LMFDB - Embedded newform 8016.2.a.x.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8016, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$64.0080822603$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 8x^{6} + 28x^{5} + 9x^{4} - 64x^{3} + 17x^{2} + 23x + 1 $ x^8 - 3x^7 - 8x^6 + 28x^5 + 9x^4 - 64x^3 + 17x^2 + 23*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 501)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$2.18779$ of defining polynomial
Character	$\chi$	$=$	8016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +2.52133 q^{5} +0.307143 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +2.52133 q^{5} +0.307143 q^{7} +1.00000 q^{9} +3.21310 q^{11} -3.88493 q^{13} -2.52133 q^{15} -4.06620 q^{17} +1.71817 q^{19} -0.307143 q^{21} -2.23889 q^{23} +1.35709 q^{25} -1.00000 q^{27} +2.71773 q^{29} -9.46958 q^{31} -3.21310 q^{33} +0.774408 q^{35} +8.47601 q^{37} +3.88493 q^{39} +1.17955 q^{41} +0.594516 q^{43} +2.52133 q^{45} -6.00069 q^{47} -6.90566 q^{49} +4.06620 q^{51} -5.37574 q^{53} +8.10129 q^{55} -1.71817 q^{57} -4.70122 q^{59} +5.56117 q^{61} +0.307143 q^{63} -9.79517 q^{65} +3.54307 q^{67} +2.23889 q^{69} -4.50819 q^{71} -8.22570 q^{73} -1.35709 q^{75} +0.986883 q^{77} -11.7396 q^{79} +1.00000 q^{81} -8.83065 q^{83} -10.2522 q^{85} -2.71773 q^{87} -8.38309 q^{89} -1.19323 q^{91} +9.46958 q^{93} +4.33207 q^{95} -6.81662 q^{97} +3.21310 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{3} + 7 q^{5} + 4 q^{7} + 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^3 + 7 * q^5 + 4 * q^7 + 8 * q^9	$ 8 q - 8 q^{3} + 7 q^{5} + 4 q^{7} + 8 q^{9} - 13 q^{11} - 7 q^{15} + 11 q^{17} - 12 q^{19} - 4 q^{21} - 7 q^{23} - 5 q^{25} - 8 q^{27} + q^{29} + 2 q^{31} + 13 q^{33} + 4 q^{35} - 9 q^{37} + 4 q^{41} - 2 q^{43} + 7 q^{45} - 17 q^{47} - 2 q^{49} - 11 q^{51} + 9 q^{53} - 7 q^{55} + 12 q^{57} - 29 q^{59} - 12 q^{61} + 4 q^{63} + 8 q^{65} + 7 q^{69} - 13 q^{71} - 20 q^{73} + 5 q^{75} - 22 q^{77} - 8 q^{79} + 8 q^{81} - 33 q^{83} - 31 q^{85} - q^{87} + 4 q^{89} - q^{91} - 2 q^{93} - 3 q^{95} - 31 q^{97} - 13 q^{99}+O(q^{100}) $ 8 * q - 8 * q^3 + 7 * q^5 + 4 * q^7 + 8 * q^9 - 13 * q^11 - 7 * q^15 + 11 * q^17 - 12 * q^19 - 4 * q^21 - 7 * q^23 - 5 * q^25 - 8 * q^27 + q^29 + 2 * q^31 + 13 * q^33 + 4 * q^35 - 9 * q^37 + 4 * q^41 - 2 * q^43 + 7 * q^45 - 17 * q^47 - 2 * q^49 - 11 * q^51 + 9 * q^53 - 7 * q^55 + 12 * q^57 - 29 * q^59 - 12 * q^61 + 4 * q^63 + 8 * q^65 + 7 * q^69 - 13 * q^71 - 20 * q^73 + 5 * q^75 - 22 * q^77 - 8 * q^79 + 8 * q^81 - 33 * q^83 - 31 * q^85 - q^87 + 4 * q^89 - q^91 - 2 * q^93 - 3 * q^95 - 31 * q^97 - 13 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	2.52133	1.12757	0.563786	−	0.825921i	$-0.309344\pi$
$5$	2.52133	1.12757	0.563786	+	0.825921i	$0.309344\pi$
$6$	0	0
$7$	0.307143	0.116089	0.0580446	−	0.998314i	$-0.481513\pi$
$7$	0.307143	0.116089	0.0580446	+	0.998314i	$0.481513\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.21310	0.968787	0.484394	−	0.874850i	$-0.339040\pi$
$11$	3.21310	0.968787	0.484394	+	0.874850i	$0.339040\pi$
$12$	0	0
$13$	−3.88493	−1.07748	−0.538742	−	0.842471i	$-0.681100\pi$
$13$	−3.88493	−1.07748	−0.538742	+	0.842471i	$0.681100\pi$
$14$	0	0
$15$	−2.52133	−0.651004
$16$	0	0
$17$	−4.06620	−0.986199	−0.493100	−	0.869973i	$-0.664136\pi$
$17$	−4.06620	−0.986199	−0.493100	+	0.869973i	$0.664136\pi$
$18$	0	0
$19$	1.71817	0.394176	0.197088	−	0.980386i	$-0.436852\pi$
$19$	1.71817	0.394176	0.197088	+	0.980386i	$0.436852\pi$
$20$	0	0
$21$	−0.307143	−0.0670242
$22$	0	0
$23$	−2.23889	−0.466842	−0.233421	−	0.972376i	$-0.574992\pi$
$23$	−2.23889	−0.466842	−0.233421	+	0.972376i	$0.574992\pi$
$24$	0	0
$25$	1.35709	0.271418
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	2.71773	0.504670	0.252335	−	0.967640i	$-0.418802\pi$
$29$	2.71773	0.504670	0.252335	+	0.967640i	$0.418802\pi$
$30$	0	0
$31$	−9.46958	−1.70079	−0.850393	−	0.526147i	$-0.823636\pi$
$31$	−9.46958	−1.70079	−0.850393	+	0.526147i	$0.823636\pi$
$32$	0	0
$33$	−3.21310	−0.559330
$34$	0	0
$35$	0.774408	0.130899
$36$	0	0
$37$	8.47601	1.39345	0.696724	−	0.717340i	$-0.254640\pi$
$37$	8.47601	1.39345	0.696724	+	0.717340i	$0.254640\pi$
$38$	0	0
$39$	3.88493	0.622086
$40$	0	0
$41$	1.17955	0.184214	0.0921072	−	0.995749i	$-0.470640\pi$
$41$	1.17955	0.184214	0.0921072	+	0.995749i	$0.470640\pi$
$42$	0	0
$43$	0.594516	0.0906628	0.0453314	−	0.998972i	$-0.485566\pi$
$43$	0.594516	0.0906628	0.0453314	+	0.998972i	$0.485566\pi$
$44$	0	0
$45$	2.52133	0.375857
$46$	0	0
$47$	−6.00069	−0.875290	−0.437645	−	0.899148i	$-0.644187\pi$
$47$	−6.00069	−0.875290	−0.437645	+	0.899148i	$0.644187\pi$
$48$	0	0
$49$	−6.90566	−0.986523
$50$	0	0
$51$	4.06620	0.569382
$52$	0	0
$53$	−5.37574	−0.738415	−0.369207	−	0.929347i	$-0.620371\pi$
