LMFDB - Embedded newform 5776.2.a.ce.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5776,2,Mod(1,5776)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5776, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5776.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5776 = 2^{4} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5776.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:	$46.1215922075$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 12x^{7} + 35x^{6} + 45x^{5} - 117x^{4} - 55x^{3} + 96x^{2} - 6x - 1 $ x^9 - 3x^8 - 12x^7 + 35x^6 + 45x^5 - 117x^4 - 55x^3 + 96x^2 - 6x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 152)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.28732$ of defining polynomial
Character	$\chi$	$=$	5776.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.28732 q^{3} +4.24329 q^{5} -3.37236 q^{7} -1.34280 q^{9} +O(q^{10})$	$q-1.28732 q^{3} +4.24329 q^{5} -3.37236 q^{7} -1.34280 q^{9} +0.495679 q^{11} +3.53330 q^{13} -5.46249 q^{15} -3.96780 q^{17} +4.34132 q^{21} -0.943313 q^{23} +13.0055 q^{25} +5.59059 q^{27} +2.66861 q^{29} +6.83360 q^{31} -0.638099 q^{33} -14.3099 q^{35} -6.73600 q^{37} -4.54851 q^{39} +2.42268 q^{41} -11.9958 q^{43} -5.69788 q^{45} -1.11794 q^{47} +4.37282 q^{49} +5.10784 q^{51} -1.01214 q^{53} +2.10331 q^{55} +6.59206 q^{59} +9.46882 q^{61} +4.52840 q^{63} +14.9928 q^{65} -10.6537 q^{67} +1.21435 q^{69} +9.42090 q^{71} +3.25475 q^{73} -16.7423 q^{75} -1.67161 q^{77} +5.75898 q^{79} -3.16851 q^{81} +7.78171 q^{83} -16.8365 q^{85} -3.43537 q^{87} -2.68649 q^{89} -11.9156 q^{91} -8.79705 q^{93} -1.43358 q^{97} -0.665596 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 3 q^{3} + 3 q^{5} - 9 q^{7} + 6 q^{9}+O(q^{10}) $ 9 * q + 3 * q^3 + 3 * q^5 - 9 * q^7 + 6 * q^9	$ 9 q + 3 q^{3} + 3 q^{5} - 9 q^{7} + 6 q^{9} - 3 q^{11} + 6 q^{13} - 3 q^{17} - 15 q^{21} - 24 q^{23} + 30 q^{25} + 12 q^{27} + 15 q^{29} + 6 q^{31} + 18 q^{33} - 15 q^{35} - 24 q^{37} - 6 q^{39} + 12 q^{41} - 9 q^{43} + 42 q^{45} + 12 q^{47} + 18 q^{49} + 12 q^{51} + 18 q^{53} - 21 q^{55} + 57 q^{59} + 15 q^{61} - 30 q^{63} + 27 q^{65} + 6 q^{67} - 3 q^{69} + 36 q^{73} + 45 q^{75} + 30 q^{77} - 3 q^{79} - 15 q^{81} - 27 q^{83} + 18 q^{85} + 18 q^{87} + 30 q^{89} - 21 q^{91} - 24 q^{93} - 3 q^{97} - 48 q^{99}+O(q^{100}) $ 9 * q + 3 * q^3 + 3 * q^5 - 9 * q^7 + 6 * q^9 - 3 * q^11 + 6 * q^13 - 3 * q^17 - 15 * q^21 - 24 * q^23 + 30 * q^25 + 12 * q^27 + 15 * q^29 + 6 * q^31 + 18 * q^33 - 15 * q^35 - 24 * q^37 - 6 * q^39 + 12 * q^41 - 9 * q^43 + 42 * q^45 + 12 * q^47 + 18 * q^49 + 12 * q^51 + 18 * q^53 - 21 * q^55 + 57 * q^59 + 15 * q^61 - 30 * q^63 + 27 * q^65 + 6 * q^67 - 3 * q^69 + 36 * q^73 + 45 * q^75 + 30 * q^77 - 3 * q^79 - 15 * q^81 - 27 * q^83 + 18 * q^85 + 18 * q^87 + 30 * q^89 - 21 * q^91 - 24 * q^93 - 3 * q^97 - 48 * 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
$3$	−1.28732	−0.743237	−0.371618	−	0.928386i	$-0.621197\pi$
$3$	−1.28732	−0.743237	−0.371618	+	0.928386i	$0.621197\pi$
$4$	0	0
$5$	4.24329	1.89766	0.948829	−	0.315792i	$-0.102270\pi$
$5$	4.24329	1.89766	0.948829	+	0.315792i	$0.102270\pi$
$6$	0	0
$7$	−3.37236	−1.27463	−0.637316	−	0.770602i	$-0.719956\pi$
$7$	−3.37236	−1.27463	−0.637316	+	0.770602i	$0.719956\pi$
$8$	0	0
$9$	−1.34280	−0.447599
$10$	0	0
$11$	0.495679	0.149453	0.0747264	−	0.997204i	$-0.476192\pi$
$11$	0.495679	0.149453	0.0747264	+	0.997204i	$0.476192\pi$
$12$	0	0
$13$	3.53330	0.979962	0.489981	−	0.871733i	$-0.337004\pi$
$13$	3.53330	0.979962	0.489981	+	0.871733i	$0.337004\pi$
$14$	0	0
$15$	−5.46249	−1.41041
$16$	0	0
$17$	−3.96780	−0.962332	−0.481166	−	0.876630i	$-0.659787\pi$
$17$	−3.96780	−0.962332	−0.481166	+	0.876630i	$0.659787\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	4.34132	0.947354
$22$	0	0
$23$	−0.943313	−0.196694	−0.0983472	−	0.995152i	$-0.531356\pi$
$23$	−0.943313	−0.196694	−0.0983472	+	0.995152i	$0.531356\pi$
$24$	0	0
$25$	13.0055	2.60110
$26$	0	0
$27$	5.59059	1.07591
$28$	0	0
$29$	2.66861	0.495549	0.247774	−	0.968818i	$-0.420301\pi$
$29$	2.66861	0.495549	0.247774	+	0.968818i	$0.420301\pi$
$30$	0	0
$31$	6.83360	1.22735	0.613675	−	0.789559i	$-0.289690\pi$
$31$	6.83360	1.22735	0.613675	+	0.789559i	$0.289690\pi$
$32$	0	0
$33$	−0.638099	−0.111079
$34$	0	0
$35$	−14.3099	−2.41882
$36$	0	0
$37$	−6.73600	−1.10739	−0.553696	−	0.832719i	$-0.686783\pi$
$37$	−6.73600	−1.10739	−0.553696	+	0.832719i	$0.686783\pi$
$38$	0	0
$39$	−4.54851	−0.728344
$40$	0	0
$41$	2.42268	0.378359	0.189180	−	0.981943i	$-0.439417\pi$
$41$	2.42268	0.378359	0.189180	+	0.981943i	$0.439417\pi$
$42$	0	0
$43$	−11.9958	−1.82935	−0.914674	−	0.404192i	$-0.867553\pi$
$43$	−11.9958	−1.82935	−0.914674	+	0.404192i	$0.867553\pi$
$44$	0	0
$45$	−5.69788	−0.849389
$46$	0	0
$47$	−1.11794	−0.163069	−0.0815344	−	0.996671i	$-0.525982\pi$
