LMFDB - Embedded newform 9968.2.a.z.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9968,2,Mod(1,9968)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9968, 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("9968.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9968 = 2^{4} \cdot 7 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9968.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:	$79.5948807348$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.135076.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 5x^{3} + 4x^{2} + 4x - 2 $ x^5 - x^4 - 5x^3 + 4x^2 + 4*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1246)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.15154$ of defining polynomial
Character	$\chi$	$=$	9968.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.283134 q^{3} -1.52242 q^{5} -1.00000 q^{7} -2.91984 q^{9} +O(q^{10})$	$q-0.283134 q^{3} -1.52242 q^{5} -1.00000 q^{7} -2.91984 q^{9} +2.81429 q^{11} -6.32866 q^{13} +0.431050 q^{15} -5.91984 q^{17} -5.27817 q^{19} +0.283134 q^{21} +2.94951 q^{23} -2.68223 q^{25} +1.67611 q^{27} -2.72807 q^{29} -7.03015 q^{31} -0.796822 q^{33} +1.52242 q^{35} +2.35665 q^{37} +1.79186 q^{39} -7.62153 q^{41} -7.90614 q^{43} +4.44522 q^{45} +4.70713 q^{47} +1.00000 q^{49} +1.67611 q^{51} +9.16290 q^{53} -4.28454 q^{55} +1.49443 q^{57} +2.80724 q^{59} -9.93978 q^{61} +2.91984 q^{63} +9.63490 q^{65} -11.9867 q^{67} -0.835107 q^{69} -3.79891 q^{71} -11.3403 q^{73} +0.759431 q^{75} -2.81429 q^{77} +6.17033 q^{79} +8.28494 q^{81} -1.89197 q^{83} +9.01249 q^{85} +0.772411 q^{87} -1.00000 q^{89} +6.32866 q^{91} +1.99048 q^{93} +8.03561 q^{95} -6.86707 q^{97} -8.21726 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 6 q^{3} - 10 q^{5} - 5 q^{7} + 7 q^{9}+O(q^{10}) $ 5 * q + 6 * q^3 - 10 * q^5 - 5 * q^7 + 7 * q^9	$ 5 q + 6 q^{3} - 10 q^{5} - 5 q^{7} + 7 q^{9} + 8 q^{11} - 8 q^{13} - 6 q^{15} - 8 q^{17} + 10 q^{19} - 6 q^{21} + 2 q^{23} + 17 q^{25} + 24 q^{27} - 10 q^{29} + 10 q^{31} + 10 q^{35} - 4 q^{37} - 24 q^{39} - 28 q^{41} + 10 q^{43} - 12 q^{45} + 10 q^{47} + 5 q^{49} + 24 q^{51} + 4 q^{53} - 4 q^{55} + 2 q^{57} + 10 q^{59} - 16 q^{61} - 7 q^{63} - 26 q^{65} - 2 q^{69} + 16 q^{71} + 10 q^{73} + 10 q^{75} - 8 q^{77} + 8 q^{79} + 41 q^{81} + 14 q^{83} + 18 q^{85} - 22 q^{87} - 5 q^{89} + 8 q^{91} + 22 q^{93} - 34 q^{95} + 6 q^{97} + 8 q^{99}+O(q^{100}) $ 5 * q + 6 * q^3 - 10 * q^5 - 5 * q^7 + 7 * q^9 + 8 * q^11 - 8 * q^13 - 6 * q^15 - 8 * q^17 + 10 * q^19 - 6 * q^21 + 2 * q^23 + 17 * q^25 + 24 * q^27 - 10 * q^29 + 10 * q^31 + 10 * q^35 - 4 * q^37 - 24 * q^39 - 28 * q^41 + 10 * q^43 - 12 * q^45 + 10 * q^47 + 5 * q^49 + 24 * q^51 + 4 * q^53 - 4 * q^55 + 2 * q^57 + 10 * q^59 - 16 * q^61 - 7 * q^63 - 26 * q^65 - 2 * q^69 + 16 * q^71 + 10 * q^73 + 10 * q^75 - 8 * q^77 + 8 * q^79 + 41 * q^81 + 14 * q^83 + 18 * q^85 - 22 * q^87 - 5 * q^89 + 8 * q^91 + 22 * q^93 - 34 * q^95 + 6 * q^97 + 8 * 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
$3$	−0.283134	−0.163468	−0.0817338	−	0.996654i	$-0.526046\pi$
$3$	−0.283134	−0.163468	−0.0817338	+	0.996654i	$0.526046\pi$
$4$	0	0
$5$	−1.52242	−0.680848	−0.340424	−	0.940272i	$-0.610571\pi$
$5$	−1.52242	−0.680848	−0.340424	+	0.940272i	$0.610571\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.91984	−0.973278
$10$	0	0
$11$	2.81429	0.848541	0.424270	−	0.905536i	$-0.360531\pi$
$11$	2.81429	0.848541	0.424270	+	0.905536i	$0.360531\pi$
$12$	0	0
$13$	−6.32866	−1.75526	−0.877628	−	0.479343i	$-0.840875\pi$
$13$	−6.32866	−1.75526	−0.877628	+	0.479343i	$0.840875\pi$
$14$	0	0
$15$	0.431050	0.111297
$16$	0	0
$17$	−5.91984	−1.43577	−0.717885	−	0.696161i	$-0.754890\pi$
$17$	−5.91984	−1.43577	−0.717885	+	0.696161i	$0.754890\pi$
$18$	0	0
$19$	−5.27817	−1.21090	−0.605448	−	0.795885i	$-0.707006\pi$
$19$	−5.27817	−1.21090	−0.605448	+	0.795885i	$0.707006\pi$
$20$	0	0
$21$	0.283134	0.0617849
$22$	0	0
$23$	2.94951	0.615015	0.307508	−	0.951546i	$-0.400505\pi$
$23$	2.94951	0.615015	0.307508	+	0.951546i	$0.400505\pi$
$24$	0	0
$25$	−2.68223	−0.536446
$26$	0	0
$27$	1.67611	0.322567
$28$	0	0
$29$	−2.72807	−0.506590	−0.253295	−	0.967389i	$-0.581514\pi$
$29$	−2.72807	−0.506590	−0.253295	+	0.967389i	$0.581514\pi$
$30$	0	0
$31$	−7.03015	−1.26265	−0.631326	−	0.775517i	$-0.717489\pi$
$31$	−7.03015	−1.26265	−0.631326	+	0.775517i	$0.717489\pi$
$32$	0	0
$33$	−0.796822	−0.138709
$34$	0	0
$35$	1.52242	0.257336
$36$	0	0
$37$	2.35665	0.387432	0.193716	−	0.981058i	$-0.437946\pi$
$37$	2.35665	0.387432	0.193716	+	0.981058i	$0.437946\pi$
$38$	0	0
$39$	1.79186	0.286927
$40$	0	0
$41$	−7.62153	−1.19028	−0.595142	−	0.803621i	$-0.702904\pi$
$41$	−7.62153	−1.19028	−0.595142	+	0.803621i	$0.702904\pi$
$42$	0	0
$43$	−7.90614	−1.20568	−0.602838	−	0.797864i	$-0.705963\pi$
$43$	−7.90614	−1.20568	−0.602838	+	0.797864i	$0.705963\pi$