$53$	−5.37574	−0.738415	−0.369207	+	0.929347i	$0.620371\pi$
$54$	0	0
$55$	8.10129	1.09238
$56$	0	0
$57$	−1.71817	−0.227577
$58$	0	0
$59$	−4.70122	−0.612047	−0.306023	−	0.952024i	$-0.598999\pi$
$59$	−4.70122	−0.612047	−0.306023	+	0.952024i	$0.598999\pi$
$60$	0	0
$61$	5.56117	0.712035	0.356017	−	0.934479i	$-0.384134\pi$
$61$	5.56117	0.712035	0.356017	+	0.934479i	$0.384134\pi$
$62$	0	0
$63$	0.307143	0.0386964
$64$	0	0
$65$	−9.79517	−1.21494
$66$	0	0
$67$	3.54307	0.432855	0.216428	−	0.976299i	$-0.430559\pi$
$67$	3.54307	0.432855	0.216428	+	0.976299i	$0.430559\pi$
$68$	0	0
$69$	2.23889	0.269531
$70$	0	0
$71$	−4.50819	−0.535023	−0.267512	−	0.963555i	$-0.586201\pi$
$71$	−4.50819	−0.535023	−0.267512	+	0.963555i	$0.586201\pi$
$72$	0	0
$73$	−8.22570	−0.962745	−0.481372	−	0.876516i	$-0.659862\pi$
$73$	−8.22570	−0.962745	−0.481372	+	0.876516i	$0.659862\pi$
$74$	0	0
$75$	−1.35709	−0.156703
$76$	0	0
$77$	0.986883	0.112466
$78$	0	0
$79$	−11.7396	−1.32081	−0.660404	−	0.750910i	$-0.729615\pi$
$79$	−11.7396	−1.32081	−0.660404	+	0.750910i	$0.729615\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−8.83065	−0.969289	−0.484645	−	0.874711i	$-0.661051\pi$
$83$	−8.83065	−0.969289	−0.484645	+	0.874711i	$0.661051\pi$
$84$	0	0
$85$	−10.2522	−1.11201
$86$	0	0
$87$	−2.71773	−0.291371
$88$	0	0
$89$	−8.38309	−0.888606	−0.444303	−	0.895877i	$-0.646549\pi$
$89$	−8.38309	−0.888606	−0.444303	+	0.895877i	$0.646549\pi$
$90$	0	0
$91$	−1.19323	−0.125084
$92$	0	0
$93$	9.46958	0.981950
$94$	0	0
$95$	4.33207	0.444461
$96$	0	0
$97$	−6.81662	−0.692123	−0.346061	−	0.938212i	$-0.612481\pi$
$97$	−6.81662	−0.692123	−0.346061	+	0.938212i	$0.612481\pi$
$98$	0	0
$99$	3.21310	0.322929
$100$	0	0
$101$	−12.5763	−1.25139	−0.625695	−	0.780068i	$-0.715185\pi$
$101$	−12.5763	−1.25139	−0.625695	+	0.780068i	$0.715185\pi$
$102$	0	0
$103$	12.7095	1.25231	0.626154	−	0.779699i	$-0.284628\pi$
$103$	12.7095	1.25231	0.626154	+	0.779699i	$0.284628\pi$
$104$	0	0
$105$	−0.774408	−0.0755745
$106$	0	0
$107$	−8.47742	−0.819543	−0.409771	−	0.912188i	$-0.634392\pi$
$107$	−8.47742	−0.819543	−0.409771	+	0.912188i	$0.634392\pi$
$108$	0	0
$109$	−5.48801	−0.525656	−0.262828	−	0.964843i	$-0.584655\pi$
$109$	−5.48801	−0.525656	−0.262828	+	0.964843i	$0.584655\pi$
$110$	0	0
$111$	−8.47601	−0.804507
$112$	0	0
$113$	15.9908	1.50428	0.752142	−	0.659001i	$-0.229021\pi$
$113$	15.9908	1.50428	0.752142	+	0.659001i	$0.229021\pi$
$114$	0	0
$115$	−5.64498	−0.526397
$116$	0	0
$117$	−3.88493	−0.359161
$118$	0	0
$119$	−1.24891	−0.114487
$120$	0	0
$121$	−0.675960	−0.0614509
$122$	0	0
$123$	−1.17955	−0.106356
$124$	0	0
$125$	−9.18497	−0.821529
$126$	0	0
$127$	15.0090	1.33184	0.665919	−	0.746024i	$-0.268040\pi$
$127$	15.0090	1.33184	0.665919	+	0.746024i	$0.268040\pi$
$128$	0	0
$129$	−0.594516	−0.0523442
$130$	0	0
$131$	−7.63090	−0.666715	−0.333357	−	0.942801i	$-0.608182\pi$
$131$	−7.63090	−0.666715	−0.333357	+	0.942801i	$0.608182\pi$
$132$	0	0
$133$	0.527725	0.0457596
$134$	0	0
$135$	−2.52133	−0.217001
$136$	0	0
$137$	10.4587	0.893549	0.446774	−	0.894647i	$-0.352573\pi$
$137$	10.4587	0.893549	0.446774	+	0.894647i	$0.352573\pi$
$138$	0	0
$139$	−16.6847	−1.41517	−0.707587	−	0.706626i	$-0.750216\pi$
$139$	−16.6847	−1.41517	−0.707587	+	0.706626i	$0.750216\pi$
$140$	0	0
$141$	6.00069	0.505349
$142$	0	0
$143$	−12.4827	−1.04385
$144$	0	0
$145$	6.85229	0.569051
$146$	0	0
$147$	6.90566	0.569569
$148$	0	0
$149$	13.6676	1.11969	0.559845	−	0.828597i	$-0.310861\pi$
$149$	13.6676	1.11969	0.559845	+	0.828597i	$0.310861\pi$
$150$	0	0
$151$	20.4423	1.66357	0.831785	−	0.555098i	$-0.187319\pi$
$151$	20.4423	1.66357	0.831785	+	0.555098i	$0.187319\pi$
$152$	0	0
$153$	−4.06620	−0.328733
$154$	0	0
$155$	−23.8759	−1.91776
$156$	0	0
$157$	−5.99947	−0.478810	−0.239405	−	0.970920i	$-0.576952\pi$
$157$	−5.99947	−0.478810	−0.239405	+	0.970920i	$0.576952\pi$
$158$	0	0
$159$	5.37574	0.426324
$160$	0	0
$161$	−0.687661	−0.0541953
$162$	0	0
$163$	12.4538	0.975455	0.487728	−	0.872996i	$-0.337826\pi$
$163$	12.4538	0.975455	0.487728	+	0.872996i	$0.337826\pi$
$164$	0	0
$165$	−8.10129	−0.630684
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	2.09265	0.160973
$170$	0	0
$171$	1.71817	0.131392
$172$	0	0
$173$	12.7594	0.970080	0.485040	−	0.874492i	$-0.338805\pi$
$173$	12.7594	0.970080	0.485040	+	0.874492i	$0.338805\pi$
$174$	0	0
$175$	0.416820	0.0315087
$176$	0	0
$177$	4.70122	0.353365
$178$	0	0
$179$	−6.10595	−0.456380	−0.228190	−	0.973617i	$-0.573281\pi$
$179$	−6.10595	−0.456380	−0.228190	+	0.973617i	$0.573281\pi$
$180$	0	0
$181$	11.1343	0.827604	0.413802	−	0.910367i	$-0.364201\pi$
$181$	11.1343	0.827604	0.413802	+	0.910367i	$0.364201\pi$
$182$	0	0
$183$	−5.56117	−0.411093
$184$	0	0
$185$	21.3708	1.57121
$186$	0	0
$187$	−13.0651	−0.955417
$188$	0	0
$189$	−0.307143	−0.0223414
$190$	0	0
