LMFDB - Embedded newform 7728.2.a.bc.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7728,2,Mod(1,7728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7728, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7728.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7728.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:	$61.7083906820$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{41}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 966)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.70156$ of defining polynomial
Character	$\chi$	$=$	7728.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +3.70156 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +3.70156 q^{5} -1.00000 q^{7} +1.00000 q^{9} +4.00000 q^{11} +5.70156 q^{13} -3.70156 q^{15} +4.00000 q^{17} +5.40312 q^{19} +1.00000 q^{21} -1.00000 q^{23} +8.70156 q^{25} -1.00000 q^{27} +0.298438 q^{29} +2.00000 q^{31} -4.00000 q^{33} -3.70156 q^{35} +4.29844 q^{37} -5.70156 q^{39} -0.298438 q^{41} +1.70156 q^{43} +3.70156 q^{45} +11.1047 q^{47} +1.00000 q^{49} -4.00000 q^{51} -9.40312 q^{53} +14.8062 q^{55} -5.40312 q^{57} +7.40312 q^{59} -2.00000 q^{61} -1.00000 q^{63} +21.1047 q^{65} -14.8062 q^{67} +1.00000 q^{69} +7.40312 q^{71} +1.40312 q^{73} -8.70156 q^{75} -4.00000 q^{77} -8.00000 q^{79} +1.00000 q^{81} -13.4031 q^{83} +14.8062 q^{85} -0.298438 q^{87} -11.4031 q^{89} -5.70156 q^{91} -2.00000 q^{93} +20.0000 q^{95} +6.29844 q^{97} +4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + q^5 - 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9} + 8 q^{11} + 5 q^{13} - q^{15} + 8 q^{17} - 2 q^{19} + 2 q^{21} - 2 q^{23} + 11 q^{25} - 2 q^{27} + 7 q^{29} + 4 q^{31} - 8 q^{33} - q^{35} + 15 q^{37} - 5 q^{39} - 7 q^{41} - 3 q^{43} + q^{45} + 3 q^{47} + 2 q^{49} - 8 q^{51} - 6 q^{53} + 4 q^{55} + 2 q^{57} + 2 q^{59} - 4 q^{61} - 2 q^{63} + 23 q^{65} - 4 q^{67} + 2 q^{69} + 2 q^{71} - 10 q^{73} - 11 q^{75} - 8 q^{77} - 16 q^{79} + 2 q^{81} - 14 q^{83} + 4 q^{85} - 7 q^{87} - 10 q^{89} - 5 q^{91} - 4 q^{93} + 40 q^{95} + 19 q^{97} + 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + q^5 - 2 * q^7 + 2 * q^9 + 8 * q^11 + 5 * q^13 - q^15 + 8 * q^17 - 2 * q^19 + 2 * q^21 - 2 * q^23 + 11 * q^25 - 2 * q^27 + 7 * q^29 + 4 * q^31 - 8 * q^33 - q^35 + 15 * q^37 - 5 * q^39 - 7 * q^41 - 3 * q^43 + q^45 + 3 * q^47 + 2 * q^49 - 8 * q^51 - 6 * q^53 + 4 * q^55 + 2 * q^57 + 2 * q^59 - 4 * q^61 - 2 * q^63 + 23 * q^65 - 4 * q^67 + 2 * q^69 + 2 * q^71 - 10 * q^73 - 11 * q^75 - 8 * q^77 - 16 * q^79 + 2 * q^81 - 14 * q^83 + 4 * q^85 - 7 * q^87 - 10 * q^89 - 5 * q^91 - 4 * q^93 + 40 * q^95 + 19 * q^97 + 8 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	3.70156	1.65539	0.827694	−	0.561179i	$-0.189652\pi$
$5$	3.70156	1.65539	0.827694	+	0.561179i	$0.189652\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	5.70156	1.58133	0.790664	−	0.612250i	$-0.209735\pi$
$13$	5.70156	1.58133	0.790664	+	0.612250i	$0.209735\pi$
$14$	0	0
$15$	−3.70156	−0.955739
$16$	0	0
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	5.40312	1.23956	0.619781	−	0.784775i	$-0.287221\pi$
$19$	5.40312	1.23956	0.619781	+	0.784775i	$0.287221\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	8.70156	1.74031
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	0.298438	0.0554185	0.0277093	−	0.999616i	$-0.491179\pi$
$29$	0.298438	0.0554185	0.0277093	+	0.999616i	$0.491179\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	0	0
$33$	−4.00000	−0.696311
$34$	0	0
$35$	−3.70156	−0.625678
$36$	0	0
$37$	4.29844	0.706659	0.353329	−	0.935499i	$-0.385049\pi$
$37$	4.29844	0.706659	0.353329	+	0.935499i	$0.385049\pi$
$38$	0	0
$39$	−5.70156	−0.912981
$40$	0	0
$41$	−0.298438	−0.0466082	−0.0233041	−	0.999728i	$-0.507419\pi$
$41$	−0.298438	−0.0466082	−0.0233041	+	0.999728i	$0.507419\pi$
$42$	0	0
$43$	1.70156	0.259486	0.129743	−	0.991548i	$-0.458585\pi$
$43$	1.70156	0.259486	0.129743	+	0.991548i	$0.458585\pi$
$44$	0	0
$45$	3.70156	0.551796
$46$	0	0
$47$	11.1047	1.61978	0.809892	−	0.586578i	$-0.199525\pi$
$47$	11.1047	1.61978	0.809892	+	0.586578i	$0.199525\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.00000	−0.560112
$52$	0	0
$53$	−9.40312	−1.29162	−0.645809	−	0.763499i	$-0.723480\pi$
$53$	−9.40312	−1.29162	−0.645809	+	0.763499i	$0.723480\pi$
$54$	0	0
$55$	14.8062	1.99647
$56$	0	0
$57$	−5.40312	−0.715661
$58$	0	0
$59$	7.40312	0.963805	0.481902	−	0.876225i	$-0.339946\pi$
$59$	7.40312	0.963805	0.481902	+	0.876225i	$0.339946\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	21.1047	2.61771
$66$	0	0
$67$	−14.8062	−1.80887	−0.904436	−	0.426610i	$-0.859708\pi$
