LMFDB - Embedded newform 8016.2.a.u.1.1

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			1.1
Root			$-1.95408$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -3.11102 q^{5} +1.66149 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.11102 q^{5} +1.66149 q^{7} +1.00000 q^{9} +3.41049 q^{11} -3.00721 q^{13} -3.11102 q^{15} +3.92724 q^{17} +2.18157 q^{19} +1.66149 q^{21} -6.61222 q^{23} +4.67846 q^{25} +1.00000 q^{27} +4.24923 q^{29} +4.74901 q^{31} +3.41049 q^{33} -5.16892 q^{35} -6.16450 q^{37} -3.00721 q^{39} -0.415489 q^{41} +4.66904 q^{43} -3.11102 q^{45} +4.14140 q^{47} -4.23947 q^{49} +3.92724 q^{51} +6.14105 q^{53} -10.6101 q^{55} +2.18157 q^{57} +0.638616 q^{59} +5.05636 q^{61} +1.66149 q^{63} +9.35549 q^{65} -1.28363 q^{67} -6.61222 q^{69} -1.84055 q^{71} -5.54682 q^{73} +4.67846 q^{75} +5.66648 q^{77} -13.5514 q^{79} +1.00000 q^{81} +11.8842 q^{83} -12.2177 q^{85} +4.24923 q^{87} -4.57866 q^{89} -4.99643 q^{91} +4.74901 q^{93} -6.78692 q^{95} -5.72903 q^{97} +3.41049 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−3.11102	−1.39129	−0.695646	−	0.718385i	$-0.744882\pi$
$5$	−3.11102	−1.39129	−0.695646	+	0.718385i	$0.744882\pi$
$6$	0	0
$7$	1.66149	0.627982	0.313991	−	0.949426i	$-0.398334\pi$
$7$	1.66149	0.627982	0.313991	+	0.949426i	$0.398334\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.41049	1.02830	0.514151	−	0.857700i	$-0.328107\pi$
$11$	3.41049	1.02830	0.514151	+	0.857700i	$0.328107\pi$
$12$	0	0
$13$	−3.00721	−0.834049	−0.417025	−	0.908895i	$-0.636927\pi$
$13$	−3.00721	−0.834049	−0.417025	+	0.908895i	$0.636927\pi$
$14$	0	0
$15$	−3.11102	−0.803262
$16$	0	0
$17$	3.92724	0.952495	0.476248	−	0.879311i	$-0.341997\pi$
$17$	3.92724	0.952495	0.476248	+	0.879311i	$0.341997\pi$
$18$	0	0
$19$	2.18157	0.500487	0.250244	−	0.968183i	$-0.419489\pi$
$19$	2.18157	0.500487	0.250244	+	0.968183i	$0.419489\pi$
$20$	0	0
$21$	1.66149	0.362566
$22$	0	0
$23$	−6.61222	−1.37874	−0.689372	−	0.724408i	$-0.742113\pi$
$23$	−6.61222	−1.37874	−0.689372	+	0.724408i	$0.742113\pi$
$24$	0	0
$25$	4.67846	0.935691
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	4.24923	0.789062	0.394531	−	0.918883i	$-0.370907\pi$
$29$	4.24923	0.789062	0.394531	+	0.918883i	$0.370907\pi$
$30$	0	0
$31$	4.74901	0.852947	0.426473	−	0.904500i	$-0.359756\pi$
$31$	4.74901	0.852947	0.426473	+	0.904500i	$0.359756\pi$
$32$	0	0
$33$	3.41049	0.593690
$34$	0	0
$35$	−5.16892	−0.873706
$36$	0	0
$37$	−6.16450	−1.01344	−0.506718	−	0.862112i	$-0.669142\pi$
$37$	−6.16450	−1.01344	−0.506718	+	0.862112i	$0.669142\pi$
$38$	0	0
$39$	−3.00721	−0.481538
$40$	0	0
$41$	−0.415489	−0.0648884	−0.0324442	−	0.999474i	$-0.510329\pi$
$41$	−0.415489	−0.0648884	−0.0324442	+	0.999474i	$0.510329\pi$
$42$	0	0
$43$	4.66904	0.712022	0.356011	−	0.934482i	$-0.384137\pi$
$43$	4.66904	0.712022	0.356011	+	0.934482i	$0.384137\pi$
$44$	0	0
$45$	−3.11102	−0.463764
$46$	0	0
$47$	4.14140	0.604085	0.302042	−	0.953294i	$-0.402332\pi$
$47$	4.14140	0.604085	0.302042	+	0.953294i	$0.402332\pi$
$48$	0	0
$49$	−4.23947	−0.605638
$50$	0	0
$51$	3.92724	0.549923
$52$	0	0
$53$	6.14105	0.843538	0.421769	−	0.906703i	$-0.361409\pi$
$53$	6.14105	0.843538	0.421769	+	0.906703i	$0.361409\pi$
$54$	0	0
$55$	−10.6101	−1.43067
$56$	0	0
$57$	2.18157	0.288956
$58$	0	0
$59$	0.638616	0.0831407	0.0415704	−	0.999136i	$-0.486764\pi$
$59$	0.638616	0.0831407	0.0415704	+	0.999136i	$0.486764\pi$
$60$	0	0
$61$	5.05636	0.647401	0.323700	−	0.946160i	$-0.395073\pi$
$61$	5.05636	0.647401	0.323700	+	0.946160i	$0.395073\pi$
$62$	0	0
$63$	1.66149	0.209327
$64$	0	0
$65$	9.35549	1.16041
$66$	0	0
$67$	−1.28363	−0.156820	−0.0784099	−	0.996921i	$-0.524984\pi$
$67$	−1.28363	−0.156820	−0.0784099	+	0.996921i	$0.524984\pi$
$68$	0	0
$69$	−6.61222	−0.796018
$70$	0	0
$71$	−1.84055	−0.218433	−0.109216	−	0.994018i	$-0.534834\pi$
$71$	−1.84055	−0.218433	−0.109216	+	0.994018i	$0.534834\pi$
$72$	0	0
$73$	−5.54682	−0.649206	−0.324603	−	0.945850i	$-0.605231\pi$
$73$	−5.54682	−0.649206	−0.324603	+	0.945850i	$0.605231\pi$
$74$	0	0
$75$	4.67846	0.540222
$76$	0	0
$77$	5.66648	0.645756
$78$	0	0
$79$	−13.5514	−1.52466	−0.762328	−	0.647191i	$-0.775944\pi$
$79$	−13.5514	−1.52466	−0.762328	+	0.647191i	$0.775944\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	11.8842	1.30446	0.652231	−	0.758021i	$-0.273833\pi$
$83$	11.8842	1.30446	0.652231	+	0.758021i	$0.273833\pi$
$84$	0	0
$85$	−12.2177	−1.32520
$86$	0	0