$191$	−0.577690	−0.0418002	−0.0209001	−	0.999782i	$-0.506653\pi$
$191$	−0.577690	−0.0418002	−0.0209001	+	0.999782i	$0.506653\pi$
$192$	0	0
$193$	−14.3256	−1.03118	−0.515590	−	0.856835i	$-0.672427\pi$
$193$	−14.3256	−1.03118	−0.515590	+	0.856835i	$0.672427\pi$
$194$	0	0
$195$	9.79517	0.701446
$196$	0	0
$197$	−12.7487	−0.908306	−0.454153	−	0.890924i	$-0.650058\pi$
$197$	−12.7487	−0.908306	−0.454153	+	0.890924i	$0.650058\pi$
$198$	0	0
$199$	−23.1945	−1.64422	−0.822108	−	0.569332i	$-0.807202\pi$
$199$	−23.1945	−1.64422	−0.822108	+	0.569332i	$0.807202\pi$
$200$	0	0
$201$	−3.54307	−0.249909
$202$	0	0
$203$	0.834733	0.0585867
$204$	0	0
$205$	2.97402	0.207715
$206$	0	0
$207$	−2.23889	−0.155614
$208$	0	0
$209$	5.52067	0.381872
$210$	0	0
$211$	19.2467	1.32500	0.662500	−	0.749062i	$-0.269496\pi$
$211$	19.2467	1.32500	0.662500	+	0.749062i	$0.269496\pi$
$212$	0	0
$213$	4.50819	0.308896
$214$	0	0
$215$	1.49897	0.102229
$216$	0	0
$217$	−2.90852	−0.197443
$218$	0	0
$219$	8.22570	0.555841
$220$	0	0
$221$	15.7969	1.06261
$222$	0	0
$223$	14.4731	0.969193	0.484596	−	0.874738i	$-0.338966\pi$
$223$	14.4731	0.969193	0.484596	+	0.874738i	$0.338966\pi$
$224$	0	0
$225$	1.35709	0.0904725
$226$	0	0
$227$	−2.81741	−0.186998	−0.0934990	−	0.995619i	$-0.529805\pi$
$227$	−2.81741	−0.186998	−0.0934990	+	0.995619i	$0.529805\pi$
$228$	0	0
$229$	−11.0231	−0.728429	−0.364215	−	0.931315i	$-0.618663\pi$
$229$	−11.0231	−0.728429	−0.364215	+	0.931315i	$0.618663\pi$
$230$	0	0
$231$	−0.986883	−0.0649322
$232$	0	0
$233$	5.84586	0.382975	0.191488	−	0.981495i	$-0.438669\pi$
$233$	5.84586	0.382975	0.191488	+	0.981495i	$0.438669\pi$
$234$	0	0
$235$	−15.1297	−0.986952
$236$	0	0
$237$	11.7396	0.762569
$238$	0	0
$239$	−12.3249	−0.797231	−0.398615	−	0.917118i	$-0.630509\pi$
$239$	−12.3249	−0.797231	−0.398615	+	0.917118i	$0.630509\pi$
$240$	0	0
$241$	−19.9006	−1.28191	−0.640954	−	0.767579i	$-0.721461\pi$
$241$	−19.9006	−1.28191	−0.640954	+	0.767579i	$0.721461\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−17.4114	−1.11238
$246$	0	0
$247$	−6.67497	−0.424718
$248$	0	0
$249$	8.83065	0.559620
$250$	0	0
$251$	−9.32458	−0.588562	−0.294281	−	0.955719i	$-0.595080\pi$
$251$	−9.32458	−0.588562	−0.294281	+	0.955719i	$0.595080\pi$
$252$	0	0
$253$	−7.19380	−0.452270
$254$	0	0
$255$	10.2522	0.642019
$256$	0	0
$257$	23.9424	1.49348	0.746742	−	0.665113i	$-0.231617\pi$
$257$	23.9424	1.49348	0.746742	+	0.665113i	$0.231617\pi$
$258$	0	0
$259$	2.60335	0.161764
$260$	0	0
$261$	2.71773	0.168223
$262$	0	0
$263$	4.09084	0.252252	0.126126	−	0.992014i	$-0.459746\pi$
$263$	4.09084	0.252252	0.126126	+	0.992014i	$0.459746\pi$
$264$	0	0
$265$	−13.5540	−0.832616
$266$	0	0
$267$	8.38309	0.513037
$268$	0	0
$269$	12.3801	0.754829	0.377414	−	0.926044i	$-0.376813\pi$
$269$	12.3801	0.754829	0.377414	+	0.926044i	$0.376813\pi$
$270$	0	0
$271$	−9.11169	−0.553496	−0.276748	−	0.960943i	$-0.589257\pi$
$271$	−9.11169	−0.553496	−0.276748	+	0.960943i	$0.589257\pi$
$272$	0	0
$273$	1.19323	0.0722175
$274$	0	0
$275$	4.36047	0.262946
$276$	0	0
$277$	9.85742	0.592275	0.296138	−	0.955145i	$-0.404301\pi$
$277$	9.85742	0.592275	0.296138	+	0.955145i	$0.404301\pi$
$278$	0	0
$279$	−9.46958	−0.566929
$280$	0	0
$281$	−12.4231	−0.741103	−0.370551	−	0.928812i	$-0.620831\pi$
$281$	−12.4231	−0.741103	−0.370551	+	0.928812i	$0.620831\pi$
$282$	0	0
$283$	29.7611	1.76911	0.884557	−	0.466433i	$-0.154461\pi$
$283$	29.7611	1.76911	0.884557	+	0.466433i	$0.154461\pi$
$284$	0	0
$285$	−4.33207	−0.256610
$286$	0	0
$287$	0.362290	0.0213853
$288$	0	0
$289$	−0.465985	−0.0274109
$290$	0	0
$291$	6.81662	0.399597
$292$	0	0
$293$	−31.8879	−1.86291	−0.931455	−	0.363855i	$-0.881460\pi$
$293$	−31.8879	−1.86291	−0.931455	+	0.363855i	$0.881460\pi$
$294$	0	0
$295$	−11.8533	−0.690126
$296$	0	0
$297$	−3.21310	−0.186443
$298$	0	0
$299$	8.69794	0.503015
$300$	0	0
$301$	0.182601	0.0105250
$302$	0	0
$303$	12.5763	0.722491
$304$	0	0
$305$	14.0215	0.802870
$306$	0	0
$307$	−24.9323	−1.42296	−0.711480	−	0.702706i	$-0.751975\pi$
$307$	−24.9323	−1.42296	−0.711480	+	0.702706i	$0.751975\pi$
$308$	0	0
$309$	−12.7095	−0.723021
$310$	0	0
$311$	7.58605	0.430165	0.215083	−	0.976596i	$-0.430998\pi$
$311$	7.58605	0.430165	0.215083	+	0.976596i	$0.430998\pi$
$312$	0	0
$313$	33.9727	1.92025	0.960126	−	0.279567i	$-0.0901910\pi$
$313$	33.9727	1.92025	0.960126	+	0.279567i	$0.0901910\pi$
$314$	0	0
$315$	0.774408	0.0436330
$316$	0	0
$317$	−9.26715	−0.520495	−0.260247	−	0.965542i	$-0.583804\pi$
$317$	−9.26715	−0.520495	−0.260247	+	0.965542i	$0.583804\pi$
$318$	0	0
$319$	8.73235	0.488918
$320$	0	0
$321$	8.47742	0.473163
$322$	0	0
$323$	−6.98644	−0.388736
$324$	0	0
$325$	−5.27219	−0.292448
$326$	0	0
$327$	5.48801	0.303488
$328$	0	0
$329$	−1.84307	−0.101612
$330$	0	0
$331$	−7.47767	−0.411010	−0.205505	−	0.978656i	$-0.565884\pi$
