LMFDB - Embedded newform 8016.2.a.u.1.3

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:	$0$
Dimension:	$5$
Coefficient field:	5.5.38569.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 5x^{3} + 4x - 1 $ x^5 - 5x^3 + 4x - 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			1.3
Root			$-1.15098$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +0.0593478 q^{5} +1.53510 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.0593478 q^{5} +1.53510 q^{7} +1.00000 q^{9} -0.726798 q^{11} -1.12339 q^{13} +0.0593478 q^{15} -6.50878 q^{17} +4.67524 q^{19} +1.53510 q^{21} +3.23604 q^{23} -4.99648 q^{25} +1.00000 q^{27} -3.01149 q^{29} +0.738105 q^{31} -0.726798 q^{33} +0.0911046 q^{35} +8.63949 q^{37} -1.12339 q^{39} +10.3776 q^{41} -7.89407 q^{43} +0.0593478 q^{45} +1.39496 q^{47} -4.64348 q^{49} -6.50878 q^{51} +13.9475 q^{53} -0.0431339 q^{55} +4.67524 q^{57} +6.92291 q^{59} -13.9518 q^{61} +1.53510 q^{63} -0.0666709 q^{65} +8.78089 q^{67} +3.23604 q^{69} +9.28259 q^{71} +9.04640 q^{73} -4.99648 q^{75} -1.11571 q^{77} +9.62559 q^{79} +1.00000 q^{81} -6.72942 q^{83} -0.386282 q^{85} -3.01149 q^{87} +7.03744 q^{89} -1.72452 q^{91} +0.738105 q^{93} +0.277465 q^{95} -0.754292 q^{97} -0.726798 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{3} - q^{5} + 4 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q + 5 * q^3 - q^5 + 4 * q^7 + 5 * q^9	$ 5 q + 5 q^{3} - q^{5} + 4 q^{7} + 5 q^{9} + 7 q^{11} - 8 q^{13} - q^{15} + 5 q^{17} + 20 q^{19} + 4 q^{21} - q^{23} - 6 q^{25} + 5 q^{27} - 5 q^{29} + 18 q^{31} + 7 q^{33} + 6 q^{35} - 17 q^{37} - 8 q^{39} + 6 q^{41} + 10 q^{43} - q^{45} + 3 q^{47} - 13 q^{49} + 5 q^{51} + 15 q^{53} + 9 q^{55} + 20 q^{57} + 17 q^{59} - 2 q^{61} + 4 q^{63} + 26 q^{65} + 24 q^{67} - q^{69} - q^{71} + 2 q^{73} - 6 q^{75} + 26 q^{77} + 6 q^{79} + 5 q^{81} + 7 q^{83} - 11 q^{85} - 5 q^{87} + 15 q^{91} + 18 q^{93} - q^{95} - 13 q^{97} + 7 q^{99}+O(q^{100}) $ 5 * q + 5 * q^3 - q^5 + 4 * q^7 + 5 * q^9 + 7 * q^11 - 8 * q^13 - q^15 + 5 * q^17 + 20 * q^19 + 4 * q^21 - q^23 - 6 * q^25 + 5 * q^27 - 5 * q^29 + 18 * q^31 + 7 * q^33 + 6 * q^35 - 17 * q^37 - 8 * q^39 + 6 * q^41 + 10 * q^43 - q^45 + 3 * q^47 - 13 * q^49 + 5 * q^51 + 15 * q^53 + 9 * q^55 + 20 * q^57 + 17 * q^59 - 2 * q^61 + 4 * q^63 + 26 * q^65 + 24 * q^67 - q^69 - q^71 + 2 * q^73 - 6 * q^75 + 26 * q^77 + 6 * q^79 + 5 * q^81 + 7 * q^83 - 11 * q^85 - 5 * q^87 + 15 * q^91 + 18 * q^93 - q^95 - 13 * q^97 + 7 * 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
$5$	0.0593478	0.0265411	0.0132706	−	0.999912i	$-0.495776\pi$
$5$	0.0593478	0.0265411	0.0132706	+	0.999912i	$0.495776\pi$
$6$	0	0
$7$	1.53510	0.580212	0.290106	−	0.956995i	$-0.406309\pi$
$7$	1.53510	0.580212	0.290106	+	0.956995i	$0.406309\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.726798	−0.219138	−0.109569	−	0.993979i	$-0.534947\pi$
$11$	−0.726798	−0.219138	−0.109569	+	0.993979i	$0.534947\pi$
$12$	0	0
$13$	−1.12339	−0.311573	−0.155787	−	0.987791i	$-0.549791\pi$
$13$	−1.12339	−0.311573	−0.155787	+	0.987791i	$0.549791\pi$
$14$	0	0
$15$	0.0593478	0.0153235
$16$	0	0
$17$	−6.50878	−1.57861	−0.789305	−	0.614001i	$-0.789559\pi$
$17$	−6.50878	−1.57861	−0.789305	+	0.614001i	$0.789559\pi$
$18$	0	0
$19$	4.67524	1.07257	0.536286	−	0.844036i	$-0.319827\pi$
$19$	4.67524	1.07257	0.536286	+	0.844036i	$0.319827\pi$
$20$	0	0
$21$	1.53510	0.334986
$22$	0	0
$23$	3.23604	0.674761	0.337380	−	0.941368i	$-0.390459\pi$
$23$	3.23604	0.674761	0.337380	+	0.941368i	$0.390459\pi$
$24$	0	0
$25$	−4.99648	−0.999296
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−3.01149	−0.559219	−0.279610	−	0.960114i	$-0.590205\pi$
$29$	−3.01149	−0.559219	−0.279610	+	0.960114i	$0.590205\pi$
$30$	0	0
$31$	0.738105	0.132568	0.0662838	−	0.997801i	$-0.478886\pi$
$31$	0.738105	0.132568	0.0662838	+	0.997801i	$0.478886\pi$
$32$	0	0
$33$	−0.726798	−0.126519
$34$	0	0
$35$	0.0911046	0.0153995
$36$	0	0
$37$	8.63949	1.42032	0.710162	−	0.704038i	$-0.248622\pi$
$37$	8.63949	1.42032	0.710162	+	0.704038i	$0.248622\pi$
$38$	0	0
$39$	−1.12339	−0.179887
$40$	0	0
$41$	10.3776	1.62071	0.810354	−	0.585940i	$-0.199275\pi$
$41$	10.3776	1.62071	0.810354	+	0.585940i	$0.199275\pi$
$42$	0	0
$43$	−7.89407	−1.20383	−0.601917	−	0.798559i	$-0.705596\pi$
$43$	−7.89407	−1.20383	−0.601917	+	0.798559i	$0.705596\pi$
$44$	0	0
$45$	0.0593478	0.00884705
$46$	0	0
$47$	1.39496	0.203475	0.101738	−	0.994811i	$-0.467560\pi$
$47$	1.39496	0.203475	0.101738	+	0.994811i	$0.467560\pi$
$48$	0	0
$49$	−4.64348	−0.663354
$50$	0	0
$51$	−6.50878	−0.911411
$52$	0	0
$53$	13.9475	1.91584	0.957919	−	0.287038i	$-0.0926707\pi$
$53$	13.9475	1.91584	0.957919	+	0.287038i	$0.0926707\pi$
$54$	0	0
$55$	−0.0431339	−0.00581617
$56$	0	0
$57$	4.67524	0.619250
$58$	0	0