$47$	−1.11794	−0.163069	−0.0815344	+	0.996671i	$0.525982\pi$
$48$	0	0
$49$	4.37282	0.624689
$50$	0	0
$51$	5.10784	0.715240
$52$	0	0
$53$	−1.01214	−0.139028	−0.0695141	−	0.997581i	$-0.522145\pi$
$53$	−1.01214	−0.139028	−0.0695141	+	0.997581i	$0.522145\pi$
$54$	0	0
$55$	2.10331	0.283610
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.59206	0.858213	0.429107	−	0.903254i	$-0.358828\pi$
$59$	6.59206	0.858213	0.429107	+	0.903254i	$0.358828\pi$
$60$	0	0
$61$	9.46882	1.21236	0.606179	−	0.795328i	$-0.292701\pi$
$61$	9.46882	1.21236	0.606179	+	0.795328i	$0.292701\pi$
$62$	0	0
$63$	4.52840	0.570524
$64$	0	0
$65$	14.9928	1.85963
$66$	0	0
$67$	−10.6537	−1.30155	−0.650777	−	0.759269i	$-0.725557\pi$
$67$	−10.6537	−1.30155	−0.650777	+	0.759269i	$0.725557\pi$
$68$	0	0
$69$	1.21435	0.146191
$70$	0	0
$71$	9.42090	1.11805	0.559027	−	0.829149i	$-0.311175\pi$
$71$	9.42090	1.11805	0.559027	+	0.829149i	$0.311175\pi$
$72$	0	0
$73$	3.25475	0.380940	0.190470	−	0.981693i	$-0.438999\pi$
$73$	3.25475	0.380940	0.190470	+	0.981693i	$0.438999\pi$
$74$	0	0
$75$	−16.7423	−1.93324
$76$	0	0
$77$	−1.67161	−0.190497
$78$	0	0
$79$	5.75898	0.647935	0.323968	−	0.946068i	$-0.394983\pi$
$79$	5.75898	0.647935	0.323968	+	0.946068i	$0.394983\pi$
$80$	0	0
$81$	−3.16851	−0.352056
$82$	0	0
$83$	7.78171	0.854153	0.427077	−	0.904215i	$-0.359543\pi$
$83$	7.78171	0.854153	0.427077	+	0.904215i	$0.359543\pi$
$84$	0	0
$85$	−16.8365	−1.82618
$86$	0	0
$87$	−3.43537	−0.368310
$88$	0	0
$89$	−2.68649	−0.284767	−0.142384	−	0.989812i	$-0.545477\pi$
$89$	−2.68649	−0.284767	−0.142384	+	0.989812i	$0.545477\pi$
$90$	0	0
$91$	−11.9156	−1.24909
$92$	0	0
$93$	−8.79705	−0.912212
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−1.43358	−0.145558	−0.0727792	−	0.997348i	$-0.523187\pi$
$97$	−1.43358	−0.145558	−0.0727792	+	0.997348i	$0.523187\pi$
$98$	0	0
$99$	−0.665596	−0.0668949
$100$	0	0
$101$	−3.72539	−0.370691	−0.185345	−	0.982673i	$-0.559340\pi$
$101$	−3.72539	−0.370691	−0.185345	+	0.982673i	$0.559340\pi$
$102$	0	0
$103$	4.56384	0.449688	0.224844	−	0.974395i	$-0.427813\pi$
$103$	4.56384	0.449688	0.224844	+	0.974395i	$0.427813\pi$
$104$	0	0
$105$	18.4215	1.79775
$106$	0	0
$107$	17.2888	1.67137	0.835684	−	0.549211i	$-0.185072\pi$
$107$	17.2888	1.67137	0.835684	+	0.549211i	$0.185072\pi$
$108$	0	0
$109$	8.56098	0.819993	0.409997	−	0.912087i	$-0.365530\pi$
$109$	8.56098	0.819993	0.409997	+	0.912087i	$0.365530\pi$
$110$	0	0
$111$	8.67142	0.823055
$112$	0	0
$113$	12.4068	1.16713	0.583566	−	0.812066i	$-0.301657\pi$
$113$	12.4068	1.16713	0.583566	+	0.812066i	$0.301657\pi$
$114$	0	0
$115$	−4.00275	−0.373259
$116$	0	0
$117$	−4.74451	−0.438630
$118$	0	0
$119$	13.3808	1.22662
$120$	0	0
$121$	−10.7543	−0.977664
$122$	0	0
$123$	−3.11877	−0.281210
$124$	0	0
$125$	33.9697	3.03834
$126$	0	0
$127$	11.2175	0.995395	0.497698	−	0.867351i	$-0.334179\pi$
$127$	11.2175	0.995395	0.497698	+	0.867351i	$0.334179\pi$
$128$	0	0
$129$	15.4425	1.35964
$130$	0	0
$131$	−5.25582	−0.459203	−0.229602	−	0.973285i	$-0.573742\pi$
$131$	−5.25582	−0.459203	−0.229602	+	0.973285i	$0.573742\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	23.7225	2.04171
$136$	0	0
$137$	−14.3613	−1.22697	−0.613486	−	0.789706i	$-0.710233\pi$
$137$	−14.3613	−1.22697	−0.613486	+	0.789706i	$0.710233\pi$
$138$	0	0
$139$	3.71164	0.314817	0.157409	−	0.987534i	$-0.449686\pi$
$139$	3.71164	0.314817	0.157409	+	0.987534i	$0.449686\pi$
$140$	0	0
$141$	1.43916	0.121199
$142$	0	0
$143$	1.75138	0.146458
$144$	0	0
$145$	11.3237	0.940381
$146$	0	0
$147$	−5.62924	−0.464292
$148$	0	0
$149$	7.82749	0.641253	0.320626	−	0.947206i	$-0.396107\pi$
$149$	7.82749	0.641253	0.320626	+	0.947206i	$0.396107\pi$
$150$	0	0
$151$	21.1437	1.72065	0.860326	−	0.509743i	$-0.170260\pi$
$151$	21.1437	1.72065	0.860326	+	0.509743i	$0.170260\pi$
$152$	0	0
$153$	5.32794	0.430739
$154$	0	0
$155$	28.9969	2.32909
$156$	0	0
$157$	−3.72651	−0.297408	−0.148704	−	0.988882i	$-0.547510\pi$
$157$	−3.72651	−0.297408	−0.148704	+	0.988882i	$0.547510\pi$
$158$	0	0
$159$	1.30295	0.103331
$160$	0	0
$161$	3.18119	0.250713
$162$	0	0
$163$	5.45975	0.427640	0.213820	−	0.976873i	$-0.431409\pi$
$163$	5.45975	0.427640	0.213820	+	0.976873i	$0.431409\pi$
$164$	0	0
$165$	−2.70764	−0.210789
$166$	0	0
$167$	−6.10192	−0.472181	−0.236090	−	0.971731i	$-0.575866\pi$
$167$	−6.10192	−0.472181	−0.236090	+	0.971731i	$0.575866\pi$
$168$	0	0
$169$	−0.515771	−0.0396747
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−22.0554	−1.67684	−0.838421	−	0.545024i	$-0.816521\pi$
$173$	−22.0554	−1.67684	−0.838421	+	0.545024i	$0.816521\pi$
$174$	0	0
$175$	−43.8593	−3.31545
$176$	0	0
$177$	−8.48612	−0.637856
$178$	0	0
$179$	−12.6008	−0.941828	−0.470914	−	0.882179i	$-0.656076\pi$
$179$	−12.6008	−0.941828	−0.470914	+	0.882179i	$0.656076\pi$
$180$	0	0
$181$	−1.34971	−0.100323	−0.0501617	−	0.998741i	$-0.515974\pi$
