LMFDB - Embedded newform 3584.2.a.l.1.3

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			1.3
Root			$1.62476$ 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)

$f(q)$	$=$	$q-0.101362 q^{3} -2.19640 q^{5} +1.00000 q^{7} -2.98973 q^{9} +O(q^{10})$	$q-0.101362 q^{3} -2.19640 q^{5} +1.00000 q^{7} -2.98973 q^{9} +0.674013 q^{11} +5.30258 q^{13} +0.222631 q^{15} +3.87523 q^{17} +0.995183 q^{19} -0.101362 q^{21} -3.36590 q^{23} -0.175829 q^{25} +0.607128 q^{27} -0.134404 q^{29} -3.11512 q^{31} -0.0683190 q^{33} -2.19640 q^{35} -2.07795 q^{37} -0.537478 q^{39} -3.76710 q^{41} +3.61268 q^{43} +6.56663 q^{45} -6.68913 q^{47} +1.00000 q^{49} -0.392799 q^{51} -2.93656 q^{53} -1.48040 q^{55} -0.100873 q^{57} +6.92085 q^{59} -4.30265 q^{61} -2.98973 q^{63} -11.6466 q^{65} +13.9311 q^{67} +0.341173 q^{69} -10.6460 q^{71} +8.62368 q^{73} +0.0178223 q^{75} +0.674013 q^{77} +7.12273 q^{79} +8.90764 q^{81} +15.1833 q^{83} -8.51155 q^{85} +0.0136234 q^{87} +0.856726 q^{89} +5.30258 q^{91} +0.315754 q^{93} -2.18582 q^{95} -11.0125 q^{97} -2.01511 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.101362	−0.0585211	−0.0292606	−	0.999572i	$-0.509315\pi$
$3$	−0.101362	−0.0585211	−0.0292606	+	0.999572i	$0.509315\pi$
$4$	0	0
$5$	−2.19640	−0.982260	−0.491130	−	0.871086i	$-0.663416\pi$
$5$	−2.19640	−0.982260	−0.491130	+	0.871086i	$0.663416\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.98973	−0.996575
$10$	0	0
$11$	0.674013	0.203223	0.101611	−	0.994824i	$-0.467600\pi$
$11$	0.674013	0.203223	0.101611	+	0.994824i	$0.467600\pi$
$12$	0	0
$13$	5.30258	1.47067	0.735335	−	0.677704i	$-0.237025\pi$
$13$	5.30258	1.47067	0.735335	+	0.677704i	$0.237025\pi$
$14$	0	0
$15$	0.222631	0.0574830
$16$	0	0
$17$	3.87523	0.939881	0.469941	−	0.882698i	$-0.344275\pi$
$17$	3.87523	0.939881	0.469941	+	0.882698i	$0.344275\pi$
$18$	0	0
$19$	0.995183	0.228311	0.114155	−	0.993463i	$-0.463584\pi$
$19$	0.995183	0.228311	0.114155	+	0.993463i	$0.463584\pi$
$20$	0	0
$21$	−0.101362	−0.0221189
$22$	0	0
$23$	−3.36590	−0.701840	−0.350920	−	0.936406i	$-0.614131\pi$
$23$	−3.36590	−0.701840	−0.350920	+	0.936406i	$0.614131\pi$
$24$	0	0
$25$	−0.175829	−0.0351657
$26$	0	0
$27$	0.607128	0.116842
$28$	0	0
$29$	−0.134404	−0.0249582	−0.0124791	−	0.999922i	$-0.503972\pi$
$29$	−0.134404	−0.0249582	−0.0124791	+	0.999922i	$0.503972\pi$
$30$	0	0
$31$	−3.11512	−0.559492	−0.279746	−	0.960074i	$-0.590250\pi$
$31$	−3.11512	−0.559492	−0.279746	+	0.960074i	$0.590250\pi$
$32$	0	0
$33$	−0.0683190	−0.0118928
$34$	0	0
$35$	−2.19640	−0.371259
$36$	0	0
$37$	−2.07795	−0.341613	−0.170807	−	0.985305i	$-0.554637\pi$
$37$	−2.07795	−0.341613	−0.170807	+	0.985305i	$0.554637\pi$
$38$	0	0
$39$	−0.537478	−0.0860653
$40$	0	0
$41$	−3.76710	−0.588321	−0.294161	−	0.955756i	$-0.595040\pi$
$41$	−3.76710	−0.588321	−0.294161	+	0.955756i	$0.595040\pi$
$42$	0	0
$43$	3.61268	0.550929	0.275464	−	0.961311i	$-0.411168\pi$
$43$	3.61268	0.550929	0.275464	+	0.961311i	$0.411168\pi$
$44$	0	0
$45$	6.56663	0.978896
$46$	0	0
$47$	−6.68913	−0.975709	−0.487855	−	0.872925i	$-0.662220\pi$