$59$	6.92291	0.901286	0.450643	−	0.892704i	$-0.351195\pi$
$59$	6.92291	0.901286	0.450643	+	0.892704i	$0.351195\pi$
$60$	0	0
$61$	−13.9518	−1.78634	−0.893171	−	0.449717i	$-0.851525\pi$
$61$	−13.9518	−1.78634	−0.893171	+	0.449717i	$0.851525\pi$
$62$	0	0
$63$	1.53510	0.193404
$64$	0	0
$65$	−0.0666709	−0.00826951
$66$	0	0
$67$	8.78089	1.07276	0.536378	−	0.843978i	$-0.319792\pi$
$67$	8.78089	1.07276	0.536378	+	0.843978i	$0.319792\pi$
$68$	0	0
$69$	3.23604	0.389573
$70$	0	0
$71$	9.28259	1.10164	0.550820	−	0.834624i	$-0.314315\pi$
$71$	9.28259	1.10164	0.550820	+	0.834624i	$0.314315\pi$
$72$	0	0
$73$	9.04640	1.05880	0.529401	−	0.848372i	$-0.322417\pi$
$73$	9.04640	1.05880	0.529401	+	0.848372i	$0.322417\pi$
$74$	0	0
$75$	−4.99648	−0.576944
$76$	0	0
$77$	−1.11571	−0.127146
$78$	0	0
$79$	9.62559	1.08296	0.541482	−	0.840713i	$-0.317864\pi$
$79$	9.62559	1.08296	0.541482	+	0.840713i	$0.317864\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.72942	−0.738650	−0.369325	−	0.929300i	$-0.620411\pi$
$83$	−6.72942	−0.738650	−0.369325	+	0.929300i	$0.620411\pi$
$84$	0	0
$85$	−0.386282	−0.0418981
$86$	0	0
$87$	−3.01149	−0.322866
$88$	0	0
$89$	7.03744	0.745967	0.372984	−	0.927838i	$-0.378335\pi$
$89$	7.03744	0.745967	0.372984	+	0.927838i	$0.378335\pi$
$90$	0	0
$91$	−1.72452	−0.180778
$92$	0	0
$93$	0.738105	0.0765380
$94$	0	0
$95$	0.277465	0.0284673
$96$	0	0
$97$	−0.754292	−0.0765867	−0.0382934	−	0.999267i	$-0.512192\pi$
$97$	−0.754292	−0.0765867	−0.0382934	+	0.999267i	$0.512192\pi$
$98$	0	0
$99$	−0.726798	−0.0730460
$100$	0	0
$101$	10.6059	1.05532	0.527661	−	0.849455i	$-0.323069\pi$
$101$	10.6059	1.05532	0.527661	+	0.849455i	$0.323069\pi$
$102$	0	0
$103$	5.52621	0.544513	0.272257	−	0.962225i	$-0.412230\pi$
$103$	5.52621	0.544513	0.272257	+	0.962225i	$0.412230\pi$
$104$	0	0
$105$	0.0911046	0.00889090
$106$	0	0
$107$	−0.273314	−0.0264222	−0.0132111	−	0.999913i	$-0.504205\pi$
$107$	−0.273314	−0.0264222	−0.0132111	+	0.999913i	$0.504205\pi$
$108$	0	0
$109$	−15.6916	−1.50298	−0.751492	−	0.659742i	$-0.770666\pi$
$109$	−15.6916	−1.50298	−0.751492	+	0.659742i	$0.770666\pi$
$110$	0	0
$111$	8.63949	0.820025
$112$	0	0
$113$	20.1726	1.89768	0.948839	−	0.315760i	$-0.102260\pi$
$113$	20.1726	1.89768	0.948839	+	0.315760i	$0.102260\pi$
$114$	0	0
$115$	0.192052	0.0179089
$116$	0	0
$117$	−1.12339	−0.103858
$118$	0	0
$119$	−9.99160	−0.915929
$120$	0	0
$121$	−10.4718	−0.951979
$122$	0	0
$123$	10.3776	0.935717
$124$	0	0
$125$	−0.593269	−0.0530636
$126$	0	0
$127$	5.57961	0.495111	0.247555	−	0.968874i	$-0.420373\pi$
$127$	5.57961	0.495111	0.247555	+	0.968874i	$0.420373\pi$
$128$	0	0
$129$	−7.89407	−0.695034
$130$	0	0
$131$	19.9390	1.74208	0.871041	−	0.491210i	$-0.163445\pi$
$131$	19.9390	1.74208	0.871041	+	0.491210i	$0.163445\pi$
$132$	0	0
$133$	7.17694	0.622319
$134$	0	0
$135$	0.0593478	0.00510784
$136$	0	0
$137$	6.58107	0.562259	0.281129	−	0.959670i	$-0.409291\pi$
$137$	6.58107	0.562259	0.281129	+	0.959670i	$0.409291\pi$
$138$	0	0
$139$	1.54196	0.130788	0.0653938	−	0.997860i	$-0.479170\pi$
$139$	1.54196	0.130788	0.0653938	+	0.997860i	$0.479170\pi$
$140$	0	0
$141$	1.39496	0.117477
$142$	0	0
$143$	0.816480	0.0682775
$144$	0	0
$145$	−0.178725	−0.0148423
$146$	0	0
$147$	−4.64348	−0.382988
$148$	0	0
$149$	1.57899	0.129356	0.0646778	−	0.997906i	$-0.479398\pi$
$149$	1.57899	0.129356	0.0646778	+	0.997906i	$0.479398\pi$
$150$	0	0
$151$	−9.20724	−0.749274	−0.374637	−	0.927171i	$-0.622233\pi$
$151$	−9.20724	−0.749274	−0.374637	+	0.927171i	$0.622233\pi$
$152$	0	0
$153$	−6.50878	−0.526204
$154$	0	0
$155$	0.0438049	0.00351850
$156$	0	0
$157$	−8.37016	−0.668012	−0.334006	−	0.942571i	$-0.608401\pi$
$157$	−8.37016	−0.668012	−0.334006	+	0.942571i	$0.608401\pi$
$158$	0	0
$159$	13.9475	1.10611
$160$	0	0
$161$	4.96763	0.391504
$162$	0	0
$163$	17.8913	1.40135	0.700676	−	0.713480i	$-0.252882\pi$
$163$	17.8913	1.40135	0.700676	+	0.713480i	$0.252882\pi$
$164$	0	0
$165$	−0.0431339	−0.00335797
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−11.7380	−0.902922
$170$	0	0
$171$	4.67524	0.357524
$172$	0	0
$173$	−8.56831	−0.651436	−0.325718	−	0.945467i	$-0.605606\pi$
$173$	−8.56831	−0.651436	−0.325718	+	0.945467i	$0.605606\pi$
$174$	0	0
$175$	−7.67008	−0.579803
$176$	0	0
$177$	6.92291	0.520358
$178$	0	0
$179$	13.3026	0.994284	0.497142	−	0.867669i	$-0.334383\pi$
$179$	13.3026	0.994284	0.497142	+	0.867669i	$0.334383\pi$
$180$	0	0
$181$	8.95629	0.665716	0.332858	−	0.942977i	$-0.391987\pi$
$181$	8.95629	0.665716	0.332858	+	0.942977i	$0.391987\pi$
$182$	0	0
$183$	−13.9518	−1.03134
$184$	0	0
$185$	0.512735	0.0376970
$186$	0	0
$187$	4.73057	0.345933
$188$	0	0
$189$	1.53510	0.111662
$190$	0	0
$191$	−6.73177	−0.487094	−0.243547	−	0.969889i	$-0.578311\pi$
$191$	−6.73177	−0.487094	−0.243547	+	0.969889i	$0.578311\pi$
$192$	0	0