$181$	−1.34971	−0.100323	−0.0501617	+	0.998741i	$0.515974\pi$
$182$	0	0
$183$	−12.1894	−0.901070
$184$	0	0
$185$	−28.5828	−2.10145
$186$	0	0
$187$	−1.96675	−0.143823
$188$	0	0
$189$	−18.8535	−1.37139
$190$	0	0
$191$	21.6982	1.57003	0.785014	−	0.619477i	$-0.212655\pi$
$191$	21.6982	1.57003	0.785014	+	0.619477i	$0.212655\pi$
$192$	0	0
$193$	25.2265	1.81584	0.907920	−	0.419143i	$-0.137669\pi$
$193$	25.2265	1.81584	0.907920	+	0.419143i	$0.137669\pi$
$194$	0	0
$195$	−19.3006	−1.38215
$196$	0	0
$197$	11.3238	0.806791	0.403395	−	0.915026i	$-0.367830\pi$
$197$	11.3238	0.806791	0.403395	+	0.915026i	$0.367830\pi$
$198$	0	0
$199$	−3.59905	−0.255130	−0.127565	−	0.991830i	$-0.540716\pi$
$199$	−3.59905	−0.255130	−0.127565	+	0.991830i	$0.540716\pi$
$200$	0	0
$201$	13.7147	0.967364
$202$	0	0
$203$	−8.99952	−0.631642
$204$	0	0
$205$	10.2801	0.717996
$206$	0	0
$207$	1.26668	0.0880402
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−16.6253	−1.14454	−0.572268	−	0.820067i	$-0.693936\pi$
$211$	−16.6253	−1.14454	−0.572268	+	0.820067i	$0.693936\pi$
$212$	0	0
$213$	−12.1277	−0.830979
$214$	0	0
$215$	−50.9018	−3.47148
$216$	0	0
$217$	−23.0454	−1.56442
$218$	0	0
$219$	−4.18992	−0.283129
$220$	0	0
$221$	−14.0194	−0.943048
$222$	0	0
$223$	10.8170	0.724357	0.362179	−	0.932109i	$-0.382033\pi$
$223$	10.8170	0.724357	0.362179	+	0.932109i	$0.382033\pi$
$224$	0	0
$225$	−17.4638	−1.16425
$226$	0	0
$227$	−9.91627	−0.658166	−0.329083	−	0.944301i	$-0.606740\pi$
$227$	−9.91627	−0.658166	−0.329083	+	0.944301i	$0.606740\pi$
$228$	0	0
$229$	22.0421	1.45658	0.728292	−	0.685267i	$-0.240315\pi$
$229$	22.0421	1.45658	0.728292	+	0.685267i	$0.240315\pi$
$230$	0	0
$231$	2.15190	0.141585
$232$	0	0
$233$	3.23440	0.211892	0.105946	−	0.994372i	$-0.466213\pi$
$233$	3.23440	0.211892	0.105946	+	0.994372i	$0.466213\pi$
$234$	0	0
$235$	−4.74376	−0.309449
$236$	0	0
$237$	−7.41367	−0.481570
$238$	0	0
$239$	4.90881	0.317524	0.158762	−	0.987317i	$-0.449250\pi$
$239$	4.90881	0.317524	0.158762	+	0.987317i	$0.449250\pi$
$240$	0	0
$241$	12.7171	0.819180	0.409590	−	0.912270i	$-0.365672\pi$
$241$	12.7171	0.819180	0.409590	+	0.912270i	$0.365672\pi$
$242$	0	0
$243$	−12.6929	−0.814248
$244$	0	0
$245$	18.5552	1.18545
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−10.0176	−0.634838
$250$	0	0
$251$	3.30616	0.208683	0.104341	−	0.994542i	$-0.466727\pi$
$251$	3.30616	0.208683	0.104341	+	0.994542i	$0.466727\pi$
$252$	0	0
$253$	−0.467580	−0.0293965
$254$	0	0
$255$	21.6740	1.35728
$256$	0	0
$257$	−16.0331	−1.00011	−0.500057	−	0.865992i	$-0.666688\pi$
$257$	−16.0331	−1.00011	−0.500057	+	0.865992i	$0.666688\pi$
$258$	0	0
$259$	22.7162	1.41152
$260$	0	0
$261$	−3.58340	−0.221807
$262$	0	0
$263$	7.47844	0.461141	0.230570	−	0.973056i	$-0.425941\pi$
$263$	7.47844	0.461141	0.230570	+	0.973056i	$0.425941\pi$
$264$	0	0
$265$	−4.29481	−0.263828
$266$	0	0
$267$	3.45838	0.211650
$268$	0	0
$269$	28.1777	1.71802	0.859011	−	0.511957i	$-0.171079\pi$
$269$	28.1777	1.71802	0.859011	+	0.511957i	$0.171079\pi$
$270$	0	0
$271$	3.85467	0.234154	0.117077	−	0.993123i	$-0.462648\pi$
$271$	3.85467	0.234154	0.117077	+	0.993123i	$0.462648\pi$
$272$	0	0
$273$	15.3392	0.928371
$274$	0	0
$275$	6.44656	0.388742
$276$	0	0
$277$	3.98501	0.239436	0.119718	−	0.992808i	$-0.461801\pi$
$277$	3.98501	0.239436	0.119718	+	0.992808i	$0.461801\pi$
$278$	0	0
$279$	−9.17613	−0.549361
$280$	0	0
$281$	−5.37064	−0.320386	−0.160193	−	0.987086i	$-0.551212\pi$
$281$	−5.37064	−0.320386	−0.160193	+	0.987086i	$0.551212\pi$
$282$	0	0
$283$	26.2986	1.56329	0.781644	−	0.623725i	$-0.214382\pi$
$283$	26.2986	1.56329	0.781644	+	0.623725i	$0.214382\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.17015	−0.482269
$288$	0	0
$289$	−1.25660	−0.0739177
$290$	0	0
$291$	1.84549	0.108184
$292$	0	0
$293$	26.2636	1.53434	0.767168	−	0.641446i	$-0.221665\pi$
$293$	26.2636	1.53434	0.767168	+	0.641446i	$0.221665\pi$
$294$	0	0
$295$	27.9720	1.62859
$296$	0	0
$297$	2.77113	0.160798
$298$	0	0
$299$	−3.33301	−0.192753
$300$	0	0
$301$	40.4543	2.33175
$302$	0	0
$303$	4.79579	0.275511
$304$	0	0
$305$	40.1790	2.30064
$306$	0	0
$307$	−5.76709	−0.329145	−0.164572	−	0.986365i	$-0.552624\pi$
$307$	−5.76709	−0.329145	−0.164572	+	0.986365i	$0.552624\pi$
$308$	0	0
$309$	−5.87514	−0.334225
$310$	0	0
$311$	−29.7715	−1.68818	−0.844092	−	0.536198i	$-0.819860\pi$
$311$	−29.7715	−1.68818	−0.844092	+	0.536198i	$0.819860\pi$
$312$	0	0
$313$	−1.07220	−0.0606045	−0.0303023	−	0.999541i	$-0.509647\pi$
$313$	−1.07220	−0.0606045	−0.0303023	+	0.999541i	$0.509647\pi$
$314$	0	0
$315$	19.2153	1.08266
$316$	0	0
$317$	−11.7437	−0.659590	−0.329795	−	0.944053i	$-0.606980\pi$
$317$	−11.7437	−0.659590	−0.329795	+	0.944053i	$0.606980\pi$
$318$	0	0
$319$	1.32277	0.0740611
$320$	0	0
$321$	−22.2562	−1.24222
$322$	0	0
$323$	0	0
$324$	0	0
$325$	45.9524	2.54898
$326$	0	0