$44$	0	0
$45$	4.44522	0.662655
$46$	0	0
$47$	4.70713	0.686606	0.343303	−	0.939225i	$-0.388454\pi$
$47$	4.70713	0.686606	0.343303	+	0.939225i	$0.388454\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.67611	0.234702
$52$	0	0
$53$	9.16290	1.25862	0.629310	−	0.777154i	$-0.283337\pi$
$53$	9.16290	1.25862	0.629310	+	0.777154i	$0.283337\pi$
$54$	0	0
$55$	−4.28454	−0.577727
$56$	0	0
$57$	1.49443	0.197942
$58$	0	0
$59$	2.80724	0.365472	0.182736	−	0.983162i	$-0.441505\pi$
$59$	2.80724	0.365472	0.182736	+	0.983162i	$0.441505\pi$
$60$	0	0
$61$	−9.93978	−1.27266	−0.636329	−	0.771418i	$-0.719548\pi$
$61$	−9.93978	−1.27266	−0.636329	+	0.771418i	$0.719548\pi$
$62$	0	0
$63$	2.91984	0.367865
$64$	0	0
$65$	9.63490	1.19506
$66$	0	0
$67$	−11.9867	−1.46441	−0.732205	−	0.681084i	$-0.761509\pi$
$67$	−11.9867	−1.46441	−0.732205	+	0.681084i	$0.761509\pi$
$68$	0	0
$69$	−0.835107	−0.100535
$70$	0	0
$71$	−3.79891	−0.450848	−0.225424	−	0.974261i	$-0.572377\pi$
$71$	−3.79891	−0.450848	−0.225424	+	0.974261i	$0.572377\pi$
$72$	0	0
$73$	−11.3403	−1.32728	−0.663640	−	0.748052i	$-0.730989\pi$
$73$	−11.3403	−1.32728	−0.663640	+	0.748052i	$0.730989\pi$
$74$	0	0
$75$	0.759431	0.0876916
$76$	0	0
$77$	−2.81429	−0.320718
$78$	0	0
$79$	6.17033	0.694216	0.347108	−	0.937825i	$-0.387164\pi$
$79$	6.17033	0.694216	0.347108	+	0.937825i	$0.387164\pi$
$80$	0	0
$81$	8.28494	0.920549
$82$	0	0
$83$	−1.89197	−0.207670	−0.103835	−	0.994595i	$-0.533111\pi$
$83$	−1.89197	−0.207670	−0.103835	+	0.994595i	$0.533111\pi$
$84$	0	0
$85$	9.01249	0.977542
$86$	0	0
$87$	0.772411	0.0828111
$88$	0	0
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	6.32866	0.663424
$92$	0	0
$93$	1.99048	0.206403
$94$	0	0
$95$	8.03561	0.824436
$96$	0	0
$97$	−6.86707	−0.697245	−0.348622	−	0.937263i	$-0.613350\pi$
$97$	−6.86707	−0.697245	−0.348622	+	0.937263i	$0.613350\pi$
$98$	0	0
$99$	−8.21726	−0.825866
$100$	0	0
$101$	−3.95924	−0.393959	−0.196980	−	0.980408i	$-0.563113\pi$
$101$	−3.95924	−0.393959	−0.196980	+	0.980408i	$0.563113\pi$
$102$	0	0
$103$	−5.97489	−0.588723	−0.294362	−	0.955694i	$-0.595107\pi$
$103$	−5.97489	−0.588723	−0.294362	+	0.955694i	$0.595107\pi$
$104$	0	0
$105$	−0.431050	−0.0420662
$106$	0	0
$107$	20.4155	1.97364	0.986820	−	0.161824i	$-0.0517378\pi$
$107$	20.4155	1.97364	0.986820	+	0.161824i	$0.0517378\pi$
$108$	0	0
$109$	−13.4986	−1.29293	−0.646466	−	0.762943i	$-0.723754\pi$
$109$	−13.4986	−1.29293	−0.646466	+	0.762943i	$0.723754\pi$
$110$	0	0
$111$	−0.667250	−0.0633325
$112$	0	0
$113$	11.4204	1.07434	0.537172	−	0.843472i	$-0.319492\pi$
$113$	11.4204	1.07434	0.537172	+	0.843472i	$0.319492\pi$
$114$	0	0
$115$	−4.49040	−0.418732
$116$	0	0
$117$	18.4787	1.70835
$118$	0	0
$119$	5.91984	0.542670
$120$	0	0
$121$	−3.07977	−0.279979
$122$	0	0
$123$	2.15792	0.194573
$124$	0	0
$125$	11.6956	1.04609
$126$	0	0
$127$	7.02928	0.623747	0.311874	−	0.950124i	$-0.399043\pi$
$127$	7.02928	0.623747	0.311874	+	0.950124i	$0.399043\pi$
$128$	0	0
$129$	2.23850	0.197089
$130$	0	0
$131$	6.11905	0.534624	0.267312	−	0.963610i	$-0.413865\pi$
$131$	6.11905	0.534624	0.267312	+	0.963610i	$0.413865\pi$
$132$	0	0
$133$	5.27817	0.457676
$134$	0	0
$135$	−2.55174	−0.219619
$136$	0	0
$137$	18.4828	1.57909	0.789544	−	0.613694i	$-0.210317\pi$
$137$	18.4828	1.57909	0.789544	+	0.613694i	$0.210317\pi$
$138$	0	0
$139$	11.8710	1.00689	0.503444	−	0.864028i	$-0.332066\pi$
$139$	11.8710	1.00689	0.503444	+	0.864028i	$0.332066\pi$
$140$	0	0
$141$	−1.33275	−0.112238
$142$	0	0
$143$	−17.8107	−1.48941
$144$	0	0
$145$	4.15328	0.344911
$146$	0	0
$147$	−0.283134	−0.0233525
$148$	0	0
$149$	−10.0751	−0.825382	−0.412691	−	0.910871i	$-0.635411\pi$
$149$	−10.0751	−0.825382	−0.412691	+	0.910871i	$0.635411\pi$
$150$	0	0
$151$	−18.7787	−1.52819	−0.764096	−	0.645103i	$-0.776814\pi$
$151$	−18.7787	−1.52819	−0.764096	+	0.645103i	$0.776814\pi$
$152$	0	0
$153$	17.2849	1.39740
$154$	0	0
$155$	10.7029	0.859674
$156$	0	0
$157$	−2.82282	−0.225285	−0.112643	−	0.993636i	$-0.535932\pi$
$157$	−2.82282	−0.225285	−0.112643	+	0.993636i	$0.535932\pi$
$158$	0	0
$159$	−2.59433	−0.205744
$160$	0	0
$161$	−2.94951	−0.232454
$162$	0	0
$163$	−11.7686	−0.921791	−0.460896	−	0.887454i	$-0.652472\pi$
$163$	−11.7686	−0.921791	−0.460896	+	0.887454i	$0.652472\pi$
$164$	0	0
$165$	1.21310	0.0944397
$166$	0	0
$167$	−22.0583	−1.70692	−0.853461	−	0.521156i	$-0.825501\pi$
$167$	−22.0583	−1.70692	−0.853461	+	0.521156i	$0.825501\pi$
$168$	0	0
$169$	27.0520	2.08092
$170$	0	0
$171$	15.4114	1.17854
$172$	0	0
$173$	−17.9857	−1.36743	−0.683714	−	0.729750i	$-0.739637\pi$
$173$	−17.9857	−1.36743	−0.683714	+	0.729750i	$0.739637\pi$
$174$	0	0
$175$	2.68223	0.202758
$176$	0	0
$177$	−0.794826	−0.0597428
$178$	0	0
$179$	−2.21109	−0.165265	−0.0826323	−	0.996580i	$-0.526333\pi$
$179$	−2.21109	−0.165265	−0.0826323	+	0.996580i	$0.526333\pi$
$180$	0	0
