LMFDB - Embedded newform 8016.2.a.u.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:	$0$
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			1.5
Root			$2.03850$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +1.84596 q^{5} +2.96293 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.84596 q^{5} +2.96293 q^{7} +1.00000 q^{9} +3.06102 q^{11} +0.260958 q^{13} +1.84596 q^{15} +4.67710 q^{17} +1.84454 q^{19} +2.96293 q^{21} -6.65892 q^{23} -1.59242 q^{25} +1.00000 q^{27} -6.07335 q^{29} +3.09808 q^{31} +3.06102 q^{33} +5.46946 q^{35} -6.26575 q^{37} +0.260958 q^{39} -2.16766 q^{41} +7.03615 q^{43} +1.84596 q^{45} +7.08133 q^{47} +1.77896 q^{49} +4.67710 q^{51} +4.74715 q^{53} +5.65052 q^{55} +1.84454 q^{57} +13.2862 q^{59} -4.09109 q^{61} +2.96293 q^{63} +0.481719 q^{65} +12.7157 q^{67} -6.65892 q^{69} +14.3364 q^{71} +7.96813 q^{73} -1.59242 q^{75} +9.06958 q^{77} -4.67403 q^{79} +1.00000 q^{81} -5.56419 q^{83} +8.63377 q^{85} -6.07335 q^{87} +11.5771 q^{89} +0.773201 q^{91} +3.09808 q^{93} +3.40495 q^{95} -1.21855 q^{97} +3.06102 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	1.84596	0.825540	0.412770	−	0.910835i	$-0.364561\pi$
$5$	1.84596	0.825540	0.412770	+	0.910835i	$0.364561\pi$
$6$	0	0
$7$	2.96293	1.11988	0.559941	−	0.828532i	$-0.310824\pi$
$7$	2.96293	1.11988	0.559941	+	0.828532i	$0.310824\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.06102	0.922931	0.461465	−	0.887158i	$-0.347324\pi$
$11$	3.06102	0.922931	0.461465	+	0.887158i	$0.347324\pi$
$12$	0	0
$13$	0.260958	0.0723768	0.0361884	−	0.999345i	$-0.488478\pi$
$13$	0.260958	0.0723768	0.0361884	+	0.999345i	$0.488478\pi$
$14$	0	0
$15$	1.84596	0.476626
$16$	0	0
$17$	4.67710	1.13436	0.567182	−	0.823592i	$-0.308034\pi$
$17$	4.67710	1.13436	0.567182	+	0.823592i	$0.308034\pi$
$18$	0	0
$19$	1.84454	0.423166	0.211583	−	0.977360i	$-0.432138\pi$
$19$	1.84454	0.423166	0.211583	+	0.977360i	$0.432138\pi$
$20$	0	0
$21$	2.96293	0.646565
$22$	0	0
$23$	−6.65892	−1.38848	−0.694240	−	0.719743i	$-0.744260\pi$
$23$	−6.65892	−1.38848	−0.694240	+	0.719743i	$0.744260\pi$
$24$	0	0
$25$	−1.59242	−0.318483
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−6.07335	−1.12779	−0.563897	−	0.825845i	$-0.690698\pi$
$29$	−6.07335	−1.12779	−0.563897	+	0.825845i	$0.690698\pi$
$30$	0	0
$31$	3.09808	0.556432	0.278216	−	0.960519i	$-0.410257\pi$
$31$	3.09808	0.556432	0.278216	+	0.960519i	$0.410257\pi$
$32$	0	0
$33$	3.06102	0.532854
$34$	0	0
$35$	5.46946	0.924508
$36$	0	0
$37$	−6.26575	−1.03008	−0.515041	−	0.857165i	$-0.672223\pi$
$37$	−6.26575	−1.03008	−0.515041	+	0.857165i	$0.672223\pi$
$38$	0	0
$39$	0.260958	0.0417868
$40$	0	0
$41$	−2.16766	−0.338532	−0.169266	−	0.985570i	$-0.554140\pi$
$41$	−2.16766	−0.338532	−0.169266	+	0.985570i	$0.554140\pi$
$42$	0	0
$43$	7.03615	1.07300	0.536501	−	0.843900i	$-0.319746\pi$
$43$	7.03615	1.07300	0.536501	+	0.843900i	$0.319746\pi$
$44$	0	0
$45$	1.84596	0.275180
$46$	0	0
$47$	7.08133	1.03292	0.516459	−	0.856312i	$-0.327250\pi$
$47$	7.08133	1.03292	0.516459	+	0.856312i	$0.327250\pi$
$48$	0	0
$49$	1.77896	0.254138
$50$	0	0
$51$	4.67710	0.654926
$52$	0	0
$53$	4.74715	0.652072	0.326036	−	0.945357i	$-0.394287\pi$
$53$	4.74715	0.652072	0.326036	+	0.945357i	$0.394287\pi$
$54$	0	0
$55$	5.65052	0.761916
$56$	0	0
$57$	1.84454	0.244315
$58$	0	0
$59$	13.2862	1.72972	0.864858	−	0.502017i	$-0.167409\pi$
$59$	13.2862	1.72972	0.864858	+	0.502017i	$0.167409\pi$
$60$	0	0
$61$	−4.09109	−0.523810	−0.261905	−	0.965094i	$-0.584351\pi$
$61$	−4.09109	−0.523810	−0.261905	+	0.965094i	$0.584351\pi$
$62$	0	0
$63$	2.96293	0.373294
$64$	0	0
$65$	0.481719	0.0597499
$66$	0	0
$67$	12.7157	1.55347	0.776737	−	0.629825i	$-0.216873\pi$
$67$	12.7157	1.55347	0.776737	+	0.629825i	$0.216873\pi$
$68$	0	0
$69$	−6.65892	−0.801640
$70$	0	0
$71$	14.3364	1.70142	0.850711	−	0.525633i	$-0.176172\pi$
$71$	14.3364	1.70142	0.850711	+	0.525633i	$0.176172\pi$
$72$	0	0
$73$	7.96813	0.932600	0.466300	−	0.884627i	$-0.345587\pi$
$73$	7.96813	0.932600	0.466300	+	0.884627i	$0.345587\pi$
$74$	0	0
$75$	−1.59242	−0.183877
$76$	0	0
$77$	9.06958	1.03357
$78$	0	0
$79$	−4.67403	−0.525869	−0.262935	−	0.964814i	$-0.584690\pi$
$79$	−4.67403	−0.525869	−0.262935	+	0.964814i	$0.584690\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−5.56419	−0.610749	−0.305374	−	0.952232i	$-0.598782\pi$
$83$	−5.56419	−0.610749	−0.305374	+	0.952232i	$0.598782\pi$
$84$	0	0
