LMFDB - Embedded newform 8016.2.a.t.1.5

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:	$5$
Coefficient field:	5.5.149169.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 6x^{3} - 3x^{2} + 4x + 1 $ x^5 - 6x^3 - 3x^2 + 4*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2004)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.81853$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +2.12560 q^{5} -0.221696 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.12560 q^{5} -0.221696 q^{7} +1.00000 q^{9} +2.12560 q^{11} -2.07866 q^{13} +2.12560 q^{15} -5.53728 q^{17} +2.18327 q^{19} -0.221696 q^{21} -8.50926 q^{23} -0.481818 q^{25} +1.00000 q^{27} -9.67698 q^{29} +3.18998 q^{31} +2.12560 q^{33} -0.471237 q^{35} -5.07948 q^{37} -2.07866 q^{39} -3.43857 q^{41} -5.53327 q^{43} +2.12560 q^{45} +11.8218 q^{47} -6.95085 q^{49} -5.53728 q^{51} +2.21498 q^{53} +4.51818 q^{55} +2.18327 q^{57} -6.60234 q^{59} +9.72573 q^{61} -0.221696 q^{63} -4.41840 q^{65} +0.214807 q^{67} -8.50926 q^{69} -12.8354 q^{71} -12.4910 q^{73} -0.481818 q^{75} -0.471237 q^{77} -7.45038 q^{79} +1.00000 q^{81} -15.9723 q^{83} -11.7701 q^{85} -9.67698 q^{87} -6.46935 q^{89} +0.460829 q^{91} +3.18998 q^{93} +4.64076 q^{95} -0.984908 q^{97} +2.12560 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{3} - 3 q^{5} + 2 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q + 5 * q^3 - 3 * q^5 + 2 * q^7 + 5 * q^9	$ 5 q + 5 q^{3} - 3 q^{5} + 2 q^{7} + 5 q^{9} - 3 q^{11} - 4 q^{13} - 3 q^{15} - 7 q^{17} + 2 q^{19} + 2 q^{21} - 13 q^{23} - 2 q^{25} + 5 q^{27} - 3 q^{29} + 12 q^{31} - 3 q^{33} - 10 q^{35} - 7 q^{37} - 4 q^{39} - 16 q^{41} - 3 q^{45} - q^{47} - 17 q^{49} - 7 q^{51} + 3 q^{53} + 23 q^{55} + 2 q^{57} - q^{59} - 22 q^{61} + 2 q^{63} - 20 q^{65} - 2 q^{67} - 13 q^{69} - 9 q^{71} - 28 q^{73} - 2 q^{75} - 10 q^{77} + 28 q^{79} + 5 q^{81} - 7 q^{83} - 11 q^{85} - 3 q^{87} - 30 q^{89} + 13 q^{91} + 12 q^{93} - 3 q^{95} - 33 q^{97} - 3 q^{99}+O(q^{100}) $ 5 * q + 5 * q^3 - 3 * q^5 + 2 * q^7 + 5 * q^9 - 3 * q^11 - 4 * q^13 - 3 * q^15 - 7 * q^17 + 2 * q^19 + 2 * q^21 - 13 * q^23 - 2 * q^25 + 5 * q^27 - 3 * q^29 + 12 * q^31 - 3 * q^33 - 10 * q^35 - 7 * q^37 - 4 * q^39 - 16 * q^41 - 3 * q^45 - q^47 - 17 * q^49 - 7 * q^51 + 3 * q^53 + 23 * q^55 + 2 * q^57 - q^59 - 22 * q^61 + 2 * q^63 - 20 * q^65 - 2 * q^67 - 13 * q^69 - 9 * q^71 - 28 * q^73 - 2 * q^75 - 10 * q^77 + 28 * q^79 + 5 * q^81 - 7 * q^83 - 11 * q^85 - 3 * q^87 - 30 * q^89 + 13 * q^91 + 12 * q^93 - 3 * q^95 - 33 * q^97 - 3 * 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
$5$	2.12560	0.950598	0.475299	−	0.879824i	$-0.342340\pi$
$5$	2.12560	0.950598	0.475299	+	0.879824i	$0.342340\pi$
$6$	0	0
$7$	−0.221696	−0.0837931	−0.0418966	−	0.999122i	$-0.513340\pi$
$7$	−0.221696	−0.0837931	−0.0418966	+	0.999122i	$0.513340\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.12560	0.640893	0.320446	−	0.947267i	$-0.396167\pi$
$11$	2.12560	0.640893	0.320446	+	0.947267i	$0.396167\pi$
$12$	0	0
$13$	−2.07866	−0.576516	−0.288258	−	0.957553i	$-0.593076\pi$
$13$	−2.07866	−0.576516	−0.288258	+	0.957553i	$0.593076\pi$
$14$	0	0
$15$	2.12560	0.548828
$16$	0	0
$17$	−5.53728	−1.34299	−0.671494	−	0.741010i	$-0.734347\pi$
$17$	−5.53728	−1.34299	−0.671494	+	0.741010i	$0.734347\pi$
$18$	0	0
$19$	2.18327	0.500876	0.250438	−	0.968133i	$-0.419425\pi$
$19$	2.18327	0.500876	0.250438	+	0.968133i	$0.419425\pi$
$20$	0	0
$21$	−0.221696	−0.0483780
$22$	0	0
$23$	−8.50926	−1.77430	−0.887152	−	0.461477i	$-0.847320\pi$
$23$	−8.50926	−1.77430	−0.887152	+	0.461477i	$0.847320\pi$
$24$	0	0
$25$	−0.481818	−0.0963636
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−9.67698	−1.79697	−0.898485	−	0.439003i	$-0.855332\pi$
$29$	−9.67698	−1.79697	−0.898485	+	0.439003i	$0.855332\pi$
$30$	0	0
$31$	3.18998	0.572938	0.286469	−	0.958089i	$-0.407518\pi$
$31$	3.18998	0.572938	0.286469	+	0.958089i	$0.407518\pi$
$32$	0	0
$33$	2.12560	0.370020
$34$	0	0
$35$	−0.471237	−0.0796536
$36$	0	0
$37$	−5.07948	−0.835061	−0.417530	−	0.908663i	$-0.637104\pi$
$37$	−5.07948	−0.835061	−0.417530	+	0.908663i	$0.637104\pi$
$38$	0	0
$39$	−2.07866	−0.332851
$40$	0	0
$41$	−3.43857	−0.537014	−0.268507	−	0.963278i	$-0.586530\pi$
$41$	−3.43857	−0.537014	−0.268507	+	0.963278i	$0.586530\pi$
$42$	0	0
$43$	−5.53327	−0.843816	−0.421908	−	0.906639i	$-0.638640\pi$
$43$	−5.53327	−0.843816	−0.421908	+	0.906639i	$0.638640\pi$
$44$	0	0
$45$	2.12560	0.316866
$46$	0	0
$47$	11.8218	1.72439	0.862195	−	0.506576i	$-0.169089\pi$
$47$	11.8218	1.72439	0.862195	+	0.506576i	$0.169089\pi$
$48$	0	0
$49$	−6.95085	−0.992979
$50$	0	0
$51$	−5.53728	−0.775375
$52$	0	0
$53$	2.21498	0.304251	0.152126	−	0.988361i	$-0.451388\pi$
$53$	2.21498	0.304251	0.152126	+	0.988361i	$0.451388\pi$
$54$	0	0
$55$	4.51818	0.609232
