LMFDB - Embedded newform 4016.2.a.i.1.3

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 17 x^{10} + 49 x^{9} + 106 x^{8} - 277 x^{7} - 317 x^{6} + 644 x^{5} + 537 x^{4} + \cdots + 104 $ x^12 - 3x^11 - 17x^10 + 49x^9 + 106x^8 - 277x^7 - 317x^6 + 644x^5 + 537x^4 - 601x^3 - 462x^2 + 136*x + 104
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2008)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.20058$ 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)

$f(q)$	$=$	$q-2.20058 q^{3} +2.09881 q^{5} -3.80569 q^{7} +1.84254 q^{9} +O(q^{10})$	$q-2.20058 q^{3} +2.09881 q^{5} -3.80569 q^{7} +1.84254 q^{9} +2.72208 q^{11} -2.57695 q^{13} -4.61859 q^{15} +5.81419 q^{17} -3.73456 q^{19} +8.37472 q^{21} +0.502178 q^{23} -0.595012 q^{25} +2.54707 q^{27} -2.43900 q^{29} +0.240541 q^{31} -5.99014 q^{33} -7.98741 q^{35} +4.24526 q^{37} +5.67079 q^{39} +0.696951 q^{41} +2.89516 q^{43} +3.86714 q^{45} -2.24693 q^{47} +7.48327 q^{49} -12.7946 q^{51} +4.84229 q^{53} +5.71311 q^{55} +8.21820 q^{57} -8.64247 q^{59} -1.05308 q^{61} -7.01215 q^{63} -5.40853 q^{65} -2.14458 q^{67} -1.10508 q^{69} +6.96304 q^{71} -0.901230 q^{73} +1.30937 q^{75} -10.3594 q^{77} +7.41932 q^{79} -11.1327 q^{81} -0.785834 q^{83} +12.2029 q^{85} +5.36721 q^{87} -13.7642 q^{89} +9.80709 q^{91} -0.529330 q^{93} -7.83813 q^{95} -2.46020 q^{97} +5.01555 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{3} + 5 q^{5} - 5 q^{7} + 7 q^{9}+O(q^{10}) $ 12 * q - 3 * q^3 + 5 * q^5 - 5 * q^7 + 7 * q^9	$ 12 q - 3 q^{3} + 5 q^{5} - 5 q^{7} + 7 q^{9} - 10 q^{11} + 3 q^{13} - 11 q^{15} + 2 q^{17} - 15 q^{19} + 3 q^{21} - 20 q^{23} - 3 q^{25} - 15 q^{27} + 6 q^{29} - 14 q^{31} - 6 q^{33} - 16 q^{35} + 5 q^{37} - 21 q^{39} - 21 q^{43} + 10 q^{45} - 27 q^{47} - 13 q^{49} - 19 q^{51} + 22 q^{53} - 24 q^{55} + q^{57} - 23 q^{59} + 4 q^{61} - 21 q^{63} - q^{65} - 26 q^{67} + 10 q^{69} - 23 q^{71} - 8 q^{73} - 16 q^{75} + 22 q^{77} - 37 q^{79} - 20 q^{81} - 30 q^{83} + 2 q^{85} - 16 q^{87} + 3 q^{89} - 8 q^{91} + 20 q^{93} - 33 q^{95} - 4 q^{97} - 15 q^{99}+O(q^{100}) $ 12 * q - 3 * q^3 + 5 * q^5 - 5 * q^7 + 7 * q^9 - 10 * q^11 + 3 * q^13 - 11 * q^15 + 2 * q^17 - 15 * q^19 + 3 * q^21 - 20 * q^23 - 3 * q^25 - 15 * q^27 + 6 * q^29 - 14 * q^31 - 6 * q^33 - 16 * q^35 + 5 * q^37 - 21 * q^39 - 21 * q^43 + 10 * q^45 - 27 * q^47 - 13 * q^49 - 19 * q^51 + 22 * q^53 - 24 * q^55 + q^57 - 23 * q^59 + 4 * q^61 - 21 * q^63 - q^65 - 26 * q^67 + 10 * q^69 - 23 * q^71 - 8 * q^73 - 16 * q^75 + 22 * q^77 - 37 * q^79 - 20 * q^81 - 30 * q^83 + 2 * q^85 - 16 * q^87 + 3 * q^89 - 8 * q^91 + 20 * q^93 - 33 * q^95 - 4 * q^97 - 15 * 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
$3$	−2.20058	−1.27050	−0.635252	−	0.772305i	$-0.719104\pi$
$3$	−2.20058	−1.27050	−0.635252	+	0.772305i	$0.719104\pi$
$4$	0	0
$5$	2.09881	0.938615	0.469307	−	0.883035i	$-0.344504\pi$
$5$	2.09881	0.938615	0.469307	+	0.883035i	$0.344504\pi$
$6$	0	0
$7$	−3.80569	−1.43842	−0.719208	−	0.694795i	$-0.755495\pi$
$7$	−3.80569	−1.43842	−0.719208	+	0.694795i	$0.755495\pi$
$8$	0	0
$9$	1.84254	0.614182
$10$	0	0
$11$	2.72208	0.820737	0.410369	−	0.911920i	$-0.365400\pi$
$11$	2.72208	0.820737	0.410369	+	0.911920i	$0.365400\pi$
$12$	0	0
$13$	−2.57695	−0.714719	−0.357359	−	0.933967i	$-0.616323\pi$
$13$	−2.57695	−0.714719	−0.357359	+	0.933967i	$0.616323\pi$
$14$	0	0
$15$	−4.61859	−1.19251
$16$	0	0
$17$	5.81419	1.41015	0.705074	−	0.709134i	$-0.250914\pi$
$17$	5.81419	1.41015	0.705074	+	0.709134i	$0.250914\pi$
$18$	0	0
$19$	−3.73456	−0.856768	−0.428384	−	0.903597i	$-0.640917\pi$
$19$	−3.73456	−0.856768	−0.428384	+	0.903597i	$0.640917\pi$
$20$	0	0
$21$	8.37472	1.82751
$22$	0	0
$23$	0.502178	0.104711	0.0523556	−	0.998629i	$-0.483327\pi$
$23$	0.502178	0.104711	0.0523556	+	0.998629i	$0.483327\pi$
$24$	0	0
$25$	−0.595012	−0.119002
$26$	0	0
$27$	2.54707	0.490184
$28$	0	0
$29$	−2.43900	−0.452911	−0.226455	−	0.974022i	$-0.572714\pi$
$29$	−2.43900	−0.452911	−0.226455	+	0.974022i	$0.572714\pi$
$30$	0	0
$31$	0.240541	0.0432025	0.0216012	−	0.999767i	$-0.493124\pi$
$31$	0.240541	0.0432025	0.0216012	+	0.999767i	$0.493124\pi$
$32$	0	0
$33$	−5.99014	−1.04275
$34$	0	0
$35$	−7.98741	−1.35012
$36$	0	0
$37$	4.24526	0.697917	0.348959	−	0.937138i	$-0.386535\pi$
$37$	4.24526	0.697917	0.348959	+	0.937138i	$0.386535\pi$
$38$	0	0
$39$	5.67079	0.908053
$40$	0	0
$41$	0.696951	0.108845	0.0544227	−	0.998518i	$-0.482668\pi$
$41$	0.696951	0.108845	0.0544227	+	0.998518i	$0.482668\pi$
$42$	0	0
$43$	2.89516	0.441508	0.220754	−	0.975330i	$-0.429148\pi$
$43$	2.89516	0.441508	0.220754	+	0.975330i	$0.429148\pi$
$44$	0	0
$45$	3.86714	0.576480
$46$	0	0
$47$	−2.24693	−0.327748	−0.163874	−	0.986481i	$-0.552399\pi$
$47$	−2.24693	−0.327748	−0.163874	+	0.986481i	$0.552399\pi$
$48$	0	0
$49$	7.48327	1.06904
$50$	0	0
