LMFDB - Embedded newform 3584.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3584,2,Mod(1,3584)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3584, 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("3584.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3584 = 2^{9} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3584.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:	$28.6183840844$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.18857984.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 9x^{4} + 16x^{3} + 6x^{2} - 12x + 2 $ x^6 - 2x^5 - 9x^4 + 16x^3 + 6x^2 - 12*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.752719$ of defining polynomial
Character	$\chi$	$=$	3584.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.53584 q^{3} +1.47134 q^{5} +1.00000 q^{7} +3.43049 q^{9} +O(q^{10})$	$q-2.53584 q^{3} +1.47134 q^{5} +1.00000 q^{7} +3.43049 q^{9} +5.35014 q^{11} -3.55212 q^{13} -3.73108 q^{15} -2.73782 q^{17} +8.61663 q^{19} -2.53584 q^{21} +6.17919 q^{23} -2.83517 q^{25} -1.09166 q^{27} +8.49543 q^{29} -10.0009 q^{31} -13.5671 q^{33} +1.47134 q^{35} -0.333865 q^{37} +9.00761 q^{39} -1.30058 q^{41} +5.82230 q^{43} +5.04741 q^{45} +9.00348 q^{47} +1.00000 q^{49} +6.94267 q^{51} +1.20997 q^{53} +7.87186 q^{55} -21.8504 q^{57} -2.71738 q^{59} -4.48221 q^{61} +3.43049 q^{63} -5.22636 q^{65} -8.45438 q^{67} -15.6694 q^{69} +2.46718 q^{71} -6.31997 q^{73} +7.18954 q^{75} +5.35014 q^{77} -11.3091 q^{79} -7.52320 q^{81} -5.41031 q^{83} -4.02825 q^{85} -21.5431 q^{87} +6.44811 q^{89} -3.55212 q^{91} +25.3606 q^{93} +12.6780 q^{95} +4.95282 q^{97} +18.3536 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{3} + 6 q^{7} + 10 q^{9}+O(q^{10}) $ 6 * q + 4 * q^3 + 6 * q^7 + 10 * q^9	$ 6 q + 4 q^{3} + 6 q^{7} + 10 q^{9} + 8 q^{11} - 8 q^{15} + 20 q^{19} + 4 q^{21} + 2 q^{25} + 16 q^{27} + 4 q^{29} - 8 q^{31} + 4 q^{33} + 20 q^{37} - 4 q^{41} + 24 q^{43} + 8 q^{47} + 6 q^{49} + 24 q^{51} + 4 q^{53} + 16 q^{55} - 4 q^{57} + 12 q^{59} - 8 q^{61} + 10 q^{63} - 8 q^{65} + 16 q^{67} - 40 q^{69} - 8 q^{71} + 16 q^{73} + 28 q^{75} + 8 q^{77} - 24 q^{79} + 10 q^{81} + 12 q^{83} - 8 q^{87} + 16 q^{89} + 24 q^{93} - 16 q^{95} + 16 q^{97} + 40 q^{99}+O(q^{100}) $ 6 * q + 4 * q^3 + 6 * q^7 + 10 * q^9 + 8 * q^11 - 8 * q^15 + 20 * q^19 + 4 * q^21 + 2 * q^25 + 16 * q^27 + 4 * q^29 - 8 * q^31 + 4 * q^33 + 20 * q^37 - 4 * q^41 + 24 * q^43 + 8 * q^47 + 6 * q^49 + 24 * q^51 + 4 * q^53 + 16 * q^55 - 4 * q^57 + 12 * q^59 - 8 * q^61 + 10 * q^63 - 8 * q^65 + 16 * q^67 - 40 * q^69 - 8 * q^71 + 16 * q^73 + 28 * q^75 + 8 * q^77 - 24 * q^79 + 10 * q^81 + 12 * q^83 - 8 * q^87 + 16 * q^89 + 24 * q^93 - 16 * q^95 + 16 * q^97 + 40 * 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
$3$	−2.53584	−1.46407	−0.732034	−	0.681268i	$-0.761429\pi$
$3$	−2.53584	−1.46407	−0.732034	+	0.681268i	$0.761429\pi$
$4$	0	0
$5$	1.47134	0.658002	0.329001	−	0.944330i	$-0.393288\pi$
$5$	1.47134	0.658002	0.329001	+	0.944330i	$0.393288\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	3.43049	1.14350
$10$	0	0
$11$	5.35014	1.61313	0.806564	−	0.591146i	$-0.201324\pi$
$11$	5.35014	1.61313	0.806564	+	0.591146i	$0.201324\pi$
$12$	0	0
$13$	−3.55212	−0.985181	−0.492590	−	0.870261i	$-0.663950\pi$
$13$	−3.55212	−0.985181	−0.492590	+	0.870261i	$0.663950\pi$
$14$	0	0
$15$	−3.73108	−0.963359
$16$	0	0
$17$	−2.73782	−0.664018	−0.332009	−	0.943276i	$-0.607727\pi$
$17$	−2.73782	−0.664018	−0.332009	+	0.943276i	$0.607727\pi$
$18$	0	0
$19$	8.61663	1.97679	0.988395	−	0.151908i	$-0.0485417\pi$
$19$	8.61663	1.97679	0.988395	+	0.151908i	$0.0485417\pi$
$20$	0	0
$21$	−2.53584	−0.553366
$22$	0	0
$23$	6.17919	1.28845	0.644225	−	0.764836i	$-0.277180\pi$
$23$	6.17919	1.28845	0.644225	+	0.764836i	$0.277180\pi$
$24$	0	0
$25$	−2.83517	−0.567034
$26$	0	0
$27$	−1.09166	−0.210090
$28$	0	0
$29$	8.49543	1.57756	0.788781	−	0.614674i	$-0.210713\pi$
$29$	8.49543	1.57756	0.788781	+	0.614674i	$0.210713\pi$
$30$	0	0
$31$	−10.0009	−1.79621	−0.898105	−	0.439782i	$-0.855056\pi$
$31$	−10.0009	−1.79621	−0.898105	+	0.439782i	$0.855056\pi$
$32$	0	0
$33$	−13.5671	−2.36173
$34$	0	0
$35$	1.47134	0.248701
$36$	0	0
$37$	−0.333865	−0.0548871	−0.0274435	−	0.999623i	$-0.508737\pi$
$37$	−0.333865	−0.0548871	−0.0274435	+	0.999623i	$0.508737\pi$
$38$	0	0
$39$	9.00761	1.44237
$40$	0	0
$41$	−1.30058	−0.203117	−0.101559	−	0.994830i	$-0.532383\pi$
$41$	−1.30058	−0.203117	−0.101559	+	0.994830i	$0.532383\pi$
$42$	0	0
$43$	5.82230	0.887892	0.443946	−	0.896053i	$-0.353578\pi$
$43$	5.82230	0.887892	0.443946	+	0.896053i	$0.353578\pi$
$44$	0	0
$45$	5.04741	0.752423
$46$	0	0
$47$	9.00348	1.31329	0.656646	−	0.754199i	$-0.271974\pi$
$47$	9.00348	1.31329	0.656646	+	0.754199i	$0.271974\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	6.94267	0.972169
$52$	0	0
