
$458$	0	0
$459$	46.8755	2.18796
$460$	0	0
$461$	8.09889	0.377203	0.188601	−	0.982054i	$-0.439605\pi$
$461$	8.09889	0.377203	0.188601	+	0.982054i	$0.439605\pi$
$462$	0	0
$463$	−13.0723	−0.607520	−0.303760	−	0.952749i	$-0.598242\pi$
$463$	−13.0723	−0.607520	−0.303760	+	0.952749i	$0.598242\pi$
$464$	0	0
$465$	−16.5121	−0.765729
$466$	0	0
$467$	11.8330	0.547566	0.273783	−	0.961792i	$-0.411725\pi$
$467$	11.8330	0.547566	0.273783	+	0.961792i	$0.411725\pi$
$468$	0	0
$469$	−1.48326	−0.0684908
$470$	0	0
$471$	26.1064	1.20292
$472$	0	0
$473$	30.2842	1.39247
$474$	0	0
$475$	−6.52226	−0.299262
$476$	0	0
$477$	24.8732	1.13887
$478$	0	0
$479$	−18.4627	−0.843581	−0.421790	−	0.906693i	$-0.638598\pi$
$479$	−18.4627	−0.843581	−0.421790	+	0.906693i	$0.638598\pi$
$480$	0	0
$481$	−3.97620	−0.181299
$482$	0	0
$483$	5.74805	0.261545
$484$	0	0
$485$	14.3997	0.653855
$486$	0	0
$487$	35.0628	1.58885	0.794424	−	0.607364i	$-0.207773\pi$
$487$	35.0628	1.58885	0.794424	+	0.607364i	$0.207773\pi$
$488$	0	0
$489$	−30.3784	−1.37376
$490$	0	0
$491$	20.0319	0.904027	0.452014	−	0.892011i	$-0.350706\pi$
$491$	20.0319	0.904027	0.452014	+	0.892011i	$0.350706\pi$
$492$	0	0
$493$	−43.8178	−1.97345
$494$	0	0
$495$	47.9995	2.15742
$496$	0	0
$497$	−2.98364	−0.133834
$498$	0	0
$499$	26.8833	1.20346	0.601731	−	0.798699i	$-0.294478\pi$
$499$	26.8833	1.20346	0.601731	+	0.798699i	$0.294478\pi$
$500$	0	0
$501$	−12.0035	−0.536278
$502$	0	0
$503$	26.8759	1.19834	0.599168	−	0.800623i	$-0.295498\pi$
$503$	26.8759	1.19834	0.599168	+	0.800623i	$0.295498\pi$
$504$	0	0
$505$	20.2399	0.900665
$506$	0	0
$507$	−29.1655	−1.29529
$508$	0	0
$509$	−33.8407	−1.49996	−0.749982	−	0.661458i	$-0.769938\pi$
$509$	−33.8407	−1.49996	−0.749982	+	0.661458i	$0.769938\pi$
$510$	0	0
$511$	2.66824	0.118036
$512$	0	0
$513$	−5.98171	−0.264099
$514$	0	0
$515$	28.3731	1.25027
$516$	0	0
$517$	5.65218	0.248583
$518$	0	0
$519$	24.1915	1.06189
$520$	0	0
$521$	6.16125	0.269929	0.134965	−	0.990850i	$-0.456908\pi$
$521$	6.16125	0.269929	0.134965	+	0.990850i	$0.456908\pi$
$522$	0	0
$523$	−16.2130	−0.708944	−0.354472	−	0.935067i	$-0.615339\pi$
$523$	−16.2130	−0.708944	−0.354472	+	0.935067i	$0.615339\pi$
$524$	0	0
$525$	−6.71479	−0.293057
$526$	0	0
$527$	12.0667	0.525635
$528$	0	0
$529$	13.1605	0.572194
$530$	0	0
$531$	−42.5706	−1.84741
$532$	0	0
$533$	−6.25080	−0.270752
$534$	0	0
$535$	27.3288	1.18153
$536$	0	0
$537$	29.0084	1.25180
$538$	0	0
$539$	18.1851	0.783290
