LMFDB - Embedded newform 5488.2.a.f.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5488,2,Mod(1,5488)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5488, 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("5488.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5488 = 2^{4} \cdot 7^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5488.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:	$43.8219006293$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 686)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.24698$ of defining polynomial
Character	$\chi$	$=$	5488.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.24698 q^{3} +0.246980 q^{5} +7.54288 q^{9} +5.04892 q^{11} -2.10992 q^{13} +0.801938 q^{15} +4.96077 q^{17} +0.506041 q^{19} +4.93900 q^{23} -4.93900 q^{25} +14.7506 q^{27} -4.66487 q^{29} +9.07606 q^{31} +16.3937 q^{33} -2.64310 q^{37} -6.85086 q^{39} -6.70171 q^{41} -3.62565 q^{43} +1.86294 q^{45} -0.917231 q^{47} +16.1075 q^{51} -3.62565 q^{53} +1.24698 q^{55} +1.64310 q^{57} -7.67456 q^{59} -13.0368 q^{61} -0.521106 q^{65} +5.46681 q^{67} +16.0368 q^{69} -11.2295 q^{71} +5.36227 q^{73} -16.0368 q^{75} +6.56465 q^{79} +25.2664 q^{81} -0.753020 q^{83} +1.22521 q^{85} -15.1468 q^{87} +9.69202 q^{89} +29.4698 q^{93} +0.124982 q^{95} +12.2784 q^{97} +38.0834 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 5 q^{3} - 4 q^{5} + 4 q^{9} + 6 q^{11} - 7 q^{13} - 2 q^{15} + 2 q^{17} + 11 q^{19} + 5 q^{23} - 5 q^{25} + 8 q^{27} - 15 q^{29} + 12 q^{31} + 17 q^{33} - 12 q^{37} - 7 q^{39} + 7 q^{41} + q^{43}+ \cdots + 36 q^{99}+O(q^{100}) $ 3 * q + 5 * q^3 - 4 * q^5 + 4 * q^9 + 6 * q^11 - 7 * q^13 - 2 * q^15 + 2 * q^17 + 11 * q^19 + 5 * q^23 - 5 * q^25 + 8 * q^27 - 15 * q^29 + 12 * q^31 + 17 * q^33 - 12 * q^37 - 7 * q^39 + 7 * q^41 + q^43 + 11 * q^45 + 4 * q^47 + 8 * q^51 + q^53 - q^55 + 9 * q^57 - 2 * q^59 - 11 * q^61 + 14 * q^65 + 13 * q^67 + 20 * q^69 - 13 * q^71 + 9 * q^73 - 20 * q^75 - 2 * q^79 + 27 * q^81 - 7 * q^83 + 2 * q^85 - 18 * q^87 + 24 * q^89 + 41 * q^93 - 24 * q^95 + 7 * q^97 + 36 * q^99

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
$3$	3.24698	1.87464	0.937322	−	0.348464i	$-0.113297\pi$
$3$	3.24698	1.87464	0.937322	+	0.348464i	$0.113297\pi$
$4$	0	0
$5$	0.246980	0.110453	0.0552263	−	0.998474i	$-0.482412\pi$
$5$	0.246980	0.110453	0.0552263	+	0.998474i	$0.482412\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	7.54288	2.51429
$10$	0	0
$11$	5.04892	1.52231	0.761153	−	0.648572i	$-0.224634\pi$
$11$	5.04892	1.52231	0.761153	+	0.648572i	$0.224634\pi$
$12$	0	0
$13$	−2.10992	−0.585185	−0.292593	−	0.956237i	$-0.594518\pi$
$13$	−2.10992	−0.585185	−0.292593	+	0.956237i	$0.594518\pi$
$14$	0	0
$15$	0.801938	0.207059
$16$	0	0
$17$	4.96077	1.20316	0.601582	−	0.798811i	$-0.294537\pi$
$17$	4.96077	1.20316	0.601582	+	0.798811i	$0.294537\pi$
$18$	0	0
$19$	0.506041	0.116094	0.0580469	−	0.998314i	$-0.481513\pi$
$19$	0.506041	0.116094	0.0580469	+	0.998314i	$0.481513\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.93900	1.02985	0.514926	−	0.857234i	$-0.327819\pi$
$23$	4.93900	1.02985	0.514926	+	0.857234i	$0.327819\pi$
$24$	0	0
$25$	−4.93900	−0.987800
$26$	0	0
$27$	14.7506	2.83876
$28$	0	0
$29$	−4.66487	−0.866245	−0.433123	−	0.901335i	$-0.642588\pi$
$29$	−4.66487	−0.866245	−0.433123	+	0.901335i	$0.642588\pi$
$30$	0	0
$31$	9.07606	1.63011	0.815055	−	0.579384i	$-0.196707\pi$
$31$	9.07606	1.63011	0.815055	+	0.579384i	$0.196707\pi$
$32$	0	0
$33$	16.3937	2.85378
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.64310	−0.434524	−0.217262	−	0.976113i	$-0.569713\pi$
$37$	−2.64310	−0.434524	−0.217262	+	0.976113i	$0.569713\pi$
$38$	0	0
$39$	−6.85086	−1.09701
$40$	0	0
$41$	−6.70171	−1.04663	−0.523316	−	0.852139i	$-0.675305\pi$
$41$	−6.70171	−1.04663	−0.523316	+	0.852139i	$0.675305\pi$
$42$	0	0
$43$	−3.62565	−0.552906	−0.276453	−	0.961027i	$-0.589159\pi$
$43$	−3.62565	−0.552906	−0.276453	+	0.961027i	$0.589159\pi$
$44$	0	0
$45$	1.86294	0.277710
$46$	0	0
$47$	−0.917231	−0.133792	−0.0668959	−	0.997760i	$-0.521310\pi$
$47$	−0.917231	−0.133792	−0.0668959	+	0.997760i	$0.521310\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	16.1075	2.25550
$52$	0	0
$53$	−3.62565	−0.498021	−0.249010	−	0.968501i	$-0.580105\pi$
$53$	−3.62565	−0.498021	−0.249010	+	0.968501i	$0.580105\pi$
$54$	0	0
$55$	1.24698	0.168143
$56$	0	0
$57$	1.64310	0.217634
$58$	0	0
$59$	−7.67456	−0.999143	−0.499572	−	0.866273i	$-0.666509\pi$
$59$	−7.67456	−0.999143	−0.499572	+	0.866273i	$0.666509\pi$
$60$	0	0
$61$	−13.0368	−1.66920	−0.834598	−	0.550860i	$-0.814300\pi$
$61$	−13.0368	−1.66920	−0.834598	+	0.550860i	$0.814300\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.521106	−0.0646353