$327$	−11.0208	−0.609449
$328$	0	0
$329$	3.77011	0.207853
$330$	0	0
$331$	11.9666	0.657741	0.328871	−	0.944375i	$-0.393332\pi$
$331$	11.9666	0.657741	0.328871	+	0.944375i	$0.393332\pi$
$332$	0	0
$333$	9.04508	0.495668
$334$	0	0
$335$	−45.2067	−2.46990
$336$	0	0
$337$	5.04164	0.274636	0.137318	−	0.990527i	$-0.456152\pi$
$337$	5.04164	0.274636	0.137318	+	0.990527i	$0.456152\pi$
$338$	0	0
$339$	−15.9715	−0.867455
$340$	0	0
$341$	3.38727	0.183431
$342$	0	0
$343$	8.85979	0.478384
$344$	0	0
$345$	5.15284	0.277420
$346$	0	0
$347$	−19.9133	−1.06900	−0.534502	−	0.845168i	$-0.679501\pi$
$347$	−19.9133	−1.06900	−0.534502	+	0.845168i	$0.679501\pi$
$348$	0	0
$349$	−0.483792	−0.0258968	−0.0129484	−	0.999916i	$-0.504122\pi$
$349$	−0.483792	−0.0258968	−0.0129484	+	0.999916i	$0.504122\pi$
$350$	0	0
$351$	19.7532	1.05435
$352$	0	0
$353$	−12.1839	−0.648481	−0.324241	−	0.945975i	$-0.605109\pi$
$353$	−12.1839	−0.648481	−0.324241	+	0.945975i	$0.605109\pi$
$354$	0	0
$355$	39.9756	2.12168
$356$	0	0
$357$	−17.2255	−0.911669
$358$	0	0
$359$	4.59475	0.242502	0.121251	−	0.992622i	$-0.461309\pi$
$359$	4.59475	0.242502	0.121251	+	0.992622i	$0.461309\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	13.8443	0.726636
$364$	0	0
$365$	13.8109	0.722893
$366$	0	0
$367$	−0.550334	−0.0287272	−0.0143636	−	0.999897i	$-0.504572\pi$
$367$	−0.550334	−0.0287272	−0.0143636	+	0.999897i	$0.504572\pi$
$368$	0	0
$369$	−3.25317	−0.169353
$370$	0	0
$371$	3.41331	0.177210
$372$	0	0
$373$	−6.66184	−0.344937	−0.172468	−	0.985015i	$-0.555174\pi$
$373$	−6.66184	−0.344937	−0.172468	+	0.985015i	$0.555174\pi$
$374$	0	0
$375$	−43.7300	−2.25821
$376$	0	0
$377$	9.42901	0.485619
$378$	0	0
$379$	3.42807	0.176088	0.0880440	−	0.996117i	$-0.471938\pi$
$379$	3.42807	0.176088	0.0880440	+	0.996117i	$0.471938\pi$
$380$	0	0
$381$	−14.4406	−0.739815
$382$	0	0
$383$	−11.9529	−0.610766	−0.305383	−	0.952230i	$-0.598784\pi$
$383$	−11.9529	−0.610766	−0.305383	+	0.952230i	$0.598784\pi$
$384$	0	0
$385$	−7.09312	−0.361499
$386$	0	0
$387$	16.1080	0.818814
$388$	0	0
$389$	25.6197	1.29897	0.649484	−	0.760375i	$-0.274985\pi$
$389$	25.6197	1.29897	0.649484	+	0.760375i	$0.274985\pi$
$390$	0	0
$391$	3.74287	0.189285
$392$	0	0
$393$	6.76595	0.341297
$394$	0	0
$395$	24.4370	1.22956
$396$	0	0
$397$	29.7374	1.49248	0.746238	−	0.665679i	$-0.231858\pi$
$397$	29.7374	1.49248	0.746238	+	0.665679i	$0.231858\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	10.1187	0.505302	0.252651	−	0.967557i	$-0.418697\pi$
$401$	10.1187	0.505302	0.252651	+	0.967557i	$0.418697\pi$
$402$	0	0
$403$	24.1452	1.20276
$404$	0	0
$405$	−13.4449	−0.668082
$406$	0	0
$407$	−3.33889	−0.165503
$408$	0	0
$409$	21.0758	1.04213	0.521065	−	0.853517i	$-0.325535\pi$
$409$	21.0758	1.04213	0.521065	+	0.853517i	$0.325535\pi$
$410$	0	0
$411$	18.4877	0.911931
$412$	0	0
$413$	−22.2308	−1.09391
$414$	0	0
$415$	33.0200	1.62089
$416$	0	0
$417$	−4.77808	−0.233984
$418$	0	0
$419$	22.9349	1.12044	0.560221	−	0.828343i	$-0.310716\pi$
$419$	22.9349	1.12044	0.560221	+	0.828343i	$0.310716\pi$
$420$	0	0
$421$	−12.7668	−0.622218	−0.311109	−	0.950374i	$-0.600700\pi$
$421$	−12.7668	−0.622218	−0.311109	+	0.950374i	$0.600700\pi$
$422$	0	0
$423$	1.50117	0.0729894
$424$	0	0
$425$	−51.6032	−2.50312
$426$	0	0
$427$	−31.9323	−1.54531
$428$	0	0
$429$	−2.25460	−0.108853
$430$	0	0
$431$	2.48237	0.119571	0.0597857	−	0.998211i	$-0.480958\pi$
$431$	2.48237	0.119571	0.0597857	+	0.998211i	$0.480958\pi$
$432$	0	0
$433$	−23.0609	−1.10824	−0.554119	−	0.832437i	$-0.686945\pi$
$433$	−23.0609	−1.10824	−0.554119	+	0.832437i	$0.686945\pi$
$434$	0	0
$435$	−14.5773	−0.698926
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−16.4717	−0.786150	−0.393075	−	0.919506i	$-0.628589\pi$
$439$	−16.4717	−0.786150	−0.393075	+	0.919506i	$0.628589\pi$
$440$	0	0
$441$	−5.87181	−0.279610
$442$	0	0
$443$	24.6027	1.16891	0.584455	−	0.811426i	$-0.301308\pi$
$443$	24.6027	1.16891	0.584455	+	0.811426i	$0.301308\pi$
$444$	0	0
$445$	−11.3996	−0.540391
$446$	0	0
$447$	−10.0765	−0.476603
$448$	0	0
$449$	−0.566576	−0.0267384	−0.0133692	−	0.999911i	$-0.504256\pi$
$449$	−0.566576	−0.0267384	−0.0133692	+	0.999911i	$0.504256\pi$
$450$	0	0
$451$	1.20087	0.0565468
$452$	0	0
$453$	−27.2188	−1.27885
$454$	0	0
$455$	−50.5612	−2.37035
$456$	0	0
$457$	−27.7632	−1.29871	−0.649353	−	0.760487i	$-0.724960\pi$
$457$	−27.7632	−1.29871	−0.649353	+	0.760487i	$0.724960\pi$
$458$	0	0
$459$	−22.1823	−1.03538
$460$	0	0
$461$	−29.3394	−1.36647	−0.683236	−	0.730198i	$-0.739428\pi$
$461$	−29.3394	−1.36647	−0.683236	+	0.730198i	$0.739428\pi$
$462$	0	0
$463$	17.4752	0.812144	0.406072	−	0.913841i	$-0.366898\pi$
$463$	17.4752	0.812144	0.406072	+	0.913841i	$0.366898\pi$
$464$	0	0
$465$	−37.3285	−1.73107
$466$	0	0
$467$	29.1407	1.34847	0.674235	−	0.738517i	$-0.264473\pi$
$467$	29.1407	1.34847	0.674235	+	0.738517i	$0.264473\pi$
