LMFDB - Embedded newform 8016.2.a.bg.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:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 2 x^{12} - 49 x^{11} + 99 x^{10} + 901 x^{9} - 1879 x^{8} - 7582 x^{7} + 16968 x^{6} + \cdots + 12144 $ x^13 - 2x^12 - 49x^11 + 99x^10 + 901x^9 - 1879x^8 - 7582x^7 + 16968x^6 + 26911x^5 - 72240x^4 - 14532x^3 + 112850x^2 - 72184x + 12144
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
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			$-3.60065$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -3.60065 q^{5} +4.85776 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -3.60065 q^{5} +4.85776 q^{7} +1.00000 q^{9} -0.783405 q^{11} -2.28883 q^{13} +3.60065 q^{15} -3.64326 q^{17} +1.90236 q^{19} -4.85776 q^{21} -0.781756 q^{23} +7.96466 q^{25} -1.00000 q^{27} -5.79522 q^{29} +5.38959 q^{31} +0.783405 q^{33} -17.4911 q^{35} -7.39337 q^{37} +2.28883 q^{39} +1.19043 q^{41} -2.44687 q^{43} -3.60065 q^{45} +0.0708292 q^{47} +16.5979 q^{49} +3.64326 q^{51} +2.21252 q^{53} +2.82076 q^{55} -1.90236 q^{57} +3.50468 q^{59} +15.0468 q^{61} +4.85776 q^{63} +8.24127 q^{65} -2.58442 q^{67} +0.781756 q^{69} +6.82762 q^{71} +8.47077 q^{73} -7.96466 q^{75} -3.80559 q^{77} +10.1377 q^{79} +1.00000 q^{81} -16.5852 q^{83} +13.1181 q^{85} +5.79522 q^{87} +0.933406 q^{89} -11.1186 q^{91} -5.38959 q^{93} -6.84973 q^{95} -15.9639 q^{97} -0.783405 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 13 q^{3} + 2 q^{5} - q^{7} + 13 q^{9}+O(q^{10}) $ 13 * q - 13 * q^3 + 2 * q^5 - q^7 + 13 * q^9	$ 13 q - 13 q^{3} + 2 q^{5} - q^{7} + 13 q^{9} - 11 q^{11} + 12 q^{13} - 2 q^{15} + 15 q^{17} - 14 q^{19} + q^{21} - 9 q^{23} + 37 q^{25} - 13 q^{27} - 3 q^{29} + 17 q^{31} + 11 q^{33} - 15 q^{35} + 16 q^{37} - 12 q^{39} + 12 q^{41} - 20 q^{43} + 2 q^{45} + 6 q^{47} + 26 q^{49} - 15 q^{51} - 12 q^{53} - 7 q^{55} + 14 q^{57} - 14 q^{59} + 24 q^{61} - q^{63} + 8 q^{65} - 3 q^{67} + 9 q^{69} - 17 q^{71} + 34 q^{73} - 37 q^{75} + 30 q^{77} - 10 q^{79} + 13 q^{81} - 44 q^{83} + 25 q^{85} + 3 q^{87} + 25 q^{89} - 29 q^{91} - 17 q^{93} + 15 q^{95} + 38 q^{97} - 11 q^{99}+O(q^{100}) $ 13 * q - 13 * q^3 + 2 * q^5 - q^7 + 13 * q^9 - 11 * q^11 + 12 * q^13 - 2 * q^15 + 15 * q^17 - 14 * q^19 + q^21 - 9 * q^23 + 37 * q^25 - 13 * q^27 - 3 * q^29 + 17 * q^31 + 11 * q^33 - 15 * q^35 + 16 * q^37 - 12 * q^39 + 12 * q^41 - 20 * q^43 + 2 * q^45 + 6 * q^47 + 26 * q^49 - 15 * q^51 - 12 * q^53 - 7 * q^55 + 14 * q^57 - 14 * q^59 + 24 * q^61 - q^63 + 8 * q^65 - 3 * q^67 + 9 * q^69 - 17 * q^71 + 34 * q^73 - 37 * q^75 + 30 * q^77 - 10 * q^79 + 13 * q^81 - 44 * q^83 + 25 * q^85 + 3 * q^87 + 25 * q^89 - 29 * q^91 - 17 * q^93 + 15 * q^95 + 38 * q^97 - 11 * 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
$5$	−3.60065	−1.61026	−0.805129	−	0.593100i	$-0.797904\pi$
$5$	−3.60065	−1.61026	−0.805129	+	0.593100i	$0.797904\pi$
$6$	0	0
$7$	4.85776	1.83606	0.918031	−	0.396509i	$-0.129778\pi$
$7$	4.85776	1.83606	0.918031	+	0.396509i	$0.129778\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.783405	−0.236205	−0.118103	−	0.993001i	$-0.537681\pi$
$11$	−0.783405	−0.236205	−0.118103	+	0.993001i	$0.537681\pi$
$12$	0	0
$13$	−2.28883	−0.634808	−0.317404	−	0.948290i	$-0.602811\pi$
$13$	−2.28883	−0.634808	−0.317404	+	0.948290i	$0.602811\pi$
$14$	0	0
$15$	3.60065	0.929683
$16$	0	0
$17$	−3.64326	−0.883619	−0.441810	−	0.897109i	$-0.645663\pi$
$17$	−3.64326	−0.883619	−0.441810	+	0.897109i	$0.645663\pi$
$18$	0	0
$19$	1.90236	0.436431	0.218216	−	0.975901i	$-0.429976\pi$
$19$	1.90236	0.436431	0.218216	+	0.975901i	$0.429976\pi$
$20$	0	0
$21$	−4.85776	−1.06005
$22$	0	0
$23$	−0.781756	−0.163007	−0.0815036	−	0.996673i	$-0.525972\pi$
$23$	−0.781756	−0.163007	−0.0815036	+	0.996673i	$0.525972\pi$
$24$	0	0
$25$	7.96466	1.59293
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−5.79522	−1.07615	−0.538073	−	0.842898i	$-0.680847\pi$
$29$	−5.79522	−1.07615	−0.538073	+	0.842898i	$0.680847\pi$
$30$	0	0
$31$	5.38959	0.967999	0.483999	−	0.875068i	$-0.339184\pi$
$31$	5.38959	0.967999	0.483999	+	0.875068i	$0.339184\pi$
$32$	0	0
$33$	0.783405	0.136373
$34$	0	0
$35$	−17.4911	−2.95653
$36$	0	0
$37$	−7.39337	−1.21546	−0.607731	−	0.794143i	$-0.707920\pi$
$37$	−7.39337	−1.21546	−0.607731	+	0.794143i	$0.707920\pi$
$38$	0	0
$39$	2.28883	0.366506
$40$	0	0
$41$	1.19043	0.185914	0.0929569	−	0.995670i	$-0.470368\pi$
$41$	1.19043	0.185914	0.0929569	+	0.995670i	$0.470368\pi$
$42$	0	0
$43$	−2.44687	−0.373143	−0.186572	−	0.982441i	$-0.559738\pi$
$43$	−2.44687	−0.373143	−0.186572	+	0.982441i	$0.559738\pi$
$44$	0	0
$45$	−3.60065	−0.536753
$46$	0	0
$47$	0.0708292	0.0103315	0.00516575	−	0.999987i	$-0.498356\pi$
$47$	0.0708292	0.0103315	0.00516575	+	0.999987i	$0.498356\pi$
$48$	0	0
$49$	16.5979	2.37112