$66$	0	0
$67$	5.46681	0.667877	0.333939	−	0.942595i	$-0.391622\pi$
$67$	5.46681	0.667877	0.333939	+	0.942595i	$0.391622\pi$
$68$	0	0
$69$	16.0368	1.93061
$70$	0	0
$71$	−11.2295	−1.33270	−0.666349	−	0.745640i	$-0.732144\pi$
$71$	−11.2295	−1.33270	−0.666349	+	0.745640i	$0.732144\pi$
$72$	0	0
$73$	5.36227	0.627607	0.313803	−	0.949488i	$-0.398397\pi$
$73$	5.36227	0.627607	0.313803	+	0.949488i	$0.398397\pi$
$74$	0	0
$75$	−16.0368	−1.85177
$76$	0	0
$77$	0	0
$78$	0	0
$79$	6.56465	0.738580	0.369290	−	0.929314i	$-0.379601\pi$
$79$	6.56465	0.738580	0.369290	+	0.929314i	$0.379601\pi$
$80$	0	0
$81$	25.2664	2.80737
$82$	0	0
$83$	−0.753020	−0.0826547	−0.0413274	−	0.999146i	$-0.513159\pi$
$83$	−0.753020	−0.0826547	−0.0413274	+	0.999146i	$0.513159\pi$
$84$	0	0
$85$	1.22521	0.132893
$86$	0	0
$87$	−15.1468	−1.62390
$88$	0	0
$89$	9.69202	1.02735	0.513676	−	0.857984i	$-0.328283\pi$
$89$	9.69202	1.02735	0.513676	+	0.857984i	$0.328283\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	29.4698	3.05588
$94$	0	0
$95$	0.124982	0.0128229
$96$	0	0
$97$	12.2784	1.24669	0.623343	−	0.781948i	$-0.285774\pi$
$97$	12.2784	1.24669	0.623343	+	0.781948i	$0.285774\pi$
$98$	0	0
$99$	38.0834	3.82752
$100$	0	0
$101$	−15.6286	−1.55511	−0.777553	−	0.628817i	$-0.783539\pi$
$101$	−15.6286	−1.55511	−0.777553	+	0.628817i	$0.783539\pi$
$102$	0	0
$103$	1.02715	0.101208	0.0506039	−	0.998719i	$-0.483885\pi$
$103$	1.02715	0.101208	0.0506039	+	0.998719i	$0.483885\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−6.83877	−0.661129	−0.330565	−	0.943783i	$-0.607239\pi$
$107$	−6.83877	−0.661129	−0.330565	+	0.943783i	$0.607239\pi$
$108$	0	0
$109$	−0.457123	−0.0437845	−0.0218922	−	0.999760i	$-0.506969\pi$
$109$	−0.457123	−0.0437845	−0.0218922	+	0.999760i	$0.506969\pi$
$110$	0	0
$111$	−8.58211	−0.814577
$112$	0	0
$113$	13.4722	1.26736	0.633678	−	0.773597i	$-0.281544\pi$
$113$	13.4722	1.26736	0.633678	+	0.773597i	$0.281544\pi$
$114$	0	0
$115$	1.21983	0.113750
$116$	0	0
$117$	−15.9148	−1.47133
$118$	0	0
$119$	0	0
$120$	0	0
$121$	14.4916	1.31742
$122$	0	0
$123$	−21.7603	−1.96206
$124$	0	0
$125$	−2.45473	−0.219558
$126$	0	0
$127$	−14.1129	−1.25232	−0.626159	−	0.779696i	$-0.715374\pi$
$127$	−14.1129	−1.25232	−0.626159	+	0.779696i	$0.715374\pi$
$128$	0	0
$129$	−11.7724	−1.03650
$130$	0	0
$131$	−4.98792	−0.435796	−0.217898	−	0.975971i	$-0.569920\pi$
$131$	−4.98792	−0.435796	−0.217898	+	0.975971i	$0.569920\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	3.64310	0.313548
$136$	0	0
$137$	−14.5308	−1.24145	−0.620725	−	0.784029i	$-0.713162\pi$
$137$	−14.5308	−1.24145	−0.620725	+	0.784029i	$0.713162\pi$
$138$	0	0
$139$	13.6649	1.15904	0.579520	−	0.814958i	$-0.303240\pi$
$139$	13.6649	1.15904	0.579520	+	0.814958i	$0.303240\pi$
$140$	0	0
$141$	−2.97823	−0.250812
$142$	0	0
$143$	−10.6528	−0.890831
$144$	0	0
$145$	−1.15213	−0.0956791
$146$	0	0
$147$	0	0
$148$	0	0
$149$	5.44504	0.446075	0.223038	−	0.974810i	$-0.428403\pi$
$149$	5.44504	0.446075	0.223038	+	0.974810i	$0.428403\pi$
$150$	0	0
$151$	6.42221	0.522632	0.261316	−	0.965253i	$-0.415844\pi$
$151$	6.42221	0.522632	0.261316	+	0.965253i	$0.415844\pi$
$152$	0	0
$153$	37.4185	3.02511
$154$	0	0
$155$	2.24160	0.180050
$156$	0	0
$157$	2.64848	0.211372	0.105686	−	0.994400i	$-0.466296\pi$
$157$	2.64848	0.211372	0.105686	+	0.994400i	$0.466296\pi$
$158$	0	0
$159$	−11.7724	−0.933612
$160$	0	0
$161$	0	0
$162$	0	0
$163$	3.01507	0.236158	0.118079	−	0.993004i	$-0.462326\pi$
$163$	3.01507	0.236158	0.118079	+	0.993004i	$0.462326\pi$
$164$	0	0
$165$	4.04892	0.315208
$166$	0	0
$167$	13.7017	1.06027	0.530135	−	0.847913i	$-0.322141\pi$
$167$	13.7017	1.06027	0.530135	+	0.847913i	$0.322141\pi$
$168$	0	0
$169$	−8.54825	−0.657558
$170$	0	0
$171$	3.81700	0.291894
$172$	0	0
$173$	5.90648	0.449061	0.224531	−	0.974467i	$-0.427915\pi$
$173$	5.90648	0.449061	0.224531	+	0.974467i	$0.427915\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−24.9191	−1.87304
$178$	0	0
$179$	−8.43727	−0.630631	−0.315316	−	0.948987i	$-0.602110\pi$
$179$	−8.43727	−0.630631	−0.315316	+	0.948987i	$0.602110\pi$
$180$	0	0
$181$	13.1347	0.976292	0.488146	−	0.872762i	$-0.337673\pi$
$181$	13.1347	0.976292	0.488146	+	0.872762i	$0.337673\pi$
$182$	0	0
$183$	−42.3303	−3.12915
$184$	0	0
$185$	−0.652793	−0.0479943
$186$	0	0
$187$	25.0465	1.83158
$188$	0	0
$189$	0	0
$190$	0	0
$191$	10.1806	0.736643	0.368321	−	0.929699i	$-0.379933\pi$
$191$	10.1806	0.736643	0.368321	+	0.929699i	$0.379933\pi$
$192$	0	0
$193$	−10.0954	−0.726686	−0.363343	−	0.931655i	$-0.618365\pi$
$193$	−10.0954	−0.726686	−0.363343	+	0.931655i	$0.618365\pi$
$194$	0	0
