LMFDB - Embedded newform 8016.2.a.w.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8016,2,Mod(1,8016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8016, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8016 = 2^{4} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8016.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.0080822603$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 7x^{5} + 11x^{4} + 12x^{3} - 14x^{2} - 6x + 3 $ x^7 - 2x^6 - 7x^5 + 11x^4 + 12x^3 - 14x^2 - 6x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 4008)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.47270$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -3.01562 q^{5} +1.84677 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.01562 q^{5} +1.84677 q^{7} +1.00000 q^{9} +0.0702134 q^{11} +1.68809 q^{13} -3.01562 q^{15} -2.17362 q^{17} +2.90752 q^{19} +1.84677 q^{21} -1.67315 q^{23} +4.09394 q^{25} +1.00000 q^{27} -4.24297 q^{29} +4.28787 q^{31} +0.0702134 q^{33} -5.56914 q^{35} -10.8489 q^{37} +1.68809 q^{39} -7.20331 q^{41} +4.33921 q^{43} -3.01562 q^{45} -11.5944 q^{47} -3.58945 q^{49} -2.17362 q^{51} +3.88527 q^{53} -0.211737 q^{55} +2.90752 q^{57} -5.40850 q^{59} +2.01512 q^{61} +1.84677 q^{63} -5.09062 q^{65} +6.98849 q^{67} -1.67315 q^{69} -2.15920 q^{71} -8.46808 q^{73} +4.09394 q^{75} +0.129668 q^{77} +10.8442 q^{79} +1.00000 q^{81} +9.35160 q^{83} +6.55479 q^{85} -4.24297 q^{87} +16.2378 q^{89} +3.11750 q^{91} +4.28787 q^{93} -8.76795 q^{95} +13.2480 q^{97} +0.0702134 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{3} - 3 q^{5} - 8 q^{7} + 7 q^{9}+O(q^{10}) $ 7 * q + 7 * q^3 - 3 * q^5 - 8 * q^7 + 7 * q^9	$ 7 q + 7 q^{3} - 3 q^{5} - 8 q^{7} + 7 q^{9} - q^{11} - 2 q^{13} - 3 q^{15} + 11 q^{17} - 2 q^{19} - 8 q^{21} - 17 q^{23} + 4 q^{25} + 7 q^{27} - 7 q^{29} - 10 q^{31} - q^{33} - 10 q^{35} - 21 q^{37} - 2 q^{39} + 8 q^{41} + 12 q^{43} - 3 q^{45} - 25 q^{47} - 7 q^{49} + 11 q^{51} - 7 q^{53} - 15 q^{55} - 2 q^{57} - 3 q^{59} - 14 q^{61} - 8 q^{63} + 4 q^{65} - 4 q^{67} - 17 q^{69} - 27 q^{71} - 12 q^{73} + 4 q^{75} + 16 q^{77} - 8 q^{79} + 7 q^{81} - 15 q^{83} - 3 q^{85} - 7 q^{87} + 14 q^{89} + 3 q^{91} - 10 q^{93} - 37 q^{95} + 3 q^{97} - q^{99}+O(q^{100}) $ 7 * q + 7 * q^3 - 3 * q^5 - 8 * q^7 + 7 * q^9 - q^11 - 2 * q^13 - 3 * q^15 + 11 * q^17 - 2 * q^19 - 8 * q^21 - 17 * q^23 + 4 * q^25 + 7 * q^27 - 7 * q^29 - 10 * q^31 - q^33 - 10 * q^35 - 21 * q^37 - 2 * q^39 + 8 * q^41 + 12 * q^43 - 3 * q^45 - 25 * q^47 - 7 * q^49 + 11 * q^51 - 7 * q^53 - 15 * q^55 - 2 * q^57 - 3 * q^59 - 14 * q^61 - 8 * q^63 + 4 * q^65 - 4 * q^67 - 17 * q^69 - 27 * q^71 - 12 * q^73 + 4 * q^75 + 16 * q^77 - 8 * q^79 + 7 * q^81 - 15 * q^83 - 3 * q^85 - 7 * q^87 + 14 * q^89 + 3 * q^91 - 10 * q^93 - 37 * q^95 + 3 * q^97 - 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
$5$	−3.01562	−1.34862	−0.674312	−	0.738446i	$-0.735560\pi$
$5$	−3.01562	−1.34862	−0.674312	+	0.738446i	$0.735560\pi$
$6$	0	0
$7$	1.84677	0.698012	0.349006	−	0.937120i	$-0.386519\pi$
$7$	1.84677	0.698012	0.349006	+	0.937120i	$0.386519\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.0702134	0.0211701	0.0105851	−	0.999944i	$-0.496631\pi$
$11$	0.0702134	0.0211701	0.0105851	+	0.999944i	$0.496631\pi$
$12$	0	0
$13$	1.68809	0.468191	0.234096	−	0.972214i	$-0.424787\pi$
$13$	1.68809	0.468191	0.234096	+	0.972214i	$0.424787\pi$
$14$	0	0
$15$	−3.01562	−0.778629
$16$	0	0
$17$	−2.17362	−0.527179	−0.263590	−	0.964635i	$-0.584907\pi$
$17$	−2.17362	−0.527179	−0.263590	+	0.964635i	$0.584907\pi$
$18$	0	0
$19$	2.90752	0.667030	0.333515	−	0.942745i	$-0.391765\pi$
$19$	2.90752	0.667030	0.333515	+	0.942745i	$0.391765\pi$
$20$	0	0
$21$	1.84677	0.402998
$22$	0	0
$23$	−1.67315	−0.348876	−0.174438	−	0.984668i	$-0.555811\pi$
$23$	−1.67315	−0.348876	−0.174438	+	0.984668i	$0.555811\pi$
$24$	0	0
$25$	4.09394	0.818788
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−4.24297	−0.787899	−0.393950	−	0.919132i	$-0.628892\pi$
$29$	−4.24297	−0.787899	−0.393950	+	0.919132i	$0.628892\pi$
$30$	0	0
$31$	4.28787	0.770124	0.385062	−	0.922891i	$-0.374180\pi$
$31$	4.28787	0.770124	0.385062	+	0.922891i	$0.374180\pi$
$32$	0	0
$33$	0.0702134	0.0122226
$34$	0	0
$35$	−5.56914	−0.941356
$36$	0	0
$37$	−10.8489	−1.78355	−0.891776	−	0.452477i	$-0.850540\pi$
$37$	−10.8489	−1.78355	−0.891776	+	0.452477i	$0.850540\pi$
$38$	0	0
$39$	1.68809	0.270310
$40$	0	0
$41$	−7.20331	−1.12497	−0.562484	−	0.826808i	$-0.690154\pi$
$41$	−7.20331	−1.12497	−0.562484	+	0.826808i	$0.690154\pi$
$42$	0	0
$43$	4.33921	0.661724	0.330862	−	0.943679i	$-0.392661\pi$
$43$	4.33921	0.661724	0.330862	+	0.943679i	$0.392661\pi$
$44$	0	0
$45$	−3.01562	−0.449541
$46$	0	0
$47$	−11.5944	−1.69122	−0.845609	−	0.533803i	$-0.820762\pi$
$47$	−11.5944	−1.69122	−0.845609	+	0.533803i	$0.820762\pi$
