LMFDB - Embedded newform 4016.2.a.k.1.14

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:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - 2 x^{16} - 28 x^{15} + 54 x^{14} + 317 x^{13} - 582 x^{12} - 1867 x^{11} + 3178 x^{10} + \cdots + 256 $ x^17 - 2x^16 - 28x^15 + 54x^14 + 317x^13 - 582x^12 - 1867x^11 + 3178x^10 + 6186x^9 - 9216x^8 - 11921x^7 + 13680x^6 + 13752x^5 - 9400x^4 - 8800x^3 + 1920x^2 + 2240x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 251)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$-2.65791$ 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)

$f(q)$	$=$	$q+2.62368 q^{3} +1.15029 q^{5} +2.51426 q^{7} +3.88372 q^{9} +O(q^{10})$	$q+2.62368 q^{3} +1.15029 q^{5} +2.51426 q^{7} +3.88372 q^{9} +6.29973 q^{11} +0.699407 q^{13} +3.01800 q^{15} +4.58455 q^{17} -7.23032 q^{19} +6.59661 q^{21} +4.43841 q^{23} -3.67683 q^{25} +2.31859 q^{27} +3.16969 q^{29} -6.86491 q^{31} +16.5285 q^{33} +2.89213 q^{35} +3.34453 q^{37} +1.83502 q^{39} +1.17827 q^{41} -8.14366 q^{43} +4.46740 q^{45} -7.94801 q^{47} -0.678520 q^{49} +12.0284 q^{51} +6.14772 q^{53} +7.24652 q^{55} -18.9701 q^{57} -3.29675 q^{59} +1.51045 q^{61} +9.76465 q^{63} +0.804522 q^{65} -4.60689 q^{67} +11.6450 q^{69} +8.22808 q^{71} -5.16971 q^{73} -9.64684 q^{75} +15.8391 q^{77} -8.65353 q^{79} -5.56790 q^{81} -4.65024 q^{83} +5.27356 q^{85} +8.31626 q^{87} +3.59697 q^{89} +1.75849 q^{91} -18.0113 q^{93} -8.31697 q^{95} -18.0684 q^{97} +24.4663 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9}+O(q^{10}) $ 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9	$ 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9} + q^{11} + 22 q^{13} + 8 q^{15} - q^{17} - 13 q^{19} + 25 q^{21} + 2 q^{23} + 32 q^{25} + 15 q^{27} + 28 q^{29} - 12 q^{31} - 16 q^{33} + 15 q^{35} + 27 q^{37} - 13 q^{39} - q^{41} - 9 q^{43} - 7 q^{45} + 20 q^{47} + 32 q^{49} + 2 q^{51} + q^{53} + 11 q^{55} - 24 q^{57} + 20 q^{59} + 59 q^{61} + 41 q^{63} - 14 q^{65} - 15 q^{67} + 38 q^{69} + 26 q^{71} + 8 q^{73} + 20 q^{75} - 33 q^{79} + 29 q^{81} + 67 q^{85} + 11 q^{87} + 11 q^{89} + 2 q^{91} + 28 q^{93} + 8 q^{95} - 10 q^{97} + 36 q^{99}+O(q^{100}) $ 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9 + q^11 + 22 * q^13 + 8 * q^15 - q^17 - 13 * q^19 + 25 * q^21 + 2 * q^23 + 32 * q^25 + 15 * q^27 + 28 * q^29 - 12 * q^31 - 16 * q^33 + 15 * q^35 + 27 * q^37 - 13 * q^39 - q^41 - 9 * q^43 - 7 * q^45 + 20 * q^47 + 32 * q^49 + 2 * q^51 + q^53 + 11 * q^55 - 24 * q^57 + 20 * q^59 + 59 * q^61 + 41 * q^63 - 14 * q^65 - 15 * q^67 + 38 * q^69 + 26 * q^71 + 8 * q^73 + 20 * q^75 - 33 * q^79 + 29 * q^81 + 67 * q^85 + 11 * q^87 + 11 * q^89 + 2 * q^91 + 28 * q^93 + 8 * q^95 - 10 * q^97 + 36 * 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
$3$	2.62368	1.51478	0.757392	−	0.652960i	$-0.226473\pi$
$3$	2.62368	1.51478	0.757392	+	0.652960i	$0.226473\pi$
$4$	0	0
$5$	1.15029	0.514426	0.257213	−	0.966355i	$-0.417196\pi$
$5$	1.15029	0.514426	0.257213	+	0.966355i	$0.417196\pi$
$6$	0	0
$7$	2.51426	0.950299	0.475150	−	0.879905i	$-0.342394\pi$
$7$	2.51426	0.950299	0.475150	+	0.879905i	$0.342394\pi$
$8$	0	0
$9$	3.88372	1.29457
$10$	0	0
$11$	6.29973	1.89944	0.949720	−	0.313102i	$-0.101368\pi$
$11$	6.29973	1.89944	0.949720	+	0.313102i	$0.101368\pi$
$12$	0	0
$13$	0.699407	0.193981	0.0969903	−	0.995285i	$-0.469078\pi$
$13$	0.699407	0.193981	0.0969903	+	0.995285i	$0.469078\pi$
$14$	0	0
$15$	3.01800	0.779244
$16$	0	0
$17$	4.58455	1.11192	0.555958	−	0.831210i	$-0.312352\pi$
$17$	4.58455	1.11192	0.555958	+	0.831210i	$0.312352\pi$
$18$	0	0
$19$	−7.23032	−1.65875	−0.829375	−	0.558693i	$-0.811303\pi$
$19$	−7.23032	−1.65875	−0.829375	+	0.558693i	$0.811303\pi$
$20$	0	0
$21$	6.59661	1.43950
$22$	0	0
$23$	4.43841	0.925472	0.462736	−	0.886496i	$-0.346868\pi$
$23$	4.43841	0.925472	0.462736	+	0.886496i	$0.346868\pi$
$24$	0	0
$25$	−3.67683	−0.735366
$26$	0	0
$27$	2.31859	0.446213
$28$	0	0
$29$	3.16969	0.588597	0.294298	−	0.955714i	$-0.404914\pi$
$29$	3.16969	0.588597	0.294298	+	0.955714i	$0.404914\pi$
$30$	0	0
$31$	−6.86491	−1.23297	−0.616487	−	0.787365i	$-0.711445\pi$
$31$	−6.86491	−1.23297	−0.616487	+	0.787365i	$0.711445\pi$
$32$	0	0
$33$	16.5285	2.87724
$34$	0	0
$35$	2.89213	0.488858
$36$	0	0
$37$	3.34453	0.549838	0.274919	−	0.961467i	$-0.411349\pi$
$37$	3.34453	0.549838	0.274919	+	0.961467i	$0.411349\pi$
$38$	0	0
$39$	1.83502	0.293839
$40$	0	0
$41$	1.17827	0.184016	0.0920078	−	0.995758i	$-0.470672\pi$
$41$	1.17827	0.184016	0.0920078	+	0.995758i	$0.470672\pi$
$42$	0	0
$43$	−8.14366	−1.24190	−0.620948	−	0.783851i	$-0.713252\pi$
$43$	−8.14366	−1.24190	−0.620948	+	0.783851i	$0.713252\pi$
$44$	0	0
$45$	4.46740	0.665961
$46$	0	0
$47$	−7.94801	−1.15934	−0.579668	−	0.814853i	$-0.696818\pi$
$47$	−7.94801	−1.15934	−0.579668	+	0.814853i	$0.696818\pi$