$53$	1.20997	0.166203	0.0831014	−	0.996541i	$-0.473517\pi$
$53$	1.20997	0.166203	0.0831014	+	0.996541i	$0.473517\pi$
$54$	0	0
$55$	7.87186	1.06144
$56$	0	0
$57$	−21.8504	−2.89416
$58$	0	0
$59$	−2.71738	−0.353773	−0.176887	−	0.984231i	$-0.556603\pi$
$59$	−2.71738	−0.353773	−0.176887	+	0.984231i	$0.556603\pi$
$60$	0	0
$61$	−4.48221	−0.573888	−0.286944	−	0.957947i	$-0.592639\pi$
$61$	−4.48221	−0.573888	−0.286944	+	0.957947i	$0.592639\pi$
$62$	0	0
$63$	3.43049	0.432201
$64$	0	0
$65$	−5.22636	−0.648250
$66$	0	0
$67$	−8.45438	−1.03287	−0.516434	−	0.856327i	$-0.672741\pi$
$67$	−8.45438	−1.03287	−0.516434	+	0.856327i	$0.672741\pi$
$68$	0	0
$69$	−15.6694	−1.88638
$70$	0	0
$71$	2.46718	0.292801	0.146400	−	0.989225i	$-0.453231\pi$
$71$	2.46718	0.292801	0.146400	+	0.989225i	$0.453231\pi$
$72$	0	0
$73$	−6.31997	−0.739696	−0.369848	−	0.929092i	$-0.620590\pi$
$73$	−6.31997	−0.739696	−0.369848	+	0.929092i	$0.620590\pi$
$74$	0	0
$75$	7.18954	0.830177
$76$	0	0
$77$	5.35014	0.609705
$78$	0	0
$79$	−11.3091	−1.27237	−0.636186	−	0.771536i	$-0.719489\pi$
$79$	−11.3091	−1.27237	−0.636186	+	0.771536i	$0.719489\pi$
$80$	0	0
$81$	−7.52320	−0.835911
$82$	0	0
$83$	−5.41031	−0.593859	−0.296929	−	0.954899i	$-0.595963\pi$
$83$	−5.41031	−0.593859	−0.296929	+	0.954899i	$0.595963\pi$
$84$	0	0
$85$	−4.02825	−0.436925
$86$	0	0
$87$	−21.5431	−2.30966
$88$	0	0
$89$	6.44811	0.683498	0.341749	−	0.939791i	$-0.388981\pi$
$89$	6.44811	0.683498	0.341749	+	0.939791i	$0.388981\pi$
$90$	0	0
$91$	−3.55212	−0.372363
$92$	0	0
$93$	25.3606	2.62977
$94$	0	0
$95$	12.6780	1.30073
$96$	0	0
$97$	4.95282	0.502883	0.251441	−	0.967872i	$-0.419095\pi$
$97$	4.95282	0.502883	0.251441	+	0.967872i	$0.419095\pi$
$98$	0	0
$99$	18.3536	1.84461
$100$	0	0
$101$	−9.20897	−0.916327	−0.458164	−	0.888868i	$-0.651493\pi$
$101$	−9.20897	−0.916327	−0.458164	+	0.888868i	$0.651493\pi$
$102$	0	0
$103$	1.26803	0.124942	0.0624712	−	0.998047i	$-0.480102\pi$
$103$	1.26803	0.124942	0.0624712	+	0.998047i	$0.480102\pi$
$104$	0	0
$105$	−3.73108	−0.364116
$106$	0	0
$107$	1.25678	0.121497	0.0607487	−	0.998153i	$-0.480651\pi$
$107$	1.25678	0.121497	0.0607487	+	0.998153i	$0.480651\pi$
$108$	0	0
$109$	11.3048	1.08280	0.541401	−	0.840764i	$-0.317894\pi$
$109$	11.3048	1.08280	0.541401	+	0.840764i	$0.317894\pi$
$110$	0	0
$111$	0.846628	0.0803584
$112$	0	0
$113$	14.9407	1.40550	0.702751	−	0.711436i	$-0.251955\pi$
$113$	14.9407	1.40550	0.702751	+	0.711436i	$0.251955\pi$
$114$	0	0
$115$	9.09166	0.847802
$116$	0	0
$117$	−12.1855	−1.12655
$118$	0	0
$119$	−2.73782	−0.250975
$120$	0	0
$121$	17.6240	1.60218
$122$	0	0
$123$	3.29807	0.297377
$124$	0	0
$125$	−11.5282	−1.03111
$126$	0	0
$127$	−15.3442	−1.36158	−0.680790	−	0.732478i	$-0.738363\pi$
$127$	−15.3442	−1.36158	−0.680790	+	0.732478i	$0.738363\pi$
$128$	0	0
$129$	−14.7644	−1.29994
$130$	0	0
$131$	6.07758	0.531001	0.265500	−	0.964111i	$-0.414463\pi$
$131$	6.07758	0.531001	0.265500	+	0.964111i	$0.414463\pi$
$132$	0	0
$133$	8.61663	0.747156
$134$	0	0
$135$	−1.60620	−0.138239
$136$	0	0
$137$	8.27393	0.706889	0.353445	−	0.935455i	$-0.385010\pi$
$137$	8.27393	0.706889	0.353445	+	0.935455i	$0.385010\pi$
$138$	0	0
$139$	22.5020	1.90859	0.954296	−	0.298863i	$-0.0966073\pi$
$139$	22.5020	1.90859	0.954296	+	0.298863i	$0.0966073\pi$
$140$	0	0
$141$	−22.8314	−1.92275
$142$	0	0
$143$	−19.0043	−1.58922
$144$	0	0
$145$	12.4996	1.03804
$146$	0	0
$147$	−2.53584	−0.209153
$148$	0	0
$149$	−4.36212	−0.357358	−0.178679	−	0.983907i	$-0.557182\pi$
$149$	−4.36212	−0.357358	−0.178679	+	0.983907i	$0.557182\pi$
$150$	0	0
$151$	16.0941	1.30972	0.654858	−	0.755752i	$-0.272728\pi$
$151$	16.0941	1.30972	0.654858	+	0.755752i	$0.272728\pi$
$152$	0	0
$153$	−9.39206	−0.759303
$154$	0	0
$155$	−14.7146	−1.18191
$156$	0	0
$157$	0.677651	0.0540824	0.0270412	−	0.999634i	$-0.491391\pi$
$157$	0.677651	0.0540824	0.0270412	+	0.999634i	$0.491391\pi$
$158$	0	0
$159$	−3.06830	−0.243332
$160$	0	0
$161$	6.17919	0.486988
$162$	0	0
$163$	−2.49003	−0.195034	−0.0975171	−	0.995234i	$-0.531090\pi$
$163$	−2.49003	−0.195034	−0.0975171	+	0.995234i	$0.531090\pi$
$164$	0	0
$165$	−19.9618	−1.55402
$166$	0	0
$167$	−21.0052	−1.62543	−0.812716	−	0.582660i	$-0.802012\pi$
$167$	−21.0052	−1.62543	−0.812716	+	0.582660i	$0.802012\pi$
$168$	0	0
$169$	−0.382445	−0.0294189
$170$	0	0
$171$	29.5593	2.26045
$172$	0	0
$173$	21.2107	1.61262	0.806310	−	0.591493i	$-0.201461\pi$
$173$	21.2107	1.61262	0.806310	+	0.591493i	$0.201461\pi$
$174$	0	0
$175$	−2.83517	−0.214319
$176$	0	0
$177$	6.89086	0.517949
$178$	0	0
$179$	5.14067	0.384232	0.192116	−	0.981372i	$-0.438465\pi$
$179$	5.14067	0.384232	0.192116	+	0.981372i	$0.438465\pi$
$180$	0	0
$181$	6.81366	0.506455	0.253228	−	0.967407i	$-0.418508\pi$
$181$	6.81366	0.506455	0.253228	+	0.967407i	$0.418508\pi$
$182$	0	0
$183$	11.3662	0.840212
$184$	0	0
$185$	−0.491228	−0.0361158
$186$	0	0
