LMFDB - Embedded newform 5776.2.a.cd.1.7

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:	$1$
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			1.7
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	$-0.378803\pi$
$3$	1.28732	0.743237	0.371618	+	0.928386i	$0.378803\pi$
$4$	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	$-0.662996\pi$
$13$	−3.53330	−0.979962	−0.489981	+	0.871733i	$0.662996\pi$
$14$	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	$-0.579699\pi$
$29$	−2.66861	−0.495549	−0.247774	+	0.968818i	$0.579699\pi$
$30$	0	0
$31$	−6.83360	−1.22735	−0.613675	−	0.789559i	$-0.710310\pi$
$31$	−6.83360	−1.22735	−0.613675	+	0.789559i	$0.710310\pi$
$32$	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	$-0.313217\pi$
$37$	6.73600	1.10739	0.553696	+	0.832719i	$0.313217\pi$
$38$	0	0
$39$	−4.54851	−0.728344
$40$	0	0
$41$	−2.42268	−0.378359	−0.189180	−	0.981943i	$-0.560583\pi$
$41$	−2.42268	−0.378359	−0.189180	+	0.981943i	$0.560583\pi$
$42$	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	$-0.477855\pi$
$53$	1.01214	0.139028	0.0695141	+	0.997581i	$0.477855\pi$
$54$	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	$-0.641172\pi$
$59$	−6.59206	−0.858213	−0.429107	+	0.903254i	$0.641172\pi$
$60$	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	$-0.274443\pi$
$67$	10.6537	1.30155	0.650777	+	0.759269i	$0.274443\pi$
$68$	0	0
$69$	−1.21435	−0.146191
$70$	0	0
$71$	−9.42090	−1.11805	−0.559027	−	0.829149i	$-0.688825\pi$
$71$	−9.42090	−1.11805	−0.559027	+	0.829149i	$0.688825\pi$
$72$	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	$-0.605017\pi$
$79$	−5.75898	−0.647935	−0.323968	+	0.946068i	$0.605017\pi$
$80$	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	$-0.454523\pi$
$89$	2.68649	0.284767	0.142384	+	0.989812i	$0.454523\pi$
$90$	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	$-0.476813\pi$
$97$	1.43358	0.145558	0.0727792	+	0.997348i	$0.476813\pi$
$98$	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	$-0.572187\pi$
$103$	−4.56384	−0.449688	−0.224844	+	0.974395i	$0.572187\pi$
$104$	0	0
$105$	−18.4215	−1.79775
$106$	0	0
$107$	−17.2888	−1.67137	−0.835684	−	0.549211i	$-0.814928\pi$
$107$	−17.2888	−1.67137	−0.835684	+	0.549211i	$0.814928\pi$
$108$	0	0
$109$	−8.56098	−0.819993	−0.409997	−	0.912087i	$-0.634470\pi$
$109$	−8.56098	−0.819993	−0.409997	+	0.912087i	$0.634470\pi$
$110$	0	0
$111$	8.67142	0.823055
$112$	0	0
$113$	−12.4068	−1.16713	−0.583566	−	0.812066i	$-0.698343\pi$
$113$	−12.4068	−1.16713	−0.583566	+	0.812066i	$0.698343\pi$
$114$	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	$-0.665821\pi$
$127$	−11.2175	−0.995395	−0.497698	+	0.867351i	$0.665821\pi$
$128$	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	$-0.829740\pi$
$151$	−21.1437	−1.72065	−0.860326	+	0.509743i	$0.829740\pi$
$152$	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	$-0.424134\pi$
$167$	6.10192	0.472181	0.236090	+	0.971731i	$0.424134\pi$
$168$	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	$-0.183479\pi$
$173$	22.0554	1.67684	0.838421	+	0.545024i	$0.183479\pi$
$174$	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	$-0.343924\pi$
$179$	12.6008	0.941828	0.470914	+	0.882179i	$0.343924\pi$
$180$	0	0
$181$	1.34971	0.100323	0.0501617	−	0.998741i	$-0.484026\pi$
$181$	1.34971	0.100323	0.0501617	+	0.998741i	$0.484026\pi$
$182$	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	$-0.862331\pi$
$193$	−25.2265	−1.81584	−0.907920	+	0.419143i	$0.862331\pi$
$194$	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	$-0.306064\pi$
$211$	16.6253	1.14454	0.572268	+	0.820067i	$0.306064\pi$
$212$	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	$-0.617967\pi$
$223$	−10.8170	−0.724357	−0.362179	+	0.932109i	$0.617967\pi$
$224$	0	0
$225$	−17.4638	−1.16425
$226$	0	0
$227$	9.91627	0.658166	0.329083	−	0.944301i	$-0.393260\pi$
$227$	9.91627	0.658166	0.329083	+	0.944301i	$0.393260\pi$
$228$	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	$-0.634328\pi$
$241$	−12.7171	−0.819180	−0.409590	+	0.912270i	$0.634328\pi$
$242$	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	$-0.333312\pi$