$48$	0	0
$49$	−0.678520	−0.0969314
$50$	0	0
$51$	12.0284	1.68431
$52$	0	0
$53$	6.14772	0.844455	0.422227	−	0.906490i	$-0.361248\pi$
$53$	6.14772	0.844455	0.422227	+	0.906490i	$0.361248\pi$
$54$	0	0
$55$	7.24652	0.977120
$56$	0	0
$57$	−18.9701	−2.51265
$58$	0	0
$59$	−3.29675	−0.429201	−0.214600	−	0.976702i	$-0.568845\pi$
$59$	−3.29675	−0.429201	−0.214600	+	0.976702i	$0.568845\pi$
$60$	0	0
$61$	1.51045	0.193394	0.0966968	−	0.995314i	$-0.469172\pi$
$61$	1.51045	0.193394	0.0966968	+	0.995314i	$0.469172\pi$
$62$	0	0
$63$	9.76465	1.23023
$64$	0	0
$65$	0.804522	0.0997887
$66$	0	0
$67$	−4.60689	−0.562821	−0.281411	−	0.959587i	$-0.590802\pi$
$67$	−4.60689	−0.562821	−0.281411	+	0.959587i	$0.590802\pi$
$68$	0	0
$69$	11.6450	1.40189
$70$	0	0
$71$	8.22808	0.976494	0.488247	−	0.872706i	$-0.337637\pi$
$71$	8.22808	0.976494	0.488247	+	0.872706i	$0.337637\pi$
$72$	0	0
$73$	−5.16971	−0.605069	−0.302535	−	0.953138i	$-0.597833\pi$
$73$	−5.16971	−0.605069	−0.302535	+	0.953138i	$0.597833\pi$
$74$	0	0
$75$	−9.64684	−1.11392
$76$	0	0
$77$	15.8391	1.80504
$78$	0	0
$79$	−8.65353	−0.973598	−0.486799	−	0.873514i	$-0.661836\pi$
$79$	−8.65353	−0.973598	−0.486799	+	0.873514i	$0.661836\pi$
$80$	0	0
$81$	−5.56790	−0.618656
$82$	0	0
$83$	−4.65024	−0.510430	−0.255215	−	0.966884i	$-0.582146\pi$
$83$	−4.65024	−0.510430	−0.255215	+	0.966884i	$0.582146\pi$
$84$	0	0
$85$	5.27356	0.571998
$86$	0	0
$87$	8.31626	0.891597
$88$	0	0
$89$	3.59697	0.381279	0.190639	−	0.981660i	$-0.438944\pi$
$89$	3.59697	0.381279	0.190639	+	0.981660i	$0.438944\pi$
$90$	0	0
$91$	1.75849	0.184340
$92$	0	0
$93$	−18.0113	−1.86769
$94$	0	0
$95$	−8.31697	−0.853304
$96$	0	0
$97$	−18.0684	−1.83456	−0.917282	−	0.398238i	$-0.869622\pi$
$97$	−18.0684	−1.83456	−0.917282	+	0.398238i	$0.869622\pi$
$98$	0	0
$99$	24.4663	2.45896
$100$	0	0
$101$	−6.97694	−0.694231	−0.347116	−	0.937822i	$-0.612839\pi$
$101$	−6.97694	−0.694231	−0.347116	+	0.937822i	$0.612839\pi$
$102$	0	0
$103$	−1.85305	−0.182587	−0.0912933	−	0.995824i	$-0.529100\pi$
$103$	−1.85305	−0.182587	−0.0912933	+	0.995824i	$0.529100\pi$
$104$	0	0
$105$	7.58802	0.740515
$106$	0	0
$107$	−6.70888	−0.648572	−0.324286	−	0.945959i	$-0.605124\pi$
$107$	−6.70888	−0.648572	−0.324286	+	0.945959i	$0.605124\pi$
$108$	0	0
$109$	−15.8205	−1.51533	−0.757664	−	0.652645i	$-0.773660\pi$
$109$	−15.8205	−1.51533	−0.757664	+	0.652645i	$0.773660\pi$
$110$	0	0
$111$	8.77500	0.832886
$112$	0	0
$113$	16.4826	1.55055	0.775275	−	0.631624i	$-0.217612\pi$
$113$	16.4826	1.55055	0.775275	+	0.631624i	$0.217612\pi$
$114$	0	0
$115$	5.10546	0.476087
$116$	0	0
$117$	2.71630	0.251122
$118$	0	0
$119$	11.5267	1.05665
$120$	0	0
$121$	28.6866	2.60787
$122$	0	0
$123$	3.09142	0.278744
$124$	0	0
$125$	−9.98088	−0.892717
$126$	0	0
$127$	−5.39892	−0.479077	−0.239538	−	0.970887i	$-0.576996\pi$
$127$	−5.39892	−0.479077	−0.239538	+	0.970887i	$0.576996\pi$
$128$	0	0
$129$	−21.3664	−1.88121
$130$	0	0
$131$	−16.4211	−1.43472	−0.717360	−	0.696703i	$-0.754650\pi$
$131$	−16.4211	−1.43472	−0.717360	+	0.696703i	$0.754650\pi$
$132$	0	0
$133$	−18.1789	−1.57631
$134$	0	0
$135$	2.66705	0.229543
$136$	0	0
$137$	−11.4638	−0.979416	−0.489708	−	0.871887i	$-0.662897\pi$
$137$	−11.4638	−0.979416	−0.489708	+	0.871887i	$0.662897\pi$
$138$	0	0
$139$	6.63997	0.563195	0.281597	−	0.959533i	$-0.409136\pi$
$139$	6.63997	0.563195	0.281597	+	0.959533i	$0.409136\pi$
$140$	0	0
$141$	−20.8531	−1.75614
$142$	0	0
$143$	4.40608	0.368455
$144$	0	0
$145$	3.64607	0.302789
$146$	0	0
$147$	−1.78022	−0.146830
$148$	0	0
$149$	22.1685	1.81611	0.908057	−	0.418846i	$-0.137565\pi$
$149$	22.1685	1.81611	0.908057	+	0.418846i	$0.137565\pi$
$150$	0	0
$151$	−4.81459	−0.391806	−0.195903	−	0.980623i	$-0.562764\pi$
$151$	−4.81459	−0.391806	−0.195903	+	0.980623i	$0.562764\pi$
$152$	0	0
$153$	17.8051	1.43946
$154$	0	0
$155$	−7.89664	−0.634273
$156$	0	0
$157$	9.72414	0.776071	0.388035	−	0.921644i	$-0.373154\pi$
$157$	9.72414	0.776071	0.388035	+	0.921644i	$0.373154\pi$
$158$	0	0
$159$	16.1297	1.27917
$160$	0	0
$161$	11.1593	0.879475
$162$	0	0
$163$	16.2697	1.27434	0.637172	−	0.770721i	$-0.280104\pi$
$163$	16.2697	1.27434	0.637172	+	0.770721i	$0.280104\pi$
$164$	0	0
$165$	19.0126	1.48013
$166$	0	0
$167$	16.4518	1.27308	0.636541	−	0.771243i	$-0.280365\pi$
$167$	16.4518	1.27308	0.636541	+	0.771243i	$0.280365\pi$
$168$	0	0
$169$	−12.5108	−0.962371
$170$	0	0
$171$	−28.0805	−2.14737
$172$	0	0
$173$	21.0702	1.60194	0.800968	−	0.598708i	$-0.204319\pi$
$173$	21.0702	1.60194	0.800968	+	0.598708i	$0.204319\pi$
$174$	0	0
$175$	−9.24449	−0.698818
$176$	0	0
$177$	−8.64964	−0.650147
$178$	0	0
$179$	−8.45297	−0.631804	−0.315902	−	0.948792i	$-0.602307\pi$
$179$	−8.45297	−0.631804	−0.315902	+	0.948792i	$0.602307\pi$
$180$	0	0
$181$	4.28715	0.318662	0.159331	−	0.987225i	$-0.449066\pi$