$47$	−6.68913	−0.975709	−0.487855	+	0.872925i	$0.662220\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.392799	−0.0550029
$52$	0	0
$53$	−2.93656	−0.403368	−0.201684	−	0.979451i	$-0.564641\pi$
$53$	−2.93656	−0.403368	−0.201684	+	0.979451i	$0.564641\pi$
$54$	0	0
$55$	−1.48040	−0.199617
$56$	0	0
$57$	−0.100873	−0.0133610
$58$	0	0
$59$	6.92085	0.901017	0.450509	−	0.892772i	$-0.351243\pi$
$59$	6.92085	0.901017	0.450509	+	0.892772i	$0.351243\pi$
$60$	0	0
$61$	−4.30265	−0.550898	−0.275449	−	0.961316i	$-0.588827\pi$
$61$	−4.30265	−0.550898	−0.275449	+	0.961316i	$0.588827\pi$
$62$	0	0
$63$	−2.98973	−0.376670
$64$	0	0
$65$	−11.6466	−1.44458
$66$	0	0
$67$	13.9311	1.70196	0.850980	−	0.525198i	$-0.176009\pi$
$67$	13.9311	1.70196	0.850980	+	0.525198i	$0.176009\pi$
$68$	0	0
$69$	0.341173	0.0410724
$70$	0	0
$71$	−10.6460	−1.26344	−0.631721	−	0.775196i	$-0.717651\pi$
$71$	−10.6460	−1.26344	−0.631721	+	0.775196i	$0.717651\pi$
$72$	0	0
$73$	8.62368	1.00932	0.504662	−	0.863317i	$-0.331617\pi$
$73$	8.62368	1.00932	0.504662	+	0.863317i	$0.331617\pi$
$74$	0	0
$75$	0.0178223	0.00205794
$76$	0	0
$77$	0.674013	0.0768109
$78$	0	0
$79$	7.12273	0.801369	0.400685	−	0.916216i	$-0.368772\pi$
$79$	7.12273	0.801369	0.400685	+	0.916216i	$0.368772\pi$
$80$	0	0
$81$	8.90764	0.989738
$82$	0	0
$83$	15.1833	1.66658	0.833291	−	0.552834i	$-0.186454\pi$
$83$	15.1833	1.66658	0.833291	+	0.552834i	$0.186454\pi$
$84$	0	0
$85$	−8.51155	−0.923207
$86$	0	0
$87$	0.0136234	0.00146058
$88$	0	0
$89$	0.856726	0.0908128	0.0454064	−	0.998969i	$-0.485542\pi$
$89$	0.856726	0.0908128	0.0454064	+	0.998969i	$0.485542\pi$
$90$	0	0
$91$	5.30258	0.555861
$92$	0	0
$93$	0.315754	0.0327421
$94$	0	0
$95$	−2.18582	−0.224260
$96$	0	0
$97$	−11.0125	−1.11815	−0.559074	−	0.829118i	$-0.688843\pi$
$97$	−11.0125	−1.11815	−0.559074	+	0.829118i	$0.688843\pi$
$98$	0	0
$99$	−2.01511	−0.202527
$100$	0	0
$101$	−0.354276	−0.0352518	−0.0176259	−	0.999845i	$-0.505611\pi$
$101$	−0.354276	−0.0352518	−0.0176259	+	0.999845i	$0.505611\pi$
$102$	0	0
$103$	16.5750	1.63318	0.816590	−	0.577218i	$-0.195862\pi$
$103$	16.5750	1.63318	0.816590	+	0.577218i	$0.195862\pi$
$104$	0	0
$105$	0.222631	0.0217265
$106$	0	0
$107$	17.7782	1.71869	0.859343	−	0.511400i	$-0.170873\pi$
$107$	17.7782	1.71869	0.859343	+	0.511400i	$0.170873\pi$
$108$	0	0
$109$	10.1967	0.976667	0.488334	−	0.872657i	$-0.337605\pi$
$109$	10.1967	0.976667	0.488334	+	0.872657i	$0.337605\pi$
$110$	0	0
$111$	0.210624	0.0199916
$112$	0	0
$113$	9.69262	0.911805	0.455902	−	0.890030i	$-0.349317\pi$
$113$	9.69262	0.911805	0.455902	+	0.890030i	$0.349317\pi$
$114$	0	0
$115$	7.39287	0.689389
$116$	0	0
$117$	−15.8533	−1.46563
$118$	0	0
$119$	3.87523	0.355242
$120$	0	0
$121$	−10.5457	−0.958701
$122$	0	0
$123$	0.381839	0.0344292
$124$	0	0
$125$	11.3682	1.01680
$126$	0	0
$127$	20.2674	1.79844	0.899220	−	0.437497i	$-0.144135\pi$
$127$	20.2674	1.79844	0.899220	+	0.437497i	$0.144135\pi$
$128$	0	0
$129$	−0.366187	−0.0322410
$130$	0	0
$131$	17.8643	1.56081	0.780406	−	0.625272i	$-0.215012\pi$
$131$	17.8643	1.56081	0.780406	+	0.625272i	$0.215012\pi$
$132$	0	0
$133$	0.995183	0.0862933
$134$	0	0