$87$	4.24923	0.455565
$88$	0	0
$89$	−4.57866	−0.485337	−0.242668	−	0.970109i	$-0.578023\pi$
$89$	−4.57866	−0.485337	−0.242668	+	0.970109i	$0.578023\pi$
$90$	0	0
$91$	−4.99643	−0.523768
$92$	0	0
$93$	4.74901	0.492449
$94$	0	0
$95$	−6.78692	−0.696323
$96$	0	0
$97$	−5.72903	−0.581694	−0.290847	−	0.956770i	$-0.593937\pi$
$97$	−5.72903	−0.581694	−0.290847	+	0.956770i	$0.593937\pi$
$98$	0	0
$99$	3.41049	0.342767
$100$	0	0
$101$	−8.50876	−0.846653	−0.423326	−	0.905977i	$-0.639138\pi$
$101$	−8.50876	−0.846653	−0.423326	+	0.905977i	$0.639138\pi$
$102$	0	0
$103$	13.1382	1.29454	0.647271	−	0.762260i	$-0.275910\pi$
$103$	13.1382	1.29454	0.647271	+	0.762260i	$0.275910\pi$
$104$	0	0
$105$	−5.16892	−0.504435
$106$	0	0
$107$	15.5067	1.49909	0.749547	−	0.661951i	$-0.230271\pi$
$107$	15.5067	1.49909	0.749547	+	0.661951i	$0.230271\pi$
$108$	0	0
$109$	6.22370	0.596122	0.298061	−	0.954547i	$-0.403660\pi$
$109$	6.22370	0.596122	0.298061	+	0.954547i	$0.403660\pi$
$110$	0	0
$111$	−6.16450	−0.585108
$112$	0	0
$113$	19.0121	1.78851	0.894253	−	0.447562i	$-0.147708\pi$
$113$	19.0121	1.78851	0.894253	+	0.447562i	$0.147708\pi$
$114$	0	0
$115$	20.5708	1.91823
$116$	0	0
$117$	−3.00721	−0.278016
$118$	0	0
$119$	6.52505	0.598150
$120$	0	0
$121$	0.631456	0.0574051
$122$	0	0
$123$	−0.415489	−0.0374634
$124$	0	0
$125$	1.00033	0.0894723
$126$	0	0
$127$	−15.2875	−1.35655	−0.678273	−	0.734810i	$-0.737271\pi$
$127$	−15.2875	−1.35655	−0.678273	+	0.734810i	$0.737271\pi$
$128$	0	0
$129$	4.66904	0.411086
$130$	0	0
$131$	−8.89251	−0.776942	−0.388471	−	0.921461i	$-0.626997\pi$
$131$	−8.89251	−0.776942	−0.388471	+	0.921461i	$0.626997\pi$
$132$	0	0
$133$	3.62465	0.314297
$134$	0	0
$135$	−3.11102	−0.267754
$136$	0	0
$137$	4.39754	0.375707	0.187854	−	0.982197i	$-0.439847\pi$
$137$	4.39754	0.375707	0.187854	+	0.982197i	$0.439847\pi$
$138$	0	0
$139$	22.4011	1.90003	0.950016	−	0.312200i	$-0.101066\pi$
$139$	22.4011	1.90003	0.950016	+	0.312200i	$0.101066\pi$
$140$	0	0
$141$	4.14140	0.348769
$142$	0	0
$143$	−10.2561	−0.857654
$144$	0	0
$145$	−13.2195	−1.09782
$146$	0	0
$147$	−4.23947	−0.349665
$148$	0	0
$149$	−18.3289	−1.50156	−0.750780	−	0.660553i	$-0.770322\pi$
$149$	−18.3289	−1.50156	−0.750780	+	0.660553i	$0.770322\pi$
$150$	0	0
$151$	15.1183	1.23031	0.615154	−	0.788407i	$-0.289094\pi$
$151$	15.1183	1.23031	0.615154	+	0.788407i	$0.289094\pi$
$152$	0	0
$153$	3.92724	0.317498
$154$	0	0
$155$	−14.7743	−1.18670
$156$	0	0
$157$	−23.7462	−1.89515	−0.947577	−	0.319527i	$-0.896476\pi$
$157$	−23.7462	−1.89515	−0.947577	+	0.319527i	$0.896476\pi$
$158$	0	0
$159$	6.14105	0.487017
$160$	0	0
$161$	−10.9861	−0.865827
$162$	0	0
$163$	3.50387	0.274444	0.137222	−	0.990540i	$-0.456183\pi$
$163$	3.50387	0.274444	0.137222	+	0.990540i	$0.456183\pi$
$164$	0	0
$165$	−10.6101	−0.825996
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−3.95671	−0.304362
$170$	0	0
$171$	2.18157	0.166829
$172$	0	0
$173$	20.4470	1.55456	0.777278	−	0.629157i	$-0.216600\pi$
$173$	20.4470	1.55456	0.777278	+	0.629157i	$0.216600\pi$
$174$	0	0
$175$	7.77318	0.587598
$176$	0	0
$177$	0.638616	0.0480013
$178$	0	0
$179$	4.22494	0.315787	0.157893	−	0.987456i	$-0.449530\pi$
$179$	4.22494	0.315787	0.157893	+	0.987456i	$0.449530\pi$
$180$	0	0
$181$	13.5816	1.00951	0.504757	−	0.863262i	$-0.331582\pi$
$181$	13.5816	1.00951	0.504757	+	0.863262i	$0.331582\pi$
$182$	0	0
$183$	5.05636	0.373777
$184$	0	0
$185$	19.1779	1.40999
$186$	0	0
$187$	13.3938	0.979453
$188$	0	0
$189$	1.66149	0.120855
$190$	0	0
$191$	0.756545	0.0547417	0.0273708	−	0.999625i	$-0.491287\pi$
$191$	0.756545	0.0547417	0.0273708	+	0.999625i	$0.491287\pi$
$192$	0	0
$193$	8.84946	0.636998	0.318499	−	0.947923i	$-0.396821\pi$
$193$	8.84946	0.636998	0.318499	+	0.947923i	$0.396821\pi$
$194$	0	0
$195$	9.35549	0.669960
$196$	0	0
$197$	4.25501	0.303157	0.151579	−	0.988445i	$-0.451564\pi$
$197$	4.25501	0.303157	0.151579	+	0.988445i	$0.451564\pi$
$198$	0	0
$199$	8.85469	0.627692	0.313846	−	0.949474i	$-0.398382\pi$
$199$	8.85469	0.627692	0.313846	+	0.949474i	$0.398382\pi$
$200$	0	0
$201$	−1.28363	−0.0905400
$202$	0	0
$203$	7.06003	0.495517
$204$	0	0
$205$	1.29259	0.0902787
$206$	0	0
$207$	−6.61222	−0.459581
$208$	0	0
$209$	7.44024	0.514652
$210$	0	0
$211$	21.5988	1.48692	0.743461	−	0.668780i	$-0.233183\pi$
$211$	21.5988	1.48692	0.743461	+	0.668780i	$0.233183\pi$
$212$	0	0
$213$	−1.84055	−0.126112
$214$	0	0
$215$	−14.5255	−0.990630
$216$	0	0
$217$	7.89040	0.535636