$540$	0	0
$541$	16.6268	0.714842	0.357421	−	0.933943i	$-0.383656\pi$
$541$	16.6268	0.714842	0.357421	+	0.933943i	$0.383656\pi$
$542$	0	0
$543$	−61.2867	−2.63006
$544$	0	0
$545$	−2.71884	−0.116462
$546$	0	0
$547$	−24.7520	−1.05832	−0.529159	−	0.848523i	$-0.677493\pi$
$547$	−24.7520	−1.05832	−0.529159	+	0.848523i	$0.677493\pi$
$548$	0	0
$549$	−66.2804	−2.82878
$550$	0	0
$551$	5.59151	0.238206
$552$	0	0
$553$	5.35295	0.227631
$554$	0	0
$555$	−23.4830	−0.996796
$556$	0	0
$557$	−19.5980	−0.830393	−0.415196	−	0.909732i	$-0.636287\pi$
$557$	−19.5980	−0.830393	−0.415196	+	0.909732i	$0.636287\pi$
$558$	0	0
$559$	19.3412	0.818044
$560$	0	0
$561$	−55.1449	−2.32822
$562$	0	0
$563$	−6.51422	−0.274542	−0.137271	−	0.990534i	$-0.543833\pi$
$563$	−6.51422	−0.274542	−0.137271	+	0.990534i	$0.543833\pi$
$564$	0	0
$565$	−15.8441	−0.666565
$566$	0	0
$567$	−0.920960	−0.0386767
$568$	0	0
$569$	13.5251	0.567001	0.283500	−	0.958972i	$-0.408504\pi$
$569$	13.5251	0.567001	0.283500	+	0.958972i	$0.408504\pi$
$570$	0	0
$571$	41.6850	1.74446	0.872232	−	0.489093i	$-0.162672\pi$
$571$	41.6850	1.74446	0.872232	+	0.489093i	$0.162672\pi$
$572$	0	0
$573$	−47.4102	−1.98059
$574$	0	0
$575$	−42.2422	−1.76162
$576$	0	0
$577$	−37.4800	−1.56031	−0.780157	−	0.625584i	$-0.784861\pi$
$577$	−37.4800	−1.56031	−0.780157	+	0.625584i	$0.784861\pi$
$578$	0	0
$579$	−70.9334	−2.94789
$580$	0	0
$581$	5.19413	0.215489
$582$	0	0
$583$	−12.5208	−0.518560
$584$	0	0
$585$	30.6552	1.26744
$586$	0	0
$587$	−20.8112	−0.858970	−0.429485	−	0.903074i	$-0.641305\pi$
$587$	−20.8112	−0.858970	−0.429485	+	0.903074i	$0.641305\pi$
$588$	0	0
$589$	−1.53981	−0.0634469
$590$	0	0
$591$	−33.0097	−1.35784
$592$	0	0
$593$	14.5392	0.597052	0.298526	−	0.954401i	$-0.403505\pi$
$593$	14.5392	0.597052	0.298526	+	0.954401i	$0.403505\pi$
$594$	0	0
$595$	8.39975	0.344356
$596$	0	0
$597$	10.5413	0.431428
$598$	0	0
$599$	0.289198	0.0118163	0.00590815	−	0.999983i	$-0.498119\pi$
$599$	0.289198	0.0118163	0.00590815	+	0.999983i	$0.498119\pi$
$600$	0	0
$601$	3.14028	0.128095	0.0640473	−	0.997947i	$-0.479599\pi$
$601$	3.14028	0.128095	0.0640473	+	0.997947i	$0.479599\pi$
$602$	0	0
$603$	23.3630	0.951416
$604$	0	0
$605$	13.9821	0.568451
$606$	0	0
$607$	−47.0210	−1.90853	−0.954263	−	0.298969i	$-0.903357\pi$
$607$	−47.0210	−1.90853	−0.954263	+	0.298969i	$0.903357\pi$
$608$	0	0
$609$	5.75657	0.233268
$610$	0	0
$611$	3.60980	0.146037
$612$	0	0
$613$	35.8608	1.44840	0.724202	−	0.689587i	$-0.242208\pi$