$67$	−14.8062	−1.80887	−0.904436	+	0.426610i	$0.859708\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	7.40312	0.878589	0.439295	−	0.898343i	$-0.355228\pi$
$71$	7.40312	0.878589	0.439295	+	0.898343i	$0.355228\pi$
$72$	0	0
$73$	1.40312	0.164223	0.0821116	−	0.996623i	$-0.473834\pi$
$73$	1.40312	0.164223	0.0821116	+	0.996623i	$0.473834\pi$
$74$	0	0
$75$	−8.70156	−1.00477
$76$	0	0
$77$	−4.00000	−0.455842
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−13.4031	−1.47118	−0.735592	−	0.677425i	$-0.763096\pi$
$83$	−13.4031	−1.47118	−0.735592	+	0.677425i	$0.763096\pi$
$84$	0	0
$85$	14.8062	1.60596
$86$	0	0
$87$	−0.298438	−0.0319959
$88$	0	0
$89$	−11.4031	−1.20873	−0.604364	−	0.796708i	$-0.706573\pi$
$89$	−11.4031	−1.20873	−0.604364	+	0.796708i	$0.706573\pi$
$90$	0	0
$91$	−5.70156	−0.597686
$92$	0	0
$93$	−2.00000	−0.207390
$94$	0	0
$95$	20.0000	2.05196
$96$	0	0
$97$	6.29844	0.639509	0.319755	−	0.947500i	$-0.396399\pi$
$97$	6.29844	0.639509	0.319755	+	0.947500i	$0.396399\pi$
$98$	0	0
$99$	4.00000	0.402015
$100$	0	0
$101$	−3.40312	−0.338624	−0.169312	−	0.985563i	$-0.554154\pi$
$101$	−3.40312	−0.338624	−0.169312	+	0.985563i	$0.554154\pi$
$102$	0	0
$103$	−2.29844	−0.226472	−0.113236	−	0.993568i	$-0.536122\pi$
$103$	−2.29844	−0.226472	−0.113236	+	0.993568i	$0.536122\pi$
$104$	0	0
$105$	3.70156	0.361235
$106$	0	0
$107$	−18.8062	−1.81807	−0.909034	−	0.416721i	$-0.863179\pi$
$107$	−18.8062	−1.81807	−0.909034	+	0.416721i	$0.863179\pi$
$108$	0	0
$109$	−7.70156	−0.737676	−0.368838	−	0.929494i	$-0.620244\pi$
$109$	−7.70156	−0.737676	−0.368838	+	0.929494i	$0.620244\pi$
$110$	0	0
$111$	−4.29844	−0.407990
$112$	0	0
$113$	7.10469	0.668353	0.334176	−	0.942511i	$-0.391542\pi$
$113$	7.10469	0.668353	0.334176	+	0.942511i	$0.391542\pi$
$114$	0	0
$115$	−3.70156	−0.345172
$116$	0	0
$117$	5.70156	0.527110
$118$	0	0
$119$	−4.00000	−0.366679
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	0.298438	0.0269092
$124$	0	0
$125$	13.7016	1.22550
$126$	0	0
$127$	5.70156	0.505932	0.252966	−	0.967475i	$-0.418594\pi$
$127$	5.70156	0.505932	0.252966	+	0.967475i	$0.418594\pi$
$128$	0	0
$129$	−1.70156	−0.149814
$130$	0	0
$131$	7.40312	0.646814	0.323407	−	0.946260i	$-0.395172\pi$
$131$	7.40312	0.646814	0.323407	+	0.946260i	$0.395172\pi$
$132$	0	0
$133$	−5.40312	−0.468510
$134$	0	0
$135$	−3.70156	−0.318580
$136$	0	0
$137$	−19.7016	−1.68322	−0.841609	−	0.540087i	$-0.818391\pi$
$137$	−19.7016	−1.68322	−0.841609	+	0.540087i	$0.818391\pi$
$138$	0	0
$139$	−17.7016	−1.50143	−0.750713	−	0.660628i	$-0.770290\pi$
$139$	−17.7016	−1.50143	−0.750713	+	0.660628i	$0.770290\pi$
$140$	0	0
$141$	−11.1047	−0.935183
$142$	0	0
$143$	22.8062	1.90715
$144$	0	0
$145$	1.10469	0.0917392
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−20.2094	−1.65562	−0.827808	−	0.561011i	$-0.810412\pi$
$149$	−20.2094	−1.65562	−0.827808	+	0.561011i	$0.810412\pi$
$150$	0	0
$151$	−23.9109	−1.94584	−0.972922	−	0.231133i	$-0.925757\pi$
$151$	−23.9109	−1.94584	−0.972922	+	0.231133i	$0.925757\pi$
$152$	0	0
$153$	4.00000	0.323381
$154$	0	0
$155$	7.40312	0.594633
$156$	0	0
$157$	−1.40312	−0.111982	−0.0559908	−	0.998431i	$-0.517832\pi$
$157$	−1.40312	−0.111982	−0.0559908	+	0.998431i	$0.517832\pi$
$158$	0	0
$159$	9.40312	0.745716
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	10.8062	0.846411	0.423205	−	0.906034i	$-0.360905\pi$
$163$	10.8062	0.846411	0.423205	+	0.906034i	$0.360905\pi$
$164$	0	0
$165$	−14.8062	−1.15266
$166$	0	0
$167$	13.4031	1.03716	0.518582	−	0.855028i	$-0.326460\pi$
$167$	13.4031	1.03716	0.518582	+	0.855028i	$0.326460\pi$
$168$	0	0
$169$	19.5078	1.50060
$170$	0	0
$171$	5.40312	0.413187
$172$	0	0
$173$	19.4031	1.47519	0.737596	−	0.675242i	$-0.235961\pi$
$173$	19.4031	1.47519	0.737596	+	0.675242i	$0.235961\pi$
$174$	0	0
$175$	−8.70156	−0.657776
$176$	0	0
$177$	−7.40312	−0.556453
$178$	0	0
$179$	25.1047	1.87641	0.938206	−	0.346077i	$-0.112486\pi$
$179$	25.1047	1.87641	0.938206	+	0.346077i	$0.112486\pi$
$180$	0	0
$181$	−16.8062	−1.24920	−0.624599	−	0.780945i	$-0.714738\pi$
$181$	−16.8062	−1.24920	−0.624599	+	0.780945i	$0.714738\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	0	0
$185$	15.9109	1.16980
$186$	0	0
$187$	16.0000	1.17004
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−22.8062	−1.65020	−0.825101	−	0.564985i	$-0.808882\pi$
$191$	−22.8062	−1.65020	−0.825101	+	0.564985i	$0.808882\pi$
$192$	0	0
$193$	−23.1047	−1.66311	−0.831556	−	0.555441i	$-0.812549\pi$
$193$	−23.1047	−1.66311	−0.831556	+	0.555441i	$0.812549\pi$
$194$	0	0
$195$	−21.1047	−1.51134
$196$	0	0
$197$	22.5078	1.60362	0.801808	−	0.597582i	$-0.203872\pi$
$197$	22.5078	1.60362	0.801808	+	0.597582i	$0.203872\pi$
$198$	0	0