$56$	0	0
$57$	2.18327	0.289181
$58$	0	0
$59$	−6.60234	−0.859551	−0.429775	−	0.902936i	$-0.641407\pi$
$59$	−6.60234	−0.859551	−0.429775	+	0.902936i	$0.641407\pi$
$60$	0	0
$61$	9.72573	1.24525	0.622626	−	0.782519i	$-0.286066\pi$
$61$	9.72573	1.24525	0.622626	+	0.782519i	$0.286066\pi$
$62$	0	0
$63$	−0.221696	−0.0279310
$64$	0	0
$65$	−4.41840	−0.548034
$66$	0	0
$67$	0.214807	0.0262429	0.0131215	−	0.999914i	$-0.495823\pi$
$67$	0.214807	0.0262429	0.0131215	+	0.999914i	$0.495823\pi$
$68$	0	0
$69$	−8.50926	−1.02439
$70$	0	0
$71$	−12.8354	−1.52329	−0.761643	−	0.647997i	$-0.775607\pi$
$71$	−12.8354	−1.52329	−0.761643	+	0.647997i	$0.775607\pi$
$72$	0	0
$73$	−12.4910	−1.46196	−0.730981	−	0.682398i	$-0.760937\pi$
$73$	−12.4910	−1.46196	−0.730981	+	0.682398i	$0.760937\pi$
$74$	0	0
$75$	−0.481818	−0.0556355
$76$	0	0
$77$	−0.471237	−0.0537024
$78$	0	0
$79$	−7.45038	−0.838233	−0.419116	−	0.907932i	$-0.637660\pi$
$79$	−7.45038	−0.838233	−0.419116	+	0.907932i	$0.637660\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−15.9723	−1.75318	−0.876591	−	0.481236i	$-0.840188\pi$
$83$	−15.9723	−1.75318	−0.876591	+	0.481236i	$0.840188\pi$
$84$	0	0
$85$	−11.7701	−1.27664
$86$	0	0
$87$	−9.67698	−1.03748
$88$	0	0
$89$	−6.46935	−0.685750	−0.342875	−	0.939381i	$-0.611401\pi$
$89$	−6.46935	−0.685750	−0.342875	+	0.939381i	$0.611401\pi$
$90$	0	0
$91$	0.460829	0.0483080
$92$	0	0
$93$	3.18998	0.330786
$94$	0	0
$95$	4.64076	0.476132
$96$	0	0
$97$	−0.984908	−0.100002	−0.0500011	−	0.998749i	$-0.515922\pi$
$97$	−0.984908	−0.100002	−0.0500011	+	0.998749i	$0.515922\pi$
$98$	0	0
$99$	2.12560	0.213631
$100$	0	0
$101$	3.30184	0.328545	0.164273	−	0.986415i	$-0.447472\pi$
$101$	3.30184	0.328545	0.164273	+	0.986415i	$0.447472\pi$
$102$	0	0
$103$	11.0295	1.08677	0.543385	−	0.839484i	$-0.317142\pi$
$103$	11.0295	1.08677	0.543385	+	0.839484i	$0.317142\pi$
$104$	0	0
$105$	−0.471237	−0.0459880
$106$	0	0
$107$	18.4356	1.78223	0.891117	−	0.453775i	$-0.149923\pi$
$107$	18.4356	1.78223	0.891117	+	0.453775i	$0.149923\pi$
$108$	0	0
$109$	−1.23544	−0.118334	−0.0591669	−	0.998248i	$-0.518844\pi$
$109$	−1.23544	−0.118334	−0.0591669	+	0.998248i	$0.518844\pi$
$110$	0	0
$111$	−5.07948	−0.482122
$112$	0	0
$113$	−16.2471	−1.52840	−0.764200	−	0.644980i	$-0.776866\pi$
$113$	−16.2471	−1.52840	−0.764200	+	0.644980i	$0.776866\pi$
$114$	0	0
$115$	−18.0873	−1.68665
$116$	0	0
$117$	−2.07866	−0.192172
$118$	0	0
$119$	1.22759	0.112533
$120$	0	0
$121$	−6.48182	−0.589256
$122$	0	0
$123$	−3.43857	−0.310045
$124$	0	0
$125$	−11.6522	−1.04220
$126$	0	0
$127$	4.17746	0.370689	0.185345	−	0.982674i	$-0.440660\pi$
$127$	4.17746	0.370689	0.185345	+	0.982674i	$0.440660\pi$
$128$	0	0
$129$	−5.53327	−0.487178
$130$	0	0
$131$	18.7764	1.64050	0.820250	−	0.572006i	$-0.193835\pi$
$131$	18.7764	1.64050	0.820250	+	0.572006i	$0.193835\pi$
$132$	0	0
$133$	−0.484022	−0.0419700
$134$	0	0
$135$	2.12560	0.182943
$136$	0	0
$137$	3.69811	0.315951	0.157976	−	0.987443i	$-0.449503\pi$
$137$	3.69811	0.315951	0.157976	+	0.987443i	$0.449503\pi$
$138$	0	0
$139$	19.8436	1.68312	0.841558	−	0.540167i	$-0.181639\pi$
$139$	19.8436	1.68312	0.841558	+	0.540167i	$0.181639\pi$
$140$	0	0
$141$	11.8218	0.995577
$142$	0	0
$143$	−4.41840	−0.369485
$144$	0	0
$145$	−20.5694	−1.70820
$146$	0	0
$147$	−6.95085	−0.573297
$148$	0	0
$149$	15.8494	1.29844	0.649219	−	0.760602i	$-0.275096\pi$
$149$	15.8494	1.29844	0.649219	+	0.760602i	$0.275096\pi$
$150$	0	0
$151$	3.34230	0.271992	0.135996	−	0.990709i	$-0.456577\pi$
$151$	3.34230	0.271992	0.135996	+	0.990709i	$0.456577\pi$
$152$	0	0
$153$	−5.53728	−0.447663
$154$	0	0
$155$	6.78064	0.544634
$156$	0	0
$157$	−1.85583	−0.148111	−0.0740555	−	0.997254i	$-0.523594\pi$
$157$	−1.85583	−0.148111	−0.0740555	+	0.997254i	$0.523594\pi$
$158$	0	0
$159$	2.21498	0.175659
$160$	0	0
$161$	1.88647	0.148675
$162$	0	0
$163$	22.6630	1.77510	0.887552	−	0.460707i	$-0.152404\pi$
$163$	22.6630	1.77510	0.887552	+	0.460707i	$0.152404\pi$
$164$	0	0
$165$	4.51818	0.351740
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−8.67919	−0.667630
$170$	0	0
$171$	2.18327	0.166959
$172$	0	0
$173$	6.18615	0.470324	0.235162	−	0.971956i	$-0.424438\pi$
$173$	6.18615	0.470324	0.235162	+	0.971956i	$0.424438\pi$
$174$	0	0
$175$	0.106817	0.00807461
$176$	0	0
$177$	−6.60234	−0.496262
$178$	0	0
$179$	−22.1116	−1.65270	−0.826349	−	0.563158i	$-0.809586\pi$
$179$	−22.1116	−1.65270	−0.826349	+	0.563158i	$0.809586\pi$
$180$	0	0
$181$	−5.15182	−0.382931	−0.191466	−	0.981499i	$-0.561324\pi$
$181$	−5.15182	−0.382931	−0.191466	+	0.981499i	$0.561324\pi$
$182$	0	0
$183$	9.72573	0.718947
$184$	0	0
$185$	−10.7969	−0.793807
$186$	0	0
$187$	−11.7701	−0.860712
$188$	0	0
$189$	−0.221696	−0.0161260
$190$	0	0
$191$	12.2339	0.885217	0.442608	−	0.896715i	$-0.354053\pi$