$331$	−7.47767	−0.411010	−0.205505	+	0.978656i	$0.565884\pi$
$332$	0	0
$333$	8.47601	0.464482
$334$	0	0
$335$	8.93324	0.488075
$336$	0	0
$337$	−10.7934	−0.587953	−0.293977	−	0.955813i	$-0.594979\pi$
$337$	−10.7934	−0.587953	−0.293977	+	0.955813i	$0.594979\pi$
$338$	0	0
$339$	−15.9908	−0.868499
$340$	0	0
$341$	−30.4267	−1.64770
$342$	0	0
$343$	−4.27103	−0.230614
$344$	0	0
$345$	5.64498	0.303916
$346$	0	0
$347$	25.3570	1.36124	0.680619	−	0.732638i	$-0.261711\pi$
$347$	25.3570	1.36124	0.680619	+	0.732638i	$0.261711\pi$
$348$	0	0
$349$	22.3397	1.19582	0.597909	−	0.801564i	$-0.295998\pi$
$349$	22.3397	1.19582	0.597909	+	0.801564i	$0.295998\pi$
$350$	0	0
$351$	3.88493	0.207362
$352$	0	0
$353$	12.5690	0.668979	0.334490	−	0.942399i	$-0.391436\pi$
$353$	12.5690	0.668979	0.334490	+	0.942399i	$0.391436\pi$
$354$	0	0
$355$	−11.3666	−0.603277
$356$	0	0
$357$	1.24891	0.0660992
$358$	0	0
$359$	−12.8341	−0.677356	−0.338678	−	0.940902i	$-0.609980\pi$
$359$	−12.8341	−0.677356	−0.338678	+	0.940902i	$0.609980\pi$
$360$	0	0
$361$	−16.0479	−0.844626
$362$	0	0
$363$	0.675960	0.0354787
$364$	0	0
$365$	−20.7397	−1.08556
$366$	0	0
$367$	−36.0617	−1.88240	−0.941202	−	0.337845i	$-0.890302\pi$
$367$	−36.0617	−1.88240	−0.941202	+	0.337845i	$0.890302\pi$
$368$	0	0
$369$	1.17955	0.0614048
$370$	0	0
$371$	−1.65112	−0.0857220
$372$	0	0
$373$	17.5507	0.908739	0.454370	−	0.890813i	$-0.349865\pi$
$373$	17.5507	0.908739	0.454370	+	0.890813i	$0.349865\pi$
$374$	0	0
$375$	9.18497	0.474310
$376$	0	0
$377$	−10.5582	−0.543774
$378$	0	0
$379$	−13.3358	−0.685015	−0.342507	−	0.939515i	$-0.611276\pi$
$379$	−13.3358	−0.685015	−0.342507	+	0.939515i	$0.611276\pi$
$380$	0	0
$381$	−15.0090	−0.768937
$382$	0	0
$383$	−15.5076	−0.792401	−0.396201	−	0.918164i	$-0.629672\pi$
$383$	−15.5076	−0.792401	−0.396201	+	0.918164i	$0.629672\pi$
$384$	0	0
$385$	2.48826	0.126813
$386$	0	0
$387$	0.594516	0.0302209
$388$	0	0
$389$	−27.9691	−1.41809	−0.709044	−	0.705165i	$-0.750873\pi$
$389$	−27.9691	−1.41809	−0.709044	+	0.705165i	$0.750873\pi$
$390$	0	0
$391$	9.10380	0.460399
$392$	0	0
$393$	7.63090	0.384928
$394$	0	0
$395$	−29.5994	−1.48931
$396$	0	0
$397$	16.2421	0.815168	0.407584	−	0.913168i	$-0.366371\pi$
$397$	16.2421	0.815168	0.407584	+	0.913168i	$0.366371\pi$
$398$	0	0
$399$	−0.527725	−0.0264193
$400$	0	0
$401$	3.93672	0.196590	0.0982951	−	0.995157i	$-0.468661\pi$
$401$	3.93672	0.196590	0.0982951	+	0.995157i	$0.468661\pi$
$402$	0	0
$403$	36.7886	1.83257
$404$	0	0
$405$	2.52133	0.125286
$406$	0	0
$407$	27.2343	1.34995
$408$	0	0
$409$	−8.32271	−0.411531	−0.205766	−	0.978601i	$-0.565968\pi$
$409$	−8.32271	−0.411531	−0.205766	+	0.978601i	$0.565968\pi$
$410$	0	0
$411$	−10.4587	−0.515890
$412$	0	0
$413$	−1.44395	−0.0710520
$414$	0	0
$415$	−22.2649	−1.09294
$416$	0	0
$417$	16.6847	0.817051
$418$	0	0
$419$	−9.46746	−0.462515	−0.231258	−	0.972893i	$-0.574284\pi$
$419$	−9.46746	−0.462515	−0.231258	+	0.972893i	$0.574284\pi$
$420$	0	0
$421$	−29.7742	−1.45111	−0.725553	−	0.688167i	$-0.758416\pi$
$421$	−29.7742	−1.45111	−0.725553	+	0.688167i	$0.758416\pi$
$422$	0	0
$423$	−6.00069	−0.291763
$424$	0	0
$425$	−5.51820	−0.267672
$426$	0	0
$427$	1.70808	0.0826596
$428$	0	0
$429$	12.4827	0.602669
$430$	0	0
$431$	28.0578	1.35150	0.675748	−	0.737133i	$-0.263821\pi$
$431$	28.0578	1.35150	0.675748	+	0.737133i	$0.263821\pi$
$432$	0	0
$433$	1.56264	0.0750957	0.0375478	−	0.999295i	$-0.488045\pi$
$433$	1.56264	0.0750957	0.0375478	+	0.999295i	$0.488045\pi$
$434$	0	0
$435$	−6.85229	−0.328542
$436$	0	0
$437$	−3.84680	−0.184018
$438$	0	0
$439$	−22.1973	−1.05942	−0.529709	−	0.848180i	$-0.677699\pi$
$439$	−22.1973	−1.05942	−0.529709	+	0.848180i	$0.677699\pi$
$440$	0	0
$441$	−6.90566	−0.328841
$442$	0	0
$443$	−4.25379	−0.202104	−0.101052	−	0.994881i	$-0.532221\pi$
$443$	−4.25379	−0.202104	−0.101052	+	0.994881i	$0.532221\pi$
$444$	0	0
$445$	−21.1365	−1.00197
$446$	0	0
$447$	−13.6676	−0.646453
$448$	0	0
$449$	8.72616	0.411813	0.205906	−	0.978572i	$-0.433986\pi$
$449$	8.72616	0.411813	0.205906	+	0.978572i	$0.433986\pi$
$450$	0	0
$451$	3.79001	0.178465
$452$	0	0
$453$	−20.4423	−0.960463
$454$	0	0
$455$	−3.00852	−0.141042
$456$	0	0
$457$	−24.0524	−1.12512	−0.562562	−	0.826755i	$-0.690184\pi$
$457$	−24.0524	−1.12512	−0.562562	+	0.826755i	$0.690184\pi$
$458$	0	0
$459$	4.06620	0.189794
$460$	0	0
$461$	29.7476	1.38548	0.692742	−	0.721186i	$-0.256403\pi$
$461$	29.7476	1.38548	0.692742	+	0.721186i	$0.256403\pi$
$462$	0	0
$463$	36.9830	1.71874	0.859372	−	0.511351i	$-0.170855\pi$
$463$	36.9830	1.71874	0.859372	+	0.511351i	$0.170855\pi$
$464$	0	0
$465$	23.8759	1.10722
$466$	0	0
$467$	−3.94329	−0.182474	−0.0912369	−	0.995829i	$-0.529082\pi$
$467$	−3.94329	−0.182474	−0.0912369	+	0.995829i	$0.529082\pi$
$468$	0	0
$469$	1.08823	0.0502498
