LMFDB - Embedded newform 2016.2.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2016,2,Mod(1,2016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2016, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2016 = 2^{5} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2016.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:	$16.0978410475$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	2016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.23607 q^{5} +1.00000 q^{7} +O(q^{10})$	$q-3.23607 q^{5} +1.00000 q^{7} -3.23607 q^{11} +4.47214 q^{13} -0.763932 q^{17} +2.47214 q^{19} +5.70820 q^{23} +5.47214 q^{25} -1.52786 q^{29} -2.47214 q^{31} -3.23607 q^{35} -4.47214 q^{37} -7.23607 q^{41} -12.9443 q^{43} +1.52786 q^{47} +1.00000 q^{49} -8.00000 q^{53} +10.4721 q^{55} -6.47214 q^{59} -4.47214 q^{61} -14.4721 q^{65} -8.00000 q^{67} +7.23607 q^{71} -14.9443 q^{73} -3.23607 q^{77} +12.9443 q^{79} +12.9443 q^{83} +2.47214 q^{85} -2.29180 q^{89} +4.47214 q^{91} -8.00000 q^{95} -6.94427 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 2 q^{7}+O(q^{10}) $ 2 * q - 2 * q^5 + 2 * q^7	$ 2 q - 2 q^{5} + 2 q^{7} - 2 q^{11} - 6 q^{17} - 4 q^{19} - 2 q^{23} + 2 q^{25} - 12 q^{29} + 4 q^{31} - 2 q^{35} - 10 q^{41} - 8 q^{43} + 12 q^{47} + 2 q^{49} - 16 q^{53} + 12 q^{55} - 4 q^{59} - 20 q^{65} - 16 q^{67} + 10 q^{71} - 12 q^{73} - 2 q^{77} + 8 q^{79} + 8 q^{83} - 4 q^{85} - 18 q^{89} - 16 q^{95} + 4 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 + 2 * q^7 - 2 * q^11 - 6 * q^17 - 4 * q^19 - 2 * q^23 + 2 * q^25 - 12 * q^29 + 4 * q^31 - 2 * q^35 - 10 * q^41 - 8 * q^43 + 12 * q^47 + 2 * q^49 - 16 * q^53 + 12 * q^55 - 4 * q^59 - 20 * q^65 - 16 * q^67 + 10 * q^71 - 12 * q^73 - 2 * q^77 + 8 * q^79 + 8 * q^83 - 4 * q^85 - 18 * q^89 - 16 * q^95 + 4 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−3.23607	−1.44721	−0.723607	−	0.690212i	$-0.757517\pi$
$5$	−3.23607	−1.44721	−0.723607	+	0.690212i	$0.757517\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.23607	−0.975711	−0.487856	−	0.872924i	$-0.662221\pi$
$11$	−3.23607	−0.975711	−0.487856	+	0.872924i	$0.662221\pi$
$12$	0	0
$13$	4.47214	1.24035	0.620174	−	0.784465i	$-0.287062\pi$
$13$	4.47214	1.24035	0.620174	+	0.784465i	$0.287062\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.763932	−0.185281	−0.0926404	−	0.995700i	$-0.529531\pi$
$17$	−0.763932	−0.185281	−0.0926404	+	0.995700i	$0.529531\pi$
$18$	0	0
$19$	2.47214	0.567147	0.283573	−	0.958951i	$-0.408480\pi$
$19$	2.47214	0.567147	0.283573	+	0.958951i	$0.408480\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.70820	1.19024	0.595121	−	0.803636i	$-0.297104\pi$
$23$	5.70820	1.19024	0.595121	+	0.803636i	$0.297104\pi$
$24$	0	0
$25$	5.47214	1.09443
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.52786	−0.283717	−0.141859	−	0.989887i	$-0.545308\pi$
$29$	−1.52786	−0.283717	−0.141859	+	0.989887i	$0.545308\pi$
$30$	0	0
$31$	−2.47214	−0.444009	−0.222004	−	0.975046i	$-0.571260\pi$
$31$	−2.47214	−0.444009	−0.222004	+	0.975046i	$0.571260\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.23607	−0.546995
$36$	0	0
$37$	−4.47214	−0.735215	−0.367607	−	0.929981i	$-0.619823\pi$
$37$	−4.47214	−0.735215	−0.367607	+	0.929981i	$0.619823\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−7.23607	−1.13008	−0.565042	−	0.825062i	$-0.691140\pi$
$41$	−7.23607	−1.13008	−0.565042	+	0.825062i	$0.691140\pi$
$42$	0	0
$43$	−12.9443	−1.97398	−0.986991	−	0.160773i	$-0.948601\pi$
$43$	−12.9443	−1.97398	−0.986991	+	0.160773i	$0.948601\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.52786	0.222862	0.111431	−	0.993772i	$-0.464457\pi$
$47$	1.52786	0.222862	0.111431	+	0.993772i	$0.464457\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−8.00000	−1.09888	−0.549442	−	0.835532i	$-0.685160\pi$
$53$	−8.00000	−1.09888	−0.549442	+	0.835532i	$0.685160\pi$
$54$	0	0
$55$	10.4721	1.41206
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−6.47214	−0.842600	−0.421300	−	0.906921i	$-0.638426\pi$
$59$	−6.47214	−0.842600	−0.421300	+	0.906921i	$0.638426\pi$
$60$	0	0
$61$	−4.47214	−0.572598	−0.286299	−	0.958140i	$-0.592425\pi$
$61$	−4.47214	−0.572598	−0.286299	+	0.958140i	$0.592425\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−14.4721	−1.79505
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	7.23607	0.858763	0.429382	−	0.903123i	$-0.358732\pi$
$71$	7.23607	0.858763	0.429382	+	0.903123i	$0.358732\pi$
$72$	0	0
$73$	−14.9443	−1.74909	−0.874547	−	0.484940i	$-0.838841\pi$