$187$	−14.6477	−1.07115
$188$	0	0
$189$	−1.09166	−0.0794065
$190$	0	0
$191$	17.1650	1.24202	0.621010	−	0.783803i	$-0.286723\pi$
$191$	17.1650	1.24202	0.621010	+	0.783803i	$0.286723\pi$
$192$	0	0
$193$	−18.9566	−1.36453	−0.682263	−	0.731107i	$-0.739004\pi$
$193$	−18.9566	−1.36453	−0.682263	+	0.731107i	$0.739004\pi$
$194$	0	0
$195$	13.2532	0.949083
$196$	0	0
$197$	−3.08800	−0.220011	−0.110005	−	0.993931i	$-0.535087\pi$
$197$	−3.08800	−0.220011	−0.110005	+	0.993931i	$0.535087\pi$
$198$	0	0
$199$	−1.49652	−0.106085	−0.0530426	−	0.998592i	$-0.516892\pi$
$199$	−1.49652	−0.106085	−0.0530426	+	0.998592i	$0.516892\pi$
$200$	0	0
$201$	21.4390	1.51219
$202$	0	0
$203$	8.49543	0.596262
$204$	0	0
$205$	−1.91360	−0.133651
$206$	0	0
$207$	21.1976	1.47334
$208$	0	0
$209$	46.1002	3.18882
$210$	0	0
$211$	20.0643	1.38129	0.690643	−	0.723196i	$-0.257328\pi$
$211$	20.0643	1.38129	0.690643	+	0.723196i	$0.257328\pi$
$212$	0	0
$213$	−6.25638	−0.428680
$214$	0	0
$215$	8.56656	0.584235
$216$	0	0
$217$	−10.0009	−0.678903
$218$	0	0
$219$	16.0264	1.08297
$220$	0	0
$221$	9.72506	0.654178
$222$	0	0
$223$	−16.1569	−1.08194	−0.540971	−	0.841041i	$-0.681943\pi$
$223$	−16.1569	−1.08194	−0.540971	+	0.841041i	$0.681943\pi$
$224$	0	0
$225$	−9.72603	−0.648402
$226$	0	0
$227$	6.81256	0.452165	0.226083	−	0.974108i	$-0.427408\pi$
$227$	6.81256	0.452165	0.226083	+	0.974108i	$0.427408\pi$
$228$	0	0
$229$	6.88613	0.455048	0.227524	−	0.973772i	$-0.426937\pi$
$229$	6.88613	0.455048	0.227524	+	0.973772i	$0.426937\pi$
$230$	0	0
$231$	−13.5671	−0.892651
$232$	0	0
$233$	0.00584477	0.000382903 0	0.000191452	−	1.00000i	$-0.499939\pi$
$233$	0.00584477	0.000382903 0	0.000191452		1.00000i	$0.499939\pi$
$234$	0	0
$235$	13.2471	0.864148
$236$	0	0
$237$	28.6781	1.86284
$238$	0	0
$239$	12.7496	0.824703	0.412351	−	0.911025i	$-0.364708\pi$
$239$	12.7496	0.824703	0.412351	+	0.911025i	$0.364708\pi$
$240$	0	0
$241$	25.1923	1.62278	0.811389	−	0.584507i	$-0.198712\pi$
$241$	25.1923	1.62278	0.811389	+	0.584507i	$0.198712\pi$
$242$	0	0
$243$	22.3526	1.43392
$244$	0	0
$245$	1.47134	0.0940002
$246$	0	0
$247$	−30.6073	−1.94749
$248$	0	0
$249$	13.7197	0.869450
$250$	0	0
$251$	19.7552	1.24694	0.623468	−	0.781849i	$-0.285723\pi$
$251$	19.7552	1.24694	0.623468	+	0.781849i	$0.285723\pi$
$252$	0	0
$253$	33.0595	2.07843
$254$	0	0
$255$	10.2150	0.639688
$256$	0	0
$257$	3.70413	0.231057	0.115529	−	0.993304i	$-0.463144\pi$
$257$	3.70413	0.231057	0.115529	+	0.993304i	$0.463144\pi$
$258$	0	0
$259$	−0.333865	−0.0207454
$260$	0	0
$261$	29.1435	1.80394
$262$	0	0
$263$	8.22781	0.507348	0.253674	−	0.967290i	$-0.418361\pi$
$263$	8.22781	0.507348	0.253674	+	0.967290i	$0.418361\pi$
$264$	0	0
$265$	1.78028	0.109362
$266$	0	0
$267$	−16.3514	−1.00069
$268$	0	0
$269$	−1.57387	−0.0959606	−0.0479803	−	0.998848i	$-0.515278\pi$
$269$	−1.57387	−0.0959606	−0.0479803	+	0.998848i	$0.515278\pi$
$270$	0	0
$271$	23.5357	1.42969	0.714845	−	0.699283i	$-0.246497\pi$
$271$	23.5357	1.42969	0.714845	+	0.699283i	$0.246497\pi$
$272$	0	0
$273$	9.00761	0.545166
$274$	0	0
$275$	−15.1686	−0.914699
$276$	0	0
$277$	6.62846	0.398265	0.199133	−	0.979973i	$-0.436188\pi$
$277$	6.62846	0.398265	0.199133	+	0.979973i	$0.436188\pi$
$278$	0	0
$279$	−34.3079	−2.05396
$280$	0	0
$281$	19.3905	1.15674	0.578370	−	0.815774i	$-0.303689\pi$
$281$	19.3905	1.15674	0.578370	+	0.815774i	$0.303689\pi$
$282$	0	0
$283$	4.65625	0.276785	0.138393	−	0.990377i	$-0.455806\pi$
$283$	4.65625	0.276785	0.138393	+	0.990377i	$0.455806\pi$
$284$	0	0
$285$	−32.1493	−1.90436
$286$	0	0
$287$	−1.30058	−0.0767710
$288$	0	0
$289$	−9.50435	−0.559080
$290$	0	0
$291$	−12.5596	−0.736255
$292$	0	0
$293$	−27.3806	−1.59959	−0.799796	−	0.600272i	$-0.795059\pi$
$293$	−27.3806	−1.59959	−0.799796	+	0.600272i	$0.795059\pi$
$294$	0	0
$295$	−3.99819	−0.232783
$296$	0	0
$297$	−5.84053	−0.338902
$298$	0	0
$299$	−21.9492	−1.26936
$300$	0	0
$301$	5.82230	0.335592
$302$	0	0
$303$	23.3525	1.34157
$304$	0	0
$305$	−6.59484	−0.377619
$306$	0	0
$307$	9.34520	0.533359	0.266679	−	0.963785i	$-0.414074\pi$
$307$	9.34520	0.533359	0.266679	+	0.963785i	$0.414074\pi$
$308$	0	0
$309$	−3.21551	−0.182924
$310$	0	0
$311$	−0.237720	−0.0134798	−0.00673992	−	0.999977i	$-0.502145\pi$
$311$	−0.237720	−0.0134798	−0.00673992	+	0.999977i	$0.502145\pi$
$312$	0	0
$313$	4.86308	0.274878	0.137439	−	0.990510i	$-0.456113\pi$
$313$	4.86308	0.274878	0.137439	+	0.990510i	$0.456113\pi$
$314$	0	0
$315$	5.04741	0.284389
$316$	0	0
$317$	−23.8299	−1.33842	−0.669210	−	0.743073i	$-0.733367\pi$
$317$	−23.8299	−1.33842	−0.669210	+	0.743073i	$0.733367\pi$
$318$	0	0
$319$	45.4518	2.54481
$320$	0	0
$321$	−3.18699	−0.177881
$322$	0	0
$323$	−23.5908	−1.31262
$324$	0	0
$325$	10.0709	0.558631
$326$	0	0
$327$	−28.6671	−1.58530
$328$	0	0
$329$	9.00348	0.496378
$330$	0	0
$331$	−8.23230	−0.452488	−0.226244	−	0.974071i	$-0.572645\pi$