$613$	35.8608	1.44840	0.724202	+	0.689587i	$0.242208\pi$
$614$	0	0
$615$	−36.9165	−1.48862
$616$	0	0
$617$	30.6368	1.23339	0.616696	−	0.787201i	$-0.288471\pi$
$617$	30.6368	1.23339	0.616696	+	0.787201i	$0.288471\pi$
$618$	0	0
$619$	−13.2349	−0.531956	−0.265978	−	0.963979i	$-0.585695\pi$
$619$	−13.2349	−0.531956	−0.265978	+	0.963979i	$0.585695\pi$
$620$	0	0
$621$	−38.7412	−1.55463
$622$	0	0
$623$	−4.77598	−0.191346
$624$	0	0
$625$	−10.7769	−0.431076
$626$	0	0
$627$	7.03694	0.281028
$628$	0	0
$629$	17.1609	0.684251
$630$	0	0
$631$	−46.4711	−1.84998	−0.924992	−	0.379987i	$-0.875928\pi$
$631$	−46.4711	−1.84998	−0.924992	+	0.379987i	$0.875928\pi$
$632$	0	0
$633$	−21.6621	−0.860992
$634$	0	0
$635$	−32.9500	−1.30758
$636$	0	0
$637$	11.6141	0.460166
$638$	0	0
$639$	46.9955	1.85911
$640$	0	0
$641$	−8.40669	−0.332044	−0.166022	−	0.986122i	$-0.553092\pi$
$641$	−8.40669	−0.332044	−0.166022	+	0.986122i	$0.553092\pi$
$642$	0	0
$643$	8.03292	0.316788	0.158394	−	0.987376i	$-0.449368\pi$
$643$	8.03292	0.316788	0.158394	+	0.987376i	$0.449368\pi$
$644$	0	0
$645$	114.227	4.49767
$646$	0	0
$647$	−26.3679	−1.03663	−0.518315	−	0.855190i	$-0.673440\pi$
$647$	−26.3679	−1.03663	−0.518315	+	0.855190i	$0.673440\pi$
$648$	0	0
$649$	21.4294	0.841179
$650$	0	0
$651$	−1.58527	−0.0621315
$652$	0	0
$653$	20.0702	0.785407	0.392703	−	0.919665i	$-0.371540\pi$
$653$	20.0702	0.785407	0.392703	+	0.919665i	$0.371540\pi$
$654$	0	0
$655$	−59.6275	−2.32984
$656$	0	0
$657$	−42.0277	−1.63966
$658$	0	0
$659$	24.5946	0.958071	0.479035	−	0.877796i	$-0.340987\pi$
$659$	24.5946	0.958071	0.479035	+	0.877796i	$0.340987\pi$
$660$	0	0
$661$	−38.3726	−1.49252	−0.746261	−	0.665654i	$-0.768153\pi$
$661$	−38.3726	−1.49252	−0.746261	+	0.665654i	$0.768153\pi$
$662$	0	0
$663$	−35.2186	−1.36778
$664$	0	0
$665$	−1.07188	−0.0415656
$666$	0	0
$667$	36.2141	1.40221
$668$	0	0
$669$	9.86547	0.381421
$670$	0	0
$671$	33.3646	1.28803
$672$	0	0
$673$	−46.8909	−1.80751	−0.903755	−	0.428050i	$-0.859201\pi$
$673$	−46.8909	−1.80751	−0.903755	+	0.428050i	$0.859201\pi$
$674$	0	0
$675$	45.2569	1.74194
$676$	0	0
$677$	47.1693	1.81286	0.906432	−	0.422353i	$-0.138796\pi$
$677$	47.1693	1.81286	0.906432	+	0.422353i	$0.138796\pi$
$678$	0	0
$679$	1.38246	0.0530540
$680$	0	0
$681$	1.79578	0.0688144
$682$	0	0
$683$	−18.1325	−0.693821	−0.346910	−	0.937898i	$-0.612769\pi$
$683$	−18.1325	−0.693821	−0.346910	+	0.937898i	$0.612769\pi$
$684$	0	0