$48$	0	0
$49$	−3.58945	−0.512779
$50$	0	0
$51$	−2.17362	−0.304367
$52$	0	0
$53$	3.88527	0.533683	0.266841	−	0.963740i	$-0.414020\pi$
$53$	3.88527	0.533683	0.266841	+	0.963740i	$0.414020\pi$
$54$	0	0
$55$	−0.211737	−0.0285505
$56$	0	0
$57$	2.90752	0.385110
$58$	0	0
$59$	−5.40850	−0.704127	−0.352063	−	0.935976i	$-0.614520\pi$
$59$	−5.40850	−0.704127	−0.352063	+	0.935976i	$0.614520\pi$
$60$	0	0
$61$	2.01512	0.258010	0.129005	−	0.991644i	$-0.458822\pi$
$61$	2.01512	0.258010	0.129005	+	0.991644i	$0.458822\pi$
$62$	0	0
$63$	1.84677	0.232671
$64$	0	0
$65$	−5.09062	−0.631414
$66$	0	0
$67$	6.98849	0.853780	0.426890	−	0.904304i	$-0.359609\pi$
$67$	6.98849	0.853780	0.426890	+	0.904304i	$0.359609\pi$
$68$	0	0
$69$	−1.67315	−0.201424
$70$	0	0
$71$	−2.15920	−0.256250	−0.128125	−	0.991758i	$-0.540896\pi$
$71$	−2.15920	−0.256250	−0.128125	+	0.991758i	$0.540896\pi$
$72$	0	0
$73$	−8.46808	−0.991114	−0.495557	−	0.868575i	$-0.665036\pi$
$73$	−8.46808	−0.991114	−0.495557	+	0.868575i	$0.665036\pi$
$74$	0	0
$75$	4.09394	0.472727
$76$	0	0
$77$	0.129668	0.0147770
$78$	0	0
$79$	10.8442	1.22007	0.610035	−	0.792374i	$-0.291155\pi$
$79$	10.8442	1.22007	0.610035	+	0.792374i	$0.291155\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	9.35160	1.02647	0.513236	−	0.858248i	$-0.328447\pi$
$83$	9.35160	1.02647	0.513236	+	0.858248i	$0.328447\pi$
$84$	0	0
$85$	6.55479	0.710967
$86$	0	0
$87$	−4.24297	−0.454894
$88$	0	0
$89$	16.2378	1.72120	0.860601	−	0.509279i	$-0.170088\pi$
$89$	16.2378	1.72120	0.860601	+	0.509279i	$0.170088\pi$
$90$	0	0
$91$	3.11750	0.326803
$92$	0	0
$93$	4.28787	0.444631
$94$	0	0
$95$	−8.76795	−0.899573
$96$	0	0
$97$	13.2480	1.34513	0.672563	−	0.740040i	$-0.265193\pi$
$97$	13.2480	1.34513	0.672563	+	0.740040i	$0.265193\pi$
$98$	0	0
$99$	0.0702134	0.00705671
$100$	0	0
$101$	−5.67392	−0.564577	−0.282288	−	0.959330i	$-0.591093\pi$
$101$	−5.67392	−0.564577	−0.282288	+	0.959330i	$0.591093\pi$
$102$	0	0
$103$	−17.5307	−1.72735	−0.863677	−	0.504046i	$-0.831844\pi$
$103$	−17.5307	−1.72735	−0.863677	+	0.504046i	$0.831844\pi$
$104$	0	0
$105$	−5.56914	−0.543492
$106$	0	0
$107$	−2.56293	−0.247768	−0.123884	−	0.992297i	$-0.539535\pi$
$107$	−2.56293	−0.247768	−0.123884	+	0.992297i	$0.539535\pi$
$108$	0	0
$109$	18.3817	1.76065	0.880325	−	0.474370i	$-0.157324\pi$
$109$	18.3817	1.76065	0.880325	+	0.474370i	$0.157324\pi$
$110$	0	0
$111$	−10.8489	−1.02973
$112$	0	0
$113$	8.64947	0.813674	0.406837	−	0.913501i	$-0.366632\pi$
$113$	8.64947	0.813674	0.406837	+	0.913501i	$0.366632\pi$
$114$	0	0
$115$	5.04558	0.470503
$116$	0	0
$117$	1.68809	0.156064
$118$	0	0
$119$	−4.01416	−0.367978
$120$	0	0
$121$	−10.9951	−0.999552
$122$	0	0
$123$	−7.20331	−0.649500
$124$	0	0
$125$	2.73233	0.244387
$126$	0	0
$127$	0.114006	0.0101164	0.00505819	−	0.999987i	$-0.498390\pi$
$127$	0.114006	0.0101164	0.00505819	+	0.999987i	$0.498390\pi$
$128$	0	0
$129$	4.33921	0.382046
$130$	0	0
$131$	−12.5617	−1.09752	−0.548759	−	0.835981i	$-0.684899\pi$
$131$	−12.5617	−1.09752	−0.548759	+	0.835981i	$0.684899\pi$
$132$	0	0
$133$	5.36950	0.465595
$134$	0	0
$135$	−3.01562	−0.259543
$136$	0	0
$137$	−6.59365	−0.563333	−0.281667	−	0.959512i	$-0.590887\pi$
$137$	−6.59365	−0.563333	−0.281667	+	0.959512i	$0.590887\pi$
$138$	0	0
$139$	−5.80181	−0.492103	−0.246051	−	0.969257i	$-0.579133\pi$
$139$	−5.80181	−0.492103	−0.246051	+	0.969257i	$0.579133\pi$
$140$	0	0
$141$	−11.5944	−0.976425
$142$	0	0
$143$	0.118526	0.00991167
$144$	0	0
$145$	12.7952	1.06258
$146$	0	0
$147$	−3.58945	−0.296053
$148$	0	0
$149$	4.34532	0.355983	0.177991	−	0.984032i	$-0.443040\pi$
$149$	4.34532	0.355983	0.177991	+	0.984032i	$0.443040\pi$
$150$	0	0
$151$	−17.8907	−1.45592	−0.727961	−	0.685619i	$-0.759532\pi$
$151$	−17.8907	−1.45592	−0.727961	+	0.685619i	$0.759532\pi$
$152$	0	0
$153$	−2.17362	−0.175726
$154$	0	0
$155$	−12.9306	−1.03861
$156$	0	0
$157$	−17.9620	−1.43352	−0.716760	−	0.697320i	$-0.754376\pi$
$157$	−17.9620	−1.43352	−0.716760	+	0.697320i	$0.754376\pi$
$158$	0	0
$159$	3.88527	0.308122
$160$	0	0
$161$	−3.08992	−0.243520
$162$	0	0
$163$	7.95764	0.623291	0.311645	−	0.950198i	$-0.399120\pi$
$163$	7.95764	0.623291	0.311645	+	0.950198i	$0.399120\pi$
$164$	0	0
$165$	−0.211737	−0.0164837
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−10.1504	−0.780797
$170$	0	0
$171$	2.90752	0.222343
$172$	0	0
$173$	−13.0823	−0.994632	−0.497316	−	0.867570i	$-0.665681\pi$
$173$	−13.0823	−0.994632	−0.497316	+	0.867570i	$0.665681\pi$
$174$	0	0
$175$	7.56055	0.571524
$176$	0	0
$177$	−5.40850	−0.406528
$178$	0	0
$179$	−10.4300	−0.779575	−0.389788	−	0.920905i	$-0.627452\pi$
$179$	−10.4300	−0.779575	−0.389788	+	0.920905i	$0.627452\pi$
$180$	0	0
$181$	−4.31004	−0.320362	−0.160181	−	0.987088i	$-0.551208\pi$
$181$	−4.31004	−0.320362	−0.160181	+	0.987088i	$0.551208\pi$