$73$	−14.9443	−1.74909	−0.874547	+	0.484940i	$0.838841\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.23607	−0.368784
$78$	0	0
$79$	12.9443	1.45634	0.728172	−	0.685394i	$-0.240370\pi$
$79$	12.9443	1.45634	0.728172	+	0.685394i	$0.240370\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	12.9443	1.42082	0.710409	−	0.703789i	$-0.248510\pi$
$83$	12.9443	1.42082	0.710409	+	0.703789i	$0.248510\pi$
$84$	0	0
$85$	2.47214	0.268141
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−2.29180	−0.242930	−0.121465	−	0.992596i	$-0.538759\pi$
$89$	−2.29180	−0.242930	−0.121465	+	0.992596i	$0.538759\pi$
$90$	0	0
$91$	4.47214	0.468807
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−8.00000	−0.820783
$96$	0	0
$97$	−6.94427	−0.705084	−0.352542	−	0.935796i	$-0.614683\pi$
$97$	−6.94427	−0.705084	−0.352542	+	0.935796i	$0.614683\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−17.7082	−1.76203	−0.881016	−	0.473086i	$-0.843140\pi$
$101$	−17.7082	−1.76203	−0.881016	+	0.473086i	$0.843140\pi$
$102$	0	0
$103$	18.4721	1.82011	0.910057	−	0.414484i	$-0.136038\pi$
$103$	18.4721	1.82011	0.910057	+	0.414484i	$0.136038\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−11.2361	−1.08623	−0.543116	−	0.839658i	$-0.682756\pi$
$107$	−11.2361	−1.08623	−0.543116	+	0.839658i	$0.682756\pi$
$108$	0	0
$109$	−10.9443	−1.04827	−0.524136	−	0.851635i	$-0.675611\pi$
$109$	−10.9443	−1.04827	−0.524136	+	0.851635i	$0.675611\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−12.9443	−1.21769	−0.608847	−	0.793287i	$-0.708368\pi$
$113$	−12.9443	−1.21769	−0.608847	+	0.793287i	$0.708368\pi$
$114$	0	0
$115$	−18.4721	−1.72254
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.763932	−0.0700295
$120$	0	0
$121$	−0.527864	−0.0479876
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.52786	−0.136656
$126$	0	0
$127$	−17.8885	−1.58735	−0.793676	−	0.608341i	$-0.791835\pi$
$127$	−17.8885	−1.58735	−0.793676	+	0.608341i	$0.791835\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	16.0000	1.39793	0.698963	−	0.715158i	$-0.253645\pi$
$131$	16.0000	1.39793	0.698963	+	0.715158i	$0.253645\pi$
$132$	0	0
$133$	2.47214	0.214361
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1.52786	−0.130534	−0.0652671	−	0.997868i	$-0.520790\pi$
$137$	−1.52786	−0.130534	−0.0652671	+	0.997868i	$0.520790\pi$
$138$	0	0
$139$	4.94427	0.419368	0.209684	−	0.977769i	$-0.432757\pi$
$139$	4.94427	0.419368	0.209684	+	0.977769i	$0.432757\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−14.4721	−1.21022
$144$	0	0
$145$	4.94427	0.410599
$146$	0	0
$147$	0	0
$148$	0	0
$149$	17.8885	1.46549	0.732743	−	0.680505i	$-0.238240\pi$
$149$	17.8885	1.46549	0.732743	+	0.680505i	$0.238240\pi$
$150$	0	0
$151$	3.05573	0.248672	0.124336	−	0.992240i	$-0.460320\pi$
$151$	3.05573	0.248672	0.124336	+	0.992240i	$0.460320\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	8.00000	0.642575
$156$	0	0
$157$	16.4721	1.31462	0.657310	−	0.753620i	$-0.271694\pi$
$157$	16.4721	1.31462	0.657310	+	0.753620i	$0.271694\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	5.70820	0.449869
$162$	0	0
$163$	−8.00000	−0.626608	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−11.4164	−0.883428	−0.441714	−	0.897156i	$-0.645629\pi$
$167$	−11.4164	−0.883428	−0.441714	+	0.897156i	$0.645629\pi$
$168$	0	0
$169$	7.00000	0.538462
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−17.7082	−1.34633	−0.673165	−	0.739492i	$-0.735066\pi$
$173$	−17.7082	−1.34633	−0.673165	+	0.739492i	$0.735066\pi$
$174$	0	0
$175$	5.47214	0.413655
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0.180340	0.0134792	0.00673962	−	0.999977i	$-0.497855\pi$
$179$	0.180340	0.0134792	0.00673962	+	0.999977i	$0.497855\pi$
$180$	0	0
$181$	−8.47214	−0.629729	−0.314864	−	0.949137i	$-0.601959\pi$
$181$	−8.47214	−0.629729	−0.314864	+	0.949137i	$0.601959\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	14.4721	1.06401
$186$	0	0
$187$	2.47214	0.180780
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−26.6525	−1.92851	−0.964253	−	0.264984i	$-0.914633\pi$
$191$	−26.6525	−1.92851	−0.964253	+	0.264984i	$0.914633\pi$
$192$	0	0
$193$	12.4721	0.897764	0.448882	−	0.893591i	$-0.351822\pi$
$193$	12.4721	0.897764	0.448882	+	0.893591i	$0.351822\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4.94427	−0.352265	−0.176132	−	0.984366i	$-0.556359\pi$
$197$	−4.94427	−0.352265	−0.176132	+	0.984366i	$0.556359\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1.52786	−0.107235
$204$	0	0