$218$	0	0
$219$	−5.54682	−0.374820
$220$	0	0
$221$	−11.8100	−0.794428
$222$	0	0
$223$	−28.7963	−1.92835	−0.964173	−	0.265274i	$-0.914538\pi$
$223$	−28.7963	−1.92835	−0.964173	+	0.265274i	$0.914538\pi$
$224$	0	0
$225$	4.67846	0.311897
$226$	0	0
$227$	27.4078	1.81912	0.909560	−	0.415573i	$-0.136419\pi$
$227$	27.4078	1.81912	0.909560	+	0.415573i	$0.136419\pi$
$228$	0	0
$229$	9.93144	0.656288	0.328144	−	0.944628i	$-0.393577\pi$
$229$	9.93144	0.656288	0.328144	+	0.944628i	$0.393577\pi$
$230$	0	0
$231$	5.66648	0.372827
$232$	0	0
$233$	26.9976	1.76867	0.884337	−	0.466849i	$-0.154611\pi$
$233$	26.9976	1.76867	0.884337	+	0.466849i	$0.154611\pi$
$234$	0	0
$235$	−12.8840	−0.840458
$236$	0	0
$237$	−13.5514	−0.880260
$238$	0	0
$239$	29.9620	1.93808	0.969040	−	0.246906i	$-0.0794138\pi$
$239$	29.9620	1.93808	0.969040	+	0.246906i	$0.0794138\pi$
$240$	0	0
$241$	−1.70837	−0.110046	−0.0550228	−	0.998485i	$-0.517523\pi$
$241$	−1.70837	−0.110046	−0.0550228	+	0.998485i	$0.517523\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	13.1891	0.842619
$246$	0	0
$247$	−6.56044	−0.417431
$248$	0	0
$249$	11.8842	0.753131
$250$	0	0
$251$	−26.3057	−1.66040	−0.830202	−	0.557463i	$-0.811775\pi$
$251$	−26.3057	−1.66040	−0.830202	+	0.557463i	$0.811775\pi$
$252$	0	0
$253$	−22.5509	−1.41777
$254$	0	0
$255$	−12.2177	−0.765104
$256$	0	0
$257$	−14.1260	−0.881153	−0.440576	−	0.897715i	$-0.645226\pi$
$257$	−14.1260	−0.881153	−0.440576	+	0.897715i	$0.645226\pi$
$258$	0	0
$259$	−10.2422	−0.636420
$260$	0	0
$261$	4.24923	0.263021
$262$	0	0
$263$	−4.59039	−0.283056	−0.141528	−	0.989934i	$-0.545201\pi$
$263$	−4.59039	−0.283056	−0.141528	+	0.989934i	$0.545201\pi$
$264$	0	0
$265$	−19.1049	−1.17361
$266$	0	0
$267$	−4.57866	−0.280209
$268$	0	0
$269$	2.27436	0.138670	0.0693349	−	0.997593i	$-0.477912\pi$
$269$	2.27436	0.138670	0.0693349	+	0.997593i	$0.477912\pi$
$270$	0	0
$271$	−18.8753	−1.14659	−0.573296	−	0.819348i	$-0.694336\pi$
$271$	−18.8753	−1.14659	−0.573296	+	0.819348i	$0.694336\pi$
$272$	0	0
$273$	−4.99643	−0.302398
$274$	0	0
$275$	15.9558	0.962173
$276$	0	0
$277$	29.2886	1.75978	0.879889	−	0.475179i	$-0.157617\pi$
$277$	29.2886	1.75978	0.879889	+	0.475179i	$0.157617\pi$
$278$	0	0
$279$	4.74901	0.284316
$280$	0	0
$281$	−18.1417	−1.08224	−0.541122	−	0.840944i	$-0.682000\pi$
$281$	−18.1417	−1.08224	−0.541122	+	0.840944i	$0.682000\pi$
$282$	0	0
$283$	13.8879	0.825551	0.412776	−	0.910833i	$-0.364559\pi$
$283$	13.8879	0.825551	0.412776	+	0.910833i	$0.364559\pi$
$284$	0	0
$285$	−6.78692	−0.402022
$286$	0	0
$287$	−0.690328	−0.0407488
$288$	0	0
$289$	−1.57680	−0.0927527
$290$	0	0
$291$	−5.72903	−0.335841
$292$	0	0
$293$	7.95986	0.465020	0.232510	−	0.972594i	$-0.425306\pi$
$293$	7.95986	0.465020	0.232510	+	0.972594i	$0.425306\pi$
$294$	0	0
$295$	−1.98675	−0.115673
$296$	0	0
$297$	3.41049	0.197897
$298$	0	0
$299$	19.8843	1.14994
$300$	0	0
$301$	7.75754	0.447137
$302$	0	0
$303$	−8.50876	−0.488815
$304$	0	0
$305$	−15.7305	−0.900723
$306$	0	0
$307$	−29.9149	−1.70733	−0.853666	−	0.520821i	$-0.825626\pi$
$307$	−29.9149	−1.70733	−0.853666	+	0.520821i	$0.825626\pi$
$308$	0	0
$309$	13.1382	0.747404
$310$	0	0
$311$	17.8528	1.01234	0.506169	−	0.862434i	$-0.331061\pi$
$311$	17.8528	1.01234	0.506169	+	0.862434i	$0.331061\pi$
$312$	0	0
$313$	−14.9079	−0.842642	−0.421321	−	0.906912i	$-0.638433\pi$
$313$	−14.9079	−0.842642	−0.421321	+	0.906912i	$0.638433\pi$
$314$	0	0
$315$	−5.16892	−0.291235
$316$	0	0
$317$	24.4010	1.37049	0.685247	−	0.728311i	$-0.259694\pi$
$317$	24.4010	1.37049	0.685247	+	0.728311i	$0.259694\pi$
$318$	0	0
$319$	14.4920	0.811395
$320$	0	0
$321$	15.5067	0.865502
$322$	0	0
$323$	8.56756	0.476712
$324$	0	0
$325$	−14.0691	−0.780412
$326$	0	0
$327$	6.22370	0.344171
$328$	0	0
$329$	6.88087	0.379355
$330$	0	0
$331$	20.7509	1.14057	0.570287	−	0.821445i	$-0.306832\pi$
$331$	20.7509	1.14057	0.570287	+	0.821445i	$0.306832\pi$
$332$	0	0
$333$	−6.16450	−0.337812
$334$	0	0
$335$	3.99339	0.218182
$336$	0	0
$337$	11.9277	0.649741	0.324871	−	0.945758i	$-0.394679\pi$
$337$	11.9277	0.649741	0.324871	+	0.945758i	$0.394679\pi$
$338$	0	0
$339$	19.0121	1.03259
$340$	0	0
$341$	16.1965	0.877087
$342$	0	0
$343$	−18.6742	−1.00831
$344$	0	0
$345$	20.5708	1.10749
$346$	0	0
$347$	−22.8349	−1.22584	−0.612921	−	0.790144i	$-0.710006\pi$
$347$	−22.8349	−1.22584	−0.612921	+	0.790144i	$0.710006\pi$
$348$	0	0
$349$	−0.290460	−0.0155480	−0.00777399	−	0.999970i	$-0.502475\pi$