$181$	4.28715	0.318662	0.159331	+	0.987225i	$0.449066\pi$
$182$	0	0
$183$	3.96295	0.292950
$184$	0	0
$185$	3.84719	0.282851
$186$	0	0
$187$	28.8814	2.11202
$188$	0	0
$189$	5.82953	0.424036
$190$	0	0
$191$	10.3921	0.751947	0.375974	−	0.926630i	$-0.377308\pi$
$191$	10.3921	0.751947	0.375974	+	0.926630i	$0.377308\pi$
$192$	0	0
$193$	12.7332	0.916557	0.458278	−	0.888809i	$-0.348466\pi$
$193$	12.7332	0.916557	0.458278	+	0.888809i	$0.348466\pi$
$194$	0	0
$195$	2.11081	0.151158
$196$	0	0
$197$	−4.12368	−0.293800	−0.146900	−	0.989151i	$-0.546930\pi$
$197$	−4.12368	−0.293800	−0.146900	+	0.989151i	$0.546930\pi$
$198$	0	0
$199$	13.7296	0.973264	0.486632	−	0.873607i	$-0.338225\pi$
$199$	13.7296	0.973264	0.486632	+	0.873607i	$0.338225\pi$
$200$	0	0
$201$	−12.0870	−0.852553
$202$	0	0
$203$	7.96941	0.559343
$204$	0	0
$205$	1.35536	0.0946624
$206$	0	0
$207$	17.2375	1.19809
$208$	0	0
$209$	−45.5491	−3.15069
$210$	0	0
$211$	−4.58825	−0.315868	−0.157934	−	0.987450i	$-0.550483\pi$
$211$	−4.58825	−0.315868	−0.157934	+	0.987450i	$0.550483\pi$
$212$	0	0
$213$	21.5879	1.47918
$214$	0	0
$215$	−9.36758	−0.638864
$216$	0	0
$217$	−17.2601	−1.17169
$218$	0	0
$219$	−13.5637	−0.916549
$220$	0	0
$221$	3.20647	0.215690
$222$	0	0
$223$	0.0912325	0.00610938	0.00305469	−	0.999995i	$-0.499028\pi$
$223$	0.0912325	0.00610938	0.00305469	+	0.999995i	$0.499028\pi$
$224$	0	0
$225$	−14.2798	−0.951984
$226$	0	0
$227$	−12.8140	−0.850497	−0.425249	−	0.905077i	$-0.639813\pi$
$227$	−12.8140	−0.850497	−0.425249	+	0.905077i	$0.639813\pi$
$228$	0	0
$229$	−16.5635	−1.09455	−0.547275	−	0.836953i	$-0.684335\pi$
$229$	−16.5635	−1.09455	−0.547275	+	0.836953i	$0.684335\pi$
$230$	0	0
$231$	41.5568	2.73424
$232$	0	0
$233$	17.8233	1.16764	0.583820	−	0.811883i	$-0.301557\pi$
$233$	17.8233	1.16764	0.583820	+	0.811883i	$0.301557\pi$
$234$	0	0
$235$	−9.14252	−0.596392
$236$	0	0
$237$	−22.7041	−1.47479
$238$	0	0
$239$	12.8815	0.833234	0.416617	−	0.909082i	$-0.363216\pi$
$239$	12.8815	0.833234	0.416617	+	0.909082i	$0.363216\pi$
$240$	0	0
$241$	0.761335	0.0490419	0.0245210	−	0.999699i	$-0.492194\pi$
$241$	0.761335	0.0490419	0.0245210	+	0.999699i	$0.492194\pi$
$242$	0	0
$243$	−21.5642	−1.38334
$244$	0	0
$245$	−0.780495	−0.0498640
$246$	0	0
$247$	−5.05694	−0.321765
$248$	0	0
$249$	−12.2007	−0.773191
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	27.9608	1.75788
$254$	0	0
$255$	13.8362	0.866454
$256$	0	0
$257$	6.12039	0.381779	0.190890	−	0.981611i	$-0.438863\pi$
$257$	6.12039	0.381779	0.190890	+	0.981611i	$0.438863\pi$
$258$	0	0
$259$	8.40901	0.522511
$260$	0	0
$261$	12.3102	0.761981
$262$	0	0
$263$	−3.47378	−0.214202	−0.107101	−	0.994248i	$-0.534157\pi$
$263$	−3.47378	−0.214202	−0.107101	+	0.994248i	$0.534157\pi$
$264$	0	0
$265$	7.07167	0.434409
$266$	0	0
$267$	9.43732	0.577555
$268$	0	0
$269$	10.7931	0.658066	0.329033	−	0.944318i	$-0.393277\pi$
$269$	10.7931	0.658066	0.329033	+	0.944318i	$0.393277\pi$
$270$	0	0
$271$	−13.8686	−0.842455	−0.421227	−	0.906955i	$-0.638401\pi$
$271$	−13.8686	−0.842455	−0.421227	+	0.906955i	$0.638401\pi$
$272$	0	0
$273$	4.61372	0.279235
$274$	0	0
$275$	−23.1630	−1.39678
$276$	0	0
$277$	20.0135	1.20250	0.601248	−	0.799062i	$-0.294670\pi$
$277$	20.0135	1.20250	0.601248	+	0.799062i	$0.294670\pi$
$278$	0	0
$279$	−26.6613	−1.59617
$280$	0	0
$281$	13.4897	0.804726	0.402363	−	0.915480i	$-0.368189\pi$
$281$	13.4897	0.804726	0.402363	+	0.915480i	$0.368189\pi$
$282$	0	0
$283$	21.9359	1.30396	0.651978	−	0.758238i	$-0.273939\pi$
$283$	21.9359	1.30396	0.651978	+	0.758238i	$0.273939\pi$
$284$	0	0
$285$	−21.8211	−1.29257
$286$	0	0
$287$	2.96248	0.174870
$288$	0	0
$289$	4.01807	0.236357
$290$	0	0
$291$	−47.4057	−2.77897
$292$	0	0
$293$	2.14505	0.125315	0.0626576	−	0.998035i	$-0.480042\pi$
$293$	2.14505	0.125315	0.0626576	+	0.998035i	$0.480042\pi$
$294$	0	0
$295$	−3.79223	−0.220792
$296$	0	0
$297$	14.6065	0.847554
$298$	0	0
$299$	3.10425	0.179524
$300$	0	0
$301$	−20.4752	−1.18017
$302$	0	0
$303$	−18.3053	−1.05161
$304$	0	0
$305$	1.73746	0.0994867
$306$	0	0
$307$	1.49142	0.0851200	0.0425600	−	0.999094i	$-0.486449\pi$
$307$	1.49142	0.0851200	0.0425600	+	0.999094i	$0.486449\pi$
$308$	0	0
$309$	−4.86182	−0.276579
$310$	0	0
$311$	−6.70155	−0.380010	−0.190005	−	0.981783i	$-0.560850\pi$
$311$	−6.70155	−0.380010	−0.190005	+	0.981783i	$0.560850\pi$
$312$	0	0
$313$	20.9053	1.18164	0.590818	−	0.806805i	$-0.298805\pi$
$313$	20.9053	1.18164	0.590818	+	0.806805i	$0.298805\pi$
$314$	0	0
$315$	11.2322	0.632862
$316$	0	0
$317$	−12.8281	−0.720499	−0.360249	−	0.932856i	$-0.617308\pi$
$317$	−12.8281	−0.720499	−0.360249	+	0.932856i	$0.617308\pi$
$318$	0	0
$319$	19.9682	1.11800
$320$	0	0
$321$	−17.6020	−0.982446
$322$	0	0
$323$	−33.1478	−1.84439
$324$	0	0
$325$	−2.57160	−0.142647
$326$	0	0
$327$	−41.5080	−2.29540
$328$	0	0