$51$	−12.7946	−1.79160
$52$	0	0
$53$	4.84229	0.665140	0.332570	−	0.943079i	$-0.392084\pi$
$53$	4.84229	0.665140	0.332570	+	0.943079i	$0.392084\pi$
$54$	0	0
$55$	5.71311	0.770356
$56$	0	0
$57$	8.21820	1.08853
$58$	0	0
$59$	−8.64247	−1.12515	−0.562577	−	0.826745i	$-0.690190\pi$
$59$	−8.64247	−1.12515	−0.562577	+	0.826745i	$0.690190\pi$
$60$	0	0
$61$	−1.05308	−0.134833	−0.0674167	−	0.997725i	$-0.521476\pi$
$61$	−1.05308	−0.134833	−0.0674167	+	0.997725i	$0.521476\pi$
$62$	0	0
$63$	−7.01215	−0.883448
$64$	0	0
$65$	−5.40853	−0.670845
$66$	0	0
$67$	−2.14458	−0.262002	−0.131001	−	0.991382i	$-0.541819\pi$
$67$	−2.14458	−0.262002	−0.131001	+	0.991382i	$0.541819\pi$
$68$	0	0
$69$	−1.10508	−0.133036
$70$	0	0
$71$	6.96304	0.826361	0.413180	−	0.910649i	$-0.364418\pi$
$71$	6.96304	0.826361	0.413180	+	0.910649i	$0.364418\pi$
$72$	0	0
$73$	−0.901230	−0.105481	−0.0527405	−	0.998608i	$-0.516796\pi$
$73$	−0.901230	−0.105481	−0.0527405	+	0.998608i	$0.516796\pi$
$74$	0	0
$75$	1.30937	0.151193
$76$	0	0
$77$	−10.3594	−1.18056
$78$	0	0
$79$	7.41932	0.834738	0.417369	−	0.908737i	$-0.362952\pi$
$79$	7.41932	0.834738	0.417369	+	0.908737i	$0.362952\pi$
$80$	0	0
$81$	−11.1327	−1.23696
$82$	0	0
$83$	−0.785834	−0.0862565	−0.0431282	−	0.999070i	$-0.513732\pi$
$83$	−0.785834	−0.0862565	−0.0431282	+	0.999070i	$0.513732\pi$
$84$	0	0
$85$	12.2029	1.32359
$86$	0	0
$87$	5.36721	0.575425
$88$	0	0
$89$	−13.7642	−1.45900	−0.729502	−	0.683979i	$-0.760248\pi$
$89$	−13.7642	−1.45900	−0.729502	+	0.683979i	$0.760248\pi$
$90$	0	0
$91$	9.80709	1.02806
$92$	0	0
$93$	−0.529330	−0.0548889
$94$	0	0
$95$	−7.83813	−0.804175
$96$	0	0
$97$	−2.46020	−0.249795	−0.124898	−	0.992170i	$-0.539860\pi$
$97$	−2.46020	−0.249795	−0.124898	+	0.992170i	$0.539860\pi$
$98$	0	0
$99$	5.01555	0.504082
$100$	0	0
$101$	4.53650	0.451399	0.225700	−	0.974197i	$-0.427533\pi$
$101$	4.53650	0.451399	0.225700	+	0.974197i	$0.427533\pi$
$102$	0	0
$103$	19.4546	1.91692	0.958462	−	0.285221i	$-0.0920670\pi$
$103$	19.4546	1.91692	0.958462	+	0.285221i	$0.0920670\pi$
$104$	0	0
$105$	17.5769	1.71533
$106$	0	0
$107$	−14.1908	−1.37188	−0.685939	−	0.727659i	$-0.740609\pi$
$107$	−14.1908	−1.37188	−0.685939	+	0.727659i	$0.740609\pi$
$108$	0	0
$109$	−14.2216	−1.36218	−0.681092	−	0.732198i	$-0.738494\pi$
$109$	−14.2216	−1.36218	−0.681092	+	0.732198i	$0.738494\pi$
$110$	0	0
$111$	−9.34204	−0.886707
$112$	0	0
$113$	−18.2135	−1.71338	−0.856690	−	0.515831i	$-0.827483\pi$
$113$	−18.2135	−1.71338	−0.856690	+	0.515831i	$0.827483\pi$
$114$	0	0
$115$	1.05397	0.0982835
$116$	0	0
$117$	−4.74815	−0.438967
$118$	0	0
$119$	−22.1270	−2.02838
$120$	0	0
$121$	−3.59030	−0.326391
$122$	0	0
$123$	−1.53370	−0.138289
$124$	0	0
$125$	−11.7428	−1.05031
$126$	0	0
$127$	−1.84673	−0.163871	−0.0819354	−	0.996638i	$-0.526110\pi$
$127$	−1.84673	−0.163871	−0.0819354	+	0.996638i	$0.526110\pi$
$128$	0	0
$129$	−6.37103	−0.560938
$130$	0	0
$131$	4.82276	0.421367	0.210683	−	0.977554i	$-0.432431\pi$
$131$	4.82276	0.421367	0.210683	+	0.977554i	$0.432431\pi$
$132$	0	0
$133$	14.2126	1.23239
$134$	0	0
$135$	5.34581	0.460094
$136$	0	0
$137$	8.83610	0.754919	0.377459	−	0.926026i	$-0.376798\pi$
$137$	8.83610	0.754919	0.377459	+	0.926026i	$0.376798\pi$
$138$	0	0
$139$	−6.39944	−0.542794	−0.271397	−	0.962468i	$-0.587486\pi$
$139$	−6.39944	−0.542794	−0.271397	+	0.962468i	$0.587486\pi$
$140$	0	0
$141$	4.94454	0.416405
$142$	0	0
$143$	−7.01467	−0.586596
$144$	0	0
$145$	−5.11898	−0.425108
$146$	0	0
$147$	−16.4675	−1.35822
$148$	0	0
$149$	3.24086	0.265502	0.132751	−	0.991149i	$-0.457619\pi$
$149$	3.24086	0.265502	0.132751	+	0.991149i	$0.457619\pi$
$150$	0	0
$151$	−9.40519	−0.765384	−0.382692	−	0.923876i	$-0.625003\pi$
$151$	−9.40519	−0.765384	−0.382692	+	0.923876i	$0.625003\pi$
$152$	0	0
$153$	10.7129	0.866087
$154$	0	0
$155$	0.504849	0.0405505
$156$	0	0
$157$	13.3716	1.06717	0.533584	−	0.845747i	$-0.320845\pi$
$157$	13.3716	1.06717	0.533584	+	0.845747i	$0.320845\pi$
$158$	0	0
$159$	−10.6558	−0.845063
$160$	0	0
$161$	−1.91113	−0.150618
$162$	0	0
$163$	13.3990	1.04949	0.524747	−	0.851258i	$-0.324160\pi$
$163$	13.3990	1.04949	0.524747	+	0.851258i	$0.324160\pi$
$164$	0	0
$165$	−12.5722	−0.978741
$166$	0	0
$167$	−3.11890	−0.241348	−0.120674	−	0.992692i	$-0.538506\pi$
$167$	−3.11890	−0.241348	−0.120674	+	0.992692i	$0.538506\pi$
$168$	0	0
$169$	−6.35931	−0.489177
$170$	0	0
$171$	−6.88110	−0.526211
$172$	0	0
$173$	20.9281	1.59113	0.795566	−	0.605867i	$-0.207174\pi$
$173$	20.9281	1.59113	0.795566	+	0.605867i	$0.207174\pi$
$174$	0	0
$175$	2.26443	0.171175
$176$	0	0
$177$	19.0184	1.42951
$178$	0	0
$179$	−2.35199	−0.175796	−0.0878979	−	0.996129i	$-0.528015\pi$
$179$	−2.35199	−0.175796	−0.0878979	+	0.996129i	$0.528015\pi$
$180$	0	0
$181$	19.3239	1.43633	0.718165	−	0.695872i	$-0.244982\pi$
$181$	19.3239	1.43633	0.718165	+	0.695872i	$0.244982\pi$