$199$	2.29844	0.162932	0.0814660	−	0.996676i	$-0.474040\pi$
$199$	2.29844	0.162932	0.0814660	+	0.996676i	$0.474040\pi$
$200$	0	0
$201$	14.8062	1.04435
$202$	0	0
$203$	−0.298438	−0.0209462
$204$	0	0
$205$	−1.10469	−0.0771546
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	21.6125	1.49497
$210$	0	0
$211$	−10.8062	−0.743933	−0.371966	−	0.928246i	$-0.621316\pi$
$211$	−10.8062	−0.743933	−0.371966	+	0.928246i	$0.621316\pi$
$212$	0	0
$213$	−7.40312	−0.507254
$214$	0	0
$215$	6.29844	0.429550
$216$	0	0
$217$	−2.00000	−0.135769
$218$	0	0
$219$	−1.40312	−0.0948143
$220$	0	0
$221$	22.8062	1.53411
$222$	0	0
$223$	22.0000	1.47323	0.736614	−	0.676313i	$-0.236423\pi$
$223$	22.0000	1.47323	0.736614	+	0.676313i	$0.236423\pi$
$224$	0	0
$225$	8.70156	0.580104
$226$	0	0
$227$	23.1047	1.53351	0.766756	−	0.641939i	$-0.221870\pi$
$227$	23.1047	1.53351	0.766756	+	0.641939i	$0.221870\pi$
$228$	0	0
$229$	−14.5969	−0.964589	−0.482294	−	0.876009i	$-0.660196\pi$
$229$	−14.5969	−0.964589	−0.482294	+	0.876009i	$0.660196\pi$
$230$	0	0
$231$	4.00000	0.263181
$232$	0	0
$233$	−26.0000	−1.70332	−0.851658	−	0.524097i	$-0.824403\pi$
$233$	−26.0000	−1.70332	−0.851658	+	0.524097i	$0.824403\pi$
$234$	0	0
$235$	41.1047	2.68137
$236$	0	0
$237$	8.00000	0.519656
$238$	0	0
$239$	−10.8062	−0.698998	−0.349499	−	0.936937i	$-0.613648\pi$
$239$	−10.8062	−0.698998	−0.349499	+	0.936937i	$0.613648\pi$
$240$	0	0
$241$	−15.9109	−1.02491	−0.512457	−	0.858713i	$-0.671264\pi$
$241$	−15.9109	−1.02491	−0.512457	+	0.858713i	$0.671264\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	3.70156	0.236484
$246$	0	0
$247$	30.8062	1.96015
$248$	0	0
$249$	13.4031	0.849388
$250$	0	0
$251$	−19.7016	−1.24355	−0.621776	−	0.783195i	$-0.713588\pi$
$251$	−19.7016	−1.24355	−0.621776	+	0.783195i	$0.713588\pi$
$252$	0	0
$253$	−4.00000	−0.251478
$254$	0	0
$255$	−14.8062	−0.927203
$256$	0	0
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	0	0
$259$	−4.29844	−0.267092
$260$	0	0
$261$	0.298438	0.0184728
$262$	0	0
$263$	−9.70156	−0.598224	−0.299112	−	0.954218i	$-0.596690\pi$
$263$	−9.70156	−0.598224	−0.299112	+	0.954218i	$0.596690\pi$
$264$	0	0
$265$	−34.8062	−2.13813
$266$	0	0
$267$	11.4031	0.697860
$268$	0	0
$269$	−10.2094	−0.622476	−0.311238	−	0.950332i	$-0.600744\pi$
$269$	−10.2094	−0.622476	−0.311238	+	0.950332i	$0.600744\pi$
$270$	0	0
$271$	−1.40312	−0.0852337	−0.0426169	−	0.999091i	$-0.513569\pi$
$271$	−1.40312	−0.0852337	−0.0426169	+	0.999091i	$0.513569\pi$
$272$	0	0
$273$	5.70156	0.345074
$274$	0	0
$275$	34.8062	2.09890
$276$	0	0
$277$	−21.4031	−1.28599	−0.642995	−	0.765871i	$-0.722308\pi$
$277$	−21.4031	−1.28599	−0.642995	+	0.765871i	$0.722308\pi$
$278$	0	0
$279$	2.00000	0.119737
$280$	0	0
$281$	14.5078	0.865463	0.432732	−	0.901523i	$-0.357550\pi$
$281$	14.5078	0.865463	0.432732	+	0.901523i	$0.357550\pi$
$282$	0	0
$283$	−4.80625	−0.285702	−0.142851	−	0.989744i	$-0.545627\pi$
$283$	−4.80625	−0.285702	−0.142851	+	0.989744i	$0.545627\pi$
$284$	0	0
$285$	−20.0000	−1.18470
$286$	0	0
$287$	0.298438	0.0176162
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−6.29844	−0.369221
$292$	0	0
$293$	−20.8062	−1.21551	−0.607757	−	0.794123i	$-0.707931\pi$
$293$	−20.8062	−1.21551	−0.607757	+	0.794123i	$0.707931\pi$
$294$	0	0
$295$	27.4031	1.59547
$296$	0	0
$297$	−4.00000	−0.232104
$298$	0	0
$299$	−5.70156	−0.329730
$300$	0	0
$301$	−1.70156	−0.0980764
$302$	0	0
$303$	3.40312	0.195504
$304$	0	0
$305$	−7.40312	−0.423902
$306$	0	0
$307$	−11.9109	−0.679793	−0.339896	−	0.940463i	$-0.610392\pi$
$307$	−11.9109	−0.679793	−0.339896	+	0.940463i	$0.610392\pi$
$308$	0	0
$309$	2.29844	0.130754
$310$	0	0
$311$	−13.4031	−0.760021	−0.380011	−	0.924982i	$-0.624080\pi$
$311$	−13.4031	−0.760021	−0.380011	+	0.924982i	$0.624080\pi$
$312$	0	0
$313$	−6.20937	−0.350974	−0.175487	−	0.984482i	$-0.556150\pi$
$313$	−6.20937	−0.350974	−0.175487	+	0.984482i	$0.556150\pi$
$314$	0	0
$315$	−3.70156	−0.208559
$316$	0	0
$317$	−8.29844	−0.466087	−0.233043	−	0.972466i	$-0.574868\pi$
$317$	−8.29844	−0.466087	−0.233043	+	0.972466i	$0.574868\pi$
$318$	0	0
$319$	1.19375	0.0668373
$320$	0	0
$321$	18.8062	1.04966
$322$	0	0
$323$	21.6125	1.20255
$324$	0	0
$325$	49.6125	2.75201
$326$	0	0
$327$	7.70156	0.425897
$328$	0	0
$329$	−11.1047	−0.612221
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	0	0
$333$	4.29844	0.235553
$334$	0	0
$335$	−54.8062	−2.99439
$336$	0	0
$337$	−4.20937	−0.229299	−0.114650	−	0.993406i	$-0.536575\pi$
$337$	−4.20937	−0.229299	−0.114650	+	0.993406i	$0.536575\pi$
$338$	0	0
$339$	−7.10469	−0.385874
$340$	0	0
$341$	8.00000	0.433224
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	3.70156	0.199285
$346$	0	0