$470$	0	0
$471$	5.99947	0.276441
$472$	0	0
$473$	1.91024	0.0878330
$474$	0	0
$475$	2.33171	0.106986
$476$	0	0
$477$	−5.37574	−0.246138
$478$	0	0
$479$	−6.73592	−0.307772	−0.153886	−	0.988089i	$-0.549179\pi$
$479$	−6.73592	−0.307772	−0.153886	+	0.988089i	$0.549179\pi$
$480$	0	0
$481$	−32.9287	−1.50142
$482$	0	0
$483$	0.687661	0.0312897
$484$	0	0
$485$	−17.1869	−0.780418
$486$	0	0
$487$	29.9495	1.35714	0.678571	−	0.734534i	$-0.262599\pi$
$487$	29.9495	1.35714	0.678571	+	0.734534i	$0.262599\pi$
$488$	0	0
$489$	−12.4538	−0.563179
$490$	0	0
$491$	42.0610	1.89819	0.949094	−	0.314992i	$-0.102002\pi$
$491$	42.0610	1.89819	0.949094	+	0.314992i	$0.102002\pi$
$492$	0	0
$493$	−11.0508	−0.497705
$494$	0	0
$495$	8.10129	0.364126
$496$	0	0
$497$	−1.38466	−0.0621105
$498$	0	0
$499$	−18.5374	−0.829848	−0.414924	−	0.909856i	$-0.636192\pi$
$499$	−18.5374	−0.829848	−0.414924	+	0.909856i	$0.636192\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	−3.82345	−0.170479	−0.0852397	−	0.996360i	$-0.527166\pi$
$503$	−3.82345	−0.170479	−0.0852397	+	0.996360i	$0.527166\pi$
$504$	0	0
$505$	−31.7090	−1.41103
$506$	0	0
$507$	−2.09265	−0.0929377
$508$	0	0
$509$	−20.7233	−0.918546	−0.459273	−	0.888295i	$-0.651890\pi$
$509$	−20.7233	−0.918546	−0.459273	+	0.888295i	$0.651890\pi$
$510$	0	0
$511$	−2.52647	−0.111764
$512$	0	0
$513$	−1.71817	−0.0758591
$514$	0	0
$515$	32.0449	1.41207
$516$	0	0
$517$	−19.2808	−0.847970
$518$	0	0
$519$	−12.7594	−0.560076
$520$	0	0
$521$	−36.9459	−1.61863	−0.809315	−	0.587375i	$-0.800161\pi$
$521$	−36.9459	−1.61863	−0.809315	+	0.587375i	$0.800161\pi$
$522$	0	0
$523$	11.8078	0.516319	0.258159	−	0.966102i	$-0.416884\pi$
$523$	11.8078	0.516319	0.258159	+	0.966102i	$0.416884\pi$
$524$	0	0
$525$	−0.416820	−0.0181915
$526$	0	0
$527$	38.5052	1.67731
$528$	0	0
$529$	−17.9874	−0.782059
$530$	0	0
$531$	−4.70122	−0.204016
$532$	0	0
$533$	−4.58245	−0.198488
$534$	0	0
$535$	−21.3743	−0.924093
$536$	0	0
$537$	6.10595	0.263491
$538$	0	0
$539$	−22.1886	−0.955731
$540$	0	0
$541$	−34.6537	−1.48988	−0.744939	−	0.667132i	$-0.767522\pi$
$541$	−34.6537	−1.48988	−0.744939	+	0.667132i	$0.767522\pi$
$542$	0	0
$543$	−11.1343	−0.477817
$544$	0	0
$545$	−13.8371	−0.592715
$546$	0	0
$547$	25.9762	1.11066	0.555330	−	0.831630i	$-0.312592\pi$
$547$	25.9762	1.11066	0.555330	+	0.831630i	$0.312592\pi$
$548$	0	0
$549$	5.56117	0.237345
$550$	0	0
$551$	4.66953	0.198929
$552$	0	0
$553$	−3.60574	−0.153332
$554$	0	0
$555$	−21.3708	−0.907140
$556$	0	0
$557$	−38.2121	−1.61910	−0.809549	−	0.587052i	$-0.800288\pi$
$557$	−38.2121	−1.61910	−0.809549	+	0.587052i	$0.800288\pi$
$558$	0	0
$559$	−2.30965	−0.0976877
$560$	0	0
$561$	13.0651	0.551611
$562$	0	0
$563$	3.05014	0.128548	0.0642741	−	0.997932i	$-0.479527\pi$
$563$	3.05014	0.128548	0.0642741	+	0.997932i	$0.479527\pi$
$564$	0	0
$565$	40.3179	1.69619
$566$	0	0
$567$	0.307143	0.0128988
$568$	0	0
$569$	−0.920294	−0.0385807	−0.0192904	−	0.999814i	$-0.506141\pi$
$569$	−0.920294	−0.0385807	−0.0192904	+	0.999814i	$0.506141\pi$
$570$	0	0
$571$	−25.0724	−1.04925	−0.524624	−	0.851334i	$-0.675794\pi$
$571$	−25.0724	−1.04925	−0.524624	+	0.851334i	$0.675794\pi$
$572$	0	0
$573$	0.577690	0.0241333
$574$	0	0
$575$	−3.03838	−0.126709
$576$	0	0
$577$	−13.2978	−0.553595	−0.276797	−	0.960928i	$-0.589273\pi$
$577$	−13.2978	−0.553595	−0.276797	+	0.960928i	$0.589273\pi$
$578$	0	0
$579$	14.3256	0.595352
$580$	0	0
$581$	−2.71227	−0.112524
$582$	0	0
$583$	−17.2728	−0.715367
$584$	0	0
$585$	−9.79517	−0.404980
$586$	0	0
$587$	−10.2538	−0.423220	−0.211610	−	0.977354i	$-0.567871\pi$
$587$	−10.2538	−0.423220	−0.211610	+	0.977354i	$0.567871\pi$
$588$	0	0
$589$	−16.2704	−0.670409
$590$	0	0
$591$	12.7487	0.524411
$592$	0	0
$593$	33.4177	1.37230	0.686150	−	0.727460i	$-0.259299\pi$
$593$	33.4177	1.37230	0.686150	+	0.727460i	$0.259299\pi$
$594$	0	0
$595$	−3.14890	−0.129092
$596$	0	0
$597$	23.1945	0.949288
$598$	0	0
$599$	11.7026	0.478156	0.239078	−	0.971000i	$-0.423155\pi$
$599$	11.7026	0.478156	0.239078	+	0.971000i	$0.423155\pi$
$600$	0	0
$601$	−27.7968	−1.13386	−0.566928	−	0.823767i	$-0.691868\pi$
$601$	−27.7968	−1.13386	−0.566928	+	0.823767i	$0.691868\pi$
$602$	0	0
$603$	3.54307	0.144285
$604$	0	0
$605$	−1.70432	−0.0692903
$606$	0	0
$607$	−12.2083	−0.495518	−0.247759	−	0.968822i	$-0.579694\pi$
$607$	−12.2083	−0.495518	−0.247759	+	0.968822i	$0.579694\pi$
$608$	0	0
$609$	−0.834733	−0.0338251
$610$	0	0
$611$	23.3122	0.943111
$612$	0	0
$613$	−37.2177	−1.50321	−0.751603	−	0.659615i	$-0.770719\pi$
$613$	−37.2177	−1.50321	−0.751603	+	0.659615i	$0.770719\pi$
$614$	0	0
$615$	−2.97402	−0.119924
$616$	0	0
$617$	39.1038	1.57426	0.787130	−	0.616787i	$-0.211566\pi$
$617$	39.1038	1.57426	0.787130	+	0.616787i	$0.211566\pi$
$618$	0	0
$619$	−38.6201	−1.55227	−0.776136	−	0.630565i	$-0.782823\pi$