$181$	13.1317	0.976069	0.488034	−	0.872824i	$-0.337714\pi$
$181$	13.1317	0.976069	0.488034	+	0.872824i	$0.337714\pi$
$182$	0	0
$183$	2.81429	0.208038
$184$	0	0
$185$	−3.58782	−0.263782
$186$	0	0
$187$	−16.6601	−1.21831
$188$	0	0
$189$	−1.67611	−0.121919
$190$	0	0
$191$	2.80584	0.203023	0.101512	−	0.994834i	$-0.467632\pi$
$191$	2.80584	0.203023	0.101512	+	0.994834i	$0.467632\pi$
$192$	0	0
$193$	5.02738	0.361878	0.180939	−	0.983494i	$-0.442086\pi$
$193$	5.02738	0.361878	0.180939	+	0.983494i	$0.442086\pi$
$194$	0	0
$195$	−2.72797	−0.195354
$196$	0	0
$197$	−11.3342	−0.807530	−0.403765	−	0.914863i	$-0.632299\pi$
$197$	−11.3342	−0.807530	−0.403765	+	0.914863i	$0.632299\pi$
$198$	0	0
$199$	15.0694	1.06824	0.534122	−	0.845407i	$-0.320642\pi$
$199$	15.0694	1.06824	0.534122	+	0.845407i	$0.320642\pi$
$200$	0	0
$201$	3.39385	0.239384
$202$	0	0
$203$	2.72807	0.191473
$204$	0	0
$205$	11.6032	0.810402
$206$	0	0
$207$	−8.61208	−0.598581
$208$	0	0
$209$	−14.8543	−1.02749
$210$	0	0
$211$	3.04632	0.209717	0.104859	−	0.994487i	$-0.466561\pi$
$211$	3.04632	0.209717	0.104859	+	0.994487i	$0.466561\pi$
$212$	0	0
$213$	1.07560	0.0736990
$214$	0	0
$215$	12.0365	0.820881
$216$	0	0
$217$	7.03015	0.477238
$218$	0	0
$219$	3.21082	0.216967
$220$	0	0
$221$	37.4646	2.52014
$222$	0	0
$223$	−0.865467	−0.0579559	−0.0289780	−	0.999580i	$-0.509225\pi$
$223$	−0.865467	−0.0579559	−0.0289780	+	0.999580i	$0.509225\pi$
$224$	0	0
$225$	7.83167	0.522111
$226$	0	0
$227$	−17.2417	−1.14437	−0.572185	−	0.820125i	$-0.693904\pi$
$227$	−17.2417	−1.14437	−0.572185	+	0.820125i	$0.693904\pi$
$228$	0	0
$229$	−22.3826	−1.47909	−0.739544	−	0.673109i	$-0.764959\pi$
$229$	−22.3826	−1.47909	−0.739544	+	0.673109i	$0.764959\pi$
$230$	0	0
$231$	0.796822	0.0524270
$232$	0	0
$233$	−6.70289	−0.439121	−0.219561	−	0.975599i	$-0.570462\pi$
$233$	−6.70289	−0.439121	−0.219561	+	0.975599i	$0.570462\pi$
$234$	0	0
$235$	−7.16624	−0.467474
$236$	0	0
$237$	−1.74703	−0.113482
$238$	0	0
$239$	8.76876	0.567204	0.283602	−	0.958942i	$-0.408471\pi$
$239$	8.76876	0.567204	0.283602	+	0.958942i	$0.408471\pi$
$240$	0	0
$241$	27.2361	1.75443	0.877215	−	0.480097i	$-0.159399\pi$
$241$	27.2361	1.75443	0.877215	+	0.480097i	$0.159399\pi$
$242$	0	0
$243$	−7.37407	−0.473047
$244$	0	0
$245$	−1.52242	−0.0972640
$246$	0	0
$247$	33.4038	2.12543
$248$	0	0
$249$	0.535681	0.0339474
$250$	0	0
$251$	24.9700	1.57609	0.788046	−	0.615616i	$-0.211093\pi$
$251$	24.9700	1.57609	0.788046	+	0.615616i	$0.211093\pi$
$252$	0	0
$253$	8.30078	0.521865
$254$	0	0
$255$	−2.55174	−0.159796
$256$	0	0
$257$	−10.9248	−0.681470	−0.340735	−	0.940159i	$-0.610676\pi$
$257$	−10.9248	−0.681470	−0.340735	+	0.940159i	$0.610676\pi$
$258$	0	0
$259$	−2.35665	−0.146435
$260$	0	0
$261$	7.96552	0.493054
$262$	0	0
$263$	−15.2838	−0.942441	−0.471220	−	0.882016i	$-0.656186\pi$
$263$	−15.2838	−0.942441	−0.471220	+	0.882016i	$0.656186\pi$
$264$	0	0
$265$	−13.9498	−0.856929
$266$	0	0
$267$	0.283134	0.0173275
$268$	0	0
$269$	14.6450	0.892924	0.446462	−	0.894803i	$-0.352684\pi$
$269$	14.6450	0.892924	0.446462	+	0.894803i	$0.352684\pi$
$270$	0	0
$271$	−12.0540	−0.732227	−0.366114	−	0.930570i	$-0.619312\pi$
$271$	−12.0540	−0.732227	−0.366114	+	0.930570i	$0.619312\pi$
$272$	0	0
$273$	−1.79186	−0.108448
$274$	0	0
$275$	−7.54858	−0.455196
$276$	0	0
$277$	−22.9946	−1.38161	−0.690807	−	0.723039i	$-0.742745\pi$
$277$	−22.9946	−1.38161	−0.690807	+	0.723039i	$0.742745\pi$
$278$	0	0
$279$	20.5269	1.22891
$280$	0	0
$281$	−9.01251	−0.537641	−0.268821	−	0.963190i	$-0.586634\pi$
$281$	−9.01251	−0.537641	−0.268821	+	0.963190i	$0.586634\pi$
$282$	0	0
$283$	13.7647	0.818225	0.409113	−	0.912484i	$-0.365838\pi$
$283$	13.7647	0.818225	0.409113	+	0.912484i	$0.365838\pi$
$284$	0	0
$285$	−2.27516	−0.134769
$286$	0	0
$287$	7.62153	0.449885
$288$	0	0
$289$	18.0444	1.06144
$290$	0	0
$291$	1.94430	0.113977
$292$	0	0
$293$	−20.4051	−1.19208	−0.596038	−	0.802956i	$-0.703259\pi$
$293$	−20.4051	−1.19208	−0.596038	+	0.802956i	$0.703259\pi$
$294$	0	0
$295$	−4.27380	−0.248831
$296$	0	0
$297$	4.71705	0.273711
$298$	0	0
$299$	−18.6665	−1.07951
$300$	0	0
$301$	7.90614	0.455702
$302$	0	0
$303$	1.12100	0.0643996
$304$	0	0
$305$	15.1325	0.866487
$306$	0	0
$307$	15.8677	0.905615	0.452808	−	0.891608i	$-0.350422\pi$
$307$	15.8677	0.905615	0.452808	+	0.891608i	$0.350422\pi$
$308$	0	0
$309$	1.69170	0.0962372
$310$	0	0
$311$	20.5290	1.16409	0.582045	−	0.813156i	$-0.302253\pi$
$311$	20.5290	1.16409	0.582045	+	0.813156i	$0.302253\pi$
$312$	0	0
$313$	2.67261	0.151065	0.0755325	−	0.997143i	$-0.475934\pi$
$313$	2.67261	0.151065	0.0755325	+	0.997143i	$0.475934\pi$
$314$	0	0
$315$	−4.44522	−0.250460
$316$	0	0
$317$	−26.1210	−1.46710	−0.733551	−	0.679635i	$-0.762138\pi$
$317$	−26.1210	−1.46710	−0.733551	+	0.679635i	$0.762138\pi$
$318$	0	0
$319$	−7.67759	−0.429863
$320$	0	0
$321$	−5.78032	−0.322626
$322$	0	0