$193$	3.59253	0.258596	0.129298	−	0.991606i	$-0.458728\pi$
$193$	3.59253	0.258596	0.129298	+	0.991606i	$0.458728\pi$
$194$	0	0
$195$	−0.0666709	−0.00477440
$196$	0	0
$197$	5.03819	0.358956	0.179478	−	0.983762i	$-0.442559\pi$
$197$	5.03819	0.358956	0.179478	+	0.983762i	$0.442559\pi$
$198$	0	0
$199$	18.8821	1.33852	0.669259	−	0.743029i	$-0.266612\pi$
$199$	18.8821	1.33852	0.669259	+	0.743029i	$0.266612\pi$
$200$	0	0
$201$	8.78089	0.619356
$202$	0	0
$203$	−4.62293	−0.324466
$204$	0	0
$205$	0.615888	0.0430155
$206$	0	0
$207$	3.23604	0.224920
$208$	0	0
$209$	−3.39795	−0.235041
$210$	0	0
$211$	−10.7029	−0.736815	−0.368408	−	0.929664i	$-0.620097\pi$
$211$	−10.7029	−0.736815	−0.368408	+	0.929664i	$0.620097\pi$
$212$	0	0
$213$	9.28259	0.636033
$214$	0	0
$215$	−0.468495	−0.0319511
$216$	0	0
$217$	1.13306	0.0769173
$218$	0	0
$219$	9.04640	0.611299
$220$	0	0
$221$	7.31192	0.491853
$222$	0	0
$223$	7.13210	0.477601	0.238800	−	0.971069i	$-0.423246\pi$
$223$	7.13210	0.477601	0.238800	+	0.971069i	$0.423246\pi$
$224$	0	0
$225$	−4.99648	−0.333099
$226$	0	0
$227$	−11.6284	−0.771803	−0.385902	−	0.922540i	$-0.626110\pi$
$227$	−11.6284	−0.771803	−0.385902	+	0.922540i	$0.626110\pi$
$228$	0	0
$229$	−12.9008	−0.852507	−0.426254	−	0.904604i	$-0.640167\pi$
$229$	−12.9008	−0.852507	−0.426254	+	0.904604i	$0.640167\pi$
$230$	0	0
$231$	−1.11571	−0.0734080
$232$	0	0
$233$	−8.14215	−0.533410	−0.266705	−	0.963778i	$-0.585935\pi$
$233$	−8.14215	−0.533410	−0.266705	+	0.963778i	$0.585935\pi$
$234$	0	0
$235$	0.0827877	0.00540047
$236$	0	0
$237$	9.62559	0.625249
$238$	0	0
$239$	−16.6237	−1.07530	−0.537648	−	0.843169i	$-0.680687\pi$
$239$	−16.6237	−1.07530	−0.537648	+	0.843169i	$0.680687\pi$
$240$	0	0
$241$	−7.91630	−0.509934	−0.254967	−	0.966950i	$-0.582065\pi$
$241$	−7.91630	−0.509934	−0.254967	+	0.966950i	$0.582065\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−0.275580	−0.0176062
$246$	0	0
$247$	−5.25213	−0.334185
$248$	0	0
$249$	−6.72942	−0.426460
$250$	0	0
$251$	21.6125	1.36417	0.682084	−	0.731274i	$-0.261074\pi$
$251$	21.6125	1.36417	0.682084	+	0.731274i	$0.261074\pi$
$252$	0	0
$253$	−2.35195	−0.147866
$254$	0	0
$255$	−0.386282	−0.0241899
$256$	0	0
$257$	−15.0536	−0.939019	−0.469509	−	0.882928i	$-0.655569\pi$
$257$	−15.0536	−0.939019	−0.469509	+	0.882928i	$0.655569\pi$
$258$	0	0
$259$	13.2625	0.824089
$260$	0	0
$261$	−3.01149	−0.186406
$262$	0	0
$263$	−27.4806	−1.69453	−0.847265	−	0.531171i	$-0.821752\pi$
$263$	−27.4806	−1.69453	−0.847265	+	0.531171i	$0.821752\pi$
$264$	0	0
$265$	0.827754	0.0508485
$266$	0	0
$267$	7.03744	0.430684
$268$	0	0
$269$	−9.53477	−0.581345	−0.290673	−	0.956823i	$-0.593879\pi$
$269$	−9.53477	−0.581345	−0.290673	+	0.956823i	$0.593879\pi$
$270$	0	0
$271$	−14.1985	−0.862494	−0.431247	−	0.902234i	$-0.641926\pi$
$271$	−14.1985	−0.862494	−0.431247	+	0.902234i	$0.641926\pi$
$272$	0	0
$273$	−1.72452	−0.104373
$274$	0	0
$275$	3.63143	0.218983
$276$	0	0
$277$	18.0059	1.08187	0.540936	−	0.841064i	$-0.318070\pi$
$277$	18.0059	1.08187	0.540936	+	0.841064i	$0.318070\pi$
$278$	0	0
$279$	0.738105	0.0441892
$280$	0	0
$281$	8.99125	0.536373	0.268186	−	0.963367i	$-0.413576\pi$
$281$	8.99125	0.536373	0.268186	+	0.963367i	$0.413576\pi$
$282$	0	0
$283$	27.2768	1.62144	0.810718	−	0.585436i	$-0.199077\pi$
$283$	27.2768	1.62144	0.810718	+	0.585436i	$0.199077\pi$
$284$	0	0
$285$	0.277465	0.0164356
$286$	0	0
$287$	15.9306	0.940355
$288$	0	0
$289$	25.3642	1.49201
$290$	0	0
$291$	−0.754292	−0.0442174
$292$	0	0
$293$	−7.59228	−0.443545	−0.221773	−	0.975098i	$-0.571184\pi$
$293$	−7.59228	−0.443545	−0.221773	+	0.975098i	$0.571184\pi$
$294$	0	0
$295$	0.410860	0.0239212
$296$	0	0
$297$	−0.726798	−0.0421731
$298$	0	0
$299$	−3.63534	−0.210237
$300$	0	0
$301$	−12.1182	−0.698479
$302$	0	0
$303$	10.6059	0.609290
$304$	0	0
$305$	−0.828007	−0.0474115
$306$	0	0
$307$	2.34933	0.134083	0.0670416	−	0.997750i	$-0.478644\pi$
$307$	2.34933	0.134083	0.0670416	+	0.997750i	$0.478644\pi$
$308$	0	0
$309$	5.52621	0.314375
$310$	0	0
$311$	9.03371	0.512255	0.256127	−	0.966643i	$-0.417553\pi$
$311$	9.03371	0.512255	0.256127	+	0.966643i	$0.417553\pi$
$312$	0	0
$313$	−4.58324	−0.259060	−0.129530	−	0.991576i	$-0.541347\pi$
$313$	−4.58324	−0.259060	−0.129530	+	0.991576i	$0.541347\pi$
$314$	0	0
$315$	0.0911046	0.00513316
$316$	0	0
$317$	−0.511404	−0.0287233	−0.0143616	−	0.999897i	$-0.504572\pi$
$317$	−0.511404	−0.0287233	−0.0143616	+	0.999897i	$0.504572\pi$
$318$	0	0
$319$	2.18874	0.122546
$320$	0	0
$321$	−0.273314	−0.0152549
$322$	0	0
$323$	−30.4301	−1.69317
$324$	0	0
$325$	5.61301	0.311354
$326$	0	0
$327$	−15.6916	−0.867748
$328$	0	0
$329$	2.14139	0.118059
$330$	0	0
$331$	−18.9587	−1.04206	−0.521031	−	0.853538i	$-0.674452\pi$
$331$	−18.9587	−1.04206	−0.521031	+	0.853538i	$0.674452\pi$
$332$	0	0