$182$	0	0
$183$	2.31739	0.171306
$184$	0	0
$185$	8.90999	0.655075
$186$	0	0
$187$	15.8267	1.15736
$188$	0	0
$189$	−9.69336	−0.705088
$190$	0	0
$191$	−21.0736	−1.52483	−0.762415	−	0.647088i	$-0.775986\pi$
$191$	−21.0736	−1.52483	−0.762415	+	0.647088i	$0.775986\pi$
$192$	0	0
$193$	−17.1984	−1.23797	−0.618985	−	0.785403i	$-0.712456\pi$
$193$	−17.1984	−1.23797	−0.618985	+	0.785403i	$0.712456\pi$
$194$	0	0
$195$	11.9019	0.852312
$196$	0	0
$197$	−11.9365	−0.850439	−0.425220	−	0.905090i	$-0.639803\pi$
$197$	−11.9365	−0.850439	−0.425220	+	0.905090i	$0.639803\pi$
$198$	0	0
$199$	−18.8842	−1.33866	−0.669332	−	0.742964i	$-0.733419\pi$
$199$	−18.8842	−1.33866	−0.669332	+	0.742964i	$0.733419\pi$
$200$	0	0
$201$	4.71932	0.332875
$202$	0	0
$203$	9.28207	0.651474
$204$	0	0
$205$	1.46277	0.102164
$206$	0	0
$207$	0.925285	0.0643117
$208$	0	0
$209$	−10.1658	−0.703181
$210$	0	0
$211$	−4.47047	−0.307760	−0.153880	−	0.988090i	$-0.549177\pi$
$211$	−4.47047	−0.307760	−0.153880	+	0.988090i	$0.549177\pi$
$212$	0	0
$213$	−15.3227	−1.04989
$214$	0	0
$215$	6.07638	0.414406
$216$	0	0
$217$	−0.915425	−0.0621431
$218$	0	0
$219$	1.98323	0.134014
$220$	0	0
$221$	−14.9829	−1.00786
$222$	0	0
$223$	−4.71227	−0.315557	−0.157779	−	0.987475i	$-0.550433\pi$
$223$	−4.71227	−0.315557	−0.157779	+	0.987475i	$0.550433\pi$
$224$	0	0
$225$	−1.09634	−0.0730891
$226$	0	0
$227$	−7.66511	−0.508751	−0.254376	−	0.967105i	$-0.581870\pi$
$227$	−7.66511	−0.508751	−0.254376	+	0.967105i	$0.581870\pi$
$228$	0	0
$229$	−23.1500	−1.52980	−0.764899	−	0.644151i	$-0.777211\pi$
$229$	−23.1500	−1.52980	−0.764899	+	0.644151i	$0.777211\pi$
$230$	0	0
$231$	22.7966	1.49991
$232$	0	0
$233$	−17.2625	−1.13090	−0.565451	−	0.824782i	$-0.691298\pi$
$233$	−17.2625	−1.13090	−0.565451	+	0.824782i	$0.691298\pi$
$234$	0	0
$235$	−4.71586	−0.307629
$236$	0	0
$237$	−16.3268	−1.06054
$238$	0	0
$239$	−7.76229	−0.502101	−0.251051	−	0.967974i	$-0.580776\pi$
$239$	−7.76229	−0.502101	−0.251051	+	0.967974i	$0.580776\pi$
$240$	0	0
$241$	−3.64359	−0.234704	−0.117352	−	0.993090i	$-0.537441\pi$
$241$	−3.64359	−0.234704	−0.117352	+	0.993090i	$0.537441\pi$
$242$	0	0
$243$	16.8571	1.08138
$244$	0	0
$245$	15.7059	1.00342
$246$	0	0
$247$	9.62380	0.612348
$248$	0	0
$249$	1.72929	0.109589
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	1.36697	0.0859404
$254$	0	0
$255$	−26.8533	−1.68162
$256$	0	0
$257$	−15.6722	−0.977607	−0.488804	−	0.872394i	$-0.662567\pi$
$257$	−15.6722	−0.977607	−0.488804	+	0.872394i	$0.662567\pi$
$258$	0	0
$259$	−16.1562	−1.00389
$260$	0	0
$261$	−4.49396	−0.278169
$262$	0	0
$263$	−16.9213	−1.04341	−0.521705	−	0.853126i	$-0.674704\pi$
$263$	−16.9213	−1.04341	−0.521705	+	0.853126i	$0.674704\pi$
$264$	0	0
$265$	10.1630	0.624310
$266$	0	0
$267$	30.2892	1.85367
$268$	0	0
$269$	−19.2617	−1.17441	−0.587205	−	0.809438i	$-0.699772\pi$
$269$	−19.2617	−1.17441	−0.587205	+	0.809438i	$0.699772\pi$
$270$	0	0
$271$	7.42478	0.451023	0.225511	−	0.974241i	$-0.427595\pi$
$271$	7.42478	0.451023	0.225511	+	0.974241i	$0.427595\pi$
$272$	0	0
$273$	−21.5813	−1.30616
$274$	0	0
$275$	−1.61967	−0.0976698
$276$	0	0
$277$	−16.7323	−1.00535	−0.502674	−	0.864476i	$-0.667650\pi$
$277$	−16.7323	−1.00535	−0.502674	+	0.864476i	$0.667650\pi$
$278$	0	0
$279$	0.443208	0.0265342
$280$	0	0
$281$	4.46483	0.266349	0.133175	−	0.991093i	$-0.457483\pi$
$281$	4.46483	0.266349	0.133175	+	0.991093i	$0.457483\pi$
$282$	0	0
$283$	−6.20610	−0.368914	−0.184457	−	0.982841i	$-0.559053\pi$
$283$	−6.20610	−0.368914	−0.184457	+	0.982841i	$0.559053\pi$
$284$	0	0
$285$	17.2484	1.02171
$286$	0	0
$287$	−2.65238	−0.156565
$288$	0	0
$289$	16.8048	0.988519
$290$	0	0
$291$	5.41386	0.317366
$292$	0	0
$293$	−6.62708	−0.387158	−0.193579	−	0.981085i	$-0.562010\pi$
$293$	−6.62708	−0.387158	−0.193579	+	0.981085i	$0.562010\pi$
$294$	0	0
$295$	−18.1389	−1.05609
$296$	0	0
$297$	6.93332	0.402312
$298$	0	0
$299$	−1.29409	−0.0748391
$300$	0	0
$301$	−11.0181	−0.635072
$302$	0	0
$303$	−9.98293	−0.573505
$304$	0	0
$305$	−2.21022	−0.126557
$306$	0	0
$307$	−11.8599	−0.676878	−0.338439	−	0.940988i	$-0.609899\pi$
$307$	−11.8599	−0.676878	−0.338439	+	0.940988i	$0.609899\pi$
$308$	0	0
$309$	−42.8115	−2.43546
$310$	0	0
$311$	−34.9285	−1.98062	−0.990308	−	0.138888i	$-0.955647\pi$
$311$	−34.9285	−1.98062	−0.990308	+	0.138888i	$0.955647\pi$
$312$	0	0
$313$	−5.60810	−0.316988	−0.158494	−	0.987360i	$-0.550664\pi$
$313$	−5.60810	−0.316988	−0.158494	+	0.987360i	$0.550664\pi$
$314$	0	0
$315$	−14.7172	−0.829218
$316$	0	0
$317$	−4.03142	−0.226427	−0.113213	−	0.993571i	$-0.536114\pi$
$317$	−4.03142	−0.226427	−0.113213	+	0.993571i	$0.536114\pi$
$318$	0	0
$319$	−6.63914	−0.371721
$320$	0	0
$321$	31.2280	1.74298
$322$	0	0
$323$	−21.7135	−1.20817
$324$	0	0
$325$	1.53332	0.0850533
$326$	0	0
$327$	31.2958	1.73066