$323$	31.2459	1.73857
$324$	0	0
$325$	16.9749	0.941600
$326$	0	0
$327$	3.82192	0.211353
$328$	0	0
$329$	−4.70713	−0.259513
$330$	0	0
$331$	−24.2456	−1.33266	−0.666330	−	0.745657i	$-0.732136\pi$
$331$	−24.2456	−1.33266	−0.666330	+	0.745657i	$0.732136\pi$
$332$	0	0
$333$	−6.88104	−0.377079
$334$	0	0
$335$	18.2488	0.997040
$336$	0	0
$337$	34.9254	1.90251	0.951253	−	0.308413i	$-0.0997979\pi$
$337$	34.9254	1.90251	0.951253	+	0.308413i	$0.0997979\pi$
$338$	0	0
$339$	−3.23352	−0.175621
$340$	0	0
$341$	−19.7849	−1.07141
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	1.27139	0.0684491
$346$	0	0
$347$	9.62659	0.516782	0.258391	−	0.966040i	$-0.416808\pi$
$347$	9.62659	0.516782	0.258391	+	0.966040i	$0.416808\pi$
$348$	0	0
$349$	−0.529255	−0.0283304	−0.0141652	−	0.999900i	$-0.504509\pi$
$349$	−0.529255	−0.0283304	−0.0141652	+	0.999900i	$0.504509\pi$
$350$	0	0
$351$	−10.6075	−0.566188
$352$	0	0
$353$	20.1579	1.07290	0.536449	−	0.843933i	$-0.319765\pi$
$353$	20.1579	1.07290	0.536449	+	0.843933i	$0.319765\pi$
$354$	0	0
$355$	5.78355	0.306959
$356$	0	0
$357$	−1.67611	−0.0887090
$358$	0	0
$359$	−17.0857	−0.901748	−0.450874	−	0.892588i	$-0.648888\pi$
$359$	−17.0857	−0.901748	−0.450874	+	0.892588i	$0.648888\pi$
$360$	0	0
$361$	8.85912	0.466269
$362$	0	0
$363$	0.871987	0.0457675
$364$	0	0
$365$	17.2647	0.903675
$366$	0	0
$367$	−31.5635	−1.64760	−0.823800	−	0.566880i	$-0.808150\pi$
$367$	−31.5635	−1.64760	−0.823800	+	0.566880i	$0.808150\pi$
$368$	0	0
$369$	22.2536	1.15848
$370$	0	0
$371$	−9.16290	−0.475714
$372$	0	0
$373$	−8.84328	−0.457888	−0.228944	−	0.973440i	$-0.573527\pi$
$373$	−8.84328	−0.457888	−0.228944	+	0.973440i	$0.573527\pi$
$374$	0	0
$375$	−3.31142	−0.171001
$376$	0	0
$377$	17.2651	0.889196
$378$	0	0
$379$	−13.4034	−0.688486	−0.344243	−	0.938881i	$-0.611864\pi$
$379$	−13.4034	−0.688486	−0.344243	+	0.938881i	$0.611864\pi$
$380$	0	0
$381$	−1.99023	−0.101963
$382$	0	0
$383$	8.44442	0.431490	0.215745	−	0.976450i	$-0.430782\pi$
$383$	8.44442	0.431490	0.215745	+	0.976450i	$0.430782\pi$
$384$	0	0
$385$	4.28454	0.218360
$386$	0	0
$387$	23.0846	1.17346
$388$	0	0
$389$	13.4954	0.684242	0.342121	−	0.939656i	$-0.388855\pi$
$389$	13.4954	0.684242	0.342121	+	0.939656i	$0.388855\pi$
$390$	0	0
$391$	−17.4606	−0.883021
$392$	0	0
$393$	−1.73251	−0.0873937
$394$	0	0
$395$	−9.39385	−0.472656
$396$	0	0
$397$	−1.61816	−0.0812133	−0.0406067	−	0.999175i	$-0.512929\pi$
$397$	−1.61816	−0.0812133	−0.0406067	+	0.999175i	$0.512929\pi$
$398$	0	0
$399$	−1.49443	−0.0748151
$400$	0	0
$401$	1.80846	0.0903100	0.0451550	−	0.998980i	$-0.485622\pi$
$401$	1.80846	0.0903100	0.0451550	+	0.998980i	$0.485622\pi$
$402$	0	0
$403$	44.4915	2.21628
$404$	0	0
$405$	−12.6132	−0.626754
$406$	0	0
$407$	6.63231	0.328751
$408$	0	0
$409$	−22.1177	−1.09365	−0.546824	−	0.837248i	$-0.684163\pi$
$409$	−22.1177	−1.09365	−0.546824	+	0.837248i	$0.684163\pi$
$410$	0	0
$411$	−5.23310	−0.258130
$412$	0	0
$413$	−2.80724	−0.138135
$414$	0	0
$415$	2.88037	0.141392
$416$	0	0
$417$	−3.36109	−0.164593
$418$	0	0
$419$	5.96489	0.291404	0.145702	−	0.989329i	$-0.453456\pi$
$419$	5.96489	0.291404	0.145702	+	0.989329i	$0.453456\pi$
$420$	0	0
$421$	6.54870	0.319164	0.159582	−	0.987185i	$-0.448985\pi$
$421$	6.54870	0.319164	0.159582	+	0.987185i	$0.448985\pi$
$422$	0	0
$423$	−13.7441	−0.668259
$424$	0	0
$425$	15.8784	0.770214
$426$	0	0
$427$	9.93978	0.481020
$428$	0	0
$429$	5.04282	0.243470
$430$	0	0
$431$	6.23620	0.300387	0.150194	−	0.988657i	$-0.452010\pi$
$431$	6.23620	0.300387	0.150194	+	0.988657i	$0.452010\pi$
$432$	0	0
$433$	−33.0255	−1.58710	−0.793551	−	0.608504i	$-0.791770\pi$
$433$	−33.0255	−1.58710	−0.793551	+	0.608504i	$0.791770\pi$
$434$	0	0
$435$	−1.17594	−0.0563818
$436$	0	0
$437$	−15.5680	−0.744720
$438$	0	0
$439$	23.2687	1.11055	0.555277	−	0.831665i	$-0.312612\pi$
$439$	23.2687	1.11055	0.555277	+	0.831665i	$0.312612\pi$
$440$	0	0
$441$	−2.91984	−0.139040
$442$	0	0
$443$	22.3258	1.06073	0.530365	−	0.847769i	$-0.322055\pi$
$443$	22.3258	1.06073	0.530365	+	0.847769i	$0.322055\pi$
$444$	0	0
$445$	1.52242	0.0721697
$446$	0	0
$447$	2.85260	0.134923
$448$	0	0
$449$	19.2679	0.909311	0.454655	−	0.890667i	$-0.349762\pi$
$449$	19.2679	0.909311	0.454655	+	0.890667i	$0.349762\pi$
$450$	0	0
$451$	−21.4492	−1.01000
$452$	0	0
$453$	5.31690	0.249810
$454$	0	0
$455$	−9.63490	−0.451691
$456$	0	0
$457$	1.73774	0.0812878	0.0406439	−	0.999174i	$-0.487059\pi$
$457$	1.73774	0.0812878	0.0406439	+	0.999174i	$0.487059\pi$
$458$	0	0
$459$	−9.92228	−0.463132
$460$	0	0
$461$	−25.7938	−1.20134	−0.600669	−	0.799498i	$-0.705099\pi$
$461$	−25.7938	−1.20134	−0.600669	+	0.799498i	$0.705099\pi$
$462$	0	0
$463$	12.4164	0.577037	0.288519	−	0.957474i	$-0.406837\pi$
$463$	12.4164	0.577037	0.288519	+	0.957474i	$0.406837\pi$
$464$	0	0
$465$	−3.03035	−0.140529
$466$	0	0