$135$	−1.33350	−0.114769
$136$	0	0
$137$	11.4894	0.981606	0.490803	−	0.871271i	$-0.336704\pi$
$137$	11.4894	0.981606	0.490803	+	0.871271i	$0.336704\pi$
$138$	0	0
$139$	0.209585	0.0177768	0.00888838	−	0.999960i	$-0.497171\pi$
$139$	0.209585	0.0177768	0.00888838	+	0.999960i	$0.497171\pi$
$140$	0	0
$141$	0.678020	0.0570996
$142$	0	0
$143$	3.57401	0.298873
$144$	0	0
$145$	0.295205	0.0245155
$146$	0	0
$147$	−0.101362	−0.00836016
$148$	0	0
$149$	−10.5895	−0.867526	−0.433763	−	0.901027i	$-0.642814\pi$
$149$	−10.5895	−0.867526	−0.433763	+	0.901027i	$0.642814\pi$
$150$	0	0
$151$	11.4820	0.934391	0.467196	−	0.884154i	$-0.345264\pi$
$151$	11.4820	0.934391	0.467196	+	0.884154i	$0.345264\pi$
$152$	0	0
$153$	−11.5859	−0.936662
$154$	0	0
$155$	6.84205	0.549567
$156$	0	0
$157$	9.98207	0.796656	0.398328	−	0.917243i	$-0.369591\pi$
$157$	9.98207	0.796656	0.398328	+	0.917243i	$0.369591\pi$
$158$	0	0
$159$	0.297654	0.0236055
$160$	0	0
$161$	−3.36590	−0.265270
$162$	0	0
$163$	−3.76859	−0.295178	−0.147589	−	0.989049i	$-0.547151\pi$
$163$	−3.76859	−0.295178	−0.147589	+	0.989049i	$0.547151\pi$
$164$	0	0
$165$	0.150056	0.0116818
$166$	0	0
$167$	8.45888	0.654568	0.327284	−	0.944926i	$-0.393867\pi$
$167$	8.45888	0.654568	0.327284	+	0.944926i	$0.393867\pi$
$168$	0	0
$169$	15.1173	1.16287
$170$	0	0
$171$	−2.97533	−0.227529
$172$	0	0
$173$	−1.41548	−0.107617	−0.0538085	−	0.998551i	$-0.517136\pi$
$173$	−1.41548	−0.107617	−0.0538085	+	0.998551i	$0.517136\pi$
$174$	0	0
$175$	−0.175829	−0.0132914
$176$	0	0
$177$	−0.701508	−0.0527286
$178$	0	0
$179$	−17.2449	−1.28894	−0.644470	−	0.764629i	$-0.722922\pi$
$179$	−17.2449	−1.28894	−0.644470	+	0.764629i	$0.722922\pi$
$180$	0	0
$181$	4.77705	0.355075	0.177538	−	0.984114i	$-0.443187\pi$
$181$	4.77705	0.355075	0.177538	+	0.984114i	$0.443187\pi$
$182$	0	0
$183$	0.436123	0.0322392
$184$	0	0
$185$	4.56401	0.335553
$186$	0	0
$187$	2.61195	0.191005
$188$	0	0
$189$	0.607128	0.0441621
$190$	0	0
$191$	−8.90148	−0.644089	−0.322044	−	0.946725i	$-0.604370\pi$
$191$	−8.90148	−0.644089	−0.322044	+	0.946725i	$0.604370\pi$
$192$	0	0
$193$	−11.9910	−0.863129	−0.431565	−	0.902082i	$-0.642038\pi$
$193$	−11.9910	−0.863129	−0.431565	+	0.902082i	$0.642038\pi$
$194$	0	0
$195$	1.18052	0.0845385
$196$	0	0
$197$	−18.5079	−1.31863	−0.659317	−	0.751865i	$-0.729154\pi$
$197$	−18.5079	−1.31863	−0.659317	+	0.751865i	$0.729154\pi$
$198$	0	0
$199$	−2.13252	−0.151170	−0.0755852	−	0.997139i	$-0.524082\pi$
$199$	−2.13252	−0.151170	−0.0755852	+	0.997139i	$0.524082\pi$
$200$	0	0
$201$	−1.41208	−0.0996006
$202$	0	0
$203$	−0.134404	−0.00943332
$204$	0	0
$205$	8.27405	0.577884
$206$	0	0
$207$	10.0631	0.699436
$208$	0	0
$209$	0.670766	0.0463979
$210$	0	0
$211$	22.7237	1.56436	0.782181	−	0.623051i	$-0.214107\pi$
$211$	22.7237	1.56436	0.782181	+	0.623051i	$0.214107\pi$
$212$	0	0
$213$	1.07909	0.0739381
$214$	0	0
$215$	−7.93489	−0.541155
$216$	0	0
$217$	−3.11512	−0.211468
$218$	0	0
$219$	−0.874109	−0.0590668
$220$	0	0
$221$	20.5487	1.38226
$222$	0	0
$223$	−1.10025	−0.0736781	−0.0368391	−	0.999321i	$-0.511729\pi$
$223$	−1.10025	−0.0736781	−0.0368391	+	0.999321i	$0.511729\pi$