$182$	0	0
$183$	2.01512	0.148962
$184$	0	0
$185$	32.7162	2.40534
$186$	0	0
$187$	−0.152617	−0.0111605
$188$	0	0
$189$	1.84677	0.134333
$190$	0	0
$191$	−18.7287	−1.35516	−0.677581	−	0.735448i	$-0.736972\pi$
$191$	−18.7287	−1.35516	−0.677581	+	0.735448i	$0.736972\pi$
$192$	0	0
$193$	15.3962	1.10824	0.554121	−	0.832436i	$-0.313054\pi$
$193$	15.3962	1.10824	0.554121	+	0.832436i	$0.313054\pi$
$194$	0	0
$195$	−5.09062	−0.364547
$196$	0	0
$197$	0.0789986	0.00562842	0.00281421	−	0.999996i	$-0.499104\pi$
$197$	0.0789986	0.00562842	0.00281421	+	0.999996i	$0.499104\pi$
$198$	0	0
$199$	−11.4542	−0.811967	−0.405983	−	0.913880i	$-0.633071\pi$
$199$	−11.4542	−0.811967	−0.405983	+	0.913880i	$0.633071\pi$
$200$	0	0
$201$	6.98849	0.492930
$202$	0	0
$203$	−7.83577	−0.549963
$204$	0	0
$205$	21.7224	1.51716
$206$	0	0
$207$	−1.67315	−0.116292
$208$	0	0
$209$	0.204146	0.0141211
$210$	0	0
$211$	−15.3500	−1.05674	−0.528370	−	0.849014i	$-0.677197\pi$
$211$	−15.3500	−1.05674	−0.528370	+	0.849014i	$0.677197\pi$
$212$	0	0
$213$	−2.15920	−0.147946
$214$	0	0
$215$	−13.0854	−0.892417
$216$	0	0
$217$	7.91869	0.537556
$218$	0	0
$219$	−8.46808	−0.572220
$220$	0	0
$221$	−3.66926	−0.246821
$222$	0	0
$223$	−24.6893	−1.65332	−0.826658	−	0.562705i	$-0.809761\pi$
$223$	−24.6893	−1.65332	−0.826658	+	0.562705i	$0.809761\pi$
$224$	0	0
$225$	4.09394	0.272929
$226$	0	0
$227$	−24.5728	−1.63096	−0.815478	−	0.578788i	$-0.803526\pi$
$227$	−24.5728	−1.63096	−0.815478	+	0.578788i	$0.803526\pi$
$228$	0	0
$229$	−8.98726	−0.593895	−0.296947	−	0.954894i	$-0.595969\pi$
$229$	−8.98726	−0.593895	−0.296947	+	0.954894i	$0.595969\pi$
$230$	0	0
$231$	0.129668	0.00853151
$232$	0	0
$233$	−22.6629	−1.48470	−0.742348	−	0.670014i	$-0.766288\pi$
$233$	−22.6629	−1.48470	−0.742348	+	0.670014i	$0.766288\pi$
$234$	0	0
$235$	34.9643	2.28082
$236$	0	0
$237$	10.8442	0.704408
$238$	0	0
$239$	−7.63603	−0.493934	−0.246967	−	0.969024i	$-0.579434\pi$
$239$	−7.63603	−0.493934	−0.246967	+	0.969024i	$0.579434\pi$
$240$	0	0
$241$	−16.3487	−1.05311	−0.526555	−	0.850141i	$-0.676516\pi$
$241$	−16.3487	−1.05311	−0.526555	+	0.850141i	$0.676516\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	10.8244	0.691546
$246$	0	0
$247$	4.90814	0.312298
$248$	0	0
$249$	9.35160	0.592633
$250$	0	0
$251$	25.3237	1.59842	0.799209	−	0.601053i	$-0.205252\pi$
$251$	25.3237	1.59842	0.799209	+	0.601053i	$0.205252\pi$
$252$	0	0
$253$	−0.117478	−0.00738575
$254$	0	0
$255$	6.55479	0.410477
$256$	0	0
$257$	15.8241	0.987082	0.493541	−	0.869723i	$-0.335702\pi$
$257$	15.8241	0.987082	0.493541	+	0.869723i	$0.335702\pi$
$258$	0	0
$259$	−20.0354	−1.24494
$260$	0	0
$261$	−4.24297	−0.262633
$262$	0	0
$263$	−19.1324	−1.17975	−0.589877	−	0.807493i	$-0.700824\pi$
$263$	−19.1324	−1.17975	−0.589877	+	0.807493i	$0.700824\pi$
$264$	0	0
$265$	−11.7165	−0.719738
$266$	0	0
$267$	16.2378	0.993737
$268$	0	0
$269$	7.78513	0.474668	0.237334	−	0.971428i	$-0.423726\pi$
$269$	7.78513	0.474668	0.237334	+	0.971428i	$0.423726\pi$
$270$	0	0
$271$	11.7908	0.716237	0.358119	−	0.933676i	$-0.383418\pi$
$271$	11.7908	0.716237	0.358119	+	0.933676i	$0.383418\pi$
$272$	0	0
$273$	3.11750	0.188680
$274$	0	0
$275$	0.287449	0.0173338
$276$	0	0
$277$	2.54189	0.152728	0.0763638	−	0.997080i	$-0.475669\pi$
$277$	2.54189	0.152728	0.0763638	+	0.997080i	$0.475669\pi$
$278$	0	0
$279$	4.28787	0.256708
$280$	0	0
$281$	−7.37140	−0.439741	−0.219870	−	0.975529i	$-0.570563\pi$
$281$	−7.37140	−0.439741	−0.219870	+	0.975529i	$0.570563\pi$
$282$	0	0
$283$	−8.42659	−0.500909	−0.250455	−	0.968128i	$-0.580580\pi$
$283$	−8.42659	−0.500909	−0.250455	+	0.968128i	$0.580580\pi$
$284$	0	0
$285$	−8.76795	−0.519369
$286$	0	0
$287$	−13.3028	−0.785241
$288$	0	0
$289$	−12.2754	−0.722082
$290$	0	0
$291$	13.2480	0.776609
$292$	0	0
$293$	17.3212	1.01192	0.505958	−	0.862558i	$-0.331139\pi$
$293$	17.3212	1.01192	0.505958	+	0.862558i	$0.331139\pi$
$294$	0	0
$295$	16.3100	0.949603
$296$	0	0
$297$	0.0702134	0.00407419
$298$	0	0
$299$	−2.82442	−0.163341
$300$	0	0
$301$	8.01351	0.461891
$302$	0	0
$303$	−5.67392	−0.325958
$304$	0	0
$305$	−6.07684	−0.347959
$306$	0	0
$307$	−6.40036	−0.365288	−0.182644	−	0.983179i	$-0.558466\pi$
$307$	−6.40036	−0.365288	−0.182644	+	0.983179i	$0.558466\pi$
$308$	0	0
$309$	−17.5307	−0.997288
$310$	0	0
$311$	−17.9862	−1.01991	−0.509953	−	0.860202i	$-0.670337\pi$
$311$	−17.9862	−1.01991	−0.509953	+	0.860202i	$0.670337\pi$
$312$	0	0
$313$	−21.4712	−1.21363	−0.606813	−	0.794844i	$-0.707552\pi$
$313$	−21.4712	−1.21363	−0.606813	+	0.794844i	$0.707552\pi$
$314$	0	0
$315$	−5.56914	−0.313785
$316$	0	0
$317$	−22.8128	−1.28130	−0.640648	−	0.767834i	$-0.721334\pi$
$317$	−22.8128	−1.28130	−0.640648	+	0.767834i	$0.721334\pi$
$318$	0	0
$319$	−0.297913	−0.0166799
$320$	0	0
$321$	−2.56293	−0.143049
$322$	0	0
$323$	−6.31982	−0.351644