$333$	8.63949	0.473441
$334$	0	0
$335$	0.521126	0.0284722
$336$	0	0
$337$	31.5383	1.71800	0.859001	−	0.511973i	$-0.171085\pi$
$337$	31.5383	1.71800	0.859001	+	0.511973i	$0.171085\pi$
$338$	0	0
$339$	20.1726	1.09562
$340$	0	0
$341$	−0.536454	−0.0290506
$342$	0	0
$343$	−17.8739	−0.965098
$344$	0	0
$345$	0.192052	0.0103397
$346$	0	0
$347$	26.9408	1.44626	0.723129	−	0.690713i	$-0.242703\pi$
$347$	26.9408	1.44626	0.723129	+	0.690713i	$0.242703\pi$
$348$	0	0
$349$	18.3800	0.983857	0.491928	−	0.870636i	$-0.336292\pi$
$349$	18.3800	0.983857	0.491928	+	0.870636i	$0.336292\pi$
$350$	0	0
$351$	−1.12339	−0.0599623
$352$	0	0
$353$	−14.2109	−0.756370	−0.378185	−	0.925730i	$-0.623452\pi$
$353$	−14.2109	−0.756370	−0.378185	+	0.925730i	$0.623452\pi$
$354$	0	0
$355$	0.550901	0.0292388
$356$	0	0
$357$	−9.99160	−0.528812
$358$	0	0
$359$	11.5677	0.610520	0.305260	−	0.952269i	$-0.401257\pi$
$359$	11.5677	0.610520	0.305260	+	0.952269i	$0.401257\pi$
$360$	0	0
$361$	2.85783	0.150412
$362$	0	0
$363$	−10.4718	−0.549625
$364$	0	0
$365$	0.536884	0.0281018
$366$	0	0
$367$	−6.06166	−0.316416	−0.158208	−	0.987406i	$-0.550572\pi$
$367$	−6.06166	−0.316416	−0.158208	+	0.987406i	$0.550572\pi$
$368$	0	0
$369$	10.3776	0.540236
$370$	0	0
$371$	21.4108	1.11159
$372$	0	0
$373$	−34.2760	−1.77474	−0.887371	−	0.461056i	$-0.847471\pi$
$373$	−34.2760	−1.77474	−0.887371	+	0.461056i	$0.847471\pi$
$374$	0	0
$375$	−0.593269	−0.0306363
$376$	0	0
$377$	3.38309	0.174238
$378$	0	0
$379$	−9.45850	−0.485851	−0.242925	−	0.970045i	$-0.578107\pi$
$379$	−9.45850	−0.485851	−0.242925	+	0.970045i	$0.578107\pi$
$380$	0	0
$381$	5.57961	0.285852
$382$	0	0
$383$	20.8981	1.06784	0.533921	−	0.845534i	$-0.320718\pi$
$383$	20.8981	1.06784	0.533921	+	0.845534i	$0.320718\pi$
$384$	0	0
$385$	−0.0662146	−0.00337461
$386$	0	0
$387$	−7.89407	−0.401278
$388$	0	0
$389$	−16.7753	−0.850542	−0.425271	−	0.905066i	$-0.639821\pi$
$389$	−16.7753	−0.850542	−0.425271	+	0.905066i	$0.639821\pi$
$390$	0	0
$391$	−21.0627	−1.06518
$392$	0	0
$393$	19.9390	1.00579
$394$	0	0
$395$	0.571257	0.0287431
$396$	0	0
$397$	5.00650	0.251269	0.125635	−	0.992077i	$-0.459903\pi$
$397$	5.00650	0.251269	0.125635	+	0.992077i	$0.459903\pi$
$398$	0	0
$399$	7.17694	0.359296
$400$	0	0
$401$	1.41427	0.0706252	0.0353126	−	0.999376i	$-0.488757\pi$
$401$	1.41427	0.0706252	0.0353126	+	0.999376i	$0.488757\pi$
$402$	0	0
$403$	−0.829183	−0.0413045
$404$	0	0
$405$	0.0593478	0.00294902
$406$	0	0
$407$	−6.27917	−0.311247
$408$	0	0
$409$	−24.7588	−1.22425	−0.612123	−	0.790763i	$-0.709684\pi$
$409$	−24.7588	−1.22425	−0.612123	+	0.790763i	$0.709684\pi$
$410$	0	0
$411$	6.58107	0.324620
$412$	0	0
$413$	10.6273	0.522937
$414$	0	0
$415$	−0.399376	−0.0196046
$416$	0	0
$417$	1.54196	0.0755103
$418$	0	0
$419$	30.6592	1.49780	0.748901	−	0.662682i	$-0.230582\pi$
$419$	30.6592	1.49780	0.748901	+	0.662682i	$0.230582\pi$
$420$	0	0
$421$	34.5471	1.68372	0.841862	−	0.539693i	$-0.181460\pi$
$421$	34.5471	1.68372	0.841862	+	0.539693i	$0.181460\pi$
$422$	0	0
$423$	1.39496	0.0678252
$424$	0	0
$425$	32.5210	1.57750
$426$	0	0
$427$	−21.4173	−1.03646
$428$	0	0
$429$	0.816480	0.0394200
$430$	0	0
$431$	−22.4664	−1.08217	−0.541084	−	0.840969i	$-0.681986\pi$
$431$	−22.4664	−1.08217	−0.541084	+	0.840969i	$0.681986\pi$
$432$	0	0
$433$	3.35851	0.161400	0.0806999	−	0.996738i	$-0.474284\pi$
$433$	3.35851	0.161400	0.0806999	+	0.996738i	$0.474284\pi$
$434$	0	0
$435$	−0.178725	−0.00856922
$436$	0	0
$437$	15.1292	0.723730
$438$	0	0
$439$	−17.9538	−0.856886	−0.428443	−	0.903569i	$-0.640938\pi$
$439$	−17.9538	−0.856886	−0.428443	+	0.903569i	$0.640938\pi$
$440$	0	0
$441$	−4.64348	−0.221118
$442$	0	0
$443$	−18.7542	−0.891041	−0.445520	−	0.895272i	$-0.646981\pi$
$443$	−18.7542	−0.891041	−0.445520	+	0.895272i	$0.646981\pi$
$444$	0	0
$445$	0.417657	0.0197988
$446$	0	0
$447$	1.57899	0.0746835
$448$	0	0
$449$	−4.79294	−0.226193	−0.113096	−	0.993584i	$-0.536077\pi$
$449$	−4.79294	−0.226193	−0.113096	+	0.993584i	$0.536077\pi$
$450$	0	0
$451$	−7.54242	−0.355159
$452$	0	0
$453$	−9.20724	−0.432594
$454$	0	0
$455$	−0.102346	−0.00479807
$456$	0	0
$457$	−29.5638	−1.38294	−0.691468	−	0.722407i	$-0.743036\pi$
$457$	−29.5638	−1.38294	−0.691468	+	0.722407i	$0.743036\pi$
$458$	0	0
$459$	−6.50878	−0.303804
$460$	0	0
$461$	7.12099	0.331657	0.165829	−	0.986155i	$-0.446970\pi$
$461$	7.12099	0.331657	0.165829	+	0.986155i	$0.446970\pi$
$462$	0	0
$463$	8.15201	0.378856	0.189428	−	0.981895i	$-0.439337\pi$
$463$	8.15201	0.378856	0.189428	+	0.981895i	$0.439337\pi$
$464$	0	0
$465$	0.0438049	0.00203140
$466$	0	0
$467$	3.56520	0.164978	0.0824889	−	0.996592i	$-0.473713\pi$
$467$	3.56520	0.164978	0.0824889	+	0.996592i	$0.473713\pi$
$468$	0	0
$469$	13.4795	0.622426
$470$	0	0
$471$	−8.37016	−0.385677
$472$	0	0
$473$	5.73739	0.263806