$50$	0	0
$51$	3.64326	0.510158
$52$	0	0
$53$	2.21252	0.303913	0.151956	−	0.988387i	$-0.451443\pi$
$53$	2.21252	0.303913	0.151956	+	0.988387i	$0.451443\pi$
$54$	0	0
$55$	2.82076	0.380352
$56$	0	0
$57$	−1.90236	−0.251974
$58$	0	0
$59$	3.50468	0.456270	0.228135	−	0.973630i	$-0.426737\pi$
$59$	3.50468	0.456270	0.228135	+	0.973630i	$0.426737\pi$
$60$	0	0
$61$	15.0468	1.92655	0.963273	−	0.268523i	$-0.0865356\pi$
$61$	15.0468	1.92655	0.963273	+	0.268523i	$0.0865356\pi$
$62$	0	0
$63$	4.85776	0.612021
$64$	0	0
$65$	8.24127	1.02220
$66$	0	0
$67$	−2.58442	−0.315737	−0.157869	−	0.987460i	$-0.550462\pi$
$67$	−2.58442	−0.315737	−0.157869	+	0.987460i	$0.550462\pi$
$68$	0	0
$69$	0.781756	0.0941123
$70$	0	0
$71$	6.82762	0.810289	0.405145	−	0.914253i	$-0.367221\pi$
$71$	6.82762	0.810289	0.405145	+	0.914253i	$0.367221\pi$
$72$	0	0
$73$	8.47077	0.991429	0.495714	−	0.868486i	$-0.334906\pi$
$73$	8.47077	0.991429	0.495714	+	0.868486i	$0.334906\pi$
$74$	0	0
$75$	−7.96466	−0.919680
$76$	0	0
$77$	−3.80559	−0.433688
$78$	0	0
$79$	10.1377	1.14058	0.570291	−	0.821443i	$-0.306830\pi$
$79$	10.1377	1.14058	0.570291	+	0.821443i	$0.306830\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−16.5852	−1.82046	−0.910232	−	0.414098i	$-0.864097\pi$
$83$	−16.5852	−1.82046	−0.910232	+	0.414098i	$0.864097\pi$
$84$	0	0
$85$	13.1181	1.42286
$86$	0	0
$87$	5.79522	0.621313
$88$	0	0
$89$	0.933406	0.0989408	0.0494704	−	0.998776i	$-0.484247\pi$
$89$	0.933406	0.0989408	0.0494704	+	0.998776i	$0.484247\pi$
$90$	0	0
$91$	−11.1186	−1.16555
$92$	0	0
$93$	−5.38959	−0.558874
$94$	0	0
$95$	−6.84973	−0.702767
$96$	0	0
$97$	−15.9639	−1.62089	−0.810446	−	0.585813i	$-0.800775\pi$
$97$	−15.9639	−1.62089	−0.810446	+	0.585813i	$0.800775\pi$
$98$	0	0
$99$	−0.783405	−0.0787351
$100$	0	0
$101$	−6.69164	−0.665843	−0.332921	−	0.942955i	$-0.608034\pi$
$101$	−6.69164	−0.665843	−0.332921	+	0.942955i	$0.608034\pi$
$102$	0	0
$103$	−15.7002	−1.54699	−0.773496	−	0.633802i	$-0.781493\pi$
$103$	−15.7002	−1.54699	−0.773496	+	0.633802i	$0.781493\pi$
$104$	0	0
$105$	17.4911	1.70696
$106$	0	0
$107$	−1.29322	−0.125020	−0.0625102	−	0.998044i	$-0.519911\pi$
$107$	−1.29322	−0.125020	−0.0625102	+	0.998044i	$0.519911\pi$
$108$	0	0
$109$	0.00600031	0.000574726 0	0.000287363	−	1.00000i	$-0.499909\pi$
$109$	0.00600031	0.000574726 0	0.000287363		1.00000i	$0.499909\pi$
$110$	0	0
$111$	7.39337	0.701748
$112$	0	0
$113$	−2.77121	−0.260694	−0.130347	−	0.991468i	$-0.541609\pi$
$113$	−2.77121	−0.260694	−0.130347	+	0.991468i	$0.541609\pi$
$114$	0	0
$115$	2.81483	0.262484
$116$	0	0
$117$	−2.28883	−0.211603
$118$	0	0
$119$	−17.6981	−1.62238
$120$	0	0
$121$	−10.3863	−0.944207
$122$	0	0
$123$	−1.19043	−0.107337
$124$	0	0
$125$	−10.6747	−0.954773
$126$	0	0
$127$	−20.0283	−1.77722	−0.888612	−	0.458660i	$-0.848330\pi$
$127$	−20.0283	−1.77722	−0.888612	+	0.458660i	$0.848330\pi$
$128$	0	0
$129$	2.44687	0.215434
$130$	0	0
$131$	13.2921	1.16134	0.580669	−	0.814140i	$-0.302791\pi$
$131$	13.2921	1.16134	0.580669	+	0.814140i	$0.302791\pi$
$132$	0	0
$133$	9.24121	0.801315
$134$	0	0
$135$	3.60065	0.309894
$136$	0	0
$137$	5.56619	0.475552	0.237776	−	0.971320i	$-0.423582\pi$
$137$	5.56619	0.475552	0.237776	+	0.971320i	$0.423582\pi$
$138$	0	0
$139$	−14.5785	−1.23653	−0.618265	−	0.785970i	$-0.712164\pi$
$139$	−14.5785	−1.23653	−0.618265	+	0.785970i	$0.712164\pi$
$140$	0	0
$141$	−0.0708292	−0.00596489
$142$	0	0
$143$	1.79308	0.149945
$144$	0	0
$145$	20.8665	1.73287
$146$	0	0
$147$	−16.5979	−1.36897
$148$	0	0
$149$	−0.839632	−0.0687854	−0.0343927	−	0.999408i	$-0.510950\pi$
$149$	−0.839632	−0.0687854	−0.0343927	+	0.999408i	$0.510950\pi$
$150$	0	0
$151$	−0.823210	−0.0669919	−0.0334959	−	0.999439i	$-0.510664\pi$
$151$	−0.823210	−0.0669919	−0.0334959	+	0.999439i	$0.510664\pi$
$152$	0	0
$153$	−3.64326	−0.294540
$154$	0	0
$155$	−19.4060	−1.55873
$156$	0	0
$157$	8.27681	0.660561	0.330281	−	0.943883i	$-0.392857\pi$
$157$	8.27681	0.660561	0.330281	+	0.943883i	$0.392857\pi$
$158$	0	0
$159$	−2.21252	−0.175464
$160$	0	0
$161$	−3.79758	−0.299291
$162$	0	0
$163$	3.21987	0.252200	0.126100	−	0.992018i	$-0.459754\pi$
$163$	3.21987	0.252200	0.126100	+	0.992018i	$0.459754\pi$
$164$	0	0
$165$	−2.82076	−0.219596
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−7.76125	−0.597019
$170$	0	0
$171$	1.90236	0.145477
$172$	0	0
$173$	8.74339	0.664748	0.332374	−	0.943148i	$-0.392150\pi$
$173$	8.74339	0.664748	0.332374	+	0.943148i	$0.392150\pi$
$174$	0	0
$175$	38.6904	2.92472
$176$	0	0
$177$	−3.50468	−0.263428
$178$	0	0
$179$	20.6471	1.54323	0.771617	−	0.636087i	$-0.219448\pi$
$179$	20.6471	1.54323	0.771617	+	0.636087i	$0.219448\pi$
$180$	0	0
$181$	23.3686	1.73697	0.868487	−	0.495713i	$-0.165093\pi$
$181$	23.3686	1.73697	0.868487	+	0.495713i	$0.165093\pi$
$182$	0	0