$467$	−18.2629	−0.845105	−0.422552	−	0.906339i	$-0.638866\pi$
$467$	−18.2629	−0.845105	−0.422552	+	0.906339i	$0.638866\pi$
$468$	0	0
$469$	11.9867	0.553495
$470$	0	0
$471$	0.799236	0.0368268
$472$	0	0
$473$	−22.2502	−1.02306
$474$	0	0
$475$	14.1573	0.649581
$476$	0	0
$477$	−26.7541	−1.22499
$478$	0	0
$479$	17.9655	0.820865	0.410432	−	0.911891i	$-0.365378\pi$
$479$	17.9655	0.820865	0.410432	+	0.911891i	$0.365378\pi$
$480$	0	0
$481$	−14.9145	−0.680042
$482$	0	0
$483$	0.835107	0.0379987
$484$	0	0
$485$	10.4546	0.474718
$486$	0	0
$487$	−10.1077	−0.458022	−0.229011	−	0.973424i	$-0.573549\pi$
$487$	−10.1077	−0.458022	−0.229011	+	0.973424i	$0.573549\pi$
$488$	0	0
$489$	3.33211	0.150683
$490$	0	0
$491$	6.94899	0.313603	0.156802	−	0.987630i	$-0.449882\pi$
$491$	6.94899	0.313603	0.156802	+	0.987630i	$0.449882\pi$
$492$	0	0
$493$	16.1497	0.727348
$494$	0	0
$495$	12.5101	0.562289
$496$	0	0
$497$	3.79891	0.170404
$498$	0	0
$499$	32.3015	1.44601	0.723007	−	0.690841i	$-0.242760\pi$
$499$	32.3015	1.44601	0.723007	+	0.690841i	$0.242760\pi$
$500$	0	0
$501$	6.24546	0.279027
$502$	0	0
$503$	−30.5663	−1.36289	−0.681443	−	0.731871i	$-0.738647\pi$
$503$	−30.5663	−1.36289	−0.681443	+	0.731871i	$0.738647\pi$
$504$	0	0
$505$	6.02764	0.268226
$506$	0	0
$507$	−7.65934	−0.340163
$508$	0	0
$509$	−17.3284	−0.768067	−0.384034	−	0.923319i	$-0.625465\pi$
$509$	−17.3284	−0.768067	−0.384034	+	0.923319i	$0.625465\pi$
$510$	0	0
$511$	11.3403	0.501664
$512$	0	0
$513$	−8.84679	−0.390595
$514$	0	0
$515$	9.09630	0.400831
$516$	0	0
$517$	13.2472	0.582613
$518$	0	0
$519$	5.09237	0.223530
$520$	0	0
$521$	−6.49405	−0.284509	−0.142255	−	0.989830i	$-0.545435\pi$
$521$	−6.49405	−0.284509	−0.142255	+	0.989830i	$0.545435\pi$
$522$	0	0
$523$	−0.554376	−0.0242411	−0.0121206	−	0.999927i	$-0.503858\pi$
$523$	−0.554376	−0.0242411	−0.0121206	+	0.999927i	$0.503858\pi$
$524$	0	0
$525$	−0.759431	−0.0331443
$526$	0	0
$527$	41.6173	1.81288
$528$	0	0
$529$	−14.3004	−0.621756
$530$	0	0
$531$	−8.19668	−0.355706
$532$	0	0
$533$	48.2341	2.08925
$534$	0	0
$535$	−31.0810	−1.34375
$536$	0	0
$537$	0.626035	0.0270154
$538$	0	0
$539$	2.81429	0.121220
$540$	0	0
$541$	−41.1860	−1.77072	−0.885362	−	0.464902i	$-0.846090\pi$
$541$	−41.1860	−1.77072	−0.885362	+	0.464902i	$0.846090\pi$
$542$	0	0
$543$	−3.71802	−0.159556
$544$	0	0
$545$	20.5506	0.880290
$546$	0	0
$547$	35.8790	1.53408	0.767038	−	0.641601i	$-0.221730\pi$
$547$	35.8790	1.53408	0.767038	+	0.641601i	$0.221730\pi$
$548$	0	0
$549$	29.0225	1.23865
$550$	0	0
$551$	14.3992	0.613428
$552$	0	0
$553$	−6.17033	−0.262389
$554$	0	0
$555$	1.01584	0.0431198
$556$	0	0
$557$	36.4641	1.54504	0.772518	−	0.634993i	$-0.218997\pi$
$557$	36.4641	1.54504	0.772518	+	0.634993i	$0.218997\pi$
$558$	0	0
$559$	50.0353	2.11627
$560$	0	0
$561$	4.71705	0.199154
$562$	0	0
$563$	20.9026	0.880939	0.440469	−	0.897768i	$-0.354812\pi$
$563$	20.9026	0.880939	0.440469	+	0.897768i	$0.354812\pi$
$564$	0	0
$565$	−17.3867	−0.731465
$566$	0	0
$567$	−8.28494	−0.347935
$568$	0	0
$569$	−9.38887	−0.393602	−0.196801	−	0.980443i	$-0.563055\pi$
$569$	−9.38887	−0.393602	−0.196801	+	0.980443i	$0.563055\pi$
$570$	0	0
$571$	4.51926	0.189125	0.0945626	−	0.995519i	$-0.469855\pi$
$571$	4.51926	0.189125	0.0945626	+	0.995519i	$0.469855\pi$
$572$	0	0
$573$	−0.794428	−0.0331877
$574$	0	0
$575$	−7.91126	−0.329923
$576$	0	0
$577$	40.1679	1.67221	0.836106	−	0.548568i	$-0.184827\pi$
$577$	40.1679	1.67221	0.836106	+	0.548568i	$0.184827\pi$
$578$	0	0
$579$	−1.42342	−0.0591554
$580$	0	0
$581$	1.89197	0.0784921
$582$	0	0
$583$	25.7871	1.06799
$584$	0	0
$585$	−28.1323	−1.16313
$586$	0	0
$587$	22.1777	0.915373	0.457687	−	0.889114i	$-0.348678\pi$
$587$	22.1777	0.915373	0.457687	+	0.889114i	$0.348678\pi$
$588$	0	0
$589$	37.1064	1.52894
$590$	0	0
$591$	3.20911	0.132005
$592$	0	0
$593$	18.8924	0.775820	0.387910	−	0.921697i	$-0.373197\pi$
$593$	18.8924	0.775820	0.387910	+	0.921697i	$0.373197\pi$
$594$	0	0
$595$	−9.01249	−0.369476
$596$	0	0
$597$	−4.26667	−0.174623
$598$	0	0
$599$	29.2882	1.19668	0.598341	−	0.801241i	$-0.295827\pi$
$599$	29.2882	1.19668	0.598341	+	0.801241i	$0.295827\pi$
$600$	0	0
$601$	26.3732	1.07579	0.537893	−	0.843013i	$-0.319221\pi$
$601$	26.3732	1.07579	0.537893	+	0.843013i	$0.319221\pi$
$602$	0	0
$603$	34.9992	1.42528
$604$	0	0
$605$	4.68871	0.190623
$606$	0	0
$607$	18.5788	0.754089	0.377044	−	0.926195i	$-0.376940\pi$
$607$	18.5788	0.754089	0.377044	+	0.926195i	$0.376940\pi$
$608$	0	0
$609$	−0.772411	−0.0312997
$610$	0	0
$611$	−29.7899	−1.20517
$612$	0	0
$613$	−42.7127	−1.72515	−0.862574	−	0.505931i	$-0.831149\pi$
$613$	−42.7127	−1.72515	−0.862574	+	0.505931i	$0.831149\pi$
$614$	0	0
$615$	−3.28526	−0.132474
$616$	0	0
$617$	5.36228	0.215877	0.107939	−	0.994158i	$-0.465575\pi$
$617$	5.36228	0.215877	0.107939	+	0.994158i	$0.465575\pi$
$618$	0	0
$619$	−34.5682	−1.38941	−0.694707	−	0.719292i	$-0.744466\pi$