$224$	0	0
$225$	0.525679	0.0350453
$226$	0	0
$227$	−2.63939	−0.175183	−0.0875913	−	0.996156i	$-0.527917\pi$
$227$	−2.63939	−0.175183	−0.0875913	+	0.996156i	$0.527917\pi$
$228$	0	0
$229$	−19.2282	−1.27064	−0.635319	−	0.772250i	$-0.719131\pi$
$229$	−19.2282	−1.27064	−0.635319	+	0.772250i	$0.719131\pi$
$230$	0	0
$231$	−0.0683190	−0.00449506
$232$	0	0
$233$	8.69975	0.569939	0.284970	−	0.958537i	$-0.408016\pi$
$233$	8.69975	0.569939	0.284970	+	0.958537i	$0.408016\pi$
$234$	0	0
$235$	14.6920	0.958400
$236$	0	0
$237$	−0.721971	−0.0468970
$238$	0	0
$239$	−1.72852	−0.111809	−0.0559044	−	0.998436i	$-0.517804\pi$
$239$	−1.72852	−0.111809	−0.0559044	+	0.998436i	$0.517804\pi$
$240$	0	0
$241$	17.6482	1.13682	0.568409	−	0.822746i	$-0.307559\pi$
$241$	17.6482	1.13682	0.568409	+	0.822746i	$0.307559\pi$
$242$	0	0
$243$	−2.72428	−0.174762
$244$	0	0
$245$	−2.19640	−0.140323
$246$	0	0
$247$	5.27704	0.335770
$248$	0	0
$249$	−1.53900	−0.0975303
$250$	0	0
$251$	−14.6099	−0.922169	−0.461084	−	0.887356i	$-0.652539\pi$
$251$	−14.6099	−0.922169	−0.461084	+	0.887356i	$0.652539\pi$
$252$	0	0
$253$	−2.26866	−0.142630
$254$	0	0
$255$	0.862744	0.0540271
$256$	0	0
$257$	−24.1929	−1.50911	−0.754556	−	0.656235i	$-0.772148\pi$
$257$	−24.1929	−1.50911	−0.754556	+	0.656235i	$0.772148\pi$
$258$	0	0
$259$	−2.07795	−0.129118
$260$	0	0
$261$	0.401832	0.0248727
$262$	0	0
$263$	−23.8158	−1.46854	−0.734271	−	0.678856i	$-0.762476\pi$
$263$	−23.8158	−1.46854	−0.734271	+	0.678856i	$0.762476\pi$
$264$	0	0
$265$	6.44986	0.396212
$266$	0	0
$267$	−0.0868391	−0.00531446
$268$	0	0
$269$	0.304478	0.0185643	0.00928216	−	0.999957i	$-0.497045\pi$
$269$	0.304478	0.0185643	0.00928216	+	0.999957i	$0.497045\pi$
$270$	0	0
$271$	−17.2018	−1.04493	−0.522467	−	0.852659i	$-0.674988\pi$
$271$	−17.2018	−1.04493	−0.522467	+	0.852659i	$0.674988\pi$
$272$	0	0
$273$	−0.537478	−0.0325296
$274$	0	0
$275$	−0.118511	−0.00714646
$276$	0	0
$277$	21.1489	1.27072	0.635358	−	0.772218i	$-0.280853\pi$
$277$	21.1489	1.27072	0.635358	+	0.772218i	$0.280853\pi$
$278$	0	0
$279$	9.31336	0.557576
$280$	0	0
$281$	11.0974	0.662018	0.331009	−	0.943628i	$-0.392611\pi$
$281$	11.0974	0.662018	0.331009	+	0.943628i	$0.392611\pi$
$282$	0	0
$283$	23.8050	1.41506	0.707530	−	0.706683i	$-0.249809\pi$
$283$	23.8050	1.41506	0.707530	+	0.706683i	$0.249809\pi$
$284$	0	0
$285$	0.221558	0.0131240
$286$	0	0
$287$	−3.76710	−0.222365
$288$	0	0
$289$	−1.98260	−0.116624
$290$	0	0
$291$	1.11624	0.0654353
$292$	0	0
$293$	−13.2713	−0.775316	−0.387658	−	0.921803i	$-0.626716\pi$
$293$	−13.2713	−0.775316	−0.387658	+	0.921803i	$0.626716\pi$
$294$	0	0
$295$	−15.2009	−0.885033
$296$	0	0
$297$	0.409212	0.0237449
$298$	0	0
$299$	−17.8480	−1.03217
$300$	0	0
$301$	3.61268	0.208231
$302$	0	0
$303$	0.0359100	0.00206298
$304$	0	0
$305$	9.45034	0.541125
$306$	0	0
$307$	14.4325	0.823705	0.411852	−	0.911251i	$-0.364882\pi$
$307$	14.4325	0.823705	0.411852	+	0.911251i	$0.364882\pi$
$308$	0	0
$309$	−1.68007	−0.0955756
$310$	0	0
$311$	−22.4304	−1.27191	−0.635956	−	0.771725i	$-0.719394\pi$
$311$	−22.4304	−1.27191	−0.635956	+	0.771725i	$0.719394\pi$