$474$	0	0
$475$	−23.3597	−1.07182
$476$	0	0
$477$	13.9475	0.638613
$478$	0	0
$479$	−24.8281	−1.13442	−0.567212	−	0.823572i	$-0.691978\pi$
$479$	−24.8281	−1.13442	−0.567212	+	0.823572i	$0.691978\pi$
$480$	0	0
$481$	−9.70555	−0.442535
$482$	0	0
$483$	4.96763	0.226035
$484$	0	0
$485$	−0.0447656	−0.00203270
$486$	0	0
$487$	−29.4523	−1.33461	−0.667306	−	0.744783i	$-0.732553\pi$
$487$	−29.4523	−1.33461	−0.667306	+	0.744783i	$0.732553\pi$
$488$	0	0
$489$	17.8913	0.809070
$490$	0	0
$491$	26.1084	1.17825	0.589127	−	0.808040i	$-0.299472\pi$
$491$	26.1084	1.17825	0.589127	+	0.808040i	$0.299472\pi$
$492$	0	0
$493$	19.6011	0.882790
$494$	0	0
$495$	−0.0431339	−0.00193872
$496$	0	0
$497$	14.2497	0.639185
$498$	0	0
$499$	22.7386	1.01792	0.508960	−	0.860790i	$-0.330030\pi$
$499$	22.7386	1.01792	0.508960	+	0.860790i	$0.330030\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	11.4082	0.508668	0.254334	−	0.967116i	$-0.418144\pi$
$503$	11.4082	0.508668	0.254334	+	0.967116i	$0.418144\pi$
$504$	0	0
$505$	0.629434	0.0280094
$506$	0	0
$507$	−11.7380	−0.521302
$508$	0	0
$509$	33.2584	1.47415	0.737077	−	0.675809i	$-0.236205\pi$
$509$	33.2584	1.47415	0.737077	+	0.675809i	$0.236205\pi$
$510$	0	0
$511$	13.8871	0.614329
$512$	0	0
$513$	4.67524	0.206417
$514$	0	0
$515$	0.327968	0.0144520
$516$	0	0
$517$	−1.01385	−0.0445892
$518$	0	0
$519$	−8.56831	−0.376107
$520$	0	0
$521$	21.6845	0.950017	0.475008	−	0.879981i	$-0.342445\pi$
$521$	21.6845	0.950017	0.475008	+	0.879981i	$0.342445\pi$
$522$	0	0
$523$	−14.4527	−0.631973	−0.315986	−	0.948764i	$-0.602335\pi$
$523$	−14.4527	−0.631973	−0.315986	+	0.948764i	$0.602335\pi$
$524$	0	0
$525$	−7.67008	−0.334750
$526$	0	0
$527$	−4.80416	−0.209273
$528$	0	0
$529$	−12.5281	−0.544698
$530$	0	0
$531$	6.92291	0.300429
$532$	0	0
$533$	−11.6581	−0.504969
$534$	0	0
$535$	−0.0162206	−0.000701276 0
$536$	0	0
$537$	13.3026	0.574050
$538$	0	0
$539$	3.37487	0.145366
$540$	0	0
$541$	28.1002	1.20812	0.604061	−	0.796938i	$-0.293548\pi$
$541$	28.1002	1.20812	0.604061	+	0.796938i	$0.293548\pi$
$542$	0	0
$543$	8.95629	0.384351
$544$	0	0
$545$	−0.931263	−0.0398909
$546$	0	0
$547$	13.7841	0.589367	0.294684	−	0.955595i	$-0.404786\pi$
$547$	13.7841	0.589367	0.294684	+	0.955595i	$0.404786\pi$
$548$	0	0
$549$	−13.9518	−0.595447
$550$	0	0
$551$	−14.0794	−0.599803
$552$	0	0
$553$	14.7762	0.628348
$554$	0	0
$555$	0.512735	0.0217644
$556$	0	0
$557$	8.80962	0.373276	0.186638	−	0.982429i	$-0.440241\pi$
$557$	8.80962	0.373276	0.186638	+	0.982429i	$0.440241\pi$
$558$	0	0
$559$	8.86814	0.375082
$560$	0	0
$561$	4.73057	0.199725
$562$	0	0
$563$	−23.8194	−1.00387	−0.501934	−	0.864906i	$-0.667378\pi$
$563$	−23.8194	−1.00387	−0.501934	+	0.864906i	$0.667378\pi$
$564$	0	0
$565$	1.19720	0.0503665
$566$	0	0
$567$	1.53510	0.0644680
$568$	0	0
$569$	23.1724	0.971436	0.485718	−	0.874116i	$-0.338558\pi$
$569$	23.1724	0.971436	0.485718	+	0.874116i	$0.338558\pi$
$570$	0	0
$571$	7.50566	0.314102	0.157051	−	0.987590i	$-0.449801\pi$
$571$	7.50566	0.314102	0.157051	+	0.987590i	$0.449801\pi$
$572$	0	0
$573$	−6.73177	−0.281224
$574$	0	0
$575$	−16.1688	−0.674285
$576$	0	0
$577$	−33.7172	−1.40366	−0.701832	−	0.712342i	$-0.747634\pi$
$577$	−33.7172	−1.40366	−0.701832	+	0.712342i	$0.747634\pi$
$578$	0	0
$579$	3.59253	0.149301
$580$	0	0
$581$	−10.3303	−0.428574
$582$	0	0
$583$	−10.1370	−0.419833
$584$	0	0
$585$	−0.0666709	−0.00275650
$586$	0	0
$587$	16.9458	0.699429	0.349715	−	0.936856i	$-0.386279\pi$
$587$	16.9458	0.699429	0.349715	+	0.936856i	$0.386279\pi$
$588$	0	0
$589$	3.45082	0.142188
$590$	0	0
$591$	5.03819	0.207243
$592$	0	0
$593$	42.1996	1.73293	0.866465	−	0.499238i	$-0.166387\pi$
$593$	42.1996	1.73293	0.866465	+	0.499238i	$0.166387\pi$
$594$	0	0
$595$	−0.592980	−0.0243098
$596$	0	0
$597$	18.8821	0.772794
$598$	0	0
$599$	26.0476	1.06428	0.532139	−	0.846657i	$-0.321388\pi$
$599$	26.0476	1.06428	0.532139	+	0.846657i	$0.321388\pi$
$600$	0	0
$601$	17.0571	0.695773	0.347887	−	0.937537i	$-0.386899\pi$
$601$	17.0571	0.695773	0.347887	+	0.937537i	$0.386899\pi$
$602$	0	0
$603$	8.78089	0.357585
$604$	0	0
$605$	−0.621476	−0.0252666
$606$	0	0
$607$	−35.7563	−1.45130	−0.725651	−	0.688063i	$-0.758461\pi$
$607$	−35.7563	−1.45130	−0.725651	+	0.688063i	$0.758461\pi$
$608$	0	0
$609$	−4.62293	−0.187330
$610$	0	0
$611$	−1.56709	−0.0633975
$612$	0	0
$613$	−5.83164	−0.235538	−0.117769	−	0.993041i	$-0.537574\pi$
$613$	−5.83164	−0.235538	−0.117769	+	0.993041i	$0.537574\pi$
$614$	0	0
$615$	0.615888	0.0248350
$616$	0	0
$617$	12.1589	0.489499	0.244750	−	0.969586i	$-0.421294\pi$
$617$	12.1589	0.489499	0.244750	+	0.969586i	$0.421294\pi$
$618$	0	0
$619$	43.5173	1.74911	0.874555	−	0.484927i	$-0.161154\pi$
$619$	43.5173	1.74911	0.874555	+	0.484927i	$0.161154\pi$
$620$	0	0
$621$	3.23604	0.129858
$622$	0	0