$347$	10.2984	0.552849	0.276425	−	0.961036i	$-0.410850\pi$
$347$	10.2984	0.552849	0.276425	+	0.961036i	$0.410850\pi$
$348$	0	0
$349$	12.5969	0.674295	0.337148	−	0.941452i	$-0.390538\pi$
$349$	12.5969	0.674295	0.337148	+	0.941452i	$0.390538\pi$
$350$	0	0
$351$	−5.70156	−0.304327
$352$	0	0
$353$	4.29844	0.228783	0.114391	−	0.993436i	$-0.463508\pi$
$353$	4.29844	0.228783	0.114391	+	0.993436i	$0.463508\pi$
$354$	0	0
$355$	27.4031	1.45441
$356$	0	0
$357$	4.00000	0.211702
$358$	0	0
$359$	2.89531	0.152809	0.0764044	−	0.997077i	$-0.475656\pi$
$359$	2.89531	0.152809	0.0764044	+	0.997077i	$0.475656\pi$
$360$	0	0
$361$	10.1938	0.536513
$362$	0	0
$363$	−5.00000	−0.262432
$364$	0	0
$365$	5.19375	0.271853
$366$	0	0
$367$	17.1047	0.892857	0.446429	−	0.894819i	$-0.352696\pi$
$367$	17.1047	0.892857	0.446429	+	0.894819i	$0.352696\pi$
$368$	0	0
$369$	−0.298438	−0.0155361
$370$	0	0
$371$	9.40312	0.488186
$372$	0	0
$373$	−0.806248	−0.0417460	−0.0208730	−	0.999782i	$-0.506645\pi$
$373$	−0.806248	−0.0417460	−0.0208730	+	0.999782i	$0.506645\pi$
$374$	0	0
$375$	−13.7016	−0.707546
$376$	0	0
$377$	1.70156	0.0876349
$378$	0	0
$379$	−13.7016	−0.703802	−0.351901	−	0.936037i	$-0.614465\pi$
$379$	−13.7016	−0.703802	−0.351901	+	0.936037i	$0.614465\pi$
$380$	0	0
$381$	−5.70156	−0.292100
$382$	0	0
$383$	−29.6125	−1.51313	−0.756564	−	0.653920i	$-0.773123\pi$
$383$	−29.6125	−1.51313	−0.756564	+	0.653920i	$0.773123\pi$
$384$	0	0
$385$	−14.8062	−0.754596
$386$	0	0
$387$	1.70156	0.0864953
$388$	0	0
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	−4.00000	−0.202289
$392$	0	0
$393$	−7.40312	−0.373438
$394$	0	0
$395$	−29.6125	−1.48997
$396$	0	0
$397$	8.59688	0.431465	0.215732	−	0.976453i	$-0.430786\pi$
$397$	8.59688	0.431465	0.215732	+	0.976453i	$0.430786\pi$
$398$	0	0
$399$	5.40312	0.270495
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	0	0
$403$	11.4031	0.568030
$404$	0	0
$405$	3.70156	0.183932
$406$	0	0
$407$	17.1938	0.852263
$408$	0	0
$409$	5.40312	0.267167	0.133584	−	0.991038i	$-0.457352\pi$
$409$	5.40312	0.267167	0.133584	+	0.991038i	$0.457352\pi$
$410$	0	0
$411$	19.7016	0.971806
$412$	0	0
$413$	−7.40312	−0.364284
$414$	0	0
$415$	−49.6125	−2.43538
$416$	0	0
$417$	17.7016	0.866849
$418$	0	0
$419$	0.209373	0.0102285	0.00511426	−	0.999987i	$-0.498372\pi$
$419$	0.209373	0.0102285	0.00511426	+	0.999987i	$0.498372\pi$
$420$	0	0
$421$	−15.7016	−0.765247	−0.382624	−	0.923904i	$-0.624979\pi$
$421$	−15.7016	−0.765247	−0.382624	+	0.923904i	$0.624979\pi$
$422$	0	0
$423$	11.1047	0.539928
$424$	0	0
$425$	34.8062	1.68835
$426$	0	0
$427$	2.00000	0.0967868
$428$	0	0
$429$	−22.8062	−1.10110
$430$	0	0
$431$	3.91093	0.188383	0.0941916	−	0.995554i	$-0.469973\pi$
$431$	3.91093	0.188383	0.0941916	+	0.995554i	$0.469973\pi$
$432$	0	0
$433$	31.3141	1.50486	0.752429	−	0.658674i	$-0.228882\pi$
$433$	31.3141	1.50486	0.752429	+	0.658674i	$0.228882\pi$
$434$	0	0
$435$	−1.10469	−0.0529657
$436$	0	0
$437$	−5.40312	−0.258466
$438$	0	0
$439$	22.0000	1.05000	0.525001	−	0.851101i	$-0.324065\pi$
$439$	22.0000	1.05000	0.525001	+	0.851101i	$0.324065\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	13.7016	0.650981	0.325490	−	0.945545i	$-0.394471\pi$
$443$	13.7016	0.650981	0.325490	+	0.945545i	$0.394471\pi$
$444$	0	0
$445$	−42.2094	−2.00092
$446$	0	0
$447$	20.2094	0.955871
$448$	0	0
$449$	−38.4187	−1.81309	−0.906546	−	0.422106i	$-0.861291\pi$
$449$	−38.4187	−1.81309	−0.906546	+	0.422106i	$0.861291\pi$
$450$	0	0
$451$	−1.19375	−0.0562116
$452$	0	0
$453$	23.9109	1.12343
$454$	0	0
$455$	−21.1047	−0.989403
$456$	0	0
$457$	28.2094	1.31958	0.659789	−	0.751451i	$-0.270645\pi$
$457$	28.2094	1.31958	0.659789	+	0.751451i	$0.270645\pi$
$458$	0	0
$459$	−4.00000	−0.186704
$460$	0	0
$461$	−2.20937	−0.102901	−0.0514504	−	0.998676i	$-0.516384\pi$
$461$	−2.20937	−0.102901	−0.0514504	+	0.998676i	$0.516384\pi$
$462$	0	0
$463$	22.2984	1.03630	0.518148	−	0.855291i	$-0.326622\pi$
$463$	22.2984	1.03630	0.518148	+	0.855291i	$0.326622\pi$
$464$	0	0
$465$	−7.40312	−0.343312
$466$	0	0
$467$	3.70156	0.171288	0.0856439	−	0.996326i	$-0.472705\pi$
$467$	3.70156	0.171288	0.0856439	+	0.996326i	$0.472705\pi$
$468$	0	0
$469$	14.8062	0.683689
$470$	0	0
$471$	1.40312	0.0646526
$472$	0	0
$473$	6.80625	0.312952
$474$	0	0
$475$	47.0156	2.15722
$476$	0	0
$477$	−9.40312	−0.430539
$478$	0	0
$479$	−16.5969	−0.758331	−0.379165	−	0.925329i	$-0.623789\pi$
$479$	−16.5969	−0.758331	−0.379165	+	0.925329i	$0.623789\pi$
$480$	0	0
$481$	24.5078	1.11746
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	23.3141	1.05864
$486$	0	0
$487$	6.29844	0.285409	0.142705	−	0.989765i	$-0.454420\pi$
$487$	6.29844	0.285409	0.142705	+	0.989765i	$0.454420\pi$