$685$	−60.5474	−2.31340
$686$	0	0
$687$	84.9542	3.24121
$688$	0	0
$689$	−7.99651	−0.304643
$690$	0	0
$691$	−36.8383	−1.40139	−0.700697	−	0.713459i	$-0.747127\pi$
$691$	−36.8383	−1.40139	−0.700697	+	0.713459i	$0.747127\pi$
$692$	0	0
$693$	4.60827	0.175054
$694$	0	0
$695$	76.5641	2.90424
$696$	0	0
$697$	26.9779	1.02186
$698$	0	0
$699$	−21.6637	−0.819397
$700$	0	0
$701$	−50.8896	−1.92207	−0.961036	−	0.276424i	$-0.910851\pi$
$701$	−50.8896	−1.92207	−0.961036	+	0.276424i	$0.910851\pi$
$702$	0	0
$703$	−2.18988	−0.0825927
$704$	0	0
$705$	21.3191	0.802922
$706$	0	0
$707$	1.94317	0.0730802
$708$	0	0
$709$	35.2993	1.32569	0.662847	−	0.748755i	$-0.269348\pi$
$709$	35.2993	1.32569	0.662847	+	0.748755i	$0.269348\pi$
$710$	0	0
$711$	−84.3148	−3.16205
$712$	0	0
$713$	−9.97278	−0.373484
$714$	0	0
$715$	−15.4314	−0.577102
$716$	0	0
$717$	−14.4470	−0.539532
$718$	0	0
$719$	1.59649	0.0595390	0.0297695	−	0.999557i	$-0.490523\pi$
$719$	1.59649	0.0595390	0.0297695	+	0.999557i	$0.490523\pi$
$720$	0	0
$721$	2.72401	0.101447
$722$	0	0
$723$	52.0220	1.93472
$724$	0	0
$725$	−42.3048	−1.57116
$726$	0	0
$727$	−38.9463	−1.44444	−0.722218	−	0.691665i	$-0.756877\pi$
$727$	−38.9463	−1.44444	−0.722218	+	0.691665i	$0.756877\pi$
$728$	0	0
$729$	−40.9874	−1.51805
$730$	0	0
$731$	−83.4748	−3.08743
$732$	0	0
$733$	36.1378	1.33478	0.667391	−	0.744708i	$-0.267411\pi$
$733$	36.1378	1.33478	0.667391	+	0.744708i	$0.267411\pi$
$734$	0	0
$735$	68.5913	2.53003
$736$	0	0
$737$	−11.7606	−0.433208
$738$	0	0
$739$	−28.8101	−1.05980	−0.529898	−	0.848061i	$-0.677770\pi$
$739$	−28.8101	−1.05980	−0.529898	+	0.848061i	$0.677770\pi$
$740$	0	0
$741$	4.49419	0.165098
$742$	0	0
$743$	28.3850	1.04135	0.520673	−	0.853756i	$-0.325681\pi$
$743$	28.3850	1.04135	0.520673	+	0.853756i	$0.325681\pi$
$744$	0	0
$745$	−17.2449	−0.631803
$746$	0	0
$747$	−81.8132	−2.99339
$748$	0	0
$749$	2.62374	0.0958695
$750$	0	0
$751$	−18.5956	−0.678562	−0.339281	−	0.940685i	$-0.610184\pi$
$751$	−18.5956	−0.678562	−0.339281	+	0.940685i	$0.610184\pi$
$752$	0	0
$753$	2.87121	0.104633
$754$	0	0
$755$	−59.8067	−2.17659
$756$	0	0
$757$	31.4705	1.14382	0.571908	−	0.820318i	$-0.306203\pi$
$757$	31.4705	1.14382	0.571908	+	0.820318i	$0.306203\pi$
$758$	0	0
$759$	45.5756	1.65429
$760$	0	0
$761$	35.8675	1.30019	0.650097	−	0.759851i	$-0.274728\pi$
$761$	35.8675	1.30019	0.650097	+	0.759851i	$0.274728\pi$
$762$	0	0
$763$	−0.261026	−0.00944979
$764$	0	0
$765$	−132.305	−4.78351
$766$	0	0