$195$	−1.69202	−0.121168
$196$	0	0
$197$	−2.38942	−0.170239	−0.0851196	−	0.996371i	$-0.527127\pi$
$197$	−2.38942	−0.170239	−0.0851196	+	0.996371i	$0.527127\pi$
$198$	0	0
$199$	19.6256	1.39123	0.695613	−	0.718417i	$-0.255133\pi$
$199$	19.6256	1.39123	0.695613	+	0.718417i	$0.255133\pi$
$200$	0	0
$201$	17.7506	1.25203
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−1.65519	−0.115603
$206$	0	0
$207$	37.2543	2.58935
$208$	0	0
$209$	2.55496	0.176730
$210$	0	0
$211$	11.0978	0.764006	0.382003	−	0.924161i	$-0.375234\pi$
$211$	11.0978	0.764006	0.382003	+	0.924161i	$0.375234\pi$
$212$	0	0
$213$	−36.4620	−2.49834
$214$	0	0
$215$	−0.895461	−0.0610699
$216$	0	0
$217$	0	0
$218$	0	0
$219$	17.4112	1.17654
$220$	0	0
$221$	−10.4668	−0.704074
$222$	0	0
$223$	−24.3250	−1.62892	−0.814460	−	0.580220i	$-0.802967\pi$
$223$	−24.3250	−1.62892	−0.814460	+	0.580220i	$0.802967\pi$
$224$	0	0
$225$	−37.2543	−2.48362
$226$	0	0
$227$	−16.5894	−1.10108	−0.550539	−	0.834810i	$-0.685578\pi$
$227$	−16.5894	−1.10108	−0.550539	+	0.834810i	$0.685578\pi$
$228$	0	0
$229$	−25.2620	−1.66936	−0.834681	−	0.550733i	$-0.814348\pi$
$229$	−25.2620	−1.66936	−0.834681	+	0.550733i	$0.814348\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4.75302	−0.311381	−0.155690	−	0.987806i	$-0.549760\pi$
$233$	−4.75302	−0.311381	−0.155690	+	0.987806i	$0.549760\pi$
$234$	0	0
$235$	−0.226537	−0.0147777
$236$	0	0
$237$	21.3153	1.38458
$238$	0	0
$239$	3.56033	0.230299	0.115149	−	0.993348i	$-0.463265\pi$
$239$	3.56033	0.230299	0.115149	+	0.993348i	$0.463265\pi$
$240$	0	0
$241$	3.70709	0.238794	0.119397	−	0.992847i	$-0.461904\pi$
$241$	3.70709	0.238794	0.119397	+	0.992847i	$0.461904\pi$
$242$	0	0
$243$	37.7875	2.42407
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.06770	−0.0679364
$248$	0	0
$249$	−2.44504	−0.154948
$250$	0	0
$251$	18.2107	1.14945	0.574726	−	0.818346i	$-0.305109\pi$
$251$	18.2107	1.14945	0.574726	+	0.818346i	$0.305109\pi$
$252$	0	0
$253$	24.9366	1.56775
$254$	0	0
$255$	3.97823	0.249126
$256$	0	0
$257$	3.99569	0.249244	0.124622	−	0.992204i	$-0.460228\pi$
$257$	3.99569	0.249244	0.124622	+	0.992204i	$0.460228\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−35.1866	−2.17799
$262$	0	0
$263$	−12.1032	−0.746316	−0.373158	−	0.927768i	$-0.621725\pi$
$263$	−12.1032	−0.746316	−0.373158	+	0.927768i	$0.621725\pi$
$264$	0	0
$265$	−0.895461	−0.0550077
$266$	0	0
$267$	31.4698	1.92592
$268$	0	0
$269$	27.8092	1.69556	0.847779	−	0.530349i	$-0.177939\pi$
$269$	27.8092	1.69556	0.847779	+	0.530349i	$0.177939\pi$
$270$	0	0
$271$	14.2905	0.868087	0.434044	−	0.900892i	$-0.357086\pi$
$271$	14.2905	0.868087	0.434044	+	0.900892i	$0.357086\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−24.9366	−1.50373
$276$	0	0
$277$	17.2325	1.03540	0.517700	−	0.855562i	$-0.326788\pi$
$277$	17.2325	1.03540	0.517700	+	0.855562i	$0.326788\pi$
$278$	0	0
$279$	68.4596	4.09857
$280$	0	0
$281$	17.6461	1.05268	0.526339	−	0.850275i	$-0.323564\pi$
$281$	17.6461	1.05268	0.526339	+	0.850275i	$0.323564\pi$
$282$	0	0
$283$	6.86294	0.407959	0.203980	−	0.978975i	$-0.434612\pi$
$283$	6.86294	0.407959	0.203980	+	0.978975i	$0.434612\pi$
$284$	0	0
$285$	0.405813	0.0240383
$286$	0	0
$287$	0	0
$288$	0	0
$289$	7.60925	0.447603
$290$	0	0
$291$	39.8678	2.33709
$292$	0	0
$293$	−11.4886	−0.671170	−0.335585	−	0.942010i	$-0.608934\pi$
$293$	−11.4886	−0.671170	−0.335585	+	0.942010i	$0.608934\pi$
$294$	0	0
$295$	−1.89546	−0.110358
$296$	0	0
$297$	74.4747	4.32146
$298$	0	0
$299$	−10.4209	−0.602655
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−50.7458	−2.91527
$304$	0	0
$305$	−3.21983	−0.184367
$306$	0	0
$307$	−12.6300	−0.720830	−0.360415	−	0.932792i	$-0.617365\pi$
$307$	−12.6300	−0.720830	−0.360415	+	0.932792i	$0.617365\pi$
$308$	0	0
$309$	3.33513	0.189729
$310$	0	0
$311$	−0.339437	−0.0192477	−0.00962387	−	0.999954i	$-0.503063\pi$
$311$	−0.339437	−0.0192477	−0.00962387	+	0.999954i	$0.503063\pi$
$312$	0	0
$313$	−21.6993	−1.22652	−0.613259	−	0.789882i	$-0.710142\pi$
$313$	−21.6993	−1.22652	−0.613259	+	0.789882i	$0.710142\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−10.7313	−0.602727	−0.301364	−	0.953509i	$-0.597442\pi$
$317$	−10.7313	−0.602727	−0.301364	+	0.953509i	$0.597442\pi$
$318$	0	0
$319$	−23.5526	−1.31869
$320$	0	0
$321$	−22.2054	−1.23938
$322$	0	0
$323$	2.51035	0.139680
$324$	0	0
$325$	10.4209	0.578046
$326$	0	0
$327$	−1.48427	−0.0820803
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−25.4969	−1.40144	−0.700719	−	0.713437i	$-0.747137\pi$
$331$	−25.4969	−1.40144	−0.700719	+	0.713437i	$0.747137\pi$
$332$	0	0
$333$	−19.9366	−1.09252
$334$	0	0
$335$	1.35019	0.0737688
$336$	0	0
$337$	17.0586	0.929241	0.464621	−	0.885510i	$-0.346191\pi$