$619$	−34.5682	−1.38941	−0.694707	+	0.719292i	$0.744466\pi$
$620$	0	0
$621$	4.94370	0.198384
$622$	0	0
$623$	1.00000	0.0400642
$624$	0	0
$625$	−4.39449	−0.175779
$626$	0	0
$627$	4.20576	0.167962
$628$	0	0
$629$	−13.9510	−0.556263
$630$	0	0
$631$	−3.28964	−0.130959	−0.0654793	−	0.997854i	$-0.520858\pi$
$631$	−3.28964	−0.130959	−0.0654793	+	0.997854i	$0.520858\pi$
$632$	0	0
$633$	−0.862517	−0.0342820
$634$	0	0
$635$	−10.7015	−0.424677
$636$	0	0
$637$	−6.32866	−0.250751
$638$	0	0
$639$	11.0922	0.438801
$640$	0	0
$641$	−33.5365	−1.32461	−0.662306	−	0.749233i	$-0.730422\pi$
$641$	−33.5365	−1.32461	−0.662306	+	0.749233i	$0.730422\pi$
$642$	0	0
$643$	40.3871	1.59271	0.796355	−	0.604829i	$-0.206759\pi$
$643$	40.3871	1.59271	0.796355	+	0.604829i	$0.206759\pi$
$644$	0	0
$645$	−3.40794	−0.134188
$646$	0	0
$647$	31.6134	1.24285	0.621425	−	0.783474i	$-0.286554\pi$
$647$	31.6134	1.24285	0.621425	+	0.783474i	$0.286554\pi$
$648$	0	0
$649$	7.90039	0.310117
$650$	0	0
$651$	−1.99048	−0.0780129
$652$	0	0
$653$	−16.0273	−0.627195	−0.313598	−	0.949556i	$-0.601534\pi$
$653$	−16.0273	−0.627195	−0.313598	+	0.949556i	$0.601534\pi$
$654$	0	0
$655$	−9.31578	−0.363998
$656$	0	0
$657$	33.1117	1.29181
$658$	0	0
$659$	−7.29424	−0.284143	−0.142072	−	0.989856i	$-0.545376\pi$
$659$	−7.29424	−0.284143	−0.142072	+	0.989856i	$0.545376\pi$
$660$	0	0
$661$	37.5660	1.46115	0.730574	−	0.682834i	$-0.239253\pi$
$661$	37.5660	1.46115	0.730574	+	0.682834i	$0.239253\pi$
$662$	0	0
$663$	−10.6075	−0.411962
$664$	0	0
$665$	−8.03561	−0.311608
$666$	0	0
$667$	−8.04648	−0.311561
$668$	0	0
$669$	0.245043	0.00947392
$670$	0	0
$671$	−27.9734	−1.07990
$672$	0	0
$673$	44.1975	1.70369	0.851845	−	0.523794i	$-0.175484\pi$
$673$	44.1975	1.70369	0.851845	+	0.523794i	$0.175484\pi$
$674$	0	0
$675$	−4.49571	−0.173040
$676$	0	0
$677$	13.0993	0.503446	0.251723	−	0.967799i	$-0.419003\pi$
$677$	13.0993	0.503446	0.251723	+	0.967799i	$0.419003\pi$
$678$	0	0
$679$	6.86707	0.263534
$680$	0	0
$681$	4.88171	0.187067
$682$	0	0
$683$	−38.9960	−1.49214	−0.746070	−	0.665867i	$-0.768062\pi$
$683$	−38.9960	−1.49214	−0.746070	+	0.665867i	$0.768062\pi$
$684$	0	0
$685$	−28.1386	−1.07512
$686$	0	0
$687$	6.33729	0.241783
$688$	0	0
$689$	−57.9889	−2.20920
$690$	0	0
$691$	−30.3793	−1.15568	−0.577842	−	0.816149i	$-0.696105\pi$
$691$	−30.3793	−1.15568	−0.577842	+	0.816149i	$0.696105\pi$
$692$	0	0
$693$	8.21726	0.312148
$694$	0	0
$695$	−18.0727	−0.685537
$696$	0	0
$697$	45.1182	1.70897
$698$	0	0
$699$	1.89782	0.0717821
$700$	0	0
$701$	−23.7607	−0.897431	−0.448715	−	0.893675i	$-0.648118\pi$
$701$	−23.7607	−0.897431	−0.448715	+	0.893675i	$0.648118\pi$
$702$	0	0
$703$	−12.4388	−0.469139
$704$	0	0
$705$	2.02901	0.0764169
$706$	0	0
$707$	3.95924	0.148903
$708$	0	0
$709$	−40.0967	−1.50586	−0.752932	−	0.658098i	$-0.771361\pi$
$709$	−40.0967	−1.50586	−0.752932	+	0.658098i	$0.771361\pi$
$710$	0	0
$711$	−18.0163	−0.675666
$712$	0	0
$713$	−20.7355	−0.776550
$714$	0	0
$715$	27.1154	1.01406
$716$	0	0
$717$	−2.48274	−0.0927195
$718$	0	0
$719$	36.7711	1.37133	0.685665	−	0.727917i	$-0.259511\pi$
$719$	36.7711	1.37133	0.685665	+	0.727917i	$0.259511\pi$
$720$	0	0
$721$	5.97489	0.222516
$722$	0	0
$723$	−7.71147	−0.286793
$724$	0	0
$725$	7.31732	0.271759
$726$	0	0
$727$	4.85895	0.180208	0.0901042	−	0.995932i	$-0.471280\pi$
$727$	4.85895	0.180208	0.0901042	+	0.995932i	$0.471280\pi$
$728$	0	0
$729$	−22.7670	−0.843221
$730$	0	0
$731$	46.8030	1.73107
$732$	0	0
$733$	−6.34626	−0.234404	−0.117202	−	0.993108i	$-0.537393\pi$
$733$	−6.34626	−0.234404	−0.117202	+	0.993108i	$0.537393\pi$
$734$	0	0
$735$	0.431050	0.0158995
$736$	0	0
$737$	−33.7341	−1.24261
$738$	0	0
$739$	7.24937	0.266672	0.133336	−	0.991071i	$-0.457431\pi$
$739$	7.24937	0.266672	0.133336	+	0.991071i	$0.457431\pi$
$740$	0	0
$741$	−9.45775	−0.347439
$742$	0	0
$743$	−3.45805	−0.126863	−0.0634317	−	0.997986i	$-0.520205\pi$
$743$	−3.45805	−0.126863	−0.0634317	+	0.997986i	$0.520205\pi$
$744$	0	0
$745$	15.3385	0.561959
$746$	0	0
$747$	5.52423	0.202121
$748$	0	0
$749$	−20.4155	−0.745965
$750$	0	0
$751$	−47.4680	−1.73213	−0.866066	−	0.499929i	$-0.833359\pi$
$751$	−47.4680	−1.73213	−0.866066	+	0.499929i	$0.833359\pi$
$752$	0	0
$753$	−7.06986	−0.257640
$754$	0	0
$755$	28.5892	1.04047
$756$	0	0
$757$	−43.8444	−1.59355	−0.796776	−	0.604275i	$-0.793463\pi$
$757$	−43.8444	−1.59355	−0.796776	+	0.604275i	$0.793463\pi$
$758$	0	0
$759$	−2.35023	−0.0853081
$760$	0	0
$761$	−24.8638	−0.901313	−0.450657	−	0.892697i	$-0.648810\pi$
$761$	−24.8638	−0.901313	−0.450657	+	0.892697i	$0.648810\pi$
$762$	0	0
$763$	13.4986	0.488683
$764$	0	0
$765$	−26.3150	−0.951420
$766$	0	0
$767$	−17.7661	−0.641496
$768$	0	0
$769$	21.4855	0.774786	0.387393	−	0.921915i	$-0.373376\pi$
$769$	21.4855	0.774786	0.387393	+	0.921915i	$0.373376\pi$
$770$	0	0
$771$	3.09318	0.111398
$772$	0	0