$468$	0	0
$469$	35.9281	1.65900
$470$	0	0
$471$	4.79722	0.221044
$472$	0	0
$473$	−5.94608	−0.273401
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.35910	0.0622289
$478$	0	0
$479$	23.8287	1.08876	0.544380	−	0.838839i	$-0.316765\pi$
$479$	23.8287	1.08876	0.544380	+	0.838839i	$0.316765\pi$
$480$	0	0
$481$	−23.8003	−1.08520
$482$	0	0
$483$	−4.09523	−0.186339
$484$	0	0
$485$	−6.08311	−0.276220
$486$	0	0
$487$	27.3616	1.23987	0.619936	−	0.784653i	$-0.287159\pi$
$487$	27.3616	1.23987	0.619936	+	0.784653i	$0.287159\pi$
$488$	0	0
$489$	−7.02846	−0.317838
$490$	0	0
$491$	−8.69962	−0.392608	−0.196304	−	0.980543i	$-0.562894\pi$
$491$	−8.69962	−0.392608	−0.196304	+	0.980543i	$0.562894\pi$
$492$	0	0
$493$	−10.5885	−0.476882
$494$	0	0
$495$	−2.82432	−0.126944
$496$	0	0
$497$	−31.7707	−1.42511
$498$	0	0
$499$	9.58344	0.429013	0.214507	−	0.976723i	$-0.431186\pi$
$499$	9.58344	0.429013	0.214507	+	0.976723i	$0.431186\pi$
$500$	0	0
$501$	7.85515	0.350942
$502$	0	0
$503$	5.37037	0.239453	0.119726	−	0.992807i	$-0.461798\pi$
$503$	5.37037	0.239453	0.119726	+	0.992807i	$0.461798\pi$
$504$	0	0
$505$	−15.8079	−0.703444
$506$	0	0
$507$	0.663965	0.0294877
$508$	0	0
$509$	−35.1070	−1.55609	−0.778046	−	0.628208i	$-0.783789\pi$
$509$	−35.1070	−1.55609	−0.778046	+	0.628208i	$0.783789\pi$
$510$	0	0
$511$	−10.9762	−0.485559
$512$	0	0
$513$	0	0
$514$	0	0
$515$	19.3657	0.853354
$516$	0	0
$517$	−0.554141	−0.0243711
$518$	0	0
$519$	28.3925	1.24629
$520$	0	0
$521$	35.2757	1.54546	0.772728	−	0.634737i	$-0.218892\pi$
$521$	35.2757	1.54546	0.772728	+	0.634737i	$0.218892\pi$
$522$	0	0
$523$	23.9207	1.04598	0.522990	−	0.852339i	$-0.324817\pi$
$523$	23.9207	1.04598	0.522990	+	0.852339i	$0.324817\pi$
$524$	0	0
$525$	56.4611	2.46417
$526$	0	0
$527$	−27.1143	−1.18112
$528$	0	0
$529$	−22.1102	−0.961311
$530$	0	0
$531$	−8.85180	−0.384135
$532$	0	0
$533$	8.56006	0.370778
$534$	0	0
$535$	73.3612	3.17168
$536$	0	0
$537$	16.2213	0.700001
$538$	0	0
$539$	2.16752	0.0933615
$540$	0	0
$541$	28.6693	1.23259	0.616294	−	0.787516i	$-0.288633\pi$
$541$	28.6693	1.23259	0.616294	+	0.787516i	$0.288633\pi$
$542$	0	0
$543$	1.73752	0.0745640
$544$	0	0
$545$	36.3267	1.55607
$546$	0	0
$547$	−17.7008	−0.756831	−0.378416	−	0.925636i	$-0.623531\pi$
$547$	−17.7008	−0.756831	−0.378416	+	0.925636i	$0.623531\pi$
$548$	0	0
$549$	−12.7147	−0.542650
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−19.4214	−0.825880
$554$	0	0
$555$	36.7954	1.56188
$556$	0	0
$557$	11.5521	0.489480	0.244740	−	0.969589i	$-0.421297\pi$
$557$	11.5521	0.489480	0.244740	+	0.969589i	$0.421297\pi$
$558$	0	0
$559$	−42.3849	−1.79269
$560$	0	0
$561$	2.53185	0.106895
$562$	0	0
$563$	6.51733	0.274673	0.137336	−	0.990524i	$-0.456146\pi$
$563$	6.51733	0.274673	0.137336	+	0.990524i	$0.456146\pi$
$564$	0	0
$565$	52.6456	2.21482
$566$	0	0
$567$	10.6854	0.448743
$568$	0	0
$569$	−8.36986	−0.350883	−0.175441	−	0.984490i	$-0.556135\pi$
$569$	−8.36986	−0.350883	−0.175441	+	0.984490i	$0.556135\pi$
$570$	0	0
$571$	−1.56658	−0.0655595	−0.0327797	−	0.999463i	$-0.510436\pi$
$571$	−1.56658	−0.0655595	−0.0327797	+	0.999463i	$0.510436\pi$
$572$	0	0
$573$	−27.9327	−1.16690
$574$	0	0
$575$	−12.2683	−0.511622
$576$	0	0
$577$	27.6062	1.14926	0.574630	−	0.818413i	$-0.305146\pi$
$577$	27.6062	1.14926	0.574630	+	0.818413i	$0.305146\pi$
$578$	0	0
$579$	−32.4746	−1.34960
$580$	0	0
$581$	−26.2427	−1.08873
$582$	0	0
$583$	−0.501697	−0.0207782
$584$	0	0
$585$	−20.1323	−0.832369
$586$	0	0
$587$	−26.4189	−1.09043	−0.545213	−	0.838298i	$-0.683551\pi$
$587$	−26.4189	−1.09043	−0.545213	+	0.838298i	$0.683551\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−14.5775	−0.599637
$592$	0	0
$593$	−36.7083	−1.50743	−0.753714	−	0.657203i	$-0.771739\pi$
$593$	−36.7083	−1.50743	−0.753714	+	0.657203i	$0.771739\pi$
$594$	0	0
$595$	56.7788	2.32770
$596$	0	0
$597$	4.63314	0.189622
$598$	0	0
$599$	−10.0573	−0.410929	−0.205464	−	0.978665i	$-0.565870\pi$
$599$	−10.0573	−0.410929	−0.205464	+	0.978665i	$0.565870\pi$
$600$	0	0
$601$	29.2159	1.19174	0.595870	−	0.803081i	$-0.296807\pi$
$601$	29.2159	1.19174	0.595870	+	0.803081i	$0.296807\pi$
$602$	0	0
$603$	14.3057	0.582575
$604$	0	0
$605$	−45.6336	−1.85527
$606$	0	0
$607$	18.9985	0.771124	0.385562	−	0.922682i	$-0.374008\pi$
$607$	18.9985	0.771124	0.385562	+	0.922682i	$0.374008\pi$
$608$	0	0
$609$	11.5853	0.469460
$610$	0	0
$611$	−3.95003	−0.159801
$612$	0	0
$613$	−13.4084	−0.541561	−0.270780	−	0.962641i	$-0.587282\pi$
$613$	−13.4084	−0.541561	−0.270780	+	0.962641i	$0.587282\pi$
$614$	0	0
$615$	−13.2339	−0.533641
$616$	0	0
$617$	30.3358	1.22127	0.610636	−	0.791911i	$-0.290914\pi$
$617$	30.3358	1.22127	0.610636	+	0.791911i	$0.290914\pi$
$618$	0	0
$619$	−30.9334	−1.24332	−0.621659	−	0.783288i	$-0.713541\pi$
$619$	−30.9334	−1.24332	−0.621659	+	0.783288i	$0.713541\pi$
$620$	0	0
$621$	−5.27368	−0.211625