$337$	17.0586	0.929241	0.464621	+	0.885510i	$0.346191\pi$
$338$	0	0
$339$	43.7439	2.37584
$340$	0	0
$341$	45.8243	2.48152
$342$	0	0
$343$	0	0
$344$	0	0
$345$	3.96077	0.213241
$346$	0	0
$347$	−24.7995	−1.33131	−0.665655	−	0.746260i	$-0.731848\pi$
$347$	−24.7995	−1.33131	−0.665655	+	0.746260i	$0.731848\pi$
$348$	0	0
$349$	−18.1002	−0.968883	−0.484441	−	0.874824i	$-0.660977\pi$
$349$	−18.1002	−0.968883	−0.484441	+	0.874824i	$0.660977\pi$
$350$	0	0
$351$	−31.1226	−1.66120
$352$	0	0
$353$	8.03923	0.427885	0.213942	−	0.976846i	$-0.431369\pi$
$353$	8.03923	0.427885	0.213942	+	0.976846i	$0.431369\pi$
$354$	0	0
$355$	−2.77346	−0.147200
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−13.5700	−0.716198	−0.358099	−	0.933684i	$-0.616575\pi$
$359$	−13.5700	−0.716198	−0.358099	+	0.933684i	$0.616575\pi$
$360$	0	0
$361$	−18.7439	−0.986522
$362$	0	0
$363$	47.0538	2.46969
$364$	0	0
$365$	1.32437	0.0693208
$366$	0	0
$367$	7.91425	0.413120	0.206560	−	0.978434i	$-0.433773\pi$
$367$	7.91425	0.413120	0.206560	+	0.978434i	$0.433773\pi$
$368$	0	0
$369$	−50.5502	−2.63154
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−6.26636	−0.324460	−0.162230	−	0.986753i	$-0.551869\pi$
$373$	−6.26636	−0.324460	−0.162230	+	0.986753i	$0.551869\pi$
$374$	0	0
$375$	−7.97046	−0.411593
$376$	0	0
$377$	9.84249	0.506914
$378$	0	0
$379$	31.5749	1.62190	0.810948	−	0.585119i	$-0.198952\pi$
$379$	31.5749	1.62190	0.810948	+	0.585119i	$0.198952\pi$
$380$	0	0
$381$	−45.8243	−2.34765
$382$	0	0
$383$	34.3056	1.75293	0.876467	−	0.481462i	$-0.159894\pi$
$383$	34.3056	1.75293	0.876467	+	0.481462i	$0.159894\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−27.3478	−1.39017
$388$	0	0
$389$	22.6896	1.15041	0.575205	−	0.818009i	$-0.304922\pi$
$389$	22.6896	1.15041	0.575205	+	0.818009i	$0.304922\pi$
$390$	0	0
$391$	24.5013	1.23908
$392$	0	0
$393$	−16.1957	−0.816963
$394$	0	0
$395$	1.62133	0.0815782
$396$	0	0
$397$	−4.76271	−0.239034	−0.119517	−	0.992832i	$-0.538135\pi$
$397$	−4.76271	−0.239034	−0.119517	+	0.992832i	$0.538135\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−11.8267	−0.590597	−0.295298	−	0.955405i	$-0.595419\pi$
$401$	−11.8267	−0.590597	−0.295298	+	0.955405i	$0.595419\pi$
$402$	0	0
$403$	−19.1497	−0.953916
$404$	0	0
$405$	6.24027	0.310082
$406$	0	0
$407$	−13.3448	−0.661478
$408$	0	0
$409$	−0.264438	−0.0130756	−0.00653781	−	0.999979i	$-0.502081\pi$
$409$	−0.264438	−0.0130756	−0.00653781	+	0.999979i	$0.502081\pi$
$410$	0	0
$411$	−47.1812	−2.32728
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−0.185981	−0.00912943
$416$	0	0
$417$	44.3696	2.17279
$418$	0	0
$419$	−32.3672	−1.58124	−0.790620	−	0.612307i	$-0.790242\pi$
$419$	−32.3672	−1.58124	−0.790620	+	0.612307i	$0.790242\pi$
$420$	0	0
$421$	−2.95646	−0.144089	−0.0720445	−	0.997401i	$-0.522952\pi$
$421$	−2.95646	−0.144089	−0.0720445	+	0.997401i	$0.522952\pi$
$422$	0	0
$423$	−6.91856	−0.336392
$424$	0	0
$425$	−24.5013	−1.18849
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−34.5894	−1.66999
$430$	0	0
$431$	−33.4252	−1.61003	−0.805017	−	0.593251i	$-0.797844\pi$
$431$	−33.4252	−1.61003	−0.805017	+	0.593251i	$0.797844\pi$
$432$	0	0
$433$	−0.670251	−0.0322102	−0.0161051	−	0.999870i	$-0.505127\pi$
$433$	−0.670251	−0.0322102	−0.0161051	+	0.999870i	$0.505127\pi$
$434$	0	0
$435$	−3.74094	−0.179364
$436$	0	0
$437$	2.49934	0.119559
$438$	0	0
$439$	−14.2911	−0.682078	−0.341039	−	0.940049i	$-0.610779\pi$
$439$	−14.2911	−0.682078	−0.341039	+	0.940049i	$0.610779\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−9.08144	−0.431472	−0.215736	−	0.976452i	$-0.569215\pi$
$443$	−9.08144	−0.431472	−0.215736	+	0.976452i	$0.569215\pi$
$444$	0	0
$445$	2.39373	0.113474
$446$	0	0
$447$	17.6799	0.836232
$448$	0	0
$449$	−19.4843	−0.919520	−0.459760	−	0.888043i	$-0.652064\pi$
$449$	−19.4843	−0.919520	−0.459760	+	0.888043i	$0.652064\pi$
$450$	0	0
$451$	−33.8364	−1.59329
$452$	0	0
$453$	20.8528	0.979749
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−32.1933	−1.50594	−0.752969	−	0.658056i	$-0.771379\pi$
$457$	−32.1933	−1.50594	−0.752969	+	0.658056i	$0.771379\pi$
$458$	0	0
$459$	73.1745	3.41549
$460$	0	0
$461$	30.9571	1.44181	0.720907	−	0.693032i	$-0.243726\pi$
$461$	30.9571	1.44181	0.720907	+	0.693032i	$0.243726\pi$
$462$	0	0
$463$	−22.0315	−1.02389	−0.511944	−	0.859019i	$-0.671075\pi$
$463$	−22.0315	−1.02389	−0.511944	+	0.859019i	$0.671075\pi$
$464$	0	0
$465$	7.27844	0.337530
$466$	0	0
$467$	−9.48965	−0.439129	−0.219564	−	0.975598i	$-0.570464\pi$
$467$	−9.48965	−0.439129	−0.219564	+	0.975598i	$0.570464\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	8.59956	0.396247
$472$	0	0
$473$	−18.3056	−0.841692