$349$	−0.290460	−0.0155480	−0.00777399	+	0.999970i	$0.502475\pi$
$350$	0	0
$351$	−3.00721	−0.160513
$352$	0	0
$353$	7.65015	0.407177	0.203588	−	0.979057i	$-0.434740\pi$
$353$	7.65015	0.407177	0.203588	+	0.979057i	$0.434740\pi$
$354$	0	0
$355$	5.72598	0.303904
$356$	0	0
$357$	6.52505	0.345342
$358$	0	0
$359$	26.4720	1.39714	0.698569	−	0.715543i	$-0.253821\pi$
$359$	26.4720	1.39714	0.698569	+	0.715543i	$0.253821\pi$
$360$	0	0
$361$	−14.2407	−0.749513
$362$	0	0
$363$	0.631456	0.0331429
$364$	0	0
$365$	17.2563	0.903235
$366$	0	0
$367$	19.2191	1.00323	0.501614	−	0.865091i	$-0.332740\pi$
$367$	19.2191	1.00323	0.501614	+	0.865091i	$0.332740\pi$
$368$	0	0
$369$	−0.415489	−0.0216295
$370$	0	0
$371$	10.2033	0.529727
$372$	0	0
$373$	−28.6523	−1.48356	−0.741780	−	0.670643i	$-0.766018\pi$
$373$	−28.6523	−1.48356	−0.741780	+	0.670643i	$0.766018\pi$
$374$	0	0
$375$	1.00033	0.0516569
$376$	0	0
$377$	−12.7783	−0.658117
$378$	0	0
$379$	20.7016	1.06337	0.531685	−	0.846942i	$-0.321559\pi$
$379$	20.7016	1.06337	0.531685	+	0.846942i	$0.321559\pi$
$380$	0	0
$381$	−15.2875	−0.783202
$382$	0	0
$383$	24.2611	1.23969	0.619843	−	0.784726i	$-0.287196\pi$
$383$	24.2611	1.23969	0.619843	+	0.784726i	$0.287196\pi$
$384$	0	0
$385$	−17.6285	−0.898434
$386$	0	0
$387$	4.66904	0.237341
$388$	0	0
$389$	12.9581	0.657004	0.328502	−	0.944503i	$-0.393456\pi$
$389$	12.9581	0.657004	0.328502	+	0.944503i	$0.393456\pi$
$390$	0	0
$391$	−25.9678	−1.31325
$392$	0	0
$393$	−8.89251	−0.448568
$394$	0	0
$395$	42.1588	2.12124
$396$	0	0
$397$	10.8214	0.543111	0.271556	−	0.962423i	$-0.412462\pi$
$397$	10.8214	0.543111	0.271556	+	0.962423i	$0.412462\pi$
$398$	0	0
$399$	3.62465	0.181460
$400$	0	0
$401$	−1.64448	−0.0821213	−0.0410607	−	0.999157i	$-0.513074\pi$
$401$	−1.64448	−0.0821213	−0.0410607	+	0.999157i	$0.513074\pi$
$402$	0	0
$403$	−14.2812	−0.711400
$404$	0	0
$405$	−3.11102	−0.154588
$406$	0	0
$407$	−21.0240	−1.04212
$408$	0	0
$409$	11.3712	0.562272	0.281136	−	0.959668i	$-0.409289\pi$
$409$	11.3712	0.562272	0.281136	+	0.959668i	$0.409289\pi$
$410$	0	0
$411$	4.39754	0.216915
$412$	0	0
$413$	1.06105	0.0522109
$414$	0	0
$415$	−36.9720	−1.81489
$416$	0	0
$417$	22.4011	1.09698
$418$	0	0
$419$	11.7928	0.576117	0.288059	−	0.957613i	$-0.406990\pi$
$419$	11.7928	0.576117	0.288059	+	0.957613i	$0.406990\pi$
$420$	0	0
$421$	27.3682	1.33385	0.666923	−	0.745127i	$-0.267611\pi$
$421$	27.3682	1.33385	0.666923	+	0.745127i	$0.267611\pi$
$422$	0	0
$423$	4.14140	0.201362
$424$	0	0
$425$	18.3734	0.891241
$426$	0	0
$427$	8.40107	0.406556
$428$	0	0
$429$	−10.2561	−0.495167
$430$	0	0
$431$	3.18587	0.153458	0.0767289	−	0.997052i	$-0.475552\pi$
$431$	3.18587	0.153458	0.0767289	+	0.997052i	$0.475552\pi$
$432$	0	0
$433$	17.6535	0.848374	0.424187	−	0.905575i	$-0.360560\pi$
$433$	17.6535	0.848374	0.424187	+	0.905575i	$0.360560\pi$
$434$	0	0
$435$	−13.2195	−0.633824
$436$	0	0
$437$	−14.4250	−0.690044
$438$	0	0
$439$	4.84685	0.231328	0.115664	−	0.993288i	$-0.463100\pi$
$439$	4.84685	0.231328	0.115664	+	0.993288i	$0.463100\pi$
$440$	0	0
$441$	−4.23947	−0.201879
$442$	0	0
$443$	−0.217883	−0.0103519	−0.00517596	−	0.999987i	$-0.501648\pi$
$443$	−0.217883	−0.0103519	−0.00517596	+	0.999987i	$0.501648\pi$
$444$	0	0
$445$	14.2443	0.675245
$446$	0	0
$447$	−18.3289	−0.866925
$448$	0	0
$449$	18.0000	0.849472	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849472	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	−1.41702	−0.0667249
$452$	0	0
$453$	15.1183	0.710319
$454$	0	0
$455$	15.5440	0.728714
$456$	0	0
$457$	−4.32309	−0.202226	−0.101113	−	0.994875i	$-0.532240\pi$
$457$	−4.32309	−0.202226	−0.101113	+	0.994875i	$0.532240\pi$
$458$	0	0
$459$	3.92724	0.183308
$460$	0	0
$461$	41.3079	1.92390	0.961951	−	0.273222i	$-0.0880893\pi$
$461$	41.3079	1.92390	0.961951	+	0.273222i	$0.0880893\pi$
$462$	0	0
$463$	−5.81362	−0.270182	−0.135091	−	0.990833i	$-0.543133\pi$
$463$	−5.81362	−0.270182	−0.135091	+	0.990833i	$0.543133\pi$
$464$	0	0
$465$	−14.7743	−0.685140
$466$	0	0
$467$	8.05188	0.372597	0.186298	−	0.982493i	$-0.440351\pi$
$467$	8.05188	0.372597	0.186298	+	0.982493i	$0.440351\pi$
$468$	0	0
$469$	−2.13273	−0.0984801
$470$	0	0
$471$	−23.7462	−1.09417
$472$	0	0
$473$	15.9237	0.732173
$474$	0	0
$475$	10.2064	0.468301
$476$	0	0
$477$	6.14105	0.281179
$478$	0	0
$479$	35.4075	1.61781	0.808904	−	0.587940i	$-0.200061\pi$
$479$	35.4075	1.61781	0.808904	+	0.587940i	$0.200061\pi$