$488$	0	0
$489$	−10.8062	−0.488675
$490$	0	0
$491$	9.61250	0.433806	0.216903	−	0.976193i	$-0.430404\pi$
$491$	9.61250	0.433806	0.216903	+	0.976193i	$0.430404\pi$
$492$	0	0
$493$	1.19375	0.0537639
$494$	0	0
$495$	14.8062	0.665491
$496$	0	0
$497$	−7.40312	−0.332076
$498$	0	0
$499$	23.4031	1.04767	0.523834	−	0.851820i	$-0.324501\pi$
$499$	23.4031	1.04767	0.523834	+	0.851820i	$0.324501\pi$
$500$	0	0
$501$	−13.4031	−0.598807
$502$	0	0
$503$	−4.59688	−0.204965	−0.102482	−	0.994735i	$-0.532678\pi$
$503$	−4.59688	−0.204965	−0.102482	+	0.994735i	$0.532678\pi$
$504$	0	0
$505$	−12.5969	−0.560554
$506$	0	0
$507$	−19.5078	−0.866372
$508$	0	0
$509$	37.6125	1.66714	0.833572	−	0.552410i	$-0.186292\pi$
$509$	37.6125	1.66714	0.833572	+	0.552410i	$0.186292\pi$
$510$	0	0
$511$	−1.40312	−0.0620706
$512$	0	0
$513$	−5.40312	−0.238554
$514$	0	0
$515$	−8.50781	−0.374899
$516$	0	0
$517$	44.4187	1.95353
$518$	0	0
$519$	−19.4031	−0.851703
$520$	0	0
$521$	−20.5969	−0.902366	−0.451183	−	0.892432i	$-0.648998\pi$
$521$	−20.5969	−0.902366	−0.451183	+	0.892432i	$0.648998\pi$
$522$	0	0
$523$	−20.8062	−0.909794	−0.454897	−	0.890544i	$-0.650324\pi$
$523$	−20.8062	−0.909794	−0.454897	+	0.890544i	$0.650324\pi$
$524$	0	0
$525$	8.70156	0.379767
$526$	0	0
$527$	8.00000	0.348485
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	7.40312	0.321268
$532$	0	0
$533$	−1.70156	−0.0737028
$534$	0	0
$535$	−69.6125	−3.00961
$536$	0	0
$537$	−25.1047	−1.08335
$538$	0	0
$539$	4.00000	0.172292
$540$	0	0
$541$	33.4031	1.43611	0.718056	−	0.695985i	$-0.245032\pi$
$541$	33.4031	1.43611	0.718056	+	0.695985i	$0.245032\pi$
$542$	0	0
$543$	16.8062	0.721225
$544$	0	0
$545$	−28.5078	−1.22114
$546$	0	0
$547$	−29.0156	−1.24062	−0.620309	−	0.784357i	$-0.712993\pi$
$547$	−29.0156	−1.24062	−0.620309	+	0.784357i	$0.712993\pi$
$548$	0	0
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	1.61250	0.0686947
$552$	0	0
$553$	8.00000	0.340195
$554$	0	0
$555$	−15.9109	−0.675382
$556$	0	0
$557$	−37.4031	−1.58482	−0.792411	−	0.609988i	$-0.791174\pi$
$557$	−37.4031	−1.58482	−0.792411	+	0.609988i	$0.791174\pi$
$558$	0	0
$559$	9.70156	0.410332
$560$	0	0
$561$	−16.0000	−0.675521
$562$	0	0
$563$	3.10469	0.130847	0.0654235	−	0.997858i	$-0.479160\pi$
$563$	3.10469	0.130847	0.0654235	+	0.997858i	$0.479160\pi$
$564$	0	0
$565$	26.2984	1.10638
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	31.7016	1.32900	0.664499	−	0.747289i	$-0.268645\pi$
$569$	31.7016	1.32900	0.664499	+	0.747289i	$0.268645\pi$
$570$	0	0
$571$	14.8062	0.619622	0.309811	−	0.950798i	$-0.399734\pi$
$571$	14.8062	0.619622	0.309811	+	0.950798i	$0.399734\pi$
$572$	0	0
$573$	22.8062	0.952745
$574$	0	0
$575$	−8.70156	−0.362880
$576$	0	0
$577$	−10.0000	−0.416305	−0.208153	−	0.978096i	$-0.566745\pi$
$577$	−10.0000	−0.416305	−0.208153	+	0.978096i	$0.566745\pi$
$578$	0	0
$579$	23.1047	0.960198
$580$	0	0
$581$	13.4031	0.556055
$582$	0	0
$583$	−37.6125	−1.55775
$584$	0	0
$585$	21.1047	0.872571
$586$	0	0
$587$	−14.8062	−0.611119	−0.305560	−	0.952173i	$-0.598844\pi$
$587$	−14.8062	−0.611119	−0.305560	+	0.952173i	$0.598844\pi$
$588$	0	0
$589$	10.8062	0.445264
$590$	0	0
$591$	−22.5078	−0.925848
$592$	0	0
$593$	43.7016	1.79461	0.897304	−	0.441413i	$-0.145523\pi$
$593$	43.7016	1.79461	0.897304	+	0.441413i	$0.145523\pi$
$594$	0	0
$595$	−14.8062	−0.606997
$596$	0	0
$597$	−2.29844	−0.0940688
$598$	0	0
$599$	−13.6125	−0.556192	−0.278096	−	0.960553i	$-0.589703\pi$
$599$	−13.6125	−0.556192	−0.278096	+	0.960553i	$0.589703\pi$
$600$	0	0
$601$	−13.4031	−0.546725	−0.273362	−	0.961911i	$-0.588136\pi$
$601$	−13.4031	−0.546725	−0.273362	+	0.961911i	$0.588136\pi$
$602$	0	0
$603$	−14.8062	−0.602957
$604$	0	0
$605$	18.5078	0.752450
$606$	0	0
$607$	39.6125	1.60782	0.803911	−	0.594750i	$-0.202749\pi$
$607$	39.6125	1.60782	0.803911	+	0.594750i	$0.202749\pi$
$608$	0	0
$609$	0.298438	0.0120933
$610$	0	0
$611$	63.3141	2.56141
$612$	0	0
$613$	3.70156	0.149505	0.0747523	−	0.997202i	$-0.476183\pi$
$613$	3.70156	0.149505	0.0747523	+	0.997202i	$0.476183\pi$
$614$	0	0
$615$	1.10469	0.0445453
$616$	0	0
$617$	−19.1938	−0.772711	−0.386356	−	0.922350i	$-0.626266\pi$
$617$	−19.1938	−0.772711	−0.386356	+	0.922350i	$0.626266\pi$
$618$	0	0
$619$	7.19375	0.289141	0.144571	−	0.989494i	$-0.453820\pi$
$619$	7.19375	0.289141	0.144571	+	0.989494i	$0.453820\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	11.4031	0.456857
$624$	0	0
$625$	7.20937	0.288375
$626$	0	0
$627$	−21.6125	−0.863120
$628$	0	0
$629$	17.1938	0.685560
$630$	0	0
$631$	29.6125	1.17885	0.589427	−	0.807821i	$-0.299353\pi$
$631$	29.6125	1.17885	0.589427	+	0.807821i	$0.299353\pi$
$632$	0	0
$633$	10.8062	0.429510