$328$	0	0
$329$	8.55111	0.471438
$330$	0	0
$331$	20.1773	1.10904	0.554521	−	0.832169i	$-0.312901\pi$
$331$	20.1773	1.10904	0.554521	+	0.832169i	$0.312901\pi$
$332$	0	0
$333$	7.82209	0.428648
$334$	0	0
$335$	−4.50106	−0.245919
$336$	0	0
$337$	−29.2768	−1.59481	−0.797405	−	0.603445i	$-0.793794\pi$
$337$	−29.2768	−1.59481	−0.797405	+	0.603445i	$0.793794\pi$
$338$	0	0
$339$	40.0802	2.17686
$340$	0	0
$341$	0.654772	0.0354579
$342$	0	0
$343$	−1.83919	−0.0993070
$344$	0	0
$345$	−2.31935	−0.124870
$346$	0	0
$347$	25.9562	1.39340	0.696700	−	0.717363i	$-0.254651\pi$
$347$	25.9562	1.39340	0.696700	+	0.717363i	$0.254651\pi$
$348$	0	0
$349$	20.8800	1.11768	0.558842	−	0.829274i	$-0.311246\pi$
$349$	20.8800	1.11768	0.558842	+	0.829274i	$0.311246\pi$
$350$	0	0
$351$	−6.56368	−0.350344
$352$	0	0
$353$	8.17677	0.435205	0.217603	−	0.976037i	$-0.430176\pi$
$353$	8.17677	0.435205	0.217603	+	0.976037i	$0.430176\pi$
$354$	0	0
$355$	14.6141	0.775634
$356$	0	0
$357$	48.6922	2.57707
$358$	0	0
$359$	−10.0268	−0.529197	−0.264598	−	0.964359i	$-0.585239\pi$
$359$	−10.0268	−0.529197	−0.264598	+	0.964359i	$0.585239\pi$
$360$	0	0
$361$	−5.05303	−0.265949
$362$	0	0
$363$	7.90073	0.414681
$364$	0	0
$365$	−1.89151	−0.0990060
$366$	0	0
$367$	−11.2119	−0.585258	−0.292629	−	0.956226i	$-0.594530\pi$
$367$	−11.2119	−0.585258	−0.292629	+	0.956226i	$0.594530\pi$
$368$	0	0
$369$	1.28416	0.0668509
$370$	0	0
$371$	−18.4283	−0.956747
$372$	0	0
$373$	−14.8325	−0.767999	−0.383999	−	0.923333i	$-0.625453\pi$
$373$	−14.8325	−0.767999	−0.383999	+	0.923333i	$0.625453\pi$
$374$	0	0
$375$	25.8411	1.33443
$376$	0	0
$377$	6.28519	0.323704
$378$	0	0
$379$	1.45201	0.0745847	0.0372924	−	0.999304i	$-0.488127\pi$
$379$	1.45201	0.0745847	0.0372924	+	0.999304i	$0.488127\pi$
$380$	0	0
$381$	4.06388	0.208199
$382$	0	0
$383$	−2.11124	−0.107880	−0.0539398	−	0.998544i	$-0.517178\pi$
$383$	−2.11124	−0.107880	−0.0539398	+	0.998544i	$0.517178\pi$
$384$	0	0
$385$	−21.7423	−1.10809
$386$	0	0
$387$	5.33447	0.271166
$388$	0	0
$389$	31.6833	1.60641	0.803204	−	0.595704i	$-0.203127\pi$
$389$	31.6833	1.60641	0.803204	+	0.595704i	$0.203127\pi$
$390$	0	0
$391$	2.91976	0.147658
$392$	0	0
$393$	−10.6129	−0.535348
$394$	0	0
$395$	15.5717	0.783498
$396$	0	0
$397$	−2.11595	−0.106197	−0.0530983	−	0.998589i	$-0.516910\pi$
$397$	−2.11595	−0.106197	−0.0530983	+	0.998589i	$0.516910\pi$
$398$	0	0
$399$	−31.2759	−1.56575
$400$	0	0
$401$	−25.8074	−1.28876	−0.644380	−	0.764705i	$-0.722884\pi$
$401$	−25.8074	−1.28876	−0.644380	+	0.764705i	$0.722884\pi$
$402$	0	0
$403$	−0.619864	−0.0308776
$404$	0	0
$405$	−23.3653	−1.16103
$406$	0	0
$407$	11.5559	0.572806
$408$	0	0
$409$	−20.6783	−1.02248	−0.511238	−	0.859439i	$-0.670813\pi$
$409$	−20.6783	−1.02248	−0.511238	+	0.859439i	$0.670813\pi$
$410$	0	0
$411$	−19.4445	−0.959128
$412$	0	0
$413$	32.8906	1.61844
$414$	0	0
$415$	−1.64931	−0.0809616
$416$	0	0
$417$	14.0825	0.689622
$418$	0	0
$419$	−22.5233	−1.10034	−0.550168	−	0.835054i	$-0.685436\pi$
$419$	−22.5233	−1.10034	−0.550168	+	0.835054i	$0.685436\pi$
$420$	0	0
$421$	20.8402	1.01569	0.507843	−	0.861449i	$-0.330443\pi$
$421$	20.8402	1.01569	0.507843	+	0.861449i	$0.330443\pi$
$422$	0	0
$423$	−4.14006	−0.201297
$424$	0	0
$425$	−3.45952	−0.167811
$426$	0	0
$427$	4.00770	0.193946
$428$	0	0
$429$	15.4363	0.745273
$430$	0	0
$431$	18.3300	0.882923	0.441462	−	0.897280i	$-0.354460\pi$
$431$	18.3300	0.882923	0.441462	+	0.897280i	$0.354460\pi$
$432$	0	0
$433$	−4.45852	−0.214263	−0.107131	−	0.994245i	$-0.534167\pi$
$433$	−4.45852	−0.214263	−0.107131	+	0.994245i	$0.534167\pi$
$434$	0	0
$435$	11.2647	0.540102
$436$	0	0
$437$	−1.87541	−0.0897133
$438$	0	0
$439$	−29.3722	−1.40186	−0.700929	−	0.713231i	$-0.747231\pi$
$439$	−29.3722	−1.40186	−0.700929	+	0.713231i	$0.747231\pi$
$440$	0	0
$441$	13.7883	0.656584
$442$	0	0
$443$	−1.67580	−0.0796197	−0.0398098	−	0.999207i	$-0.512675\pi$
$443$	−1.67580	−0.0796197	−0.0398098	+	0.999207i	$0.512675\pi$
$444$	0	0
$445$	−28.8884	−1.36944
$446$	0	0
$447$	−7.13177	−0.337321
$448$	0	0
$449$	−14.2036	−0.670310	−0.335155	−	0.942163i	$-0.608789\pi$
$449$	−14.2036	−0.670310	−0.335155	+	0.942163i	$0.608789\pi$
$450$	0	0
$451$	1.89715	0.0893335
$452$	0	0
$453$	20.6969	0.972423
$454$	0	0
$455$	20.5832	0.964954
$456$	0	0
$457$	−11.8582	−0.554703	−0.277352	−	0.960768i	$-0.589457\pi$
$457$	−11.8582	−0.554703	−0.277352	+	0.960768i	$0.589457\pi$
$458$	0	0
$459$	14.8092	0.691232
$460$	0	0
$461$	−17.9041	−0.833875	−0.416937	−	0.908935i	$-0.636897\pi$
$461$	−17.9041	−0.833875	−0.416937	+	0.908935i	$0.636897\pi$
$462$	0	0
$463$	17.7445	0.824655	0.412327	−	0.911036i	$-0.364716\pi$
$463$	17.7445	0.824655	0.412327	+	0.911036i	$0.364716\pi$
$464$	0	0
$465$	−1.11096	−0.0515196
$466$	0	0
$467$	33.3421	1.54289	0.771444	−	0.636297i	$-0.219535\pi$