$623$	10.8032	0.432819
$624$	0	0
$625$	24.9472	0.997887
$626$	0	0
$627$	−3.39795	−0.135701
$628$	0	0
$629$	−56.2326	−2.24214
$630$	0	0
$631$	8.33034	0.331626	0.165813	−	0.986157i	$-0.446975\pi$
$631$	8.33034	0.331626	0.165813	+	0.986157i	$0.446975\pi$
$632$	0	0
$633$	−10.7029	−0.425400
$634$	0	0
$635$	0.331138	0.0131408
$636$	0	0
$637$	5.21645	0.206683
$638$	0	0
$639$	9.28259	0.367214
$640$	0	0
$641$	18.6439	0.736392	0.368196	−	0.929748i	$-0.379976\pi$
$641$	18.6439	0.736392	0.368196	+	0.929748i	$0.379976\pi$
$642$	0	0
$643$	50.0060	1.97204	0.986022	−	0.166616i	$-0.0532840\pi$
$643$	50.0060	1.97204	0.986022	+	0.166616i	$0.0532840\pi$
$644$	0	0
$645$	−0.468495	−0.0184470
$646$	0	0
$647$	28.7022	1.12840	0.564201	−	0.825638i	$-0.309184\pi$
$647$	28.7022	1.12840	0.564201	+	0.825638i	$0.309184\pi$
$648$	0	0
$649$	−5.03156	−0.197506
$650$	0	0
$651$	1.13306	0.0444082
$652$	0	0
$653$	3.90284	0.152730	0.0763650	−	0.997080i	$-0.475669\pi$
$653$	3.90284	0.152730	0.0763650	+	0.997080i	$0.475669\pi$
$654$	0	0
$655$	1.18334	0.0462369
$656$	0	0
$657$	9.04640	0.352934
$658$	0	0
$659$	−37.9732	−1.47923	−0.739614	−	0.673032i	$-0.764992\pi$
$659$	−37.9732	−1.47923	−0.739614	+	0.673032i	$0.764992\pi$
$660$	0	0
$661$	−14.0911	−0.548080	−0.274040	−	0.961718i	$-0.588360\pi$
$661$	−14.0911	−0.548080	−0.274040	+	0.961718i	$0.588360\pi$
$662$	0	0
$663$	7.31192	0.283971
$664$	0	0
$665$	0.425935	0.0165171
$666$	0	0
$667$	−9.74529	−0.377339
$668$	0	0
$669$	7.13210	0.275743
$670$	0	0
$671$	10.1401	0.391455
$672$	0	0
$673$	−2.00760	−0.0773874	−0.0386937	−	0.999251i	$-0.512320\pi$
$673$	−2.00760	−0.0773874	−0.0386937	+	0.999251i	$0.512320\pi$
$674$	0	0
$675$	−4.99648	−0.192315
$676$	0	0
$677$	−14.0908	−0.541554	−0.270777	−	0.962642i	$-0.587281\pi$
$677$	−14.0908	−0.541554	−0.270777	+	0.962642i	$0.587281\pi$
$678$	0	0
$679$	−1.15791	−0.0444365
$680$	0	0
$681$	−11.6284	−0.445601
$682$	0	0
$683$	−15.3972	−0.589158	−0.294579	−	0.955627i	$-0.595179\pi$
$683$	−15.3972	−0.589158	−0.294579	+	0.955627i	$0.595179\pi$
$684$	0	0
$685$	0.390572	0.0149230
$686$	0	0
$687$	−12.9008	−0.492195
$688$	0	0
$689$	−15.6685	−0.596924
$690$	0	0
$691$	−8.96256	−0.340952	−0.170476	−	0.985362i	$-0.554530\pi$
$691$	−8.96256	−0.340952	−0.170476	+	0.985362i	$0.554530\pi$
$692$	0	0
$693$	−1.11571	−0.0423821
$694$	0	0
$695$	0.0915121	0.00347125
$696$	0	0
$697$	−67.5455	−2.55847
$698$	0	0
$699$	−8.14215	−0.307964
$700$	0	0
$701$	−51.4198	−1.94210	−0.971049	−	0.238881i	$-0.923219\pi$
$701$	−51.4198	−1.94210	−0.971049	+	0.238881i	$0.923219\pi$
$702$	0	0
$703$	40.3917	1.52340
$704$	0	0
$705$	0.0827877	0.00311796
$706$	0	0
$707$	16.2810	0.612310
$708$	0	0
$709$	14.1777	0.532456	0.266228	−	0.963910i	$-0.414223\pi$
$709$	14.1777	0.532456	0.266228	+	0.963910i	$0.414223\pi$
$710$	0	0
$711$	9.62559	0.360988
$712$	0	0
$713$	2.38854	0.0894514
$714$	0	0
$715$	0.0484563	0.00181216
$716$	0	0
$717$	−16.6237	−0.620822
$718$	0	0
$719$	−18.9176	−0.705507	−0.352754	−	0.935716i	$-0.614755\pi$
$719$	−18.9176	−0.705507	−0.352754	+	0.935716i	$0.614755\pi$
$720$	0	0
$721$	8.48326	0.315933
$722$	0	0
$723$	−7.91630	−0.294410
$724$	0	0
$725$	15.0468	0.558826
$726$	0	0
$727$	4.52329	0.167760	0.0838798	−	0.996476i	$-0.473269\pi$
$727$	4.52329	0.167760	0.0838798	+	0.996476i	$0.473269\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	51.3807	1.90038
$732$	0	0
$733$	−29.3400	−1.08370	−0.541849	−	0.840476i	$-0.682276\pi$
$733$	−29.3400	−1.08370	−0.541849	+	0.840476i	$0.682276\pi$
$734$	0	0
$735$	−0.275580	−0.0101649
$736$	0	0
$737$	−6.38193	−0.235082
$738$	0	0
$739$	44.8214	1.64878	0.824391	−	0.566021i	$-0.191518\pi$
$739$	44.8214	1.64878	0.824391	+	0.566021i	$0.191518\pi$
$740$	0	0
$741$	−5.25213	−0.192942
$742$	0	0
$743$	−20.6928	−0.759145	−0.379573	−	0.925162i	$-0.623929\pi$
$743$	−20.6928	−0.759145	−0.379573	+	0.925162i	$0.623929\pi$
$744$	0	0
$745$	0.0937093	0.00343324
$746$	0	0
$747$	−6.72942	−0.246217
$748$	0	0
$749$	−0.419563	−0.0153305
$750$	0	0
$751$	21.3810	0.780205	0.390103	−	0.920771i	$-0.372440\pi$
$751$	21.3810	0.780205	0.390103	+	0.920771i	$0.372440\pi$
$752$	0	0
$753$	21.6125	0.787603
$754$	0	0
$755$	−0.546429	−0.0198866
$756$	0	0
$757$	8.80621	0.320067	0.160034	−	0.987112i	$-0.448840\pi$
$757$	8.80621	0.320067	0.160034	+	0.987112i	$0.448840\pi$
$758$	0	0
$759$	−2.35195	−0.0853702
$760$	0	0
$761$	−11.7609	−0.426333	−0.213166	−	0.977016i	$-0.568378\pi$
$761$	−11.7609	−0.426333	−0.213166	+	0.977016i	$0.568378\pi$
$762$	0	0
$763$	−24.0881	−0.872049
$764$	0	0
$765$	−0.386282	−0.0139660
$766$	0	0
$767$	−7.77715	−0.280817
$768$	0	0
$769$	−51.4539	−1.85548	−0.927738	−	0.373232i	$-0.878250\pi$
$769$	−51.4539	−1.85548	−0.927738	+	0.373232i	$0.878250\pi$
$770$	0	0
$771$	−15.0536	−0.542143
$772$	0	0