$773$	29.8108	1.07222	0.536110	−	0.844148i	$-0.319893\pi$
$773$	29.8108	1.07222	0.536110	+	0.844148i	$0.319893\pi$
$774$	0	0
$775$	18.8565	0.677345
$776$	0	0
$777$	0.667250	0.0239374
$778$	0	0
$779$	40.2278	1.44131
$780$	0	0
$781$	−10.6912	−0.382563
$782$	0	0
$783$	−4.57254	−0.163409
$784$	0	0
$785$	4.29752	0.153385
$786$	0	0
$787$	−37.3359	−1.33088	−0.665441	−	0.746451i	$-0.731756\pi$
$787$	−37.3359	−1.33088	−0.665441	+	0.746451i	$0.731756\pi$
$788$	0	0
$789$	4.32737	0.154059
$790$	0	0
$791$	−11.4204	−0.406064
$792$	0	0
$793$	62.9055	2.23384
$794$	0	0
$795$	3.94966	0.140080
$796$	0	0
$797$	16.5159	0.585022	0.292511	−	0.956262i	$-0.405509\pi$
$797$	16.5159	0.585022	0.292511	+	0.956262i	$0.405509\pi$
$798$	0	0
$799$	−27.8655	−0.985809
$800$	0	0
$801$	2.91984	0.103167
$802$	0	0
$803$	−31.9148	−1.12625
$804$	0	0
$805$	4.49040	0.158266
$806$	0	0
$807$	−4.14651	−0.145964
$808$	0	0
$809$	−18.0395	−0.634237	−0.317118	−	0.948386i	$-0.602715\pi$
$809$	−18.0395	−0.634237	−0.317118	+	0.948386i	$0.602715\pi$
$810$	0	0
$811$	−42.1972	−1.48174	−0.740872	−	0.671646i	$-0.765588\pi$
$811$	−42.1972	−1.48174	−0.740872	+	0.671646i	$0.765588\pi$
$812$	0	0
$813$	3.41290	0.119695
$814$	0	0
$815$	17.9168	0.627600
$816$	0	0
$817$	41.7300	1.45995
$818$	0	0
$819$	−18.4787	−0.645696
$820$	0	0
$821$	25.1379	0.877319	0.438660	−	0.898653i	$-0.355453\pi$
$821$	25.1379	0.877319	0.438660	+	0.898653i	$0.355453\pi$
$822$	0	0
$823$	33.6549	1.17314	0.586569	−	0.809900i	$-0.300478\pi$
$823$	33.6549	1.17314	0.586569	+	0.809900i	$0.300478\pi$
$824$	0	0
$825$	2.13726	0.0744099
$826$	0	0
$827$	8.47416	0.294675	0.147338	−	0.989086i	$-0.452930\pi$
$827$	8.47416	0.294675	0.147338	+	0.989086i	$0.452930\pi$
$828$	0	0
$829$	−22.3732	−0.777055	−0.388527	−	0.921437i	$-0.627016\pi$
$829$	−22.3732	−0.777055	−0.388527	+	0.921437i	$0.627016\pi$
$830$	0	0
$831$	6.51057	0.225849
$832$	0	0
$833$	−5.91984	−0.205110
$834$	0	0
$835$	33.5821	1.16215
$836$	0	0
$837$	−11.7833	−0.407290
$838$	0	0
$839$	9.14804	0.315826	0.157913	−	0.987453i	$-0.449524\pi$
$839$	9.14804	0.315826	0.157913	+	0.987453i	$0.449524\pi$
$840$	0	0
$841$	−21.5576	−0.743366
$842$	0	0
$843$	2.55175	0.0878869
$844$	0	0
$845$	−41.1845	−1.41679
$846$	0	0
$847$	3.07977	0.105822
$848$	0	0
$849$	−3.89725	−0.133753
$850$	0	0
$851$	6.95098	0.238276
$852$	0	0
$853$	−33.6390	−1.15178	−0.575889	−	0.817528i	$-0.695344\pi$
$853$	−33.6390	−1.15178	−0.575889	+	0.817528i	$0.695344\pi$
$854$	0	0
$855$	−23.4627	−0.802406
$856$	0	0
$857$	−25.4442	−0.869158	−0.434579	−	0.900634i	$-0.643103\pi$
$857$	−25.4442	−0.869158	−0.434579	+	0.900634i	$0.643103\pi$
$858$	0	0
$859$	8.14294	0.277834	0.138917	−	0.990304i	$-0.455638\pi$
$859$	8.14294	0.277834	0.138917	+	0.990304i	$0.455638\pi$
$860$	0	0
$861$	−2.15792	−0.0735416
$862$	0	0
$863$	3.67074	0.124954	0.0624768	−	0.998046i	$-0.480100\pi$
$863$	3.67074	0.124954	0.0624768	+	0.998046i	$0.480100\pi$
$864$	0	0
$865$	27.3818	0.931011
$866$	0	0
$867$	−5.10900	−0.173511
$868$	0	0
$869$	17.3651	0.589071
$870$	0	0
$871$	75.8599	2.57041
$872$	0	0
$873$	20.0507	0.678613
$874$	0	0
$875$	−11.6956	−0.395383
$876$	0	0
$877$	4.24480	0.143337	0.0716683	−	0.997429i	$-0.477168\pi$
$877$	4.24480	0.143337	0.0716683	+	0.997429i	$0.477168\pi$
$878$	0	0
$879$	5.77737	0.194866
$880$	0	0
$881$	8.10456	0.273049	0.136525	−	0.990637i	$-0.456407\pi$
$881$	8.10456	0.273049	0.136525	+	0.990637i	$0.456407\pi$
$882$	0	0
$883$	−16.6835	−0.561444	−0.280722	−	0.959789i	$-0.590574\pi$
$883$	−16.6835	−0.561444	−0.280722	+	0.959789i	$0.590574\pi$
$884$	0	0
$885$	1.21006	0.0406757
$886$	0	0
$887$	48.8830	1.64133	0.820666	−	0.571408i	$-0.193603\pi$
$887$	48.8830	1.64133	0.820666	+	0.571408i	$0.193603\pi$
$888$	0	0
$889$	−7.02928	−0.235754
$890$	0	0
$891$	23.3162	0.781123
$892$	0	0
$893$	−24.8451	−0.831408
$894$	0	0
$895$	3.36621	0.112520
$896$	0	0
$897$	5.28511	0.176465
$898$	0	0
$899$	19.1788	0.639648
$900$	0	0
$901$	−54.2428	−1.80709
$902$	0	0
$903$	−2.23850	−0.0744926
$904$	0	0
$905$	−19.9919	−0.664554
$906$	0	0
$907$	−9.40259	−0.312208	−0.156104	−	0.987741i	$-0.549893\pi$
$907$	−9.40259	−0.312208	−0.156104	+	0.987741i	$0.549893\pi$
$908$	0	0
$909$	11.5603	0.383432
$910$	0	0
$911$	−21.6058	−0.715832	−0.357916	−	0.933754i	$-0.616513\pi$
$911$	−21.6058	−0.715832	−0.357916	+	0.933754i	$0.616513\pi$
$912$	0	0
$913$	−5.32455	−0.176217
$914$	0	0
$915$	−4.28454	−0.141642
$916$	0	0
$917$	−6.11905	−0.202069
$918$	0	0
$919$	25.7910	0.850765	0.425383	−	0.905014i	$-0.360140\pi$
$919$	25.7910	0.850765	0.425383	+	0.905014i	$0.360140\pi$
$920$	0	0
$921$	−4.49268	−0.148039
$922$	0	0
$923$	24.0420	0.791353
$924$	0	0
$925$	−6.32109	−0.207836
$926$	0	0
$927$	17.4457	0.572992
$928$	0	0
$929$	5.68373	0.186477	0.0932386	−	0.995644i	$-0.470278\pi$
$929$	5.68373	0.186477	0.0932386	+	0.995644i	$0.470278\pi$
$930$	0	0