$85$	8.63377	0.936463
$86$	0	0
$87$	−6.07335	−0.651132
$88$	0	0
$89$	11.5771	1.22717	0.613584	−	0.789629i	$-0.289727\pi$
$89$	11.5771	1.22717	0.613584	+	0.789629i	$0.289727\pi$
$90$	0	0
$91$	0.773201	0.0810535
$92$	0	0
$93$	3.09808	0.321256
$94$	0	0
$95$	3.40495	0.349340
$96$	0	0
$97$	−1.21855	−0.123725	−0.0618626	−	0.998085i	$-0.519704\pi$
$97$	−1.21855	−0.123725	−0.0618626	+	0.998085i	$0.519704\pi$
$98$	0	0
$99$	3.06102	0.307644
$100$	0	0
$101$	−14.5781	−1.45058	−0.725288	−	0.688446i	$-0.758293\pi$
$101$	−14.5781	−1.45058	−0.725288	+	0.688446i	$0.758293\pi$
$102$	0	0
$103$	−2.86124	−0.281926	−0.140963	−	0.990015i	$-0.545020\pi$
$103$	−2.86124	−0.281926	−0.140963	+	0.990015i	$0.545020\pi$
$104$	0	0
$105$	5.46946	0.533765
$106$	0	0
$107$	−16.4084	−1.58626	−0.793130	−	0.609053i	$-0.791550\pi$
$107$	−16.4084	−1.58626	−0.793130	+	0.609053i	$0.791550\pi$
$108$	0	0
$109$	1.14920	0.110074	0.0550369	−	0.998484i	$-0.482472\pi$
$109$	1.14920	0.110074	0.0550369	+	0.998484i	$0.482472\pi$
$110$	0	0
$111$	−6.26575	−0.594718
$112$	0	0
$113$	−12.3390	−1.16076	−0.580380	−	0.814346i	$-0.697096\pi$
$113$	−12.3390	−1.16076	−0.580380	+	0.814346i	$0.697096\pi$
$114$	0	0
$115$	−12.2921	−1.14625
$116$	0	0
$117$	0.260958	0.0241256
$118$	0	0
$119$	13.8579	1.27036
$120$	0	0
$121$	−1.63019	−0.148199
$122$	0	0
$123$	−2.16766	−0.195452
$124$	0	0
$125$	−12.1694	−1.08846
$126$	0	0
$127$	−0.757786	−0.0672426	−0.0336213	−	0.999435i	$-0.510704\pi$
$127$	−0.757786	−0.0672426	−0.0336213	+	0.999435i	$0.510704\pi$
$128$	0	0
$129$	7.03615	0.619498
$130$	0	0
$131$	2.32234	0.202903	0.101452	−	0.994840i	$-0.467651\pi$
$131$	2.32234	0.202903	0.101452	+	0.994840i	$0.467651\pi$
$132$	0	0
$133$	5.46524	0.473896
$134$	0	0
$135$	1.84596	0.158875
$136$	0	0
$137$	0.0466887	0.00398889	0.00199444	−	0.999998i	$-0.499365\pi$
$137$	0.0466887	0.00398889	0.00199444	+	0.999998i	$0.499365\pi$
$138$	0	0
$139$	−15.6251	−1.32531	−0.662654	−	0.748926i	$-0.730570\pi$
$139$	−15.6251	−1.32531	−0.662654	+	0.748926i	$0.730570\pi$
$140$	0	0
$141$	7.08133	0.596355
$142$	0	0
$143$	0.798797	0.0667988
$144$	0	0
$145$	−11.2112	−0.931039
$146$	0	0
$147$	1.77896	0.146726
$148$	0	0
$149$	11.6532	0.954671	0.477335	−	0.878721i	$-0.341603\pi$
$149$	11.6532	0.954671	0.477335	+	0.878721i	$0.341603\pi$
$150$	0	0
$151$	−11.2843	−0.918300	−0.459150	−	0.888359i	$-0.651846\pi$
$151$	−11.2843	−0.918300	−0.459150	+	0.888359i	$0.651846\pi$
$152$	0	0
$153$	4.67710	0.378122
$154$	0	0
$155$	5.71895	0.459357
$156$	0	0
$157$	14.1874	1.13227	0.566137	−	0.824311i	$-0.308437\pi$
$157$	14.1874	1.13227	0.566137	+	0.824311i	$0.308437\pi$
$158$	0	0
$159$	4.74715	0.376474
$160$	0	0
$161$	−19.7299	−1.55494
$162$	0	0
$163$	14.6056	1.14400	0.571999	−	0.820254i	$-0.306168\pi$
$163$	14.6056	1.14400	0.571999	+	0.820254i	$0.306168\pi$
$164$	0	0
$165$	5.65052	0.439893
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−12.9319	−0.994762
$170$	0	0
$171$	1.84454	0.141055
$172$	0	0
$173$	3.91688	0.297795	0.148898	−	0.988853i	$-0.452428\pi$
$173$	3.91688	0.297795	0.148898	+	0.988853i	$0.452428\pi$
$174$	0	0
$175$	−4.71822	−0.356664
$176$	0	0
$177$	13.2862	0.998652
$178$	0	0
$179$	8.91646	0.666448	0.333224	−	0.942848i	$-0.391863\pi$
$179$	8.91646	0.666448	0.333224	+	0.942848i	$0.391863\pi$
$180$	0	0
$181$	−2.77606	−0.206343	−0.103171	−	0.994664i	$-0.532899\pi$
$181$	−2.77606	−0.206343	−0.103171	+	0.994664i	$0.532899\pi$
$182$	0	0
$183$	−4.09109	−0.302422
$184$	0	0
$185$	−11.5663	−0.850374
$186$	0	0
$187$	14.3167	1.04694
$188$	0	0
$189$	2.96293	0.215522
$190$	0	0
$191$	−17.3322	−1.25412	−0.627058	−	0.778972i	$-0.715741\pi$
$191$	−17.3322	−1.25412	−0.627058	+	0.778972i	$0.715741\pi$
$192$	0	0
$193$	−9.25598	−0.666260	−0.333130	−	0.942881i	$-0.608105\pi$
$193$	−9.25598	−0.666260	−0.333130	+	0.942881i	$0.608105\pi$
$194$	0	0
$195$	0.481719	0.0344966
$196$	0	0
$197$	−5.66776	−0.403811	−0.201906	−	0.979405i	$-0.564713\pi$
$197$	−5.66776	−0.403811	−0.201906	+	0.979405i	$0.564713\pi$
$198$	0	0
$199$	−4.18870	−0.296929	−0.148465	−	0.988918i	$-0.547433\pi$
$199$	−4.18870	−0.296929	−0.148465	+	0.988918i	$0.547433\pi$
$200$	0	0
$201$	12.7157	0.896899
$202$	0	0
$203$	−17.9949	−1.26300
$204$	0	0
$205$	−4.00143	−0.279472
$206$	0	0
$207$	−6.65892	−0.462827
$208$	0	0
$209$	5.64616	0.390553
$210$	0	0
$211$	13.0460	0.898122	0.449061	−	0.893501i	$-0.351759\pi$
$211$	13.0460	0.898122	0.449061	+	0.893501i	$0.351759\pi$