$622$	0	0
$623$	9.05982	0.362974
$624$	0	0
$625$	79.1158	3.16463
$626$	0	0
$627$	0	0
$628$	0	0
$629$	26.7271	1.06568
$630$	0	0
$631$	−3.23694	−0.128861	−0.0644303	−	0.997922i	$-0.520523\pi$
$631$	−3.23694	−0.128861	−0.0644303	+	0.997922i	$0.520523\pi$
$632$	0	0
$633$	21.4022	0.850661
$634$	0	0
$635$	47.5993	1.88892
$636$	0	0
$637$	15.4505	0.612171
$638$	0	0
$639$	−12.6503	−0.500440
$640$	0	0
$641$	40.3249	1.59274	0.796370	−	0.604810i	$-0.206751\pi$
$641$	40.3249	1.59274	0.796370	+	0.604810i	$0.206751\pi$
$642$	0	0
$643$	−2.35532	−0.0928847	−0.0464424	−	0.998921i	$-0.514788\pi$
$643$	−2.35532	−0.0928847	−0.0464424	+	0.998921i	$0.514788\pi$
$644$	0	0
$645$	65.5271	2.58013
$646$	0	0
$647$	24.9858	0.982293	0.491147	−	0.871077i	$-0.336578\pi$
$647$	24.9858	0.982293	0.491147	+	0.871077i	$0.336578\pi$
$648$	0	0
$649$	3.26754	0.128262
$650$	0	0
$651$	29.6668	1.16274
$652$	0	0
$653$	−9.06915	−0.354903	−0.177452	−	0.984130i	$-0.556785\pi$
$653$	−9.06915	−0.354903	−0.177452	+	0.984130i	$0.556785\pi$
$654$	0	0
$655$	−22.3020	−0.871411
$656$	0	0
$657$	−4.37047	−0.170508
$658$	0	0
$659$	−5.50488	−0.214440	−0.107220	−	0.994235i	$-0.534195\pi$
$659$	−5.50488	−0.214440	−0.107220	+	0.994235i	$0.534195\pi$
$660$	0	0
$661$	16.0572	0.624551	0.312275	−	0.949992i	$-0.398909\pi$
$661$	16.0572	0.624551	0.312275	+	0.949992i	$0.398909\pi$
$662$	0	0
$663$	18.0475	0.700908
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2.51734	−0.0974716
$668$	0	0
$669$	−13.9249	−0.538369
$670$	0	0
$671$	4.69349	0.181190
$672$	0	0
$673$	17.7397	0.683817	0.341909	−	0.939733i	$-0.388927\pi$
$673$	17.7397	0.683817	0.341909	+	0.939733i	$0.388927\pi$
$674$	0	0
$675$	72.7085	2.79855
$676$	0	0
$677$	−23.0056	−0.884179	−0.442089	−	0.896971i	$-0.645763\pi$
$677$	−23.0056	−0.884179	−0.442089	+	0.896971i	$0.645763\pi$
$678$	0	0
$679$	4.83456	0.185533
$680$	0	0
$681$	12.7654	0.489173
$682$	0	0
$683$	33.5439	1.28352	0.641760	−	0.766905i	$-0.278204\pi$
$683$	33.5439	1.28352	0.641760	+	0.766905i	$0.278204\pi$
$684$	0	0
$685$	−60.9393	−2.32837
$686$	0	0
$687$	−28.3753	−1.08259
$688$	0	0
$689$	−3.57620	−0.136242
$690$	0	0
$691$	−29.5346	−1.12355	−0.561774	−	0.827291i	$-0.689881\pi$
$691$	−29.5346	−1.12355	−0.561774	+	0.827291i	$0.689881\pi$
$692$	0	0
$693$	2.24463	0.0852664
$694$	0	0
$695$	15.7496	0.597415
$696$	0	0
$697$	−9.61270	−0.364107
$698$	0	0
$699$	−4.16372	−0.157486
$700$	0	0
$701$	−43.5717	−1.64568	−0.822840	−	0.568273i	$-0.807612\pi$
$701$	−43.5717	−1.64568	−0.822840	+	0.568273i	$0.807612\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	6.10675	0.229994
$706$	0	0
$707$	12.5634	0.472494
$708$	0	0
$709$	2.69687	0.101283	0.0506416	−	0.998717i	$-0.483873\pi$
$709$	2.69687	0.101283	0.0506416	+	0.998717i	$0.483873\pi$
$710$	0	0
$711$	−7.73313	−0.290015
$712$	0	0
$713$	−6.44622	−0.241413
$714$	0	0
$715$	7.43163	0.277927
$716$	0	0
$717$	−6.31923	−0.235996
$718$	0	0
$719$	8.98821	0.335204	0.167602	−	0.985855i	$-0.446398\pi$
$719$	8.98821	0.335204	0.167602	+	0.985855i	$0.446398\pi$
$720$	0	0
$721$	−15.3909	−0.573188
$722$	0	0
$723$	−16.3710	−0.608845
$724$	0	0
$725$	34.7067	1.28897
$726$	0	0
$727$	14.7805	0.548179	0.274090	−	0.961704i	$-0.411624\pi$
$727$	14.7805	0.548179	0.274090	+	0.961704i	$0.411624\pi$
$728$	0	0
$729$	25.8454	0.957235
$730$	0	0
$731$	47.5970	1.76044
$732$	0	0
$733$	7.93257	0.292996	0.146498	−	0.989211i	$-0.453200\pi$
$733$	7.93257	0.292996	0.146498	+	0.989211i	$0.453200\pi$
$734$	0	0
$735$	−23.8865	−0.881067
$736$	0	0
$737$	−5.28081	−0.194521
$738$	0	0
$739$	−51.2271	−1.88442	−0.942211	−	0.335021i	$-0.891256\pi$
$739$	−51.2271	−1.88442	−0.942211	+	0.335021i	$0.891256\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−18.8285	−0.690751	−0.345376	−	0.938465i	$-0.612248\pi$
$743$	−18.8285	−0.690751	−0.345376	+	0.938465i	$0.612248\pi$
$744$	0	0
$745$	33.2143	1.21688
$746$	0	0
$747$	−10.4493	−0.382318
$748$	0	0
$749$	−58.3039	−2.13038
$750$	0	0
$751$	−30.5850	−1.11606	−0.558031	−	0.829820i	$-0.688443\pi$
$751$	−30.5850	−1.11606	−0.558031	+	0.829820i	$0.688443\pi$
$752$	0	0
$753$	−4.25609	−0.155101
$754$	0	0
$755$	89.7190	3.26521
$756$	0	0
$757$	9.71680	0.353163	0.176582	−	0.984286i	$-0.443496\pi$
$757$	9.71680	0.353163	0.176582	+	0.984286i	$0.443496\pi$
$758$	0	0
$759$	0.601927	0.0218486
$760$	0	0
$761$	−30.5544	−1.10760	−0.553798	−	0.832651i	$-0.686822\pi$
$761$	−30.5544	−1.10760	−0.553798	+	0.832651i	$0.686822\pi$
$762$	0	0
$763$	−28.8707	−1.04519
$764$	0	0
$765$	22.6080	0.817394
$766$	0	0
$767$	23.2917	0.841016
$768$	0	0
$769$	−21.7530	−0.784434	−0.392217	−	0.919873i	$-0.628292\pi$
$769$	−21.7530	−0.784434	−0.392217	+	0.919873i	$0.628292\pi$
$770$	0	0
$771$	20.6397	0.743322
$772$	0	0
$773$	7.05573	0.253777	0.126888	−	0.991917i	$-0.459501\pi$
$773$	7.05573	0.253777	0.126888	+	0.991917i	$0.459501\pi$
$774$	0	0