$931$	−5.27817	−0.172985
$932$	0	0
$933$	−5.81245	−0.190291
$934$	0	0
$935$	25.3638	0.829484
$936$	0	0
$937$	−1.18573	−0.0387361	−0.0193680	−	0.999812i	$-0.506165\pi$
$937$	−1.18573	−0.0387361	−0.0193680	+	0.999812i	$0.506165\pi$
$938$	0	0
$939$	−0.756708	−0.0246942
$940$	0	0
$941$	−56.4467	−1.84011	−0.920055	−	0.391790i	$-0.871856\pi$
$941$	−56.4467	−1.84011	−0.920055	+	0.391790i	$0.871856\pi$
$942$	0	0
$943$	−22.4798	−0.732042
$944$	0	0
$945$	2.55174	0.0830082
$946$	0	0
$947$	−59.4281	−1.93115	−0.965577	−	0.260116i	$-0.916239\pi$
$947$	−59.4281	−1.93115	−0.965577	+	0.260116i	$0.916239\pi$
$948$	0	0
$949$	71.7688	2.32971
$950$	0	0
$951$	7.39575	0.239824
$952$	0	0
$953$	−2.39444	−0.0775635	−0.0387817	−	0.999248i	$-0.512348\pi$
$953$	−2.39444	−0.0775635	−0.0387817	+	0.999248i	$0.512348\pi$
$954$	0	0
$955$	−4.27167	−0.138228
$956$	0	0
$957$	2.17379	0.0702686
$958$	0	0
$959$	−18.4828	−0.596839
$960$	0	0
$961$	18.4230	0.594291
$962$	0	0
$963$	−59.6098	−1.92090
$964$	0	0
$965$	−7.65379	−0.246384
$966$	0	0
$967$	−15.4892	−0.498099	−0.249049	−	0.968491i	$-0.580118\pi$
$967$	−15.4892	−0.498099	−0.249049	+	0.968491i	$0.580118\pi$
$968$	0	0
$969$	−8.84679	−0.284200
$970$	0	0
$971$	−19.1625	−0.614955	−0.307477	−	0.951555i	$-0.599485\pi$
$971$	−19.1625	−0.614955	−0.307477	+	0.951555i	$0.599485\pi$
$972$	0	0
$973$	−11.8710	−0.380568
$974$	0	0
$975$	−4.80619	−0.153921
$976$	0	0
$977$	−14.8874	−0.476290	−0.238145	−	0.971230i	$-0.576539\pi$
$977$	−14.8874	−0.476290	−0.238145	+	0.971230i	$0.576539\pi$
$978$	0	0
$979$	−2.81429	−0.0899451
$980$	0	0
$981$	39.4137	1.25838
$982$	0	0
$983$	−41.8849	−1.33592	−0.667961	−	0.744196i	$-0.732833\pi$
$983$	−41.8849	−1.33592	−0.667961	+	0.744196i	$0.732833\pi$
$984$	0	0
$985$	17.2555	0.549805
$986$	0	0
$987$	1.33275	0.0424219
$988$	0	0
$989$	−23.3192	−0.741509
$990$	0	0
$991$	−25.4020	−0.806922	−0.403461	−	0.914997i	$-0.632193\pi$
$991$	−25.4020	−0.806922	−0.403461	+	0.914997i	$0.632193\pi$
$992$	0	0
$993$	6.86476	0.217847
$994$	0	0
$995$	−22.9421	−0.727312
$996$	0	0
$997$	50.6555	1.60428	0.802138	−	0.597139i	$-0.203696\pi$
$997$	50.6555	1.60428	0.802138	+	0.597139i	$0.203696\pi$
$998$	0	0
$999$	3.95001	0.124973

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9968.2.a.z.1.2		5
4.3	odd	2		1246.2.a.n.1.4	✓	5
28.27	even	2		8722.2.a.x.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1246.2.a.n.1.4	✓	5	4.3	odd	2
8722.2.a.x.1.2		5	28.27	even	2
9968.2.a.z.1.2		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:	\( N \)	\(=\)	\( 9968 = 2^{4} \cdot 7 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9968.a (trivial)

\(f(q)\)	\(=\)	\(q-0.283134 q^{3} -1.52242 q^{5} -1.00000 q^{7} -2.91984 q^{9} +O(q^{10})\)	\(q-0.283134 q^{3} -1.52242 q^{5} -1.00000 q^{7} -2.91984 q^{9} +2.81429 q^{11} -6.32866 q^{13} +0.431050 q^{15} -5.91984 q^{17} -5.27817 q^{19} +0.283134 q^{21} +2.94951 q^{23} -2.68223 q^{25} +1.67611 q^{27} -2.72807 q^{29} -7.03015 q^{31} -0.796822 q^{33} +1.52242 q^{35} +2.35665 q^{37} +1.79186 q^{39} -7.62153 q^{41} -7.90614 q^{43} +4.44522 q^{45} +4.70713 q^{47} +1.00000 q^{49} +1.67611 q^{51} +9.16290 q^{53} -4.28454 q^{55} +1.49443 q^{57} +2.80724 q^{59} -9.93978 q^{61} +2.91984 q^{63} +9.63490 q^{65} -11.9867 q^{67} -0.835107 q^{69} -3.79891 q^{71} -11.3403 q^{73} +0.759431 q^{75} -2.81429 q^{77} +6.17033 q^{79} +8.28494 q^{81} -1.89197 q^{83} +9.01249 q^{85} +0.772411 q^{87} -1.00000 q^{89} +6.32866 q^{91} +1.99048 q^{93} +8.03561 q^{95} -6.86707 q^{97} -8.21726 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 6 q^{3} - 10 q^{5} - 5 q^{7} + 7 q^{9}+O(q^{10}) \) 5 * q + 6 * q^3 - 10 * q^5 - 5 * q^7 + 7 * q^9	\( 5 q + 6 q^{3} - 10 q^{5} - 5 q^{7} + 7 q^{9} + 8 q^{11} - 8 q^{13} - 6 q^{15} - 8 q^{17} + 10 q^{19} - 6 q^{21} + 2 q^{23} + 17 q^{25} + 24 q^{27} - 10 q^{29} + 10 q^{31} + 10 q^{35} - 4 q^{37} - 24 q^{39} - 28 q^{41} + 10 q^{43} - 12 q^{45} + 10 q^{47} + 5 q^{49} + 24 q^{51} + 4 q^{53} - 4 q^{55} + 2 q^{57} + 10 q^{59} - 16 q^{61} - 7 q^{63} - 26 q^{65} - 2 q^{69} + 16 q^{71} + 10 q^{73} + 10 q^{75} - 8 q^{77} + 8 q^{79} + 41 q^{81} + 14 q^{83} + 18 q^{85} - 22 q^{87} - 5 q^{89} + 8 q^{91} + 22 q^{93} - 34 q^{95} + 6 q^{97} + 8 q^{99}+O(q^{100}) \) 5 * q + 6 * q^3 - 10 * q^5 - 5 * q^7 + 7 * q^9 + 8 * q^11 - 8 * q^13 - 6 * q^15 - 8 * q^17 + 10 * q^19 - 6 * q^21 + 2 * q^23 + 17 * q^25 + 24 * q^27 - 10 * q^29 + 10 * q^31 + 10 * q^35 - 4 * q^37 - 24 * q^39 - 28 * q^41 + 10 * q^43 - 12 * q^45 + 10 * q^47 + 5 * q^49 + 24 * q^51 + 4 * q^53 - 4 * q^55 + 2 * q^57 + 10 * q^59 - 16 * q^61 - 7 * q^63 - 26 * q^65 - 2 * q^69 + 16 * q^71 + 10 * q^73 + 10 * q^75 - 8 * q^77 + 8 * q^79 + 41 * q^81 + 14 * q^83 + 18 * q^85 - 22 * q^87 - 5 * q^89 + 8 * q^91 + 22 * q^93 - 34 * q^95 + 6 * q^97 + 8 * q^99

Introduction

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