$212$	0	0
$213$	14.3364	0.982317
$214$	0	0
$215$	12.9885	0.885806
$216$	0	0
$217$	9.17941	0.623139
$218$	0	0
$219$	7.96813	0.538437
$220$	0	0
$221$	1.22053	0.0821017
$222$	0	0
$223$	1.34116	0.0898110	0.0449055	−	0.998991i	$-0.485701\pi$
$223$	1.34116	0.0898110	0.0449055	+	0.998991i	$0.485701\pi$
$224$	0	0
$225$	−1.59242	−0.106161
$226$	0	0
$227$	−25.3544	−1.68283	−0.841416	−	0.540388i	$-0.818277\pi$
$227$	−25.3544	−1.68283	−0.841416	+	0.540388i	$0.818277\pi$
$228$	0	0
$229$	−3.01873	−0.199484	−0.0997418	−	0.995013i	$-0.531802\pi$
$229$	−3.01873	−0.199484	−0.0997418	+	0.995013i	$0.531802\pi$
$230$	0	0
$231$	9.06958	0.596734
$232$	0	0
$233$	14.5996	0.956453	0.478226	−	0.878237i	$-0.341280\pi$
$233$	14.5996	0.956453	0.478226	+	0.878237i	$0.341280\pi$
$234$	0	0
$235$	13.0719	0.852715
$236$	0	0
$237$	−4.67403	−0.303611
$238$	0	0
$239$	2.55770	0.165444	0.0827220	−	0.996573i	$-0.473639\pi$
$239$	2.55770	0.165444	0.0827220	+	0.996573i	$0.473639\pi$
$240$	0	0
$241$	−3.17421	−0.204469	−0.102234	−	0.994760i	$-0.532599\pi$
$241$	−3.17421	−0.204469	−0.102234	+	0.994760i	$0.532599\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	3.28390	0.209801
$246$	0	0
$247$	0.481347	0.0306274
$248$	0	0
$249$	−5.56419	−0.352616
$250$	0	0
$251$	−11.5791	−0.730865	−0.365433	−	0.930838i	$-0.619079\pi$
$251$	−11.5791	−0.730865	−0.365433	+	0.930838i	$0.619079\pi$
$252$	0	0
$253$	−20.3831	−1.28147
$254$	0	0
$255$	8.63377	0.540667
$256$	0	0
$257$	0.706928	0.0440970	0.0220485	−	0.999757i	$-0.492981\pi$
$257$	0.706928	0.0440970	0.0220485	+	0.999757i	$0.492981\pi$
$258$	0	0
$259$	−18.5650	−1.15357
$260$	0	0
$261$	−6.07335	−0.375931
$262$	0	0
$263$	18.4391	1.13700	0.568502	−	0.822682i	$-0.307523\pi$
$263$	18.4391	1.13700	0.568502	+	0.822682i	$0.307523\pi$
$264$	0	0
$265$	8.76307	0.538311
$266$	0	0
$267$	11.5771	0.708506
$268$	0	0
$269$	30.4675	1.85764	0.928819	−	0.370533i	$-0.120825\pi$
$269$	30.4675	1.85764	0.928819	+	0.370533i	$0.120825\pi$
$270$	0	0
$271$	10.7782	0.654730	0.327365	−	0.944898i	$-0.393839\pi$
$271$	10.7782	0.654730	0.327365	+	0.944898i	$0.393839\pi$
$272$	0	0
$273$	0.773201	0.0467963
$274$	0	0
$275$	−4.87441	−0.293938
$276$	0	0
$277$	−10.7465	−0.645693	−0.322847	−	0.946451i	$-0.604640\pi$
$277$	−10.7465	−0.645693	−0.322847	+	0.946451i	$0.604640\pi$
$278$	0	0
$279$	3.09808	0.185477
$280$	0	0
$281$	20.8992	1.24674	0.623372	−	0.781926i	$-0.285762\pi$
$281$	20.8992	1.24674	0.623372	+	0.781926i	$0.285762\pi$
$282$	0	0
$283$	30.2167	1.79619	0.898097	−	0.439797i	$-0.144950\pi$
$283$	30.2167	1.79619	0.898097	+	0.439797i	$0.144950\pi$
$284$	0	0
$285$	3.40495	0.201692
$286$	0	0
$287$	−6.42264	−0.379116
$288$	0	0
$289$	4.87531	0.286783
$290$	0	0
$291$	−1.21855	−0.0714328
$292$	0	0
$293$	−24.4441	−1.42804	−0.714019	−	0.700126i	$-0.753127\pi$
$293$	−24.4441	−1.42804	−0.714019	+	0.700126i	$0.753127\pi$
$294$	0	0
$295$	24.5258	1.42795
$296$	0	0
$297$	3.06102	0.177618
$298$	0	0
$299$	−1.73770	−0.100494
$300$	0	0
$301$	20.8476	1.20164
$302$	0	0
$303$	−14.5781	−0.837490
$304$	0	0
$305$	−7.55200	−0.432426
$306$	0	0
$307$	−14.0634	−0.802642	−0.401321	−	0.915937i	$-0.631449\pi$
$307$	−14.0634	−0.802642	−0.401321	+	0.915937i	$0.631449\pi$
$308$	0	0
$309$	−2.86124	−0.162770
$310$	0	0
$311$	3.87923	0.219971	0.109985	−	0.993933i	$-0.464920\pi$
$311$	3.87923	0.219971	0.109985	+	0.993933i	$0.464920\pi$
$312$	0	0
$313$	15.5885	0.881114	0.440557	−	0.897725i	$-0.354781\pi$
$313$	15.5885	0.881114	0.440557	+	0.897725i	$0.354781\pi$
$314$	0	0
$315$	5.46946	0.308169
$316$	0	0
$317$	8.05190	0.452240	0.226120	−	0.974099i	$-0.427396\pi$
$317$	8.05190	0.452240	0.226120	+	0.974099i	$0.427396\pi$
$318$	0	0
$319$	−18.5906	−1.04088
$320$	0	0
$321$	−16.4084	−0.915827
$322$	0	0
$323$	8.62709	0.480024
$324$	0	0
$325$	−0.415554	−0.0230508
$326$	0	0
$327$	1.14920	0.0635511
$328$	0	0
$329$	20.9815	1.15675
$330$	0	0
$331$	−20.0887	−1.10417	−0.552087	−	0.833786i	$-0.686168\pi$
$331$	−20.0887	−1.10417	−0.552087	+	0.833786i	$0.686168\pi$
$332$	0	0
$333$	−6.26575	−0.343361
$334$	0	0
$335$	23.4728	1.28246
$336$	0	0
$337$	−27.7164	−1.50981	−0.754903	−	0.655836i	$-0.772316\pi$
$337$	−27.7164	−1.50981	−0.754903	+	0.655836i	$0.772316\pi$
$338$	0	0
$339$	−12.3390	−0.670165
$340$	0	0
$341$	9.48328	0.513548
$342$	0	0
$343$	−15.4696	−0.835278
$344$	0	0
$345$	−12.2921	−0.661786
$346$	0	0
$347$	−28.5050	−1.53023	−0.765115	−	0.643894i	$-0.777318\pi$