$329$	−19.9833	−1.10172
$330$	0	0
$331$	−13.3489	−0.733723	−0.366862	−	0.930275i	$-0.619568\pi$
$331$	−13.3489	−0.733723	−0.366862	+	0.930275i	$0.619568\pi$
$332$	0	0
$333$	12.9892	0.711805
$334$	0	0
$335$	−5.29927	−0.289530
$336$	0	0
$337$	12.8628	0.700681	0.350340	−	0.936622i	$-0.386066\pi$
$337$	12.8628	0.700681	0.350340	+	0.936622i	$0.386066\pi$
$338$	0	0
$339$	43.2450	2.34875
$340$	0	0
$341$	−43.2470	−2.34196
$342$	0	0
$343$	−19.3058	−1.04241
$344$	0	0
$345$	13.3951	0.721168
$346$	0	0
$347$	2.39781	0.128721	0.0643607	−	0.997927i	$-0.479499\pi$
$347$	2.39781	0.128721	0.0643607	+	0.997927i	$0.479499\pi$
$348$	0	0
$349$	35.2451	1.88663	0.943313	−	0.331903i	$-0.107691\pi$
$349$	35.2451	1.88663	0.943313	+	0.331903i	$0.107691\pi$
$350$	0	0
$351$	1.62164	0.0865566
$352$	0	0
$353$	−11.5147	−0.612868	−0.306434	−	0.951892i	$-0.599136\pi$
$353$	−11.5147	−0.612868	−0.306434	+	0.951892i	$0.599136\pi$
$354$	0	0
$355$	9.46469	0.502333
$356$	0	0
$357$	30.2425	1.60060
$358$	0	0
$359$	16.1553	0.852642	0.426321	−	0.904572i	$-0.359809\pi$
$359$	16.1553	0.852642	0.426321	+	0.904572i	$0.359809\pi$
$360$	0	0
$361$	33.2776	1.75145
$362$	0	0
$363$	75.2645	3.95036
$364$	0	0
$365$	−5.94668	−0.311263
$366$	0	0
$367$	1.07079	0.0558946	0.0279473	−	0.999609i	$-0.491103\pi$
$367$	1.07079	0.0558946	0.0279473	+	0.999609i	$0.491103\pi$
$368$	0	0
$369$	4.57608	0.238221
$370$	0	0
$371$	15.4569	0.802485
$372$	0	0
$373$	−12.7804	−0.661742	−0.330871	−	0.943676i	$-0.607343\pi$
$373$	−12.7804	−0.661742	−0.330871	+	0.943676i	$0.607343\pi$
$374$	0	0
$375$	−26.1867	−1.35227
$376$	0	0
$377$	2.21690	0.114176
$378$	0	0
$379$	−17.5543	−0.901703	−0.450851	−	0.892599i	$-0.648880\pi$
$379$	−17.5543	−0.901703	−0.450851	+	0.892599i	$0.648880\pi$
$380$	0	0
$381$	−14.1651	−0.725698
$382$	0	0
$383$	10.9209	0.558030	0.279015	−	0.960287i	$-0.409992\pi$
$383$	10.9209	0.558030	0.279015	+	0.960287i	$0.409992\pi$
$384$	0	0
$385$	18.2196	0.928557
$386$	0	0
$387$	−31.6277	−1.60772
$388$	0	0
$389$	−2.85338	−0.144672	−0.0723361	−	0.997380i	$-0.523045\pi$
$389$	−2.85338	−0.144672	−0.0723361	+	0.997380i	$0.523045\pi$
$390$	0	0
$391$	20.3481	1.02905
$392$	0	0
$393$	−43.0838	−2.17329
$394$	0	0
$395$	−9.95407	−0.500844
$396$	0	0
$397$	−18.3951	−0.923225	−0.461613	−	0.887082i	$-0.652729\pi$
$397$	−18.3951	−0.923225	−0.461613	+	0.887082i	$0.652729\pi$
$398$	0	0
$399$	−47.6956	−2.38777
$400$	0	0
$401$	−11.0050	−0.549564	−0.274782	−	0.961506i	$-0.588606\pi$
$401$	−11.0050	−0.549564	−0.274782	+	0.961506i	$0.588606\pi$
$402$	0	0
$403$	−4.80137	−0.239173
$404$	0	0
$405$	−6.40471	−0.318252
$406$	0	0
$407$	21.0697	1.04438
$408$	0	0
$409$	−13.5283	−0.668930	−0.334465	−	0.942408i	$-0.608556\pi$
$409$	−13.5283	−0.668930	−0.334465	+	0.942408i	$0.608556\pi$
$410$	0	0
$411$	−30.0773	−1.48360
$412$	0	0
$413$	−8.28888	−0.407869
$414$	0	0
$415$	−5.34912	−0.262578
$416$	0	0
$417$	17.4212	0.853119
$418$	0	0
$419$	−14.1141	−0.689518	−0.344759	−	0.938691i	$-0.612039\pi$
$419$	−14.1141	−0.689518	−0.344759	+	0.938691i	$0.612039\pi$
$420$	0	0
$421$	37.9882	1.85143	0.925716	−	0.378219i	$-0.123463\pi$
$421$	37.9882	1.85143	0.925716	+	0.378219i	$0.123463\pi$
$422$	0	0
$423$	−30.8678	−1.50084
$424$	0	0
$425$	−16.8566	−0.817665
$426$	0	0
$427$	3.79766	0.183782
$428$	0	0
$429$	11.5601	0.558129
$430$	0	0
$431$	1.89014	0.0910450	0.0455225	−	0.998963i	$-0.485505\pi$
$431$	1.89014	0.0910450	0.0455225	+	0.998963i	$0.485505\pi$
$432$	0	0
$433$	8.69728	0.417965	0.208982	−	0.977919i	$-0.432985\pi$
$433$	8.69728	0.417965	0.208982	+	0.977919i	$0.432985\pi$
$434$	0	0
$435$	9.56612	0.458661
$436$	0	0
$437$	−32.0911	−1.53513
$438$	0	0
$439$	−11.2331	−0.536129	−0.268064	−	0.963401i	$-0.586384\pi$
$439$	−11.2331	−0.536129	−0.268064	+	0.963401i	$0.586384\pi$
$440$	0	0
$441$	−2.63518	−0.125485
$442$	0	0
$443$	18.8903	0.897505	0.448753	−	0.893656i	$-0.351868\pi$
$443$	18.8903	0.897505	0.448753	+	0.893656i	$0.351868\pi$
$444$	0	0
$445$	4.13757	0.196140
$446$	0	0
$447$	58.1631	2.75102
$448$	0	0
$449$	−3.47576	−0.164031	−0.0820157	−	0.996631i	$-0.526136\pi$
$449$	−3.47576	−0.164031	−0.0820157	+	0.996631i	$0.526136\pi$
$450$	0	0
$451$	7.42281	0.349526
$452$	0	0
$453$	−12.6320	−0.593501
$454$	0	0
$455$	2.02277	0.0948291
$456$	0	0
$457$	−14.6495	−0.685273	−0.342637	−	0.939468i	$-0.611320\pi$
$457$	−14.6495	−0.685273	−0.342637	+	0.939468i	$0.611320\pi$
$458$	0	0
$459$	10.6297	0.496151
$460$	0	0
$461$	−31.0557	−1.44641	−0.723203	−	0.690635i	$-0.757331\pi$
$461$	−31.0557	−1.44641	−0.723203	+	0.690635i	$0.757331\pi$
$462$	0	0
$463$	−12.3099	−0.572089	−0.286045	−	0.958216i	$-0.592341\pi$
$463$	−12.3099	−0.572089	−0.286045	+	0.958216i	$0.592341\pi$
$464$	0	0
$465$	−20.7183	−0.960787
$466$	0	0
$467$	9.59342	0.443930	0.221965	−	0.975055i	$-0.428753\pi$
$467$	9.59342	0.443930	0.221965	+	0.975055i	$0.428753\pi$