$767$	13.6861	0.494175
$768$	0	0
$769$	−26.5760	−0.958356	−0.479178	−	0.877718i	$-0.659065\pi$
$769$	−26.5760	−0.958356	−0.479178	+	0.877718i	$0.659065\pi$
$770$	0	0
$771$	−67.8748	−2.44445
$772$	0	0
$773$	−35.4566	−1.27528	−0.637642	−	0.770333i	$-0.720090\pi$
$773$	−35.4566	−1.27528	−0.637642	+	0.770333i	$0.720090\pi$
$774$	0	0
$775$	11.6501	0.418483
$776$	0	0
$777$	−2.25452	−0.0808804
$778$	0	0
$779$	−3.44260	−0.123344
$780$	0	0
$781$	−23.6569	−0.846510
$782$	0	0
$783$	−38.7986	−1.38655
$784$	0	0
$785$	−31.5297	−1.12534
$786$	0	0
$787$	−18.1045	−0.645356	−0.322678	−	0.946509i	$-0.604583\pi$
$787$	−18.1045	−0.645356	−0.322678	+	0.946509i	$0.604583\pi$
$788$	0	0
$789$	34.5427	1.22975
$790$	0	0
$791$	−1.52113	−0.0540853
$792$	0	0
$793$	21.3085	0.756688
$794$	0	0
$795$	−47.2264	−1.67495
$796$	0	0
$797$	−9.07704	−0.321525	−0.160763	−	0.986993i	$-0.551395\pi$
$797$	−9.07704	−0.321525	−0.160763	+	0.986993i	$0.551395\pi$
$798$	0	0
$799$	−15.5796	−0.551166
$800$	0	0
$801$	75.2269	2.65801
$802$	0	0
$803$	21.1561	0.746584
$804$	0	0
$805$	−6.94213	−0.244678
$806$	0	0
$807$	−40.8961	−1.43961
$808$	0	0
$809$	−5.89944	−0.207413	−0.103707	−	0.994608i	$-0.533070\pi$
$809$	−5.89944	−0.207413	−0.103707	+	0.994608i	$0.533070\pi$
$810$	0	0
$811$	−22.2806	−0.782379	−0.391190	−	0.920310i	$-0.627936\pi$
$811$	−22.2806	−0.782379	−0.391190	+	0.920310i	$0.627936\pi$
$812$	0	0
$813$	7.93818	0.278404
$814$	0	0
$815$	36.6892	1.28517
$816$	0	0
$817$	10.6521	0.372669
$818$	0	0
$819$	2.94310	0.102840
$820$	0	0
$821$	−17.7621	−0.619903	−0.309951	−	0.950752i	$-0.600313\pi$
$821$	−17.7621	−0.619903	−0.309951	+	0.950752i	$0.600313\pi$
$822$	0	0
$823$	29.5547	1.03021	0.515105	−	0.857127i	$-0.327753\pi$
$823$	29.5547	1.03021	0.515105	+	0.857127i	$0.327753\pi$
$824$	0	0
$825$	−53.2407	−1.85360
$826$	0	0
$827$	14.6943	0.510971	0.255485	−	0.966813i	$-0.417765\pi$
$827$	14.6943	0.510971	0.255485	+	0.966813i	$0.417765\pi$
$828$	0	0
$829$	−6.37265	−0.221331	−0.110666	−	0.993858i	$-0.535298\pi$
$829$	−6.37265	−0.221331	−0.110666	+	0.993858i	$0.535298\pi$
$830$	0	0
$831$	57.2802	1.98703
$832$	0	0
$833$	−50.1253	−1.73674
$834$	0	0
$835$	14.4971	0.501693
$836$	0	0
$837$	10.6845	0.369311
$838$	0	0
$839$	33.3204	1.15035	0.575174	−	0.818031i	$-0.304934\pi$
$839$	33.3204	1.15035	0.575174	+	0.818031i	$0.304934\pi$
$840$	0	0
$841$	7.26775	0.250612
$842$	0	0
$843$	51.4244	1.77115
$844$	0	0
$845$	35.2243	1.21175
$846$	0	0