$775$	88.8744	3.19246
$776$	0	0
$777$	−29.2432	−1.04909
$778$	0	0
$779$	0	0
$780$	0	0
$781$	4.66974	0.167096
$782$	0	0
$783$	14.9191	0.533165
$784$	0	0
$785$	−15.8127	−0.564378
$786$	0	0
$787$	−5.96411	−0.212597	−0.106299	−	0.994334i	$-0.533900\pi$
$787$	−5.96411	−0.212597	−0.106299	+	0.994334i	$0.533900\pi$
$788$	0	0
$789$	−9.62718	−0.342737
$790$	0	0
$791$	−41.8401	−1.48766
$792$	0	0
$793$	33.4562	1.18807
$794$	0	0
$795$	5.52881	0.196087
$796$	0	0
$797$	−46.5731	−1.64970	−0.824852	−	0.565349i	$-0.808742\pi$
$797$	−46.5731	−1.64970	−0.824852	+	0.565349i	$0.808742\pi$
$798$	0	0
$799$	4.43577	0.156926
$800$	0	0
$801$	3.60741	0.127462
$802$	0	0
$803$	1.61331	0.0569325
$804$	0	0
$805$	13.4987	0.475768
$806$	0	0
$807$	−36.2738	−1.27690
$808$	0	0
$809$	−8.60120	−0.302402	−0.151201	−	0.988503i	$-0.548314\pi$
$809$	−8.60120	−0.302402	−0.151201	+	0.988503i	$0.548314\pi$
$810$	0	0
$811$	24.2531	0.851642	0.425821	−	0.904807i	$-0.359985\pi$
$811$	24.2531	0.851642	0.425821	+	0.904807i	$0.359985\pi$
$812$	0	0
$813$	−4.96220	−0.174032
$814$	0	0
$815$	23.1673	0.811515
$816$	0	0
$817$	0	0
$818$	0	0
$819$	16.0002	0.559092
$820$	0	0
$821$	27.5514	0.961551	0.480776	−	0.876844i	$-0.340355\pi$
$821$	27.5514	0.961551	0.480776	+	0.876844i	$0.340355\pi$
$822$	0	0
$823$	−56.0530	−1.95389	−0.976943	−	0.213501i	$-0.931513\pi$
$823$	−56.0530	−1.95389	−0.976943	+	0.213501i	$0.931513\pi$
$824$	0	0
$825$	−8.29881	−0.288927
$826$	0	0
$827$	46.0033	1.59969	0.799846	−	0.600205i	$-0.204914\pi$
$827$	46.0033	1.59969	0.799846	+	0.600205i	$0.204914\pi$
$828$	0	0
$829$	11.1441	0.387050	0.193525	−	0.981095i	$-0.438008\pi$
$829$	11.1441	0.387050	0.193525	+	0.981095i	$0.438008\pi$
$830$	0	0
$831$	−5.13001	−0.177958
$832$	0	0
$833$	−17.3505	−0.601158
$834$	0	0
$835$	−25.8922	−0.896037
$836$	0	0
$837$	38.2038	1.32052
$838$	0	0
$839$	−20.5363	−0.708993	−0.354497	−	0.935057i	$-0.615348\pi$
$839$	−20.5363	−0.708993	−0.354497	+	0.935057i	$0.615348\pi$
$840$	0	0
$841$	−21.8785	−0.754432
$842$	0	0
$843$	6.91375	0.238122
$844$	0	0
$845$	−2.18857	−0.0752890
$846$	0	0
$847$	36.2674	1.24616
$848$	0	0
$849$	−33.8548	−1.16189
$850$	0	0
$851$	6.35416	0.217818
$852$	0	0
$853$	−3.61575	−0.123801	−0.0619005	−	0.998082i	$-0.519716\pi$
$853$	−3.61575	−0.123801	−0.0619005	+	0.998082i	$0.519716\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−21.4396	−0.732364	−0.366182	−	0.930543i	$-0.619335\pi$
$857$	−21.4396	−0.732364	−0.366182	+	0.930543i	$0.619335\pi$
$858$	0	0
$859$	−33.7794	−1.15254	−0.576268	−	0.817260i	$-0.695492\pi$
$859$	−33.7794	−1.15254	−0.576268	+	0.817260i	$0.695492\pi$
$860$	0	0
$861$	10.5176	0.358440
$862$	0	0
$863$	15.1565	0.515932	0.257966	−	0.966154i	$-0.416948\pi$
$863$	15.1565	0.515932	0.257966	+	0.966154i	$0.416948\pi$
$864$	0	0
$865$	−93.5875	−3.18207
$866$	0	0
$867$	1.61765	0.0549383
$868$	0	0
$869$	2.85460	0.0968357
$870$	0	0
$871$	−37.6427	−1.27547
$872$	0	0
$873$	1.92501	0.0651517
$874$	0	0
$875$	−114.558	−3.87277
$876$	0	0
$877$	21.0686	0.711436	0.355718	−	0.934593i	$-0.384236\pi$
$877$	21.0686	0.711436	0.355718	+	0.934593i	$0.384236\pi$
$878$	0	0
$879$	−33.8098	−1.14038
$880$	0	0
$881$	34.0725	1.14793	0.573966	−	0.818879i	$-0.305404\pi$
$881$	34.0725	1.14793	0.573966	+	0.818879i	$0.305404\pi$
$882$	0	0
$883$	12.7188	0.428022	0.214011	−	0.976831i	$-0.431347\pi$
$883$	12.7188	0.428022	0.214011	+	0.976831i	$0.431347\pi$
$884$	0	0
$885$	−36.0091	−1.21043
$886$	0	0
$887$	−36.3560	−1.22072	−0.610358	−	0.792126i	$-0.708974\pi$
$887$	−36.3560	−1.22072	−0.610358	+	0.792126i	$0.708974\pi$
$888$	0	0
$889$	−37.8296	−1.26876
$890$	0	0
$891$	−1.57056	−0.0526158
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−53.4689	−1.78727
$896$	0	0
$897$	4.29067	0.143261
$898$	0	0
$899$	18.2362	0.608212
$900$	0	0
$901$	4.01597	0.133791
$902$	0	0
$903$	−52.0778	−1.73304
$904$	0	0
$905$	−5.72722	−0.190379
$906$	0	0
$907$	−42.8289	−1.42211	−0.711055	−	0.703136i	$-0.751782\pi$
$907$	−42.8289	−1.42211	−0.711055	+	0.703136i	$0.751782\pi$
$908$	0	0
$909$	5.00245	0.165921
$910$	0	0
$911$	59.2711	1.96374	0.981870	−	0.189555i	$-0.0607047\pi$
$911$	59.2711	1.96374	0.981870	+	0.189555i	$0.0607047\pi$
$912$	0	0
$913$	3.85723	0.127656
$914$	0	0
$915$	−51.7234	−1.70992
$916$	0	0
$917$	17.7245	0.585316
$918$	0	0
$919$	24.4914	0.807898	0.403949	−	0.914781i	$-0.367637\pi$
$919$	24.4914	0.807898	0.403949	+	0.914781i	$0.367637\pi$
$920$	0	0
$921$	7.42411	0.244633
$922$	0	0
$923$	33.2869	1.09565
$924$	0	0
$925$	−87.6052	−2.88044
$926$	0	0
$927$	−6.12831	−0.201280
$928$	0	0
$929$	−42.2030	−1.38464	−0.692318	−	0.721592i	$-0.743411\pi$
$929$	−42.2030	−1.38464	−0.692318	+	0.721592i	$0.743411\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	38.3255	1.25472
$934$	0	0
$935$	−8.34550	−0.272927
$936$	0	0
$937$	−18.8075	−0.614414	−0.307207	−	0.951643i	$-0.599394\pi$