$474$	0	0
$475$	−2.49934	−0.114677
$476$	0	0
$477$	−27.3478	−1.25217
$478$	0	0
$479$	−8.59717	−0.392815	−0.196407	−	0.980522i	$-0.562928\pi$
$479$	−8.59717	−0.392815	−0.196407	+	0.980522i	$0.562928\pi$
$480$	0	0
$481$	5.57673	0.254277
$482$	0	0
$483$	0	0
$484$	0	0
$485$	3.03252	0.137700
$486$	0	0
$487$	−25.7313	−1.16599	−0.582997	−	0.812474i	$-0.698120\pi$
$487$	−25.7313	−1.16599	−0.582997	+	0.812474i	$0.698120\pi$
$488$	0	0
$489$	9.78986	0.442713
$490$	0	0
$491$	17.3448	0.782761	0.391380	−	0.920229i	$-0.371998\pi$
$491$	17.3448	0.782761	0.391380	+	0.920229i	$0.371998\pi$
$492$	0	0
$493$	−23.1414	−1.04224
$494$	0	0
$495$	9.40581	0.422760
$496$	0	0
$497$	0	0
$498$	0	0
$499$	36.0984	1.61599	0.807994	−	0.589191i	$-0.200553\pi$
$499$	36.0984	1.61599	0.807994	+	0.589191i	$0.200553\pi$
$500$	0	0
$501$	44.4892	1.98763
$502$	0	0
$503$	7.33513	0.327057	0.163529	−	0.986539i	$-0.447712\pi$
$503$	7.33513	0.327057	0.163529	+	0.986539i	$0.447712\pi$
$504$	0	0
$505$	−3.85995	−0.171766
$506$	0	0
$507$	−27.7560	−1.23269
$508$	0	0
$509$	2.26098	0.100216	0.0501081	−	0.998744i	$-0.484043\pi$
$509$	2.26098	0.100216	0.0501081	+	0.998744i	$0.484043\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	7.46442	0.329562
$514$	0	0
$515$	0.253684	0.0111787
$516$	0	0
$517$	−4.63102	−0.203672
$518$	0	0
$519$	19.1782	0.841830
$520$	0	0
$521$	9.96316	0.436494	0.218247	−	0.975894i	$-0.429966\pi$
$521$	9.96316	0.436494	0.218247	+	0.975894i	$0.429966\pi$
$522$	0	0
$523$	31.3139	1.36926	0.684632	−	0.728889i	$-0.259963\pi$
$523$	31.3139	1.36926	0.684632	+	0.728889i	$0.259963\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	45.0243	1.96129
$528$	0	0
$529$	1.39373	0.0605970
$530$	0	0
$531$	−57.8883	−2.51214
$532$	0	0
$533$	14.1400	0.612473
$534$	0	0
$535$	−1.68904	−0.0730235
$536$	0	0
$537$	−27.3957	−1.18221
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−30.2620	−1.30107	−0.650533	−	0.759478i	$-0.725455\pi$
$541$	−30.2620	−1.30107	−0.650533	+	0.759478i	$0.725455\pi$
$542$	0	0
$543$	42.6480	1.83020
$544$	0	0
$545$	−0.112900	−0.00483611
$546$	0	0
$547$	−23.2760	−0.995212	−0.497606	−	0.867403i	$-0.665787\pi$
$547$	−23.2760	−0.995212	−0.497606	+	0.867403i	$0.665787\pi$
$548$	0	0
$549$	−98.3352	−4.19685
$550$	0	0
$551$	−2.36062	−0.100566
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−2.11960	−0.0899722
$556$	0	0
$557$	−21.7536	−0.921730	−0.460865	−	0.887470i	$-0.652461\pi$
$557$	−21.7536	−0.921730	−0.460865	+	0.887470i	$0.652461\pi$
$558$	0	0
$559$	7.64981	0.323552
$560$	0	0
$561$	81.3256	3.43357
$562$	0	0
$563$	37.3226	1.57296	0.786479	−	0.617617i	$-0.211902\pi$
$563$	37.3226	1.57296	0.786479	+	0.617617i	$0.211902\pi$
$564$	0	0
$565$	3.32736	0.139983
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−21.1357	−0.886056	−0.443028	−	0.896508i	$-0.646096\pi$
$569$	−21.1357	−0.886056	−0.443028	+	0.896508i	$0.646096\pi$
$570$	0	0
$571$	−46.1420	−1.93098	−0.965491	−	0.260438i	$-0.916133\pi$
$571$	−46.1420	−1.93098	−0.965491	+	0.260438i	$0.916133\pi$
$572$	0	0
$573$	33.0562	1.38094
$574$	0	0
$575$	−24.3937	−1.01729
$576$	0	0
$577$	−30.0834	−1.25239	−0.626193	−	0.779668i	$-0.715388\pi$
$577$	−30.0834	−1.25239	−0.626193	+	0.779668i	$0.715388\pi$
$578$	0	0
$579$	−32.7797	−1.36228
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−18.3056	−0.758140
$584$	0	0
$585$	−3.93064	−0.162512
$586$	0	0
$587$	7.40880	0.305794	0.152897	−	0.988242i	$-0.451140\pi$
$587$	7.40880	0.305794	0.152897	+	0.988242i	$0.451140\pi$
$588$	0	0
$589$	4.59286	0.189245
$590$	0	0
$591$	−7.75840	−0.319138
$592$	0	0
$593$	9.53558	0.391579	0.195790	−	0.980646i	$-0.437273\pi$
$593$	9.53558	0.391579	0.195790	+	0.980646i	$0.437273\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	63.7241	2.60805
$598$	0	0
$599$	8.30021	0.339137	0.169569	−	0.985518i	$-0.445763\pi$
$599$	8.30021	0.339137	0.169569	+	0.985518i	$0.445763\pi$
$600$	0	0
$601$	−46.9342	−1.91449	−0.957243	−	0.289284i	$-0.906583\pi$
$601$	−46.9342	−1.91449	−0.957243	+	0.289284i	$0.906583\pi$
$602$	0	0
$603$	41.2355	1.67924
$604$	0	0
$605$	3.57912	0.145512
$606$	0	0
$607$	−29.3056	−1.18948	−0.594739	−	0.803919i	$-0.702744\pi$
$607$	−29.3056	−1.18948	−0.594739	+	0.803919i	$0.702744\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.93528	0.0782931
$612$	0	0
$613$	−13.5163	−0.545919	−0.272960	−	0.962026i	$-0.588003\pi$
$613$	−13.5163	−0.545919	−0.272960	+	0.962026i	$0.588003\pi$
$614$	0	0
$615$	−5.37435	−0.216715
$616$	0	0
$617$	−18.4698	−0.743566	−0.371783	−	0.928320i	$-0.621253\pi$
$617$	−18.4698	−0.743566	−0.371783	+	0.928320i	$0.621253\pi$
$618$	0	0
$619$	13.2959	0.534407	0.267204	−	0.963640i	$-0.413900\pi$