$205$	23.4164	1.63547
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−8.00000	−0.553372
$210$	0	0
$211$	−24.0000	−1.65223	−0.826114	−	0.563503i	$-0.809453\pi$
$211$	−24.0000	−1.65223	−0.826114	+	0.563503i	$0.809453\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	41.8885	2.85677
$216$	0	0
$217$	−2.47214	−0.167820
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−3.41641	−0.229812
$222$	0	0
$223$	12.9443	0.866813	0.433406	−	0.901199i	$-0.357312\pi$
$223$	12.9443	0.866813	0.433406	+	0.901199i	$0.357312\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	16.3607	1.08590	0.542948	−	0.839766i	$-0.317308\pi$
$227$	16.3607	1.08590	0.542948	+	0.839766i	$0.317308\pi$
$228$	0	0
$229$	12.4721	0.824182	0.412091	−	0.911143i	$-0.364799\pi$
$229$	12.4721	0.824182	0.412091	+	0.911143i	$0.364799\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−24.3607	−1.59592	−0.797961	−	0.602710i	$-0.794088\pi$
$233$	−24.3607	−1.59592	−0.797961	+	0.602710i	$0.794088\pi$
$234$	0	0
$235$	−4.94427	−0.322529
$236$	0	0
$237$	0	0
$238$	0	0
$239$	8.76393	0.566892	0.283446	−	0.958988i	$-0.408522\pi$
$239$	8.76393	0.566892	0.283446	+	0.958988i	$0.408522\pi$
$240$	0	0
$241$	7.88854	0.508146	0.254073	−	0.967185i	$-0.418230\pi$
$241$	7.88854	0.508146	0.254073	+	0.967185i	$0.418230\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−3.23607	−0.206745
$246$	0	0
$247$	11.0557	0.703459
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−3.41641	−0.215642	−0.107821	−	0.994170i	$-0.534387\pi$
$251$	−3.41641	−0.215642	−0.107821	+	0.994170i	$0.534387\pi$
$252$	0	0
$253$	−18.4721	−1.16133
$254$	0	0
$255$	0	0
$256$	0	0
$257$	29.7082	1.85315	0.926573	−	0.376114i	$-0.122740\pi$
$257$	29.7082	1.85315	0.926573	+	0.376114i	$0.122740\pi$
$258$	0	0
$259$	−4.47214	−0.277885
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−28.1803	−1.73767	−0.868837	−	0.495098i	$-0.835132\pi$
$263$	−28.1803	−1.73767	−0.868837	+	0.495098i	$0.835132\pi$
$264$	0	0
$265$	25.8885	1.59032
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−3.23607	−0.197307	−0.0986533	−	0.995122i	$-0.531453\pi$
$269$	−3.23607	−0.197307	−0.0986533	+	0.995122i	$0.531453\pi$
$270$	0	0
$271$	7.41641	0.450515	0.225257	−	0.974299i	$-0.427678\pi$
$271$	7.41641	0.450515	0.225257	+	0.974299i	$0.427678\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−17.7082	−1.06784
$276$	0	0
$277$	−11.5279	−0.692642	−0.346321	−	0.938116i	$-0.612569\pi$
$277$	−11.5279	−0.692642	−0.346321	+	0.938116i	$0.612569\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	11.4164	0.681046	0.340523	−	0.940236i	$-0.389396\pi$
$281$	11.4164	0.681046	0.340523	+	0.940236i	$0.389396\pi$
$282$	0	0
$283$	−2.47214	−0.146953	−0.0734766	−	0.997297i	$-0.523409\pi$
$283$	−2.47214	−0.146953	−0.0734766	+	0.997297i	$0.523409\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−7.23607	−0.427132
$288$	0	0
$289$	−16.4164	−0.965671
$290$	0	0
$291$	0	0
$292$	0	0
$293$	32.1803	1.88000	0.939998	−	0.341181i	$-0.110827\pi$
$293$	32.1803	1.88000	0.939998	+	0.341181i	$0.110827\pi$
$294$	0	0
$295$	20.9443	1.21942
$296$	0	0
$297$	0	0
$298$	0	0
$299$	25.5279	1.47631
$300$	0	0
$301$	−12.9443	−0.746095
$302$	0	0
$303$	0	0
$304$	0	0
$305$	14.4721	0.828672
$306$	0	0
$307$	−18.4721	−1.05426	−0.527130	−	0.849785i	$-0.676732\pi$
$307$	−18.4721	−1.05426	−0.527130	+	0.849785i	$0.676732\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	11.4164	0.647365	0.323683	−	0.946166i	$-0.395079\pi$
$311$	11.4164	0.647365	0.323683	+	0.946166i	$0.395079\pi$
$312$	0	0
$313$	−15.8885	−0.898074	−0.449037	−	0.893513i	$-0.648233\pi$
$313$	−15.8885	−0.898074	−0.449037	+	0.893513i	$0.648233\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	20.9443	1.17635	0.588174	−	0.808735i	$-0.299847\pi$
$317$	20.9443	1.17635	0.588174	+	0.808735i	$0.299847\pi$
$318$	0	0
$319$	4.94427	0.276826
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.88854	−0.105081
$324$	0	0
$325$	24.4721	1.35747
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.52786	0.0842339
$330$	0	0
$331$	17.8885	0.983243	0.491622	−	0.870809i	$-0.336404\pi$
$331$	17.8885	0.983243	0.491622	+	0.870809i	$0.336404\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	25.8885	1.41444
$336$	0	0
$337$	−11.8885	−0.647610	−0.323805	−	0.946124i	$-0.604962\pi$
$337$	−11.8885	−0.647610	−0.323805	+	0.946124i	$0.604962\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	8.00000	0.433224
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−20.7639	−1.11467	−0.557333	−	0.830289i	$-0.688175\pi$