$467$	33.3421	1.54289	0.771444	+	0.636297i	$0.219535\pi$
$468$	0	0
$469$	8.16161	0.376868
$470$	0	0
$471$	−29.4252	−1.35584
$472$	0	0
$473$	7.88085	0.362362
$474$	0	0
$475$	2.22211	0.101958
$476$	0	0
$477$	8.92214	0.408517
$478$	0	0
$479$	−1.92588	−0.0879959	−0.0439979	−	0.999032i	$-0.514010\pi$
$479$	−1.92588	−0.0879959	−0.0439979	+	0.999032i	$0.514010\pi$
$480$	0	0
$481$	−10.9399	−0.498814
$482$	0	0
$483$	4.20560	0.191361
$484$	0	0
$485$	−5.16348	−0.234462
$486$	0	0
$487$	24.6191	1.11560	0.557800	−	0.829976i	$-0.311646\pi$
$487$	24.6191	1.11560	0.557800	+	0.829976i	$0.311646\pi$
$488$	0	0
$489$	−29.4856	−1.33339
$490$	0	0
$491$	9.59465	0.433000	0.216500	−	0.976283i	$-0.430536\pi$
$491$	9.59465	0.433000	0.216500	+	0.976283i	$0.430536\pi$
$492$	0	0
$493$	−14.1808	−0.638671
$494$	0	0
$495$	10.5267	0.473138
$496$	0	0
$497$	−26.4992	−1.18865
$498$	0	0
$499$	−17.2553	−0.772451	−0.386226	−	0.922404i	$-0.626221\pi$
$499$	−17.2553	−0.772451	−0.386226	+	0.922404i	$0.626221\pi$
$500$	0	0
$501$	6.86338	0.306633
$502$	0	0
$503$	38.1113	1.69930	0.849650	−	0.527347i	$-0.176813\pi$
$503$	38.1113	1.69930	0.849650	+	0.527347i	$0.176813\pi$
$504$	0	0
$505$	9.52124	0.423690
$506$	0	0
$507$	13.9942	0.621502
$508$	0	0
$509$	4.46254	0.197798	0.0988992	−	0.995097i	$-0.468468\pi$
$509$	4.46254	0.197798	0.0988992	+	0.995097i	$0.468468\pi$
$510$	0	0
$511$	3.42980	0.151725
$512$	0	0
$513$	−9.51220	−0.419974
$514$	0	0
$515$	40.8315	1.79925
$516$	0	0
$517$	−6.11631	−0.268995
$518$	0	0
$519$	−46.0539	−2.02154
$520$	0	0
$521$	−13.1756	−0.577235	−0.288618	−	0.957444i	$-0.593196\pi$
$521$	−13.1756	−0.577235	−0.288618	+	0.957444i	$0.593196\pi$
$522$	0	0
$523$	−40.1404	−1.75522	−0.877608	−	0.479380i	$-0.840862\pi$
$523$	−40.1404	−1.75522	−0.877608	+	0.479380i	$0.840862\pi$
$524$	0	0
$525$	−4.98306	−0.217479
$526$	0	0
$527$	1.39855	0.0609219
$528$	0	0
$529$	−22.7478	−0.989036
$530$	0	0
$531$	−15.9241	−0.691049
$532$	0	0
$533$	−1.79601	−0.0777939
$534$	0	0
$535$	−29.7838	−1.28767
$536$	0	0
$537$	5.17573	0.223349
$538$	0	0
$539$	20.3700	0.877400
$540$	0	0
$541$	−5.88624	−0.253069	−0.126535	−	0.991962i	$-0.540385\pi$
$541$	−5.88624	−0.253069	−0.126535	+	0.991962i	$0.540385\pi$
$542$	0	0
$543$	−42.5237	−1.82486
$544$	0	0
$545$	−29.8484	−1.27856
$546$	0	0
$547$	26.1048	1.11616	0.558079	−	0.829788i	$-0.311538\pi$
$547$	26.1048	1.11616	0.558079	+	0.829788i	$0.311538\pi$
$548$	0	0
$549$	−1.94035	−0.0828122
$550$	0	0
$551$	9.10860	0.388039
$552$	0	0
$553$	−28.2356	−1.20070
$554$	0	0
$555$	−19.6071	−0.832276
$556$	0	0
$557$	15.8086	0.669834	0.334917	−	0.942248i	$-0.391292\pi$
$557$	15.8086	0.669834	0.334917	+	0.942248i	$0.391292\pi$
$558$	0	0
$559$	−7.46070	−0.315554
$560$	0	0
$561$	−34.8278	−1.47043
$562$	0	0
$563$	31.4117	1.32384	0.661922	−	0.749572i	$-0.269741\pi$
$563$	31.4117	1.32384	0.661922	+	0.749572i	$0.269741\pi$
$564$	0	0
$565$	−38.2266	−1.60820
$566$	0	0
$567$	42.3675	1.77927
$568$	0	0
$569$	27.5476	1.15486	0.577428	−	0.816442i	$-0.304057\pi$
$569$	27.5476	1.15486	0.577428	+	0.816442i	$0.304057\pi$
$570$	0	0
$571$	−2.64843	−0.110834	−0.0554168	−	0.998463i	$-0.517649\pi$
$571$	−2.64843	−0.110834	−0.0554168	+	0.998463i	$0.517649\pi$
$572$	0	0
$573$	46.3740	1.93730
$574$	0	0
$575$	−0.298802	−0.0124609
$576$	0	0
$577$	40.0716	1.66820	0.834101	−	0.551611i	$-0.185987\pi$
$577$	40.0716	1.66820	0.834101	+	0.551611i	$0.185987\pi$
$578$	0	0
$579$	37.8465	1.57285
$580$	0	0
$581$	2.99064	0.124073
$582$	0	0
$583$	13.1811	0.545905
$584$	0	0
$585$	−9.96545	−0.412021
$586$	0	0
$587$	−38.5967	−1.59306	−0.796528	−	0.604602i	$-0.793332\pi$
$587$	−38.5967	−1.59306	−0.796528	+	0.604602i	$0.793332\pi$
$588$	0	0
$589$	−0.898316	−0.0370145
$590$	0	0
$591$	26.2672	1.08049
$592$	0	0
$593$	20.6283	0.847105	0.423552	−	0.905872i	$-0.360783\pi$
$593$	20.6283	0.847105	0.423552	+	0.905872i	$0.360783\pi$
$594$	0	0
$595$	−46.4403	−1.90387
$596$	0	0
$597$	41.5561	1.70078
$598$	0	0
$599$	−7.14902	−0.292101	−0.146050	−	0.989277i	$-0.546656\pi$
$599$	−7.14902	−0.292101	−0.146050	+	0.989277i	$0.546656\pi$
$600$	0	0
$601$	−0.915052	−0.0373258	−0.0186629	−	0.999826i	$-0.505941\pi$
$601$	−0.915052	−0.0373258	−0.0186629	+	0.999826i	$0.505941\pi$
$602$	0	0
$603$	−3.95149	−0.160917
$604$	0	0
$605$	−7.53534	−0.306355
$606$	0	0
$607$	33.6586	1.36616	0.683079	−	0.730344i	$-0.260640\pi$
$607$	33.6586	1.36616	0.683079	+	0.730344i	$0.260640\pi$
$608$	0	0
$609$	−20.4259	−0.827700
$610$	0	0
$611$	5.79023	0.234248
$612$	0	0
$613$	−14.0662	−0.568128	−0.284064	−	0.958805i	$-0.591683\pi$
$613$	−14.0662	−0.568128	−0.284064	+	0.958805i	$0.591683\pi$
$614$	0	0
$615$	−3.21893	−0.129800
$616$	0	0
$617$	5.66666	0.228131	0.114066	−	0.993473i	$-0.463613\pi$
$617$	5.66666	0.228131	0.114066	+	0.993473i	$0.463613\pi$
$618$	0	0
$619$	17.1826	0.690628	0.345314	−	0.938487i	$-0.387772\pi$