$619$	13.2959	0.534407	0.267204	+	0.963640i	$0.413900\pi$
$620$	0	0
$621$	72.8534	2.92350
$622$	0	0
$623$	0	0
$624$	0	0
$625$	24.0887	0.963549
$626$	0	0
$627$	8.29590	0.331306
$628$	0	0
$629$	−13.1118	−0.522803
$630$	0	0
$631$	30.7318	1.22342	0.611708	−	0.791084i	$-0.290483\pi$
$631$	30.7318	1.22342	0.611708	+	0.791084i	$0.290483\pi$
$632$	0	0
$633$	36.0344	1.43224
$634$	0	0
$635$	−3.48560	−0.138322
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−84.7029	−3.35079
$640$	0	0
$641$	3.69633	0.145996	0.0729982	−	0.997332i	$-0.476743\pi$
$641$	3.69633	0.145996	0.0729982	+	0.997332i	$0.476743\pi$
$642$	0	0
$643$	−10.4340	−0.411478	−0.205739	−	0.978607i	$-0.565960\pi$
$643$	−10.4340	−0.411478	−0.205739	+	0.978607i	$0.565960\pi$
$644$	0	0
$645$	−2.90754	−0.114484
$646$	0	0
$647$	−45.3424	−1.78259	−0.891297	−	0.453419i	$-0.850204\pi$
$647$	−45.3424	−1.78259	−0.891297	+	0.453419i	$0.850204\pi$
$648$	0	0
$649$	−38.7482	−1.52100
$650$	0	0
$651$	0	0
$652$	0	0
$653$	26.5719	1.03984	0.519920	−	0.854215i	$-0.325962\pi$
$653$	26.5719	1.03984	0.519920	+	0.854215i	$0.325962\pi$
$654$	0	0
$655$	−1.23191	−0.0481349
$656$	0	0
$657$	40.4470	1.57799
$658$	0	0
$659$	24.1497	0.940740	0.470370	−	0.882469i	$-0.344120\pi$
$659$	24.1497	0.940740	0.470370	+	0.882469i	$0.344120\pi$
$660$	0	0
$661$	2.02416	0.0787308	0.0393654	−	0.999225i	$-0.487466\pi$
$661$	2.02416	0.0787308	0.0393654	+	0.999225i	$0.487466\pi$
$662$	0	0
$663$	−33.9855	−1.31989
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−23.0398	−0.892105
$668$	0	0
$669$	−78.9827	−3.05365
$670$	0	0
$671$	−65.8219	−2.54103
$672$	0	0
$673$	−44.8219	−1.72776	−0.863879	−	0.503700i	$-0.831972\pi$
$673$	−44.8219	−1.72776	−0.863879	+	0.503700i	$0.831972\pi$
$674$	0	0
$675$	−72.8534	−2.80413
$676$	0	0
$677$	28.7313	1.10423	0.552116	−	0.833767i	$-0.313821\pi$
$677$	28.7313	1.10423	0.552116	+	0.833767i	$0.313821\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−53.8654	−2.06413
$682$	0	0
$683$	−3.54719	−0.135729	−0.0678647	−	0.997695i	$-0.521619\pi$
$683$	−3.54719	−0.135729	−0.0678647	+	0.997695i	$0.521619\pi$
$684$	0	0
$685$	−3.58881	−0.137121
$686$	0	0
$687$	−82.0253	−3.12946
$688$	0	0
$689$	7.64981	0.291435
$690$	0	0
$691$	37.7265	1.43518	0.717591	−	0.696465i	$-0.245245\pi$
$691$	37.7265	1.43518	0.717591	+	0.696465i	$0.245245\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3.37495	0.128019
$696$	0	0
$697$	−33.2457	−1.25927
$698$	0	0
$699$	−15.4330	−0.583728
$700$	0	0
$701$	−44.1312	−1.66681	−0.833406	−	0.552661i	$-0.813613\pi$
$701$	−44.1312	−1.66681	−0.833406	+	0.552661i	$0.813613\pi$
$702$	0	0
$703$	−1.33752	−0.0504455
$704$	0	0
$705$	−0.735562	−0.0277029
$706$	0	0
$707$	0	0
$708$	0	0
$709$	10.9758	0.412206	0.206103	−	0.978530i	$-0.433922\pi$
$709$	10.9758	0.412206	0.206103	+	0.978530i	$0.433922\pi$
$710$	0	0
$711$	49.5163	1.85701
$712$	0	0
$713$	44.8267	1.67877
$714$	0	0
$715$	−2.63102	−0.0983947
$716$	0	0
$717$	11.5603	0.431729
$718$	0	0
$719$	16.2218	0.604969	0.302485	−	0.953154i	$-0.402184\pi$
$719$	16.2218	0.604969	0.302485	+	0.953154i	$0.402184\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	12.0368	0.447655
$724$	0	0
$725$	23.0398	0.855677
$726$	0	0
$727$	15.8025	0.586083	0.293042	−	0.956100i	$-0.405333\pi$
$727$	15.8025	0.586083	0.293042	+	0.956100i	$0.405333\pi$
$728$	0	0
$729$	46.8961	1.73689
$730$	0	0
$731$	−17.9860	−0.665236
$732$	0	0
$733$	47.9874	1.77246	0.886228	−	0.463249i	$-0.153316\pi$
$733$	47.9874	1.77246	0.886228	+	0.463249i	$0.153316\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	27.6015	1.01671
$738$	0	0
$739$	−22.2513	−0.818527	−0.409263	−	0.912416i	$-0.634214\pi$
$739$	−22.2513	−0.818527	−0.409263	+	0.912416i	$0.634214\pi$
$740$	0	0
$741$	−3.46681	−0.127357
$742$	0	0
$743$	9.69501	0.355675	0.177838	−	0.984060i	$-0.443090\pi$
$743$	9.69501	0.355675	0.177838	+	0.984060i	$0.443090\pi$
$744$	0	0
$745$	1.34481	0.0492702
$746$	0	0
$747$	−5.67994	−0.207818
$748$	0	0
$749$	0	0
$750$	0	0
$751$	10.6280	0.387823	0.193911	−	0.981019i	$-0.437883\pi$
$751$	10.6280	0.387823	0.193911	+	0.981019i	$0.437883\pi$
$752$	0	0
$753$	59.1299	2.15481
$754$	0	0
$755$	1.58615	0.0577261
$756$	0	0
$757$	−41.0823	−1.49316	−0.746581	−	0.665295i	$-0.768306\pi$
$757$	−41.0823	−1.49316	−0.746581	+	0.665295i	$0.768306\pi$
$758$	0	0
$759$	80.9687	2.93898
$760$	0	0
$761$	23.1414	0.838874	0.419437	−	0.907784i	$-0.362227\pi$
$761$	23.1414	0.838874	0.419437	+	0.907784i	$0.362227\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	9.24160	0.334131
$766$	0	0
$767$	16.1927	0.584684
$768$	0	0
$769$	38.7928	1.39891	0.699453	−	0.714679i	$-0.253427\pi$
$769$	38.7928	1.39891	0.699453	+	0.714679i	$0.253427\pi$