$183$	−15.0468	−1.11229
$184$	0	0
$185$	26.6209	1.95721
$186$	0	0
$187$	2.85414	0.208716
$188$	0	0
$189$	−4.85776	−0.353350
$190$	0	0
$191$	−18.3180	−1.32544	−0.662721	−	0.748866i	$-0.730599\pi$
$191$	−18.3180	−1.32544	−0.662721	+	0.748866i	$0.730599\pi$
$192$	0	0
$193$	8.36969	0.602463	0.301232	−	0.953551i	$-0.402602\pi$
$193$	8.36969	0.602463	0.301232	+	0.953551i	$0.402602\pi$
$194$	0	0
$195$	−8.24127	−0.590170
$196$	0	0
$197$	12.1914	0.868599	0.434300	−	0.900768i	$-0.356996\pi$
$197$	12.1914	0.868599	0.434300	+	0.900768i	$0.356996\pi$
$198$	0	0
$199$	−3.39186	−0.240443	−0.120221	−	0.992747i	$-0.538360\pi$
$199$	−3.39186	−0.240443	−0.120221	+	0.992747i	$0.538360\pi$
$200$	0	0
$201$	2.58442	0.182291
$202$	0	0
$203$	−28.1518	−1.97587
$204$	0	0
$205$	−4.28631	−0.299369
$206$	0	0
$207$	−0.781756	−0.0543358
$208$	0	0
$209$	−1.49032	−0.103087
$210$	0	0
$211$	19.3307	1.33078	0.665389	−	0.746497i	$-0.268266\pi$
$211$	19.3307	1.33078	0.665389	+	0.746497i	$0.268266\pi$
$212$	0	0
$213$	−6.82762	−0.467821
$214$	0	0
$215$	8.81030	0.600857
$216$	0	0
$217$	26.1813	1.77731
$218$	0	0
$219$	−8.47077	−0.572402
$220$	0	0
$221$	8.33880	0.560928
$222$	0	0
$223$	−11.5418	−0.772896	−0.386448	−	0.922311i	$-0.626298\pi$
$223$	−11.5418	−0.772896	−0.386448	+	0.922311i	$0.626298\pi$
$224$	0	0
$225$	7.96466	0.530977
$226$	0	0
$227$	12.1495	0.806392	0.403196	−	0.915114i	$-0.367899\pi$
$227$	12.1495	0.806392	0.403196	+	0.915114i	$0.367899\pi$
$228$	0	0
$229$	23.6217	1.56097	0.780483	−	0.625177i	$-0.214973\pi$
$229$	23.6217	1.56097	0.780483	+	0.625177i	$0.214973\pi$
$230$	0	0
$231$	3.80559	0.250390
$232$	0	0
$233$	7.08707	0.464290	0.232145	−	0.972681i	$-0.425426\pi$
$233$	7.08707	0.464290	0.232145	+	0.972681i	$0.425426\pi$
$234$	0	0
$235$	−0.255031	−0.0166364
$236$	0	0
$237$	−10.1377	−0.658516
$238$	0	0
$239$	30.5104	1.97355	0.986776	−	0.162091i	$-0.0518239\pi$
$239$	30.5104	1.97355	0.986776	+	0.162091i	$0.0518239\pi$
$240$	0	0
$241$	10.7672	0.693578	0.346789	−	0.937943i	$-0.387272\pi$
$241$	10.7672	0.693578	0.346789	+	0.937943i	$0.387272\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−59.7630	−3.81812
$246$	0	0
$247$	−4.35418	−0.277050
$248$	0	0
$249$	16.5852	1.05105
$250$	0	0
$251$	16.4758	1.03994	0.519972	−	0.854183i	$-0.325942\pi$
$251$	16.4758	1.03994	0.519972	+	0.854183i	$0.325942\pi$
$252$	0	0
$253$	0.612431	0.0385032
$254$	0	0
$255$	−13.1181	−0.821486
$256$	0	0
$257$	−14.1522	−0.882790	−0.441395	−	0.897313i	$-0.645516\pi$
$257$	−14.1522	−0.882790	−0.441395	+	0.897313i	$0.645516\pi$
$258$	0	0
$259$	−35.9153	−2.23167
$260$	0	0
$261$	−5.79522	−0.358715
$262$	0	0
$263$	1.37436	0.0847469	0.0423734	−	0.999102i	$-0.486508\pi$
$263$	1.37436	0.0847469	0.0423734	+	0.999102i	$0.486508\pi$
$264$	0	0
$265$	−7.96650	−0.489378
$266$	0	0
$267$	−0.933406	−0.0571235
$268$	0	0
$269$	1.22578	0.0747372	0.0373686	−	0.999302i	$-0.488102\pi$
$269$	1.22578	0.0747372	0.0373686	+	0.999302i	$0.488102\pi$
$270$	0	0
$271$	−19.2991	−1.17234	−0.586168	−	0.810189i	$-0.699364\pi$
$271$	−19.2991	−1.17234	−0.586168	+	0.810189i	$0.699364\pi$
$272$	0	0
$273$	11.1186	0.672928
$274$	0	0
$275$	−6.23955	−0.376259
$276$	0	0
$277$	13.5577	0.814601	0.407301	−	0.913294i	$-0.366470\pi$
$277$	13.5577	0.814601	0.407301	+	0.913294i	$0.366470\pi$
$278$	0	0
$279$	5.38959	0.322666
$280$	0	0
$281$	16.5306	0.986134	0.493067	−	0.869991i	$-0.335876\pi$
$281$	16.5306	0.986134	0.493067	+	0.869991i	$0.335876\pi$
$282$	0	0
$283$	21.1153	1.25517	0.627587	−	0.778546i	$-0.284043\pi$
$283$	21.1153	1.25517	0.627587	+	0.778546i	$0.284043\pi$
$284$	0	0
$285$	6.84973	0.405743
$286$	0	0
$287$	5.78282	0.341349
$288$	0	0
$289$	−3.72669	−0.219217
$290$	0	0
$291$	15.9639	0.935823
$292$	0	0
$293$	−14.3562	−0.838698	−0.419349	−	0.907825i	$-0.637742\pi$
$293$	−14.3562	−0.838698	−0.419349	+	0.907825i	$0.637742\pi$
$294$	0	0
$295$	−12.6191	−0.734712
$296$	0	0
$297$	0.783405	0.0454578
$298$	0	0
$299$	1.78931	0.103478
$300$	0	0
$301$	−11.8863	−0.685114
$302$	0	0
$303$	6.69164	0.384424
$304$	0	0
$305$	−54.1782	−3.10224
$306$	0	0
$307$	23.3658	1.33356	0.666778	−	0.745257i	$-0.267673\pi$
$307$	23.3658	1.33356	0.666778	+	0.745257i	$0.267673\pi$
$308$	0	0
$309$	15.7002	0.893156
$310$	0	0
$311$	12.7290	0.721794	0.360897	−	0.932606i	$-0.382471\pi$
$311$	12.7290	0.721794	0.360897	+	0.932606i	$0.382471\pi$
$312$	0	0
$313$	9.72833	0.549878	0.274939	−	0.961462i	$-0.411342\pi$
$313$	9.72833	0.549878	0.274939	+	0.961462i	$0.411342\pi$
$314$	0	0
$315$	−17.4911	−0.985511
$316$	0	0
$317$	−14.6156	−0.820892	−0.410446	−	0.911885i	$-0.634627\pi$
$317$	−14.6156	−0.820892	−0.410446	+	0.911885i	$0.634627\pi$
$318$	0	0
$319$	4.54000	0.254191
$320$	0	0
$321$	1.29322	0.0721806
$322$	0	0
$323$	−6.93078	−0.385639
$324$	0	0
$325$	−18.2298	−1.01121
$326$	0	0