$324$	0	0
$325$	6.91093	0.383349
$326$	0	0
$327$	18.3817	1.01651
$328$	0	0
$329$	−21.4122	−1.18049
$330$	0	0
$331$	22.1607	1.21806	0.609031	−	0.793147i	$-0.291559\pi$
$331$	22.1607	1.21806	0.609031	+	0.793147i	$0.291559\pi$
$332$	0	0
$333$	−10.8489	−0.594517
$334$	0	0
$335$	−21.0746	−1.15143
$336$	0	0
$337$	32.6497	1.77854	0.889271	−	0.457380i	$-0.151212\pi$
$337$	32.6497	1.77854	0.889271	+	0.457380i	$0.151212\pi$
$338$	0	0
$339$	8.64947	0.469775
$340$	0	0
$341$	0.301066	0.0163036
$342$	0	0
$343$	−19.5562	−1.05594
$344$	0	0
$345$	5.04558	0.271645
$346$	0	0
$347$	21.5493	1.15683	0.578414	−	0.815744i	$-0.303672\pi$
$347$	21.5493	1.15683	0.578414	+	0.815744i	$0.303672\pi$
$348$	0	0
$349$	−1.96612	−0.105244	−0.0526219	−	0.998615i	$-0.516758\pi$
$349$	−1.96612	−0.105244	−0.0526219	+	0.998615i	$0.516758\pi$
$350$	0	0
$351$	1.68809	0.0901035
$352$	0	0
$353$	0.411409	0.0218971	0.0109485	−	0.999940i	$-0.496515\pi$
$353$	0.411409	0.0218971	0.0109485	+	0.999940i	$0.496515\pi$
$354$	0	0
$355$	6.51131	0.345585
$356$	0	0
$357$	−4.01416	−0.212452
$358$	0	0
$359$	−17.1536	−0.905334	−0.452667	−	0.891680i	$-0.649527\pi$
$359$	−17.1536	−0.905334	−0.452667	+	0.891680i	$0.649527\pi$
$360$	0	0
$361$	−10.5464	−0.555071
$362$	0	0
$363$	−10.9951	−0.577092
$364$	0	0
$365$	25.5365	1.33664
$366$	0	0
$367$	23.3616	1.21947	0.609733	−	0.792607i	$-0.291277\pi$
$367$	23.3616	1.21947	0.609733	+	0.792607i	$0.291277\pi$
$368$	0	0
$369$	−7.20331	−0.374989
$370$	0	0
$371$	7.17519	0.372517
$372$	0	0
$373$	10.3568	0.536257	0.268129	−	0.963383i	$-0.413595\pi$
$373$	10.3568	0.536257	0.268129	+	0.963383i	$0.413595\pi$
$374$	0	0
$375$	2.73233	0.141097
$376$	0	0
$377$	−7.16250	−0.368888
$378$	0	0
$379$	−1.75887	−0.0903471	−0.0451735	−	0.998979i	$-0.514384\pi$
$379$	−1.75887	−0.0903471	−0.0451735	+	0.998979i	$0.514384\pi$
$380$	0	0
$381$	0.114006	0.00584069
$382$	0	0
$383$	−2.23977	−0.114447	−0.0572233	−	0.998361i	$-0.518225\pi$
$383$	−2.23977	−0.114447	−0.0572233	+	0.998361i	$0.518225\pi$
$384$	0	0
$385$	−0.391028	−0.0199286
$386$	0	0
$387$	4.33921	0.220575
$388$	0	0
$389$	10.8084	0.548008	0.274004	−	0.961729i	$-0.411652\pi$
$389$	10.8084	0.548008	0.274004	+	0.961729i	$0.411652\pi$
$390$	0	0
$391$	3.63679	0.183920
$392$	0	0
$393$	−12.5617	−0.633652
$394$	0	0
$395$	−32.7020	−1.64542
$396$	0	0
$397$	−11.3168	−0.567975	−0.283987	−	0.958828i	$-0.591657\pi$
$397$	−11.3168	−0.567975	−0.283987	+	0.958828i	$0.591657\pi$
$398$	0	0
$399$	5.36950	0.268811
$400$	0	0
$401$	−15.5327	−0.775667	−0.387834	−	0.921729i	$-0.626776\pi$
$401$	−15.5327	−0.775667	−0.387834	+	0.921729i	$0.626776\pi$
$402$	0	0
$403$	7.23830	0.360565
$404$	0	0
$405$	−3.01562	−0.149847
$406$	0	0
$407$	−0.761739	−0.0377580
$408$	0	0
$409$	−16.0150	−0.791890	−0.395945	−	0.918274i	$-0.629583\pi$
$409$	−16.0150	−0.791890	−0.395945	+	0.918274i	$0.629583\pi$
$410$	0	0
$411$	−6.59365	−0.325241
$412$	0	0
$413$	−9.98824	−0.491489
$414$	0	0
$415$	−28.2008	−1.38432
$416$	0	0
$417$	−5.80181	−0.284116
$418$	0	0
$419$	4.60543	0.224990	0.112495	−	0.993652i	$-0.464116\pi$
$419$	4.60543	0.224990	0.112495	+	0.993652i	$0.464116\pi$
$420$	0	0
$421$	−23.7515	−1.15758	−0.578789	−	0.815478i	$-0.696474\pi$
$421$	−23.7515	−1.15758	−0.578789	+	0.815478i	$0.696474\pi$
$422$	0	0
$423$	−11.5944	−0.563739
$424$	0	0
$425$	−8.89865	−0.431648
$426$	0	0
$427$	3.72146	0.180094
$428$	0	0
$429$	0.118526	0.00572250
$430$	0	0
$431$	8.50119	0.409488	0.204744	−	0.978816i	$-0.434364\pi$
$431$	8.50119	0.409488	0.204744	+	0.978816i	$0.434364\pi$
$432$	0	0
$433$	27.1642	1.30543	0.652715	−	0.757604i	$-0.273630\pi$
$433$	27.1642	1.30543	0.652715	+	0.757604i	$0.273630\pi$
$434$	0	0
$435$	12.7952	0.613481
$436$	0	0
$437$	−4.86471	−0.232711
$438$	0	0
$439$	−31.9205	−1.52348	−0.761740	−	0.647882i	$-0.775655\pi$
$439$	−31.9205	−1.52348	−0.761740	+	0.647882i	$0.775655\pi$
$440$	0	0
$441$	−3.58945	−0.170926
$442$	0	0
$443$	−4.34160	−0.206276	−0.103138	−	0.994667i	$-0.532888\pi$
$443$	−4.34160	−0.206276	−0.103138	+	0.994667i	$0.532888\pi$
$444$	0	0
$445$	−48.9669	−2.32126
$446$	0	0
$447$	4.34532	0.205527
$448$	0	0
$449$	−10.9358	−0.516093	−0.258046	−	0.966133i	$-0.583079\pi$
$449$	−10.9358	−0.516093	−0.258046	+	0.966133i	$0.583079\pi$
$450$	0	0
$451$	−0.505768	−0.0238157
$452$	0	0
$453$	−17.8907	−0.840577
$454$	0	0
$455$	−9.40120	−0.440735
$456$	0	0
$457$	23.9782	1.12165	0.560826	−	0.827933i	$-0.310484\pi$
$457$	23.9782	1.12165	0.560826	+	0.827933i	$0.310484\pi$
$458$	0	0
$459$	−2.17362	−0.101456
$460$	0	0
$461$	6.39210	0.297710	0.148855	−	0.988859i	$-0.452441\pi$
$461$	6.39210	0.297710	0.148855	+	0.988859i	$0.452441\pi$
$462$	0	0
$463$	−19.8982	−0.924748	−0.462374	−	0.886685i	$-0.653002\pi$
$463$	−19.8982	−0.924748	−0.462374	+	0.886685i	$0.653002\pi$
$464$	0	0
$465$	−12.9306	−0.599640
$466$	0	0