$331$	−8.23230	−0.452488	−0.226244	+	0.974071i	$0.572645\pi$
$332$	0	0
$333$	−1.14532	−0.0627632
$334$	0	0
$335$	−12.4392	−0.679628
$336$	0	0
$337$	−20.2587	−1.10356	−0.551780	−	0.833990i	$-0.686051\pi$
$337$	−20.2587	−1.10356	−0.551780	+	0.833990i	$0.686051\pi$
$338$	0	0
$339$	−37.8872	−2.05775
$340$	0	0
$341$	−53.5061	−2.89752
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−23.0550	−1.24124
$346$	0	0
$347$	21.7190	1.16594	0.582968	−	0.812495i	$-0.301891\pi$
$347$	21.7190	1.16594	0.582968	+	0.812495i	$0.301891\pi$
$348$	0	0
$349$	−15.6559	−0.838043	−0.419021	−	0.907976i	$-0.637627\pi$
$349$	−15.6559	−0.838043	−0.419021	+	0.907976i	$0.637627\pi$
$350$	0	0
$351$	3.87770	0.206977
$352$	0	0
$353$	15.1663	0.807219	0.403610	−	0.914931i	$-0.367755\pi$
$353$	15.1663	0.807219	0.403610	+	0.914931i	$0.367755\pi$
$354$	0	0
$355$	3.63005	0.192663
$356$	0	0
$357$	6.94267	0.367445
$358$	0	0
$359$	35.1850	1.85699	0.928496	−	0.371342i	$-0.121102\pi$
$359$	35.1850	1.85699	0.928496	+	0.371342i	$0.121102\pi$
$360$	0	0
$361$	55.2462	2.90770
$362$	0	0
$363$	−44.6918	−2.34571
$364$	0	0
$365$	−9.29880	−0.486721
$366$	0	0
$367$	10.1677	0.530751	0.265376	−	0.964145i	$-0.414504\pi$
$367$	10.1677	0.530751	0.265376	+	0.964145i	$0.414504\pi$
$368$	0	0
$369$	−4.46164	−0.232264
$370$	0	0
$371$	1.20997	0.0628187
$372$	0	0
$373$	−11.0935	−0.574400	−0.287200	−	0.957871i	$-0.592724\pi$
$373$	−11.0935	−0.574400	−0.287200	+	0.957871i	$0.592724\pi$
$374$	0	0
$375$	29.2336	1.50962
$376$	0	0
$377$	−30.1768	−1.55418
$378$	0	0
$379$	36.0994	1.85430	0.927152	−	0.374686i	$-0.122249\pi$
$379$	36.0994	1.85430	0.927152	+	0.374686i	$0.122249\pi$
$380$	0	0
$381$	38.9105	1.99345
$382$	0	0
$383$	−7.18325	−0.367047	−0.183523	−	0.983015i	$-0.558750\pi$
$383$	−7.18325	−0.367047	−0.183523	+	0.983015i	$0.558750\pi$
$384$	0	0
$385$	7.87186	0.401187
$386$	0	0
$387$	19.9734	1.01530
$388$	0	0
$389$	25.6497	1.30049	0.650245	−	0.759725i	$-0.274666\pi$
$389$	25.6497	1.30049	0.650245	+	0.759725i	$0.274666\pi$
$390$	0	0
$391$	−16.9175	−0.855554
$392$	0	0
$393$	−15.4118	−0.777422
$394$	0	0
$395$	−16.6395	−0.837223
$396$	0	0
$397$	−28.1738	−1.41400	−0.707002	−	0.707212i	$-0.749953\pi$
$397$	−28.1738	−1.41400	−0.707002	+	0.707212i	$0.749953\pi$
$398$	0	0
$399$	−21.8504	−1.09389
$400$	0	0
$401$	10.2803	0.513371	0.256686	−	0.966495i	$-0.417369\pi$
$401$	10.2803	0.513371	0.256686	+	0.966495i	$0.417369\pi$
$402$	0	0
$403$	35.5243	1.76959
$404$	0	0
$405$	−11.0692	−0.550031
$406$	0	0
$407$	−1.78623	−0.0885399
$408$	0	0
$409$	−19.0375	−0.941343	−0.470672	−	0.882309i	$-0.655988\pi$
$409$	−19.0375	−0.941343	−0.470672	+	0.882309i	$0.655988\pi$
$410$	0	0
$411$	−20.9814	−1.03493
$412$	0	0
$413$	−2.71738	−0.133714
$414$	0	0
$415$	−7.96039	−0.390760
$416$	0	0
$417$	−57.0614	−2.79431
$418$	0	0
$419$	0.952050	0.0465107	0.0232553	−	0.999730i	$-0.492597\pi$
$419$	0.952050	0.0465107	0.0232553	+	0.999730i	$0.492597\pi$
$420$	0	0
$421$	−27.1640	−1.32389	−0.661946	−	0.749552i	$-0.730269\pi$
$421$	−27.1640	−1.32389	−0.661946	+	0.749552i	$0.730269\pi$
$422$	0	0
$423$	30.8864	1.50175
$424$	0	0
$425$	7.76218	0.376521
$426$	0	0
$427$	−4.48221	−0.216909
$428$	0	0
$429$	48.1920	2.32673
$430$	0	0
$431$	−36.4598	−1.75621	−0.878104	−	0.478470i	$-0.841191\pi$
$431$	−36.4598	−1.75621	−0.878104	+	0.478470i	$0.841191\pi$
$432$	0	0
$433$	−18.9245	−0.909456	−0.454728	−	0.890630i	$-0.650263\pi$
$433$	−18.9245	−0.909456	−0.454728	+	0.890630i	$0.650263\pi$
$434$	0	0
$435$	−31.6971	−1.51976
$436$	0	0
$437$	53.2437	2.54699
$438$	0	0
$439$	20.9378	0.999306	0.499653	−	0.866226i	$-0.333461\pi$
$439$	20.9378	0.999306	0.499653	+	0.866226i	$0.333461\pi$
$440$	0	0
$441$	3.43049	0.163357
$442$	0	0
$443$	28.7793	1.36734	0.683672	−	0.729790i	$-0.260382\pi$
$443$	28.7793	1.36734	0.683672	+	0.729790i	$0.260382\pi$
$444$	0	0
$445$	9.48734	0.449743
$446$	0	0
$447$	11.0616	0.523197
$448$	0	0
$449$	−22.1940	−1.04740	−0.523700	−	0.851903i	$-0.675449\pi$
$449$	−22.1940	−1.04740	−0.523700	+	0.851903i	$0.675449\pi$
$450$	0	0
$451$	−6.95831	−0.327654
$452$	0	0
$453$	−40.8120	−1.91751
$454$	0	0
$455$	−5.22636	−0.245016
$456$	0	0
$457$	24.6211	1.15173	0.575863	−	0.817546i	$-0.304666\pi$
$457$	24.6211	1.15173	0.575863	+	0.817546i	$0.304666\pi$
$458$	0	0
$459$	2.98876	0.139504
$460$	0	0
$461$	18.4057	0.857239	0.428619	−	0.903485i	$-0.359000\pi$
$461$	18.4057	0.857239	0.428619	+	0.903485i	$0.359000\pi$
$462$	0	0
$463$	−40.8454	−1.89825	−0.949123	−	0.314905i	$-0.898027\pi$
$463$	−40.8454	−1.89825	−0.949123	+	0.314905i	$0.898027\pi$
$464$	0	0
$465$	37.3140	1.73040
$466$	0	0
$467$	−7.56396	−0.350018	−0.175009	−	0.984567i	$-0.555996\pi$
$467$	−7.56396	−0.350018	−0.175009	+	0.984567i	$0.555996\pi$
$468$	0	0
$469$	−8.45438	−0.390387
$470$	0	0
$471$	−1.71841	−0.0791804
$472$	0	0
$473$	31.1501	1.43228