$347$	−20.7639	−1.11467	−0.557333	+	0.830289i	$0.688175\pi$
$348$	0	0
$349$	−20.4721	−1.09585	−0.547924	−	0.836528i	$-0.684582\pi$
$349$	−20.4721	−1.09585	−0.547924	+	0.836528i	$0.684582\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	12.1803	0.648294	0.324147	−	0.946007i	$-0.394923\pi$
$353$	12.1803	0.648294	0.324147	+	0.946007i	$0.394923\pi$
$354$	0	0
$355$	−23.4164	−1.24281
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−16.7639	−0.884766	−0.442383	−	0.896826i	$-0.645867\pi$
$359$	−16.7639	−0.884766	−0.442383	+	0.896826i	$0.645867\pi$
$360$	0	0
$361$	−12.8885	−0.678344
$362$	0	0
$363$	0	0
$364$	0	0
$365$	48.3607	2.53131
$366$	0	0
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−8.00000	−0.415339
$372$	0	0
$373$	−11.8885	−0.615565	−0.307783	−	0.951457i	$-0.599587\pi$
$373$	−11.8885	−0.615565	−0.307783	+	0.951457i	$0.599587\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6.83282	−0.351908
$378$	0	0
$379$	24.0000	1.23280	0.616399	−	0.787434i	$-0.288591\pi$
$379$	24.0000	1.23280	0.616399	+	0.787434i	$0.288591\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	32.0000	1.63512	0.817562	−	0.575841i	$-0.195325\pi$
$383$	32.0000	1.63512	0.817562	+	0.575841i	$0.195325\pi$
$384$	0	0
$385$	10.4721	0.533709
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1.52786	−0.0774658	−0.0387329	−	0.999250i	$-0.512332\pi$
$389$	−1.52786	−0.0774658	−0.0387329	+	0.999250i	$0.512332\pi$
$390$	0	0
$391$	−4.36068	−0.220529
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−41.8885	−2.10764
$396$	0	0
$397$	0.472136	0.0236958	0.0118479	−	0.999930i	$-0.496229\pi$
$397$	0.472136	0.0236958	0.0118479	+	0.999930i	$0.496229\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	17.5279	0.875300	0.437650	−	0.899145i	$-0.355811\pi$
$401$	17.5279	0.875300	0.437650	+	0.899145i	$0.355811\pi$
$402$	0	0
$403$	−11.0557	−0.550725
$404$	0	0
$405$	0	0
$406$	0	0
$407$	14.4721	0.717357
$408$	0	0
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−6.47214	−0.318473
$414$	0	0
$415$	−41.8885	−2.05623
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−6.47214	−0.316185	−0.158092	−	0.987424i	$-0.550534\pi$
$419$	−6.47214	−0.316185	−0.158092	+	0.987424i	$0.550534\pi$
$420$	0	0
$421$	−21.4164	−1.04377	−0.521886	−	0.853015i	$-0.674771\pi$
$421$	−21.4164	−1.04377	−0.521886	+	0.853015i	$0.674771\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−4.18034	−0.202776
$426$	0	0
$427$	−4.47214	−0.216422
$428$	0	0
$429$	0	0
$430$	0	0
$431$	9.12461	0.439517	0.219759	−	0.975554i	$-0.429473\pi$
$431$	9.12461	0.439517	0.219759	+	0.975554i	$0.429473\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	14.1115	0.675042
$438$	0	0
$439$	−22.8328	−1.08975	−0.544875	−	0.838517i	$-0.683423\pi$
$439$	−22.8328	−1.08975	−0.544875	+	0.838517i	$0.683423\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−7.81966	−0.371523	−0.185762	−	0.982595i	$-0.559475\pi$
$443$	−7.81966	−0.371523	−0.185762	+	0.982595i	$0.559475\pi$
$444$	0	0
$445$	7.41641	0.351571
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−12.9443	−0.610878	−0.305439	−	0.952212i	$-0.598803\pi$
$449$	−12.9443	−0.610878	−0.305439	+	0.952212i	$0.598803\pi$
$450$	0	0
$451$	23.4164	1.10264
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−14.4721	−0.678464
$456$	0	0
$457$	−0.111456	−0.00521370	−0.00260685	−	0.999997i	$-0.500830\pi$
$457$	−0.111456	−0.00521370	−0.00260685	+	0.999997i	$0.500830\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−9.70820	−0.452156	−0.226078	−	0.974109i	$-0.572590\pi$
$461$	−9.70820	−0.452156	−0.226078	+	0.974109i	$0.572590\pi$
$462$	0	0
$463$	17.8885	0.831351	0.415676	−	0.909513i	$-0.363545\pi$
$463$	17.8885	0.831351	0.415676	+	0.909513i	$0.363545\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	25.5279	1.18129	0.590644	−	0.806932i	$-0.298874\pi$
$467$	25.5279	1.18129	0.590644	+	0.806932i	$0.298874\pi$
$468$	0	0
$469$	−8.00000	−0.369406
$470$	0	0
$471$	0	0
$472$	0	0
$473$	41.8885	1.92604
$474$	0	0
$475$	13.5279	0.620701
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−27.4164	−1.25269	−0.626344	−	0.779547i	$-0.715449\pi$
$479$	−27.4164	−1.25269	−0.626344	+	0.779547i	$0.715449\pi$
$480$	0	0
$481$	−20.0000	−0.911922
$482$	0	0
$483$	0	0
$484$	0	0
$485$	22.4721	1.02041
$486$	0	0
$487$	22.8328	1.03465	0.517327	−	0.855788i	$-0.326927\pi$
$487$	22.8328	1.03465	0.517327	+	0.855788i	$0.326927\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	42.0689	1.89854	0.949271	−	0.314459i	$-0.101823\pi$