$480$	0	0
$481$	18.5379	0.845256
$482$	0	0
$483$	−10.9861	−0.499885
$484$	0	0
$485$	17.8231	0.809306
$486$	0	0
$487$	−13.8182	−0.626164	−0.313082	−	0.949726i	$-0.601361\pi$
$487$	−13.8182	−0.626164	−0.313082	+	0.949726i	$0.601361\pi$
$488$	0	0
$489$	3.50387	0.158450
$490$	0	0
$491$	−6.13156	−0.276713	−0.138357	−	0.990382i	$-0.544182\pi$
$491$	−6.13156	−0.276713	−0.138357	+	0.990382i	$0.544182\pi$
$492$	0	0
$493$	16.6877	0.751578
$494$	0	0
$495$	−10.6101	−0.476889
$496$	0	0
$497$	−3.05804	−0.137172
$498$	0	0
$499$	−12.7349	−0.570092	−0.285046	−	0.958514i	$-0.592009\pi$
$499$	−12.7349	−0.570092	−0.285046	+	0.958514i	$0.592009\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	−33.7280	−1.50386	−0.751928	−	0.659245i	$-0.770876\pi$
$503$	−33.7280	−1.50386	−0.751928	+	0.659245i	$0.770876\pi$
$504$	0	0
$505$	26.4709	1.17794
$506$	0	0
$507$	−3.95671	−0.175724
$508$	0	0
$509$	19.7796	0.876714	0.438357	−	0.898801i	$-0.355561\pi$
$509$	19.7796	0.876714	0.438357	+	0.898801i	$0.355561\pi$
$510$	0	0
$511$	−9.21596	−0.407690
$512$	0	0
$513$	2.18157	0.0963188
$514$	0	0
$515$	−40.8731	−1.80108
$516$	0	0
$517$	14.1242	0.621182
$518$	0	0
$519$	20.4470	0.897523
$520$	0	0
$521$	4.84514	0.212269	0.106135	−	0.994352i	$-0.466153\pi$
$521$	4.84514	0.212269	0.106135	+	0.994352i	$0.466153\pi$
$522$	0	0
$523$	−13.7364	−0.600650	−0.300325	−	0.953837i	$-0.597095\pi$
$523$	−13.7364	−0.600650	−0.300325	+	0.953837i	$0.597095\pi$
$524$	0	0
$525$	7.77318	0.339250
$526$	0	0
$527$	18.6505	0.812428
$528$	0	0
$529$	20.7215	0.900935
$530$	0	0
$531$	0.638616	0.0277136
$532$	0	0
$533$	1.24946	0.0541201
$534$	0	0
$535$	−48.2418	−2.08568
$536$	0	0
$537$	4.22494	0.182319
$538$	0	0
$539$	−14.4587	−0.622779
$540$	0	0
$541$	9.87731	0.424659	0.212329	−	0.977198i	$-0.431895\pi$
$541$	9.87731	0.424659	0.212329	+	0.977198i	$0.431895\pi$
$542$	0	0
$543$	13.5816	0.582843
$544$	0	0
$545$	−19.3621	−0.829380
$546$	0	0
$547$	−11.7160	−0.500939	−0.250469	−	0.968125i	$-0.580585\pi$
$547$	−11.7160	−0.500939	−0.250469	+	0.968125i	$0.580585\pi$
$548$	0	0
$549$	5.05636	0.215800
$550$	0	0
$551$	9.27001	0.394916
$552$	0	0
$553$	−22.5155	−0.957457
$554$	0	0
$555$	19.1779	0.814056
$556$	0	0
$557$	24.9413	1.05680	0.528398	−	0.848997i	$-0.322793\pi$
$557$	24.9413	1.05680	0.528398	+	0.848997i	$0.322793\pi$
$558$	0	0
$559$	−14.0408	−0.593861
$560$	0	0
$561$	13.3938	0.565487
$562$	0	0
$563$	−16.3043	−0.687143	−0.343572	−	0.939127i	$-0.611637\pi$
$563$	−16.3043	−0.687143	−0.343572	+	0.939127i	$0.611637\pi$
$564$	0	0
$565$	−59.1470	−2.48833
$566$	0	0
$567$	1.66149	0.0697758
$568$	0	0
$569$	−22.5479	−0.945258	−0.472629	−	0.881262i	$-0.656695\pi$
$569$	−22.5479	−0.945258	−0.472629	+	0.881262i	$0.656695\pi$
$570$	0	0
$571$	35.5482	1.48765	0.743823	−	0.668377i	$-0.233011\pi$
$571$	35.5482	1.48765	0.743823	+	0.668377i	$0.233011\pi$
$572$	0	0
$573$	0.756545	0.0316051
$574$	0	0
$575$	−30.9350	−1.29008
$576$	0	0
$577$	−21.2559	−0.884897	−0.442448	−	0.896794i	$-0.645890\pi$
$577$	−21.2559	−0.884897	−0.442448	+	0.896794i	$0.645890\pi$
$578$	0	0
$579$	8.84946	0.367771
$580$	0	0
$581$	19.7454	0.819179
$582$	0	0
$583$	20.9440	0.867412
$584$	0	0
$585$	9.35549	0.386802
$586$	0	0
$587$	14.0631	0.580448	0.290224	−	0.956959i	$-0.406270\pi$
$587$	14.0631	0.580448	0.290224	+	0.956959i	$0.406270\pi$
$588$	0	0
$589$	10.3603	0.426889
$590$	0	0
$591$	4.25501	0.175028
$592$	0	0
$593$	23.4411	0.962612	0.481306	−	0.876553i	$-0.340163\pi$
$593$	23.4411	0.962612	0.481306	+	0.876553i	$0.340163\pi$
$594$	0	0
$595$	−20.2996	−0.832201
$596$	0	0
$597$	8.85469	0.362398
$598$	0	0
$599$	33.5151	1.36939	0.684695	−	0.728829i	$-0.259935\pi$
$599$	33.5151	1.36939	0.684695	+	0.728829i	$0.259935\pi$
$600$	0	0
$601$	−11.7328	−0.478591	−0.239296	−	0.970947i	$-0.576916\pi$
$601$	−11.7328	−0.478591	−0.239296	+	0.970947i	$0.576916\pi$
$602$	0	0
$603$	−1.28363	−0.0522733
$604$	0	0
$605$	−1.96447	−0.0798672
$606$	0	0
$607$	29.3259	1.19030	0.595151	−	0.803614i	$-0.297092\pi$
$607$	29.3259	1.19030	0.595151	+	0.803614i	$0.297092\pi$
$608$	0	0
$609$	7.06003	0.286087
$610$	0	0
$611$	−12.4540	−0.503836
$612$	0	0
$613$	−10.4917	−0.423754	−0.211877	−	0.977296i	$-0.567958\pi$
$613$	−10.4917	−0.423754	−0.211877	+	0.977296i	$0.567958\pi$
$614$	0	0
$615$	1.29259	0.0521224
$616$	0	0
$617$	−37.2093	−1.49799	−0.748996	−	0.662575i	$-0.769464\pi$
$617$	−37.2093	−1.49799	−0.748996	+	0.662575i	$0.769464\pi$
$618$	0	0