$619$	17.1826	0.690628	0.345314	+	0.938487i	$0.387772\pi$
$620$	0	0
$621$	1.27908	0.0513278
$622$	0	0
$623$	52.3823	2.09865
$624$	0	0
$625$	−21.6709	−0.866836
$626$	0	0
$627$	22.3706	0.893395
$628$	0	0
$629$	24.6828	0.984167
$630$	0	0
$631$	−15.8703	−0.631787	−0.315894	−	0.948795i	$-0.602304\pi$
$631$	−15.8703	−0.631787	−0.315894	+	0.948795i	$0.602304\pi$
$632$	0	0
$633$	9.83763	0.391011
$634$	0	0
$635$	−3.87593	−0.153812
$636$	0	0
$637$	−19.2841	−0.764062
$638$	0	0
$639$	12.8297	0.507536
$640$	0	0
$641$	11.4768	0.453305	0.226652	−	0.973976i	$-0.427222\pi$
$641$	11.4768	0.453305	0.226652	+	0.973976i	$0.427222\pi$
$642$	0	0
$643$	12.4220	0.489877	0.244938	−	0.969539i	$-0.421232\pi$
$643$	12.4220	0.489877	0.244938	+	0.969539i	$0.421232\pi$
$644$	0	0
$645$	−13.3716	−0.526505
$646$	0	0
$647$	−19.6357	−0.771958	−0.385979	−	0.922508i	$-0.626136\pi$
$647$	−19.6357	−0.771958	−0.385979	+	0.922508i	$0.626136\pi$
$648$	0	0
$649$	−23.5255	−0.923456
$650$	0	0
$651$	2.01446	0.0789531
$652$	0	0
$653$	32.7940	1.28333	0.641664	−	0.766986i	$-0.278244\pi$
$653$	32.7940	1.28333	0.641664	+	0.766986i	$0.278244\pi$
$654$	0	0
$655$	10.1220	0.395501
$656$	0	0
$657$	−1.66056	−0.0647845
$658$	0	0
$659$	−39.6815	−1.54577	−0.772887	−	0.634544i	$-0.781188\pi$
$659$	−39.6815	−1.54577	−0.772887	+	0.634544i	$0.781188\pi$
$660$	0	0
$661$	34.1412	1.32794	0.663969	−	0.747760i	$-0.268871\pi$
$661$	34.1412	1.32794	0.663969	+	0.747760i	$0.268871\pi$
$662$	0	0
$663$	32.9711	1.28049
$664$	0	0
$665$	29.8295	1.15674
$666$	0	0
$667$	−1.22481	−0.0474248
$668$	0	0
$669$	10.3697	0.400917
$670$	0	0
$671$	−2.86657	−0.110663
$672$	0	0
$673$	−36.7855	−1.41798	−0.708988	−	0.705221i	$-0.750848\pi$
$673$	−36.7855	−1.41798	−0.708988	+	0.705221i	$0.750848\pi$
$674$	0	0
$675$	−1.51554	−0.0583331
$676$	0	0
$677$	15.7752	0.606292	0.303146	−	0.952944i	$-0.401963\pi$
$677$	15.7752	0.606292	0.303146	+	0.952944i	$0.401963\pi$
$678$	0	0
$679$	9.36276	0.359310
$680$	0	0
$681$	16.8677	0.646371
$682$	0	0
$683$	−15.0199	−0.574721	−0.287360	−	0.957823i	$-0.592778\pi$
$683$	−15.0199	−0.574721	−0.287360	+	0.957823i	$0.592778\pi$
$684$	0	0
$685$	18.5453	0.708578
$686$	0	0
$687$	50.9435	1.94361
$688$	0	0
$689$	−12.4784	−0.475388
$690$	0	0
$691$	27.3782	1.04152	0.520758	−	0.853704i	$-0.325649\pi$
$691$	27.3782	1.04152	0.520758	+	0.853704i	$0.325649\pi$
$692$	0	0
$693$	−19.0876	−0.725079
$694$	0	0
$695$	−13.4312	−0.509474
$696$	0	0
$697$	4.05221	0.153488
$698$	0	0
$699$	37.9874	1.43682
$700$	0	0
$701$	−19.4812	−0.735794	−0.367897	−	0.929867i	$-0.619922\pi$
$701$	−19.4812	−0.735794	−0.367897	+	0.929867i	$0.619922\pi$
$702$	0	0
$703$	−15.8542	−0.597953
$704$	0	0
$705$	10.3776	0.390844
$706$	0	0
$707$	−17.2645	−0.649299
$708$	0	0
$709$	34.2515	1.28634	0.643172	−	0.765722i	$-0.277618\pi$
$709$	34.2515	1.28634	0.643172	+	0.765722i	$0.277618\pi$
$710$	0	0
$711$	13.6704	0.512681
$712$	0	0
$713$	0.120794	0.00452379
$714$	0	0
$715$	−14.7224	−0.550588
$716$	0	0
$717$	17.0815	0.637922
$718$	0	0
$719$	−10.2768	−0.383260	−0.191630	−	0.981467i	$-0.561377\pi$
$719$	−10.2768	−0.383260	−0.191630	+	0.981467i	$0.561377\pi$
$720$	0	0
$721$	−74.0384	−2.75733
$722$	0	0
$723$	8.01801	0.298193
$724$	0	0
$725$	1.45123	0.0538975
$726$	0	0
$727$	−21.0114	−0.779269	−0.389634	−	0.920970i	$-0.627399\pi$
$727$	−21.0114	−0.779269	−0.389634	+	0.920970i	$0.627399\pi$
$728$	0	0
$729$	−3.69734	−0.136939
$730$	0	0
$731$	16.8330	0.622592
$732$	0	0
$733$	34.7020	1.28175	0.640874	−	0.767646i	$-0.278572\pi$
$733$	34.7020	1.28175	0.640874	+	0.767646i	$0.278572\pi$
$734$	0	0
$735$	−34.5622	−1.27484
$736$	0	0
$737$	−5.83772	−0.215035
$738$	0	0
$739$	−6.41119	−0.235839	−0.117920	−	0.993023i	$-0.537623\pi$
$739$	−6.41119	−0.235839	−0.117920	+	0.993023i	$0.537623\pi$
$740$	0	0
$741$	−21.1779	−0.777991
$742$	0	0
$743$	7.06714	0.259268	0.129634	−	0.991562i	$-0.458620\pi$
$743$	7.06714	0.259268	0.129634	+	0.991562i	$0.458620\pi$
$744$	0	0
$745$	6.80194	0.249204
$746$	0	0
$747$	−1.44793	−0.0529771
$748$	0	0
$749$	54.0059	1.97333
$750$	0	0
$751$	0.441269	0.0161021	0.00805106	−	0.999968i	$-0.497437\pi$
$751$	0.441269	0.0161021	0.00805106	+	0.999968i	$0.497437\pi$
$752$	0	0
$753$	2.20058	0.0801935
$754$	0	0
$755$	−19.7397	−0.718400
$756$	0	0
$757$	44.4187	1.61443	0.807213	−	0.590261i	$-0.200975\pi$
$757$	44.4187	1.61443	0.807213	+	0.590261i	$0.200975\pi$
$758$	0	0
$759$	−3.00812	−0.109188
$760$	0	0
$761$	−15.1484	−0.549129	−0.274564	−	0.961569i	$-0.588534\pi$
$761$	−15.1484	−0.549129	−0.274564	+	0.961569i	$0.588534\pi$
$762$	0	0
$763$	54.1230	1.95939
$764$	0	0
$765$	22.4843	0.812922
$766$	0	0
$767$	22.2713	0.804168
$768$	0	0
$769$	13.8807	0.500552	0.250276	−	0.968175i	$-0.419479\pi$
$769$	13.8807	0.500552	0.250276	+	0.968175i	$0.419479\pi$
$770$	0	0
$771$	34.4880	1.24205