$257$	16.0331	1.00011	0.500057	+	0.865992i	$0.333312\pi$
$258$	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	$-0.828921\pi$
$269$	−28.1777	−1.71802	−0.859011	+	0.511957i	$0.828921\pi$
$270$	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	$-0.448788\pi$
$281$	5.37064	0.320386	0.160193	+	0.987086i	$0.448788\pi$
$282$	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	$-0.778335\pi$
$293$	−26.2636	−1.53434	−0.767168	+	0.641446i	$0.778335\pi$
$294$	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	$-0.447376\pi$
$307$	5.76709	0.329145	0.164572	+	0.986365i	$0.447376\pi$
$308$	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	$-0.393020\pi$
$317$	11.7437	0.659590	0.329795	+	0.944053i	$0.393020\pi$
$318$	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	$-0.606668\pi$
$331$	−11.9666	−0.657741	−0.328871	+	0.944375i	$0.606668\pi$
$332$	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	$-0.543848\pi$
$337$	−5.04164	−0.274636	−0.137318	+	0.990527i	$0.543848\pi$
$338$	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	$-0.444826\pi$
$373$	6.66184	0.344937	0.172468	+	0.985015i	$0.444826\pi$
$374$	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	$-0.528062\pi$
$379$	−3.42807	−0.176088	−0.0880440	+	0.996117i	$0.528062\pi$
$380$	0	0
$381$	−14.4406	−0.739815
$382$	0	0
$383$	11.9529	0.610766	0.305383	−	0.952230i	$-0.401216\pi$
$383$	11.9529	0.610766	0.305383	+	0.952230i	$0.401216\pi$
$384$	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	$-0.581303\pi$
$401$	−10.1187	−0.505302	−0.252651	+	0.967557i	$0.581303\pi$
$402$	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	$-0.674465\pi$
$409$	−21.0758	−1.04213	−0.521065	+	0.853517i	$0.674465\pi$
$410$	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	$-0.399300\pi$
$421$	12.7668	0.622218	0.311109	+	0.950374i	$0.399300\pi$
$422$	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	$-0.519042\pi$
$431$	−2.48237	−0.119571	−0.0597857	+	0.998211i	$0.519042\pi$
$432$	0	0
$433$	23.0609	1.10824	0.554119	−	0.832437i	$-0.313055\pi$
$433$	23.0609	1.10824	0.554119	+	0.832437i	$0.313055\pi$
$434$	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	$-0.371411\pi$
$439$	16.4717	0.786150	0.393075	+	0.919506i	$0.371411\pi$
$440$	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	$-0.495744\pi$
$449$	0.566576	0.0267384	0.0133692	+	0.999911i	$0.495744\pi$
$450$	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	$-0.712841\pi$
$487$	−27.3616	−1.23987	−0.619936	+	0.784653i	$0.712841\pi$
$488$	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	$-0.216211\pi$
$509$	35.1070	1.55609	0.778046	+	0.628208i	$0.216211\pi$
$510$	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	$-0.781108\pi$
$521$	−35.2757	−1.54546	−0.772728	+	0.634737i	$0.781108\pi$
$522$	0	0
$523$	−23.9207	−1.04598	−0.522990	−	0.852339i	$-0.675183\pi$
$523$	−23.9207	−1.04598	−0.522990	+	0.852339i	$0.675183\pi$
$524$	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	$-0.376469\pi$
$547$	17.7008	0.756831	0.378416	+	0.925636i	$0.376469\pi$
$548$	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	$-0.543854\pi$
$563$	−6.51733	−0.274673	−0.137336	+	0.990524i	$0.543854\pi$
$564$	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	$-0.443865\pi$
$569$	8.36986	0.350883	0.175441	+	0.984490i	$0.443865\pi$
$570$	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	$-0.434130\pi$
$599$	10.0573	0.410929	0.205464	+	0.978665i	$0.434130\pi$
$600$	0	0
$601$	−29.2159	−1.19174	−0.595870	−	0.803081i	$-0.703193\pi$
$601$	−29.2159	−1.19174	−0.595870	+	0.803081i	$0.703193\pi$
$602$	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	$-0.625992\pi$
$607$	−18.9985	−0.771124	−0.385562	+	0.922682i	$0.625992\pi$
$608$	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	$-0.793249\pi$
$641$	−40.3249	−1.59274	−0.796370	+	0.604810i	$0.793249\pi$
$642$	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	$-0.465805\pi$
$659$	5.50488	0.214440	0.107220	+	0.994235i	$0.465805\pi$
$660$	0	0
$661$	−16.0572	−0.624551	−0.312275	−	0.949992i	$-0.601091\pi$