$347$	−28.5050	−1.53023	−0.765115	+	0.643894i	$0.777318\pi$
$348$	0	0
$349$	−19.9171	−1.06614	−0.533068	−	0.846072i	$-0.678961\pi$
$349$	−19.9171	−1.06614	−0.533068	+	0.846072i	$0.678961\pi$
$350$	0	0
$351$	0.260958	0.0139289
$352$	0	0
$353$	16.6594	0.886692	0.443346	−	0.896351i	$-0.353791\pi$
$353$	16.6594	0.886692	0.443346	+	0.896351i	$0.353791\pi$
$354$	0	0
$355$	26.4646	1.40459
$356$	0	0
$357$	13.8579	0.733440
$358$	0	0
$359$	0.769506	0.0406130	0.0203065	−	0.999794i	$-0.493536\pi$
$359$	0.769506	0.0406130	0.0203065	+	0.999794i	$0.493536\pi$
$360$	0	0
$361$	−15.5977	−0.820931
$362$	0	0
$363$	−1.63019	−0.0855626
$364$	0	0
$365$	14.7089	0.769898
$366$	0	0
$367$	−15.9342	−0.831757	−0.415878	−	0.909420i	$-0.636526\pi$
$367$	−15.9342	−0.831757	−0.415878	+	0.909420i	$0.636526\pi$
$368$	0	0
$369$	−2.16766	−0.112844
$370$	0	0
$371$	14.0655	0.730244
$372$	0	0
$373$	37.0735	1.91959	0.959797	−	0.280696i	$-0.0905654\pi$
$373$	37.0735	1.91959	0.959797	+	0.280696i	$0.0905654\pi$
$374$	0	0
$375$	−12.1694	−0.628423
$376$	0	0
$377$	−1.58489	−0.0816261
$378$	0	0
$379$	−3.21536	−0.165162	−0.0825809	−	0.996584i	$-0.526316\pi$
$379$	−3.21536	−0.165162	−0.0825809	+	0.996584i	$0.526316\pi$
$380$	0	0
$381$	−0.757786	−0.0388226
$382$	0	0
$383$	29.3117	1.49776	0.748878	−	0.662707i	$-0.230593\pi$
$383$	29.3117	1.49776	0.748878	+	0.662707i	$0.230593\pi$
$384$	0	0
$385$	16.7421	0.853257
$386$	0	0
$387$	7.03615	0.357667
$388$	0	0
$389$	23.8124	1.20734	0.603668	−	0.797235i	$-0.293705\pi$
$389$	23.8124	1.20734	0.603668	+	0.797235i	$0.293705\pi$
$390$	0	0
$391$	−31.1445	−1.57504
$392$	0	0
$393$	2.32234	0.117146
$394$	0	0
$395$	−8.62809	−0.434126
$396$	0	0
$397$	−24.7244	−1.24088	−0.620441	−	0.784253i	$-0.713046\pi$
$397$	−24.7244	−1.24088	−0.620441	+	0.784253i	$0.713046\pi$
$398$	0	0
$399$	5.46524	0.273604
$400$	0	0
$401$	−17.3880	−0.868314	−0.434157	−	0.900837i	$-0.642954\pi$
$401$	−17.3880	−0.868314	−0.434157	+	0.900837i	$0.642954\pi$
$402$	0	0
$403$	0.808470	0.0402728
$404$	0	0
$405$	1.84596	0.0917267
$406$	0	0
$407$	−19.1795	−0.950695
$408$	0	0
$409$	31.4738	1.55628	0.778141	−	0.628090i	$-0.216163\pi$
$409$	31.4738	1.55628	0.778141	+	0.628090i	$0.216163\pi$
$410$	0	0
$411$	0.0466887	0.00230298
$412$	0	0
$413$	39.3661	1.93708
$414$	0	0
$415$	−10.2713	−0.504198
$416$	0	0
$417$	−15.6251	−0.765167
$418$	0	0
$419$	26.6383	1.30137	0.650684	−	0.759349i	$-0.274482\pi$
$419$	26.6383	1.30137	0.650684	+	0.759349i	$0.274482\pi$
$420$	0	0
$421$	−27.2396	−1.32758	−0.663788	−	0.747921i	$-0.731052\pi$
$421$	−27.2396	−1.32758	−0.663788	+	0.747921i	$0.731052\pi$
$422$	0	0
$423$	7.08133	0.344306
$424$	0	0
$425$	−7.44790	−0.361276
$426$	0	0
$427$	−12.1216	−0.586606
$428$	0	0
$429$	0.798797	0.0385663
$430$	0	0
$431$	−18.3948	−0.886046	−0.443023	−	0.896510i	$-0.646094\pi$
$431$	−18.3948	−0.886046	−0.443023	+	0.896510i	$0.646094\pi$
$432$	0	0
$433$	−29.2595	−1.40612	−0.703060	−	0.711130i	$-0.748184\pi$
$433$	−29.2595	−1.40612	−0.703060	+	0.711130i	$0.748184\pi$
$434$	0	0
$435$	−11.2112	−0.537536
$436$	0	0
$437$	−12.2826	−0.587558
$438$	0	0
$439$	−8.72283	−0.416318	−0.208159	−	0.978095i	$-0.566747\pi$
$439$	−8.72283	−0.416318	−0.208159	+	0.978095i	$0.566747\pi$
$440$	0	0
$441$	1.77896	0.0847126
$442$	0	0
$443$	2.86578	0.136157	0.0680786	−	0.997680i	$-0.478313\pi$
$443$	2.86578	0.136157	0.0680786	+	0.997680i	$0.478313\pi$
$444$	0	0
$445$	21.3709	1.01308
$446$	0	0
$447$	11.6532	0.551179
$448$	0	0
$449$	−10.8112	−0.510212	−0.255106	−	0.966913i	$-0.582110\pi$
$449$	−10.8112	−0.510212	−0.255106	+	0.966913i	$0.582110\pi$
$450$	0	0
$451$	−6.63525	−0.312442
$452$	0	0
$453$	−11.2843	−0.530181
$454$	0	0
$455$	1.42730	0.0669129
$456$	0	0
$457$	39.1244	1.83016	0.915081	−	0.403269i	$-0.132126\pi$
$457$	39.1244	1.83016	0.915081	+	0.403269i	$0.132126\pi$
$458$	0	0
$459$	4.67710	0.218309
$460$	0	0
$461$	3.70794	0.172696	0.0863480	−	0.996265i	$-0.472480\pi$
$461$	3.70794	0.172696	0.0863480	+	0.996265i	$0.472480\pi$
$462$	0	0
$463$	−19.4262	−0.902812	−0.451406	−	0.892319i	$-0.649077\pi$
$463$	−19.4262	−0.902812	−0.451406	+	0.892319i	$0.649077\pi$
$464$	0	0
$465$	5.71895	0.265210
$466$	0	0
$467$	−6.53678	−0.302486	−0.151243	−	0.988497i	$-0.548328\pi$
$467$	−6.53678	−0.302486	−0.151243	+	0.988497i	$0.548328\pi$
$468$	0	0
$469$	37.6759	1.73971
$470$	0	0
$471$	14.1874	0.653719
$472$	0	0
$473$	21.5378	0.990307