$467$	−6.50454	−0.300994	−0.150497	−	0.988610i	$-0.548087\pi$
$467$	−6.50454	−0.300994	−0.150497	+	0.988610i	$0.548087\pi$
$468$	0	0
$469$	12.9061	0.595949
$470$	0	0
$471$	−17.9620	−0.827643
$472$	0	0
$473$	0.304671	0.0140088
$474$	0	0
$475$	11.9032	0.546156
$476$	0	0
$477$	3.88527	0.177894
$478$	0	0
$479$	−1.93092	−0.0882259	−0.0441130	−	0.999027i	$-0.514046\pi$
$479$	−1.93092	−0.0882259	−0.0441130	+	0.999027i	$0.514046\pi$
$480$	0	0
$481$	−18.3139	−0.835044
$482$	0	0
$483$	−3.08992	−0.140596
$484$	0	0
$485$	−39.9508	−1.81407
$486$	0	0
$487$	−22.3754	−1.01392	−0.506962	−	0.861968i	$-0.669232\pi$
$487$	−22.3754	−1.01392	−0.506962	+	0.861968i	$0.669232\pi$
$488$	0	0
$489$	7.95764	0.359857
$490$	0	0
$491$	−0.000619001 0	−2.79351e−5 0	−1.39676e−5	−	1.00000i	$-0.500004\pi$
$491$	−0.000619001 0	−2.79351e−5 0	−1.39676e−5		1.00000i	$0.500004\pi$
$492$	0	0
$493$	9.22259	0.415364
$494$	0	0
$495$	−0.211737	−0.00951685
$496$	0	0
$497$	−3.98754	−0.178865
$498$	0	0
$499$	38.6573	1.73054	0.865269	−	0.501309i	$-0.167148\pi$
$499$	38.6573	1.73054	0.865269	+	0.501309i	$0.167148\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	−23.2786	−1.03794	−0.518971	−	0.854792i	$-0.673685\pi$
$503$	−23.2786	−1.03794	−0.518971	+	0.854792i	$0.673685\pi$
$504$	0	0
$505$	17.1104	0.761402
$506$	0	0
$507$	−10.1504	−0.450793
$508$	0	0
$509$	14.8713	0.659157	0.329578	−	0.944128i	$-0.393093\pi$
$509$	14.8713	0.659157	0.329578	+	0.944128i	$0.393093\pi$
$510$	0	0
$511$	−15.6386	−0.691810
$512$	0	0
$513$	2.90752	0.128370
$514$	0	0
$515$	52.8659	2.32955
$516$	0	0
$517$	−0.814082	−0.0358033
$518$	0	0
$519$	−13.0823	−0.574251
$520$	0	0
$521$	10.4692	0.458663	0.229331	−	0.973348i	$-0.426346\pi$
$521$	10.4692	0.458663	0.229331	+	0.973348i	$0.426346\pi$
$522$	0	0
$523$	6.32882	0.276740	0.138370	−	0.990381i	$-0.455814\pi$
$523$	6.32882	0.276740	0.138370	+	0.990381i	$0.455814\pi$
$524$	0	0
$525$	7.56055	0.329969
$526$	0	0
$527$	−9.32018	−0.405993
$528$	0	0
$529$	−20.2006	−0.878286
$530$	0	0
$531$	−5.40850	−0.234709
$532$	0	0
$533$	−12.1598	−0.526700
$534$	0	0
$535$	7.72882	0.334146
$536$	0	0
$537$	−10.4300	−0.450088
$538$	0	0
$539$	−0.252027	−0.0108556
$540$	0	0
$541$	11.5446	0.496340	0.248170	−	0.968716i	$-0.420171\pi$
$541$	11.5446	0.496340	0.248170	+	0.968716i	$0.420171\pi$
$542$	0	0
$543$	−4.31004	−0.184961
$544$	0	0
$545$	−55.4322	−2.37446
$546$	0	0
$547$	20.7041	0.885243	0.442622	−	0.896708i	$-0.354048\pi$
$547$	20.7041	0.885243	0.442622	+	0.896708i	$0.354048\pi$
$548$	0	0
$549$	2.01512	0.0860034
$550$	0	0
$551$	−12.3365	−0.525552
$552$	0	0
$553$	20.0268	0.851625
$554$	0	0
$555$	32.7162	1.38872
$556$	0	0
$557$	34.1956	1.44891	0.724457	−	0.689320i	$-0.242091\pi$
$557$	34.1956	1.44891	0.724457	+	0.689320i	$0.242091\pi$
$558$	0	0
$559$	7.32497	0.309813
$560$	0	0
$561$	−0.152617	−0.00644349
$562$	0	0
$563$	−13.4440	−0.566598	−0.283299	−	0.959032i	$-0.591429\pi$
$563$	−13.4440	−0.566598	−0.283299	+	0.959032i	$0.591429\pi$
$564$	0	0
$565$	−26.0835	−1.09734
$566$	0	0
$567$	1.84677	0.0775569
$568$	0	0
$569$	1.42022	0.0595386	0.0297693	−	0.999557i	$-0.490523\pi$
$569$	1.42022	0.0595386	0.0297693	+	0.999557i	$0.490523\pi$
$570$	0	0
$571$	19.0276	0.796280	0.398140	−	0.917325i	$-0.369656\pi$
$571$	19.0276	0.796280	0.398140	+	0.917325i	$0.369656\pi$
$572$	0	0
$573$	−18.7287	−0.782404
$574$	0	0
$575$	−6.84977	−0.285655
$576$	0	0
$577$	30.5919	1.27356	0.636778	−	0.771047i	$-0.280267\pi$
$577$	30.5919	1.27356	0.636778	+	0.771047i	$0.280267\pi$
$578$	0	0
$579$	15.3962	0.639844
$580$	0	0
$581$	17.2702	0.716490
$582$	0	0
$583$	0.272798	0.0112981
$584$	0	0
$585$	−5.09062	−0.210471
$586$	0	0
$587$	−32.6169	−1.34624	−0.673121	−	0.739532i	$-0.735047\pi$
$587$	−32.6169	−1.34624	−0.673121	+	0.739532i	$0.735047\pi$
$588$	0	0
$589$	12.4670	0.513696
$590$	0	0
$591$	0.0789986	0.00324957
$592$	0	0
$593$	16.2145	0.665849	0.332924	−	0.942953i	$-0.391965\pi$
$593$	16.2145	0.665849	0.332924	+	0.942953i	$0.391965\pi$
$594$	0	0
$595$	12.1052	0.496264
$596$	0	0
$597$	−11.4542	−0.468789
$598$	0	0
$599$	−20.5160	−0.838263	−0.419131	−	0.907926i	$-0.637665\pi$
$599$	−20.5160	−0.838263	−0.419131	+	0.907926i	$0.637665\pi$
$600$	0	0
$601$	21.1781	0.863874	0.431937	−	0.901904i	$-0.357830\pi$
$601$	21.1781	0.863874	0.431937	+	0.901904i	$0.357830\pi$
$602$	0	0
$603$	6.98849	0.284593
$604$	0	0
$605$	33.1569	1.34802
$606$	0	0
$607$	0.963530	0.0391085	0.0195542	−	0.999809i	$-0.493775\pi$
$607$	0.963530	0.0391085	0.0195542	+	0.999809i	$0.493775\pi$
$608$	0	0
$609$	−7.83577	−0.317522
$610$	0	0
$611$	−19.5724	−0.791813
$612$	0	0
$613$	−37.3703	−1.50937	−0.754685	−	0.656087i	$-0.772210\pi$
$613$	−37.3703	−1.50937	−0.754685	+	0.656087i	$0.772210\pi$
$614$	0	0
$615$	21.7224	0.875932
$616$	0	0
$617$	−17.9204	−0.721448	−0.360724	−	0.932673i	$-0.617470\pi$