$468$	0	0
$469$	−11.5829	−0.534849
$470$	0	0
$471$	25.5131	1.17558
$472$	0	0
$473$	−51.3028	−2.35891
$474$	0	0
$475$	26.5847	1.21979
$476$	0	0
$477$	23.8760	1.09321
$478$	0	0
$479$	14.5463	0.664639	0.332320	−	0.943167i	$-0.392169\pi$
$479$	14.5463	0.664639	0.332320	+	0.943167i	$0.392169\pi$
$480$	0	0
$481$	2.33919	0.106658
$482$	0	0
$483$	29.2784	1.33222
$484$	0	0
$485$	−20.7839	−0.943747
$486$	0	0
$487$	18.1844	0.824015	0.412007	−	0.911180i	$-0.364828\pi$
$487$	18.1844	0.824015	0.412007	+	0.911180i	$0.364828\pi$
$488$	0	0
$489$	42.6867	1.93036
$490$	0	0
$491$	−27.1596	−1.22570	−0.612848	−	0.790201i	$-0.709976\pi$
$491$	−27.1596	−1.22570	−0.612848	+	0.790201i	$0.709976\pi$
$492$	0	0
$493$	14.5316	0.654470
$494$	0	0
$495$	28.1434	1.26495
$496$	0	0
$497$	20.6875	0.927961
$498$	0	0
$499$	−18.4154	−0.824386	−0.412193	−	0.911096i	$-0.635237\pi$
$499$	−18.4154	−0.824386	−0.412193	+	0.911096i	$0.635237\pi$
$500$	0	0
$501$	43.1644	1.92844
$502$	0	0
$503$	−40.1500	−1.79020	−0.895100	−	0.445865i	$-0.852896\pi$
$503$	−40.1500	−1.79020	−0.895100	+	0.445865i	$0.852896\pi$
$504$	0	0
$505$	−8.02551	−0.357130
$506$	0	0
$507$	−32.8245	−1.45779
$508$	0	0
$509$	11.0878	0.491456	0.245728	−	0.969339i	$-0.420973\pi$
$509$	11.0878	0.491456	0.245728	+	0.969339i	$0.420973\pi$
$510$	0	0
$511$	−12.9980	−0.574997
$512$	0	0
$513$	−16.7641	−0.740155
$514$	0	0
$515$	−2.13155	−0.0939273
$516$	0	0
$517$	−50.0703	−2.20209
$518$	0	0
$519$	55.2815	2.42659
$520$	0	0
$521$	−11.2398	−0.492423	−0.246212	−	0.969216i	$-0.579186\pi$
$521$	−11.2398	−0.492423	−0.246212	+	0.969216i	$0.579186\pi$
$522$	0	0
$523$	−12.8526	−0.562004	−0.281002	−	0.959707i	$-0.590667\pi$
$523$	−12.8526	−0.562004	−0.281002	+	0.959707i	$0.590667\pi$
$524$	0	0
$525$	−24.2546	−1.05856
$526$	0	0
$527$	−31.4725	−1.37096
$528$	0	0
$529$	−3.30055	−0.143502
$530$	0	0
$531$	−12.8037	−0.555631
$532$	0	0
$533$	0.824094	0.0356955
$534$	0	0
$535$	−7.71716	−0.333642
$536$	0	0
$537$	−22.1779	−0.957047
$538$	0	0
$539$	−4.27449	−0.184115
$540$	0	0
$541$	40.7501	1.75198	0.875991	−	0.482328i	$-0.160209\pi$
$541$	40.7501	1.75198	0.875991	+	0.482328i	$0.160209\pi$
$542$	0	0
$543$	11.2481	0.482704
$544$	0	0
$545$	−18.1982	−0.779524
$546$	0	0
$547$	30.4796	1.30321	0.651606	−	0.758558i	$-0.274096\pi$
$547$	30.4796	1.30321	0.651606	+	0.758558i	$0.274096\pi$
$548$	0	0
$549$	5.86617	0.250362
$550$	0	0
$551$	−22.9179	−0.976335
$552$	0	0
$553$	−21.7572	−0.925209
$554$	0	0
$555$	10.0938	0.428458
$556$	0	0
$557$	17.2478	0.730812	0.365406	−	0.930848i	$-0.380930\pi$
$557$	17.2478	0.730812	0.365406	+	0.930848i	$0.380930\pi$
$558$	0	0
$559$	−5.69574	−0.240904
$560$	0	0
$561$	75.7756	3.19925
$562$	0	0
$563$	7.43193	0.313218	0.156609	−	0.987661i	$-0.449944\pi$
$563$	7.43193	0.313218	0.156609	+	0.987661i	$0.449944\pi$
$564$	0	0
$565$	18.9597	0.797642
$566$	0	0
$567$	−13.9991	−0.587908
$568$	0	0
$569$	25.6087	1.07357	0.536786	−	0.843719i	$-0.319638\pi$
$569$	25.6087	1.07357	0.536786	+	0.843719i	$0.319638\pi$
$570$	0	0
$571$	−5.04995	−0.211334	−0.105667	−	0.994402i	$-0.533698\pi$
$571$	−5.04995	−0.211334	−0.105667	+	0.994402i	$0.533698\pi$
$572$	0	0
$573$	27.2656	1.13904
$574$	0	0
$575$	−16.3193	−0.680561
$576$	0	0
$577$	−20.2333	−0.842322	−0.421161	−	0.906986i	$-0.638377\pi$
$577$	−20.2333	−0.842322	−0.421161	+	0.906986i	$0.638377\pi$
$578$	0	0
$579$	33.4079	1.38839
$580$	0	0
$581$	−11.6919	−0.485061
$582$	0	0
$583$	38.7290	1.60399
$584$	0	0
$585$	3.12453	0.129184
$586$	0	0
$587$	18.6139	0.768279	0.384140	−	0.923275i	$-0.374498\pi$
$587$	18.6139	0.768279	0.384140	+	0.923275i	$0.374498\pi$
$588$	0	0
$589$	49.6355	2.04519
$590$	0	0
$591$	−10.8192	−0.445043
$592$	0	0
$593$	2.15937	0.0886748	0.0443374	−	0.999017i	$-0.485882\pi$
$593$	2.15937	0.0886748	0.0443374	+	0.999017i	$0.485882\pi$
$594$	0	0
$595$	13.2591	0.543569
$596$	0	0
$597$	36.0221	1.47429
$598$	0	0
$599$	16.3514	0.668101	0.334050	−	0.942555i	$-0.391584\pi$
$599$	16.3514	0.668101	0.334050	+	0.942555i	$0.391584\pi$
$600$	0	0
$601$	−39.7825	−1.62276	−0.811380	−	0.584519i	$-0.801283\pi$
$601$	−39.7825	−1.62276	−0.811380	+	0.584519i	$0.801283\pi$
$602$	0	0
$603$	−17.8919	−0.728613
$604$	0	0
$605$	32.9979	1.34156
$606$	0	0
$607$	28.5774	1.15992	0.579959	−	0.814645i	$-0.303068\pi$
$607$	28.5774	1.15992	0.579959	+	0.814645i	$0.303068\pi$
$608$	0	0
$609$	20.9092	0.847284
$610$	0	0
$611$	−5.55890	−0.224889
$612$	0	0
$613$	−17.3087	−0.699093	−0.349547	−	0.936919i	$-0.613664\pi$
$613$	−17.3087	−0.699093	−0.349547	+	0.936919i	$0.613664\pi$
$614$	0	0
$615$	3.55603	0.143393
$616$	0	0
$617$	11.9753	0.482107	0.241054	−	0.970512i	$-0.422507\pi$
$617$	11.9753	0.482107	0.241054	+	0.970512i	$0.422507\pi$
$618$	0	0
$619$	−32.9447	−1.32416	−0.662080	−	0.749433i	$-0.730326\pi$