$634$	0	0
$635$	21.1047	0.837514
$636$	0	0
$637$	5.70156	0.225904
$638$	0	0
$639$	7.40312	0.292863
$640$	0	0
$641$	−20.7172	−0.818280	−0.409140	−	0.912472i	$-0.634171\pi$
$641$	−20.7172	−0.818280	−0.409140	+	0.912472i	$0.634171\pi$
$642$	0	0
$643$	−0.806248	−0.0317953	−0.0158977	−	0.999874i	$-0.505061\pi$
$643$	−0.806248	−0.0317953	−0.0158977	+	0.999874i	$0.505061\pi$
$644$	0	0
$645$	−6.29844	−0.248001
$646$	0	0
$647$	6.59688	0.259350	0.129675	−	0.991557i	$-0.458607\pi$
$647$	6.59688	0.259350	0.129675	+	0.991557i	$0.458607\pi$
$648$	0	0
$649$	29.6125	1.16239
$650$	0	0
$651$	2.00000	0.0783862
$652$	0	0
$653$	7.10469	0.278028	0.139014	−	0.990290i	$-0.455607\pi$
$653$	7.10469	0.278028	0.139014	+	0.990290i	$0.455607\pi$
$654$	0	0
$655$	27.4031	1.07073
$656$	0	0
$657$	1.40312	0.0547411
$658$	0	0
$659$	−8.00000	−0.311636	−0.155818	−	0.987786i	$-0.549801\pi$
$659$	−8.00000	−0.311636	−0.155818	+	0.987786i	$0.549801\pi$
$660$	0	0
$661$	42.4187	1.64990	0.824949	−	0.565207i	$-0.191204\pi$
$661$	42.4187	1.64990	0.824949	+	0.565207i	$0.191204\pi$
$662$	0	0
$663$	−22.8062	−0.885721
$664$	0	0
$665$	−20.0000	−0.775567
$666$	0	0
$667$	−0.298438	−0.0115556
$668$	0	0
$669$	−22.0000	−0.850569
$670$	0	0
$671$	−8.00000	−0.308837
$672$	0	0
$673$	7.70156	0.296873	0.148437	−	0.988922i	$-0.452576\pi$
$673$	7.70156	0.296873	0.148437	+	0.988922i	$0.452576\pi$
$674$	0	0
$675$	−8.70156	−0.334923
$676$	0	0
$677$	−50.4187	−1.93775	−0.968875	−	0.247551i	$-0.920374\pi$
$677$	−50.4187	−1.93775	−0.968875	+	0.247551i	$0.920374\pi$
$678$	0	0
$679$	−6.29844	−0.241712
$680$	0	0
$681$	−23.1047	−0.885374
$682$	0	0
$683$	−33.6125	−1.28615	−0.643073	−	0.765805i	$-0.722341\pi$
$683$	−33.6125	−1.28615	−0.643073	+	0.765805i	$0.722341\pi$
$684$	0	0
$685$	−72.9266	−2.78638
$686$	0	0
$687$	14.5969	0.556906
$688$	0	0
$689$	−53.6125	−2.04247
$690$	0	0
$691$	0.507811	0.0193180	0.00965901	−	0.999953i	$-0.496925\pi$
$691$	0.507811	0.0193180	0.00965901	+	0.999953i	$0.496925\pi$
$692$	0	0
$693$	−4.00000	−0.151947
$694$	0	0
$695$	−65.5234	−2.48545
$696$	0	0
$697$	−1.19375	−0.0452166
$698$	0	0
$699$	26.0000	0.983410
$700$	0	0
$701$	47.0156	1.77576	0.887878	−	0.460079i	$-0.152179\pi$
$701$	47.0156	1.77576	0.887878	+	0.460079i	$0.152179\pi$
$702$	0	0
$703$	23.2250	0.875947
$704$	0	0
$705$	−41.1047	−1.54809
$706$	0	0
$707$	3.40312	0.127988
$708$	0	0
$709$	23.1938	0.871060	0.435530	−	0.900174i	$-0.356561\pi$
$709$	23.1938	0.871060	0.435530	+	0.900174i	$0.356561\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	0	0
$713$	−2.00000	−0.0749006
$714$	0	0
$715$	84.4187	3.15708
$716$	0	0
$717$	10.8062	0.403567
$718$	0	0
$719$	5.91093	0.220441	0.110220	−	0.993907i	$-0.464844\pi$
$719$	5.91093	0.220441	0.110220	+	0.993907i	$0.464844\pi$
$720$	0	0
$721$	2.29844	0.0855983
$722$	0	0
$723$	15.9109	0.591734
$724$	0	0
$725$	2.59688	0.0964455
$726$	0	0
$727$	12.0000	0.445055	0.222528	−	0.974926i	$-0.428569\pi$
$727$	12.0000	0.445055	0.222528	+	0.974926i	$0.428569\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	6.80625	0.251738
$732$	0	0
$733$	52.2094	1.92840	0.964199	−	0.265181i	$-0.0854318\pi$
$733$	52.2094	1.92840	0.964199	+	0.265181i	$0.0854318\pi$
$734$	0	0
$735$	−3.70156	−0.136534
$736$	0	0
$737$	−59.2250	−2.18158
$738$	0	0
$739$	29.0156	1.06736	0.533678	−	0.845687i	$-0.320809\pi$
$739$	29.0156	1.06736	0.533678	+	0.845687i	$0.320809\pi$
$740$	0	0
$741$	−30.8062	−1.13170
$742$	0	0
$743$	−32.0000	−1.17397	−0.586983	−	0.809599i	$-0.699684\pi$
$743$	−32.0000	−1.17397	−0.586983	+	0.809599i	$0.699684\pi$
$744$	0	0
$745$	−74.8062	−2.74069
$746$	0	0
$747$	−13.4031	−0.490395
$748$	0	0
$749$	18.8062	0.687165
$750$	0	0
$751$	−29.6125	−1.08058	−0.540288	−	0.841480i	$-0.681685\pi$
$751$	−29.6125	−1.08058	−0.540288	+	0.841480i	$0.681685\pi$
$752$	0	0
$753$	19.7016	0.717965
$754$	0	0
$755$	−88.5078	−3.22113
$756$	0	0
$757$	−26.4187	−0.960206	−0.480103	−	0.877212i	$-0.659401\pi$
$757$	−26.4187	−0.960206	−0.480103	+	0.877212i	$0.659401\pi$
$758$	0	0
$759$	4.00000	0.145191
$760$	0	0
$761$	−35.6125	−1.29095	−0.645476	−	0.763781i	$-0.723341\pi$
$761$	−35.6125	−1.29095	−0.645476	+	0.763781i	$0.723341\pi$
$762$	0	0
$763$	7.70156	0.278815
$764$	0	0
$765$	14.8062	0.535321
$766$	0	0
$767$	42.2094	1.52409
$768$	0	0
$769$	15.9109	0.573763	0.286881	−	0.957966i	$-0.407381\pi$
$769$	15.9109	0.573763	0.286881	+	0.957966i	$0.407381\pi$
$770$	0	0
$771$	−2.00000	−0.0720282
$772$	0	0
$773$	−7.10469	−0.255538	−0.127769	−	0.991804i	$-0.540782\pi$
$773$	−7.10469	−0.255538	−0.127769	+	0.991804i	$0.540782\pi$
$774$	0	0
$775$	17.4031	0.625139
$776$	0	0
$777$	4.29844	0.154206
$778$	0	0
$779$	−1.61250	−0.0577737
$780$	0	0