$617$	−17.9204	−0.721448	−0.360724	+	0.932673i	$0.617470\pi$
$618$	0	0
$619$	−0.709375	−0.0285122	−0.0142561	−	0.999898i	$-0.504538\pi$
$619$	−0.709375	−0.0285122	−0.0142561	+	0.999898i	$0.504538\pi$
$620$	0	0
$621$	−1.67315	−0.0671412
$622$	0	0
$623$	29.9874	1.20142
$624$	0	0
$625$	−28.7094	−1.14837
$626$	0	0
$627$	0.204146	0.00815282
$628$	0	0
$629$	23.5814	0.940252
$630$	0	0
$631$	−12.1195	−0.482472	−0.241236	−	0.970467i	$-0.577553\pi$
$631$	−12.1195	−0.482472	−0.241236	+	0.970467i	$0.577553\pi$
$632$	0	0
$633$	−15.3500	−0.610109
$634$	0	0
$635$	−0.343798	−0.0136432
$636$	0	0
$637$	−6.05931	−0.240079
$638$	0	0
$639$	−2.15920	−0.0854166
$640$	0	0
$641$	15.8998	0.628003	0.314002	−	0.949422i	$-0.398330\pi$
$641$	15.8998	0.628003	0.314002	+	0.949422i	$0.398330\pi$
$642$	0	0
$643$	23.8562	0.940798	0.470399	−	0.882454i	$-0.344110\pi$
$643$	23.8562	0.940798	0.470399	+	0.882454i	$0.344110\pi$
$644$	0	0
$645$	−13.0854	−0.515237
$646$	0	0
$647$	−2.97952	−0.117137	−0.0585685	−	0.998283i	$-0.518654\pi$
$647$	−2.97952	−0.117137	−0.0585685	+	0.998283i	$0.518654\pi$
$648$	0	0
$649$	−0.379749	−0.0149065
$650$	0	0
$651$	7.91869	0.310358
$652$	0	0
$653$	7.97720	0.312172	0.156086	−	0.987743i	$-0.450112\pi$
$653$	7.97720	0.312172	0.156086	+	0.987743i	$0.450112\pi$
$654$	0	0
$655$	37.8811	1.48014
$656$	0	0
$657$	−8.46808	−0.330371
$658$	0	0
$659$	29.9289	1.16586	0.582931	−	0.812521i	$-0.301906\pi$
$659$	29.9289	1.16586	0.582931	+	0.812521i	$0.301906\pi$
$660$	0	0
$661$	15.2290	0.592339	0.296170	−	0.955135i	$-0.404291\pi$
$661$	15.2290	0.592339	0.296170	+	0.955135i	$0.404291\pi$
$662$	0	0
$663$	−3.66926	−0.142502
$664$	0	0
$665$	−16.1924	−0.627913
$666$	0	0
$667$	7.09912	0.274879
$668$	0	0
$669$	−24.6893	−0.954542
$670$	0	0
$671$	0.141489	0.00546211
$672$	0	0
$673$	29.3603	1.13176	0.565878	−	0.824489i	$-0.308537\pi$
$673$	29.3603	1.13176	0.565878	+	0.824489i	$0.308537\pi$
$674$	0	0
$675$	4.09394	0.157576
$676$	0	0
$677$	0.142354	0.00547111	0.00273556	−	0.999996i	$-0.499129\pi$
$677$	0.142354	0.00547111	0.00273556	+	0.999996i	$0.499129\pi$
$678$	0	0
$679$	24.4659	0.938915
$680$	0	0
$681$	−24.5728	−0.941633
$682$	0	0
$683$	37.2505	1.42535	0.712676	−	0.701494i	$-0.247483\pi$
$683$	37.2505	1.42535	0.712676	+	0.701494i	$0.247483\pi$
$684$	0	0
$685$	19.8839	0.759725
$686$	0	0
$687$	−8.98726	−0.342885
$688$	0	0
$689$	6.55868	0.249866
$690$	0	0
$691$	−36.4586	−1.38695	−0.693475	−	0.720481i	$-0.743921\pi$
$691$	−36.4586	−1.38695	−0.693475	+	0.720481i	$0.743921\pi$
$692$	0	0
$693$	0.129668	0.00492567
$694$	0	0
$695$	17.4960	0.663662
$696$	0	0
$697$	15.6572	0.593060
$698$	0	0
$699$	−22.6629	−0.857190
$700$	0	0
$701$	−35.2147	−1.33004	−0.665020	−	0.746826i	$-0.731577\pi$
$701$	−35.2147	−1.33004	−0.665020	+	0.746826i	$0.731577\pi$
$702$	0	0
$703$	−31.5434	−1.18968
$704$	0	0
$705$	34.9643	1.31683
$706$	0	0
$707$	−10.4784	−0.394081
$708$	0	0
$709$	−38.7522	−1.45537	−0.727684	−	0.685912i	$-0.759403\pi$
$709$	−38.7522	−1.45537	−0.727684	+	0.685912i	$0.759403\pi$
$710$	0	0
$711$	10.8442	0.406690
$712$	0	0
$713$	−7.17425	−0.268678
$714$	0	0
$715$	−0.357430	−0.0133671
$716$	0	0
$717$	−7.63603	−0.285173
$718$	0	0
$719$	28.2855	1.05487	0.527435	−	0.849596i	$-0.323154\pi$
$719$	28.2855	1.05487	0.527435	+	0.849596i	$0.323154\pi$
$720$	0	0
$721$	−32.3752	−1.20571
$722$	0	0
$723$	−16.3487	−0.608013
$724$	0	0
$725$	−17.3705	−0.645122
$726$	0	0
$727$	−22.0460	−0.817642	−0.408821	−	0.912615i	$-0.634060\pi$
$727$	−22.0460	−0.817642	−0.408821	+	0.912615i	$0.634060\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−9.43179	−0.348847
$732$	0	0
$733$	13.1625	0.486167	0.243084	−	0.970005i	$-0.421841\pi$
$733$	13.1625	0.486167	0.243084	+	0.970005i	$0.421841\pi$
$734$	0	0
$735$	10.8244	0.399264
$736$	0	0
$737$	0.490685	0.0180746
$738$	0	0
$739$	9.86634	0.362939	0.181470	−	0.983397i	$-0.441915\pi$
$739$	9.86634	0.362939	0.181470	+	0.983397i	$0.441915\pi$
$740$	0	0
$741$	4.90814	0.180305
$742$	0	0
$743$	−5.70369	−0.209248	−0.104624	−	0.994512i	$-0.533364\pi$
$743$	−5.70369	−0.209248	−0.104624	+	0.994512i	$0.533364\pi$
$744$	0	0
$745$	−13.1038	−0.480087
$746$	0	0
$747$	9.35160	0.342157
$748$	0	0
$749$	−4.73314	−0.172945
$750$	0	0
$751$	25.2515	0.921439	0.460719	−	0.887546i	$-0.347591\pi$
$751$	25.2515	0.921439	0.460719	+	0.887546i	$0.347591\pi$
$752$	0	0
$753$	25.3237	0.922847
$754$	0	0
$755$	53.9514	1.96349
$756$	0	0
$757$	26.6032	0.966910	0.483455	−	0.875369i	$-0.339382\pi$
$757$	26.6032	0.966910	0.483455	+	0.875369i	$0.339382\pi$
$758$	0	0
$759$	−0.117478	−0.00426416
$760$	0	0
$761$	−38.0804	−1.38041	−0.690207	−	0.723612i	$-0.742481\pi$
$761$	−38.0804	−1.38041	−0.690207	+	0.723612i	$0.742481\pi$
$762$	0	0
$763$	33.9468	1.22896
$764$	0	0
$765$	6.55479	0.236989
$766$	0	0
$767$	−9.13003	−0.329666
$768$	0	0