$619$	−32.9447	−1.32416	−0.662080	+	0.749433i	$0.730326\pi$
$620$	0	0
$621$	10.2908	0.412957
$622$	0	0
$623$	9.04371	0.362329
$624$	0	0
$625$	6.90324	0.276130
$626$	0	0
$627$	−119.506	−4.77262
$628$	0	0
$629$	15.3332	0.611374
$630$	0	0
$631$	−2.95086	−0.117472	−0.0587360	−	0.998274i	$-0.518707\pi$
$631$	−2.95086	−0.117472	−0.0587360	+	0.998274i	$0.518707\pi$
$632$	0	0
$633$	−12.0381	−0.478472
$634$	0	0
$635$	−6.21033	−0.246449
$636$	0	0
$637$	−0.474562	−0.0188028
$638$	0	0
$639$	31.9555	1.26414
$640$	0	0
$641$	18.1419	0.716560	0.358280	−	0.933614i	$-0.383363\pi$
$641$	18.1419	0.716560	0.358280	+	0.933614i	$0.383363\pi$
$642$	0	0
$643$	40.9334	1.61425	0.807127	−	0.590378i	$-0.201021\pi$
$643$	40.9334	1.61425	0.807127	+	0.590378i	$0.201021\pi$
$644$	0	0
$645$	−24.5776	−0.967741
$646$	0	0
$647$	−12.9579	−0.509428	−0.254714	−	0.967016i	$-0.581981\pi$
$647$	−12.9579	−0.509428	−0.254714	+	0.967016i	$0.581981\pi$
$648$	0	0
$649$	−20.7686	−0.815241
$650$	0	0
$651$	−45.2851	−1.77486
$652$	0	0
$653$	−0.0158713	−0.000621093 0	−0.000310546	−	1.00000i	$-0.500099\pi$
$653$	−0.0158713	−0.000621093 0	−0.000310546		1.00000i	$0.500099\pi$
$654$	0	0
$655$	−18.8891	−0.738057
$656$	0	0
$657$	−20.0777	−0.783306
$658$	0	0
$659$	29.2333	1.13877	0.569385	−	0.822071i	$-0.307182\pi$
$659$	29.2333	1.13877	0.569385	+	0.822071i	$0.307182\pi$
$660$	0	0
$661$	32.7524	1.27392	0.636961	−	0.770896i	$-0.280191\pi$
$661$	32.7524	1.27392	0.636961	+	0.770896i	$0.280191\pi$
$662$	0	0
$663$	8.41275	0.326724
$664$	0	0
$665$	−20.9110	−0.810894
$666$	0	0
$667$	14.0684	0.544730
$668$	0	0
$669$	0.239365	0.00925439
$670$	0	0
$671$	9.51544	0.367340
$672$	0	0
$673$	−28.0615	−1.08169	−0.540846	−	0.841122i	$-0.681896\pi$
$673$	−28.0615	−1.08169	−0.540846	+	0.841122i	$0.681896\pi$
$674$	0	0
$675$	−8.52506	−0.328130
$676$	0	0
$677$	−2.05737	−0.0790713	−0.0395357	−	0.999218i	$-0.512588\pi$
$677$	−2.05737	−0.0790713	−0.0395357	+	0.999218i	$0.512588\pi$
$678$	0	0
$679$	−45.4285	−1.74339
$680$	0	0
$681$	−33.6200	−1.28832
$682$	0	0
$683$	39.9521	1.52873	0.764363	−	0.644786i	$-0.223054\pi$
$683$	39.9521	1.52873	0.764363	+	0.644786i	$0.223054\pi$
$684$	0	0
$685$	−13.1867	−0.503837
$686$	0	0
$687$	−43.4575	−1.65801
$688$	0	0
$689$	4.29976	0.163808
$690$	0	0
$691$	−17.9711	−0.683654	−0.341827	−	0.939763i	$-0.611046\pi$
$691$	−17.9711	−0.683654	−0.341827	+	0.939763i	$0.611046\pi$
$692$	0	0
$693$	61.5146	2.33675
$694$	0	0
$695$	7.63790	0.289722
$696$	0	0
$697$	5.40186	0.204610
$698$	0	0
$699$	46.7626	1.76872
$700$	0	0
$701$	46.5074	1.75656	0.878281	−	0.478146i	$-0.158691\pi$
$701$	46.5074	1.75656	0.878281	+	0.478146i	$0.158691\pi$
$702$	0	0
$703$	−24.1821	−0.912044
$704$	0	0
$705$	−23.9871	−0.903406
$706$	0	0
$707$	−17.5418	−0.659727
$708$	0	0
$709$	−49.0363	−1.84160	−0.920798	−	0.390041i	$-0.872461\pi$
$709$	−49.0363	−1.84160	−0.920798	+	0.390041i	$0.872461\pi$
$710$	0	0
$711$	−33.6078	−1.26039
$712$	0	0
$713$	−30.4692	−1.14108
$714$	0	0
$715$	5.06827	0.189543
$716$	0	0
$717$	33.7969	1.26217
$718$	0	0
$719$	−26.5201	−0.989032	−0.494516	−	0.869169i	$-0.664655\pi$
$719$	−26.5201	−0.989032	−0.494516	+	0.869169i	$0.664655\pi$
$720$	0	0
$721$	−4.65905	−0.173512
$722$	0	0
$723$	1.99750	0.0742879
$724$	0	0
$725$	−11.6544	−0.432834
$726$	0	0
$727$	−32.6887	−1.21236	−0.606179	−	0.795329i	$-0.707298\pi$
$727$	−32.6887	−1.21236	−0.606179	+	0.795329i	$0.707298\pi$
$728$	0	0
$729$	−39.8739	−1.47681
$730$	0	0
$731$	−37.3350	−1.38088
$732$	0	0
$733$	12.4066	0.458248	0.229124	−	0.973397i	$-0.426414\pi$
$733$	12.4066	0.458248	0.229124	+	0.973397i	$0.426414\pi$
$734$	0	0
$735$	−2.04777	−0.0755332
$736$	0	0
$737$	−29.0222	−1.06904
$738$	0	0
$739$	21.1186	0.776862	0.388431	−	0.921478i	$-0.373017\pi$
$739$	21.1186	0.776862	0.388431	+	0.921478i	$0.373017\pi$
$740$	0	0
$741$	−13.2678	−0.487405
$742$	0	0
$743$	40.8627	1.49911	0.749554	−	0.661944i	$-0.230268\pi$
$743$	40.8627	1.49911	0.749554	+	0.661944i	$0.230268\pi$
$744$	0	0
$745$	25.5002	0.934256
$746$	0	0
$747$	−18.0602	−0.660788
$748$	0	0
$749$	−16.8678	−0.616337
$750$	0	0
$751$	−14.3790	−0.524699	−0.262349	−	0.964973i	$-0.584497\pi$
$751$	−14.3790	−0.524699	−0.262349	+	0.964973i	$0.584497\pi$
$752$	0	0
$753$	−2.62368	−0.0956123
$754$	0	0
$755$	−5.53818	−0.201555
$756$	0	0
$757$	−18.6835	−0.679064	−0.339532	−	0.940595i	$-0.610269\pi$
$757$	−18.6835	−0.679064	−0.339532	+	0.940595i	$0.610269\pi$
$758$	0	0
$759$	73.3602	2.66281
$760$	0	0
$761$	−21.8685	−0.792732	−0.396366	−	0.918092i	$-0.629729\pi$
$761$	−21.8685	−0.792732	−0.396366	+	0.918092i	$0.629729\pi$
$762$	0	0
$763$	−39.7768	−1.44002
$764$	0	0
$765$	20.4810	0.740493
$766$	0	0
$767$	−2.30577	−0.0832567
$768$	0	0
$769$	7.85128	0.283124	0.141562	−	0.989929i	$-0.454788\pi$
$769$	7.85128	0.283124	0.141562	+	0.989929i	$0.454788\pi$
$770$	0	0
$771$	16.0580	0.578313