$191$	12.2339	0.885217	0.442608	+	0.896715i	$0.354053\pi$
$192$	0	0
$193$	20.6509	1.48649	0.743244	−	0.669021i	$-0.233286\pi$
$193$	20.6509	1.48649	0.743244	+	0.669021i	$0.233286\pi$
$194$	0	0
$195$	−4.41840	−0.316408
$196$	0	0
$197$	19.1452	1.36404	0.682020	−	0.731334i	$-0.261102\pi$
$197$	19.1452	1.36404	0.682020	+	0.731334i	$0.261102\pi$
$198$	0	0
$199$	−12.0432	−0.853720	−0.426860	−	0.904318i	$-0.640380\pi$
$199$	−12.0432	−0.853720	−0.426860	+	0.904318i	$0.640380\pi$
$200$	0	0
$201$	0.214807	0.0151513
$202$	0	0
$203$	2.14535	0.150574
$204$	0	0
$205$	−7.30902	−0.510484
$206$	0	0
$207$	−8.50926	−0.591435
$208$	0	0
$209$	4.64076	0.321008
$210$	0	0
$211$	−18.1845	−1.25187	−0.625936	−	0.779874i	$-0.715283\pi$
$211$	−18.1845	−1.25187	−0.625936	+	0.779874i	$0.715283\pi$
$212$	0	0
$213$	−12.8354	−0.879469
$214$	0	0
$215$	−11.7615	−0.802130
$216$	0	0
$217$	−0.707206	−0.0480083
$218$	0	0
$219$	−12.4910	−0.844064
$220$	0	0
$221$	11.5101	0.774254
$222$	0	0
$223$	15.5932	1.04420	0.522100	−	0.852884i	$-0.325149\pi$
$223$	15.5932	1.04420	0.522100	+	0.852884i	$0.325149\pi$
$224$	0	0
$225$	−0.481818	−0.0321212
$226$	0	0
$227$	−14.4872	−0.961547	−0.480774	−	0.876845i	$-0.659644\pi$
$227$	−14.4872	−0.961547	−0.480774	+	0.876845i	$0.659644\pi$
$228$	0	0
$229$	−24.9771	−1.65053	−0.825266	−	0.564744i	$-0.808975\pi$
$229$	−24.9771	−1.65053	−0.825266	+	0.564744i	$0.808975\pi$
$230$	0	0
$231$	−0.471237	−0.0310051
$232$	0	0
$233$	19.3461	1.26740	0.633702	−	0.773577i	$-0.281535\pi$
$233$	19.3461	1.26740	0.633702	+	0.773577i	$0.281535\pi$
$234$	0	0
$235$	25.1285	1.63920
$236$	0	0
$237$	−7.45038	−0.483954
$238$	0	0
$239$	0.789530	0.0510705	0.0255352	−	0.999674i	$-0.491871\pi$
$239$	0.789530	0.0510705	0.0255352	+	0.999674i	$0.491871\pi$
$240$	0	0
$241$	−11.7550	−0.757208	−0.378604	−	0.925559i	$-0.623596\pi$
$241$	−11.7550	−0.757208	−0.378604	+	0.925559i	$0.623596\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−14.7747	−0.943924
$246$	0	0
$247$	−4.53827	−0.288763
$248$	0	0
$249$	−15.9723	−1.01220
$250$	0	0
$251$	31.2564	1.97289	0.986443	−	0.164103i	$-0.0524731\pi$
$251$	31.2564	1.97289	0.986443	+	0.164103i	$0.0524731\pi$
$252$	0	0
$253$	−18.0873	−1.13714
$254$	0	0
$255$	−11.7701	−0.737069
$256$	0	0
$257$	4.93107	0.307592	0.153796	−	0.988103i	$-0.450850\pi$
$257$	4.93107	0.307592	0.153796	+	0.988103i	$0.450850\pi$
$258$	0	0
$259$	1.12610	0.0699723
$260$	0	0
$261$	−9.67698	−0.598990
$262$	0	0
$263$	−28.3288	−1.74683	−0.873415	−	0.486976i	$-0.838100\pi$
$263$	−28.3288	−1.74683	−0.873415	+	0.486976i	$0.838100\pi$
$264$	0	0
$265$	4.70817	0.289220
$266$	0	0
$267$	−6.46935	−0.395918
$268$	0	0
$269$	13.4412	0.819526	0.409763	−	0.912192i	$-0.365611\pi$
$269$	13.4412	0.819526	0.409763	+	0.912192i	$0.365611\pi$
$270$	0	0
$271$	9.08921	0.552130	0.276065	−	0.961139i	$-0.410970\pi$
$271$	9.08921	0.552130	0.276065	+	0.961139i	$0.410970\pi$
$272$	0	0
$273$	0.460829	0.0278907
$274$	0	0
$275$	−1.02415	−0.0617587
$276$	0	0
$277$	−15.7966	−0.949124	−0.474562	−	0.880222i	$-0.657394\pi$
$277$	−15.7966	−0.949124	−0.474562	+	0.880222i	$0.657394\pi$
$278$	0	0
$279$	3.18998	0.190979
$280$	0	0
$281$	−4.10473	−0.244868	−0.122434	−	0.992477i	$-0.539070\pi$
$281$	−4.10473	−0.244868	−0.122434	+	0.992477i	$0.539070\pi$
$282$	0	0
$283$	−23.4844	−1.39600	−0.698000	−	0.716098i	$-0.745927\pi$
$283$	−23.4844	−1.39600	−0.698000	+	0.716098i	$0.745927\pi$
$284$	0	0
$285$	4.64076	0.274895
$286$	0	0
$287$	0.762315	0.0449981
$288$	0	0
$289$	13.6615	0.803617
$290$	0	0
$291$	−0.984908	−0.0577363
$292$	0	0
$293$	18.3264	1.07064	0.535321	−	0.844649i	$-0.320191\pi$
$293$	18.3264	1.07064	0.535321	+	0.844649i	$0.320191\pi$
$294$	0	0
$295$	−14.0339	−0.817087
$296$	0	0
$297$	2.12560	0.123340
$298$	0	0
$299$	17.6878	1.02291
$300$	0	0
$301$	1.22670	0.0707060
$302$	0	0
$303$	3.30184	0.189686
$304$	0	0
$305$	20.6730	1.18373
$306$	0	0
$307$	25.0476	1.42954	0.714770	−	0.699359i	$-0.246531\pi$
$307$	25.0476	1.42954	0.714770	+	0.699359i	$0.246531\pi$
$308$	0	0
$309$	11.0295	0.627447
$310$	0	0
$311$	−11.2135	−0.635861	−0.317931	−	0.948114i	$-0.602988\pi$
$311$	−11.2135	−0.635861	−0.317931	+	0.948114i	$0.602988\pi$
$312$	0	0
$313$	2.39275	0.135246	0.0676232	−	0.997711i	$-0.478458\pi$
$313$	2.39275	0.135246	0.0676232	+	0.997711i	$0.478458\pi$
$314$	0	0
$315$	−0.471237	−0.0265512
$316$	0	0
$317$	32.6943	1.83629	0.918147	−	0.396239i	$-0.129685\pi$
$317$	32.6943	1.83629	0.918147	+	0.396239i	$0.129685\pi$
$318$	0	0
$319$	−20.5694	−1.15167
$320$	0	0
$321$	18.4356	1.02897
$322$	0	0
$323$	−12.0894	−0.672671
$324$	0	0
$325$	1.00153	0.0555551
$326$	0	0
$327$	−1.23544	−0.0683200
$328$	0	0
$329$	−2.62085	−0.144492
$330$	0	0
$331$	17.2684	0.949154	0.474577	−	0.880214i	$-0.342601\pi$
$331$	17.2684	0.949154	0.474577	+	0.880214i	$0.342601\pi$