$327$	−0.00600031	−0.000331818 0
$328$	0	0
$329$	0.344071	0.0189693
$330$	0	0
$331$	−7.54681	−0.414810	−0.207405	−	0.978255i	$-0.566502\pi$
$331$	−7.54681	−0.414810	−0.207405	+	0.978255i	$0.566502\pi$
$332$	0	0
$333$	−7.39337	−0.405154
$334$	0	0
$335$	9.30559	0.508419
$336$	0	0
$337$	7.59790	0.413884	0.206942	−	0.978353i	$-0.433649\pi$
$337$	7.59790	0.413884	0.206942	+	0.978353i	$0.433649\pi$
$338$	0	0
$339$	2.77121	0.150512
$340$	0	0
$341$	−4.22223	−0.228647
$342$	0	0
$343$	46.6241	2.51747
$344$	0	0
$345$	−2.81483	−0.151545
$346$	0	0
$347$	18.8244	1.01055	0.505274	−	0.862959i	$-0.331391\pi$
$347$	18.8244	1.01055	0.505274	+	0.862959i	$0.331391\pi$
$348$	0	0
$349$	13.3873	0.716605	0.358303	−	0.933605i	$-0.383356\pi$
$349$	13.3873	0.716605	0.358303	+	0.933605i	$0.383356\pi$
$350$	0	0
$351$	2.28883	0.122169
$352$	0	0
$353$	−9.06818	−0.482651	−0.241325	−	0.970444i	$-0.577582\pi$
$353$	−9.06818	−0.482651	−0.241325	+	0.970444i	$0.577582\pi$
$354$	0	0
$355$	−24.5838	−1.30477
$356$	0	0
$357$	17.6981	0.936681
$358$	0	0
$359$	33.6393	1.77541	0.887707	−	0.460409i	$-0.152297\pi$
$359$	33.6393	1.77541	0.887707	+	0.460409i	$0.152297\pi$
$360$	0	0
$361$	−15.3810	−0.809528
$362$	0	0
$363$	10.3863	0.545138
$364$	0	0
$365$	−30.5003	−1.59646
$366$	0	0
$367$	0.524318	0.0273692	0.0136846	−	0.999906i	$-0.495644\pi$
$367$	0.524318	0.0273692	0.0136846	+	0.999906i	$0.495644\pi$
$368$	0	0
$369$	1.19043	0.0619713
$370$	0	0
$371$	10.7479	0.558003
$372$	0	0
$373$	21.2711	1.10138	0.550688	−	0.834711i	$-0.314365\pi$
$373$	21.2711	1.10138	0.550688	+	0.834711i	$0.314365\pi$
$374$	0	0
$375$	10.6747	0.551238
$376$	0	0
$377$	13.2643	0.683145
$378$	0	0
$379$	15.1229	0.776813	0.388407	−	0.921488i	$-0.373026\pi$
$379$	15.1229	0.776813	0.388407	+	0.921488i	$0.373026\pi$
$380$	0	0
$381$	20.0283	1.02608
$382$	0	0
$383$	19.4932	0.996054	0.498027	−	0.867162i	$-0.334058\pi$
$383$	19.4932	0.996054	0.498027	+	0.867162i	$0.334058\pi$
$384$	0	0
$385$	13.7026	0.698349
$386$	0	0
$387$	−2.44687	−0.124381
$388$	0	0
$389$	−8.32263	−0.421974	−0.210987	−	0.977489i	$-0.567668\pi$
$389$	−8.32263	−0.421974	−0.210987	+	0.977489i	$0.567668\pi$
$390$	0	0
$391$	2.84814	0.144036
$392$	0	0
$393$	−13.2921	−0.670499
$394$	0	0
$395$	−36.5023	−1.83663
$396$	0	0
$397$	−2.62043	−0.131516	−0.0657578	−	0.997836i	$-0.520946\pi$
$397$	−2.62043	−0.131516	−0.0657578	+	0.997836i	$0.520946\pi$
$398$	0	0
$399$	−9.24121	−0.462639
$400$	0	0
$401$	−18.1038	−0.904061	−0.452030	−	0.892003i	$-0.649300\pi$
$401$	−18.1038	−0.904061	−0.452030	+	0.892003i	$0.649300\pi$
$402$	0	0
$403$	−12.3359	−0.614493
$404$	0	0
$405$	−3.60065	−0.178918
$406$	0	0
$407$	5.79200	0.287099
$408$	0	0
$409$	12.4295	0.614600	0.307300	−	0.951613i	$-0.400574\pi$
$409$	12.4295	0.614600	0.307300	+	0.951613i	$0.400574\pi$
$410$	0	0
$411$	−5.56619	−0.274560
$412$	0	0
$413$	17.0249	0.837740
$414$	0	0
$415$	59.7175	2.93142
$416$	0	0
$417$	14.5785	0.713910
$418$	0	0
$419$	5.91586	0.289009	0.144504	−	0.989504i	$-0.453841\pi$
$419$	5.91586	0.289009	0.144504	+	0.989504i	$0.453841\pi$
$420$	0	0
$421$	−12.0216	−0.585896	−0.292948	−	0.956128i	$-0.594636\pi$
$421$	−12.0216	−0.585896	−0.292948	+	0.956128i	$0.594636\pi$
$422$	0	0
$423$	0.0708292	0.00344383
$424$	0	0
$425$	−29.0173	−1.40755
$426$	0	0
$427$	73.0938	3.53726
$428$	0	0
$429$	−1.79308	−0.0865708
$430$	0	0
$431$	14.6512	0.705724	0.352862	−	0.935675i	$-0.385209\pi$
$431$	14.6512	0.705724	0.352862	+	0.935675i	$0.385209\pi$
$432$	0	0
$433$	19.9063	0.956634	0.478317	−	0.878187i	$-0.341247\pi$
$433$	19.9063	0.956634	0.478317	+	0.878187i	$0.341247\pi$
$434$	0	0
$435$	−20.8665	−1.00047
$436$	0	0
$437$	−1.48718	−0.0711415
$438$	0	0
$439$	30.8325	1.47156	0.735778	−	0.677223i	$-0.236817\pi$
$439$	30.8325	1.47156	0.735778	+	0.677223i	$0.236817\pi$
$440$	0	0
$441$	16.5979	0.790374
$442$	0	0
$443$	−30.0346	−1.42699	−0.713494	−	0.700661i	$-0.752889\pi$
$443$	−30.0346	−1.42699	−0.713494	+	0.700661i	$0.752889\pi$
$444$	0	0
$445$	−3.36086	−0.159320
$446$	0	0
$447$	0.839632	0.0397133
$448$	0	0
$449$	−5.30369	−0.250296	−0.125148	−	0.992138i	$-0.539941\pi$
$449$	−5.30369	−0.250296	−0.125148	+	0.992138i	$0.539941\pi$
$450$	0	0
$451$	−0.932588	−0.0439138
$452$	0	0
$453$	0.823210	0.0386778
$454$	0	0
$455$	40.0342	1.87683
$456$	0	0
$457$	19.6992	0.921491	0.460745	−	0.887532i	$-0.347582\pi$
$457$	19.6992	0.921491	0.460745	+	0.887532i	$0.347582\pi$
$458$	0	0
$459$	3.64326	0.170053
$460$	0	0
$461$	24.4557	1.13901	0.569507	−	0.821986i	$-0.307134\pi$
$461$	24.4557	1.13901	0.569507	+	0.821986i	$0.307134\pi$
$462$	0	0
$463$	−24.0218	−1.11639	−0.558194	−	0.829711i	$-0.688505\pi$
$463$	−24.0218	−1.11639	−0.558194	+	0.829711i	$0.688505\pi$
$464$	0	0
$465$	19.4060	0.899932
$466$	0	0
$467$	−32.6160	−1.50929	−0.754644	−	0.656134i	$-0.772191\pi$