$772$	0	0
$773$	41.0681	1.47712	0.738558	−	0.674190i	$-0.235507\pi$
$773$	41.0681	1.47712	0.738558	+	0.674190i	$0.235507\pi$
$774$	0	0
$775$	25.2411	0.906687
$776$	0	0
$777$	22.0626	0.791491
$778$	0	0
$779$	−8.51931	−0.305236
$780$	0	0
$781$	51.8347	1.85479
$782$	0	0
$783$	7.34921	0.262639
$784$	0	0
$785$	11.1856	0.399231
$786$	0	0
$787$	−42.5345	−1.51619	−0.758096	−	0.652143i	$-0.773870\pi$
$787$	−42.5345	−1.51619	−0.758096	+	0.652143i	$0.773870\pi$
$788$	0	0
$789$	−9.11410	−0.324470
$790$	0	0
$791$	41.4414	1.47349
$792$	0	0
$793$	1.05642	0.0375146
$794$	0	0
$795$	18.5538	0.658036
$796$	0	0
$797$	33.5901	1.18982	0.594912	−	0.803791i	$-0.297187\pi$
$797$	33.5901	1.18982	0.594912	+	0.803791i	$0.297187\pi$
$798$	0	0
$799$	−36.4380	−1.28908
$800$	0	0
$801$	13.9696	0.493592
$802$	0	0
$803$	−32.5678	−1.14929
$804$	0	0
$805$	12.8364	0.452425
$806$	0	0
$807$	28.3176	0.996828
$808$	0	0
$809$	−23.4068	−0.822940	−0.411470	−	0.911423i	$-0.634984\pi$
$809$	−23.4068	−0.822940	−0.411470	+	0.911423i	$0.634984\pi$
$810$	0	0
$811$	−14.1515	−0.496928	−0.248464	−	0.968641i	$-0.579926\pi$
$811$	−14.1515	−0.496928	−0.248464	+	0.968641i	$0.579926\pi$
$812$	0	0
$813$	−36.3867	−1.27614
$814$	0	0
$815$	18.7149	0.655556
$816$	0	0
$817$	58.8813	2.06000
$818$	0	0
$819$	6.82947	0.238641
$820$	0	0
$821$	−7.42484	−0.259129	−0.129564	−	0.991571i	$-0.541358\pi$
$821$	−7.42484	−0.259129	−0.129564	+	0.991571i	$0.541358\pi$
$822$	0	0
$823$	1.63364	0.0569452	0.0284726	−	0.999595i	$-0.490936\pi$
$823$	1.63364	0.0569452	0.0284726	+	0.999595i	$0.490936\pi$
$824$	0	0
$825$	−60.7725	−2.11583
$826$	0	0
$827$	−46.8226	−1.62818	−0.814090	−	0.580739i	$-0.802764\pi$
$827$	−46.8226	−1.62818	−0.814090	+	0.580739i	$0.802764\pi$
$828$	0	0
$829$	10.0372	0.348605	0.174303	−	0.984692i	$-0.444233\pi$
$829$	10.0372	0.348605	0.174303	+	0.984692i	$0.444233\pi$
$830$	0	0
$831$	52.5092	1.82152
$832$	0	0
$833$	−3.11071	−0.107780
$834$	0	0
$835$	18.9244	0.654906
$836$	0	0
$837$	−15.9169	−0.550168
$838$	0	0
$839$	−14.9971	−0.517756	−0.258878	−	0.965910i	$-0.583353\pi$
$839$	−14.9971	−0.517756	−0.258878	+	0.965910i	$0.583353\pi$
$840$	0	0
$841$	−18.9531	−0.653554
$842$	0	0
$843$	35.3926	1.21899
$844$	0	0
$845$	−14.3911	−0.495069
$846$	0	0
$847$	72.1254	2.47826
$848$	0	0
$849$	57.5530	1.97521
$850$	0	0
$851$	14.8444	0.508860
$852$	0	0
$853$	43.8473	1.50130	0.750652	−	0.660698i	$-0.229739\pi$
$853$	43.8473	1.50130	0.750652	+	0.660698i	$0.229739\pi$
$854$	0	0
$855$	−32.3008	−1.10466
$856$	0	0
$857$	−12.7227	−0.434598	−0.217299	−	0.976105i	$-0.569725\pi$
$857$	−12.7227	−0.434598	−0.217299	+	0.976105i	$0.569725\pi$
$858$	0	0
$859$	−10.7352	−0.366281	−0.183140	−	0.983087i	$-0.558626\pi$
$859$	−10.7352	−0.366281	−0.183140	+	0.983087i	$0.558626\pi$
$860$	0	0
$861$	7.77262	0.264890
$862$	0	0
$863$	−17.1108	−0.582458	−0.291229	−	0.956653i	$-0.594064\pi$
$863$	−17.1108	−0.582458	−0.291229	+	0.956653i	$0.594064\pi$
$864$	0	0
$865$	24.2368	0.824077
$866$	0	0
$867$	10.5421	0.358030
$868$	0	0
$869$	−54.5149	−1.84929
$870$	0	0
$871$	−3.22209	−0.109176
$872$	0	0
$873$	−70.1724	−2.37498
$874$	0	0
$875$	−25.0945	−0.848348
$876$	0	0
$877$	−26.8816	−0.907726	−0.453863	−	0.891071i	$-0.649954\pi$
$877$	−26.8816	−0.907726	−0.453863	+	0.891071i	$0.649954\pi$
$878$	0	0
$879$	5.62793	0.189825
$880$	0	0
$881$	48.8555	1.64598	0.822992	−	0.568053i	$-0.192303\pi$
$881$	48.8555	1.64598	0.822992	+	0.568053i	$0.192303\pi$
$882$	0	0
$883$	−26.8833	−0.904695	−0.452347	−	0.891842i	$-0.649413\pi$
$883$	−26.8833	−0.904695	−0.452347	+	0.891842i	$0.649413\pi$
$884$	0	0
$885$	−9.94960	−0.334452
$886$	0	0
$887$	2.58557	0.0868150	0.0434075	−	0.999057i	$-0.486179\pi$
$887$	2.58557	0.0868150	0.0434075	+	0.999057i	$0.486179\pi$
$888$	0	0
$889$	−13.5743	−0.455266
$890$	0	0
$891$	−35.0763	−1.17510
$892$	0	0
$893$	57.4667	1.92305
$894$	0	0
$895$	−9.72337	−0.325016
$896$	0	0
$897$	8.14458	0.271940
$898$	0	0
$899$	−21.7596	−0.725724
$900$	0	0
$901$	28.1845	0.938963
$902$	0	0
$903$	−53.7206	−1.78771
$904$	0	0
$905$	4.93147	0.163928
$906$	0	0
$907$	42.0643	1.39672	0.698360	−	0.715746i	$-0.253913\pi$
$907$	42.0643	1.39672	0.698360	+	0.715746i	$0.253913\pi$
$908$	0	0
$909$	−27.0964	−0.898732
$910$	0	0
$911$	59.0365	1.95597	0.977983	−	0.208685i	$-0.0669184\pi$
$911$	59.0365	1.95597	0.977983	+	0.208685i	$0.0669184\pi$
$912$	0	0
$913$	−29.2952	−0.969530
$914$	0	0
$915$	4.55855	0.150701
$916$	0	0
$917$	−41.2869	−1.36341
$918$	0	0
$919$	−5.57673	−0.183959	−0.0919797	−	0.995761i	$-0.529319\pi$
$919$	−5.57673	−0.183959	−0.0919797	+	0.995761i	$0.529319\pi$
$920$	0	0
$921$	3.91302	0.128938
$922$	0	0
$923$	5.75478	0.189421
$924$	0	0
$925$	−12.2973	−0.404332
$926$	0	0
$927$	−7.19673	−0.236372
$928$	0	0
$929$	−39.9616	−1.31110	−0.655550	−	0.755152i	$-0.727563\pi$