$770$	0	0
$771$	12.9739	0.467244
$772$	0	0
$773$	28.5327	1.02625	0.513125	−	0.858314i	$-0.328488\pi$
$773$	28.5327	1.02625	0.513125	+	0.858314i	$0.328488\pi$
$774$	0	0
$775$	−44.8267	−1.61022
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3.39134	−0.121507
$780$	0	0
$781$	−56.6969	−2.02878
$782$	0	0
$783$	−68.8098	−2.45906
$784$	0	0
$785$	0.654121	0.0233466
$786$	0	0
$787$	33.5706	1.19666	0.598332	−	0.801249i	$-0.295831\pi$
$787$	33.5706	1.19666	0.598332	+	0.801249i	$0.295831\pi$
$788$	0	0
$789$	−39.2989	−1.39908
$790$	0	0
$791$	0	0
$792$	0	0
$793$	27.5066	0.976789
$794$	0	0
$795$	−2.90754	−0.103120
$796$	0	0
$797$	−20.0224	−0.709228	−0.354614	−	0.935013i	$-0.615388\pi$
$797$	−20.0224	−0.709228	−0.354614	+	0.935013i	$0.615388\pi$
$798$	0	0
$799$	−4.55017	−0.160974
$800$	0	0
$801$	73.1057	2.58306
$802$	0	0
$803$	27.0737	0.955409
$804$	0	0
$805$	0	0
$806$	0	0
$807$	90.2960	3.17857
$808$	0	0
$809$	−11.8039	−0.415002	−0.207501	−	0.978235i	$-0.566533\pi$
$809$	−11.8039	−0.415002	−0.207501	+	0.978235i	$0.566533\pi$
$810$	0	0
$811$	−4.55496	−0.159946	−0.0799731	−	0.996797i	$-0.525483\pi$
$811$	−4.55496	−0.159946	−0.0799731	+	0.996797i	$0.525483\pi$
$812$	0	0
$813$	46.4010	1.62736
$814$	0	0
$815$	0.744660	0.0260843
$816$	0	0
$817$	−1.83472	−0.0641889
$818$	0	0
$819$	0	0
$820$	0	0
$821$	31.6980	1.10627	0.553134	−	0.833092i	$-0.313432\pi$
$821$	31.6980	1.10627	0.553134	+	0.833092i	$0.313432\pi$
$822$	0	0
$823$	47.4577	1.65427	0.827136	−	0.562002i	$-0.189969\pi$
$823$	47.4577	1.65427	0.827136	+	0.562002i	$0.189969\pi$
$824$	0	0
$825$	−80.9687	−2.81897
$826$	0	0
$827$	−34.4704	−1.19865	−0.599326	−	0.800505i	$-0.704565\pi$
$827$	−34.4704	−1.19865	−0.599326	+	0.800505i	$0.704565\pi$
$828$	0	0
$829$	−32.6243	−1.13309	−0.566545	−	0.824031i	$-0.691720\pi$
$829$	−32.6243	−1.13309	−0.566545	+	0.824031i	$0.691720\pi$
$830$	0	0
$831$	55.9536	1.94101
$832$	0	0
$833$	0	0
$834$	0	0
$835$	3.38404	0.117110
$836$	0	0
$837$	133.878	4.62749
$838$	0	0
$839$	6.24698	0.215670	0.107835	−	0.994169i	$-0.465608\pi$
$839$	6.24698	0.215670	0.107835	+	0.994169i	$0.465608\pi$
$840$	0	0
$841$	−7.23895	−0.249619
$842$	0	0
$843$	57.2965	1.97340
$844$	0	0
$845$	−2.11124	−0.0726290
$846$	0	0
$847$	0	0
$848$	0	0
$849$	22.2838	0.764779
$850$	0	0
$851$	−13.0543	−0.447495
$852$	0	0
$853$	−7.59658	−0.260102	−0.130051	−	0.991507i	$-0.541514\pi$
$853$	−7.59658	−0.260102	−0.130051	+	0.991507i	$0.541514\pi$
$854$	0	0
$855$	0.942722	0.0322404
$856$	0	0
$857$	43.3521	1.48088	0.740440	−	0.672123i	$-0.234617\pi$
$857$	43.3521	1.48088	0.740440	+	0.672123i	$0.234617\pi$
$858$	0	0
$859$	−42.0388	−1.43434	−0.717172	−	0.696896i	$-0.754564\pi$
$859$	−42.0388	−1.43434	−0.717172	+	0.696896i	$0.754564\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	42.9511	1.46207	0.731036	−	0.682339i	$-0.239037\pi$
$863$	42.9511	1.46207	0.731036	+	0.682339i	$0.239037\pi$
$864$	0	0
$865$	1.45878	0.0496000
$866$	0	0
$867$	24.7071	0.839097
$868$	0	0
$869$	33.1444	1.12435
$870$	0	0
$871$	−11.5345	−0.390832
$872$	0	0
$873$	92.6147	3.13453
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−35.8133	−1.20933	−0.604664	−	0.796480i	$-0.706693\pi$
$877$	−35.8133	−1.20933	−0.604664	+	0.796480i	$0.706693\pi$
$878$	0	0
$879$	−37.3032	−1.25821
$880$	0	0
$881$	−3.75600	−0.126543	−0.0632715	−	0.997996i	$-0.520153\pi$
$881$	−3.75600	−0.126543	−0.0632715	+	0.997996i	$0.520153\pi$
$882$	0	0
$883$	7.84979	0.264166	0.132083	−	0.991239i	$-0.457833\pi$
$883$	7.84979	0.264166	0.132083	+	0.991239i	$0.457833\pi$
$884$	0	0
$885$	−6.15452	−0.206882
$886$	0	0
$887$	48.9667	1.64414	0.822071	−	0.569385i	$-0.192819\pi$
$887$	48.9667	1.64414	0.822071	+	0.569385i	$0.192819\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	127.568	4.27368
$892$	0	0
$893$	−0.464156	−0.0155324
$894$	0	0
$895$	−2.08383	−0.0696549
$896$	0	0
$897$	−33.8364	−1.12976
$898$	0	0
$899$	−42.3387	−1.41207
$900$	0	0
$901$	−17.9860	−0.599201
$902$	0	0
$903$	0	0
$904$	0	0
$905$	3.24400	0.107834
$906$	0	0
$907$	8.71810	0.289480	0.144740	−	0.989470i	$-0.453765\pi$
$907$	8.71810	0.289480	0.144740	+	0.989470i	$0.453765\pi$
$908$	0	0
$909$	−117.885	−3.90999
$910$	0	0
$911$	42.5187	1.40871	0.704354	−	0.709849i	$-0.251237\pi$
$911$	42.5187	1.40871	0.704354	+	0.709849i	$0.251237\pi$
$912$	0	0
$913$	−3.80194	−0.125826
$914$	0	0
$915$	−10.4547	−0.345623
$916$	0	0
$917$	0	0
$918$	0	0
$919$	23.1570	0.763880	0.381940	−	0.924187i	$-0.375256\pi$
$919$	23.1570	0.763880	0.381940	+	0.924187i	$0.375256\pi$
$920$	0	0
$921$	−41.0092	−1.35130
$922$	0	0
$923$	23.6933	0.779876
$924$	0	0
$925$	13.0543	0.429223
$926$	0	0
$927$	7.74764	0.254466
$928$	0	0