$772$	0	0
$773$	43.2713	1.55636	0.778180	−	0.628041i	$-0.216143\pi$
$773$	43.2713	1.55636	0.778180	+	0.628041i	$0.216143\pi$
$774$	0	0
$775$	−0.143125	−0.00514120
$776$	0	0
$777$	35.5529	1.27545
$778$	0	0
$779$	−2.60281	−0.0932553
$780$	0	0
$781$	18.9539	0.678225
$782$	0	0
$783$	−6.21230	−0.222009
$784$	0	0
$785$	28.0643	1.00166
$786$	0	0
$787$	−15.0475	−0.536385	−0.268193	−	0.963365i	$-0.586426\pi$
$787$	−15.0475	−0.536385	−0.268193	+	0.963365i	$0.586426\pi$
$788$	0	0
$789$	37.2366	1.32566
$790$	0	0
$791$	69.3149	2.46455
$792$	0	0
$793$	2.71374	0.0963679
$794$	0	0
$795$	−22.3645	−0.793189
$796$	0	0
$797$	−9.72906	−0.344621	−0.172311	−	0.985043i	$-0.555123\pi$
$797$	−9.72906	−0.344621	−0.172311	+	0.985043i	$0.555123\pi$
$798$	0	0
$799$	−13.0641	−0.462173
$800$	0	0
$801$	−25.3612	−0.896093
$802$	0	0
$803$	−2.45322	−0.0865721
$804$	0	0
$805$	−4.01110	−0.141373
$806$	0	0
$807$	42.3870	1.49209
$808$	0	0
$809$	4.44577	0.156305	0.0781525	−	0.996941i	$-0.475098\pi$
$809$	4.44577	0.156305	0.0781525	+	0.996941i	$0.475098\pi$
$810$	0	0
$811$	3.67708	0.129120	0.0645600	−	0.997914i	$-0.479436\pi$
$811$	3.67708	0.129120	0.0645600	+	0.997914i	$0.479436\pi$
$812$	0	0
$813$	−16.3388	−0.573027
$814$	0	0
$815$	28.1220	0.985071
$816$	0	0
$817$	−10.8122	−0.378270
$818$	0	0
$819$	18.0700	0.631417
$820$	0	0
$821$	−30.4062	−1.06118	−0.530592	−	0.847627i	$-0.678031\pi$
$821$	−30.4062	−1.06118	−0.530592	+	0.847627i	$0.678031\pi$
$822$	0	0
$823$	51.8751	1.80825	0.904127	−	0.427264i	$-0.140523\pi$
$823$	51.8751	1.80825	0.904127	+	0.427264i	$0.140523\pi$
$824$	0	0
$825$	3.56421	0.124090
$826$	0	0
$827$	−22.0212	−0.765754	−0.382877	−	0.923799i	$-0.625067\pi$
$827$	−22.0212	−0.765754	−0.382877	+	0.923799i	$0.625067\pi$
$828$	0	0
$829$	17.7074	0.615002	0.307501	−	0.951548i	$-0.400507\pi$
$829$	17.7074	0.615002	0.307501	+	0.951548i	$0.400507\pi$
$830$	0	0
$831$	36.8208	1.27730
$832$	0	0
$833$	43.5092	1.50750
$834$	0	0
$835$	−6.54597	−0.226533
$836$	0	0
$837$	0.612675	0.0211772
$838$	0	0
$839$	−31.6761	−1.09358	−0.546790	−	0.837270i	$-0.684150\pi$
$839$	−31.6761	−1.09358	−0.546790	+	0.837270i	$0.684150\pi$
$840$	0	0
$841$	−23.0513	−0.794872
$842$	0	0
$843$	−9.82521	−0.338398
$844$	0	0
$845$	−13.3470	−0.459149
$846$	0	0
$847$	13.6636	0.469485
$848$	0	0
$849$	13.6570	0.468707
$850$	0	0
$851$	2.13188	0.0730798
$852$	0	0
$853$	43.3452	1.48411	0.742055	−	0.670338i	$-0.233851\pi$
$853$	43.3452	1.48411	0.742055	+	0.670338i	$0.233851\pi$
$854$	0	0
$855$	−14.4421	−0.493909
$856$	0	0
$857$	−9.82002	−0.335446	−0.167723	−	0.985834i	$-0.553641\pi$
$857$	−9.82002	−0.335446	−0.167723	+	0.985834i	$0.553641\pi$
$858$	0	0
$859$	−2.06240	−0.0703681	−0.0351841	−	0.999381i	$-0.511202\pi$
$859$	−2.06240	−0.0703681	−0.0351841	+	0.999381i	$0.511202\pi$
$860$	0	0
$861$	5.83677	0.198917
$862$	0	0
$863$	33.9387	1.15529	0.577644	−	0.816289i	$-0.303972\pi$
$863$	33.9387	1.15529	0.577644	+	0.816289i	$0.303972\pi$
$864$	0	0
$865$	43.9240	1.49346
$866$	0	0
$867$	−36.9803	−1.25592
$868$	0	0
$869$	20.1960	0.685101
$870$	0	0
$871$	5.52649	0.187258
$872$	0	0
$873$	−4.53303	−0.153420
$874$	0	0
$875$	44.6896	1.51079
$876$	0	0
$877$	−33.4604	−1.12988	−0.564939	−	0.825133i	$-0.691100\pi$
$877$	−33.4604	−1.12988	−0.564939	+	0.825133i	$0.691100\pi$
$878$	0	0
$879$	14.5834	0.491886
$880$	0	0
$881$	25.2446	0.850511	0.425255	−	0.905073i	$-0.360184\pi$
$881$	25.2446	0.850511	0.425255	+	0.905073i	$0.360184\pi$
$882$	0	0
$883$	27.7070	0.932413	0.466207	−	0.884676i	$-0.345620\pi$
$883$	27.7070	0.932413	0.466207	+	0.884676i	$0.345620\pi$
$884$	0	0
$885$	39.9160	1.34176
$886$	0	0
$887$	−44.5082	−1.49444	−0.747220	−	0.664577i	$-0.768612\pi$
$887$	−44.5082	−1.49444	−0.747220	+	0.664577i	$0.768612\pi$
$888$	0	0
$889$	7.02808	0.235714
$890$	0	0
$891$	−30.3040	−1.01522
$892$	0	0
$893$	8.39129	0.280804
$894$	0	0
$895$	−4.93637	−0.165005
$896$	0	0
$897$	2.84774	0.0950834
$898$	0	0
$899$	−0.586679	−0.0195669
$900$	0	0
$901$	28.1540	0.937946
$902$	0	0
$903$	24.2462	0.806862
$904$	0	0
$905$	40.5570	1.34816
$906$	0	0
$907$	55.1823	1.83230	0.916149	−	0.400837i	$-0.131281\pi$
$907$	55.1823	1.83230	0.916149	+	0.400837i	$0.131281\pi$
$908$	0	0
$909$	8.35871	0.277241
$910$	0	0
$911$	−10.8027	−0.357911	−0.178955	−	0.983857i	$-0.557272\pi$
$911$	−10.8027	−0.357911	−0.178955	+	0.983857i	$0.557272\pi$
$912$	0	0
$913$	−2.13910	−0.0707939
$914$	0	0
$915$	4.86375	0.160791
$916$	0	0
$917$	−18.3539	−0.606100
$918$	0	0
$919$	−0.657994	−0.0217052	−0.0108526	−	0.999941i	$-0.503455\pi$
$919$	−0.657994	−0.0217052	−0.0108526	+	0.999941i	$0.503455\pi$
$920$	0	0
$921$	26.0985	0.859976
$922$	0	0
$923$	−17.9434	−0.590615
$924$	0	0
$925$	−2.52598	−0.0830539
$926$	0	0
$927$	35.8461	1.17734
$928$	0	0
$929$	39.9819	1.31176	0.655882	−	0.754864i	$-0.272297\pi$