$937$	−18.8075	−0.614414	−0.307207	+	0.951643i	$0.599394\pi$
$938$	0	0
$939$	1.38027	0.0450435
$940$	0	0
$941$	−31.0177	−1.01115	−0.505575	−	0.862783i	$-0.668719\pi$
$941$	−31.0177	−1.01115	−0.505575	+	0.862783i	$0.668719\pi$
$942$	0	0
$943$	−2.28535	−0.0744211
$944$	0	0
$945$	−80.0008	−2.60243
$946$	0	0
$947$	35.6500	1.15847	0.579235	−	0.815160i	$-0.303351\pi$
$947$	35.6500	1.15847	0.579235	+	0.815160i	$0.303351\pi$
$948$	0	0
$949$	11.5000	0.373307
$950$	0	0
$951$	15.1179	0.490232
$952$	0	0
$953$	18.4122	0.596430	0.298215	−	0.954499i	$-0.403609\pi$
$953$	18.4122	0.596430	0.298215	+	0.954499i	$0.403609\pi$
$954$	0	0
$955$	92.0719	2.97938
$956$	0	0
$957$	−1.70284	−0.0550449
$958$	0	0
$959$	48.4316	1.56394
$960$	0	0
$961$	15.6980	0.506389
$962$	0	0
$963$	−23.2153	−0.748102
$964$	0	0
$965$	107.043	3.44584
$966$	0	0
$967$	24.7891	0.797164	0.398582	−	0.917133i	$-0.369502\pi$
$967$	24.7891	0.797164	0.398582	+	0.917133i	$0.369502\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1.23409	0.0396038	0.0198019	−	0.999804i	$-0.493696\pi$
$971$	1.23409	0.0396038	0.0198019	+	0.999804i	$0.493696\pi$
$972$	0	0
$973$	−12.5170	−0.401276
$974$	0	0
$975$	−59.1557	−1.89450
$976$	0	0
$977$	−47.2144	−1.51052	−0.755261	−	0.655424i	$-0.772490\pi$
$977$	−47.2144	−1.51052	−0.755261	+	0.655424i	$0.772490\pi$
$978$	0	0
$979$	−1.33164	−0.0425593
$980$	0	0
$981$	−11.4957	−0.367028
$982$	0	0
$983$	−29.5778	−0.943386	−0.471693	−	0.881763i	$-0.656357\pi$
$983$	−29.5778	−0.943386	−0.471693	+	0.881763i	$0.656357\pi$
$984$	0	0
$985$	48.0504	1.53101
$986$	0	0
$987$	−4.85335	−0.154484
$988$	0	0
$989$	11.3158	0.359823
$990$	0	0
$991$	27.4180	0.870961	0.435480	−	0.900198i	$-0.356579\pi$
$991$	27.4180	0.870961	0.435480	+	0.900198i	$0.356579\pi$
$992$	0	0
$993$	−15.4048	−0.488858
$994$	0	0
$995$	−15.2718	−0.484149
$996$	0	0
$997$	−3.08324	−0.0976472	−0.0488236	−	0.998807i	$-0.515547\pi$
$997$	−3.08324	−0.0976472	−0.0488236	+	0.998807i	$0.515547\pi$
$998$	0	0
$999$	−37.6582	−1.19145

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5776.2.a.ce.1.3		9
4.3	odd	2		2888.2.a.x.1.7		9
19.3	odd	18		304.2.u.f.161.3		18
19.13	odd	18		304.2.u.f.17.3		18
19.18	odd	2		5776.2.a.cd.1.7		9
76.3	even	18		152.2.q.c.9.1	✓	18
76.51	even	18		152.2.q.c.17.1	yes	18
76.75	even	2		2888.2.a.y.1.3		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.2.q.c.9.1	✓	18	76.3	even	18
152.2.q.c.17.1	yes	18	76.51	even	18
304.2.u.f.17.3		18	19.13	odd	18
304.2.u.f.161.3		18	19.3	odd	18
2888.2.a.x.1.7		9	4.3	odd	2
2888.2.a.y.1.3		9	76.75	even	2
5776.2.a.cd.1.7		9	19.18	odd	2
5776.2.a.ce.1.3		9	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:	\( N \)	\(=\)	\( 5776 = 2^{4} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5776.a (trivial)

\(f(q)\)	\(=\)	\(q-1.28732 q^{3} +4.24329 q^{5} -3.37236 q^{7} -1.34280 q^{9} +O(q^{10})\)	\(q-1.28732 q^{3} +4.24329 q^{5} -3.37236 q^{7} -1.34280 q^{9} +0.495679 q^{11} +3.53330 q^{13} -5.46249 q^{15} -3.96780 q^{17} +4.34132 q^{21} -0.943313 q^{23} +13.0055 q^{25} +5.59059 q^{27} +2.66861 q^{29} +6.83360 q^{31} -0.638099 q^{33} -14.3099 q^{35} -6.73600 q^{37} -4.54851 q^{39} +2.42268 q^{41} -11.9958 q^{43} -5.69788 q^{45} -1.11794 q^{47} +4.37282 q^{49} +5.10784 q^{51} -1.01214 q^{53} +2.10331 q^{55} +6.59206 q^{59} +9.46882 q^{61} +4.52840 q^{63} +14.9928 q^{65} -10.6537 q^{67} +1.21435 q^{69} +9.42090 q^{71} +3.25475 q^{73} -16.7423 q^{75} -1.67161 q^{77} +5.75898 q^{79} -3.16851 q^{81} +7.78171 q^{83} -16.8365 q^{85} -3.43537 q^{87} -2.68649 q^{89} -11.9156 q^{91} -8.79705 q^{93} -1.43358 q^{97} -0.665596 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 3 q^{3} + 3 q^{5} - 9 q^{7} + 6 q^{9}+O(q^{10}) \) 9 * q + 3 * q^3 + 3 * q^5 - 9 * q^7 + 6 * q^9	\( 9 q + 3 q^{3} + 3 q^{5} - 9 q^{7} + 6 q^{9} - 3 q^{11} + 6 q^{13} - 3 q^{17} - 15 q^{21} - 24 q^{23} + 30 q^{25} + 12 q^{27} + 15 q^{29} + 6 q^{31} + 18 q^{33} - 15 q^{35} - 24 q^{37} - 6 q^{39} + 12 q^{41} - 9 q^{43} + 42 q^{45} + 12 q^{47} + 18 q^{49} + 12 q^{51} + 18 q^{53} - 21 q^{55} + 57 q^{59} + 15 q^{61} - 30 q^{63} + 27 q^{65} + 6 q^{67} - 3 q^{69} + 36 q^{73} + 45 q^{75} + 30 q^{77} - 3 q^{79} - 15 q^{81} - 27 q^{83} + 18 q^{85} + 18 q^{87} + 30 q^{89} - 21 q^{91} - 24 q^{93} - 3 q^{97} - 48 q^{99}+O(q^{100}) \) 9 * q + 3 * q^3 + 3 * q^5 - 9 * q^7 + 6 * q^9 - 3 * q^11 + 6 * q^13 - 3 * q^17 - 15 * q^21 - 24 * q^23 + 30 * q^25 + 12 * q^27 + 15 * q^29 + 6 * q^31 + 18 * q^33 - 15 * q^35 - 24 * q^37 - 6 * q^39 + 12 * q^41 - 9 * q^43 + 42 * q^45 + 12 * q^47 + 18 * q^49 + 12 * q^51 + 18 * q^53 - 21 * q^55 + 57 * q^59 + 15 * q^61 - 30 * q^63 + 27 * q^65 + 6 * q^67 - 3 * q^69 + 36 * q^73 + 45 * q^75 + 30 * q^77 - 3 * q^79 - 15 * q^81 - 27 * q^83 + 18 * q^85 + 18 * q^87 + 30 * q^89 - 21 * q^91 - 24 * q^93 - 3 * q^97 - 48 * q^99

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