$847$	1.34237	0.0461244
$848$	0	0
$849$	93.1061	3.19539
$850$	0	0
$851$	−14.1830	−0.486186
$852$	0	0
$853$	−26.9320	−0.922133	−0.461066	−	0.887366i	$-0.652533\pi$
$853$	−26.9320	−0.922133	−0.461066	+	0.887366i	$0.652533\pi$
$854$	0	0
$855$	16.8832	0.577395
$856$	0	0
$857$	30.6148	1.04578	0.522892	−	0.852399i	$-0.324853\pi$
$857$	30.6148	1.04578	0.522892	+	0.852399i	$0.324853\pi$
$858$	0	0
$859$	−15.9639	−0.544681	−0.272341	−	0.962201i	$-0.587798\pi$
$859$	−15.9639	−0.544681	−0.272341	+	0.962201i	$0.587798\pi$
$860$	0	0
$861$	−3.54422	−0.120787
$862$	0	0
$863$	29.0226	0.987940	0.493970	−	0.869479i	$-0.335545\pi$
$863$	29.0226	0.987940	0.493970	+	0.869479i	$0.335545\pi$
$864$	0	0
$865$	−29.2170	−0.993407
$866$	0	0
$867$	103.190	3.50452
$868$	0	0
$869$	42.4429	1.43978
$870$	0	0
$871$	−7.51100	−0.254500
$872$	0	0
$873$	−21.7753	−0.736982
$874$	0	0
$875$	2.33744	0.0790200
$876$	0	0
$877$	16.1626	0.545772	0.272886	−	0.962046i	$-0.412022\pi$
$877$	16.1626	0.545772	0.272886	+	0.962046i	$0.412022\pi$
$878$	0	0
$879$	−39.5061	−1.33251
$880$	0	0
$881$	0.881726	0.0297061	0.0148530	−	0.999890i	$-0.495272\pi$
$881$	0.881726	0.0297061	0.0148530	+	0.999890i	$0.495272\pi$
$882$	0	0
$883$	−15.7531	−0.530134	−0.265067	−	0.964230i	$-0.585394\pi$
$883$	−15.7531	−0.530134	−0.265067	+	0.964230i	$0.585394\pi$
$884$	0	0
$885$	80.8282	2.71701
$886$	0	0
$887$	17.2960	0.580742	0.290371	−	0.956914i	$-0.406221\pi$
$887$	17.2960	0.580742	0.290371	+	0.956914i	$0.406221\pi$
$888$	0	0
$889$	−3.16341	−0.106097
$890$	0	0
$891$	−7.30218	−0.244632
$892$	0	0
$893$	1.98809	0.0665287
$894$	0	0
$895$	−35.0346	−1.17108
$896$	0	0
$897$	29.1071	0.971859
$898$	0	0
$899$	−9.98756	−0.333104
$900$	0	0
$901$	34.5122	1.14977
$902$	0	0
$903$	10.9665	0.364942
$904$	0	0
$905$	74.0183	2.46045
$906$	0	0
$907$	51.1547	1.69856	0.849282	−	0.527940i	$-0.177035\pi$
$907$	51.1547	1.69856	0.849282	+	0.527940i	$0.177035\pi$
$908$	0	0
$909$	−30.6070	−1.01517
$910$	0	0
$911$	41.1823	1.36443	0.682216	−	0.731151i	$-0.261016\pi$
$911$	41.1823	1.36443	0.682216	+	0.731151i	$0.261016\pi$
$912$	0	0
$913$	41.1836	1.36298
$914$	0	0
$915$	125.846	4.16033
$916$	0	0
$917$	−5.72464	−0.189044
$918$	0	0
$919$	−11.2266	−0.370333	−0.185166	−	0.982707i	$-0.559282\pi$
$919$	−11.2266	−0.370333	−0.185166	+	0.982707i	$0.559282\pi$
$920$	0	0
$921$	14.8337	0.488789
$922$	0	0
$923$	−15.1086	−0.497307
$924$	0	0
$925$	16.5684	0.544764
$926$	0	0
$927$	−42.9061	−1.40922
$928$	0	0