$619$	−1.82194	−0.0732299	−0.0366149	−	0.999329i	$-0.511658\pi$
$619$	−1.82194	−0.0732299	−0.0366149	+	0.999329i	$0.511658\pi$
$620$	0	0
$621$	−6.61222	−0.265339
$622$	0	0
$623$	−7.60738	−0.304783
$624$	0	0
$625$	−26.5043	−1.06017
$626$	0	0
$627$	7.44024	0.297134
$628$	0	0
$629$	−24.2094	−0.965294
$630$	0	0
$631$	−36.4466	−1.45092	−0.725458	−	0.688267i	$-0.758372\pi$
$631$	−36.4466	−1.45092	−0.725458	+	0.688267i	$0.758372\pi$
$632$	0	0
$633$	21.5988	0.858475
$634$	0	0
$635$	47.5597	1.88735
$636$	0	0
$637$	12.7490	0.505132
$638$	0	0
$639$	−1.84055	−0.0728110
$640$	0	0
$641$	−18.4692	−0.729488	−0.364744	−	0.931108i	$-0.618843\pi$
$641$	−18.4692	−0.729488	−0.364744	+	0.931108i	$0.618843\pi$
$642$	0	0
$643$	13.5587	0.534703	0.267351	−	0.963599i	$-0.413852\pi$
$643$	13.5587	0.534703	0.267351	+	0.963599i	$0.413852\pi$
$644$	0	0
$645$	−14.5255	−0.571940
$646$	0	0
$647$	−21.6428	−0.850866	−0.425433	−	0.904990i	$-0.639878\pi$
$647$	−21.6428	−0.850866	−0.425433	+	0.904990i	$0.639878\pi$
$648$	0	0
$649$	2.17800	0.0854938
$650$	0	0
$651$	7.89040	0.309249
$652$	0	0
$653$	17.7976	0.696474	0.348237	−	0.937407i	$-0.386780\pi$
$653$	17.7976	0.696474	0.348237	+	0.937407i	$0.386780\pi$
$654$	0	0
$655$	27.6648	1.08095
$656$	0	0
$657$	−5.54682	−0.216402
$658$	0	0
$659$	10.3429	0.402904	0.201452	−	0.979498i	$-0.435434\pi$
$659$	10.3429	0.402904	0.201452	+	0.979498i	$0.435434\pi$
$660$	0	0
$661$	−18.5557	−0.721732	−0.360866	−	0.932618i	$-0.617519\pi$
$661$	−18.5557	−0.721732	−0.360866	+	0.932618i	$0.617519\pi$
$662$	0	0
$663$	−11.8100	−0.458663
$664$	0	0
$665$	−11.2764	−0.437279
$666$	0	0
$667$	−28.0969	−1.08792
$668$	0	0
$669$	−28.7963	−1.11333
$670$	0	0
$671$	17.2447	0.665724
$672$	0	0
$673$	−15.8898	−0.612506	−0.306253	−	0.951950i	$-0.599075\pi$
$673$	−15.8898	−0.612506	−0.306253	+	0.951950i	$0.599075\pi$
$674$	0	0
$675$	4.67846	0.180074
$676$	0	0
$677$	−30.1310	−1.15803	−0.579014	−	0.815318i	$-0.696562\pi$
$677$	−30.1310	−1.15803	−0.579014	+	0.815318i	$0.696562\pi$
$678$	0	0
$679$	−9.51869	−0.365294
$680$	0	0
$681$	27.4078	1.05027
$682$	0	0
$683$	−16.1576	−0.618253	−0.309127	−	0.951021i	$-0.600037\pi$
$683$	−16.1576	−0.618253	−0.309127	+	0.951021i	$0.600037\pi$
$684$	0	0
$685$	−13.6808	−0.522718
$686$	0	0
$687$	9.93144	0.378908
$688$	0	0
$689$	−18.4674	−0.703552
$690$	0	0
$691$	−5.09647	−0.193879	−0.0969395	−	0.995290i	$-0.530905\pi$
$691$	−5.09647	−0.193879	−0.0969395	+	0.995290i	$0.530905\pi$
$692$	0	0
$693$	5.66648	0.215252
$694$	0	0
$695$	−69.6902	−2.64350
$696$	0	0
$697$	−1.63172	−0.0618059
$698$	0	0
$699$	26.9976	1.02114
$700$	0	0
$701$	−34.4598	−1.30153	−0.650765	−	0.759279i	$-0.725552\pi$
$701$	−34.4598	−1.30153	−0.650765	+	0.759279i	$0.725552\pi$
$702$	0	0
$703$	−13.4483	−0.507212
$704$	0	0
$705$	−12.8840	−0.485239
$706$	0	0
$707$	−14.1372	−0.531683
$708$	0	0
$709$	−18.4496	−0.692890	−0.346445	−	0.938070i	$-0.612611\pi$
$709$	−18.4496	−0.692890	−0.346445	+	0.938070i	$0.612611\pi$
$710$	0	0
$711$	−13.5514	−0.508219
$712$	0	0
$713$	−31.4015	−1.17600
$714$	0	0
$715$	31.9068	1.19325
$716$	0	0
$717$	29.9620	1.11895
$718$	0	0
$719$	45.3442	1.69105	0.845526	−	0.533934i	$-0.179287\pi$
$719$	45.3442	1.69105	0.845526	+	0.533934i	$0.179287\pi$
$720$	0	0
$721$	21.8289	0.812949
$722$	0	0
$723$	−1.70837	−0.0635349
$724$	0	0
$725$	19.8798	0.738319
$726$	0	0
$727$	−41.4593	−1.53764	−0.768819	−	0.639466i	$-0.779156\pi$
$727$	−41.4593	−1.53764	−0.768819	+	0.639466i	$0.779156\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	18.3364	0.678197
$732$	0	0
$733$	2.15796	0.0797061	0.0398530	−	0.999206i	$-0.487311\pi$
$733$	2.15796	0.0797061	0.0398530	+	0.999206i	$0.487311\pi$
$734$	0	0
$735$	13.1891	0.486486
$736$	0	0
$737$	−4.37780	−0.161258
$738$	0	0
$739$	36.4961	1.34253	0.671265	−	0.741217i	$-0.265751\pi$
$739$	36.4961	1.34253	0.671265	+	0.741217i	$0.265751\pi$
$740$	0	0
$741$	−6.56044	−0.241004
$742$	0	0
$743$	−13.5544	−0.497265	−0.248632	−	0.968598i	$-0.579981\pi$
$743$	−13.5544	−0.497265	−0.248632	+	0.968598i	$0.579981\pi$
$744$	0	0
$745$	57.0215	2.08911
$746$	0	0
$747$	11.8842	0.434820
$748$	0	0
$749$	25.7642	0.941405
$750$	0	0
$751$	11.4530	0.417925	0.208963	−	0.977924i	$-0.432991\pi$
$751$	11.4530	0.417925	0.208963	+	0.977924i	$0.432991\pi$
$752$	0	0
$753$	−26.3057	−0.958634
$754$	0	0
$755$	−47.0333	−1.71172
$756$	0	0
$757$	−36.3640	−1.32167	−0.660835	−	0.750531i	$-0.729798\pi$
$757$	−36.3640	−1.32167	−0.660835	+	0.750531i	$0.729798\pi$