$332$	0	0
$333$	−5.07948	−0.278354
$334$	0	0
$335$	0.456595	0.0249465
$336$	0	0
$337$	28.1993	1.53611	0.768057	−	0.640382i	$-0.221224\pi$
$337$	28.1993	1.53611	0.768057	+	0.640382i	$0.221224\pi$
$338$	0	0
$339$	−16.2471	−0.882422
$340$	0	0
$341$	6.78064	0.367192
$342$	0	0
$343$	3.09285	0.166998
$344$	0	0
$345$	−18.0873	−0.973788
$346$	0	0
$347$	−7.97291	−0.428008	−0.214004	−	0.976833i	$-0.568651\pi$
$347$	−7.97291	−0.428008	−0.214004	+	0.976833i	$0.568651\pi$
$348$	0	0
$349$	24.6252	1.31815	0.659077	−	0.752075i	$-0.270947\pi$
$349$	24.6252	1.31815	0.659077	+	0.752075i	$0.270947\pi$
$350$	0	0
$351$	−2.07866	−0.110950
$352$	0	0
$353$	2.38038	0.126695	0.0633473	−	0.997992i	$-0.479822\pi$
$353$	2.38038	0.126695	0.0633473	+	0.997992i	$0.479822\pi$
$354$	0	0
$355$	−27.2830	−1.44803
$356$	0	0
$357$	1.22759	0.0649711
$358$	0	0
$359$	−19.3781	−1.02274	−0.511369	−	0.859361i	$-0.670862\pi$
$359$	−19.3781	−1.02274	−0.511369	+	0.859361i	$0.670862\pi$
$360$	0	0
$361$	−14.2333	−0.749123
$362$	0	0
$363$	−6.48182	−0.340207
$364$	0	0
$365$	−26.5509	−1.38974
$366$	0	0
$367$	−9.87368	−0.515402	−0.257701	−	0.966225i	$-0.582965\pi$
$367$	−9.87368	−0.515402	−0.257701	+	0.966225i	$0.582965\pi$
$368$	0	0
$369$	−3.43857	−0.179005
$370$	0	0
$371$	−0.491052	−0.0254941
$372$	0	0
$373$	−30.4976	−1.57911	−0.789553	−	0.613683i	$-0.789687\pi$
$373$	−30.4976	−1.57911	−0.789553	+	0.613683i	$0.789687\pi$
$374$	0	0
$375$	−11.6522	−0.601715
$376$	0	0
$377$	20.1151	1.03598
$378$	0	0
$379$	−15.8226	−0.812752	−0.406376	−	0.913706i	$-0.633208\pi$
$379$	−15.8226	−0.812752	−0.406376	+	0.913706i	$0.633208\pi$
$380$	0	0
$381$	4.17746	0.214018
$382$	0	0
$383$	−10.6600	−0.544702	−0.272351	−	0.962198i	$-0.587801\pi$
$383$	−10.6600	−0.544702	−0.272351	+	0.962198i	$0.587801\pi$
$384$	0	0
$385$	−1.00166	−0.0510494
$386$	0	0
$387$	−5.53327	−0.281272
$388$	0	0
$389$	−13.8665	−0.703060	−0.351530	−	0.936177i	$-0.614338\pi$
$389$	−13.8665	−0.703060	−0.351530	+	0.936177i	$0.614338\pi$
$390$	0	0
$391$	47.1182	2.38287
$392$	0	0
$393$	18.7764	0.947143
$394$	0	0
$395$	−15.8365	−0.796822
$396$	0	0
$397$	35.7439	1.79393	0.896966	−	0.442099i	$-0.145766\pi$
$397$	35.7439	1.79393	0.896966	+	0.442099i	$0.145766\pi$
$398$	0	0
$399$	−0.484022	−0.0242314
$400$	0	0
$401$	−30.5873	−1.52745	−0.763727	−	0.645539i	$-0.776633\pi$
$401$	−30.5873	−1.52745	−0.763727	+	0.645539i	$0.776633\pi$
$402$	0	0
$403$	−6.63088	−0.330308
$404$	0	0
$405$	2.12560	0.105622
$406$	0	0
$407$	−10.7969	−0.535184
$408$	0	0
$409$	−0.0787983	−0.00389633	−0.00194816	−	0.999998i	$-0.500620\pi$
$409$	−0.0787983	−0.00389633	−0.00194816	+	0.999998i	$0.500620\pi$
$410$	0	0
$411$	3.69811	0.182415
$412$	0	0
$413$	1.46371	0.0720245
$414$	0	0
$415$	−33.9506	−1.66657
$416$	0	0
$417$	19.8436	0.971747
$418$	0	0
$419$	−27.0551	−1.32173	−0.660865	−	0.750505i	$-0.729811\pi$
$419$	−27.0551	−1.32173	−0.660865	+	0.750505i	$0.729811\pi$
$420$	0	0
$421$	−39.3526	−1.91793	−0.958964	−	0.283529i	$-0.908495\pi$
$421$	−39.3526	−1.91793	−0.958964	+	0.283529i	$0.908495\pi$
$422$	0	0
$423$	11.8218	0.574797
$424$	0	0
$425$	2.66796	0.129415
$426$	0	0
$427$	−2.15615	−0.104344
$428$	0	0
$429$	−4.41840	−0.213322
$430$	0	0
$431$	1.03817	0.0500068	0.0250034	−	0.999687i	$-0.492040\pi$
$431$	1.03817	0.0500068	0.0250034	+	0.999687i	$0.492040\pi$
$432$	0	0
$433$	20.6291	0.991370	0.495685	−	0.868502i	$-0.334917\pi$
$433$	20.6291	0.991370	0.495685	+	0.868502i	$0.334917\pi$
$434$	0	0
$435$	−20.5694	−0.986228
$436$	0	0
$437$	−18.5780	−0.888707
$438$	0	0
$439$	26.1283	1.24704	0.623518	−	0.781809i	$-0.285703\pi$
$439$	26.1283	1.24704	0.623518	+	0.781809i	$0.285703\pi$
$440$	0	0
$441$	−6.95085	−0.330993
$442$	0	0
$443$	−21.8113	−1.03629	−0.518144	−	0.855293i	$-0.673377\pi$
$443$	−21.8113	−1.03629	−0.518144	+	0.855293i	$0.673377\pi$
$444$	0	0
$445$	−13.7513	−0.651872
$446$	0	0
$447$	15.8494	0.749653
$448$	0	0
$449$	−35.4070	−1.67096	−0.835479	−	0.549522i	$-0.814810\pi$
$449$	−35.4070	−1.67096	−0.835479	+	0.549522i	$0.814810\pi$
$450$	0	0
$451$	−7.30902	−0.344168
$452$	0	0
$453$	3.34230	0.157035
$454$	0	0
$455$	0.979540	0.0459215
$456$	0	0
$457$	18.1615	0.849559	0.424780	−	0.905297i	$-0.360352\pi$
$457$	18.1615	0.849559	0.424780	+	0.905297i	$0.360352\pi$
$458$	0	0
$459$	−5.53728	−0.258458
$460$	0	0
$461$	2.47013	0.115046	0.0575228	−	0.998344i	$-0.481680\pi$
$461$	2.47013	0.115046	0.0575228	+	0.998344i	$0.481680\pi$
$462$	0	0
$463$	24.8116	1.15309	0.576545	−	0.817065i	$-0.304400\pi$
$463$	24.8116	1.15309	0.576545	+	0.817065i	$0.304400\pi$
$464$	0	0
$465$	6.78064	0.314444
$466$	0	0
$467$	21.4827	0.994102	0.497051	−	0.867721i	$-0.334416\pi$
$467$	21.4827	0.994102	0.497051	+	0.867721i	$0.334416\pi$
$468$	0	0
$469$	−0.0476219	−0.00219898
$470$	0	0
$471$	−1.85583	−0.0855120