$773$	−32.0957	−1.15440	−0.577201	−	0.816602i	$-0.695855\pi$
$773$	−32.0957	−1.15440	−0.577201	+	0.816602i	$0.695855\pi$
$774$	0	0
$775$	−3.68793	−0.132474
$776$	0	0
$777$	13.2625	0.475788
$778$	0	0
$779$	48.5177	1.73833
$780$	0	0
$781$	−6.74657	−0.241411
$782$	0	0
$783$	−3.01149	−0.107622
$784$	0	0
$785$	−0.496751	−0.0177298
$786$	0	0
$787$	−45.4030	−1.61844	−0.809220	−	0.587505i	$-0.800110\pi$
$787$	−45.4030	−1.61844	−0.809220	+	0.587505i	$0.800110\pi$
$788$	0	0
$789$	−27.4806	−0.978337
$790$	0	0
$791$	30.9669	1.10106
$792$	0	0
$793$	15.6733	0.556576
$794$	0	0
$795$	0.827754	0.0293574
$796$	0	0
$797$	−0.658673	−0.0233314	−0.0116657	−	0.999932i	$-0.503713\pi$
$797$	−0.658673	−0.0233314	−0.0116657	+	0.999932i	$0.503713\pi$
$798$	0	0
$799$	−9.07947	−0.321209
$800$	0	0
$801$	7.03744	0.248656
$802$	0	0
$803$	−6.57491	−0.232023
$804$	0	0
$805$	0.294818	0.0103910
$806$	0	0
$807$	−9.53477	−0.335640
$808$	0	0
$809$	−5.32511	−0.187221	−0.0936105	−	0.995609i	$-0.529841\pi$
$809$	−5.32511	−0.187221	−0.0936105	+	0.995609i	$0.529841\pi$
$810$	0	0
$811$	12.2598	0.430501	0.215251	−	0.976559i	$-0.430943\pi$
$811$	12.2598	0.430501	0.215251	+	0.976559i	$0.430943\pi$
$812$	0	0
$813$	−14.1985	−0.497961
$814$	0	0
$815$	1.06181	0.0371935
$816$	0	0
$817$	−36.9066	−1.29120
$818$	0	0
$819$	−1.72452	−0.0602595
$820$	0	0
$821$	−7.24036	−0.252690	−0.126345	−	0.991986i	$-0.540325\pi$
$821$	−7.24036	−0.252690	−0.126345	+	0.991986i	$0.540325\pi$
$822$	0	0
$823$	−14.5934	−0.508694	−0.254347	−	0.967113i	$-0.581861\pi$
$823$	−14.5934	−0.508694	−0.254347	+	0.967113i	$0.581861\pi$
$824$	0	0
$825$	3.63143	0.126430
$826$	0	0
$827$	−41.5316	−1.44419	−0.722097	−	0.691792i	$-0.756822\pi$
$827$	−41.5316	−1.44419	−0.722097	+	0.691792i	$0.756822\pi$
$828$	0	0
$829$	−4.01369	−0.139401	−0.0697006	−	0.997568i	$-0.522204\pi$
$829$	−4.01369	−0.139401	−0.0697006	+	0.997568i	$0.522204\pi$
$830$	0	0
$831$	18.0059	0.624619
$832$	0	0
$833$	30.2234	1.04718
$834$	0	0
$835$	0.0593478	0.00205382
$836$	0	0
$837$	0.738105	0.0255127
$838$	0	0
$839$	−7.96730	−0.275062	−0.137531	−	0.990497i	$-0.543917\pi$
$839$	−7.96730	−0.275062	−0.137531	+	0.990497i	$0.543917\pi$
$840$	0	0
$841$	−19.9309	−0.687274
$842$	0	0
$843$	8.99125	0.309675
$844$	0	0
$845$	−0.696624	−0.0239646
$846$	0	0
$847$	−16.0752	−0.552349
$848$	0	0
$849$	27.2768	0.936137
$850$	0	0
$851$	27.9577	0.958379
$852$	0	0
$853$	31.0202	1.06211	0.531056	−	0.847337i	$-0.321795\pi$
$853$	31.0202	1.06211	0.531056	+	0.847337i	$0.321795\pi$
$854$	0	0
$855$	0.277465	0.00948910
$856$	0	0
$857$	14.6173	0.499318	0.249659	−	0.968334i	$-0.419681\pi$
$857$	14.6173	0.499318	0.249659	+	0.968334i	$0.419681\pi$
$858$	0	0
$859$	−32.1234	−1.09604	−0.548018	−	0.836467i	$-0.684617\pi$
$859$	−32.1234	−1.09604	−0.548018	+	0.836467i	$0.684617\pi$
$860$	0	0
$861$	15.9306	0.542914
$862$	0	0
$863$	−40.7770	−1.38806	−0.694032	−	0.719944i	$-0.744167\pi$
$863$	−40.7770	−1.38806	−0.694032	+	0.719944i	$0.744167\pi$
$864$	0	0
$865$	−0.508510	−0.0172899
$866$	0	0
$867$	25.3642	0.861413
$868$	0	0
$869$	−6.99586	−0.237318
$870$	0	0
$871$	−9.86439	−0.334242
$872$	0	0
$873$	−0.754292	−0.0255289
$874$	0	0
$875$	−0.910725	−0.0307881
$876$	0	0
$877$	−37.5229	−1.26706	−0.633528	−	0.773719i	$-0.718394\pi$
$877$	−37.5229	−1.26706	−0.633528	+	0.773719i	$0.718394\pi$
$878$	0	0
$879$	−7.59228	−0.256081
$880$	0	0
$881$	38.0924	1.28336	0.641682	−	0.766971i	$-0.278237\pi$
$881$	38.0924	1.28336	0.641682	+	0.766971i	$0.278237\pi$
$882$	0	0
$883$	−19.0442	−0.640890	−0.320445	−	0.947267i	$-0.603832\pi$
$883$	−19.0442	−0.640890	−0.320445	+	0.947267i	$0.603832\pi$
$884$	0	0
$885$	0.410860	0.0138109
$886$	0	0
$887$	−27.3252	−0.917491	−0.458745	−	0.888568i	$-0.651701\pi$
$887$	−27.3252	−0.917491	−0.458745	+	0.888568i	$0.651701\pi$
$888$	0	0
$889$	8.56525	0.287269
$890$	0	0
$891$	−0.726798	−0.0243487
$892$	0	0
$893$	6.52176	0.218242
$894$	0	0
$895$	0.789481	0.0263894
$896$	0	0
$897$	−3.63534	−0.121381
$898$	0	0
$899$	−2.22280	−0.0741344
$900$	0	0
$901$	−90.7813	−3.02436
$902$	0	0
$903$	−12.1182	−0.403267
$904$	0	0
$905$	0.531536	0.0176689
$906$	0	0
$907$	54.3781	1.80559	0.902797	−	0.430067i	$-0.141510\pi$
$907$	54.3781	1.80559	0.902797	+	0.430067i	$0.141510\pi$
$908$	0	0
$909$	10.6059	0.351774
$910$	0	0
$911$	39.5068	1.30892	0.654460	−	0.756097i	$-0.272896\pi$
$911$	39.5068	1.30892	0.654460	+	0.756097i	$0.272896\pi$
$912$	0	0
$913$	4.89093	0.161866
$914$	0	0
$915$	−0.828007	−0.0273731
$916$	0	0
$917$	30.6084	1.01078
$918$	0	0
$919$	−39.2491	−1.29471	−0.647354	−	0.762190i	$-0.724124\pi$
$919$	−39.2491	−1.29471	−0.647354	+	0.762190i	$0.724124\pi$
$920$	0	0
$921$	2.34933	0.0774130
$922$	0	0
$923$	−10.4280	−0.343242
$924$	0	0
$925$	−43.1670	−1.41932
$926$	0	0
$927$	5.52621	0.181504
$928$	0	0