$619$	−38.6201	−1.55227	−0.776136	+	0.630565i	$0.782823\pi$
$620$	0	0
$621$	2.23889	0.0898437
$622$	0	0
$623$	−2.57481	−0.103158
$624$	0	0
$625$	−29.9438	−1.19775
$626$	0	0
$627$	−5.52067	−0.220474
$628$	0	0
$629$	−34.4652	−1.37422
$630$	0	0
$631$	−40.6058	−1.61649	−0.808245	−	0.588846i	$-0.799582\pi$
$631$	−40.6058	−1.61649	−0.808245	+	0.588846i	$0.799582\pi$
$632$	0	0
$633$	−19.2467	−0.764989
$634$	0	0
$635$	37.8427	1.50174
$636$	0	0
$637$	26.8280	1.06296
$638$	0	0
$639$	−4.50819	−0.178341
$640$	0	0
$641$	12.7806	0.504802	0.252401	−	0.967623i	$-0.418780\pi$
$641$	12.7806	0.504802	0.252401	+	0.967623i	$0.418780\pi$
$642$	0	0
$643$	−19.8520	−0.782887	−0.391444	−	0.920202i	$-0.628024\pi$
$643$	−19.8520	−0.782887	−0.391444	+	0.920202i	$0.628024\pi$
$644$	0	0
$645$	−1.49897	−0.0590218
$646$	0	0
$647$	−45.5357	−1.79019	−0.895097	−	0.445872i	$-0.852894\pi$
$647$	−45.5357	−1.79019	−0.895097	+	0.445872i	$0.852894\pi$
$648$	0	0
$649$	−15.1055	−0.592943
$650$	0	0
$651$	2.90852	0.113994
$652$	0	0
$653$	4.91900	0.192495	0.0962476	−	0.995357i	$-0.469316\pi$
$653$	4.91900	0.192495	0.0962476	+	0.995357i	$0.469316\pi$
$654$	0	0
$655$	−19.2400	−0.751769
$656$	0	0
$657$	−8.22570	−0.320915
$658$	0	0
$659$	−37.7879	−1.47201	−0.736004	−	0.676977i	$-0.763290\pi$
$659$	−37.7879	−1.47201	−0.736004	+	0.676977i	$0.763290\pi$
$660$	0	0
$661$	−12.1106	−0.471048	−0.235524	−	0.971869i	$-0.575681\pi$
$661$	−12.1106	−0.471048	−0.235524	+	0.971869i	$0.575681\pi$
$662$	0	0
$663$	−15.7969	−0.613501
$664$	0	0
$665$	1.33057	0.0515972
$666$	0	0
$667$	−6.08471	−0.235601
$668$	0	0
$669$	−14.4731	−0.559564
$670$	0	0
$671$	17.8686	0.689810
$672$	0	0
$673$	32.3792	1.24813	0.624064	−	0.781374i	$-0.285481\pi$
$673$	32.3792	1.24813	0.624064	+	0.781374i	$0.285481\pi$
$674$	0	0
$675$	−1.35709	−0.0522343
$676$	0	0
$677$	18.0772	0.694763	0.347382	−	0.937724i	$-0.387071\pi$
$677$	18.0772	0.694763	0.347382	+	0.937724i	$0.387071\pi$
$678$	0	0
$679$	−2.09368	−0.0803480
$680$	0	0
$681$	2.81741	0.107963
$682$	0	0
$683$	2.08654	0.0798394	0.0399197	−	0.999203i	$-0.487290\pi$
$683$	2.08654	0.0798394	0.0399197	+	0.999203i	$0.487290\pi$
$684$	0	0
$685$	26.3698	1.00754
$686$	0	0
$687$	11.0231	0.420559
$688$	0	0
$689$	20.8844	0.795631
$690$	0	0
$691$	5.24236	0.199429	0.0997143	−	0.995016i	$-0.468207\pi$
$691$	5.24236	0.199429	0.0997143	+	0.995016i	$0.468207\pi$
$692$	0	0
$693$	0.986883	0.0374886
$694$	0	0
$695$	−42.0675	−1.59571
$696$	0	0
$697$	−4.79628	−0.181672
$698$	0	0
$699$	−5.84586	−0.221111
$700$	0	0
$701$	21.6239	0.816724	0.408362	−	0.912820i	$-0.366100\pi$
$701$	21.6239	0.816724	0.408362	+	0.912820i	$0.366100\pi$
$702$	0	0
$703$	14.5632	0.549263
$704$	0	0
$705$	15.1297	0.569817
$706$	0	0
$707$	−3.86273	−0.145273
$708$	0	0
$709$	7.16341	0.269028	0.134514	−	0.990912i	$-0.457053\pi$
$709$	7.16341	0.269028	0.134514	+	0.990912i	$0.457053\pi$
$710$	0	0
$711$	−11.7396	−0.440270
$712$	0	0
$713$	21.2014	0.793998
$714$	0	0
$715$	−31.4729	−1.17702
$716$	0	0
$717$	12.3249	0.460281
$718$	0	0
$719$	42.5159	1.58558	0.792788	−	0.609497i	$-0.208629\pi$
$719$	42.5159	1.58558	0.792788	+	0.609497i	$0.208629\pi$
$720$	0	0
$721$	3.90365	0.145380
$722$	0	0
$723$	19.9006	0.740110
$724$	0	0
$725$	3.68820	0.136976
$726$	0	0
$727$	30.6067	1.13514	0.567570	−	0.823325i	$-0.307884\pi$
$727$	30.6067	1.13514	0.567570	+	0.823325i	$0.307884\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−2.41742	−0.0894116
$732$	0	0
$733$	−37.5596	−1.38730	−0.693648	−	0.720314i	$-0.743998\pi$
$733$	−37.5596	−1.38730	−0.693648	+	0.720314i	$0.743998\pi$
$734$	0	0
$735$	17.4114	0.642230
$736$	0	0
$737$	11.3843	0.419345
$738$	0	0
$739$	−5.61048	−0.206385	−0.103192	−	0.994661i	$-0.532906\pi$
$739$	−5.61048	−0.206385	−0.103192	+	0.994661i	$0.532906\pi$
$740$	0	0
$741$	6.67497	0.245211
$742$	0	0
$743$	27.4547	1.00721	0.503607	−	0.863933i	$-0.332006\pi$
$743$	27.4547	1.00721	0.503607	+	0.863933i	$0.332006\pi$
$744$	0	0
$745$	34.4604	1.26253
$746$	0	0
$747$	−8.83065	−0.323096
$748$	0	0
$749$	−2.60378	−0.0951401
$750$	0	0
$751$	−0.659942	−0.0240816	−0.0120408	−	0.999928i	$-0.503833\pi$
$751$	−0.659942	−0.0240816	−0.0120408	+	0.999928i	$0.503833\pi$
$752$	0	0
$753$	9.32458	0.339807
$754$	0	0
$755$	51.5417	1.87579
$756$	0	0
$757$	20.9489	0.761399	0.380700	−	0.924699i	$-0.375683\pi$
$757$	20.9489	0.761399	0.380700	+	0.924699i	$0.375683\pi$
$758$	0	0
$759$	7.19380	0.261118
$760$	0	0
$761$	−19.8568	−0.719807	−0.359903	−	0.932990i	$-0.617190\pi$
$761$	−19.8568	−0.719807	−0.359903	+	0.932990i	$0.617190\pi$
$762$	0	0
$763$	−1.68561	−0.0610230
$764$	0	0
$765$	−10.2522	−0.370670
$766$	0	0
$767$	18.2639	0.659471
$768$	0	0
$769$	35.9479	1.29632	0.648158	−	0.761506i	$-0.275540\pi$
$769$	35.9479	1.29632	0.648158	+	0.761506i	$0.275540\pi$
$770$	0	0