$474$	0	0
$475$	−2.93727	−0.134771
$476$	0	0
$477$	4.74715	0.217357
$478$	0	0
$479$	−32.9962	−1.50764	−0.753818	−	0.657083i	$-0.771790\pi$
$479$	−32.9962	−1.50764	−0.753818	+	0.657083i	$0.771790\pi$
$480$	0	0
$481$	−1.63510	−0.0745540
$482$	0	0
$483$	−19.7299	−0.897743
$484$	0	0
$485$	−2.24940	−0.102140
$486$	0	0
$487$	1.22191	0.0553701	0.0276850	−	0.999617i	$-0.491186\pi$
$487$	1.22191	0.0553701	0.0276850	+	0.999617i	$0.491186\pi$
$488$	0	0
$489$	14.6056	0.660488
$490$	0	0
$491$	29.4393	1.32858	0.664289	−	0.747476i	$-0.268735\pi$
$491$	29.4393	1.32858	0.664289	+	0.747476i	$0.268735\pi$
$492$	0	0
$493$	−28.4057	−1.27933
$494$	0	0
$495$	5.65052	0.253972
$496$	0	0
$497$	42.4779	1.90539
$498$	0	0
$499$	27.7305	1.24139	0.620695	−	0.784052i	$-0.286851\pi$
$499$	27.7305	1.24139	0.620695	+	0.784052i	$0.286851\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	−17.3251	−0.772489	−0.386245	−	0.922396i	$-0.626228\pi$
$503$	−17.3251	−0.772489	−0.386245	+	0.922396i	$0.626228\pi$
$504$	0	0
$505$	−26.9106	−1.19751
$506$	0	0
$507$	−12.9319	−0.574326
$508$	0	0
$509$	−34.8323	−1.54391	−0.771956	−	0.635675i	$-0.780722\pi$
$509$	−34.8323	−1.54391	−0.771956	+	0.635675i	$0.780722\pi$
$510$	0	0
$511$	23.6090	1.04440
$512$	0	0
$513$	1.84454	0.0814383
$514$	0	0
$515$	−5.28174	−0.232741
$516$	0	0
$517$	21.6760	0.953312
$518$	0	0
$519$	3.91688	0.171932
$520$	0	0
$521$	−29.7449	−1.30315	−0.651573	−	0.758586i	$-0.725891\pi$
$521$	−29.7449	−1.30315	−0.651573	+	0.758586i	$0.725891\pi$
$522$	0	0
$523$	5.08561	0.222378	0.111189	−	0.993799i	$-0.464534\pi$
$523$	5.08561	0.222378	0.111189	+	0.993799i	$0.464534\pi$
$524$	0	0
$525$	−4.71822	−0.205920
$526$	0	0
$527$	14.4901	0.631197
$528$	0	0
$529$	21.3412	0.927879
$530$	0	0
$531$	13.2862	0.576572
$532$	0	0
$533$	−0.565669	−0.0245019
$534$	0	0
$535$	−30.2893	−1.30952
$536$	0	0
$537$	8.91646	0.384774
$538$	0	0
$539$	5.44544	0.234552
$540$	0	0
$541$	−40.5232	−1.74223	−0.871114	−	0.491081i	$-0.836602\pi$
$541$	−40.5232	−1.74223	−0.871114	+	0.491081i	$0.836602\pi$
$542$	0	0
$543$	−2.77606	−0.119132
$544$	0	0
$545$	2.12139	0.0908703
$546$	0	0
$547$	3.92991	0.168031	0.0840155	−	0.996464i	$-0.473225\pi$
$547$	3.92991	0.168031	0.0840155	+	0.996464i	$0.473225\pi$
$548$	0	0
$549$	−4.09109	−0.174603
$550$	0	0
$551$	−11.2025	−0.477244
$552$	0	0
$553$	−13.8488	−0.588912
$554$	0	0
$555$	−11.5663	−0.490964
$556$	0	0
$557$	21.6907	0.919065	0.459532	−	0.888161i	$-0.348017\pi$
$557$	21.6907	0.919065	0.459532	+	0.888161i	$0.348017\pi$
$558$	0	0
$559$	1.83614	0.0776605
$560$	0	0
$561$	14.3167	0.604451
$562$	0	0
$563$	35.3085	1.48808	0.744038	−	0.668138i	$-0.232908\pi$
$563$	35.3085	1.48808	0.744038	+	0.668138i	$0.232908\pi$
$564$	0	0
$565$	−22.7774	−0.958254
$566$	0	0
$567$	2.96293	0.124431
$568$	0	0
$569$	10.2765	0.430814	0.215407	−	0.976524i	$-0.430892\pi$
$569$	10.2765	0.430814	0.215407	+	0.976524i	$0.430892\pi$
$570$	0	0
$571$	7.73146	0.323552	0.161776	−	0.986828i	$-0.448278\pi$
$571$	7.73146	0.323552	0.161776	+	0.986828i	$0.448278\pi$
$572$	0	0
$573$	−17.3322	−0.724065
$574$	0	0
$575$	10.6038	0.442208
$576$	0	0
$577$	−26.3312	−1.09618	−0.548091	−	0.836419i	$-0.684645\pi$
$577$	−26.3312	−1.09618	−0.548091	+	0.836419i	$0.684645\pi$
$578$	0	0
$579$	−9.25598	−0.384665
$580$	0	0
$581$	−16.4863	−0.683967
$582$	0	0
$583$	14.5311	0.601817
$584$	0	0
$585$	0.481719	0.0199166
$586$	0	0
$587$	−22.2371	−0.917824	−0.458912	−	0.888482i	$-0.651761\pi$
$587$	−22.2371	−0.917824	−0.458912	+	0.888482i	$0.651761\pi$
$588$	0	0
$589$	5.71453	0.235463
$590$	0	0
$591$	−5.66776	−0.233140
$592$	0	0
$593$	1.63803	0.0672660	0.0336330	−	0.999434i	$-0.489292\pi$
$593$	1.63803	0.0672660	0.0336330	+	0.999434i	$0.489292\pi$
$594$	0	0
$595$	25.5813	1.04873
$596$	0	0
$597$	−4.18870	−0.171432
$598$	0	0
$599$	38.6920	1.58091	0.790456	−	0.612519i	$-0.209844\pi$
$599$	38.6920	1.58091	0.790456	+	0.612519i	$0.209844\pi$
$600$	0	0
$601$	−45.8727	−1.87119	−0.935593	−	0.353082i	$-0.885134\pi$
$601$	−45.8727	−1.87119	−0.935593	+	0.353082i	$0.885134\pi$
$602$	0	0
$603$	12.7157	0.517825
$604$	0	0
$605$	−3.00926	−0.122344
$606$	0	0
$607$	32.8360	1.33277	0.666386	−	0.745607i	$-0.267840\pi$
$607$	32.8360	1.33277	0.666386	+	0.745607i	$0.267840\pi$
$608$	0	0
$609$	−17.9949	−0.729192
$610$	0	0
$611$	1.84793	0.0747593
$612$	0	0
$613$	−28.3070	−1.14331	−0.571654	−	0.820495i	$-0.693698\pi$
$613$	−28.3070	−1.14331	−0.571654	+	0.820495i	$0.693698\pi$