$472$	0	0
$473$	−11.7615	−0.540796
$474$	0	0
$475$	−1.05194	−0.0482662
$476$	0	0
$477$	2.21498	0.101417
$478$	0	0
$479$	3.90720	0.178524	0.0892622	−	0.996008i	$-0.471549\pi$
$479$	3.90720	0.178524	0.0892622	+	0.996008i	$0.471549\pi$
$480$	0	0
$481$	10.5585	0.481425
$482$	0	0
$483$	1.88647	0.0858373
$484$	0	0
$485$	−2.09352	−0.0950619
$486$	0	0
$487$	2.72645	0.123547	0.0617737	−	0.998090i	$-0.480324\pi$
$487$	2.72645	0.123547	0.0617737	+	0.998090i	$0.480324\pi$
$488$	0	0
$489$	22.6630	1.02486
$490$	0	0
$491$	−32.0967	−1.44850	−0.724252	−	0.689536i	$-0.757815\pi$
$491$	−32.0967	−1.44850	−0.724252	+	0.689536i	$0.757815\pi$
$492$	0	0
$493$	53.5842	2.41331
$494$	0	0
$495$	4.51818	0.203077
$496$	0	0
$497$	2.84556	0.127641
$498$	0	0
$499$	37.1880	1.66476	0.832381	−	0.554204i	$-0.186977\pi$
$499$	37.1880	1.66476	0.832381	+	0.554204i	$0.186977\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	−20.9373	−0.933550	−0.466775	−	0.884376i	$-0.654584\pi$
$503$	−20.9373	−0.933550	−0.466775	+	0.884376i	$0.654584\pi$
$504$	0	0
$505$	7.01840	0.312315
$506$	0	0
$507$	−8.67919	−0.385456
$508$	0	0
$509$	25.2811	1.12056	0.560282	−	0.828302i	$-0.310693\pi$
$509$	25.2811	1.12056	0.560282	+	0.828302i	$0.310693\pi$
$510$	0	0
$511$	2.76920	0.122502
$512$	0	0
$513$	2.18327	0.0963937
$514$	0	0
$515$	23.4443	1.03308
$516$	0	0
$517$	25.1285	1.10515
$518$	0	0
$519$	6.18615	0.271542
$520$	0	0
$521$	−18.8514	−0.825893	−0.412947	−	0.910755i	$-0.635500\pi$
$521$	−18.8514	−0.825893	−0.412947	+	0.910755i	$0.635500\pi$
$522$	0	0
$523$	0.790161	0.0345513	0.0172757	−	0.999851i	$-0.494501\pi$
$523$	0.790161	0.0345513	0.0172757	+	0.999851i	$0.494501\pi$
$524$	0	0
$525$	0.106817	0.00466188
$526$	0	0
$527$	−17.6638	−0.769449
$528$	0	0
$529$	49.4076	2.14815
$530$	0	0
$531$	−6.60234	−0.286517
$532$	0	0
$533$	7.14760	0.309597
$534$	0	0
$535$	39.1867	1.69419
$536$	0	0
$537$	−22.1116	−0.954186
$538$	0	0
$539$	−14.7747	−0.636393
$540$	0	0
$541$	−13.9752	−0.600840	−0.300420	−	0.953807i	$-0.597127\pi$
$541$	−13.9752	−0.600840	−0.300420	+	0.953807i	$0.597127\pi$
$542$	0	0
$543$	−5.15182	−0.221086
$544$	0	0
$545$	−2.62606	−0.112488
$546$	0	0
$547$	−7.27623	−0.311109	−0.155555	−	0.987827i	$-0.549716\pi$
$547$	−7.27623	−0.311109	−0.155555	+	0.987827i	$0.549716\pi$
$548$	0	0
$549$	9.72573	0.415084
$550$	0	0
$551$	−21.1275	−0.900060
$552$	0	0
$553$	1.65172	0.0702382
$554$	0	0
$555$	−10.7969	−0.458305
$556$	0	0
$557$	2.84469	0.120533	0.0602667	−	0.998182i	$-0.480805\pi$
$557$	2.84469	0.120533	0.0602667	+	0.998182i	$0.480805\pi$
$558$	0	0
$559$	11.5018	0.486473
$560$	0	0
$561$	−11.7701	−0.496932
$562$	0	0
$563$	−14.6772	−0.618570	−0.309285	−	0.950969i	$-0.600090\pi$
$563$	−14.6772	−0.618570	−0.309285	+	0.950969i	$0.600090\pi$
$564$	0	0
$565$	−34.5349	−1.45289
$566$	0	0
$567$	−0.221696	−0.00931035
$568$	0	0
$569$	−20.9857	−0.879766	−0.439883	−	0.898055i	$-0.644980\pi$
$569$	−20.9857	−0.879766	−0.439883	+	0.898055i	$0.644980\pi$
$570$	0	0
$571$	−47.1954	−1.97507	−0.987533	−	0.157412i	$-0.949685\pi$
$571$	−47.1954	−1.97507	−0.987533	+	0.157412i	$0.949685\pi$
$572$	0	0
$573$	12.2339	0.511080
$574$	0	0
$575$	4.09992	0.170978
$576$	0	0
$577$	8.43182	0.351021	0.175511	−	0.984478i	$-0.443842\pi$
$577$	8.43182	0.351021	0.175511	+	0.984478i	$0.443842\pi$
$578$	0	0
$579$	20.6509	0.858224
$580$	0	0
$581$	3.54098	0.146905
$582$	0	0
$583$	4.70817	0.194992
$584$	0	0
$585$	−4.41840	−0.182678
$586$	0	0
$587$	−2.17179	−0.0896395	−0.0448198	−	0.998995i	$-0.514271\pi$
$587$	−2.17179	−0.0896395	−0.0448198	+	0.998995i	$0.514271\pi$
$588$	0	0
$589$	6.96460	0.286971
$590$	0	0
$591$	19.1452	0.787528
$592$	0	0
$593$	−17.0848	−0.701587	−0.350794	−	0.936453i	$-0.614088\pi$
$593$	−17.0848	−0.701587	−0.350794	+	0.936453i	$0.614088\pi$
$594$	0	0
$595$	2.60937	0.106974
$596$	0	0
$597$	−12.0432	−0.492896
$598$	0	0
$599$	35.0599	1.43251	0.716255	−	0.697839i	$-0.245855\pi$
$599$	35.0599	1.43251	0.716255	+	0.697839i	$0.245855\pi$
$600$	0	0
$601$	−10.3582	−0.422519	−0.211259	−	0.977430i	$-0.567756\pi$
$601$	−10.3582	−0.422519	−0.211259	+	0.977430i	$0.567756\pi$
$602$	0	0
$603$	0.214807	0.00874764
$604$	0	0
$605$	−13.7778	−0.560146
$606$	0	0
$607$	10.3560	0.420339	0.210169	−	0.977665i	$-0.432598\pi$
$607$	10.3560	0.420339	0.210169	+	0.977665i	$0.432598\pi$
$608$	0	0
$609$	2.14535	0.0869338
$610$	0	0
$611$	−24.5735	−0.994138
$612$	0	0
$613$	26.4256	1.06732	0.533660	−	0.845699i	$-0.320816\pi$
$613$	26.4256	1.06732	0.533660	+	0.845699i	$0.320816\pi$
$614$	0	0
$615$	−7.30902	−0.294728
$616$	0	0
$617$	−42.3118	−1.70341	−0.851705	−	0.524022i	$-0.824431\pi$
$617$	−42.3118	−1.70341	−0.851705	+	0.524022i	$0.824431\pi$
$618$	0	0
$619$	−11.5350	−0.463629	−0.231815	−	0.972760i	$-0.574466\pi$
$619$	−11.5350	−0.463629	−0.231815	+	0.972760i	$0.574466\pi$