$661$	−16.0572	−0.624551	−0.312275	+	0.949992i	$0.601091\pi$
$662$	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	$-0.611073\pi$
$673$	−17.7397	−0.683817	−0.341909	+	0.939733i	$0.611073\pi$
$674$	0	0
$675$	−72.7085	−2.79855
$676$	0	0
$677$	23.0056	0.884179	0.442089	−	0.896971i	$-0.354237\pi$
$677$	23.0056	0.884179	0.442089	+	0.896971i	$0.354237\pi$
$678$	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	$-0.721796\pi$
$683$	−33.5439	−1.28352	−0.641760	+	0.766905i	$0.721796\pi$
$684$	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	$-0.387752\pi$
$743$	18.8285	0.690751	0.345376	+	0.938465i	$0.387752\pi$
$744$	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	$-0.311557\pi$
$751$	30.5850	1.11606	0.558031	+	0.829820i	$0.311557\pi$
$752$	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	$-0.540499\pi$
$773$	−7.05573	−0.253777	−0.126888	+	0.991917i	$0.540499\pi$
$774$	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	$-0.466100\pi$
$787$	5.96411	0.212597	0.106299	+	0.994334i	$0.466100\pi$
$788$	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	$-0.191258\pi$
$797$	46.5731	1.64970	0.824852	+	0.565349i	$0.191258\pi$
$798$	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	$-0.640015\pi$
$811$	−24.2531	−0.851642	−0.425821	+	0.904807i	$0.640015\pi$
$812$	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	$-0.795086\pi$
$827$	−46.0033	−1.59969	−0.799846	+	0.600205i	$0.795086\pi$
$828$	0	0
$829$	−11.1441	−0.387050	−0.193525	−	0.981095i	$-0.561992\pi$
$829$	−11.1441	−0.387050	−0.193525	+	0.981095i	$0.561992\pi$
$830$	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	$-0.384652\pi$
$839$	20.5363	0.708993	0.354497	+	0.935057i	$0.384652\pi$
$840$	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	$-0.380665\pi$
$857$	21.4396	0.732364	0.366182	+	0.930543i	$0.380665\pi$
$858$	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	$-0.583052\pi$
$863$	−15.1565	−0.515932	−0.257966	+	0.966154i	$0.583052\pi$
$864$	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	$-0.615764\pi$
$877$	−21.0686	−0.711436	−0.355718	+	0.934593i	$0.615764\pi$
$878$	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	$-0.291026\pi$
$887$	36.3560	1.22072	0.610358	+	0.792126i	$0.291026\pi$
$888$	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	$-0.248218\pi$
$907$	42.8289	1.42211	0.711055	+	0.703136i	$0.248218\pi$
$908$	0	0
$909$	5.00245	0.165921
$910$	0	0
$911$	−59.2711	−1.96374	−0.981870	−	0.189555i	$-0.939295\pi$
$911$	−59.2711	−1.96374	−0.981870	+	0.189555i	$0.939295\pi$
$912$	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	$-0.331281\pi$
$941$	31.0177	1.01115	0.505575	+	0.862783i	$0.331281\pi$
$942$	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	$-0.596391\pi$
$953$	−18.4122	−0.596430	−0.298215	+	0.954499i	$0.596391\pi$
$954$	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	$-0.506304\pi$
$971$	−1.23409	−0.0396038	−0.0198019	+	0.999804i	$0.506304\pi$
$972$	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	$-0.227510\pi$
$977$	47.2144	1.51052	0.755261	+	0.655424i	$0.227510\pi$
$978$	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	$-0.343643\pi$
$983$	29.5778	0.943386	0.471693	+	0.881763i	$0.343643\pi$
$984$	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	$-0.643421\pi$
$991$	−27.4180	−0.870961	−0.435480	+	0.900198i	$0.643421\pi$
$992$	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.cd.1.7		9
4.3	odd	2		2888.2.a.y.1.3		9
19.6	even	9		304.2.u.f.17.3		18
19.16	even	9		304.2.u.f.161.3		18
19.18	odd	2		5776.2.a.ce.1.3		9
76.35	odd	18		152.2.q.c.9.1	✓	18
76.63	odd	18		152.2.q.c.17.1	yes	18
76.75	even	2		2888.2.a.x.1.7		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.2.q.c.9.1	✓	18	76.35	odd	18
152.2.q.c.17.1	yes	18	76.63	odd	18
304.2.u.f.17.3		18	19.6	even	9
304.2.u.f.161.3		18	19.16	even	9
2888.2.a.x.1.7		9	76.75	even	2
2888.2.a.y.1.3		9	4.3	odd	2
5776.2.a.cd.1.7		9	1.1	even	1	trivial
5776.2.a.ce.1.3		9	19.18	odd	2

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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