$614$	0	0
$615$	−4.00143	−0.161353
$616$	0	0
$617$	4.94743	0.199176	0.0995880	−	0.995029i	$-0.468248\pi$
$617$	4.94743	0.199176	0.0995880	+	0.995029i	$0.468248\pi$
$618$	0	0
$619$	17.3828	0.698674	0.349337	−	0.936997i	$-0.386407\pi$
$619$	17.3828	0.698674	0.349337	+	0.936997i	$0.386407\pi$
$620$	0	0
$621$	−6.65892	−0.267213
$622$	0	0
$623$	34.3021	1.37429
$624$	0	0
$625$	−14.5021	−0.580085
$626$	0	0
$627$	5.64616	0.225486
$628$	0	0
$629$	−29.3056	−1.16849
$630$	0	0
$631$	23.8623	0.949944	0.474972	−	0.880001i	$-0.342458\pi$
$631$	23.8623	0.949944	0.474972	+	0.880001i	$0.342458\pi$
$632$	0	0
$633$	13.0460	0.518531
$634$	0	0
$635$	−1.39885	−0.0555115
$636$	0	0
$637$	0.464235	0.0183937
$638$	0	0
$639$	14.3364	0.567141
$640$	0	0
$641$	46.8859	1.85188	0.925941	−	0.377668i	$-0.123274\pi$
$641$	46.8859	1.85188	0.925941	+	0.377668i	$0.123274\pi$
$642$	0	0
$643$	−35.3769	−1.39513	−0.697564	−	0.716522i	$-0.745733\pi$
$643$	−35.3769	−1.39513	−0.697564	+	0.716522i	$0.745733\pi$
$644$	0	0
$645$	12.9885	0.511421
$646$	0	0
$647$	−16.1524	−0.635016	−0.317508	−	0.948256i	$-0.602846\pi$
$647$	−16.1524	−0.635016	−0.317508	+	0.948256i	$0.602846\pi$
$648$	0	0
$649$	40.6693	1.59641
$650$	0	0
$651$	9.17941	0.359769
$652$	0	0
$653$	14.9296	0.584242	0.292121	−	0.956381i	$-0.405639\pi$
$653$	14.9296	0.584242	0.292121	+	0.956381i	$0.405639\pi$
$654$	0	0
$655$	4.28695	0.167505
$656$	0	0
$657$	7.96813	0.310867
$658$	0	0
$659$	45.7257	1.78122	0.890610	−	0.454768i	$-0.150278\pi$
$659$	45.7257	1.78122	0.890610	+	0.454768i	$0.150278\pi$
$660$	0	0
$661$	−1.06498	−0.0414229	−0.0207115	−	0.999785i	$-0.506593\pi$
$661$	−1.06498	−0.0414229	−0.0207115	+	0.999785i	$0.506593\pi$
$662$	0	0
$663$	1.22053	0.0474014
$664$	0	0
$665$	10.0886	0.391220
$666$	0	0
$667$	40.4420	1.56592
$668$	0	0
$669$	1.34116	0.0518524
$670$	0	0
$671$	−12.5229	−0.483440
$672$	0	0
$673$	−7.83726	−0.302104	−0.151052	−	0.988526i	$-0.548266\pi$
$673$	−7.83726	−0.302104	−0.151052	+	0.988526i	$0.548266\pi$
$674$	0	0
$675$	−1.59242	−0.0612922
$676$	0	0
$677$	18.2812	0.702602	0.351301	−	0.936263i	$-0.385739\pi$
$677$	18.2812	0.702602	0.351301	+	0.936263i	$0.385739\pi$
$678$	0	0
$679$	−3.61049	−0.138558
$680$	0	0
$681$	−25.3544	−0.971583
$682$	0	0
$683$	45.5044	1.74118	0.870590	−	0.492010i	$-0.163738\pi$
$683$	45.5044	1.74118	0.870590	+	0.492010i	$0.163738\pi$
$684$	0	0
$685$	0.0861857	0.00329299
$686$	0	0
$687$	−3.01873	−0.115172
$688$	0	0
$689$	1.23881	0.0471948
$690$	0	0
$691$	−21.0891	−0.802268	−0.401134	−	0.916019i	$-0.631384\pi$
$691$	−21.0891	−0.802268	−0.401134	+	0.916019i	$0.631384\pi$
$692$	0	0
$693$	9.06958	0.344525
$694$	0	0
$695$	−28.8435	−1.09409
$696$	0	0
$697$	−10.1384	−0.384019
$698$	0	0
$699$	14.5996	0.552208
$700$	0	0
$701$	−6.42411	−0.242635	−0.121318	−	0.992614i	$-0.538712\pi$
$701$	−6.42411	−0.242635	−0.121318	+	0.992614i	$0.538712\pi$
$702$	0	0
$703$	−11.5574	−0.435896
$704$	0	0
$705$	13.0719	0.492315
$706$	0	0
$707$	−43.1939	−1.62447
$708$	0	0
$709$	41.6671	1.56484	0.782421	−	0.622750i	$-0.213985\pi$
$709$	41.6671	1.56484	0.782421	+	0.622750i	$0.213985\pi$
$710$	0	0
$711$	−4.67403	−0.175290
$712$	0	0
$713$	−20.6299	−0.772596
$714$	0	0
$715$	1.47455	0.0551451
$716$	0	0
$717$	2.55770	0.0955192
$718$	0	0
$719$	−20.3139	−0.757579	−0.378789	−	0.925483i	$-0.623660\pi$
$719$	−20.3139	−0.757579	−0.378789	+	0.925483i	$0.623660\pi$
$720$	0	0
$721$	−8.47766	−0.315724
$722$	0	0
$723$	−3.17421	−0.118050
$724$	0	0
$725$	9.67132	0.359184
$726$	0	0
$727$	−8.95521	−0.332130	−0.166065	−	0.986115i	$-0.553106\pi$
$727$	−8.95521	−0.332130	−0.166065	+	0.986115i	$0.553106\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	32.9088	1.21718
$732$	0	0
$733$	41.2894	1.52506	0.762530	−	0.646953i	$-0.223957\pi$
$733$	41.2894	1.52506	0.762530	+	0.646953i	$0.223957\pi$
$734$	0	0
$735$	3.28390	0.121129
$736$	0	0
$737$	38.9231	1.43375
$738$	0	0
$739$	−43.9709	−1.61750	−0.808748	−	0.588155i	$-0.799854\pi$
$739$	−43.9709	−1.61750	−0.808748	+	0.588155i	$0.799854\pi$
$740$	0	0
$741$	0.481347	0.0176827
$742$	0	0
$743$	−0.975352	−0.0357822	−0.0178911	−	0.999840i	$-0.505695\pi$
$743$	−0.975352	−0.0357822	−0.0178911	+	0.999840i	$0.505695\pi$
$744$	0	0
$745$	21.5115	0.788119
$746$	0	0
$747$	−5.56419	−0.203583
$748$	0	0
$749$	−48.6170	−1.77642
$750$	0	0
$751$	−19.2204	−0.701363	−0.350682	−	0.936495i	$-0.614050\pi$