$758$	0	0
$759$	−22.5509	−0.818547
$760$	0	0
$761$	48.0790	1.74286	0.871431	−	0.490519i	$-0.163193\pi$
$761$	48.0790	1.74286	0.871431	+	0.490519i	$0.163193\pi$
$762$	0	0
$763$	10.3406	0.374354
$764$	0	0
$765$	−12.2177	−0.441733
$766$	0	0
$767$	−1.92045	−0.0693435
$768$	0	0
$769$	25.8507	0.932202	0.466101	−	0.884732i	$-0.345658\pi$
$769$	25.8507	0.932202	0.466101	+	0.884732i	$0.345658\pi$
$770$	0	0
$771$	−14.1260	−0.508734
$772$	0	0
$773$	−24.7942	−0.891787	−0.445893	−	0.895086i	$-0.647114\pi$
$773$	−24.7942	−0.891787	−0.445893	+	0.895086i	$0.647114\pi$
$774$	0	0
$775$	22.2180	0.798095
$776$	0	0
$777$	−10.2422	−0.367438
$778$	0	0
$779$	−0.906419	−0.0324758
$780$	0	0
$781$	−6.27717	−0.224615
$782$	0	0
$783$	4.24923	0.151855
$784$	0	0
$785$	73.8750	2.63671
$786$	0	0
$787$	48.2257	1.71906	0.859530	−	0.511085i	$-0.170756\pi$
$787$	48.2257	1.71906	0.859530	+	0.511085i	$0.170756\pi$
$788$	0	0
$789$	−4.59039	−0.163422
$790$	0	0
$791$	31.5883	1.12315
$792$	0	0
$793$	−15.2055	−0.539964
$794$	0	0
$795$	−19.1049	−0.677582
$796$	0	0
$797$	−32.9956	−1.16876	−0.584382	−	0.811479i	$-0.698663\pi$
$797$	−32.9956	−1.16876	−0.584382	+	0.811479i	$0.698663\pi$
$798$	0	0
$799$	16.2643	0.575388
$800$	0	0
$801$	−4.57866	−0.161779
$802$	0	0
$803$	−18.9174	−0.667580
$804$	0	0
$805$	34.1780	1.20462
$806$	0	0
$807$	2.27436	0.0800611
$808$	0	0
$809$	13.2744	0.466704	0.233352	−	0.972392i	$-0.425030\pi$
$809$	13.2744	0.466704	0.233352	+	0.972392i	$0.425030\pi$
$810$	0	0
$811$	−24.1766	−0.848954	−0.424477	−	0.905439i	$-0.639542\pi$
$811$	−24.1766	−0.848954	−0.424477	+	0.905439i	$0.639542\pi$
$812$	0	0
$813$	−18.8753	−0.661985
$814$	0	0
$815$	−10.9006	−0.381831
$816$	0	0
$817$	10.1858	0.356358
$818$	0	0
$819$	−4.99643	−0.174589
$820$	0	0
$821$	−41.4110	−1.44526	−0.722628	−	0.691237i	$-0.757066\pi$
$821$	−41.4110	−1.44526	−0.722628	+	0.691237i	$0.757066\pi$
$822$	0	0
$823$	21.6187	0.753581	0.376791	−	0.926298i	$-0.377028\pi$
$823$	21.6187	0.753581	0.376791	+	0.926298i	$0.377028\pi$
$824$	0	0
$825$	15.9558	0.555511
$826$	0	0
$827$	−27.6816	−0.962582	−0.481291	−	0.876561i	$-0.659832\pi$
$827$	−27.6816	−0.962582	−0.481291	+	0.876561i	$0.659832\pi$
$828$	0	0
$829$	−56.5229	−1.96312	−0.981561	−	0.191151i	$-0.938778\pi$
$829$	−56.5229	−1.96312	−0.981561	+	0.191151i	$0.938778\pi$
$830$	0	0
$831$	29.2886	1.01601
$832$	0	0
$833$	−16.6494	−0.576868
$834$	0	0
$835$	−3.11102	−0.107661
$836$	0	0
$837$	4.74901	0.164150
$838$	0	0
$839$	−47.0176	−1.62323	−0.811613	−	0.584195i	$-0.801410\pi$
$839$	−47.0176	−1.62323	−0.811613	+	0.584195i	$0.801410\pi$
$840$	0	0
$841$	−10.9440	−0.377380
$842$	0	0
$843$	−18.1417	−0.624834
$844$	0	0
$845$	12.3094	0.423456
$846$	0	0
$847$	1.04916	0.0360494
$848$	0	0
$849$	13.8879	0.476632
$850$	0	0
$851$	40.7610	1.39727
$852$	0	0
$853$	−23.3308	−0.798832	−0.399416	−	0.916770i	$-0.630787\pi$
$853$	−23.3308	−0.798832	−0.399416	+	0.916770i	$0.630787\pi$
$854$	0	0
$855$	−6.78692	−0.232108
$856$	0	0
$857$	11.9927	0.409662	0.204831	−	0.978797i	$-0.434335\pi$
$857$	11.9927	0.409662	0.204831	+	0.978797i	$0.434335\pi$
$858$	0	0
$859$	−45.9331	−1.56722	−0.783608	−	0.621255i	$-0.786623\pi$
$859$	−45.9331	−1.56722	−0.783608	+	0.621255i	$0.786623\pi$
$860$	0	0
$861$	−0.690328	−0.0235263
$862$	0	0
$863$	−20.8209	−0.708752	−0.354376	−	0.935103i	$-0.615307\pi$
$863$	−20.8209	−0.708752	−0.354376	+	0.935103i	$0.615307\pi$
$864$	0	0
$865$	−63.6110	−2.16284
$866$	0	0
$867$	−1.57680	−0.0535508
$868$	0	0
$869$	−46.2171	−1.56781
$870$	0	0
$871$	3.86013	0.130795
$872$	0	0
$873$	−5.72903	−0.193898
$874$	0	0
$875$	1.66203	0.0561870
$876$	0	0
$877$	11.6586	0.393682	0.196841	−	0.980435i	$-0.436932\pi$
$877$	11.6586	0.393682	0.196841	+	0.980435i	$0.436932\pi$
$878$	0	0
$879$	7.95986	0.268479
$880$	0	0
$881$	−46.5313	−1.56768	−0.783840	−	0.620963i	$-0.786742\pi$
$881$	−46.5313	−1.56768	−0.783840	+	0.620963i	$0.786742\pi$
$882$	0	0
$883$	−8.17021	−0.274950	−0.137475	−	0.990505i	$-0.543899\pi$
$883$	−8.17021	−0.274950	−0.137475	+	0.990505i	$0.543899\pi$
$884$	0	0
$885$	−1.98675	−0.0667838
$886$	0	0
$887$	−12.5590	−0.421690	−0.210845	−	0.977520i	$-0.567621\pi$
$887$	−12.5590	−0.421690	−0.210845	+	0.977520i	$0.567621\pi$
$888$	0	0
$889$	−25.3999	−0.851887
$890$	0	0
$891$	3.41049	0.114256
$892$	0	0
$893$	9.03476	0.302337
$894$	0	0
$895$	−13.1439	−0.439351
$896$	0	0
$897$	19.8843	0.663918
$898$	0	0
$899$	20.1796	0.673028