$474$	0	0
$475$	−24.4296	−1.12091
$476$	0	0
$477$	4.15081	0.190052
$478$	0	0
$479$	−37.8779	−1.73069	−0.865343	−	0.501181i	$-0.832899\pi$
$479$	−37.8779	−1.73069	−0.865343	+	0.501181i	$0.832899\pi$
$480$	0	0
$481$	1.18593	0.0540737
$482$	0	0
$483$	−15.6694	−0.712984
$484$	0	0
$485$	7.28727	0.330898
$486$	0	0
$487$	18.7938	0.851630	0.425815	−	0.904810i	$-0.359987\pi$
$487$	18.7938	0.851630	0.425815	+	0.904810i	$0.359987\pi$
$488$	0	0
$489$	6.31432	0.285543
$490$	0	0
$491$	−11.9445	−0.539045	−0.269523	−	0.962994i	$-0.586866\pi$
$491$	−11.9445	−0.539045	−0.269523	+	0.962994i	$0.586866\pi$
$492$	0	0
$493$	−23.2589	−1.04753
$494$	0	0
$495$	27.0043	1.21376
$496$	0	0
$497$	2.46718	0.110668
$498$	0	0
$499$	−16.0387	−0.717992	−0.358996	−	0.933339i	$-0.616881\pi$
$499$	−16.0387	−0.717992	−0.358996	+	0.933339i	$0.616881\pi$
$500$	0	0
$501$	53.2659	2.37975
$502$	0	0
$503$	−2.25982	−0.100760	−0.0503801	−	0.998730i	$-0.516043\pi$
$503$	−2.25982	−0.100760	−0.0503801	+	0.998730i	$0.516043\pi$
$504$	0	0
$505$	−13.5495	−0.602945
$506$	0	0
$507$	0.969820	0.0430712
$508$	0	0
$509$	−19.8827	−0.881285	−0.440642	−	0.897683i	$-0.645249\pi$
$509$	−19.8827	−0.881285	−0.440642	+	0.897683i	$0.645249\pi$
$510$	0	0
$511$	−6.31997	−0.279579
$512$	0	0
$513$	−9.40642	−0.415303
$514$	0	0
$515$	1.86569	0.0822123
$516$	0	0
$517$	48.1699	2.11851
$518$	0	0
$519$	−53.7870	−2.36099
$520$	0	0
$521$	−14.9222	−0.653754	−0.326877	−	0.945067i	$-0.605996\pi$
$521$	−14.9222	−0.653754	−0.326877	+	0.945067i	$0.605996\pi$
$522$	0	0
$523$	−16.2480	−0.710474	−0.355237	−	0.934776i	$-0.615600\pi$
$523$	−16.2480	−0.710474	−0.355237	+	0.934776i	$0.615600\pi$
$524$	0	0
$525$	7.18954	0.313777
$526$	0	0
$527$	27.3806	1.19272
$528$	0	0
$529$	15.1823	0.660101
$530$	0	0
$531$	−9.32197	−0.404539
$532$	0	0
$533$	4.61983	0.200107
$534$	0	0
$535$	1.84914	0.0799455
$536$	0	0
$537$	−13.0359	−0.562542
$538$	0	0
$539$	5.35014	0.230447
$540$	0	0
$541$	−39.7623	−1.70952	−0.854758	−	0.519027i	$-0.826295\pi$
$541$	−39.7623	−1.70952	−0.854758	+	0.519027i	$0.826295\pi$
$542$	0	0
$543$	−17.2784	−0.741485
$544$	0	0
$545$	16.6331	0.712486
$546$	0	0
$547$	36.7280	1.57038	0.785189	−	0.619256i	$-0.212566\pi$
$547$	36.7280	1.57038	0.785189	+	0.619256i	$0.212566\pi$
$548$	0	0
$549$	−15.3762	−0.656240
$550$	0	0
$551$	73.2020	3.11851
$552$	0	0
$553$	−11.3091	−0.480912
$554$	0	0
$555$	1.24568	0.0528760
$556$	0	0
$557$	−25.0967	−1.06338	−0.531691	−	0.846939i	$-0.678443\pi$
$557$	−25.0967	−1.06338	−0.531691	+	0.846939i	$0.678443\pi$
$558$	0	0
$559$	−20.6815	−0.874734
$560$	0	0
$561$	37.1443	1.56823
$562$	0	0
$563$	27.5781	1.16228	0.581138	−	0.813805i	$-0.302608\pi$
$563$	27.5781	1.16228	0.581138	+	0.813805i	$0.302608\pi$
$564$	0	0
$565$	21.9828	0.924822
$566$	0	0
$567$	−7.52320	−0.315945
$568$	0	0
$569$	−25.4129	−1.06536	−0.532682	−	0.846315i	$-0.678816\pi$
$569$	−25.4129	−1.06536	−0.532682	+	0.846315i	$0.678816\pi$
$570$	0	0
$571$	−6.29583	−0.263472	−0.131736	−	0.991285i	$-0.542055\pi$
$571$	−6.29583	−0.263472	−0.131736	+	0.991285i	$0.542055\pi$
$572$	0	0
$573$	−43.5278	−1.81840
$574$	0	0
$575$	−17.5190	−0.730594
$576$	0	0
$577$	8.39622	0.349539	0.174770	−	0.984609i	$-0.444082\pi$
$577$	8.39622	0.349539	0.174770	+	0.984609i	$0.444082\pi$
$578$	0	0
$579$	48.0709	1.99776
$580$	0	0
$581$	−5.41031	−0.224458
$582$	0	0
$583$	6.47354	0.268106
$584$	0	0
$585$	−17.9290	−0.741273
$586$	0	0
$587$	−36.7104	−1.51520	−0.757601	−	0.652718i	$-0.773629\pi$
$587$	−36.7104	−1.51520	−0.757601	+	0.652718i	$0.773629\pi$
$588$	0	0
$589$	−86.1737	−3.55073
$590$	0	0
$591$	7.83068	0.322111
$592$	0	0
$593$	−37.3627	−1.53430	−0.767152	−	0.641466i	$-0.778327\pi$
$593$	−37.3627	−1.53430	−0.767152	+	0.641466i	$0.778327\pi$
$594$	0	0
$595$	−4.02825	−0.165142
$596$	0	0
$597$	3.79493	0.155316
$598$	0	0
$599$	18.2380	0.745185	0.372592	−	0.927995i	$-0.378469\pi$
$599$	18.2380	0.745185	0.372592	+	0.927995i	$0.378469\pi$
$600$	0	0
$601$	11.2357	0.458314	0.229157	−	0.973389i	$-0.426403\pi$
$601$	11.2357	0.458314	0.229157	+	0.973389i	$0.426403\pi$
$602$	0	0
$603$	−29.0027	−1.18108
$604$	0	0
$605$	25.9309	1.05424
$606$	0	0
$607$	16.1984	0.657473	0.328737	−	0.944422i	$-0.393377\pi$
$607$	16.1984	0.657473	0.328737	+	0.944422i	$0.393377\pi$
$608$	0	0
$609$	−21.5431	−0.872969
$610$	0	0
$611$	−31.9814	−1.29383
$612$	0	0
$613$	23.5593	0.951550	0.475775	−	0.879567i	$-0.342168\pi$
$613$	23.5593	0.951550	0.475775	+	0.879567i	$0.342168\pi$
$614$	0	0
$615$	4.85257	0.195675
$616$	0	0
$617$	−17.5014	−0.704580	−0.352290	−	0.935891i	$-0.614597\pi$
$617$	−17.5014	−0.704580	−0.352290	+	0.935891i	$0.614597\pi$
$618$	0	0
$619$	22.7093	0.912764	0.456382	−	0.889784i	$-0.349145\pi$
$619$	22.7093	0.912764	0.456382	+	0.889784i	$0.349145\pi$
$620$	0	0
$621$	−6.74556	−0.270690
$622$	0	0
$623$	6.44811	0.258338
$624$	0	0
$625$	−2.78596	−0.111439