$771$	−23.9424	−0.862264
$772$	0	0
$773$	34.6512	1.24632	0.623158	−	0.782096i	$-0.285849\pi$
$773$	34.6512	1.24632	0.623158	+	0.782096i	$0.285849\pi$
$774$	0	0
$775$	−12.8511	−0.461624
$776$	0	0
$777$	−2.60335	−0.0933946
$778$	0	0
$779$	2.02667	0.0726128
$780$	0	0
$781$	−14.4853	−0.518324
$782$	0	0
$783$	−2.71773	−0.0971238
$784$	0	0
$785$	−15.1266	−0.539892
$786$	0	0
$787$	−39.0453	−1.39182	−0.695908	−	0.718131i	$-0.744998\pi$
$787$	−39.0453	−1.39182	−0.695908	+	0.718131i	$0.744998\pi$
$788$	0	0
$789$	−4.09084	−0.145638
$790$	0	0
$791$	4.91145	0.174631
$792$	0	0
$793$	−21.6047	−0.767206
$794$	0	0
$795$	13.5540	0.480711
$796$	0	0
$797$	−24.7440	−0.876478	−0.438239	−	0.898858i	$-0.644398\pi$
$797$	−24.7440	−0.876478	−0.438239	+	0.898858i	$0.644398\pi$
$798$	0	0
$799$	24.4000	0.863210
$800$	0	0
$801$	−8.38309	−0.296202
$802$	0	0
$803$	−26.4300	−0.932695
$804$	0	0
$805$	−1.73382	−0.0611091
$806$	0	0
$807$	−12.3801	−0.435801
$808$	0	0
$809$	39.0752	1.37381	0.686905	−	0.726747i	$-0.258969\pi$
$809$	39.0752	1.37381	0.686905	+	0.726747i	$0.258969\pi$
$810$	0	0
$811$	−43.4283	−1.52497	−0.762487	−	0.647004i	$-0.776022\pi$
$811$	−43.4283	−1.52497	−0.762487	+	0.647004i	$0.776022\pi$
$812$	0	0
$813$	9.11169	0.319561
$814$	0	0
$815$	31.4000	1.09990
$816$	0	0
$817$	1.02148	0.0357371
$818$	0	0
$819$	−1.19323	−0.0416948
$820$	0	0
$821$	53.6805	1.87346	0.936732	−	0.350048i	$-0.113835\pi$
$821$	53.6805	1.87346	0.936732	+	0.350048i	$0.113835\pi$
$822$	0	0
$823$	−0.514926	−0.0179492	−0.00897460	−	0.999960i	$-0.502857\pi$
$823$	−0.514926	−0.0179492	−0.00897460	+	0.999960i	$0.502857\pi$
$824$	0	0
$825$	−4.36047	−0.151812
$826$	0	0
$827$	−28.2701	−0.983048	−0.491524	−	0.870864i	$-0.663560\pi$
$827$	−28.2701	−0.983048	−0.491524	+	0.870864i	$0.663560\pi$
$828$	0	0
$829$	0.635408	0.0220686	0.0110343	−	0.999939i	$-0.496488\pi$
$829$	0.635408	0.0220686	0.0110343	+	0.999939i	$0.496488\pi$
$830$	0	0
$831$	−9.85742	−0.341950
$832$	0	0
$833$	28.0798	0.972909
$834$	0	0
$835$	2.52133	0.0872541
$836$	0	0
$837$	9.46958	0.327317
$838$	0	0
$839$	23.0900	0.797154	0.398577	−	0.917135i	$-0.369504\pi$
$839$	23.0900	0.797154	0.398577	+	0.917135i	$0.369504\pi$
$840$	0	0
$841$	−21.6139	−0.745308
$842$	0	0
$843$	12.4231	0.427876
$844$	0	0
$845$	5.27624	0.181508
$846$	0	0
$847$	−0.207617	−0.00713379
$848$	0	0
$849$	−29.7611	−1.02140
$850$	0	0
$851$	−18.9769	−0.650519
$852$	0	0
$853$	6.20149	0.212335	0.106168	−	0.994348i	$-0.466142\pi$
$853$	6.20149	0.212335	0.106168	+	0.994348i	$0.466142\pi$
$854$	0	0
$855$	4.33207	0.148154
$856$	0	0
$857$	−18.0053	−0.615049	−0.307525	−	0.951540i	$-0.599501\pi$
$857$	−18.0053	−0.615049	−0.307525	+	0.951540i	$0.599501\pi$
$858$	0	0
$859$	41.6881	1.42238	0.711189	−	0.703000i	$-0.248157\pi$
$859$	41.6881	1.42238	0.711189	+	0.703000i	$0.248157\pi$
$860$	0	0
$861$	−0.362290	−0.0123468
$862$	0	0
$863$	−50.2330	−1.70995	−0.854975	−	0.518669i	$-0.826428\pi$
$863$	−50.2330	−1.70995	−0.854975	+	0.518669i	$0.826428\pi$
$864$	0	0
$865$	32.1706	1.09383
$866$	0	0
$867$	0.465985	0.0158257
$868$	0	0
$869$	−37.7206	−1.27958
$870$	0	0
$871$	−13.7646	−0.466395
$872$	0	0
$873$	−6.81662	−0.230708
$874$	0	0
$875$	−2.82110	−0.0953706
$876$	0	0
$877$	29.4504	0.994470	0.497235	−	0.867616i	$-0.334349\pi$
$877$	29.4504	0.994470	0.497235	+	0.867616i	$0.334349\pi$
$878$	0	0
$879$	31.8879	1.07555
$880$	0	0
$881$	−31.5149	−1.06176	−0.530882	−	0.847446i	$-0.678139\pi$
$881$	−31.5149	−1.06176	−0.530882	+	0.847446i	$0.678139\pi$
$882$	0	0
$883$	36.1738	1.21734	0.608672	−	0.793422i	$-0.291702\pi$
$883$	36.1738	1.21734	0.608672	+	0.793422i	$0.291702\pi$
$884$	0	0
$885$	11.8533	0.398445
$886$	0	0
$887$	55.5902	1.86653	0.933267	−	0.359182i	$-0.116944\pi$
$887$	55.5902	1.86653	0.933267	+	0.359182i	$0.116944\pi$
$888$	0	0
$889$	4.60993	0.154612
$890$	0	0
$891$	3.21310	0.107643
$892$	0	0
$893$	−10.3102	−0.345018
$894$	0	0
$895$	−15.3951	−0.514601
$896$	0	0
$897$	−8.69794	−0.290416
$898$	0	0
$899$	−25.7358	−0.858336
$900$	0	0
$901$	21.8589	0.728224
$902$	0	0
$903$	−0.182601	−0.00607660
$904$	0	0
$905$	28.0731	0.933183
$906$	0	0
$907$	−38.7064	−1.28523	−0.642613	−	0.766191i	$-0.722150\pi$
$907$	−38.7064	−1.28523	−0.642613	+	0.766191i	$0.722150\pi$
$908$	0	0
$909$	−12.5763	−0.417130
$910$	0	0
$911$	26.2647	0.870189	0.435094	−	0.900385i	$-0.356715\pi$
$911$	26.2647	0.870189	0.435094	+	0.900385i	$0.356715\pi$
$912$	0	0
$913$	−28.3738	−0.939035
$914$	0	0
$915$	−14.0215	−0.463537
$916$	0	0
$917$	−2.34378	−0.0773984
$918$	0	0
$919$	−34.9367	−1.15246	−0.576229	−	0.817289i	$-0.695476\pi$
$919$	−34.9367	−1.15246	−0.576229	+	0.817289i	$0.695476\pi$
$920$	0	0
$921$	24.9323	0.821547
$922$	0	0
$923$	17.5140	0.576479
$924$	0	0
$925$	11.5027	0.378206
$926$	0	0
$927$	12.7095	0.417436
$928$	0	0