$312$	0	0
$313$	−24.6010	−1.39053	−0.695266	−	0.718753i	$-0.744713\pi$
$313$	−24.6010	−1.39053	−0.695266	+	0.718753i	$0.744713\pi$
$314$	0	0
$315$	6.56663	0.369988
$316$	0	0
$317$	30.9518	1.73842	0.869212	−	0.494439i	$-0.164626\pi$
$317$	30.9518	1.73842	0.869212	+	0.494439i	$0.164626\pi$
$318$	0	0
$319$	−0.0905901	−0.00507207
$320$	0	0
$321$	−1.80203	−0.100579
$322$	0	0
$323$	3.85656	0.214585
$324$	0	0
$325$	−0.932345	−0.0517172
$326$	0	0
$327$	−1.03355	−0.0571557
$328$	0	0
$329$	−6.68913	−0.368783
$330$	0	0
$331$	33.4297	1.83746	0.918731	−	0.394884i	$-0.129215\pi$
$331$	33.4297	1.83746	0.918731	+	0.394884i	$0.129215\pi$
$332$	0	0
$333$	6.21251	0.340443
$334$	0	0
$335$	−30.5984	−1.67177
$336$	0	0
$337$	27.1717	1.48014	0.740069	−	0.672530i	$-0.234793\pi$
$337$	27.1717	1.48014	0.740069	+	0.672530i	$0.234793\pi$
$338$	0	0
$339$	−0.982459	−0.0533599
$340$	0	0
$341$	−2.09963	−0.113701
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−0.749353	−0.0403438
$346$	0	0
$347$	25.0784	1.34628	0.673139	−	0.739516i	$-0.264946\pi$
$347$	25.0784	1.34628	0.673139	+	0.739516i	$0.264946\pi$
$348$	0	0
$349$	−23.7113	−1.26924	−0.634618	−	0.772826i	$-0.718843\pi$
$349$	−23.7113	−1.26924	−0.634618	+	0.772826i	$0.718843\pi$
$350$	0	0
$351$	3.21934	0.171836
$352$	0	0
$353$	−20.6382	−1.09846	−0.549230	−	0.835672i	$-0.685079\pi$
$353$	−20.6382	−1.09846	−0.549230	+	0.835672i	$0.685079\pi$
$354$	0	0
$355$	23.3828	1.24103
$356$	0	0
$357$	−0.392799	−0.0207891
$358$	0	0
$359$	27.2976	1.44071	0.720355	−	0.693605i	$-0.243979\pi$
$359$	27.2976	1.44071	0.720355	+	0.693605i	$0.243979\pi$
$360$	0	0
$361$	−18.0096	−0.947874
$362$	0	0
$363$	1.06893	0.0561042
$364$	0	0
$365$	−18.9410	−0.991419
$366$	0	0
$367$	28.7125	1.49878	0.749390	−	0.662128i	$-0.230347\pi$
$367$	28.7125	1.49878	0.749390	+	0.662128i	$0.230347\pi$
$368$	0	0
$369$	11.2626	0.586307
$370$	0	0
$371$	−2.93656	−0.152459
$372$	0	0
$373$	29.1078	1.50715	0.753573	−	0.657364i	$-0.228329\pi$
$373$	29.1078	1.50715	0.753573	+	0.657364i	$0.228329\pi$
$374$	0	0
$375$	−1.15230	−0.0595044
$376$	0	0
$377$	−0.712688	−0.0367053
$378$	0	0
$379$	−19.5975	−1.00666	−0.503328	−	0.864096i	$-0.667891\pi$
$379$	−19.5975	−1.00666	−0.503328	+	0.864096i	$0.667891\pi$
$380$	0	0
$381$	−2.05433	−0.105247
$382$	0	0
$383$	19.5193	0.997389	0.498694	−	0.866778i	$-0.333813\pi$
$383$	19.5193	0.997389	0.498694	+	0.866778i	$0.333813\pi$
$384$	0	0
$385$	−1.48040	−0.0754482
$386$	0	0
$387$	−10.8009	−0.549042
$388$	0	0
$389$	10.7701	0.546065	0.273033	−	0.962005i	$-0.411973\pi$
$389$	10.7701	0.546065	0.273033	+	0.962005i	$0.411973\pi$
$390$	0	0
$391$	−13.0437	−0.659646
$392$	0	0
$393$	−1.81076	−0.0913405
$394$	0	0
$395$	−15.6444	−0.787153
$396$	0	0
$397$	−29.9845	−1.50488	−0.752440	−	0.658661i	$-0.771123\pi$
$397$	−29.9845	−1.50488	−0.752440	+	0.658661i	$0.771123\pi$
$398$	0	0
$399$	−0.100873	−0.00504998
$400$	0	0
$401$	34.4739	1.72155	0.860773	−	0.508989i	$-0.169981\pi$
$401$	34.4739	1.72155	0.860773	+	0.508989i	$0.169981\pi$
$402$	0	0
$403$	−16.5182	−0.822829
$404$	0	0
$405$	−19.5647	−0.972179
$406$	0	0