$751$	−19.2204	−0.701363	−0.350682	+	0.936495i	$0.614050\pi$
$752$	0	0
$753$	−11.5791	−0.421965
$754$	0	0
$755$	−20.8303	−0.758094
$756$	0	0
$757$	41.8117	1.51967	0.759836	−	0.650115i	$-0.225279\pi$
$757$	41.8117	1.51967	0.759836	+	0.650115i	$0.225279\pi$
$758$	0	0
$759$	−20.3831	−0.739858
$760$	0	0
$761$	31.4947	1.14168	0.570841	−	0.821061i	$-0.306617\pi$
$761$	31.4947	1.14168	0.570841	+	0.821061i	$0.306617\pi$
$762$	0	0
$763$	3.40501	0.123270
$764$	0	0
$765$	8.63377	0.312154
$766$	0	0
$767$	3.46714	0.125191
$768$	0	0
$769$	−24.9982	−0.901457	−0.450729	−	0.892661i	$-0.648836\pi$
$769$	−24.9982	−0.901457	−0.450729	+	0.892661i	$0.648836\pi$
$770$	0	0
$771$	0.706928	0.0254594
$772$	0	0
$773$	−46.2114	−1.66211	−0.831055	−	0.556190i	$-0.812263\pi$
$773$	−46.2114	−1.66211	−0.831055	+	0.556190i	$0.812263\pi$
$774$	0	0
$775$	−4.93344	−0.177214
$776$	0	0
$777$	−18.5650	−0.666015
$778$	0	0
$779$	−3.99833	−0.143255
$780$	0	0
$781$	43.8841	1.57030
$782$	0	0
$783$	−6.07335	−0.217044
$784$	0	0
$785$	26.1894	0.934738
$786$	0	0
$787$	−5.29787	−0.188849	−0.0944243	−	0.995532i	$-0.530101\pi$
$787$	−5.29787	−0.188849	−0.0944243	+	0.995532i	$0.530101\pi$
$788$	0	0
$789$	18.4391	0.656450
$790$	0	0
$791$	−36.5598	−1.29992
$792$	0	0
$793$	−1.06760	−0.0379117
$794$	0	0
$795$	8.76307	0.310794
$796$	0	0
$797$	−14.9240	−0.528636	−0.264318	−	0.964436i	$-0.585147\pi$
$797$	−14.9240	−0.528636	−0.264318	+	0.964436i	$0.585147\pi$
$798$	0	0
$799$	33.1201	1.17171
$800$	0	0
$801$	11.5771	0.409056
$802$	0	0
$803$	24.3906	0.860725
$804$	0	0
$805$	−36.4207	−1.28366
$806$	0	0
$807$	30.4675	1.07251
$808$	0	0
$809$	16.4205	0.577314	0.288657	−	0.957433i	$-0.406791\pi$
$809$	16.4205	0.577314	0.288657	+	0.957433i	$0.406791\pi$
$810$	0	0
$811$	35.8845	1.26007	0.630037	−	0.776565i	$-0.283040\pi$
$811$	35.8845	1.26007	0.630037	+	0.776565i	$0.283040\pi$
$812$	0	0
$813$	10.7782	0.378008
$814$	0	0
$815$	26.9614	0.944416
$816$	0	0
$817$	12.9784	0.454058
$818$	0	0
$819$	0.773201	0.0270178
$820$	0	0
$821$	−7.58815	−0.264828	−0.132414	−	0.991194i	$-0.542273\pi$
$821$	−7.58815	−0.264828	−0.132414	+	0.991194i	$0.542273\pi$
$822$	0	0
$823$	−47.8928	−1.66944	−0.834718	−	0.550677i	$-0.814370\pi$
$823$	−47.8928	−1.66944	−0.834718	+	0.550677i	$0.814370\pi$
$824$	0	0
$825$	−4.87441	−0.169705
$826$	0	0
$827$	7.33812	0.255172	0.127586	−	0.991828i	$-0.459277\pi$
$827$	7.33812	0.255172	0.127586	+	0.991828i	$0.459277\pi$
$828$	0	0
$829$	−25.6070	−0.889368	−0.444684	−	0.895688i	$-0.646684\pi$
$829$	−25.6070	−0.889368	−0.444684	+	0.895688i	$0.646684\pi$
$830$	0	0
$831$	−10.7465	−0.372791
$832$	0	0
$833$	8.32040	0.288285
$834$	0	0
$835$	1.84596	0.0638822
$836$	0	0
$837$	3.09808	0.107085
$838$	0	0
$839$	26.3391	0.909327	0.454664	−	0.890663i	$-0.349759\pi$
$839$	26.3391	0.909327	0.454664	+	0.890663i	$0.349759\pi$
$840$	0	0
$841$	7.88564	0.271919
$842$	0	0
$843$	20.8992	0.719808
$844$	0	0
$845$	−23.8718	−0.821216
$846$	0	0
$847$	−4.83013	−0.165965
$848$	0	0
$849$	30.2167	1.03703
$850$	0	0
$851$	41.7231	1.43025
$852$	0	0
$853$	−26.2465	−0.898662	−0.449331	−	0.893365i	$-0.648338\pi$
$853$	−26.2465	−0.898662	−0.449331	+	0.893365i	$0.648338\pi$
$854$	0	0
$855$	3.40495	0.116447
$856$	0	0
$857$	2.57976	0.0881229	0.0440615	−	0.999029i	$-0.485970\pi$
$857$	2.57976	0.0881229	0.0440615	+	0.999029i	$0.485970\pi$
$858$	0	0
$859$	10.6694	0.364035	0.182018	−	0.983295i	$-0.441737\pi$
$859$	10.6694	0.364035	0.182018	+	0.983295i	$0.441737\pi$
$860$	0	0
$861$	−6.42264	−0.218883
$862$	0	0
$863$	−20.7958	−0.707898	−0.353949	−	0.935265i	$-0.615161\pi$
$863$	−20.7958	−0.707898	−0.353949	+	0.935265i	$0.615161\pi$
$864$	0	0
$865$	7.23043	0.245842
$866$	0	0
$867$	4.87531	0.165574
$868$	0	0
$869$	−14.3073	−0.485341
$870$	0	0
$871$	3.31828	0.112436
$872$	0	0
$873$	−1.21855	−0.0412417
$874$	0	0
$875$	−36.0570	−1.21895
$876$	0	0
$877$	−33.8266	−1.14224	−0.571122	−	0.820865i	$-0.693492\pi$
$877$	−33.8266	−1.14224	−0.571122	+	0.820865i	$0.693492\pi$
$878$	0	0
$879$	−24.4441	−0.824479
$880$	0	0
$881$	4.52081	0.152310	0.0761550	−	0.997096i	$-0.475736\pi$
$881$	4.52081	0.152310	0.0761550	+	0.997096i	$0.475736\pi$
$882$	0	0
$883$	−37.0884	−1.24812	−0.624061	−	0.781375i	$-0.714518\pi$
$883$	−37.0884	−1.24812	−0.624061	+	0.781375i	$0.714518\pi$
$884$	0	0
$885$	24.5258	0.824427
$886$	0	0
$887$	−28.0114	−0.940530	−0.470265	−	0.882525i	$-0.655842\pi$
$887$	−28.0114	−0.940530	−0.470265	+	0.882525i	$0.655842\pi$