$491$	42.0689	1.89854	0.949271	+	0.314459i	$0.101823\pi$
$492$	0	0
$493$	1.16718	0.0525673
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7.23607	0.324582
$498$	0	0
$499$	−28.9443	−1.29572	−0.647862	−	0.761758i	$-0.724337\pi$
$499$	−28.9443	−1.29572	−0.647862	+	0.761758i	$0.724337\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	28.9443	1.29056	0.645281	−	0.763946i	$-0.276740\pi$
$503$	28.9443	1.29056	0.645281	+	0.763946i	$0.276740\pi$
$504$	0	0
$505$	57.3050	2.55004
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−5.12461	−0.227144	−0.113572	−	0.993530i	$-0.536229\pi$
$509$	−5.12461	−0.227144	−0.113572	+	0.993530i	$0.536229\pi$
$510$	0	0
$511$	−14.9443	−0.661096
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−59.7771	−2.63409
$516$	0	0
$517$	−4.94427	−0.217449
$518$	0	0
$519$	0	0
$520$	0	0
$521$	25.1246	1.10073	0.550365	−	0.834924i	$-0.314489\pi$
$521$	25.1246	1.10073	0.550365	+	0.834924i	$0.314489\pi$
$522$	0	0
$523$	16.0000	0.699631	0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000	0.699631	0.349816	+	0.936819i	$0.386244\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.88854	0.0822663
$528$	0	0
$529$	9.58359	0.416678
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−32.3607	−1.40170
$534$	0	0
$535$	36.3607	1.57201
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−3.23607	−0.139387
$540$	0	0
$541$	38.3607	1.64925	0.824627	−	0.565677i	$-0.191385\pi$
$541$	38.3607	1.64925	0.824627	+	0.565677i	$0.191385\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	35.4164	1.51707
$546$	0	0
$547$	38.8328	1.66037	0.830186	−	0.557487i	$-0.188234\pi$
$547$	38.8328	1.66037	0.830186	+	0.557487i	$0.188234\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−3.77709	−0.160909
$552$	0	0
$553$	12.9443	0.550446
$554$	0	0
$555$	0	0
$556$	0	0
$557$	20.9443	0.887437	0.443719	−	0.896166i	$-0.353659\pi$
$557$	20.9443	0.887437	0.443719	+	0.896166i	$0.353659\pi$
$558$	0	0
$559$	−57.8885	−2.44842
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−29.3050	−1.23506	−0.617528	−	0.786549i	$-0.711866\pi$
$563$	−29.3050	−1.23506	−0.617528	+	0.786549i	$0.711866\pi$
$564$	0	0
$565$	41.8885	1.76226
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−17.5279	−0.734806	−0.367403	−	0.930062i	$-0.619753\pi$
$569$	−17.5279	−0.734806	−0.367403	+	0.930062i	$0.619753\pi$
$570$	0	0
$571$	28.9443	1.21128	0.605640	−	0.795739i	$-0.292917\pi$
$571$	28.9443	1.21128	0.605640	+	0.795739i	$0.292917\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	31.2361	1.30263
$576$	0	0
$577$	−30.0000	−1.24892	−0.624458	−	0.781058i	$-0.714680\pi$
$577$	−30.0000	−1.24892	−0.624458	+	0.781058i	$0.714680\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	12.9443	0.537019
$582$	0	0
$583$	25.8885	1.07219
$584$	0	0
$585$	0	0
$586$	0	0
$587$	19.4164	0.801401	0.400700	−	0.916209i	$-0.368767\pi$
$587$	19.4164	0.801401	0.400700	+	0.916209i	$0.368767\pi$
$588$	0	0
$589$	−6.11146	−0.251818
$590$	0	0
$591$	0	0
$592$	0	0
$593$	17.1246	0.703224	0.351612	−	0.936146i	$-0.385634\pi$
$593$	17.1246	0.703224	0.351612	+	0.936146i	$0.385634\pi$
$594$	0	0
$595$	2.47214	0.101348
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−24.7639	−1.01183	−0.505913	−	0.862584i	$-0.668844\pi$
$599$	−24.7639	−1.01183	−0.505913	+	0.862584i	$0.668844\pi$
$600$	0	0
$601$	0.111456	0.00454639	0.00227320	−	0.999997i	$-0.499276\pi$
$601$	0.111456	0.00454639	0.00227320	+	0.999997i	$0.499276\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.70820	0.0694484
$606$	0	0
$607$	−28.9443	−1.17481	−0.587406	−	0.809292i	$-0.699851\pi$
$607$	−28.9443	−1.17481	−0.587406	+	0.809292i	$0.699851\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	6.83282	0.276426
$612$	0	0
$613$	12.8328	0.518313	0.259156	−	0.965835i	$-0.416556\pi$
$613$	12.8328	0.518313	0.259156	+	0.965835i	$0.416556\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	46.4721	1.87090	0.935449	−	0.353462i	$-0.114996\pi$
$617$	46.4721	1.87090	0.935449	+	0.353462i	$0.114996\pi$
$618$	0	0
$619$	11.0557	0.444367	0.222184	−	0.975005i	$-0.428682\pi$
$619$	11.0557	0.444367	0.222184	+	0.975005i	$0.428682\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−2.29180	−0.0918189
$624$	0	0
$625$	−22.4164	−0.896656
$626$	0	0
$627$	0	0
$628$	0	0
$629$	3.41641	0.136221
$630$	0	0
$631$	17.8885	0.712132	0.356066	−	0.934461i	$-0.384118\pi$
$631$	17.8885	0.712132	0.356066	+	0.934461i	$0.384118\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	57.8885	2.29724