$626$	0	0
$627$	−116.903	−4.66865
$628$	0	0
$629$	0.914061	0.0364460
$630$	0	0
$631$	1.05047	0.0418185	0.0209092	−	0.999781i	$-0.493344\pi$
$631$	1.05047	0.0418185	0.0209092	+	0.999781i	$0.493344\pi$
$632$	0	0
$633$	−50.8799	−2.02230
$634$	0	0
$635$	−22.5765	−0.895922
$636$	0	0
$637$	−3.55212	−0.140740
$638$	0	0
$639$	8.46365	0.334817
$640$	0	0
$641$	−5.48096	−0.216485	−0.108242	−	0.994125i	$-0.534522\pi$
$641$	−5.48096	−0.216485	−0.108242	+	0.994125i	$0.534522\pi$
$642$	0	0
$643$	−1.22391	−0.0482662	−0.0241331	−	0.999709i	$-0.507683\pi$
$643$	−1.22391	−0.0482662	−0.0241331	+	0.999709i	$0.507683\pi$
$644$	0	0
$645$	−21.7234	−0.855360
$646$	0	0
$647$	17.7538	0.697975	0.348987	−	0.937127i	$-0.386526\pi$
$647$	17.7538	0.697975	0.348987	+	0.937127i	$0.386526\pi$
$648$	0	0
$649$	−14.5384	−0.570682
$650$	0	0
$651$	25.3606	0.993961
$652$	0	0
$653$	27.0233	1.05751	0.528753	−	0.848776i	$-0.322660\pi$
$653$	27.0233	1.05751	0.528753	+	0.848776i	$0.322660\pi$
$654$	0	0
$655$	8.94216	0.349399
$656$	0	0
$657$	−21.6806	−0.845841
$658$	0	0
$659$	29.2572	1.13970	0.569849	−	0.821750i	$-0.307002\pi$
$659$	29.2572	1.13970	0.569849	+	0.821750i	$0.307002\pi$
$660$	0	0
$661$	1.24828	0.0485523	0.0242762	−	0.999705i	$-0.492272\pi$
$661$	1.24828	0.0485523	0.0242762	+	0.999705i	$0.492272\pi$
$662$	0	0
$663$	−24.6612	−0.957762
$664$	0	0
$665$	12.6780	0.491630
$666$	0	0
$667$	52.4949	2.03261
$668$	0	0
$669$	40.9712	1.58404
$670$	0	0
$671$	−23.9805	−0.925756
$672$	0	0
$673$	16.0287	0.617861	0.308931	−	0.951085i	$-0.400029\pi$
$673$	16.0287	0.617861	0.308931	+	0.951085i	$0.400029\pi$
$674$	0	0
$675$	3.09504	0.119128
$676$	0	0
$677$	3.28162	0.126123	0.0630614	−	0.998010i	$-0.479914\pi$
$677$	3.28162	0.126123	0.0630614	+	0.998010i	$0.479914\pi$
$678$	0	0
$679$	4.95282	0.190072
$680$	0	0
$681$	−17.2756	−0.662001
$682$	0	0
$683$	−18.1046	−0.692755	−0.346377	−	0.938095i	$-0.612588\pi$
$683$	−18.1046	−0.692755	−0.346377	+	0.938095i	$0.612588\pi$
$684$	0	0
$685$	12.1737	0.465134
$686$	0	0
$687$	−17.4621	−0.666222
$688$	0	0
$689$	−4.29797	−0.163740
$690$	0	0
$691$	16.3412	0.621648	0.310824	−	0.950467i	$-0.399395\pi$
$691$	16.3412	0.621648	0.310824	+	0.950467i	$0.399395\pi$
$692$	0	0
$693$	18.3536	0.697196
$694$	0	0
$695$	33.1080	1.25586
$696$	0	0
$697$	3.56076	0.134873
$698$	0	0
$699$	−0.0148214	−0.000560597 0
$700$	0	0
$701$	5.84196	0.220648	0.110324	−	0.993896i	$-0.464811\pi$
$701$	5.84196	0.220648	0.110324	+	0.993896i	$0.464811\pi$
$702$	0	0
$703$	−2.87679	−0.108500
$704$	0	0
$705$	−33.5927	−1.26517
$706$	0	0
$707$	−9.20897	−0.346339
$708$	0	0
$709$	13.7622	0.516851	0.258426	−	0.966031i	$-0.416796\pi$
$709$	13.7622	0.516851	0.258426	+	0.966031i	$0.416796\pi$
$710$	0	0
$711$	−38.7958	−1.45495
$712$	0	0
$713$	−61.7972	−2.31432
$714$	0	0
$715$	−27.9618	−1.04571
$716$	0	0
$717$	−32.3309	−1.20742
$718$	0	0
$719$	−25.5551	−0.953045	−0.476522	−	0.879162i	$-0.658103\pi$
$719$	−25.5551	−0.953045	−0.476522	+	0.879162i	$0.658103\pi$
$720$	0	0
$721$	1.26803	0.0472238
$722$	0	0
$723$	−63.8836	−2.37586
$724$	0	0
$725$	−24.0860	−0.894531
$726$	0	0
$727$	29.7659	1.10395	0.551977	−	0.833859i	$-0.313874\pi$
$727$	29.7659	1.10395	0.551977	+	0.833859i	$0.313874\pi$
$728$	0	0
$729$	−34.1131	−1.26345
$730$	0	0
$731$	−15.9404	−0.589577
$732$	0	0
$733$	−32.2469	−1.19107	−0.595533	−	0.803331i	$-0.703059\pi$
$733$	−32.2469	−1.19107	−0.595533	+	0.803331i	$0.703059\pi$
$734$	0	0
$735$	−3.73108	−0.137623
$736$	0	0
$737$	−45.2322	−1.66615
$738$	0	0
$739$	−8.37793	−0.308187	−0.154094	−	0.988056i	$-0.549246\pi$
$739$	−8.37793	−0.308187	−0.154094	+	0.988056i	$0.549246\pi$
$740$	0	0
$741$	77.6152	2.85127
$742$	0	0
$743$	−25.4881	−0.935068	−0.467534	−	0.883975i	$-0.654857\pi$
$743$	−25.4881	−0.935068	−0.467534	+	0.883975i	$0.654857\pi$
$744$	0	0
$745$	−6.41814	−0.235142
$746$	0	0
$747$	−18.5600	−0.679076
$748$	0	0
$749$	1.25678	0.0459217
$750$	0	0
$751$	−27.7490	−1.01258	−0.506288	−	0.862365i	$-0.668983\pi$
$751$	−27.7490	−1.01258	−0.506288	+	0.862365i	$0.668983\pi$
$752$	0	0
$753$	−50.0960	−1.82560
$754$	0	0
$755$	23.6798	0.861795
$756$	0	0
$757$	12.5446	0.455942	0.227971	−	0.973668i	$-0.426791\pi$
$757$	12.5446	0.455942	0.227971	+	0.973668i	$0.426791\pi$
$758$	0	0
$759$	−83.8337	−3.04297
$760$	0	0
$761$	36.9458	1.33929	0.669643	−	0.742683i	$-0.266447\pi$
$761$	36.9458	1.33929	0.669643	+	0.742683i	$0.266447\pi$
$762$	0	0
$763$	11.3048	0.409261
$764$	0	0
$765$	−13.8189	−0.499623
$766$	0	0
$767$	9.65248	0.348531
$768$	0	0
$769$	−34.5020	−1.24417	−0.622087	−	0.782948i	$-0.713715\pi$
$769$	−34.5020	−1.24417	−0.622087	+	0.782948i	$0.713715\pi$
$770$	0	0
$771$	−9.39308	−0.338283
$772$	0	0
$773$	12.1677	0.437641	0.218820	−	0.975765i	$-0.429779\pi$
$773$	12.1677	0.437641	0.218820	+	0.975765i	$0.429779\pi$
$774$	0	0
$775$	28.3542	1.01851
$776$	0	0