$467$	−32.6160	−1.50929	−0.754644	+	0.656134i	$0.772191\pi$
$468$	0	0
$469$	−12.5545	−0.579713
$470$	0	0
$471$	−8.27681	−0.381375
$472$	0	0
$473$	1.91689	0.0881385
$474$	0	0
$475$	15.1516	0.695205
$476$	0	0
$477$	2.21252	0.101304
$478$	0	0
$479$	11.9826	0.547500	0.273750	−	0.961801i	$-0.411736\pi$
$479$	11.9826	0.547500	0.273750	+	0.961801i	$0.411736\pi$
$480$	0	0
$481$	16.9222	0.771585
$482$	0	0
$483$	3.79758	0.172796
$484$	0	0
$485$	57.4805	2.61006
$486$	0	0
$487$	−1.58389	−0.0717728	−0.0358864	−	0.999356i	$-0.511425\pi$
$487$	−1.58389	−0.0717728	−0.0358864	+	0.999356i	$0.511425\pi$
$488$	0	0
$489$	−3.21987	−0.145607
$490$	0	0
$491$	−4.01554	−0.181219	−0.0906095	−	0.995887i	$-0.528881\pi$
$491$	−4.01554	−0.181219	−0.0906095	+	0.995887i	$0.528881\pi$
$492$	0	0
$493$	21.1135	0.950903
$494$	0	0
$495$	2.82076	0.126784
$496$	0	0
$497$	33.1669	1.48774
$498$	0	0
$499$	10.1751	0.455501	0.227751	−	0.973720i	$-0.426863\pi$
$499$	10.1751	0.455501	0.227751	+	0.973720i	$0.426863\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	2.26034	0.100784	0.0503918	−	0.998730i	$-0.483953\pi$
$503$	2.26034	0.100784	0.0503918	+	0.998730i	$0.483953\pi$
$504$	0	0
$505$	24.0942	1.07218
$506$	0	0
$507$	7.76125	0.344689
$508$	0	0
$509$	32.1893	1.42677	0.713383	−	0.700774i	$-0.247162\pi$
$509$	32.1893	1.42677	0.713383	+	0.700774i	$0.247162\pi$
$510$	0	0
$511$	41.1490	1.82032
$512$	0	0
$513$	−1.90236	−0.0839912
$514$	0	0
$515$	56.5310	2.49106
$516$	0	0
$517$	−0.0554879	−0.00244036
$518$	0	0
$519$	−8.74339	−0.383792
$520$	0	0
$521$	27.2358	1.19322	0.596611	−	0.802530i	$-0.296513\pi$
$521$	27.2358	1.19322	0.596611	+	0.802530i	$0.296513\pi$
$522$	0	0
$523$	−0.145872	−0.00637853	−0.00318927	−	0.999995i	$-0.501015\pi$
$523$	−0.145872	−0.00637853	−0.00318927	+	0.999995i	$0.501015\pi$
$524$	0	0
$525$	−38.6904	−1.68859
$526$	0	0
$527$	−19.6356	−0.855342
$528$	0	0
$529$	−22.3889	−0.973429
$530$	0	0
$531$	3.50468	0.152090
$532$	0	0
$533$	−2.72469	−0.118019
$534$	0	0
$535$	4.65643	0.201315
$536$	0	0
$537$	−20.6471	−0.890987
$538$	0	0
$539$	−13.0028	−0.560072
$540$	0	0
$541$	−35.7512	−1.53707	−0.768533	−	0.639810i	$-0.779013\pi$
$541$	−35.7512	−1.53707	−0.768533	+	0.639810i	$0.779013\pi$
$542$	0	0
$543$	−23.3686	−1.00284
$544$	0	0
$545$	−0.0216050	−0.000925457 0
$546$	0	0
$547$	−10.4338	−0.446116	−0.223058	−	0.974805i	$-0.571604\pi$
$547$	−10.4338	−0.446116	−0.223058	+	0.974805i	$0.571604\pi$
$548$	0	0
$549$	15.0468	0.642182
$550$	0	0
$551$	−11.0246	−0.469664
$552$	0	0
$553$	49.2466	2.09418
$554$	0	0
$555$	−26.6209	−1.13000
$556$	0	0
$557$	−10.8646	−0.460350	−0.230175	−	0.973149i	$-0.573930\pi$
$557$	−10.8646	−0.460350	−0.230175	+	0.973149i	$0.573930\pi$
$558$	0	0
$559$	5.60046	0.236874
$560$	0	0
$561$	−2.85414	−0.120502
$562$	0	0
$563$	−24.8441	−1.04705	−0.523527	−	0.852009i	$-0.675384\pi$
$563$	−24.8441	−1.04705	−0.523527	+	0.852009i	$0.675384\pi$
$564$	0	0
$565$	9.97815	0.419784
$566$	0	0
$567$	4.85776	0.204007
$568$	0	0
$569$	6.52556	0.273566	0.136783	−	0.990601i	$-0.456324\pi$
$569$	6.52556	0.273566	0.136783	+	0.990601i	$0.456324\pi$
$570$	0	0
$571$	−28.6005	−1.19689	−0.598446	−	0.801163i	$-0.704215\pi$
$571$	−28.6005	−1.19689	−0.598446	+	0.801163i	$0.704215\pi$
$572$	0	0
$573$	18.3180	0.765245
$574$	0	0
$575$	−6.22642	−0.259659
$576$	0	0
$577$	−9.16872	−0.381699	−0.190849	−	0.981619i	$-0.561124\pi$
$577$	−9.16872	−0.381699	−0.190849	+	0.981619i	$0.561124\pi$
$578$	0	0
$579$	−8.36969	−0.347832
$580$	0	0
$581$	−80.5671	−3.34249
$582$	0	0
$583$	−1.73330	−0.0717859
$584$	0	0
$585$	8.24127	0.340735
$586$	0	0
$587$	−29.9963	−1.23808	−0.619040	−	0.785359i	$-0.712478\pi$
$587$	−29.9963	−1.23808	−0.619040	+	0.785359i	$0.712478\pi$
$588$	0	0
$589$	10.2529	0.422465
$590$	0	0
$591$	−12.1914	−0.501486
$592$	0	0
$593$	23.7026	0.973349	0.486675	−	0.873583i	$-0.338210\pi$
$593$	23.7026	0.973349	0.486675	+	0.873583i	$0.338210\pi$
$594$	0	0
$595$	63.7245	2.61245
$596$	0	0
$597$	3.39186	0.138820
$598$	0	0
$599$	35.0587	1.43246	0.716229	−	0.697865i	$-0.245867\pi$
$599$	35.0587	1.43246	0.716229	+	0.697865i	$0.245867\pi$
$600$	0	0
$601$	44.3852	1.81051	0.905254	−	0.424870i	$-0.139680\pi$
$601$	44.3852	1.81051	0.905254	+	0.424870i	$0.139680\pi$
$602$	0	0
$603$	−2.58442	−0.105246
$604$	0	0
$605$	37.3973	1.52042
$606$	0	0
$607$	20.8130	0.844773	0.422386	−	0.906416i	$-0.361193\pi$
$607$	20.8130	0.844773	0.422386	+	0.906416i	$0.361193\pi$
$608$	0	0
$609$	28.1518	1.14077
$610$	0	0
$611$	−0.162116	−0.00655851
$612$	0	0
$613$	−17.3968	−0.702650	−0.351325	−	0.936254i	$-0.614269\pi$
$613$	−17.3968	−0.702650	−0.351325	+	0.936254i	$0.614269\pi$
$614$	0	0
$615$	4.28631	0.172841
$616$	0	0
$617$	36.8295	1.48270	0.741349	−	0.671119i	$-0.234186\pi$
$617$	36.8295	1.48270	0.741349	+	0.671119i	$0.234186\pi$