$900$	0	0
$901$	24.1174	0.803466
$902$	0	0
$903$	7.75754	0.258155
$904$	0	0
$905$	−42.2527	−1.40453
$906$	0	0
$907$	2.57441	0.0854817	0.0427409	−	0.999086i	$-0.486391\pi$
$907$	2.57441	0.0854817	0.0427409	+	0.999086i	$0.486391\pi$
$908$	0	0
$909$	−8.50876	−0.282218
$910$	0	0
$911$	−5.14201	−0.170363	−0.0851813	−	0.996365i	$-0.527147\pi$
$911$	−5.14201	−0.170363	−0.0851813	+	0.996365i	$0.527147\pi$
$912$	0	0
$913$	40.5310	1.34138
$914$	0	0
$915$	−15.7305	−0.520033
$916$	0	0
$917$	−14.7748	−0.487906
$918$	0	0
$919$	56.5388	1.86504	0.932522	−	0.361114i	$-0.117603\pi$
$919$	56.5388	1.86504	0.932522	+	0.361114i	$0.117603\pi$
$920$	0	0
$921$	−29.9149	−0.985729
$922$	0	0
$923$	5.53491	0.182184
$924$	0	0
$925$	−28.8403	−0.948264
$926$	0	0
$927$	13.1382	0.431514
$928$	0	0
$929$	17.1015	0.561083	0.280542	−	0.959842i	$-0.409486\pi$
$929$	17.1015	0.561083	0.280542	+	0.959842i	$0.409486\pi$
$930$	0	0
$931$	−9.24871	−0.303114
$932$	0	0
$933$	17.8528	0.584473
$934$	0	0
$935$	−41.6685	−1.36270
$936$	0	0
$937$	33.4491	1.09273	0.546367	−	0.837546i	$-0.316010\pi$
$937$	33.4491	1.09273	0.546367	+	0.837546i	$0.316010\pi$
$938$	0	0
$939$	−14.9079	−0.486500
$940$	0	0
$941$	−7.89860	−0.257487	−0.128743	−	0.991678i	$-0.541094\pi$
$941$	−7.89860	−0.257487	−0.128743	+	0.991678i	$0.541094\pi$
$942$	0	0
$943$	2.74730	0.0894645
$944$	0	0
$945$	−5.16892	−0.168145
$946$	0	0
$947$	47.2433	1.53520	0.767600	−	0.640929i	$-0.221451\pi$
$947$	47.2433	1.53520	0.767600	+	0.640929i	$0.221451\pi$
$948$	0	0
$949$	16.6804	0.541470
$950$	0	0
$951$	24.4010	0.791255
$952$	0	0
$953$	−45.2508	−1.46582	−0.732909	−	0.680326i	$-0.761838\pi$
$953$	−45.2508	−1.46582	−0.732909	+	0.680326i	$0.761838\pi$
$954$	0	0
$955$	−2.35363	−0.0761616
$956$	0	0
$957$	14.4920	0.468459
$958$	0	0
$959$	7.30645	0.235938
$960$	0	0
$961$	−8.44693	−0.272482
$962$	0	0
$963$	15.5067	0.499698
$964$	0	0
$965$	−27.5309	−0.886250
$966$	0	0
$967$	−7.33846	−0.235989	−0.117995	−	0.993014i	$-0.537647\pi$
$967$	−7.33846	−0.235989	−0.117995	+	0.993014i	$0.537647\pi$
$968$	0	0
$969$	8.56756	0.275230
$970$	0	0
$971$	13.8585	0.444739	0.222369	−	0.974962i	$-0.428621\pi$
$971$	13.8585	0.444739	0.222369	+	0.974962i	$0.428621\pi$
$972$	0	0
$973$	37.2190	1.19319
$974$	0	0
$975$	−14.0691	−0.450571
$976$	0	0
$977$	−36.5787	−1.17026	−0.585128	−	0.810941i	$-0.698956\pi$
$977$	−36.5787	−1.17026	−0.585128	+	0.810941i	$0.698956\pi$
$978$	0	0
$979$	−15.6155	−0.499073
$980$	0	0
$981$	6.22370	0.198707
$982$	0	0
$983$	−28.0293	−0.893996	−0.446998	−	0.894535i	$-0.647507\pi$
$983$	−28.0293	−0.893996	−0.446998	+	0.894535i	$0.647507\pi$
$984$	0	0
$985$	−13.2374	−0.421780
$986$	0	0
$987$	6.88087	0.219021
$988$	0	0
$989$	−30.8727	−0.981696
$990$	0	0
$991$	12.0422	0.382534	0.191267	−	0.981538i	$-0.438740\pi$
$991$	12.0422	0.382534	0.191267	+	0.981538i	$0.438740\pi$
$992$	0	0
$993$	20.7509	0.658511
$994$	0	0
$995$	−27.5471	−0.873302
$996$	0	0
$997$	−55.4540	−1.75625	−0.878123	−	0.478434i	$-0.841205\pi$
$997$	−55.4540	−1.75625	−0.878123	+	0.478434i	$0.841205\pi$
$998$	0	0
$999$	−6.16450	−0.195036

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.u.1.1		5
4.3	odd	2		501.2.a.c.1.5	✓	5
12.11	even	2		1503.2.a.c.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
501.2.a.c.1.5	✓	5	4.3	odd	2
1503.2.a.c.1.1		5	12.11	even	2
8016.2.a.u.1.1		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.11102 q^{5} +1.66149 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.11102 q^{5} +1.66149 q^{7} +1.00000 q^{9} +3.41049 q^{11} -3.00721 q^{13} -3.11102 q^{15} +3.92724 q^{17} +2.18157 q^{19} +1.66149 q^{21} -6.61222 q^{23} +4.67846 q^{25} +1.00000 q^{27} +4.24923 q^{29} +4.74901 q^{31} +3.41049 q^{33} -5.16892 q^{35} -6.16450 q^{37} -3.00721 q^{39} -0.415489 q^{41} +4.66904 q^{43} -3.11102 q^{45} +4.14140 q^{47} -4.23947 q^{49} +3.92724 q^{51} +6.14105 q^{53} -10.6101 q^{55} +2.18157 q^{57} +0.638616 q^{59} +5.05636 q^{61} +1.66149 q^{63} +9.35549 q^{65} -1.28363 q^{67} -6.61222 q^{69} -1.84055 q^{71} -5.54682 q^{73} +4.67846 q^{75} +5.66648 q^{77} -13.5514 q^{79} +1.00000 q^{81} +11.8842 q^{83} -12.2177 q^{85} +4.24923 q^{87} -4.57866 q^{89} -4.99643 q^{91} +4.74901 q^{93} -6.78692 q^{95} -5.72903 q^{97} +3.41049 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} - q^{5} + 4 q^{7} + 5 q^{9}+O(q^{10}) \) 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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