$407$	−1.40057	−0.0694235
$408$	0	0
$409$	−14.7405	−0.728869	−0.364435	−	0.931229i	$-0.618738\pi$
$409$	−14.7405	−0.728869	−0.364435	+	0.931229i	$0.618738\pi$
$410$	0	0
$411$	−1.16458	−0.0574447
$412$	0	0
$413$	6.92085	0.340553
$414$	0	0
$415$	−33.3486	−1.63702
$416$	0	0
$417$	−0.0212438	−0.00104032
$418$	0	0
$419$	−5.42033	−0.264800	−0.132400	−	0.991196i	$-0.542268\pi$
$419$	−5.42033	−0.264800	−0.132400	+	0.991196i	$0.542268\pi$
$420$	0	0
$421$	9.72201	0.473822	0.236911	−	0.971531i	$-0.423865\pi$
$421$	9.72201	0.473822	0.236911	+	0.971531i	$0.423865\pi$
$422$	0	0
$423$	19.9987	0.972368
$424$	0	0
$425$	−0.681376	−0.0330516
$426$	0	0
$427$	−4.30265	−0.208220
$428$	0	0
$429$	−0.362267	−0.0174904
$430$	0	0
$431$	12.1520	0.585341	0.292670	−	0.956213i	$-0.405456\pi$
$431$	12.1520	0.585341	0.292670	+	0.956213i	$0.405456\pi$
$432$	0	0
$433$	30.0836	1.44573	0.722864	−	0.690991i	$-0.242825\pi$
$433$	30.0836	1.44573	0.722864	+	0.690991i	$0.242825\pi$
$434$	0	0
$435$	−0.0299225	−0.00143467
$436$	0	0
$437$	−3.34969	−0.160238
$438$	0	0
$439$	−0.195719	−0.00934117	−0.00467058	−	0.999989i	$-0.501487\pi$
$439$	−0.195719	−0.00934117	−0.00467058	+	0.999989i	$0.501487\pi$
$440$	0	0
$441$	−2.98973	−0.142368
$442$	0	0
$443$	−28.1256	−1.33629	−0.668144	−	0.744032i	$-0.732911\pi$
$443$	−28.1256	−1.33629	−0.668144	+	0.744032i	$0.732911\pi$
$444$	0	0
$445$	−1.88171	−0.0892017
$446$	0	0
$447$	1.07337	0.0507686
$448$	0	0
$449$	−12.7288	−0.600709	−0.300355	−	0.953828i	$-0.597105\pi$
$449$	−12.7288	−0.600709	−0.300355	+	0.953828i	$0.597105\pi$
$450$	0	0
$451$	−2.53907	−0.119560
$452$	0	0
$453$	−1.16383	−0.0546816
$454$	0	0
$455$	−11.6466	−0.546000
$456$	0	0
$457$	3.64289	0.170407	0.0852035	−	0.996364i	$-0.472846\pi$
$457$	3.64289	0.170407	0.0852035	+	0.996364i	$0.472846\pi$
$458$	0	0
$459$	2.35276	0.109817
$460$	0	0
$461$	−11.4883	−0.535064	−0.267532	−	0.963549i	$-0.586208\pi$
$461$	−11.4883	−0.535064	−0.267532	+	0.963549i	$0.586208\pi$
$462$	0	0
$463$	−7.98371	−0.371035	−0.185517	−	0.982641i	$-0.559396\pi$
$463$	−7.98371	−0.371035	−0.185517	+	0.982641i	$0.559396\pi$
$464$	0	0
$465$	−0.693521	−0.0321613
$466$	0	0
$467$	27.5689	1.27573	0.637867	−	0.770146i	$-0.279817\pi$
$467$	27.5689	1.27573	0.637867	+	0.770146i	$0.279817\pi$
$468$	0	0
$469$	13.9311	0.643280
$470$	0	0
$471$	−1.01180	−0.0466212
$472$	0	0
$473$	2.43499	0.111961
$474$	0	0
$475$	−0.174982	−0.00802871
$476$	0	0
$477$	8.77951	0.401986
$478$	0	0
$479$	−12.9612	−0.592210	−0.296105	−	0.955155i	$-0.595688\pi$
$479$	−12.9612	−0.592210	−0.296105	+	0.955155i	$0.595688\pi$
$480$	0	0
$481$	−11.0185	−0.502401
$482$	0	0
$483$	0.341173	0.0155239
$484$	0	0
$485$	24.1878	1.09831
$486$	0	0
$487$	8.86309	0.401625	0.200812	−	0.979630i	$-0.435642\pi$
$487$	8.86309	0.401625	0.200812	+	0.979630i	$0.435642\pi$
$488$	0	0
$489$	0.381990	0.0172742
$490$	0	0
$491$	−17.5466	−0.791865	−0.395933	−	0.918280i	$-0.629579\pi$
$491$	−17.5466	−0.791865	−0.395933	+	0.918280i	$0.629579\pi$
$492$	0	0
$493$	−0.520847	−0.0234578
$494$	0	0
$495$	4.42599	0.198934
$496$	0	0
$497$	−10.6460	−0.477536
$498$	0	0