$636$	0	0
$637$	4.47214	0.177192
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−50.2492	−1.98473	−0.992363	−	0.123356i	$-0.960634\pi$
$641$	−50.2492	−1.98473	−0.992363	+	0.123356i	$0.960634\pi$
$642$	0	0
$643$	−23.4164	−0.923453	−0.461726	−	0.887022i	$-0.652770\pi$
$643$	−23.4164	−0.923453	−0.461726	+	0.887022i	$0.652770\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−30.4721	−1.19798	−0.598992	−	0.800755i	$-0.704432\pi$
$647$	−30.4721	−1.19798	−0.598992	+	0.800755i	$0.704432\pi$
$648$	0	0
$649$	20.9443	0.822135
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−11.4164	−0.446759	−0.223379	−	0.974732i	$-0.571709\pi$
$653$	−11.4164	−0.446759	−0.223379	+	0.974732i	$0.571709\pi$
$654$	0	0
$655$	−51.7771	−2.02310
$656$	0	0
$657$	0	0
$658$	0	0
$659$	30.6525	1.19405	0.597025	−	0.802222i	$-0.296349\pi$
$659$	30.6525	1.19405	0.597025	+	0.802222i	$0.296349\pi$
$660$	0	0
$661$	24.4721	0.951856	0.475928	−	0.879484i	$-0.342112\pi$
$661$	24.4721	0.951856	0.475928	+	0.879484i	$0.342112\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−8.00000	−0.310227
$666$	0	0
$667$	−8.72136	−0.337692
$668$	0	0
$669$	0	0
$670$	0	0
$671$	14.4721	0.558691
$672$	0	0
$673$	43.3050	1.66928	0.834642	−	0.550793i	$-0.185675\pi$
$673$	43.3050	1.66928	0.834642	+	0.550793i	$0.185675\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	31.0132	1.19193	0.595966	−	0.803010i	$-0.296769\pi$
$677$	31.0132	1.19193	0.595966	+	0.803010i	$0.296769\pi$
$678$	0	0
$679$	−6.94427	−0.266497
$680$	0	0
$681$	0	0
$682$	0	0
$683$	8.18034	0.313012	0.156506	−	0.987677i	$-0.449977\pi$
$683$	8.18034	0.313012	0.156506	+	0.987677i	$0.449977\pi$
$684$	0	0
$685$	4.94427	0.188911
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−35.7771	−1.36300
$690$	0	0
$691$	−41.8885	−1.59352	−0.796758	−	0.604299i	$-0.793453\pi$
$691$	−41.8885	−1.59352	−0.796758	+	0.604299i	$0.793453\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−16.0000	−0.606915
$696$	0	0
$697$	5.52786	0.209383
$698$	0	0
$699$	0	0
$700$	0	0
$701$	11.4164	0.431192	0.215596	−	0.976483i	$-0.430831\pi$
$701$	11.4164	0.431192	0.215596	+	0.976483i	$0.430831\pi$
$702$	0	0
$703$	−11.0557	−0.416975
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−17.7082	−0.665986
$708$	0	0
$709$	−2.94427	−0.110574	−0.0552872	−	0.998470i	$-0.517607\pi$
$709$	−2.94427	−0.110574	−0.0552872	+	0.998470i	$0.517607\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−14.1115	−0.528478
$714$	0	0
$715$	46.8328	1.75145
$716$	0	0
$717$	0	0
$718$	0	0
$719$	19.0557	0.710659	0.355329	−	0.934741i	$-0.384369\pi$
$719$	19.0557	0.710659	0.355329	+	0.934741i	$0.384369\pi$
$720$	0	0
$721$	18.4721	0.687938
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−8.36068	−0.310508
$726$	0	0
$727$	12.3607	0.458432	0.229216	−	0.973376i	$-0.426384\pi$
$727$	12.3607	0.458432	0.229216	+	0.973376i	$0.426384\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	9.88854	0.365741
$732$	0	0
$733$	−47.3050	−1.74725	−0.873624	−	0.486601i	$-0.838236\pi$
$733$	−47.3050	−1.74725	−0.873624	+	0.486601i	$0.838236\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	25.8885	0.953617
$738$	0	0
$739$	40.0000	1.47142	0.735712	−	0.677295i	$-0.236848\pi$
$739$	40.0000	1.47142	0.735712	+	0.677295i	$0.236848\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−49.4853	−1.81544	−0.907720	−	0.419577i	$-0.862178\pi$
$743$	−49.4853	−1.81544	−0.907720	+	0.419577i	$0.862178\pi$
$744$	0	0
$745$	−57.8885	−2.12087
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−11.2361	−0.410557
$750$	0	0
$751$	−24.0000	−0.875772	−0.437886	−	0.899030i	$-0.644273\pi$
$751$	−24.0000	−0.875772	−0.437886	+	0.899030i	$0.644273\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−9.88854	−0.359881
$756$	0	0
$757$	−18.9443	−0.688541	−0.344271	−	0.938870i	$-0.611874\pi$
$757$	−18.9443	−0.688541	−0.344271	+	0.938870i	$0.611874\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−12.1803	−0.441537	−0.220768	−	0.975326i	$-0.570857\pi$
$761$	−12.1803	−0.441537	−0.220768	+	0.975326i	$0.570857\pi$
$762$	0	0
$763$	−10.9443	−0.396209
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−28.9443	−1.04512
$768$	0	0
$769$	27.8885	1.00569	0.502843	−	0.864378i	$-0.332287\pi$
$769$	27.8885	1.00569	0.502843	+	0.864378i	$0.332287\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−20.7639	−0.746827	−0.373413	−	0.927665i	$-0.621813\pi$
$773$	−20.7639	−0.746827	−0.373413	+	0.927665i	$0.621813\pi$
$774$	0	0
$775$	−13.5279	−0.485935