$618$	0	0
$619$	−41.4504	−1.66603	−0.833016	−	0.553249i	$-0.813388\pi$
$619$	−41.4504	−1.66603	−0.833016	+	0.553249i	$0.813388\pi$
$620$	0	0
$621$	0.781756	0.0313708
$622$	0	0
$623$	4.53426	0.181661
$624$	0	0
$625$	−1.38751	−0.0555004
$626$	0	0
$627$	1.49032	0.0595176
$628$	0	0
$629$	26.9359	1.07401
$630$	0	0
$631$	34.3932	1.36917	0.684586	−	0.728932i	$-0.259983\pi$
$631$	34.3932	1.36917	0.684586	+	0.728932i	$0.259983\pi$
$632$	0	0
$633$	−19.3307	−0.768325
$634$	0	0
$635$	72.1148	2.86179
$636$	0	0
$637$	−37.9897	−1.50521
$638$	0	0
$639$	6.82762	0.270096
$640$	0	0
$641$	−50.1958	−1.98261	−0.991307	−	0.131567i	$-0.957999\pi$
$641$	−50.1958	−1.98261	−0.991307	+	0.131567i	$0.957999\pi$
$642$	0	0
$643$	−22.1392	−0.873083	−0.436541	−	0.899684i	$-0.643797\pi$
$643$	−22.1392	−0.873083	−0.436541	+	0.899684i	$0.643797\pi$
$644$	0	0
$645$	−8.81030	−0.346905
$646$	0	0
$647$	21.0276	0.826682	0.413341	−	0.910576i	$-0.364362\pi$
$647$	21.0276	0.826682	0.413341	+	0.910576i	$0.364362\pi$
$648$	0	0
$649$	−2.74558	−0.107773
$650$	0	0
$651$	−26.1813	−1.02613
$652$	0	0
$653$	32.1566	1.25838	0.629192	−	0.777250i	$-0.283386\pi$
$653$	32.1566	1.25838	0.629192	+	0.777250i	$0.283386\pi$
$654$	0	0
$655$	−47.8602	−1.87005
$656$	0	0
$657$	8.47077	0.330476
$658$	0	0
$659$	−17.7137	−0.690027	−0.345014	−	0.938598i	$-0.612126\pi$
$659$	−17.7137	−0.690027	−0.345014	+	0.938598i	$0.612126\pi$
$660$	0	0
$661$	−32.5254	−1.26509	−0.632545	−	0.774524i	$-0.717990\pi$
$661$	−32.5254	−1.26509	−0.632545	+	0.774524i	$0.717990\pi$
$662$	0	0
$663$	−8.33880	−0.323852
$664$	0	0
$665$	−33.2743	−1.29032
$666$	0	0
$667$	4.53045	0.175420
$668$	0	0
$669$	11.5418	0.446232
$670$	0	0
$671$	−11.7877	−0.455061
$672$	0	0
$673$	21.4777	0.827906	0.413953	−	0.910298i	$-0.364148\pi$
$673$	21.4777	0.827906	0.413953	+	0.910298i	$0.364148\pi$
$674$	0	0
$675$	−7.96466	−0.306560
$676$	0	0
$677$	7.37246	0.283347	0.141673	−	0.989913i	$-0.454752\pi$
$677$	7.37246	0.283347	0.141673	+	0.989913i	$0.454752\pi$
$678$	0	0
$679$	−77.5490	−2.97606
$680$	0	0
$681$	−12.1495	−0.465571
$682$	0	0
$683$	14.6019	0.558727	0.279363	−	0.960185i	$-0.409877\pi$
$683$	14.6019	0.558727	0.279363	+	0.960185i	$0.409877\pi$
$684$	0	0
$685$	−20.0419	−0.765761
$686$	0	0
$687$	−23.6217	−0.901224
$688$	0	0
$689$	−5.06408	−0.192926
$690$	0	0
$691$	−9.88343	−0.375983	−0.187992	−	0.982171i	$-0.560198\pi$
$691$	−9.88343	−0.375983	−0.187992	+	0.982171i	$0.560198\pi$
$692$	0	0
$693$	−3.80559	−0.144563
$694$	0	0
$695$	52.4919	1.99113
$696$	0	0
$697$	−4.33704	−0.164277
$698$	0	0
$699$	−7.08707	−0.268058
$700$	0	0
$701$	−12.0603	−0.455512	−0.227756	−	0.973718i	$-0.573139\pi$
$701$	−12.0603	−0.455512	−0.227756	+	0.973718i	$0.573139\pi$
$702$	0	0
$703$	−14.0649	−0.530466
$704$	0	0
$705$	0.255031	0.00960502
$706$	0	0
$707$	−32.5064	−1.22253
$708$	0	0
$709$	31.8075	1.19456	0.597279	−	0.802034i	$-0.296249\pi$
$709$	31.8075	1.19456	0.597279	+	0.802034i	$0.296249\pi$
$710$	0	0
$711$	10.1377	0.380194
$712$	0	0
$713$	−4.21334	−0.157791
$714$	0	0
$715$	−6.45625	−0.241450
$716$	0	0
$717$	−30.5104	−1.13943
$718$	0	0
$719$	27.9494	1.04234	0.521168	−	0.853454i	$-0.325496\pi$
$719$	27.9494	1.04234	0.521168	+	0.853454i	$0.325496\pi$
$720$	0	0
$721$	−76.2681	−2.84037
$722$	0	0
$723$	−10.7672	−0.400438
$724$	0	0
$725$	−46.1570	−1.71423
$726$	0	0
$727$	20.6044	0.764173	0.382087	−	0.924127i	$-0.375206\pi$
$727$	20.6044	0.764173	0.382087	+	0.924127i	$0.375206\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	8.91456	0.329717
$732$	0	0
$733$	23.2340	0.858168	0.429084	−	0.903265i	$-0.358836\pi$
$733$	23.2340	0.858168	0.429084	+	0.903265i	$0.358836\pi$
$734$	0	0
$735$	59.7630	2.20439
$736$	0	0
$737$	2.02465	0.0745789
$738$	0	0
$739$	12.2336	0.450022	0.225011	−	0.974356i	$-0.427758\pi$
$739$	12.2336	0.450022	0.225011	+	0.974356i	$0.427758\pi$
$740$	0	0
$741$	4.35418	0.159955
$742$	0	0
$743$	2.31177	0.0848105	0.0424053	−	0.999100i	$-0.486498\pi$
$743$	2.31177	0.0848105	0.0424053	+	0.999100i	$0.486498\pi$
$744$	0	0
$745$	3.02322	0.110762
$746$	0	0
$747$	−16.5852	−0.606822
$748$	0	0
$749$	−6.28216	−0.229545
$750$	0	0
$751$	−3.19637	−0.116637	−0.0583185	−	0.998298i	$-0.518574\pi$
$751$	−3.19637	−0.116637	−0.0583185	+	0.998298i	$0.518574\pi$
$752$	0	0
$753$	−16.4758	−0.600412
$754$	0	0
$755$	2.96409	0.107874
$756$	0	0
$757$	39.6654	1.44167	0.720833	−	0.693109i	$-0.243760\pi$
$757$	39.6654	1.44167	0.720833	+	0.693109i	$0.243760\pi$
$758$	0	0
$759$	−0.612431	−0.0222298
$760$	0	0
$761$	−28.6313	−1.03788	−0.518941	−	0.854810i	$-0.673674\pi$
$761$	−28.6313	−1.03788	−0.518941	+	0.854810i	$0.673674\pi$
$762$	0	0
$763$	0.0291481	0.00105523
$764$	0	0
$765$	13.1181	0.474285
$766$	0	0
$767$	−8.02161	−0.289644
$768$	0	0
$769$	−13.9728	−0.503872	−0.251936	−	0.967744i	$-0.581067\pi$