$499$	−0.153293	−0.00686231	−0.00343116	−	0.999994i	$-0.501092\pi$
$499$	−0.153293	−0.00686231	−0.00343116	+	0.999994i	$0.501092\pi$
$500$	0	0
$501$	−0.857406	−0.0383061
$502$	0	0
$503$	15.5136	0.691719	0.345860	−	0.938286i	$-0.387587\pi$
$503$	15.5136	0.691719	0.345860	+	0.938286i	$0.387587\pi$
$504$	0	0
$505$	0.778133	0.0346264
$506$	0	0
$507$	−1.53232	−0.0680526
$508$	0	0
$509$	−27.0386	−1.19846	−0.599232	−	0.800575i	$-0.704527\pi$
$509$	−27.0386	−1.19846	−0.599232	+	0.800575i	$0.704527\pi$
$510$	0	0
$511$	8.62368	0.381489
$512$	0	0
$513$	0.604204	0.0266762
$514$	0	0
$515$	−36.4053	−1.60421
$516$	0	0
$517$	−4.50856	−0.198286
$518$	0	0
$519$	0.143475	0.00629787
$520$	0	0
$521$	−15.7310	−0.689190	−0.344595	−	0.938752i	$-0.611984\pi$
$521$	−15.7310	−0.689190	−0.344595	+	0.938752i	$0.611984\pi$
$522$	0	0
$523$	−43.4999	−1.90212	−0.951059	−	0.309009i	$-0.900003\pi$
$523$	−43.4999	−1.90212	−0.951059	+	0.309009i	$0.900003\pi$
$524$	0	0
$525$	0.0178223	0.000777827 0
$526$	0	0
$527$	−12.0718	−0.525856
$528$	0	0
$529$	−11.6707	−0.507421
$530$	0	0
$531$	−20.6914	−0.897932
$532$	0	0
$533$	−19.9753	−0.865227
$534$	0	0
$535$	−39.0481	−1.68820
$536$	0	0
$537$	1.74797	0.0754303
$538$	0	0
$539$	0.674013	0.0290318
$540$	0	0
$541$	−30.1374	−1.29571	−0.647853	−	0.761765i	$-0.724333\pi$
$541$	−30.1374	−1.29571	−0.647853	+	0.761765i	$0.724333\pi$
$542$	0	0
$543$	−0.484209	−0.0207794
$544$	0	0
$545$	−22.3960	−0.959341
$546$	0	0
$547$	22.1918	0.948852	0.474426	−	0.880295i	$-0.342656\pi$
$547$	22.1918	0.948852	0.474426	+	0.880295i	$0.342656\pi$
$548$	0	0
$549$	12.8637	0.549011
$550$	0	0
$551$	−0.133757	−0.00569823
$552$	0	0
$553$	7.12273	0.302889
$554$	0	0
$555$	−0.462616	−0.0196369
$556$	0	0
$557$	4.64011	0.196608	0.0983038	−	0.995156i	$-0.468658\pi$
$557$	4.64011	0.196608	0.0983038	+	0.995156i	$0.468658\pi$
$558$	0	0
$559$	19.1565	0.810235
$560$	0	0
$561$	−0.264752	−0.0111778
$562$	0	0
$563$	−42.9470	−1.81000	−0.905000	−	0.425411i	$-0.860130\pi$
$563$	−42.9470	−1.81000	−0.905000	+	0.425411i	$0.860130\pi$
$564$	0	0
$565$	−21.2889	−0.895629
$566$	0	0
$567$	8.90764	0.374086
$568$	0	0
$569$	28.2672	1.18502	0.592512	−	0.805561i	$-0.298136\pi$
$569$	28.2672	1.18502	0.592512	+	0.805561i	$0.298136\pi$
$570$	0	0
$571$	22.6294	0.947012	0.473506	−	0.880791i	$-0.342988\pi$
$571$	22.6294	0.947012	0.473506	+	0.880791i	$0.342988\pi$
$572$	0	0
$573$	0.902268	0.0376928
$574$	0	0
$575$	0.591822	0.0246807
$576$	0	0
$577$	12.2701	0.510809	0.255405	−	0.966834i	$-0.417791\pi$
$577$	12.2701	0.510809	0.255405	+	0.966834i	$0.417791\pi$
$578$	0	0
$579$	1.21542	0.0505113
$580$	0	0
$581$	15.1833	0.629909
$582$	0	0
$583$	−1.97928	−0.0819734
$584$	0	0
$585$	34.8201	1.43963
$586$	0	0
$587$	21.0007	0.866793	0.433396	−	0.901203i	$-0.357315\pi$
$587$	21.0007	0.866793	0.433396	+	0.901203i	$0.357315\pi$
$588$	0	0
$589$	−3.10012	−0.127738
$590$	0	0
$591$	1.87599	0.0771679
$592$	0	0
$593$	4.30581	0.176819	0.0884093	−	0.996084i	$-0.471822\pi$
$593$	4.30581	0.176819	0.0884093	+	0.996084i	$0.471822\pi$
$594$	0	0
$595$	−8.51155	−0.348940
$596$	0	0
$597$	0.216156	0.00884666
$598$	0	0