$929$	42.3874	1.39068	0.695342	−	0.718679i	$-0.255253\pi$
$929$	42.3874	1.39068	0.695342	+	0.718679i	$0.255253\pi$
$930$	0	0
$931$	−11.8651	−0.388863
$932$	0	0
$933$	−7.58605	−0.248356
$934$	0	0
$935$	−32.9415	−1.07730
$936$	0	0
$937$	28.2653	0.923389	0.461694	−	0.887039i	$-0.347242\pi$
$937$	28.2653	0.923389	0.461694	+	0.887039i	$0.347242\pi$
$938$	0	0
$939$	−33.9727	−1.10866
$940$	0	0
$941$	−0.0565076	−0.00184209	−0.000921047	−	1.00000i	$-0.500293\pi$
$941$	−0.0565076	−0.00184209	−0.000921047		1.00000i	$0.500293\pi$
$942$	0	0
$943$	−2.64088	−0.0859989
$944$	0	0
$945$	−0.774408	−0.0251915
$946$	0	0
$947$	−8.51163	−0.276591	−0.138295	−	0.990391i	$-0.544162\pi$
$947$	−8.51163	−0.276591	−0.138295	+	0.990391i	$0.544162\pi$
$948$	0	0
$949$	31.9562	1.03734
$950$	0	0
$951$	9.26715	0.300508
$952$	0	0
$953$	0.957694	0.0310228	0.0155114	−	0.999880i	$-0.495062\pi$
$953$	0.957694	0.0310228	0.0155114	+	0.999880i	$0.495062\pi$
$954$	0	0
$955$	−1.45654	−0.0471327
$956$	0	0
$957$	−8.73235	−0.282277
$958$	0	0
$959$	3.21232	0.103731
$960$	0	0
$961$	58.6730	1.89268
$962$	0	0
$963$	−8.47742	−0.273181
$964$	0	0
$965$	−36.1196	−1.16273
$966$	0	0
$967$	44.5543	1.43277	0.716384	−	0.697706i	$-0.245796\pi$
$967$	44.5543	1.43277	0.716384	+	0.697706i	$0.245796\pi$
$968$	0	0
$969$	6.98644	0.224437
$970$	0	0
$971$	11.6867	0.375044	0.187522	−	0.982260i	$-0.439954\pi$
$971$	11.6867	0.375044	0.187522	+	0.982260i	$0.439954\pi$
$972$	0	0
$973$	−5.12458	−0.164286
$974$	0	0
$975$	5.27219	0.168845
$976$	0	0
$977$	−54.3110	−1.73756	−0.868782	−	0.495195i	$-0.835097\pi$
$977$	−54.3110	−1.73756	−0.868782	+	0.495195i	$0.835097\pi$
$978$	0	0
$979$	−26.9357	−0.860870
$980$	0	0
$981$	−5.48801	−0.175219
$982$	0	0
$983$	−14.4971	−0.462384	−0.231192	−	0.972908i	$-0.574263\pi$
$983$	−14.4971	−0.462384	−0.231192	+	0.972908i	$0.574263\pi$
$984$	0	0
$985$	−32.1436	−1.02418
$986$	0	0
$987$	1.84307	0.0586656
$988$	0	0
$989$	−1.33106	−0.0423252
$990$	0	0
$991$	−25.4623	−0.808837	−0.404419	−	0.914574i	$-0.632526\pi$
$991$	−25.4623	−0.808837	−0.404419	+	0.914574i	$0.632526\pi$
$992$	0	0
$993$	7.47767	0.237297
$994$	0	0
$995$	−58.4810	−1.85397
$996$	0	0
$997$	−42.1516	−1.33496	−0.667478	−	0.744630i	$-0.732626\pi$
$997$	−42.1516	−1.33496	−0.667478	+	0.744630i	$0.732626\pi$
$998$	0	0
$999$	−8.47601	−0.268169

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.x.1.7		8
4.3	odd	2		501.2.a.e.1.7	✓	8
12.11	even	2		1503.2.a.e.1.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
501.2.a.e.1.7	✓	8	4.3	odd	2
1503.2.a.e.1.2		8	12.11	even	2
8016.2.a.x.1.7		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.52133 q^{5} +0.307143 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +2.52133 q^{5} +0.307143 q^{7} +1.00000 q^{9} +3.21310 q^{11} -3.88493 q^{13} -2.52133 q^{15} -4.06620 q^{17} +1.71817 q^{19} -0.307143 q^{21} -2.23889 q^{23} +1.35709 q^{25} -1.00000 q^{27} +2.71773 q^{29} -9.46958 q^{31} -3.21310 q^{33} +0.774408 q^{35} +8.47601 q^{37} +3.88493 q^{39} +1.17955 q^{41} +0.594516 q^{43} +2.52133 q^{45} -6.00069 q^{47} -6.90566 q^{49} +4.06620 q^{51} -5.37574 q^{53} +8.10129 q^{55} -1.71817 q^{57} -4.70122 q^{59} +5.56117 q^{61} +0.307143 q^{63} -9.79517 q^{65} +3.54307 q^{67} +2.23889 q^{69} -4.50819 q^{71} -8.22570 q^{73} -1.35709 q^{75} +0.986883 q^{77} -11.7396 q^{79} +1.00000 q^{81} -8.83065 q^{83} -10.2522 q^{85} -2.71773 q^{87} -8.38309 q^{89} -1.19323 q^{91} +9.46958 q^{93} +4.33207 q^{95} -6.81662 q^{97} +3.21310 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{3} + 7 q^{5} + 4 q^{7} + 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^3 + 7 * q^5 + 4 * q^7 + 8 * q^9	\( 8 q - 8 q^{3} + 7 q^{5} + 4 q^{7} + 8 q^{9} - 13 q^{11} - 7 q^{15} + 11 q^{17} - 12 q^{19} - 4 q^{21} - 7 q^{23} - 5 q^{25} - 8 q^{27} + q^{29} + 2 q^{31} + 13 q^{33} + 4 q^{35} - 9 q^{37} + 4 q^{41} - 2 q^{43} + 7 q^{45} - 17 q^{47} - 2 q^{49} - 11 q^{51} + 9 q^{53} - 7 q^{55} + 12 q^{57} - 29 q^{59} - 12 q^{61} + 4 q^{63} + 8 q^{65} + 7 q^{69} - 13 q^{71} - 20 q^{73} + 5 q^{75} - 22 q^{77} - 8 q^{79} + 8 q^{81} - 33 q^{83} - 31 q^{85} - q^{87} + 4 q^{89} - q^{91} - 2 q^{93} - 3 q^{95} - 31 q^{97} - 13 q^{99}+O(q^{100}) \) 8 * q - 8 * q^3 + 7 * q^5 + 4 * q^7 + 8 * q^9 - 13 * q^11 - 7 * q^15 + 11 * q^17 - 12 * q^19 - 4 * q^21 - 7 * q^23 - 5 * q^25 - 8 * q^27 + q^29 + 2 * q^31 + 13 * q^33 + 4 * q^35 - 9 * q^37 + 4 * q^41 - 2 * q^43 + 7 * q^45 - 17 * q^47 - 2 * q^49 - 11 * q^51 + 9 * q^53 - 7 * q^55 + 12 * q^57 - 29 * q^59 - 12 * q^61 + 4 * q^63 + 8 * q^65 + 7 * q^69 - 13 * q^71 - 20 * q^73 + 5 * q^75 - 22 * q^77 - 8 * q^79 + 8 * q^81 - 33 * q^83 - 31 * q^85 - q^87 + 4 * q^89 - q^91 - 2 * q^93 - 3 * q^95 - 31 * q^97 - 13 * q^99

Introduction

L-functions

Modular forms

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