$769$	−13.9728	−0.503872	−0.251936	+	0.967744i	$0.581067\pi$
$770$	0	0
$771$	14.1522	0.509679
$772$	0	0
$773$	49.5495	1.78217	0.891086	−	0.453835i	$-0.149944\pi$
$773$	49.5495	1.78217	0.891086	+	0.453835i	$0.149944\pi$
$774$	0	0
$775$	42.9262	1.54196
$776$	0	0
$777$	35.9153	1.28845
$778$	0	0
$779$	2.26462	0.0811386
$780$	0	0
$781$	−5.34879	−0.191395
$782$	0	0
$783$	5.79522	0.207104
$784$	0	0
$785$	−29.8019	−1.06367
$786$	0	0
$787$	55.1125	1.96455	0.982275	−	0.187448i	$-0.0600216\pi$
$787$	55.1125	1.96455	0.982275	+	0.187448i	$0.0600216\pi$
$788$	0	0
$789$	−1.37436	−0.0489286
$790$	0	0
$791$	−13.4619	−0.478650
$792$	0	0
$793$	−34.4396	−1.22299
$794$	0	0
$795$	7.96650	0.282543
$796$	0	0
$797$	47.0379	1.66617	0.833083	−	0.553148i	$-0.186573\pi$
$797$	47.0379	1.66617	0.833083	+	0.553148i	$0.186573\pi$
$798$	0	0
$799$	−0.258049	−0.00912911
$800$	0	0
$801$	0.933406	0.0329803
$802$	0	0
$803$	−6.63604	−0.234181
$804$	0	0
$805$	13.6738	0.481937
$806$	0	0
$807$	−1.22578	−0.0431495
$808$	0	0
$809$	36.0404	1.26711	0.633557	−	0.773696i	$-0.281594\pi$
$809$	36.0404	1.26711	0.633557	+	0.773696i	$0.281594\pi$
$810$	0	0
$811$	−9.87847	−0.346880	−0.173440	−	0.984844i	$-0.555488\pi$
$811$	−9.87847	−0.346880	−0.173440	+	0.984844i	$0.555488\pi$
$812$	0	0
$813$	19.2991	0.676849
$814$	0	0
$815$	−11.5936	−0.406106
$816$	0	0
$817$	−4.65482	−0.162851
$818$	0	0
$819$	−11.1186	−0.388515
$820$	0	0
$821$	33.2827	1.16158	0.580788	−	0.814055i	$-0.302745\pi$
$821$	33.2827	1.16158	0.580788	+	0.814055i	$0.302745\pi$
$822$	0	0
$823$	−50.0524	−1.74472	−0.872358	−	0.488867i	$-0.837410\pi$
$823$	−50.0524	−1.74472	−0.872358	+	0.488867i	$0.837410\pi$
$824$	0	0
$825$	6.23955	0.217233
$826$	0	0
$827$	40.7439	1.41680	0.708402	−	0.705809i	$-0.249416\pi$
$827$	40.7439	1.41680	0.708402	+	0.705809i	$0.249416\pi$
$828$	0	0
$829$	32.5331	1.12992	0.564961	−	0.825118i	$-0.308891\pi$
$829$	32.5331	1.12992	0.564961	+	0.825118i	$0.308891\pi$
$830$	0	0
$831$	−13.5577	−0.470310
$832$	0	0
$833$	−60.4702	−2.09517
$834$	0	0
$835$	−3.60065	−0.124606
$836$	0	0
$837$	−5.38959	−0.186291
$838$	0	0
$839$	−14.9288	−0.515401	−0.257700	−	0.966225i	$-0.582965\pi$
$839$	−14.9288	−0.515401	−0.257700	+	0.966225i	$0.582965\pi$
$840$	0	0
$841$	4.58460	0.158090
$842$	0	0
$843$	−16.5306	−0.569345
$844$	0	0
$845$	27.9455	0.961355
$846$	0	0
$847$	−50.4541	−1.73362
$848$	0	0
$849$	−21.1153	−0.724675
$850$	0	0
$851$	5.77981	0.198129
$852$	0	0
$853$	44.7233	1.53130	0.765648	−	0.643260i	$-0.222419\pi$
$853$	44.7233	1.53130	0.765648	+	0.643260i	$0.222419\pi$
$854$	0	0
$855$	−6.84973	−0.234256
$856$	0	0
$857$	−29.1460	−0.995608	−0.497804	−	0.867289i	$-0.665860\pi$
$857$	−29.1460	−0.995608	−0.497804	+	0.867289i	$0.665860\pi$
$858$	0	0
$859$	−16.0575	−0.547873	−0.273937	−	0.961748i	$-0.588326\pi$
$859$	−16.0575	−0.547873	−0.273937	+	0.961748i	$0.588326\pi$
$860$	0	0
$861$	−5.78282	−0.197078
$862$	0	0
$863$	37.7571	1.28527	0.642634	−	0.766174i	$-0.277842\pi$
$863$	37.7571	1.28527	0.642634	+	0.766174i	$0.277842\pi$
$864$	0	0
$865$	−31.4819	−1.07042
$866$	0	0
$867$	3.72669	0.126565
$868$	0	0
$869$	−7.94194	−0.269412
$870$	0	0
$871$	5.91531	0.200433
$872$	0	0
$873$	−15.9639	−0.540298
$874$	0	0
$875$	−51.8551	−1.75302
$876$	0	0
$877$	4.14356	0.139918	0.0699590	−	0.997550i	$-0.477713\pi$
$877$	4.14356	0.139918	0.0699590	+	0.997550i	$0.477713\pi$
$878$	0	0
$879$	14.3562	0.484223
$880$	0	0
$881$	−15.5386	−0.523507	−0.261754	−	0.965135i	$-0.584301\pi$
$881$	−15.5386	−0.523507	−0.261754	+	0.965135i	$0.584301\pi$
$882$	0	0
$883$	39.8472	1.34097	0.670483	−	0.741925i	$-0.266087\pi$
$883$	39.8472	1.34097	0.670483	+	0.741925i	$0.266087\pi$
$884$	0	0
$885$	12.6191	0.424186
$886$	0	0
$887$	11.3968	0.382666	0.191333	−	0.981525i	$-0.438719\pi$
$887$	11.3968	0.382666	0.191333	+	0.981525i	$0.438719\pi$
$888$	0	0
$889$	−97.2927	−3.26309
$890$	0	0
$891$	−0.783405	−0.0262450
$892$	0	0
$893$	0.134743	0.00450899
$894$	0	0
$895$	−74.3428	−2.48501
$896$	0	0
$897$	−1.78931	−0.0597432
$898$	0	0
$899$	−31.2339	−1.04171
$900$	0	0
$901$	−8.06077	−0.268543
$902$	0	0
$903$	11.8863	0.395551
$904$	0	0
$905$	−84.1420	−2.79698
$906$	0	0
$907$	−33.3058	−1.10590	−0.552951	−	0.833214i	$-0.686498\pi$
$907$	−33.3058	−1.10590	−0.552951	+	0.833214i	$0.686498\pi$
$908$	0	0
$909$	−6.69164	−0.221948
$910$	0	0
$911$	33.9755	1.12566	0.562830	−	0.826573i	$-0.309713\pi$
$911$	33.9755	1.12566	0.562830	+	0.826573i	$0.309713\pi$
$912$	0	0
$913$	12.9929	0.430004
$914$	0	0
$915$	54.1782	1.79108
$916$	0	0
$917$	64.5699	2.13229
$918$	0	0
$919$	−23.2442	−0.766756	−0.383378	−	0.923592i	$-0.625239\pi$
$919$	−23.2442	−0.766756	−0.383378	+	0.923592i	$0.625239\pi$
$920$	0	0
$921$	−23.3658	−0.769928
$922$	0	0
$923$	−15.6273	−0.514378
$924$	0	0
$925$	−58.8857	−1.93615
$926$	0	0