$620$	0	0
$621$	−8.50926	−0.341465
$622$	0	0
$623$	1.43423	0.0574611
$624$	0	0
$625$	−22.3588	−0.894350
$626$	0	0
$627$	4.64076	0.185334
$628$	0	0
$629$	28.1265	1.12148
$630$	0	0
$631$	−20.9442	−0.833776	−0.416888	−	0.908958i	$-0.636879\pi$
$631$	−20.9442	−0.833776	−0.416888	+	0.908958i	$0.636879\pi$
$632$	0	0
$633$	−18.1845	−0.722769
$634$	0	0
$635$	8.87961	0.352377
$636$	0	0
$637$	14.4484	0.572468
$638$	0	0
$639$	−12.8354	−0.507762
$640$	0	0
$641$	−46.7978	−1.84840	−0.924201	−	0.381908i	$-0.875267\pi$
$641$	−46.7978	−1.84840	−0.924201	+	0.381908i	$0.875267\pi$
$642$	0	0
$643$	37.3967	1.47478	0.737390	−	0.675467i	$-0.236058\pi$
$643$	37.3967	1.47478	0.737390	+	0.675467i	$0.236058\pi$
$644$	0	0
$645$	−11.7615	−0.463110
$646$	0	0
$647$	−6.14478	−0.241576	−0.120788	−	0.992678i	$-0.538542\pi$
$647$	−6.14478	−0.241576	−0.120788	+	0.992678i	$0.538542\pi$
$648$	0	0
$649$	−14.0339	−0.550880
$650$	0	0
$651$	−0.707206	−0.0277176
$652$	0	0
$653$	−8.28774	−0.324324	−0.162162	−	0.986764i	$-0.551847\pi$
$653$	−8.28774	−0.324324	−0.162162	+	0.986764i	$0.551847\pi$
$654$	0	0
$655$	39.9111	1.55946
$656$	0	0
$657$	−12.4910	−0.487321
$658$	0	0
$659$	−17.1321	−0.667371	−0.333686	−	0.942684i	$-0.608292\pi$
$659$	−17.1321	−0.667371	−0.333686	+	0.942684i	$0.608292\pi$
$660$	0	0
$661$	13.6715	0.531761	0.265880	−	0.964006i	$-0.414337\pi$
$661$	13.6715	0.531761	0.265880	+	0.964006i	$0.414337\pi$
$662$	0	0
$663$	11.5101	0.447016
$664$	0	0
$665$	−1.02884	−0.0398966
$666$	0	0
$667$	82.3440	3.18837
$668$	0	0
$669$	15.5932	0.602870
$670$	0	0
$671$	20.6730	0.798074
$672$	0	0
$673$	−29.7525	−1.14687	−0.573437	−	0.819250i	$-0.694390\pi$
$673$	−29.7525	−1.14687	−0.573437	+	0.819250i	$0.694390\pi$
$674$	0	0
$675$	−0.481818	−0.0185452
$676$	0	0
$677$	37.2609	1.43205	0.716026	−	0.698074i	$-0.245959\pi$
$677$	37.2609	1.43205	0.716026	+	0.698074i	$0.245959\pi$
$678$	0	0
$679$	0.218350	0.00837950
$680$	0	0
$681$	−14.4872	−0.555150
$682$	0	0
$683$	−45.9067	−1.75657	−0.878286	−	0.478135i	$-0.841313\pi$
$683$	−45.9067	−1.75657	−0.878286	+	0.478135i	$0.841313\pi$
$684$	0	0
$685$	7.86072	0.300343
$686$	0	0
$687$	−24.9771	−0.952935
$688$	0	0
$689$	−4.60418	−0.175405
$690$	0	0
$691$	−10.4030	−0.395750	−0.197875	−	0.980227i	$-0.563404\pi$
$691$	−10.4030	−0.395750	−0.197875	+	0.980227i	$0.563404\pi$
$692$	0	0
$693$	−0.471237	−0.0179008
$694$	0	0
$695$	42.1797	1.59997
$696$	0	0
$697$	19.0403	0.721203
$698$	0	0
$699$	19.3461	0.731736
$700$	0	0
$701$	2.09973	0.0793057	0.0396528	−	0.999214i	$-0.487375\pi$
$701$	2.09973	0.0793057	0.0396528	+	0.999214i	$0.487375\pi$
$702$	0	0
$703$	−11.0899	−0.418262
$704$	0	0
$705$	25.1285	0.946394
$706$	0	0
$707$	−0.732004	−0.0275299
$708$	0	0
$709$	−44.1091	−1.65655	−0.828275	−	0.560321i	$-0.810678\pi$
$709$	−44.1091	−1.65655	−0.828275	+	0.560321i	$0.810678\pi$
$710$	0	0
$711$	−7.45038	−0.279411
$712$	0	0
$713$	−27.1444	−1.01657
$714$	0	0
$715$	−9.39175	−0.351231
$716$	0	0
$717$	0.789530	0.0294856
$718$	0	0
$719$	9.03699	0.337023	0.168511	−	0.985700i	$-0.446104\pi$
$719$	9.03699	0.337023	0.168511	+	0.985700i	$0.446104\pi$
$720$	0	0
$721$	−2.44520	−0.0910638
$722$	0	0
$723$	−11.7550	−0.437174
$724$	0	0
$725$	4.66254	0.173163
$726$	0	0
$727$	−25.1230	−0.931759	−0.465879	−	0.884848i	$-0.654262\pi$
$727$	−25.1230	−0.931759	−0.465879	+	0.884848i	$0.654262\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	30.6393	1.13324
$732$	0	0
$733$	29.2768	1.08137	0.540683	−	0.841227i	$-0.318166\pi$
$733$	29.2768	1.08137	0.540683	+	0.841227i	$0.318166\pi$
$734$	0	0
$735$	−14.7747	−0.544974
$736$	0	0
$737$	0.456595	0.0168189
$738$	0	0
$739$	−47.8374	−1.75973	−0.879863	−	0.475228i	$-0.842366\pi$
$739$	−47.8374	−1.75973	−0.879863	+	0.475228i	$0.842366\pi$
$740$	0	0
$741$	−4.53827	−0.166717
$742$	0	0
$743$	17.7810	0.652321	0.326160	−	0.945314i	$-0.394245\pi$
$743$	17.7810	0.652321	0.326160	+	0.945314i	$0.394245\pi$
$744$	0	0
$745$	33.6896	1.23429
$746$	0	0
$747$	−15.9723	−0.584394
$748$	0	0
$749$	−4.08709	−0.149339
$750$	0	0
$751$	21.4708	0.783480	0.391740	−	0.920076i	$-0.371873\pi$
$751$	21.4708	0.783480	0.391740	+	0.920076i	$0.371873\pi$
$752$	0	0
$753$	31.2564	1.13905
$754$	0	0
$755$	7.10439	0.258555
$756$	0	0
$757$	48.4916	1.76246	0.881229	−	0.472689i	$-0.156717\pi$
$757$	48.4916	1.76246	0.881229	+	0.472689i	$0.156717\pi$
$758$	0	0
$759$	−18.0873	−0.656528
$760$	0	0
$761$	−0.182652	−0.00662112	−0.00331056	−	0.999995i	$-0.501054\pi$
$761$	−0.182652	−0.00662112	−0.00331056	+	0.999995i	$0.501054\pi$
$762$	0	0
$763$	0.273892	0.00991556
$764$	0	0
$765$	−11.7701	−0.425547
$766$	0	0
$767$	13.7240	0.495544
$768$	0	0
$769$	−6.09075	−0.219638	−0.109819	−	0.993952i	$-0.535027\pi$
$769$	−6.09075	−0.219638	−0.109819	+	0.993952i	$0.535027\pi$
$770$	0	0