$769$	5.50542	0.198531	0.0992653	−	0.995061i	$-0.468351\pi$
$769$	5.50542	0.198531	0.0992653	+	0.995061i	$0.468351\pi$
$770$	0	0
$771$	15.8241	0.569892
$772$	0	0
$773$	−17.0250	−0.612345	−0.306173	−	0.951976i	$-0.599048\pi$
$773$	−17.0250	−0.612345	−0.306173	+	0.951976i	$0.599048\pi$
$774$	0	0
$775$	17.5543	0.630568
$776$	0	0
$777$	−20.0354	−0.718767
$778$	0	0
$779$	−20.9437	−0.750387
$780$	0	0
$781$	−0.151605	−0.00542484
$782$	0	0
$783$	−4.24297	−0.151631
$784$	0	0
$785$	54.1664	1.93328
$786$	0	0
$787$	−43.9181	−1.56551	−0.782756	−	0.622329i	$-0.786187\pi$
$787$	−43.9181	−1.56551	−0.782756	+	0.622329i	$0.786187\pi$
$788$	0	0
$789$	−19.1324	−0.681131
$790$	0	0
$791$	15.9736	0.567955
$792$	0	0
$793$	3.40170	0.120798
$794$	0	0
$795$	−11.7165	−0.415541
$796$	0	0
$797$	55.5931	1.96921	0.984605	−	0.174795i	$-0.0559264\pi$
$797$	55.5931	1.96921	0.984605	+	0.174795i	$0.0559264\pi$
$798$	0	0
$799$	25.2018	0.891575
$800$	0	0
$801$	16.2378	0.573734
$802$	0	0
$803$	−0.594572	−0.0209820
$804$	0	0
$805$	9.31801	0.328417
$806$	0	0
$807$	7.78513	0.274050
$808$	0	0
$809$	39.9154	1.40335	0.701675	−	0.712497i	$-0.252436\pi$
$809$	39.9154	1.40335	0.701675	+	0.712497i	$0.252436\pi$
$810$	0	0
$811$	40.6843	1.42862	0.714309	−	0.699830i	$-0.246741\pi$
$811$	40.6843	1.42862	0.714309	+	0.699830i	$0.246741\pi$
$812$	0	0
$813$	11.7908	0.413520
$814$	0	0
$815$	−23.9972	−0.840585
$816$	0	0
$817$	12.6163	0.441390
$818$	0	0
$819$	3.11750	0.108934
$820$	0	0
$821$	−23.4731	−0.819217	−0.409608	−	0.912261i	$-0.634335\pi$
$821$	−23.4731	−0.819217	−0.409608	+	0.912261i	$0.634335\pi$
$822$	0	0
$823$	6.51601	0.227134	0.113567	−	0.993530i	$-0.463772\pi$
$823$	6.51601	0.227134	0.113567	+	0.993530i	$0.463772\pi$
$824$	0	0
$825$	0.287449	0.0100077
$826$	0	0
$827$	−45.4145	−1.57922	−0.789609	−	0.613611i	$-0.789717\pi$
$827$	−45.4145	−1.57922	−0.789609	+	0.613611i	$0.789717\pi$
$828$	0	0
$829$	32.9723	1.14518	0.572588	−	0.819843i	$-0.305939\pi$
$829$	32.9723	1.14518	0.572588	+	0.819843i	$0.305939\pi$
$830$	0	0
$831$	2.54189	0.0881773
$832$	0	0
$833$	7.80209	0.270326
$834$	0	0
$835$	−3.01562	−0.104360
$836$	0	0
$837$	4.28787	0.148210
$838$	0	0
$839$	27.4949	0.949230	0.474615	−	0.880193i	$-0.342587\pi$
$839$	27.4949	0.949230	0.474615	+	0.880193i	$0.342587\pi$
$840$	0	0
$841$	−10.9972	−0.379215
$842$	0	0
$843$	−7.37140	−0.253884
$844$	0	0
$845$	30.6096	1.05300
$846$	0	0
$847$	−20.3053	−0.697699
$848$	0	0
$849$	−8.42659	−0.289200
$850$	0	0
$851$	18.1519	0.622238
$852$	0	0
$853$	1.91679	0.0656298	0.0328149	−	0.999461i	$-0.489553\pi$
$853$	1.91679	0.0656298	0.0328149	+	0.999461i	$0.489553\pi$
$854$	0	0
$855$	−8.76795	−0.299858
$856$	0	0
$857$	−0.386484	−0.0132021	−0.00660103	−	0.999978i	$-0.502101\pi$
$857$	−0.386484	−0.0132021	−0.00660103	+	0.999978i	$0.502101\pi$
$858$	0	0
$859$	21.7433	0.741873	0.370937	−	0.928658i	$-0.379037\pi$
$859$	21.7433	0.741873	0.370937	+	0.928658i	$0.379037\pi$
$860$	0	0
$861$	−13.3028	−0.453359
$862$	0	0
$863$	40.4063	1.37545	0.687723	−	0.725973i	$-0.258610\pi$
$863$	40.4063	1.37545	0.687723	+	0.725973i	$0.258610\pi$
$864$	0	0
$865$	39.4513	1.34138
$866$	0	0
$867$	−12.2754	−0.416894
$868$	0	0
$869$	0.761410	0.0258291
$870$	0	0
$871$	11.7972	0.399732
$872$	0	0
$873$	13.2480	0.448376
$874$	0	0
$875$	5.04598	0.170585
$876$	0	0
$877$	−47.3266	−1.59811	−0.799053	−	0.601261i	$-0.794665\pi$
$877$	−47.3266	−1.59811	−0.799053	+	0.601261i	$0.794665\pi$
$878$	0	0
$879$	17.3212	0.584229
$880$	0	0
$881$	40.0078	1.34790	0.673949	−	0.738778i	$-0.264597\pi$
$881$	40.0078	1.34790	0.673949	+	0.738778i	$0.264597\pi$
$882$	0	0
$883$	−14.2412	−0.479255	−0.239627	−	0.970865i	$-0.577025\pi$
$883$	−14.2412	−0.479255	−0.239627	+	0.970865i	$0.577025\pi$
$884$	0	0
$885$	16.3100	0.548253
$886$	0	0
$887$	−16.2299	−0.544948	−0.272474	−	0.962163i	$-0.587842\pi$
$887$	−16.2299	−0.544948	−0.272474	+	0.962163i	$0.587842\pi$
$888$	0	0
$889$	0.210542	0.00706136
$890$	0	0
$891$	0.0702134	0.00235224
$892$	0	0
$893$	−33.7109	−1.12809
$894$	0	0
$895$	31.4529	1.05135
$896$	0	0
$897$	−2.82442	−0.0943048
$898$	0	0
$899$	−18.1933	−0.606780
$900$	0	0
$901$	−8.44509	−0.281347
$902$	0	0
$903$	8.01351	0.266673
$904$	0	0
$905$	12.9974	0.432049
$906$	0	0
$907$	−14.4413	−0.479516	−0.239758	−	0.970833i	$-0.577068\pi$
$907$	−14.4413	−0.479516	−0.239758	+	0.970833i	$0.577068\pi$
$908$	0	0
$909$	−5.67392	−0.188192
$910$	0	0
$911$	53.7207	1.77984	0.889922	−	0.456112i	$-0.150758\pi$
$911$	53.7207	1.77984	0.889922	+	0.456112i	$0.150758\pi$
$912$	0	0
$913$	0.656607	0.0217305
$914$	0	0
$915$	−6.07684	−0.200894
$916$	0	0
$917$	−23.1985	−0.766081
$918$	0	0
$919$	−29.4559	−0.971661	−0.485831	−	0.874053i	$-0.661483\pi$
$919$	−29.4559	−0.971661	−0.485831	+	0.874053i	$0.661483\pi$
$920$	0	0
$921$	−6.40036	−0.210899
$922$	0	0
$923$	−3.64492	−0.119974