$777$	0.846628	0.0303726
$778$	0	0
$779$	−11.2066	−0.401520
$780$	0	0
$781$	13.1998	0.472325
$782$	0	0
$783$	−9.27412	−0.331430
$784$	0	0
$785$	0.997052	0.0355863
$786$	0	0
$787$	−24.2744	−0.865289	−0.432644	−	0.901565i	$-0.642419\pi$
$787$	−24.2744	−0.865289	−0.432644	+	0.901565i	$0.642419\pi$
$788$	0	0
$789$	−20.8644	−0.742793
$790$	0	0
$791$	14.9407	0.531230
$792$	0	0
$793$	15.9214	0.565384
$794$	0	0
$795$	−4.51451	−0.160113
$796$	0	0
$797$	−4.58429	−0.162384	−0.0811919	−	0.996698i	$-0.525873\pi$
$797$	−4.58429	−0.162384	−0.0811919	+	0.996698i	$0.525873\pi$
$798$	0	0
$799$	−24.6499	−0.872050
$800$	0	0
$801$	22.1202	0.781578
$802$	0	0
$803$	−33.8127	−1.19323
$804$	0	0
$805$	9.09166	0.320439
$806$	0	0
$807$	3.99109	0.140493
$808$	0	0
$809$	−31.9465	−1.12318	−0.561589	−	0.827416i	$-0.689810\pi$
$809$	−31.9465	−1.12318	−0.561589	+	0.827416i	$0.689810\pi$
$810$	0	0
$811$	−9.10676	−0.319781	−0.159891	−	0.987135i	$-0.551114\pi$
$811$	−9.10676	−0.319781	−0.159891	+	0.987135i	$0.551114\pi$
$812$	0	0
$813$	−59.6827	−2.09316
$814$	0	0
$815$	−3.66367	−0.128333
$816$	0	0
$817$	50.1686	1.75518
$818$	0	0
$819$	−12.1855	−0.425796
$820$	0	0
$821$	28.5354	0.995891	0.497946	−	0.867208i	$-0.334088\pi$
$821$	28.5354	0.995891	0.497946	+	0.867208i	$0.334088\pi$
$822$	0	0
$823$	−7.58038	−0.264235	−0.132118	−	0.991234i	$-0.542178\pi$
$823$	−7.58038	−0.264235	−0.132118	+	0.991234i	$0.542178\pi$
$824$	0	0
$825$	38.4651	1.33918
$826$	0	0
$827$	−17.7460	−0.617087	−0.308544	−	0.951210i	$-0.599842\pi$
$827$	−17.7460	−0.617087	−0.308544	+	0.951210i	$0.599842\pi$
$828$	0	0
$829$	10.6545	0.370046	0.185023	−	0.982734i	$-0.440764\pi$
$829$	10.6545	0.370046	0.185023	+	0.982734i	$0.440764\pi$
$830$	0	0
$831$	−16.8087	−0.583088
$832$	0	0
$833$	−2.73782	−0.0948598
$834$	0	0
$835$	−30.9057	−1.06954
$836$	0	0
$837$	10.9175	0.377365
$838$	0	0
$839$	−34.1821	−1.18010	−0.590048	−	0.807368i	$-0.700891\pi$
$839$	−34.1821	−1.18010	−0.590048	+	0.807368i	$0.700891\pi$
$840$	0	0
$841$	43.1724	1.48870
$842$	0	0
$843$	−49.1713	−1.69355
$844$	0	0
$845$	−0.562705	−0.0193577
$846$	0	0
$847$	17.6240	0.605569
$848$	0	0
$849$	−11.8075	−0.405233
$850$	0	0
$851$	−2.06301	−0.0707192
$852$	0	0
$853$	−28.6019	−0.979312	−0.489656	−	0.871916i	$-0.662878\pi$
$853$	−28.6019	−0.979312	−0.489656	+	0.871916i	$0.662878\pi$
$854$	0	0
$855$	43.4916	1.48738
$856$	0	0
$857$	−4.18332	−0.142899	−0.0714497	−	0.997444i	$-0.522763\pi$
$857$	−4.18332	−0.142899	−0.0714497	+	0.997444i	$0.522763\pi$
$858$	0	0
$859$	25.9454	0.885245	0.442622	−	0.896708i	$-0.354048\pi$
$859$	25.9454	0.885245	0.442622	+	0.896708i	$0.354048\pi$
$860$	0	0
$861$	3.29807	0.112398
$862$	0	0
$863$	−2.24075	−0.0762759	−0.0381379	−	0.999272i	$-0.512143\pi$
$863$	−2.24075	−0.0762759	−0.0381379	+	0.999272i	$0.512143\pi$
$864$	0	0
$865$	31.2081	1.06111
$866$	0	0
$867$	24.1015	0.818531
$868$	0	0
$869$	−60.5053	−2.05250
$870$	0	0
$871$	30.0310	1.01756
$872$	0	0
$873$	16.9906	0.575045
$874$	0	0
$875$	−11.5282	−0.389723
$876$	0	0
$877$	30.6413	1.03468	0.517341	−	0.855779i	$-0.326922\pi$
$877$	30.6413	1.03468	0.517341	+	0.855779i	$0.326922\pi$
$878$	0	0
$879$	69.4328	2.34191
$880$	0	0
$881$	39.8271	1.34181	0.670905	−	0.741543i	$-0.265906\pi$
$881$	39.8271	1.34181	0.670905	+	0.741543i	$0.265906\pi$
$882$	0	0
$883$	2.90916	0.0979010	0.0489505	−	0.998801i	$-0.484412\pi$
$883$	2.90916	0.0979010	0.0489505	+	0.998801i	$0.484412\pi$
$884$	0	0
$885$	10.1388	0.340811
$886$	0	0
$887$	−12.3473	−0.414580	−0.207290	−	0.978279i	$-0.566464\pi$
$887$	−12.3473	−0.414580	−0.207290	+	0.978279i	$0.566464\pi$
$888$	0	0
$889$	−15.3442	−0.514629
$890$	0	0
$891$	−40.2502	−1.34843
$892$	0	0
$893$	77.5796	2.59610
$894$	0	0
$895$	7.56366	0.252825
$896$	0	0
$897$	55.6597	1.85842
$898$	0	0
$899$	−84.9617	−2.83363
$900$	0	0
$901$	−3.31269	−0.110362
$902$	0	0
$903$	−14.7644	−0.491329
$904$	0	0
$905$	10.0252	0.333248
$906$	0	0
$907$	28.0705	0.932067	0.466034	−	0.884767i	$-0.345683\pi$
$907$	28.0705	0.932067	0.466034	+	0.884767i	$0.345683\pi$
$908$	0	0
$909$	−31.5913	−1.04782
$910$	0	0
$911$	1.04576	0.0346474	0.0173237	−	0.999850i	$-0.494485\pi$
$911$	1.04576	0.0346474	0.0173237	+	0.999850i	$0.494485\pi$
$912$	0	0
$913$	−28.9459	−0.957971
$914$	0	0
$915$	16.7235	0.552861
$916$	0	0
$917$	6.07758	0.200699
$918$	0	0
$919$	−1.28887	−0.0425159	−0.0212579	−	0.999774i	$-0.506767\pi$
$919$	−1.28887	−0.0425159	−0.0212579	+	0.999774i	$0.506767\pi$
$920$	0	0
$921$	−23.6979	−0.780874
$922$	0	0
$923$	−8.76372	−0.288461
$924$	0	0
$925$	0.946564	0.0311228
$926$	0	0
$927$	4.34996	0.142871
$928$	0	0
$929$	−5.26769	−0.172827	−0.0864137	−	0.996259i	$-0.527541\pi$
$929$	−5.26769	−0.172827	−0.0864137	+	0.996259i	$0.527541\pi$
$930$	0	0
$931$	8.61663	0.282398
$932$	0	0
$933$	0.602819	0.0197354
$934$	0	0
$935$	−21.5517	−0.704817
$936$	0	0