$929$	24.0549	0.789215	0.394608	−	0.918850i	$-0.370881\pi$
$929$	24.0549	0.789215	0.394608	+	0.918850i	$0.370881\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−1.10215	−0.0360827
$934$	0	0
$935$	6.18598	0.202303
$936$	0	0
$937$	−7.52840	−0.245942	−0.122971	−	0.992410i	$-0.539242\pi$
$937$	−7.52840	−0.245942	−0.122971	+	0.992410i	$0.539242\pi$
$938$	0	0
$939$	−70.4572	−2.29929
$940$	0	0
$941$	−18.4383	−0.601073	−0.300536	−	0.953770i	$-0.597166\pi$
$941$	−18.4383	−0.601073	−0.300536	+	0.953770i	$0.597166\pi$
$942$	0	0
$943$	−33.0998	−1.07788
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−3.52840	−0.114658	−0.0573288	−	0.998355i	$-0.518258\pi$
$947$	−3.52840	−0.114658	−0.0573288	+	0.998355i	$0.518258\pi$
$948$	0	0
$949$	−11.3139	−0.367266
$950$	0	0
$951$	−34.8442	−1.12990
$952$	0	0
$953$	11.0935	0.359354	0.179677	−	0.983726i	$-0.442495\pi$
$953$	11.0935	0.359354	0.179677	+	0.983726i	$0.442495\pi$
$954$	0	0
$955$	2.51440	0.0813641
$956$	0	0
$957$	−76.4747	−2.47208
$958$	0	0
$959$	0	0
$960$	0	0
$961$	51.3749	1.65726
$962$	0	0
$963$	−51.5840	−1.66227
$964$	0	0
$965$	−2.49337	−0.0802644
$966$	0	0
$967$	23.6418	0.760268	0.380134	−	0.924931i	$-0.375878\pi$
$967$	23.6418	0.760268	0.380134	+	0.924931i	$0.375878\pi$
$968$	0	0
$969$	8.15106	0.261850
$970$	0	0
$971$	34.3957	1.10381	0.551904	−	0.833907i	$-0.313901\pi$
$971$	34.3957	1.10381	0.551904	+	0.833907i	$0.313901\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	33.8364	1.08363
$976$	0	0
$977$	−13.5681	−0.434082	−0.217041	−	0.976162i	$-0.569641\pi$
$977$	−13.5681	−0.434082	−0.217041	+	0.976162i	$0.569641\pi$
$978$	0	0
$979$	48.9342	1.56394
$980$	0	0
$981$	−3.44803	−0.110087
$982$	0	0
$983$	30.4644	0.971664	0.485832	−	0.874052i	$-0.338517\pi$
$983$	30.4644	0.971664	0.485832	+	0.874052i	$0.338517\pi$
$984$	0	0
$985$	−0.590138	−0.0188034
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−17.9071	−0.569412
$990$	0	0
$991$	−58.9947	−1.87403	−0.937015	−	0.349290i	$-0.886423\pi$
$991$	−58.9947	−1.87403	−0.937015	+	0.349290i	$0.886423\pi$
$992$	0	0
$993$	−82.7881	−2.62720
$994$	0	0
$995$	4.84713	0.153664
$996$	0	0
$997$	−22.0838	−0.699402	−0.349701	−	0.936861i	$-0.613717\pi$
$997$	−22.0838	−0.699402	−0.349701	+	0.936861i	$0.613717\pi$
$998$	0	0
$999$	−38.9874	−1.23351

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5488.2.a.f.1.3		3
4.3	odd	2		686.2.a.c.1.1	✓	3
7.6	odd	2		5488.2.a.a.1.1		3
12.11	even	2		6174.2.a.e.1.1		3
28.3	even	6		686.2.c.a.667.1		6
28.11	odd	6		686.2.c.b.667.3		6
28.19	even	6		686.2.c.a.361.1		6
28.23	odd	6		686.2.c.b.361.3		6
28.27	even	2		686.2.a.d.1.3	yes	3
84.83	odd	2		6174.2.a.c.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
686.2.a.c.1.1	✓	3	4.3	odd	2
686.2.a.d.1.3	yes	3	28.27	even	2
686.2.c.a.361.1		6	28.19	even	6
686.2.c.a.667.1		6	28.3	even	6
686.2.c.b.361.3		6	28.23	odd	6
686.2.c.b.667.3		6	28.11	odd	6
5488.2.a.a.1.1		3	7.6	odd	2
5488.2.a.f.1.3		3	1.1	even	1	trivial
6174.2.a.c.1.3		3	84.83	odd	2
6174.2.a.e.1.1		3	12.11	even	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:	\( N \)	\(=\)	\( 5488 = 2^{4} \cdot 7^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5488.a (trivial)

\(f(q)\)	\(=\)	\(q+3.24698 q^{3} +0.246980 q^{5} +7.54288 q^{9} +5.04892 q^{11} -2.10992 q^{13} +0.801938 q^{15} +4.96077 q^{17} +0.506041 q^{19} +4.93900 q^{23} -4.93900 q^{25} +14.7506 q^{27} -4.66487 q^{29} +9.07606 q^{31} +16.3937 q^{33} -2.64310 q^{37} -6.85086 q^{39} -6.70171 q^{41} -3.62565 q^{43} +1.86294 q^{45} -0.917231 q^{47} +16.1075 q^{51} -3.62565 q^{53} +1.24698 q^{55} +1.64310 q^{57} -7.67456 q^{59} -13.0368 q^{61} -0.521106 q^{65} +5.46681 q^{67} +16.0368 q^{69} -11.2295 q^{71} +5.36227 q^{73} -16.0368 q^{75} +6.56465 q^{79} +25.2664 q^{81} -0.753020 q^{83} +1.22521 q^{85} -15.1468 q^{87} +9.69202 q^{89} +29.4698 q^{93} +0.124982 q^{95} +12.2784 q^{97} +38.0834 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 5 q^{3} - 4 q^{5} + 4 q^{9} + 6 q^{11} - 7 q^{13} - 2 q^{15} + 2 q^{17} + 11 q^{19} + 5 q^{23} - 5 q^{25} + 8 q^{27} - 15 q^{29} + 12 q^{31} + 17 q^{33} - 12 q^{37} - 7 q^{39} + 7 q^{41} + q^{43}+ \cdots + 36 q^{99}+O(q^{100}) \) 3 * q + 5 * q^3 - 4 * q^5 + 4 * q^9 + 6 * q^11 - 7 * q^13 - 2 * q^15 + 2 * q^17 + 11 * q^19 + 5 * q^23 - 5 * q^25 + 8 * q^27 - 15 * q^29 + 12 * q^31 + 17 * q^33 - 12 * q^37 - 7 * q^39 + 7 * q^41 + q^43 + 11 * q^45 + 4 * q^47 + 8 * q^51 + q^53 - q^55 + 9 * q^57 - 2 * q^59 - 11 * q^61 + 14 * q^65 + 13 * q^67 + 20 * q^69 - 13 * q^71 + 9 * q^73 - 20 * q^75 - 2 * q^79 + 27 * q^81 - 7 * q^83 + 2 * q^85 - 18 * q^87 + 24 * q^89 + 41 * q^93 - 24 * q^95 + 7 * q^97 + 36 * q^99

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