$929$	6.86226	0.225143	0.112572	−	0.993644i	$-0.464091\pi$
$929$	6.86226	0.225143	0.112572	+	0.993644i	$0.464091\pi$
$930$	0	0
$931$	6.39640	0.209634
$932$	0	0
$933$	−12.0325	−0.393926
$934$	0	0
$935$	66.6006	2.17807
$936$	0	0
$937$	36.5399	1.19371	0.596853	−	0.802350i	$-0.296417\pi$
$937$	36.5399	1.19371	0.596853	+	0.802350i	$0.296417\pi$
$938$	0	0
$939$	36.7354	1.19881
$940$	0	0
$941$	−52.3177	−1.70551	−0.852754	−	0.522313i	$-0.825069\pi$
$941$	−52.3177	−1.70551	−0.852754	+	0.522313i	$0.825069\pi$
$942$	0	0
$943$	−22.2964	−0.726071
$944$	0	0
$945$	7.43759	0.241945
$946$	0	0
$947$	−46.2260	−1.50214	−0.751072	−	0.660221i	$-0.770463\pi$
$947$	−46.2260	−1.50214	−0.751072	+	0.660221i	$0.770463\pi$
$948$	0	0
$949$	13.5115	0.438602
$950$	0	0
$951$	79.4744	2.57713
$952$	0	0
$953$	−20.6238	−0.668072	−0.334036	−	0.942560i	$-0.608411\pi$
$953$	−20.6238	−0.668072	−0.334036	+	0.942560i	$0.608411\pi$
$954$	0	0
$955$	57.2591	1.85286
$956$	0	0
$957$	45.6431	1.47543
$958$	0	0
$959$	−5.81294	−0.187710
$960$	0	0
$961$	−28.2496	−0.911277
$962$	0	0
$963$	−41.3268	−1.33174
$964$	0	0
$965$	85.6690	2.75778
$966$	0	0
$967$	−50.7100	−1.63072	−0.815362	−	0.578951i	$-0.803462\pi$
$967$	−50.7100	−1.63072	−0.815362	+	0.578951i	$0.803462\pi$
$968$	0	0
$969$	−19.3965	−0.623106
$970$	0	0
$971$	−19.5575	−0.627630	−0.313815	−	0.949484i	$-0.601607\pi$
$971$	−19.5575	−0.627630	−0.313815	+	0.949484i	$0.601607\pi$
$972$	0	0
$973$	7.35066	0.235651
$974$	0	0
$975$	−34.0025	−1.08895
$976$	0	0
$977$	28.4633	0.910621	0.455310	−	0.890333i	$-0.349528\pi$
$977$	28.4633	0.910621	0.455310	+	0.890333i	$0.349528\pi$
$978$	0	0
$979$	−37.8682	−1.21027
$980$	0	0
$981$	4.11145	0.131269
$982$	0	0
$983$	−7.14552	−0.227907	−0.113953	−	0.993486i	$-0.536351\pi$
$983$	−7.14552	−0.227907	−0.113953	+	0.993486i	$0.536351\pi$
$984$	0	0
$985$	39.8671	1.27027
$986$	0	0
$987$	2.04677	0.0651494
$988$	0	0
$989$	68.9894	2.19373
$990$	0	0
$991$	43.8421	1.39269	0.696345	−	0.717707i	$-0.254808\pi$
$991$	43.8421	1.39269	0.696345	+	0.717707i	$0.254808\pi$
$992$	0	0
$993$	1.08995	0.0345884
$994$	0	0
$995$	−12.7312	−0.403606
$996$	0	0
$997$	11.0637	0.350392	0.175196	−	0.984534i	$-0.443944\pi$
$997$	11.0637	0.350392	0.175196	+	0.984534i	$0.443944\pi$
$998$	0	0
$999$	15.1952	0.480755

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.f.1.6		6
4.3	odd	2		502.2.a.e.1.1	✓	6
12.11	even	2		4518.2.a.x.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