$929$	39.9819	1.31176	0.655882	+	0.754864i	$0.272297\pi$
$930$	0	0
$931$	−27.9468	−0.915918
$932$	0	0
$933$	76.8630	2.51638
$934$	0	0
$935$	33.2171	1.08632
$936$	0	0
$937$	−29.7694	−0.972525	−0.486262	−	0.873813i	$-0.661640\pi$
$937$	−29.7694	−0.972525	−0.486262	+	0.873813i	$0.661640\pi$
$938$	0	0
$939$	12.3411	0.402735
$940$	0	0
$941$	31.2486	1.01868	0.509338	−	0.860567i	$-0.329890\pi$
$941$	31.2486	1.01868	0.509338	+	0.860567i	$0.329890\pi$
$942$	0	0
$943$	0.349993	0.0113973
$944$	0	0
$945$	−20.3445	−0.661806
$946$	0	0
$947$	9.26651	0.301121	0.150561	−	0.988601i	$-0.451892\pi$
$947$	9.26651	0.301121	0.150561	+	0.988601i	$0.451892\pi$
$948$	0	0
$949$	2.32243	0.0753892
$950$	0	0
$951$	8.87145	0.287676
$952$	0	0
$953$	45.0873	1.46052	0.730261	−	0.683169i	$-0.239399\pi$
$953$	45.0873	1.46052	0.730261	+	0.683169i	$0.239399\pi$
$954$	0	0
$955$	−44.2293	−1.43123
$956$	0	0
$957$	14.6099	0.472273
$958$	0	0
$959$	−33.6274	−1.08589
$960$	0	0
$961$	−30.9421	−0.998134
$962$	0	0
$963$	−26.1472	−0.842583
$964$	0	0
$965$	−36.0961	−1.16198
$966$	0	0
$967$	26.0430	0.837486	0.418743	−	0.908105i	$-0.362471\pi$
$967$	26.0430	0.837486	0.418743	+	0.908105i	$0.362471\pi$
$968$	0	0
$969$	47.7822	1.53499
$970$	0	0
$971$	−23.8180	−0.764357	−0.382179	−	0.924088i	$-0.624826\pi$
$971$	−23.8180	−0.764357	−0.382179	+	0.924088i	$0.624826\pi$
$972$	0	0
$973$	24.3543	0.780763
$974$	0	0
$975$	−3.37419	−0.108061
$976$	0	0
$977$	28.3780	0.907891	0.453946	−	0.891029i	$-0.350016\pi$
$977$	28.3780	0.907891	0.453946	+	0.891029i	$0.350016\pi$
$978$	0	0
$979$	−37.4673	−1.19746
$980$	0	0
$981$	−26.2040	−0.836628
$982$	0	0
$983$	−45.0582	−1.43713	−0.718566	−	0.695458i	$-0.755201\pi$
$983$	−45.0582	−1.43713	−0.718566	+	0.695458i	$0.755201\pi$
$984$	0	0
$985$	−25.0524	−0.798235
$986$	0	0
$987$	−18.8174	−0.598964
$988$	0	0
$989$	1.45389	0.0462309
$990$	0	0
$991$	−28.1998	−0.895795	−0.447897	−	0.894085i	$-0.647827\pi$
$991$	−28.1998	−0.895795	−0.447897	+	0.894085i	$0.647827\pi$
$992$	0	0
$993$	−44.4017	−1.40904
$994$	0	0
$995$	−39.6342	−1.25649
$996$	0	0
$997$	−0.652340	−0.0206598	−0.0103299	−	0.999947i	$-0.503288\pi$
$997$	−0.652340	−0.0206598	−0.0103299	+	0.999947i	$0.503288\pi$
$998$	0	0
$999$	10.8130	0.342108

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.i.1.3		12
4.3	odd	2		2008.2.a.b.1.10	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.b.1.10	✓	12	4.3	odd	2
4016.2.a.i.1.3		12	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:	\( N \)	\(=\)	\( 4016 = 2^{4} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4016.a (trivial)

\(f(q)\)	\(=\)	\(q-2.20058 q^{3} +2.09881 q^{5} -3.80569 q^{7} +1.84254 q^{9} +O(q^{10})\)	\(q-2.20058 q^{3} +2.09881 q^{5} -3.80569 q^{7} +1.84254 q^{9} +2.72208 q^{11} -2.57695 q^{13} -4.61859 q^{15} +5.81419 q^{17} -3.73456 q^{19} +8.37472 q^{21} +0.502178 q^{23} -0.595012 q^{25} +2.54707 q^{27} -2.43900 q^{29} +0.240541 q^{31} -5.99014 q^{33} -7.98741 q^{35} +4.24526 q^{37} +5.67079 q^{39} +0.696951 q^{41} +2.89516 q^{43} +3.86714 q^{45} -2.24693 q^{47} +7.48327 q^{49} -12.7946 q^{51} +4.84229 q^{53} +5.71311 q^{55} +8.21820 q^{57} -8.64247 q^{59} -1.05308 q^{61} -7.01215 q^{63} -5.40853 q^{65} -2.14458 q^{67} -1.10508 q^{69} +6.96304 q^{71} -0.901230 q^{73} +1.30937 q^{75} -10.3594 q^{77} +7.41932 q^{79} -11.1327 q^{81} -0.785834 q^{83} +12.2029 q^{85} +5.36721 q^{87} -13.7642 q^{89} +9.80709 q^{91} -0.529330 q^{93} -7.83813 q^{95} -2.46020 q^{97} +5.01555 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{3} + 5 q^{5} - 5 q^{7} + 7 q^{9}+O(q^{10}) \) 12 * q - 3 * q^3 + 5 * q^5 - 5 * q^7 + 7 * q^9	\( 12 q - 3 q^{3} + 5 q^{5} - 5 q^{7} + 7 q^{9} - 10 q^{11} + 3 q^{13} - 11 q^{15} + 2 q^{17} - 15 q^{19} + 3 q^{21} - 20 q^{23} - 3 q^{25} - 15 q^{27} + 6 q^{29} - 14 q^{31} - 6 q^{33} - 16 q^{35} + 5 q^{37} - 21 q^{39} - 21 q^{43} + 10 q^{45} - 27 q^{47} - 13 q^{49} - 19 q^{51} + 22 q^{53} - 24 q^{55} + q^{57} - 23 q^{59} + 4 q^{61} - 21 q^{63} - q^{65} - 26 q^{67} + 10 q^{69} - 23 q^{71} - 8 q^{73} - 16 q^{75} + 22 q^{77} - 37 q^{79} - 20 q^{81} - 30 q^{83} + 2 q^{85} - 16 q^{87} + 3 q^{89} - 8 q^{91} + 20 q^{93} - 33 q^{95} - 4 q^{97} - 15 q^{99}+O(q^{100}) \) 12 * q - 3 * q^3 + 5 * q^5 - 5 * q^7 + 7 * q^9 - 10 * q^11 + 3 * q^13 - 11 * q^15 + 2 * q^17 - 15 * q^19 + 3 * q^21 - 20 * q^23 - 3 * q^25 - 15 * q^27 + 6 * q^29 - 14 * q^31 - 6 * q^33 - 16 * q^35 + 5 * q^37 - 21 * q^39 - 21 * q^43 + 10 * q^45 - 27 * q^47 - 13 * q^49 - 19 * q^51 + 22 * q^53 - 24 * q^55 + q^57 - 23 * q^59 + 4 * q^61 - 21 * q^63 - q^65 - 26 * q^67 + 10 * q^69 - 23 * q^71 - 8 * q^73 - 16 * q^75 + 22 * q^77 - 37 * q^79 - 20 * q^81 - 30 * q^83 + 2 * q^85 - 16 * q^87 + 3 * q^89 - 8 * q^91 + 20 * q^93 - 33 * q^95 - 4 * q^97 - 15 * q^99

Introduction

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