$924$	0	0
$925$	−44.4148	−1.46035
$926$	0	0
$927$	−17.5307	−0.575785
$928$	0	0
$929$	−34.9105	−1.14538	−0.572688	−	0.819773i	$-0.694100\pi$
$929$	−34.9105	−1.14538	−0.572688	+	0.819773i	$0.694100\pi$
$930$	0	0
$931$	−10.4364	−0.342039
$932$	0	0
$933$	−17.9862	−0.588843
$934$	0	0
$935$	0.460234	0.0150513
$936$	0	0
$937$	42.0982	1.37529	0.687644	−	0.726048i	$-0.258645\pi$
$937$	42.0982	1.37529	0.687644	+	0.726048i	$0.258645\pi$
$938$	0	0
$939$	−21.4712	−0.700688
$940$	0	0
$941$	16.7714	0.546732	0.273366	−	0.961910i	$-0.411863\pi$
$941$	16.7714	0.546732	0.273366	+	0.961910i	$0.411863\pi$
$942$	0	0
$943$	12.0522	0.392474
$944$	0	0
$945$	−5.56914	−0.181164
$946$	0	0
$947$	32.4958	1.05597	0.527985	−	0.849254i	$-0.322948\pi$
$947$	32.4958	1.05597	0.527985	+	0.849254i	$0.322948\pi$
$948$	0	0
$949$	−14.2949	−0.464031
$950$	0	0
$951$	−22.8128	−0.739757
$952$	0	0
$953$	37.5661	1.21689	0.608443	−	0.793597i	$-0.291794\pi$
$953$	37.5661	1.21689	0.608443	+	0.793597i	$0.291794\pi$
$954$	0	0
$955$	56.4786	1.82761
$956$	0	0
$957$	−0.297913	−0.00963016
$958$	0	0
$959$	−12.1769	−0.393214
$960$	0	0
$961$	−12.6142	−0.406909
$962$	0	0
$963$	−2.56293	−0.0825894
$964$	0	0
$965$	−46.4290	−1.49460
$966$	0	0
$967$	5.46025	0.175590	0.0877949	−	0.996139i	$-0.472018\pi$
$967$	5.46025	0.175590	0.0877949	+	0.996139i	$0.472018\pi$
$968$	0	0
$969$	−6.31982	−0.203022
$970$	0	0
$971$	34.4319	1.10497	0.552487	−	0.833522i	$-0.313679\pi$
$971$	34.4319	1.10497	0.552487	+	0.833522i	$0.313679\pi$
$972$	0	0
$973$	−10.7146	−0.343494
$974$	0	0
$975$	6.91093	0.221327
$976$	0	0
$977$	13.1807	0.421689	0.210844	−	0.977520i	$-0.432379\pi$
$977$	13.1807	0.421689	0.210844	+	0.977520i	$0.432379\pi$
$978$	0	0
$979$	1.14011	0.0364381
$980$	0	0
$981$	18.3817	0.586884
$982$	0	0
$983$	−10.2408	−0.326632	−0.163316	−	0.986574i	$-0.552219\pi$
$983$	−10.2408	−0.326632	−0.163316	+	0.986574i	$0.552219\pi$
$984$	0	0
$985$	−0.238229	−0.00759062
$986$	0	0
$987$	−21.4122	−0.681557
$988$	0	0
$989$	−7.26015	−0.230859
$990$	0	0
$991$	−27.8098	−0.883409	−0.441704	−	0.897161i	$-0.645626\pi$
$991$	−27.8098	−0.883409	−0.441704	+	0.897161i	$0.645626\pi$
$992$	0	0
$993$	22.1607	0.703248
$994$	0	0
$995$	34.5415	1.09504
$996$	0	0
$997$	11.4323	0.362065	0.181033	−	0.983477i	$-0.442056\pi$
$997$	11.4323	0.362065	0.181033	+	0.983477i	$0.442056\pi$
$998$	0	0
$999$	−10.8489	−0.343245

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.w.1.2		7
4.3	odd	2		4008.2.a.g.1.2	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.g.1.2	✓	7	4.3	odd	2
8016.2.a.w.1.2		7	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.01562 q^{5} +1.84677 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.01562 q^{5} +1.84677 q^{7} +1.00000 q^{9} +0.0702134 q^{11} +1.68809 q^{13} -3.01562 q^{15} -2.17362 q^{17} +2.90752 q^{19} +1.84677 q^{21} -1.67315 q^{23} +4.09394 q^{25} +1.00000 q^{27} -4.24297 q^{29} +4.28787 q^{31} +0.0702134 q^{33} -5.56914 q^{35} -10.8489 q^{37} +1.68809 q^{39} -7.20331 q^{41} +4.33921 q^{43} -3.01562 q^{45} -11.5944 q^{47} -3.58945 q^{49} -2.17362 q^{51} +3.88527 q^{53} -0.211737 q^{55} +2.90752 q^{57} -5.40850 q^{59} +2.01512 q^{61} +1.84677 q^{63} -5.09062 q^{65} +6.98849 q^{67} -1.67315 q^{69} -2.15920 q^{71} -8.46808 q^{73} +4.09394 q^{75} +0.129668 q^{77} +10.8442 q^{79} +1.00000 q^{81} +9.35160 q^{83} +6.55479 q^{85} -4.24297 q^{87} +16.2378 q^{89} +3.11750 q^{91} +4.28787 q^{93} -8.76795 q^{95} +13.2480 q^{97} +0.0702134 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{3} - 3 q^{5} - 8 q^{7} + 7 q^{9}+O(q^{10}) \) 7 * q + 7 * q^3 - 3 * q^5 - 8 * q^7 + 7 * q^9	\( 7 q + 7 q^{3} - 3 q^{5} - 8 q^{7} + 7 q^{9} - q^{11} - 2 q^{13} - 3 q^{15} + 11 q^{17} - 2 q^{19} - 8 q^{21} - 17 q^{23} + 4 q^{25} + 7 q^{27} - 7 q^{29} - 10 q^{31} - q^{33} - 10 q^{35} - 21 q^{37} - 2 q^{39} + 8 q^{41} + 12 q^{43} - 3 q^{45} - 25 q^{47} - 7 q^{49} + 11 q^{51} - 7 q^{53} - 15 q^{55} - 2 q^{57} - 3 q^{59} - 14 q^{61} - 8 q^{63} + 4 q^{65} - 4 q^{67} - 17 q^{69} - 27 q^{71} - 12 q^{73} + 4 q^{75} + 16 q^{77} - 8 q^{79} + 7 q^{81} - 15 q^{83} - 3 q^{85} - 7 q^{87} + 14 q^{89} + 3 q^{91} - 10 q^{93} - 37 q^{95} + 3 q^{97} - q^{99}+O(q^{100}) \) 7 * q + 7 * q^3 - 3 * q^5 - 8 * q^7 + 7 * q^9 - q^11 - 2 * q^13 - 3 * q^15 + 11 * q^17 - 2 * q^19 - 8 * q^21 - 17 * q^23 + 4 * q^25 + 7 * q^27 - 7 * q^29 - 10 * q^31 - q^33 - 10 * q^35 - 21 * q^37 - 2 * q^39 + 8 * q^41 + 12 * q^43 - 3 * q^45 - 25 * q^47 - 7 * q^49 + 11 * q^51 - 7 * q^53 - 15 * q^55 - 2 * q^57 - 3 * q^59 - 14 * q^61 - 8 * q^63 + 4 * q^65 - 4 * q^67 - 17 * q^69 - 27 * q^71 - 12 * q^73 + 4 * q^75 + 16 * q^77 - 8 * q^79 + 7 * q^81 - 15 * q^83 - 3 * q^85 - 7 * q^87 + 14 * q^89 + 3 * q^91 - 10 * q^93 - 37 * q^95 + 3 * q^97 - q^99

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