$927$	−15.7002	−0.515664
$928$	0	0
$929$	9.87584	0.324016	0.162008	−	0.986789i	$-0.448203\pi$
$929$	9.87584	0.324016	0.162008	+	0.986789i	$0.448203\pi$
$930$	0	0
$931$	31.5751	1.03483
$932$	0	0
$933$	−12.7290	−0.416728
$934$	0	0
$935$	−10.2768	−0.336086
$936$	0	0
$937$	22.8528	0.746570	0.373285	−	0.927717i	$-0.378231\pi$
$937$	22.8528	0.746570	0.373285	+	0.927717i	$0.378231\pi$
$938$	0	0
$939$	−9.72833	−0.317472
$940$	0	0
$941$	−44.8358	−1.46161	−0.730803	−	0.682589i	$-0.760854\pi$
$941$	−44.8358	−1.46161	−0.730803	+	0.682589i	$0.760854\pi$
$942$	0	0
$943$	−0.930624	−0.0303053
$944$	0	0
$945$	17.4911	0.568985
$946$	0	0
$947$	23.2347	0.755027	0.377513	−	0.926004i	$-0.376779\pi$
$947$	23.2347	0.755027	0.377513	+	0.926004i	$0.376779\pi$
$948$	0	0
$949$	−19.3882	−0.629367
$950$	0	0
$951$	14.6156	0.473942
$952$	0	0
$953$	13.1951	0.427431	0.213715	−	0.976896i	$-0.431443\pi$
$953$	13.1951	0.427431	0.213715	+	0.976896i	$0.431443\pi$
$954$	0	0
$955$	65.9566	2.13430
$956$	0	0
$957$	−4.54000	−0.146757
$958$	0	0
$959$	27.0392	0.873142
$960$	0	0
$961$	−1.95233	−0.0629785
$962$	0	0
$963$	−1.29322	−0.0416735
$964$	0	0
$965$	−30.1363	−0.970122
$966$	0	0
$967$	18.7083	0.601618	0.300809	−	0.953684i	$-0.402743\pi$
$967$	18.7083	0.601618	0.300809	+	0.953684i	$0.402743\pi$
$968$	0	0
$969$	6.93078	0.222649
$970$	0	0
$971$	8.51320	0.273202	0.136601	−	0.990626i	$-0.456382\pi$
$971$	8.51320	0.273202	0.136601	+	0.990626i	$0.456382\pi$
$972$	0	0
$973$	−70.8187	−2.27034
$974$	0	0
$975$	18.2298	0.583820
$976$	0	0
$977$	7.05179	0.225607	0.112803	−	0.993617i	$-0.464017\pi$
$977$	7.05179	0.225607	0.112803	+	0.993617i	$0.464017\pi$
$978$	0	0
$979$	−0.731235	−0.0233704
$980$	0	0
$981$	0.00600031	0.000191575 0
$982$	0	0
$983$	10.4440	0.333112	0.166556	−	0.986032i	$-0.446735\pi$
$983$	10.4440	0.333112	0.166556	+	0.986032i	$0.446735\pi$
$984$	0	0
$985$	−43.8968	−1.39867
$986$	0	0
$987$	−0.344071	−0.0109519
$988$	0	0
$989$	1.91285	0.0608251
$990$	0	0
$991$	31.4692	0.999654	0.499827	−	0.866125i	$-0.333397\pi$
$991$	31.4692	0.999654	0.499827	+	0.866125i	$0.333397\pi$
$992$	0	0
$993$	7.54681	0.239491
$994$	0	0
$995$	12.2129	0.387175
$996$	0	0
$997$	28.6281	0.906661	0.453331	−	0.891342i	$-0.350236\pi$
$997$	28.6281	0.906661	0.453331	+	0.891342i	$0.350236\pi$
$998$	0	0
$999$	7.39337	0.233916

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.bg.1.2		13
4.3	odd	2		4008.2.a.m.1.2	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.m.1.2	✓	13	4.3	odd	2
8016.2.a.bg.1.2		13	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.60065 q^{5} +4.85776 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -3.60065 q^{5} +4.85776 q^{7} +1.00000 q^{9} -0.783405 q^{11} -2.28883 q^{13} +3.60065 q^{15} -3.64326 q^{17} +1.90236 q^{19} -4.85776 q^{21} -0.781756 q^{23} +7.96466 q^{25} -1.00000 q^{27} -5.79522 q^{29} +5.38959 q^{31} +0.783405 q^{33} -17.4911 q^{35} -7.39337 q^{37} +2.28883 q^{39} +1.19043 q^{41} -2.44687 q^{43} -3.60065 q^{45} +0.0708292 q^{47} +16.5979 q^{49} +3.64326 q^{51} +2.21252 q^{53} +2.82076 q^{55} -1.90236 q^{57} +3.50468 q^{59} +15.0468 q^{61} +4.85776 q^{63} +8.24127 q^{65} -2.58442 q^{67} +0.781756 q^{69} +6.82762 q^{71} +8.47077 q^{73} -7.96466 q^{75} -3.80559 q^{77} +10.1377 q^{79} +1.00000 q^{81} -16.5852 q^{83} +13.1181 q^{85} +5.79522 q^{87} +0.933406 q^{89} -11.1186 q^{91} -5.38959 q^{93} -6.84973 q^{95} -15.9639 q^{97} -0.783405 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 13 q^{3} + 2 q^{5} - q^{7} + 13 q^{9}+O(q^{10}) \) 13 * q - 13 * q^3 + 2 * q^5 - q^7 + 13 * q^9	\( 13 q - 13 q^{3} + 2 q^{5} - q^{7} + 13 q^{9} - 11 q^{11} + 12 q^{13} - 2 q^{15} + 15 q^{17} - 14 q^{19} + q^{21} - 9 q^{23} + 37 q^{25} - 13 q^{27} - 3 q^{29} + 17 q^{31} + 11 q^{33} - 15 q^{35} + 16 q^{37} - 12 q^{39} + 12 q^{41} - 20 q^{43} + 2 q^{45} + 6 q^{47} + 26 q^{49} - 15 q^{51} - 12 q^{53} - 7 q^{55} + 14 q^{57} - 14 q^{59} + 24 q^{61} - q^{63} + 8 q^{65} - 3 q^{67} + 9 q^{69} - 17 q^{71} + 34 q^{73} - 37 q^{75} + 30 q^{77} - 10 q^{79} + 13 q^{81} - 44 q^{83} + 25 q^{85} + 3 q^{87} + 25 q^{89} - 29 q^{91} - 17 q^{93} + 15 q^{95} + 38 q^{97} - 11 q^{99}+O(q^{100}) \) 13 * q - 13 * q^3 + 2 * q^5 - q^7 + 13 * q^9 - 11 * q^11 + 12 * q^13 - 2 * q^15 + 15 * q^17 - 14 * q^19 + q^21 - 9 * q^23 + 37 * q^25 - 13 * q^27 - 3 * q^29 + 17 * q^31 + 11 * q^33 - 15 * q^35 + 16 * q^37 - 12 * q^39 + 12 * q^41 - 20 * q^43 + 2 * q^45 + 6 * q^47 + 26 * q^49 - 15 * q^51 - 12 * q^53 - 7 * q^55 + 14 * q^57 - 14 * q^59 + 24 * q^61 - q^63 + 8 * q^65 - 3 * q^67 + 9 * q^69 - 17 * q^71 + 34 * q^73 - 37 * q^75 + 30 * q^77 - 10 * q^79 + 13 * q^81 - 44 * q^83 + 25 * q^85 + 3 * q^87 + 25 * q^89 - 29 * q^91 - 17 * q^93 + 15 * q^95 + 38 * q^97 - 11 * q^99

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