502.2.a.e.1.1	✓	6	4.3	odd	2
4016.2.a.f.1.6		6	1.1	even	1	trivial
4518.2.a.x.1.6		6	12.11	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:	\( N \)	\(=\)	\( 4016 = 2^{4} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4016.a (trivial)

\(f(q)\)	\(=\)	\(q+2.87121 q^{3} -3.46767 q^{5} -0.332919 q^{7} +5.24384 q^{9} +O(q^{10})\)	\(q+2.87121 q^{3} -3.46767 q^{5} -0.332919 q^{7} +5.24384 q^{9} -2.63967 q^{11} -1.68584 q^{13} -9.95640 q^{15} +7.27596 q^{17} -0.928472 q^{19} -0.955880 q^{21} -6.01336 q^{23} +7.02472 q^{25} +6.44253 q^{27} -6.02227 q^{29} +1.65844 q^{31} -7.57905 q^{33} +1.15445 q^{35} +2.35858 q^{37} -4.84041 q^{39} +3.70781 q^{41} -11.4727 q^{43} -18.1839 q^{45} -2.14124 q^{47} -6.88917 q^{49} +20.8908 q^{51} +4.74332 q^{53} +9.15351 q^{55} -2.66584 q^{57} -8.11822 q^{59} -12.6397 q^{61} -1.74577 q^{63} +5.84595 q^{65} +4.45533 q^{67} -17.2656 q^{69} +8.96205 q^{71} -8.01468 q^{73} +20.1694 q^{75} +0.878797 q^{77} -16.0788 q^{79} +2.76632 q^{81} -15.6018 q^{83} -25.2306 q^{85} -17.2912 q^{87} +14.3458 q^{89} +0.561250 q^{91} +4.76172 q^{93} +3.21963 q^{95} -4.15255 q^{97} -13.8420 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{3} - q^{5} - 6 q^{7} + 15 q^{9}+O(q^{10}) \) 6 * q - q^3 - q^5 - 6 * q^7 + 15 * q^9	\( 6 q - q^{3} - q^{5} - 6 q^{7} + 15 q^{9} - q^{11} - 5 q^{13} - 2 q^{15} + 8 q^{17} - 3 q^{19} - 14 q^{21} - 18 q^{23} - q^{25} - 16 q^{27} + q^{29} - 6 q^{31} - 16 q^{33} - 6 q^{35} - 13 q^{37} + 6 q^{39} + 4 q^{41} + 5 q^{43} - 23 q^{45} - 8 q^{47} + 8 q^{49} + 16 q^{51} - 3 q^{53} + 30 q^{55} - 24 q^{57} + 5 q^{59} - 61 q^{61} + 27 q^{63} - q^{65} + 13 q^{67} - 21 q^{69} - 22 q^{71} + 6 q^{73} + 30 q^{75} + 4 q^{77} - 28 q^{79} + 2 q^{81} - 14 q^{83} - 16 q^{85} + 24 q^{87} + 18 q^{89} + 16 q^{91} + 27 q^{93} - 20 q^{95} + 16 q^{97} - 5 q^{99}+O(q^{100}) \) 6 * q - q^3 - q^5 - 6 * q^7 + 15 * q^9 - q^11 - 5 * q^13 - 2 * q^15 + 8 * q^17 - 3 * q^19 - 14 * q^21 - 18 * q^23 - q^25 - 16 * q^27 + q^29 - 6 * q^31 - 16 * q^33 - 6 * q^35 - 13 * q^37 + 6 * q^39 + 4 * q^41 + 5 * q^43 - 23 * q^45 - 8 * q^47 + 8 * q^49 + 16 * q^51 - 3 * q^53 + 30 * q^55 - 24 * q^57 + 5 * q^59 - 61 * q^61 + 27 * q^63 - q^65 + 13 * q^67 - 21 * q^69 - 22 * q^71 + 6 * q^73 + 30 * q^75 + 4 * q^77 - 28 * q^79 + 2 * q^81 - 14 * q^83 - 16 * q^85 + 24 * q^87 + 18 * q^89 + 16 * q^91 + 27 * q^93 - 20 * q^95 + 16 * q^97 - 5 * q^99

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