$937$	37.7934	1.23466	0.617329	−	0.786705i	$-0.288215\pi$
$937$	37.7934	1.23466	0.617329	+	0.786705i	$0.288215\pi$
$938$	0	0
$939$	−12.3320	−0.402440
$940$	0	0
$941$	−19.6472	−0.640481	−0.320241	−	0.947336i	$-0.603764\pi$
$941$	−19.6472	−0.640481	−0.320241	+	0.947336i	$0.603764\pi$
$942$	0	0
$943$	−8.03655	−0.261706
$944$	0	0
$945$	−1.60620	−0.0522496
$946$	0	0
$947$	9.77421	0.317619	0.158810	−	0.987309i	$-0.449234\pi$
$947$	9.77421	0.317619	0.158810	+	0.987309i	$0.449234\pi$
$948$	0	0
$949$	22.4493	0.728735
$950$	0	0
$951$	60.4289	1.95954
$952$	0	0
$953$	58.8098	1.90504	0.952519	−	0.304480i	$-0.0984825\pi$
$953$	58.8098	1.90504	0.952519	+	0.304480i	$0.0984825\pi$
$954$	0	0
$955$	25.2556	0.817251
$956$	0	0
$957$	−115.259	−3.72578
$958$	0	0
$959$	8.27393	0.267179
$960$	0	0
$961$	69.0174	2.22637
$962$	0	0
$963$	4.31137	0.138932
$964$	0	0
$965$	−27.8915	−0.897860
$966$	0	0
$967$	−45.9820	−1.47868	−0.739340	−	0.673332i	$-0.764862\pi$
$967$	−45.9820	−1.47868	−0.739340	+	0.673332i	$0.764862\pi$
$968$	0	0
$969$	59.8224	1.92177
$970$	0	0
$971$	−18.1060	−0.581048	−0.290524	−	0.956868i	$-0.593830\pi$
$971$	−18.1060	−0.581048	−0.290524	+	0.956868i	$0.593830\pi$
$972$	0	0
$973$	22.5020	0.721380
$974$	0	0
$975$	−25.5381	−0.817874
$976$	0	0
$977$	20.6440	0.660461	0.330231	−	0.943900i	$-0.392873\pi$
$977$	20.6440	0.660461	0.330231	+	0.943900i	$0.392873\pi$
$978$	0	0
$979$	34.4983	1.10257
$980$	0	0
$981$	38.7810	1.23818
$982$	0	0
$983$	−57.7075	−1.84058	−0.920291	−	0.391234i	$-0.872048\pi$
$983$	−57.7075	−1.84058	−0.920291	+	0.391234i	$0.872048\pi$
$984$	0	0
$985$	−4.54349	−0.144767
$986$	0	0
$987$	−22.8314	−0.726731
$988$	0	0
$989$	35.9771	1.14400
$990$	0	0
$991$	−15.8234	−0.502647	−0.251323	−	0.967903i	$-0.580866\pi$
$991$	−15.8234	−0.502647	−0.251323	+	0.967903i	$0.580866\pi$
$992$	0	0
$993$	20.8758	0.662474
$994$	0	0
$995$	−2.20188	−0.0698043
$996$	0	0
$997$	41.9024	1.32706	0.663530	−	0.748149i	$-0.269057\pi$
$997$	41.9024	1.32706	0.663530	+	0.748149i	$0.269057\pi$
$998$	0	0
$999$	0.364467	0.0115312

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3584.2.a.l.1.1	yes	6
4.3	odd	2		3584.2.a.e.1.6	✓	6
8.3	odd	2		3584.2.a.k.1.1	yes	6
8.5	even	2		3584.2.a.f.1.6	yes	6
16.3	odd	4		3584.2.b.k.1793.10		12
16.5	even	4		3584.2.b.i.1793.10		12
16.11	odd	4		3584.2.b.k.1793.3		12
16.13	even	4		3584.2.b.i.1793.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3584.2.a.e.1.6	✓	6	4.3	odd	2
3584.2.a.f.1.6	yes	6	8.5	even	2
3584.2.a.k.1.1	yes	6	8.3	odd	2
3584.2.a.l.1.1	yes	6	1.1	even	1	trivial
3584.2.b.i.1793.3		12	16.13	even	4
3584.2.b.i.1793.10		12	16.5	even	4
3584.2.b.k.1793.3		12	16.11	odd	4
3584.2.b.k.1793.10		12	16.3	odd	4

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:	\( N \)	\(=\)	\( 3584 = 2^{9} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3584.a (trivial)

\(f(q)\)	\(=\)	\(q-2.53584 q^{3} +1.47134 q^{5} +1.00000 q^{7} +3.43049 q^{9} +O(q^{10})\)	\(q-2.53584 q^{3} +1.47134 q^{5} +1.00000 q^{7} +3.43049 q^{9} +5.35014 q^{11} -3.55212 q^{13} -3.73108 q^{15} -2.73782 q^{17} +8.61663 q^{19} -2.53584 q^{21} +6.17919 q^{23} -2.83517 q^{25} -1.09166 q^{27} +8.49543 q^{29} -10.0009 q^{31} -13.5671 q^{33} +1.47134 q^{35} -0.333865 q^{37} +9.00761 q^{39} -1.30058 q^{41} +5.82230 q^{43} +5.04741 q^{45} +9.00348 q^{47} +1.00000 q^{49} +6.94267 q^{51} +1.20997 q^{53} +7.87186 q^{55} -21.8504 q^{57} -2.71738 q^{59} -4.48221 q^{61} +3.43049 q^{63} -5.22636 q^{65} -8.45438 q^{67} -15.6694 q^{69} +2.46718 q^{71} -6.31997 q^{73} +7.18954 q^{75} +5.35014 q^{77} -11.3091 q^{79} -7.52320 q^{81} -5.41031 q^{83} -4.02825 q^{85} -21.5431 q^{87} +6.44811 q^{89} -3.55212 q^{91} +25.3606 q^{93} +12.6780 q^{95} +4.95282 q^{97} +18.3536 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{3} + 6 q^{7} + 10 q^{9}+O(q^{10}) \) 6 * q + 4 * q^3 + 6 * q^7 + 10 * q^9	\( 6 q + 4 q^{3} + 6 q^{7} + 10 q^{9} + 8 q^{11} - 8 q^{15} + 20 q^{19} + 4 q^{21} + 2 q^{25} + 16 q^{27} + 4 q^{29} - 8 q^{31} + 4 q^{33} + 20 q^{37} - 4 q^{41} + 24 q^{43} + 8 q^{47} + 6 q^{49} + 24 q^{51} + 4 q^{53} + 16 q^{55} - 4 q^{57} + 12 q^{59} - 8 q^{61} + 10 q^{63} - 8 q^{65} + 16 q^{67} - 40 q^{69} - 8 q^{71} + 16 q^{73} + 28 q^{75} + 8 q^{77} - 24 q^{79} + 10 q^{81} + 12 q^{83} - 8 q^{87} + 16 q^{89} + 24 q^{93} - 16 q^{95} + 16 q^{97} + 40 q^{99}+O(q^{100}) \) 6 * q + 4 * q^3 + 6 * q^7 + 10 * q^9 + 8 * q^11 - 8 * q^15 + 20 * q^19 + 4 * q^21 + 2 * q^25 + 16 * q^27 + 4 * q^29 - 8 * q^31 + 4 * q^33 + 20 * q^37 - 4 * q^41 + 24 * q^43 + 8 * q^47 + 6 * q^49 + 24 * q^51 + 4 * q^53 + 16 * q^55 - 4 * q^57 + 12 * q^59 - 8 * q^61 + 10 * q^63 - 8 * q^65 + 16 * q^67 - 40 * q^69 - 8 * q^71 + 16 * q^73 + 28 * q^75 + 8 * q^77 - 24 * q^79 + 10 * q^81 + 12 * q^83 - 8 * q^87 + 16 * q^89 + 24 * q^93 - 16 * q^95 + 16 * q^97 + 40 * q^99

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