$929$	−3.32230	−0.109001	−0.0545006	−	0.998514i	$-0.517357\pi$
$929$	−3.32230	−0.109001	−0.0545006	+	0.998514i	$0.517357\pi$
$930$	0	0
$931$	−21.7094	−0.711495
$932$	0	0
$933$	9.03371	0.295750
$934$	0	0
$935$	0.280749	0.00918147
$936$	0	0
$937$	2.16258	0.0706484	0.0353242	−	0.999376i	$-0.488754\pi$
$937$	2.16258	0.0706484	0.0353242	+	0.999376i	$0.488754\pi$
$938$	0	0
$939$	−4.58324	−0.149568
$940$	0	0
$941$	−22.0637	−0.719255	−0.359628	−	0.933096i	$-0.617096\pi$
$941$	−22.0637	−0.719255	−0.359628	+	0.933096i	$0.617096\pi$
$942$	0	0
$943$	33.5823	1.09359
$944$	0	0
$945$	0.0911046	0.00296363
$946$	0	0
$947$	−54.8587	−1.78267	−0.891334	−	0.453347i	$-0.850230\pi$
$947$	−54.8587	−1.78267	−0.891334	+	0.453347i	$0.850230\pi$
$948$	0	0
$949$	−10.1627	−0.329894
$950$	0	0
$951$	−0.511404	−0.0165834
$952$	0	0
$953$	35.6512	1.15486	0.577428	−	0.816441i	$-0.304056\pi$
$953$	35.6512	1.15486	0.577428	+	0.816441i	$0.304056\pi$
$954$	0	0
$955$	−0.399516	−0.0129280
$956$	0	0
$957$	2.18874	0.0707521
$958$	0	0
$959$	10.1026	0.326229
$960$	0	0
$961$	−30.4552	−0.982426
$962$	0	0
$963$	−0.273314	−0.00880741
$964$	0	0
$965$	0.213209	0.00686344
$966$	0	0
$967$	−54.3232	−1.74692	−0.873458	−	0.486899i	$-0.838128\pi$
$967$	−54.3232	−1.74692	−0.873458	+	0.486899i	$0.838128\pi$
$968$	0	0
$969$	−30.4301	−0.977555
$970$	0	0
$971$	−7.33160	−0.235282	−0.117641	−	0.993056i	$-0.537533\pi$
$971$	−7.33160	−0.235282	−0.117641	+	0.993056i	$0.537533\pi$
$972$	0	0
$973$	2.36706	0.0758845
$974$	0	0
$975$	5.61301	0.179760
$976$	0	0
$977$	4.90524	0.156933	0.0784663	−	0.996917i	$-0.474998\pi$
$977$	4.90524	0.156933	0.0784663	+	0.996917i	$0.474998\pi$
$978$	0	0
$979$	−5.11480	−0.163470
$980$	0	0
$981$	−15.6916	−0.500995
$982$	0	0
$983$	45.3449	1.44628	0.723138	−	0.690703i	$-0.242699\pi$
$983$	45.3449	1.44628	0.723138	+	0.690703i	$0.242699\pi$
$984$	0	0
$985$	0.299005	0.00952710
$986$	0	0
$987$	2.14139	0.0681613
$988$	0	0
$989$	−25.5455	−0.812300
$990$	0	0
$991$	−62.1348	−1.97378	−0.986889	−	0.161402i	$-0.948398\pi$
$991$	−62.1348	−1.97378	−0.986889	+	0.161402i	$0.948398\pi$
$992$	0	0
$993$	−18.9587	−0.601635
$994$	0	0
$995$	1.12061	0.0355258
$996$	0	0
$997$	28.6058	0.905955	0.452978	−	0.891522i	$-0.350362\pi$
$997$	28.6058	0.905955	0.452978	+	0.891522i	$0.350362\pi$
$998$	0	0
$999$	8.63949	0.273342

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.u.1.3		5
4.3	odd	2		501.2.a.c.1.4	✓	5
12.11	even	2		1503.2.a.c.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
501.2.a.c.1.4	✓	5	4.3	odd	2
1503.2.a.c.1.2		5	12.11	even	2
8016.2.a.u.1.3		5	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.0593478 q^{5} +1.53510 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.0593478 q^{5} +1.53510 q^{7} +1.00000 q^{9} -0.726798 q^{11} -1.12339 q^{13} +0.0593478 q^{15} -6.50878 q^{17} +4.67524 q^{19} +1.53510 q^{21} +3.23604 q^{23} -4.99648 q^{25} +1.00000 q^{27} -3.01149 q^{29} +0.738105 q^{31} -0.726798 q^{33} +0.0911046 q^{35} +8.63949 q^{37} -1.12339 q^{39} +10.3776 q^{41} -7.89407 q^{43} +0.0593478 q^{45} +1.39496 q^{47} -4.64348 q^{49} -6.50878 q^{51} +13.9475 q^{53} -0.0431339 q^{55} +4.67524 q^{57} +6.92291 q^{59} -13.9518 q^{61} +1.53510 q^{63} -0.0666709 q^{65} +8.78089 q^{67} +3.23604 q^{69} +9.28259 q^{71} +9.04640 q^{73} -4.99648 q^{75} -1.11571 q^{77} +9.62559 q^{79} +1.00000 q^{81} -6.72942 q^{83} -0.386282 q^{85} -3.01149 q^{87} +7.03744 q^{89} -1.72452 q^{91} +0.738105 q^{93} +0.277465 q^{95} -0.754292 q^{97} -0.726798 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} - q^{5} + 4 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q + 5 * q^3 - q^5 + 4 * q^7 + 5 * q^9	\( 5 q + 5 q^{3} - q^{5} + 4 q^{7} + 5 q^{9} + 7 q^{11} - 8 q^{13} - q^{15} + 5 q^{17} + 20 q^{19} + 4 q^{21} - q^{23} - 6 q^{25} + 5 q^{27} - 5 q^{29} + 18 q^{31} + 7 q^{33} + 6 q^{35} - 17 q^{37} - 8 q^{39} + 6 q^{41} + 10 q^{43} - q^{45} + 3 q^{47} - 13 q^{49} + 5 q^{51} + 15 q^{53} + 9 q^{55} + 20 q^{57} + 17 q^{59} - 2 q^{61} + 4 q^{63} + 26 q^{65} + 24 q^{67} - q^{69} - q^{71} + 2 q^{73} - 6 q^{75} + 26 q^{77} + 6 q^{79} + 5 q^{81} + 7 q^{83} - 11 q^{85} - 5 q^{87} + 15 q^{91} + 18 q^{93} - q^{95} - 13 q^{97} + 7 q^{99}+O(q^{100}) \) 5 * q + 5 * q^3 - q^5 + 4 * q^7 + 5 * q^9 + 7 * q^11 - 8 * q^13 - q^15 + 5 * q^17 + 20 * q^19 + 4 * q^21 - q^23 - 6 * q^25 + 5 * q^27 - 5 * q^29 + 18 * q^31 + 7 * q^33 + 6 * q^35 - 17 * q^37 - 8 * q^39 + 6 * q^41 + 10 * q^43 - q^45 + 3 * q^47 - 13 * q^49 + 5 * q^51 + 15 * q^53 + 9 * q^55 + 20 * q^57 + 17 * q^59 - 2 * q^61 + 4 * q^63 + 26 * q^65 + 24 * q^67 - q^69 - q^71 + 2 * q^73 - 6 * q^75 + 26 * q^77 + 6 * q^79 + 5 * q^81 + 7 * q^83 - 11 * q^85 - 5 * q^87 + 15 * q^91 + 18 * q^93 - q^95 - 13 * q^97 + 7 * q^99

Introduction

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