$781$	29.6125	1.05962
$782$	0	0
$783$	−0.298438	−0.0106653
$784$	0	0
$785$	−5.19375	−0.185373
$786$	0	0
$787$	18.5969	0.662907	0.331454	−	0.943472i	$-0.392461\pi$
$787$	18.5969	0.662907	0.331454	+	0.943472i	$0.392461\pi$
$788$	0	0
$789$	9.70156	0.345385
$790$	0	0
$791$	−7.10469	−0.252614
$792$	0	0
$793$	−11.4031	−0.404937
$794$	0	0
$795$	34.8062	1.23445
$796$	0	0
$797$	15.7016	0.556178	0.278089	−	0.960555i	$-0.410299\pi$
$797$	15.7016	0.556178	0.278089	+	0.960555i	$0.410299\pi$
$798$	0	0
$799$	44.4187	1.57142
$800$	0	0
$801$	−11.4031	−0.402910
$802$	0	0
$803$	5.61250	0.198061
$804$	0	0
$805$	3.70156	0.130463
$806$	0	0
$807$	10.2094	0.359387
$808$	0	0
$809$	31.0156	1.09045	0.545226	−	0.838289i	$-0.316444\pi$
$809$	31.0156	1.09045	0.545226	+	0.838289i	$0.316444\pi$
$810$	0	0
$811$	−52.5078	−1.84380	−0.921899	−	0.387430i	$-0.873363\pi$
$811$	−52.5078	−1.84380	−0.921899	+	0.387430i	$0.873363\pi$
$812$	0	0
$813$	1.40312	0.0492097
$814$	0	0
$815$	40.0000	1.40114
$816$	0	0
$817$	9.19375	0.321649
$818$	0	0
$819$	−5.70156	−0.199229
$820$	0	0
$821$	27.6125	0.963683	0.481841	−	0.876258i	$-0.339968\pi$
$821$	27.6125	0.963683	0.481841	+	0.876258i	$0.339968\pi$
$822$	0	0
$823$	13.7016	0.477606	0.238803	−	0.971068i	$-0.423245\pi$
$823$	13.7016	0.477606	0.238803	+	0.971068i	$0.423245\pi$
$824$	0	0
$825$	−34.8062	−1.21180
$826$	0	0
$827$	47.4031	1.64837	0.824184	−	0.566322i	$-0.191634\pi$
$827$	47.4031	1.64837	0.824184	+	0.566322i	$0.191634\pi$
$828$	0	0
$829$	55.4031	1.92423	0.962115	−	0.272644i	$-0.0878981\pi$
$829$	55.4031	1.92423	0.962115	+	0.272644i	$0.0878981\pi$
$830$	0	0
$831$	21.4031	0.742466
$832$	0	0
$833$	4.00000	0.138592
$834$	0	0
$835$	49.6125	1.71691
$836$	0	0
$837$	−2.00000	−0.0691301
$838$	0	0
$839$	−42.2094	−1.45723	−0.728615	−	0.684924i	$-0.759835\pi$
$839$	−42.2094	−1.45723	−0.728615	+	0.684924i	$0.759835\pi$
$840$	0	0
$841$	−28.9109	−0.996929
$842$	0	0
$843$	−14.5078	−0.499676
$844$	0	0
$845$	72.2094	2.48408
$846$	0	0
$847$	−5.00000	−0.171802
$848$	0	0
$849$	4.80625	0.164950
$850$	0	0
$851$	−4.29844	−0.147349
$852$	0	0
$853$	9.10469	0.311739	0.155869	−	0.987778i	$-0.450182\pi$
$853$	9.10469	0.311739	0.155869	+	0.987778i	$0.450182\pi$
$854$	0	0
$855$	20.0000	0.683986
$856$	0	0
$857$	−0.298438	−0.0101944	−0.00509722	−	0.999987i	$-0.501623\pi$
$857$	−0.298438	−0.0101944	−0.00509722	+	0.999987i	$0.501623\pi$
$858$	0	0
$859$	15.4922	0.528587	0.264293	−	0.964442i	$-0.414861\pi$
$859$	15.4922	0.528587	0.264293	+	0.964442i	$0.414861\pi$
$860$	0	0
$861$	−0.298438	−0.0101707
$862$	0	0
$863$	6.38750	0.217433	0.108717	−	0.994073i	$-0.465326\pi$
$863$	6.38750	0.217433	0.108717	+	0.994073i	$0.465326\pi$
$864$	0	0
$865$	71.8219	2.44202
$866$	0	0
$867$	1.00000	0.0339618
$868$	0	0
$869$	−32.0000	−1.08553
$870$	0	0
$871$	−84.4187	−2.86042
$872$	0	0
$873$	6.29844	0.213170
$874$	0	0
$875$	−13.7016	−0.463197
$876$	0	0
$877$	−17.4031	−0.587662	−0.293831	−	0.955857i	$-0.594930\pi$
$877$	−17.4031	−0.587662	−0.293831	+	0.955857i	$0.594930\pi$
$878$	0	0
$879$	20.8062	0.701777
$880$	0	0
$881$	6.20937	0.209199	0.104600	−	0.994514i	$-0.466644\pi$
$881$	6.20937	0.209199	0.104600	+	0.994514i	$0.466644\pi$
$882$	0	0
$883$	24.5969	0.827751	0.413875	−	0.910334i	$-0.364175\pi$
$883$	24.5969	0.827751	0.413875	+	0.910334i	$0.364175\pi$
$884$	0	0
$885$	−27.4031	−0.921146
$886$	0	0
$887$	28.2094	0.947178	0.473589	−	0.880746i	$-0.342958\pi$
$887$	28.2094	0.947178	0.473589	+	0.880746i	$0.342958\pi$
$888$	0	0
$889$	−5.70156	−0.191224
$890$	0	0
$891$	4.00000	0.134005
$892$	0	0
$893$	60.0000	2.00782
$894$	0	0
$895$	92.9266	3.10619
$896$	0	0
$897$	5.70156	0.190370
$898$	0	0
$899$	0.596876	0.0199069
$900$	0	0
$901$	−37.6125	−1.25305
$902$	0	0
$903$	1.70156	0.0566244
$904$	0	0
$905$	−62.2094	−2.06791
$906$	0	0
$907$	−20.5078	−0.680951	−0.340475	−	0.940253i	$-0.610588\pi$
$907$	−20.5078	−0.680951	−0.340475	+	0.940253i	$0.610588\pi$
$908$	0	0
$909$	−3.40312	−0.112875
$910$	0	0
$911$	−53.1047	−1.75944	−0.879718	−	0.475495i	$-0.842269\pi$
$911$	−53.1047	−1.75944	−0.879718	+	0.475495i	$0.842269\pi$
$912$	0	0
$913$	−53.6125	−1.77431
$914$	0	0
$915$	7.40312	0.244740
$916$	0	0
$917$	−7.40312	−0.244473
$918$	0	0
$919$	52.4187	1.72913	0.864567	−	0.502517i	$-0.167593\pi$
$919$	52.4187	1.72913	0.864567	+	0.502517i	$0.167593\pi$
$920$	0	0
$921$	11.9109	0.392479
$922$	0	0
$923$	42.2094	1.38934
$924$	0	0
$925$	37.4031	1.22981
$926$	0	0
$927$	−2.29844	−0.0754906
$928$	0	0
$929$	−45.9109	−1.50629	−0.753144	−	0.657855i	$-0.771464\pi$
$929$	−45.9109	−1.50629	−0.753144	+	0.657855i	$0.771464\pi$
$930$	0	0
$931$	5.40312	0.177080
$932$	0	0
$933$	13.4031	0.438799
$934$	0	0