$929$	−39.9616	−1.31110	−0.655550	+	0.755152i	$0.727563\pi$
$930$	0	0
$931$	4.90592	0.160785
$932$	0	0
$933$	−17.5827	−0.575633
$934$	0	0
$935$	33.2220	1.08648
$936$	0	0
$937$	−26.0767	−0.851888	−0.425944	−	0.904749i	$-0.640058\pi$
$937$	−26.0767	−0.851888	−0.425944	+	0.904749i	$0.640058\pi$
$938$	0	0
$939$	54.8489	1.78993
$940$	0	0
$941$	−45.6025	−1.48660	−0.743299	−	0.668959i	$-0.766740\pi$
$941$	−45.6025	−1.48660	−0.743299	+	0.668959i	$0.766740\pi$
$942$	0	0
$943$	5.22966	0.170301
$944$	0	0
$945$	6.70565	0.218135
$946$	0	0
$947$	35.2548	1.14563	0.572814	−	0.819685i	$-0.305852\pi$
$947$	35.2548	1.14563	0.572814	+	0.819685i	$0.305852\pi$
$948$	0	0
$949$	−3.61574	−0.117372
$950$	0	0
$951$	−33.6569	−1.09140
$952$	0	0
$953$	−20.6865	−0.670103	−0.335051	−	0.942200i	$-0.608754\pi$
$953$	−20.6865	−0.670103	−0.335051	+	0.942200i	$0.608754\pi$
$954$	0	0
$955$	11.9540	0.386821
$956$	0	0
$957$	52.3902	1.69353
$958$	0	0
$959$	−28.8228	−0.930738
$960$	0	0
$961$	16.1269	0.520224
$962$	0	0
$963$	−26.0554	−0.839622
$964$	0	0
$965$	14.6469	0.471500
$966$	0	0
$967$	−31.8565	−1.02444	−0.512219	−	0.858855i	$-0.671176\pi$
$967$	−31.8565	−1.02444	−0.512219	+	0.858855i	$0.671176\pi$
$968$	0	0
$969$	−86.9692	−2.79385
$970$	0	0
$971$	2.02520	0.0649919	0.0324960	−	0.999472i	$-0.489654\pi$
$971$	2.02520	0.0649919	0.0324960	+	0.999472i	$0.489654\pi$
$972$	0	0
$973$	16.6946	0.535204
$974$	0	0
$975$	−6.74707	−0.216079
$976$	0	0
$977$	−3.33996	−0.106855	−0.0534274	−	0.998572i	$-0.517015\pi$
$977$	−3.33996	−0.106855	−0.0534274	+	0.998572i	$0.517015\pi$
$978$	0	0
$979$	22.6600	0.724215
$980$	0	0
$981$	−61.4423	−1.96170
$982$	0	0
$983$	−8.40905	−0.268207	−0.134104	−	0.990967i	$-0.542815\pi$
$983$	−8.40905	−0.268207	−0.134104	+	0.990967i	$0.542815\pi$
$984$	0	0
$985$	−4.74343	−0.151138
$986$	0	0
$987$	−52.4299	−1.66886
$988$	0	0
$989$	−36.1449	−1.14934
$990$	0	0
$991$	1.60494	0.0509825	0.0254912	−	0.999675i	$-0.491885\pi$
$991$	1.60494	0.0509825	0.0254912	+	0.999675i	$0.491885\pi$
$992$	0	0
$993$	−35.0234	−1.11143
$994$	0	0
$995$	15.7930	0.500672
$996$	0	0
$997$	−20.8631	−0.660742	−0.330371	−	0.943851i	$-0.607174\pi$
$997$	−20.8631	−0.660742	−0.330371	+	0.943851i	$0.607174\pi$
$998$	0	0
$999$	7.75460	0.245345

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.k.1.14		17
4.3	odd	2		251.2.a.b.1.1	✓	17
12.11	even	2		2259.2.a.k.1.17		17
20.19	odd	2		6275.2.a.e.1.17		17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
251.2.a.b.1.1	✓	17	4.3	odd	2
2259.2.a.k.1.17		17	12.11	even	2
4016.2.a.k.1.14		17	1.1	even	1	trivial
6275.2.a.e.1.17		17	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.62368 q^{3} +1.15029 q^{5} +2.51426 q^{7} +3.88372 q^{9} +O(q^{10})\)	\(q+2.62368 q^{3} +1.15029 q^{5} +2.51426 q^{7} +3.88372 q^{9} +6.29973 q^{11} +0.699407 q^{13} +3.01800 q^{15} +4.58455 q^{17} -7.23032 q^{19} +6.59661 q^{21} +4.43841 q^{23} -3.67683 q^{25} +2.31859 q^{27} +3.16969 q^{29} -6.86491 q^{31} +16.5285 q^{33} +2.89213 q^{35} +3.34453 q^{37} +1.83502 q^{39} +1.17827 q^{41} -8.14366 q^{43} +4.46740 q^{45} -7.94801 q^{47} -0.678520 q^{49} +12.0284 q^{51} +6.14772 q^{53} +7.24652 q^{55} -18.9701 q^{57} -3.29675 q^{59} +1.51045 q^{61} +9.76465 q^{63} +0.804522 q^{65} -4.60689 q^{67} +11.6450 q^{69} +8.22808 q^{71} -5.16971 q^{73} -9.64684 q^{75} +15.8391 q^{77} -8.65353 q^{79} -5.56790 q^{81} -4.65024 q^{83} +5.27356 q^{85} +8.31626 q^{87} +3.59697 q^{89} +1.75849 q^{91} -18.0113 q^{93} -8.31697 q^{95} -18.0684 q^{97} +24.4663 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9}+O(q^{10}) \) 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9	\( 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9} + q^{11} + 22 q^{13} + 8 q^{15} - q^{17} - 13 q^{19} + 25 q^{21} + 2 q^{23} + 32 q^{25} + 15 q^{27} + 28 q^{29} - 12 q^{31} - 16 q^{33} + 15 q^{35} + 27 q^{37} - 13 q^{39} - q^{41} - 9 q^{43} - 7 q^{45} + 20 q^{47} + 32 q^{49} + 2 q^{51} + q^{53} + 11 q^{55} - 24 q^{57} + 20 q^{59} + 59 q^{61} + 41 q^{63} - 14 q^{65} - 15 q^{67} + 38 q^{69} + 26 q^{71} + 8 q^{73} + 20 q^{75} - 33 q^{79} + 29 q^{81} + 67 q^{85} + 11 q^{87} + 11 q^{89} + 2 q^{91} + 28 q^{93} + 8 q^{95} - 10 q^{97} + 36 q^{99}+O(q^{100}) \) 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9 + q^11 + 22 * q^13 + 8 * q^15 - q^17 - 13 * q^19 + 25 * q^21 + 2 * q^23 + 32 * q^25 + 15 * q^27 + 28 * q^29 - 12 * q^31 - 16 * q^33 + 15 * q^35 + 27 * q^37 - 13 * q^39 - q^41 - 9 * q^43 - 7 * q^45 + 20 * q^47 + 32 * q^49 + 2 * q^51 + q^53 + 11 * q^55 - 24 * q^57 + 20 * q^59 + 59 * q^61 + 41 * q^63 - 14 * q^65 - 15 * q^67 + 38 * q^69 + 26 * q^71 + 8 * q^73 + 20 * q^75 - 33 * q^79 + 29 * q^81 + 67 * q^85 + 11 * q^87 + 11 * q^89 + 2 * q^91 + 28 * q^93 + 8 * q^95 - 10 * q^97 + 36 * q^99

Introduction

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