$771$	4.93107	0.177588
$772$	0	0
$773$	−9.39591	−0.337947	−0.168974	−	0.985621i	$-0.554045\pi$
$773$	−9.39591	−0.337947	−0.168974	+	0.985621i	$0.554045\pi$
$774$	0	0
$775$	−1.53699	−0.0552104
$776$	0	0
$777$	1.12610	0.0403986
$778$	0	0
$779$	−7.50731	−0.268977
$780$	0	0
$781$	−27.2830	−0.976263
$782$	0	0
$783$	−9.67698	−0.345827
$784$	0	0
$785$	−3.94475	−0.140794
$786$	0	0
$787$	44.3464	1.58078	0.790389	−	0.612605i	$-0.209878\pi$
$787$	44.3464	1.58078	0.790389	+	0.612605i	$0.209878\pi$
$788$	0	0
$789$	−28.3288	−1.00853
$790$	0	0
$791$	3.60192	0.128069
$792$	0	0
$793$	−20.2165	−0.717907
$794$	0	0
$795$	4.70817	0.166981
$796$	0	0
$797$	−52.1209	−1.84622	−0.923109	−	0.384538i	$-0.874361\pi$
$797$	−52.1209	−1.84622	−0.923109	+	0.384538i	$0.874361\pi$
$798$	0	0
$799$	−65.4608	−2.31584
$800$	0	0
$801$	−6.46935	−0.228583
$802$	0	0
$803$	−26.5509	−0.936961
$804$	0	0
$805$	4.00988	0.141330
$806$	0	0
$807$	13.4412	0.473153
$808$	0	0
$809$	−15.8448	−0.557073	−0.278537	−	0.960426i	$-0.589849\pi$
$809$	−15.8448	−0.557073	−0.278537	+	0.960426i	$0.589849\pi$
$810$	0	0
$811$	−47.0530	−1.65225	−0.826127	−	0.563483i	$-0.809461\pi$
$811$	−47.0530	−1.65225	−0.826127	+	0.563483i	$0.809461\pi$
$812$	0	0
$813$	9.08921	0.318772
$814$	0	0
$815$	48.1725	1.68741
$816$	0	0
$817$	−12.0806	−0.422648
$818$	0	0
$819$	0.460829	0.0161027
$820$	0	0
$821$	24.9098	0.869356	0.434678	−	0.900586i	$-0.356862\pi$
$821$	24.9098	0.869356	0.434678	+	0.900586i	$0.356862\pi$
$822$	0	0
$823$	−26.9393	−0.939045	−0.469523	−	0.882920i	$-0.655574\pi$
$823$	−26.9393	−0.939045	−0.469523	+	0.882920i	$0.655574\pi$
$824$	0	0
$825$	−1.02415	−0.0356564
$826$	0	0
$827$	−6.07808	−0.211355	−0.105678	−	0.994400i	$-0.533701\pi$
$827$	−6.07808	−0.211355	−0.105678	+	0.994400i	$0.533701\pi$
$828$	0	0
$829$	−15.7278	−0.546249	−0.273124	−	0.961979i	$-0.588057\pi$
$829$	−15.7278	−0.546249	−0.273124	+	0.961979i	$0.588057\pi$
$830$	0	0
$831$	−15.7966	−0.547977
$832$	0	0
$833$	38.4888	1.33356
$834$	0	0
$835$	−2.12560	−0.0735595
$836$	0	0
$837$	3.18998	0.110262
$838$	0	0
$839$	23.2845	0.803870	0.401935	−	0.915668i	$-0.368338\pi$
$839$	23.2845	0.803870	0.401935	+	0.915668i	$0.368338\pi$
$840$	0	0
$841$	64.6440	2.22910
$842$	0	0
$843$	−4.10473	−0.141374
$844$	0	0
$845$	−18.4485	−0.634648
$846$	0	0
$847$	1.43699	0.0493756
$848$	0	0
$849$	−23.4844	−0.805981
$850$	0	0
$851$	43.2226	1.48165
$852$	0	0
$853$	20.9765	0.718220	0.359110	−	0.933295i	$-0.383080\pi$
$853$	20.9765	0.718220	0.359110	+	0.933295i	$0.383080\pi$
$854$	0	0
$855$	4.64076	0.158711
$856$	0	0
$857$	−17.9102	−0.611800	−0.305900	−	0.952064i	$-0.598957\pi$
$857$	−17.9102	−0.611800	−0.305900	+	0.952064i	$0.598957\pi$
$858$	0	0
$859$	−0.572629	−0.0195379	−0.00976893	−	0.999952i	$-0.503110\pi$
$859$	−0.572629	−0.0195379	−0.00976893	+	0.999952i	$0.503110\pi$
$860$	0	0
$861$	0.762315	0.0259796
$862$	0	0
$863$	20.3880	0.694017	0.347008	−	0.937862i	$-0.387198\pi$
$863$	20.3880	0.694017	0.347008	+	0.937862i	$0.387198\pi$
$864$	0	0
$865$	13.1493	0.447089
$866$	0	0
$867$	13.6615	0.463969
$868$	0	0
$869$	−15.8365	−0.537218
$870$	0	0
$871$	−0.446511	−0.0151294
$872$	0	0
$873$	−0.984908	−0.0333341
$874$	0	0
$875$	2.58324	0.0873293
$876$	0	0
$877$	−35.5316	−1.19982	−0.599909	−	0.800069i	$-0.704796\pi$
$877$	−35.5316	−1.19982	−0.599909	+	0.800069i	$0.704796\pi$
$878$	0	0
$879$	18.3264	0.618135
$880$	0	0
$881$	−23.1356	−0.779458	−0.389729	−	0.920930i	$-0.627431\pi$
$881$	−23.1356	−0.779458	−0.389729	+	0.920930i	$0.627431\pi$
$882$	0	0
$883$	29.7695	1.00182	0.500912	−	0.865498i	$-0.332998\pi$
$883$	29.7695	1.00182	0.500912	+	0.865498i	$0.332998\pi$
$884$	0	0
$885$	−14.0339	−0.471746
$886$	0	0
$887$	−33.5567	−1.12672	−0.563362	−	0.826210i	$-0.690492\pi$
$887$	−33.5567	−1.12672	−0.563362	+	0.826210i	$0.690492\pi$
$888$	0	0
$889$	−0.926125	−0.0310612
$890$	0	0
$891$	2.12560	0.0712103
$892$	0	0
$893$	25.8102	0.863707
$894$	0	0
$895$	−47.0004	−1.57105
$896$	0	0
$897$	17.6878	0.590580
$898$	0	0
$899$	−30.8694	−1.02955
$900$	0	0
$901$	−12.2650	−0.408606
$902$	0	0
$903$	1.22670	0.0408221
$904$	0	0
$905$	−10.9507	−0.364014
$906$	0	0
$907$	1.20646	0.0400597	0.0200298	−	0.999799i	$-0.493624\pi$
$907$	1.20646	0.0400597	0.0200298	+	0.999799i	$0.493624\pi$
$908$	0	0
$909$	3.30184	0.109515
$910$	0	0
$911$	−28.8600	−0.956173	−0.478086	−	0.878313i	$-0.658669\pi$
$911$	−28.8600	−0.956173	−0.478086	+	0.878313i	$0.658669\pi$
$912$	0	0
$913$	−33.9506	−1.12360
$914$	0	0
$915$	20.6730	0.683429
$916$	0	0
$917$	−4.16264	−0.137463
$918$	0	0
$919$	−33.3352	−1.09963	−0.549814	−	0.835287i	$-0.685301\pi$
$919$	−33.3352	−1.09963	−0.549814	+	0.835287i	$0.685301\pi$
$920$	0	0
$921$	25.0476	0.825346
$922$	0	0
$923$	26.6804	0.878198
$924$	0	0
$925$	2.44738	0.0804694
$926$	0	0
$927$	11.0295	0.362257
$928$	0	0