$935$	59.2250	1.93686
$936$	0	0
$937$	25.1047	0.820134	0.410067	−	0.912055i	$-0.365505\pi$
$937$	25.1047	0.820134	0.410067	+	0.912055i	$0.365505\pi$
$938$	0	0
$939$	6.20937	0.202635
$940$	0	0
$941$	25.3141	0.825215	0.412607	−	0.910909i	$-0.364618\pi$
$941$	25.3141	0.825215	0.412607	+	0.910909i	$0.364618\pi$
$942$	0	0
$943$	0.298438	0.00971847
$944$	0	0
$945$	3.70156	0.120412
$946$	0	0
$947$	−15.9109	−0.517036	−0.258518	−	0.966006i	$-0.583234\pi$
$947$	−15.9109	−0.517036	−0.258518	+	0.966006i	$0.583234\pi$
$948$	0	0
$949$	8.00000	0.259691
$950$	0	0
$951$	8.29844	0.269095
$952$	0	0
$953$	61.2250	1.98327	0.991636	−	0.129066i	$-0.0411978\pi$
$953$	61.2250	1.98327	0.991636	+	0.129066i	$0.0411978\pi$
$954$	0	0
$955$	−84.4187	−2.73173
$956$	0	0
$957$	−1.19375	−0.0385885
$958$	0	0
$959$	19.7016	0.636197
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	−18.8062	−0.606023
$964$	0	0
$965$	−85.5234	−2.75310
$966$	0	0
$967$	14.8062	0.476137	0.238068	−	0.971248i	$-0.423486\pi$
$967$	14.8062	0.476137	0.238068	+	0.971248i	$0.423486\pi$
$968$	0	0
$969$	−21.6125	−0.694293
$970$	0	0
$971$	−49.8219	−1.59886	−0.799430	−	0.600759i	$-0.794865\pi$
$971$	−49.8219	−1.59886	−0.799430	+	0.600759i	$0.794865\pi$
$972$	0	0
$973$	17.7016	0.567486
$974$	0	0
$975$	−49.6125	−1.58887
$976$	0	0
$977$	16.7172	0.534830	0.267415	−	0.963581i	$-0.413831\pi$
$977$	16.7172	0.534830	0.267415	+	0.963581i	$0.413831\pi$
$978$	0	0
$979$	−45.6125	−1.45778
$980$	0	0
$981$	−7.70156	−0.245892
$982$	0	0
$983$	−42.2094	−1.34627	−0.673135	−	0.739520i	$-0.735053\pi$
$983$	−42.2094	−1.34627	−0.673135	+	0.739520i	$0.735053\pi$
$984$	0	0
$985$	83.3141	2.65461
$986$	0	0
$987$	11.1047	0.353466
$988$	0	0
$989$	−1.70156	−0.0541065
$990$	0	0
$991$	−28.0000	−0.889449	−0.444725	−	0.895667i	$-0.646698\pi$
$991$	−28.0000	−0.889449	−0.444725	+	0.895667i	$0.646698\pi$
$992$	0	0
$993$	−20.0000	−0.634681
$994$	0	0
$995$	8.50781	0.269716
$996$	0	0
$997$	−23.4031	−0.741184	−0.370592	−	0.928796i	$-0.620845\pi$
$997$	−23.4031	−0.741184	−0.370592	+	0.928796i	$0.620845\pi$
$998$	0	0
$999$	−4.29844	−0.135997

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.bc.1.2		2
4.3	odd	2		966.2.a.n.1.2	✓	2
12.11	even	2		2898.2.a.bb.1.1		2
28.27	even	2		6762.2.a.bo.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
966.2.a.n.1.2	✓	2	4.3	odd	2
2898.2.a.bb.1.1		2	12.11	even	2
6762.2.a.bo.1.1		2	28.27	even	2
7728.2.a.bc.1.2		2	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:	\( N \)	\(=\)	\( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7728.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.70156 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +3.70156 q^{5} -1.00000 q^{7} +1.00000 q^{9} +4.00000 q^{11} +5.70156 q^{13} -3.70156 q^{15} +4.00000 q^{17} +5.40312 q^{19} +1.00000 q^{21} -1.00000 q^{23} +8.70156 q^{25} -1.00000 q^{27} +0.298438 q^{29} +2.00000 q^{31} -4.00000 q^{33} -3.70156 q^{35} +4.29844 q^{37} -5.70156 q^{39} -0.298438 q^{41} +1.70156 q^{43} +3.70156 q^{45} +11.1047 q^{47} +1.00000 q^{49} -4.00000 q^{51} -9.40312 q^{53} +14.8062 q^{55} -5.40312 q^{57} +7.40312 q^{59} -2.00000 q^{61} -1.00000 q^{63} +21.1047 q^{65} -14.8062 q^{67} +1.00000 q^{69} +7.40312 q^{71} +1.40312 q^{73} -8.70156 q^{75} -4.00000 q^{77} -8.00000 q^{79} +1.00000 q^{81} -13.4031 q^{83} +14.8062 q^{85} -0.298438 q^{87} -11.4031 q^{89} -5.70156 q^{91} -2.00000 q^{93} +20.0000 q^{95} +6.29844 q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + q^5 - 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9} + 8 q^{11} + 5 q^{13} - q^{15} + 8 q^{17} - 2 q^{19} + 2 q^{21} - 2 q^{23} + 11 q^{25} - 2 q^{27} + 7 q^{29} + 4 q^{31} - 8 q^{33} - q^{35} + 15 q^{37} - 5 q^{39} - 7 q^{41} - 3 q^{43} + q^{45} + 3 q^{47} + 2 q^{49} - 8 q^{51} - 6 q^{53} + 4 q^{55} + 2 q^{57} + 2 q^{59} - 4 q^{61} - 2 q^{63} + 23 q^{65} - 4 q^{67} + 2 q^{69} + 2 q^{71} - 10 q^{73} - 11 q^{75} - 8 q^{77} - 16 q^{79} + 2 q^{81} - 14 q^{83} + 4 q^{85} - 7 q^{87} - 10 q^{89} - 5 q^{91} - 4 q^{93} + 40 q^{95} + 19 q^{97} + 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + q^5 - 2 * q^7 + 2 * q^9 + 8 * q^11 + 5 * q^13 - q^15 + 8 * q^17 - 2 * q^19 + 2 * q^21 - 2 * q^23 + 11 * q^25 - 2 * q^27 + 7 * q^29 + 4 * q^31 - 8 * q^33 - q^35 + 15 * q^37 - 5 * q^39 - 7 * q^41 - 3 * q^43 + q^45 + 3 * q^47 + 2 * q^49 - 8 * q^51 - 6 * q^53 + 4 * q^55 + 2 * q^57 + 2 * q^59 - 4 * q^61 - 2 * q^63 + 23 * q^65 - 4 * q^67 + 2 * q^69 + 2 * q^71 - 10 * q^73 - 11 * q^75 - 8 * q^77 - 16 * q^79 + 2 * q^81 - 14 * q^83 + 4 * q^85 - 7 * q^87 - 10 * q^89 - 5 * q^91 - 4 * q^93 + 40 * q^95 + 19 * q^97 + 8 * q^99

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