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−17.8885	−0.640924
$780$	0	0
$781$	−23.4164	−0.837905
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−53.3050	−1.90254
$786$	0	0
$787$	−1.16718	−0.0416056	−0.0208028	−	0.999784i	$-0.506622\pi$
$787$	−1.16718	−0.0416056	−0.0208028	+	0.999784i	$0.506622\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−12.9443	−0.460245
$792$	0	0
$793$	−20.0000	−0.710221
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.34752	0.0477318	0.0238659	−	0.999715i	$-0.492403\pi$
$797$	1.34752	0.0477318	0.0238659	+	0.999715i	$0.492403\pi$
$798$	0	0
$799$	−1.16718	−0.0412920
$800$	0	0
$801$	0	0
$802$	0	0
$803$	48.3607	1.70661
$804$	0	0
$805$	−18.4721	−0.651057
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−32.0000	−1.12506	−0.562530	−	0.826777i	$-0.690172\pi$
$809$	−32.0000	−1.12506	−0.562530	+	0.826777i	$0.690172\pi$
$810$	0	0
$811$	6.11146	0.214602	0.107301	−	0.994227i	$-0.465779\pi$
$811$	6.11146	0.214602	0.107301	+	0.994227i	$0.465779\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	25.8885	0.906836
$816$	0	0
$817$	−32.0000	−1.11954
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−36.9443	−1.28936	−0.644682	−	0.764451i	$-0.723010\pi$
$821$	−36.9443	−1.28936	−0.644682	+	0.764451i	$0.723010\pi$
$822$	0	0
$823$	19.0557	0.664241	0.332120	−	0.943237i	$-0.392236\pi$
$823$	19.0557	0.664241	0.332120	+	0.943237i	$0.392236\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	15.8197	0.550103	0.275052	−	0.961429i	$-0.411305\pi$
$827$	15.8197	0.550103	0.275052	+	0.961429i	$0.411305\pi$
$828$	0	0
$829$	31.5279	1.09501	0.547504	−	0.836803i	$-0.315578\pi$
$829$	31.5279	1.09501	0.547504	+	0.836803i	$0.315578\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−0.763932	−0.0264687
$834$	0	0
$835$	36.9443	1.27851
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−14.4721	−0.499634	−0.249817	−	0.968293i	$-0.580370\pi$
$839$	−14.4721	−0.499634	−0.249817	+	0.968293i	$0.580370\pi$
$840$	0	0
$841$	−26.6656	−0.919505
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−22.6525	−0.779269
$846$	0	0
$847$	−0.527864	−0.0181376
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−25.5279	−0.875084
$852$	0	0
$853$	−18.5836	−0.636290	−0.318145	−	0.948042i	$-0.603060\pi$
$853$	−18.5836	−0.636290	−0.318145	+	0.948042i	$0.603060\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−12.1803	−0.416072	−0.208036	−	0.978121i	$-0.566707\pi$
$857$	−12.1803	−0.416072	−0.208036	+	0.978121i	$0.566707\pi$
$858$	0	0
$859$	34.4721	1.17617	0.588087	−	0.808798i	$-0.299881\pi$
$859$	34.4721	1.17617	0.588087	+	0.808798i	$0.299881\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−44.1803	−1.50392	−0.751958	−	0.659211i	$-0.770890\pi$
$863$	−44.1803	−1.50392	−0.751958	+	0.659211i	$0.770890\pi$
$864$	0	0
$865$	57.3050	1.94843
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−41.8885	−1.42097
$870$	0	0
$871$	−35.7771	−1.21226
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1.52786	−0.0516512
$876$	0	0
$877$	−8.83282	−0.298263	−0.149131	−	0.988817i	$-0.547648\pi$
$877$	−8.83282	−0.298263	−0.149131	+	0.988817i	$0.547648\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1.12461	0.0378891	0.0189446	−	0.999821i	$-0.493969\pi$
$881$	1.12461	0.0378891	0.0189446	+	0.999821i	$0.493969\pi$
$882$	0	0
$883$	−12.9443	−0.435609	−0.217805	−	0.975992i	$-0.569890\pi$
$883$	−12.9443	−0.435609	−0.217805	+	0.975992i	$0.569890\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	30.4721	1.02315	0.511577	−	0.859237i	$-0.329061\pi$
$887$	30.4721	1.02315	0.511577	+	0.859237i	$0.329061\pi$
$888$	0	0
$889$	−17.8885	−0.599963
$890$	0	0
$891$	0	0
$892$	0	0
$893$	3.77709	0.126395
$894$	0	0
$895$	−0.583592	−0.0195073
$896$	0	0
$897$	0	0
$898$	0	0
$899$	3.77709	0.125973
$900$	0	0
$901$	6.11146	0.203602
$902$	0	0
$903$	0	0
$904$	0	0
$905$	27.4164	0.911352
$906$	0	0
$907$	12.9443	0.429807	0.214904	−	0.976635i	$-0.431056\pi$
$907$	12.9443	0.429807	0.214904	+	0.976635i	$0.431056\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	25.1246	0.832416	0.416208	−	0.909270i	$-0.363359\pi$
$911$	25.1246	0.832416	0.416208	+	0.909270i	$0.363359\pi$
$912$	0	0
$913$	−41.8885	−1.38631
$914$	0	0
$915$	0	0
$916$	0	0
$917$	16.0000	0.528367
$918$	0	0
$919$	−44.9443	−1.48257	−0.741287	−	0.671188i	$-0.765784\pi$
$919$	−44.9443	−1.48257	−0.741287	+	0.671188i	$0.765784\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	32.3607	1.06516
$924$	0	0
$925$	−24.4721	−0.804639