$888$	0	0
$889$	−2.24527	−0.0753039
$890$	0	0
$891$	3.06102	0.102548
$892$	0	0
$893$	13.0618	0.437095
$894$	0	0
$895$	16.4595	0.550179
$896$	0	0
$897$	−1.73770	−0.0580201
$898$	0	0
$899$	−18.8158	−0.627541
$900$	0	0
$901$	22.2029	0.739687
$902$	0	0
$903$	20.8476	0.693765
$904$	0	0
$905$	−5.12450	−0.170344
$906$	0	0
$907$	29.6037	0.982976	0.491488	−	0.870884i	$-0.336453\pi$
$907$	29.6037	0.982976	0.491488	+	0.870884i	$0.336453\pi$
$908$	0	0
$909$	−14.5781	−0.483525
$910$	0	0
$911$	−47.3438	−1.56857	−0.784284	−	0.620401i	$-0.786970\pi$
$911$	−47.3438	−1.56857	−0.784284	+	0.620401i	$0.786970\pi$
$912$	0	0
$913$	−17.0321	−0.563679
$914$	0	0
$915$	−7.55200	−0.249661
$916$	0	0
$917$	6.88092	0.227228
$918$	0	0
$919$	−38.4397	−1.26801	−0.634004	−	0.773330i	$-0.718590\pi$
$919$	−38.4397	−1.26801	−0.634004	+	0.773330i	$0.718590\pi$
$920$	0	0
$921$	−14.0634	−0.463406
$922$	0	0
$923$	3.74121	0.123143
$924$	0	0
$925$	9.97768	0.328064
$926$	0	0
$927$	−2.86124	−0.0939754
$928$	0	0
$929$	6.77877	0.222404	0.111202	−	0.993798i	$-0.464530\pi$
$929$	6.77877	0.222404	0.111202	+	0.993798i	$0.464530\pi$
$930$	0	0
$931$	3.28136	0.107542
$932$	0	0
$933$	3.87923	0.127000
$934$	0	0
$935$	26.4281	0.864291
$936$	0	0
$937$	−9.71842	−0.317487	−0.158743	−	0.987320i	$-0.550744\pi$
$937$	−9.71842	−0.317487	−0.158743	+	0.987320i	$0.550744\pi$
$938$	0	0
$939$	15.5885	0.508712
$940$	0	0
$941$	39.1367	1.27582	0.637910	−	0.770111i	$-0.279799\pi$
$941$	39.1367	1.27582	0.637910	+	0.770111i	$0.279799\pi$
$942$	0	0
$943$	14.4343	0.470045
$944$	0	0
$945$	5.46946	0.177922
$946$	0	0
$947$	−42.2393	−1.37259	−0.686296	−	0.727322i	$-0.740765\pi$
$947$	−42.2393	−1.37259	−0.686296	+	0.727322i	$0.740765\pi$
$948$	0	0
$949$	2.07935	0.0674986
$950$	0	0
$951$	8.05190	0.261101
$952$	0	0
$953$	10.9849	0.355836	0.177918	−	0.984045i	$-0.443064\pi$
$953$	10.9849	0.355836	0.177918	+	0.984045i	$0.443064\pi$
$954$	0	0
$955$	−31.9947	−1.03532
$956$	0	0
$957$	−18.5906	−0.600950
$958$	0	0
$959$	0.138336	0.00446709
$960$	0	0
$961$	−21.4019	−0.690383
$962$	0	0
$963$	−16.4084	−0.528753
$964$	0	0
$965$	−17.0862	−0.550024
$966$	0	0
$967$	−18.6674	−0.600302	−0.300151	−	0.953892i	$-0.597037\pi$
$967$	−18.6674	−0.600302	−0.300151	+	0.953892i	$0.597037\pi$
$968$	0	0
$969$	8.62709	0.277142
$970$	0	0
$971$	−51.4472	−1.65102	−0.825510	−	0.564387i	$-0.809112\pi$
$971$	−51.4472	−1.65102	−0.825510	+	0.564387i	$0.809112\pi$
$972$	0	0
$973$	−46.2962	−1.48419
$974$	0	0
$975$	−0.415554	−0.0133084
$976$	0	0
$977$	−29.9888	−0.959426	−0.479713	−	0.877425i	$-0.659259\pi$
$977$	−29.9888	−0.959426	−0.479713	+	0.877425i	$0.659259\pi$
$978$	0	0
$979$	35.4376	1.13259
$980$	0	0
$981$	1.14920	0.0366913
$982$	0	0
$983$	8.14730	0.259859	0.129929	−	0.991523i	$-0.458525\pi$
$983$	8.14730	0.259859	0.129929	+	0.991523i	$0.458525\pi$
$984$	0	0
$985$	−10.4625	−0.333362
$986$	0	0
$987$	20.9815	0.667848
$988$	0	0
$989$	−46.8531	−1.48984
$990$	0	0
$991$	49.9061	1.58532	0.792661	−	0.609663i	$-0.208695\pi$
$991$	49.9061	1.58532	0.792661	+	0.609663i	$0.208695\pi$
$992$	0	0
$993$	−20.0887	−0.637495
$994$	0	0
$995$	−7.73219	−0.245127
$996$	0	0
$997$	10.9233	0.345943	0.172971	−	0.984927i	$-0.444663\pi$
$997$	10.9233	0.345943	0.172971	+	0.984927i	$0.444663\pi$
$998$	0	0
$999$	−6.26575	−0.198239

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.84596 q^{5} +2.96293 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.84596 q^{5} +2.96293 q^{7} +1.00000 q^{9} +3.06102 q^{11} +0.260958 q^{13} +1.84596 q^{15} +4.67710 q^{17} +1.84454 q^{19} +2.96293 q^{21} -6.65892 q^{23} -1.59242 q^{25} +1.00000 q^{27} -6.07335 q^{29} +3.09808 q^{31} +3.06102 q^{33} +5.46946 q^{35} -6.26575 q^{37} +0.260958 q^{39} -2.16766 q^{41} +7.03615 q^{43} +1.84596 q^{45} +7.08133 q^{47} +1.77896 q^{49} +4.67710 q^{51} +4.74715 q^{53} +5.65052 q^{55} +1.84454 q^{57} +13.2862 q^{59} -4.09109 q^{61} +2.96293 q^{63} +0.481719 q^{65} +12.7157 q^{67} -6.65892 q^{69} +14.3364 q^{71} +7.96813 q^{73} -1.59242 q^{75} +9.06958 q^{77} -4.67403 q^{79} +1.00000 q^{81} -5.56419 q^{83} +8.63377 q^{85} -6.07335 q^{87} +11.5771 q^{89} +0.773201 q^{91} +3.09808 q^{93} +3.40495 q^{95} -1.21855 q^{97} +3.06102 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} - q^{5} + 4 q^{7} + 5 q^{9}+O(q^{10}) \) 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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