$929$	36.7007	1.20411	0.602056	−	0.798454i	$-0.294348\pi$
$929$	36.7007	1.20411	0.602056	+	0.798454i	$0.294348\pi$
$930$	0	0
$931$	−15.1756	−0.497360
$932$	0	0
$933$	−11.2135	−0.367115
$934$	0	0
$935$	−25.0184	−0.818191
$936$	0	0
$937$	7.42782	0.242656	0.121328	−	0.992612i	$-0.461285\pi$
$937$	7.42782	0.242656	0.121328	+	0.992612i	$0.461285\pi$
$938$	0	0
$939$	2.39275	0.0780846
$940$	0	0
$941$	−6.94417	−0.226374	−0.113187	−	0.993574i	$-0.536106\pi$
$941$	−6.94417	−0.226374	−0.113187	+	0.993574i	$0.536106\pi$
$942$	0	0
$943$	29.2597	0.952826
$944$	0	0
$945$	−0.471237	−0.0153293
$946$	0	0
$947$	15.8796	0.516019	0.258009	−	0.966142i	$-0.416933\pi$
$947$	15.8796	0.516019	0.258009	+	0.966142i	$0.416933\pi$
$948$	0	0
$949$	25.9645	0.842844
$950$	0	0
$951$	32.6943	1.06019
$952$	0	0
$953$	59.2310	1.91868	0.959340	−	0.282252i	$-0.0910813\pi$
$953$	59.2310	1.91868	0.959340	+	0.282252i	$0.0910813\pi$
$954$	0	0
$955$	26.0045	0.841485
$956$	0	0
$957$	−20.5694	−0.664915
$958$	0	0
$959$	−0.819856	−0.0264745
$960$	0	0
$961$	−20.8240	−0.671742
$962$	0	0
$963$	18.4356	0.594078
$964$	0	0
$965$	43.8957	1.41305
$966$	0	0
$967$	30.8083	0.990730	0.495365	−	0.868685i	$-0.335034\pi$
$967$	30.8083	0.990730	0.495365	+	0.868685i	$0.335034\pi$
$968$	0	0
$969$	−12.0894	−0.388367
$970$	0	0
$971$	−42.6220	−1.36780	−0.683902	−	0.729573i	$-0.739719\pi$
$971$	−42.6220	−1.36780	−0.683902	+	0.729573i	$0.739719\pi$
$972$	0	0
$973$	−4.39925	−0.141033
$974$	0	0
$975$	1.00153	0.0320748
$976$	0	0
$977$	−6.13049	−0.196132	−0.0980659	−	0.995180i	$-0.531266\pi$
$977$	−6.13049	−0.196132	−0.0980659	+	0.995180i	$0.531266\pi$
$978$	0	0
$979$	−13.7513	−0.439492
$980$	0	0
$981$	−1.23544	−0.0394446
$982$	0	0
$983$	33.9342	1.08233	0.541167	−	0.840915i	$-0.317983\pi$
$983$	33.9342	1.08233	0.541167	+	0.840915i	$0.317983\pi$
$984$	0	0
$985$	40.6951	1.29665
$986$	0	0
$987$	−2.62085	−0.0834226
$988$	0	0
$989$	47.0841	1.49719
$990$	0	0
$991$	14.1117	0.448274	0.224137	−	0.974558i	$-0.428044\pi$
$991$	14.1117	0.448274	0.224137	+	0.974558i	$0.428044\pi$
$992$	0	0
$993$	17.2684	0.547995
$994$	0	0
$995$	−25.5991	−0.811545
$996$	0	0
$997$	16.7835	0.531537	0.265769	−	0.964037i	$-0.414374\pi$
$997$	16.7835	0.531537	0.265769	+	0.964037i	$0.414374\pi$
$998$	0	0
$999$	−5.07948	−0.160707

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.t.1.5		5
4.3	odd	2		2004.2.a.a.1.5	✓	5
12.11	even	2		6012.2.a.e.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2004.2.a.a.1.5	✓	5	4.3	odd	2
6012.2.a.e.1.1		5	12.11	even	2
8016.2.a.t.1.5		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.12560 q^{5} -0.221696 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.12560 q^{5} -0.221696 q^{7} +1.00000 q^{9} +2.12560 q^{11} -2.07866 q^{13} +2.12560 q^{15} -5.53728 q^{17} +2.18327 q^{19} -0.221696 q^{21} -8.50926 q^{23} -0.481818 q^{25} +1.00000 q^{27} -9.67698 q^{29} +3.18998 q^{31} +2.12560 q^{33} -0.471237 q^{35} -5.07948 q^{37} -2.07866 q^{39} -3.43857 q^{41} -5.53327 q^{43} +2.12560 q^{45} +11.8218 q^{47} -6.95085 q^{49} -5.53728 q^{51} +2.21498 q^{53} +4.51818 q^{55} +2.18327 q^{57} -6.60234 q^{59} +9.72573 q^{61} -0.221696 q^{63} -4.41840 q^{65} +0.214807 q^{67} -8.50926 q^{69} -12.8354 q^{71} -12.4910 q^{73} -0.481818 q^{75} -0.471237 q^{77} -7.45038 q^{79} +1.00000 q^{81} -15.9723 q^{83} -11.7701 q^{85} -9.67698 q^{87} -6.46935 q^{89} +0.460829 q^{91} +3.18998 q^{93} +4.64076 q^{95} -0.984908 q^{97} +2.12560 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} - 3 q^{5} + 2 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q + 5 * q^3 - 3 * q^5 + 2 * q^7 + 5 * q^9	\( 5 q + 5 q^{3} - 3 q^{5} + 2 q^{7} + 5 q^{9} - 3 q^{11} - 4 q^{13} - 3 q^{15} - 7 q^{17} + 2 q^{19} + 2 q^{21} - 13 q^{23} - 2 q^{25} + 5 q^{27} - 3 q^{29} + 12 q^{31} - 3 q^{33} - 10 q^{35} - 7 q^{37} - 4 q^{39} - 16 q^{41} - 3 q^{45} - q^{47} - 17 q^{49} - 7 q^{51} + 3 q^{53} + 23 q^{55} + 2 q^{57} - q^{59} - 22 q^{61} + 2 q^{63} - 20 q^{65} - 2 q^{67} - 13 q^{69} - 9 q^{71} - 28 q^{73} - 2 q^{75} - 10 q^{77} + 28 q^{79} + 5 q^{81} - 7 q^{83} - 11 q^{85} - 3 q^{87} - 30 q^{89} + 13 q^{91} + 12 q^{93} - 3 q^{95} - 33 q^{97} - 3 q^{99}+O(q^{100}) \) 5 * q + 5 * q^3 - 3 * q^5 + 2 * q^7 + 5 * q^9 - 3 * q^11 - 4 * q^13 - 3 * q^15 - 7 * q^17 + 2 * q^19 + 2 * q^21 - 13 * q^23 - 2 * q^25 + 5 * q^27 - 3 * q^29 + 12 * q^31 - 3 * q^33 - 10 * q^35 - 7 * q^37 - 4 * q^39 - 16 * q^41 - 3 * q^45 - q^47 - 17 * q^49 - 7 * q^51 + 3 * q^53 + 23 * q^55 + 2 * q^57 - q^59 - 22 * q^61 + 2 * q^63 - 20 * q^65 - 2 * q^67 - 13 * q^69 - 9 * q^71 - 28 * q^73 - 2 * q^75 - 10 * q^77 + 28 * q^79 + 5 * q^81 - 7 * q^83 - 11 * q^85 - 3 * q^87 - 30 * q^89 + 13 * q^91 + 12 * q^93 - 3 * q^95 - 33 * q^97 - 3 * q^99

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