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−55.2361	−1.81224	−0.906118	−	0.423024i	$-0.860968\pi$
$929$	−55.2361	−1.81224	−0.906118	+	0.423024i	$0.860968\pi$
$930$	0	0
$931$	2.47214	0.0810210
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−8.00000	−0.261628
$936$	0	0
$937$	14.9443	0.488208	0.244104	−	0.969749i	$-0.421506\pi$
$937$	14.9443	0.488208	0.244104	+	0.969749i	$0.421506\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	50.4296	1.64396	0.821978	−	0.569519i	$-0.192870\pi$
$941$	50.4296	1.64396	0.821978	+	0.569519i	$0.192870\pi$
$942$	0	0
$943$	−41.3050	−1.34507
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−29.1246	−0.946423	−0.473211	−	0.880949i	$-0.656905\pi$
$947$	−29.1246	−0.946423	−0.473211	+	0.880949i	$0.656905\pi$
$948$	0	0
$949$	−66.8328	−2.16949
$950$	0	0
$951$	0	0
$952$	0	0
$953$	19.7771	0.640643	0.320321	−	0.947309i	$-0.396209\pi$
$953$	19.7771	0.640643	0.320321	+	0.947309i	$0.396209\pi$
$954$	0	0
$955$	86.2492	2.79096
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−1.52786	−0.0493373
$960$	0	0
$961$	−24.8885	−0.802856
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−40.3607	−1.29926
$966$	0	0
$967$	−11.7771	−0.378726	−0.189363	−	0.981907i	$-0.560642\pi$
$967$	−11.7771	−0.378726	−0.189363	+	0.981907i	$0.560642\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−9.88854	−0.317338	−0.158669	−	0.987332i	$-0.550720\pi$
$971$	−9.88854	−0.317338	−0.158669	+	0.987332i	$0.550720\pi$
$972$	0	0
$973$	4.94427	0.158506
$974$	0	0
$975$	0	0
$976$	0	0
$977$	50.2492	1.60762	0.803808	−	0.594889i	$-0.202804\pi$
$977$	50.2492	1.60762	0.803808	+	0.594889i	$0.202804\pi$
$978$	0	0
$979$	7.41641	0.237029
$980$	0	0
$981$	0	0
$982$	0	0
$983$	12.9443	0.412858	0.206429	−	0.978462i	$-0.433816\pi$
$983$	12.9443	0.412858	0.206429	+	0.978462i	$0.433816\pi$
$984$	0	0
$985$	16.0000	0.509802
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−73.8885	−2.34952
$990$	0	0
$991$	−9.16718	−0.291205	−0.145603	−	0.989343i	$-0.546512\pi$
$991$	−9.16718	−0.291205	−0.145603	+	0.989343i	$0.546512\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−25.8885	−0.820722
$996$	0	0
$997$	50.3607	1.59494	0.797469	−	0.603359i	$-0.206172\pi$
$997$	50.3607	1.59494	0.797469	+	0.603359i	$0.206172\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2016.2.a.q.1.1	yes	2
3.2	odd	2		2016.2.a.v.1.2	yes	2
4.3	odd	2		2016.2.a.p.1.1	✓	2
8.3	odd	2		4032.2.a.bu.1.2		2
8.5	even	2		4032.2.a.bx.1.2		2
12.11	even	2		2016.2.a.u.1.2	yes	2
24.5	odd	2		4032.2.a.bp.1.1		2
24.11	even	2		4032.2.a.bo.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2016.2.a.p.1.1	✓	2	4.3	odd	2
2016.2.a.q.1.1	yes	2	1.1	even	1	trivial
2016.2.a.u.1.2	yes	2	12.11	even	2
2016.2.a.v.1.2	yes	2	3.2	odd	2
4032.2.a.bo.1.1		2	24.11	even	2
4032.2.a.bp.1.1		2	24.5	odd	2
4032.2.a.bu.1.2		2	8.3	odd	2
4032.2.a.bx.1.2		2	8.5	even	2

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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:	\( N \)	\(=\)	\( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2016.a (trivial)

\(f(q)\)	\(=\)	\(q-3.23607 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q-3.23607 q^{5} +1.00000 q^{7} -3.23607 q^{11} +4.47214 q^{13} -0.763932 q^{17} +2.47214 q^{19} +5.70820 q^{23} +5.47214 q^{25} -1.52786 q^{29} -2.47214 q^{31} -3.23607 q^{35} -4.47214 q^{37} -7.23607 q^{41} -12.9443 q^{43} +1.52786 q^{47} +1.00000 q^{49} -8.00000 q^{53} +10.4721 q^{55} -6.47214 q^{59} -4.47214 q^{61} -14.4721 q^{65} -8.00000 q^{67} +7.23607 q^{71} -14.9443 q^{73} -3.23607 q^{77} +12.9443 q^{79} +12.9443 q^{83} +2.47214 q^{85} -2.29180 q^{89} +4.47214 q^{91} -8.00000 q^{95} -6.94427 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 2 q^{7}+O(q^{10}) \) 2 * q - 2 * q^5 + 2 * q^7	\( 2 q - 2 q^{5} + 2 q^{7} - 2 q^{11} - 6 q^{17} - 4 q^{19} - 2 q^{23} + 2 q^{25} - 12 q^{29} + 4 q^{31} - 2 q^{35} - 10 q^{41} - 8 q^{43} + 12 q^{47} + 2 q^{49} - 16 q^{53} + 12 q^{55} - 4 q^{59} - 20 q^{65} - 16 q^{67} + 10 q^{71} - 12 q^{73} - 2 q^{77} + 8 q^{79} + 8 q^{83} - 4 q^{85} - 18 q^{89} - 16 q^{95} + 4 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 + 2 * q^7 - 2 * q^11 - 6 * q^17 - 4 * q^19 - 2 * q^23 + 2 * q^25 - 12 * q^29 + 4 * q^31 - 2 * q^35 - 10 * q^41 - 8 * q^43 + 12 * q^47 + 2 * q^49 - 16 * q^53 + 12 * q^55 - 4 * q^59 - 20 * q^65 - 16 * q^67 + 10 * q^71 - 12 * q^73 - 2 * q^77 + 8 * q^79 + 8 * q^83 - 4 * q^85 - 18 * q^89 - 16 * q^95 + 4 * q^97

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