$599$	2.42319	0.0990087	0.0495044	−	0.998774i	$-0.484236\pi$
$599$	2.42319	0.0990087	0.0495044	+	0.998774i	$0.484236\pi$
$600$	0	0
$601$	6.18973	0.252485	0.126242	−	0.991999i	$-0.459708\pi$
$601$	6.18973	0.252485	0.126242	+	0.991999i	$0.459708\pi$
$602$	0	0
$603$	−41.6503	−1.69613
$604$	0	0
$605$	23.1626	0.941693
$606$	0	0
$607$	−8.80892	−0.357543	−0.178772	−	0.983891i	$-0.557212\pi$
$607$	−8.80892	−0.357543	−0.178772	+	0.983891i	$0.557212\pi$
$608$	0	0
$609$	0.0136234	0.000552049 0
$610$	0	0
$611$	−35.4696	−1.43495
$612$	0	0
$613$	19.7718	0.798577	0.399289	−	0.916825i	$-0.369257\pi$
$613$	19.7718	0.798577	0.399289	+	0.916825i	$0.369257\pi$
$614$	0	0
$615$	−0.838670	−0.0338185
$616$	0	0
$617$	12.9420	0.521026	0.260513	−	0.965470i	$-0.416108\pi$
$617$	12.9420	0.521026	0.260513	+	0.965470i	$0.416108\pi$
$618$	0	0
$619$	17.1723	0.690213	0.345107	−	0.938563i	$-0.387843\pi$
$619$	17.1723	0.690213	0.345107	+	0.938563i	$0.387843\pi$
$620$	0	0
$621$	−2.04354	−0.0820042
$622$	0	0
$623$	0.856726	0.0343240
$624$	0	0
$625$	−24.0899	−0.963598
$626$	0	0
$627$	−0.0679899	−0.00271526
$628$	0	0
$629$	−8.05254	−0.321076
$630$	0	0
$631$	13.7312	0.546629	0.273314	−	0.961925i	$-0.411880\pi$
$631$	13.7312	0.546629	0.273314	+	0.961925i	$0.411880\pi$
$632$	0	0
$633$	−2.30331	−0.0915482
$634$	0	0
$635$	−44.5153	−1.76654
$636$	0	0
$637$	5.30258	0.210096
$638$	0	0
$639$	31.8285	1.25912
$640$	0	0
$641$	−11.7414	−0.463759	−0.231879	−	0.972745i	$-0.574487\pi$
$641$	−11.7414	−0.463759	−0.231879	+	0.972745i	$0.574487\pi$
$642$	0	0
$643$	−2.59560	−0.102361	−0.0511803	−	0.998689i	$-0.516298\pi$
$643$	−2.59560	−0.102361	−0.0511803	+	0.998689i	$0.516298\pi$
$644$	0	0
$645$	0.804293	0.0316690
$646$	0	0
$647$	29.0272	1.14118	0.570588	−	0.821236i	$-0.306715\pi$
$647$	29.0272	1.14118	0.570588	+	0.821236i	$0.306715\pi$
$648$	0	0
$649$	4.66474	0.183107
$650$	0	0
$651$	0.315754	0.0123754
$652$	0	0
$653$	39.2320	1.53527	0.767633	−	0.640890i	$-0.221435\pi$
$653$	39.2320	1.53527	0.767633	+	0.640890i	$0.221435\pi$
$654$	0	0
$655$	−39.2372	−1.53312
$656$	0	0
$657$	−25.7824	−1.00587
$658$	0	0
$659$	−18.2689	−0.711655	−0.355828	−	0.934552i	$-0.615801\pi$
$659$	−18.2689	−0.711655	−0.355828	+	0.934552i	$0.615801\pi$
$660$	0	0
$661$	39.5936	1.54001	0.770007	−	0.638036i	$-0.220253\pi$
$661$	39.5936	1.54001	0.770007	+	0.638036i	$0.220253\pi$
$662$	0	0
$663$	−2.08285	−0.0808911
$664$	0	0
$665$	−2.18582	−0.0847625
$666$	0	0
$667$	0.452392	0.0175167
$668$	0	0
$669$	0.111523	0.00431173
$670$	0	0
$671$	−2.90004	−0.111955
$672$	0	0
$673$	−8.38015	−0.323031	−0.161516	−	0.986870i	$-0.551638\pi$
$673$	−8.38015	−0.323031	−0.161516	+	0.986870i	$0.551638\pi$
$674$	0	0
$675$	−0.106750	−0.00410883
$676$	0	0
$677$	−20.5355	−0.789243	−0.394621	−	0.918844i	$-0.629124\pi$
$677$	−20.5355	−0.789243	−0.394621	+	0.918844i	$0.629124\pi$
$678$	0	0
$679$	−11.0125	−0.422620
$680$	0	0
$681$	0.267533	0.0102519
$682$	0	0
$683$	−24.2522	−0.927983	−0.463991	−	0.885840i	$-0.653583\pi$
$683$	−24.2522	−0.927983	−0.463991	+	0.885840i	$0.653583\pi$
$684$	0	0
$685$	−25.2353	−0.964192
$686$	0	0
$687$	1.94900	0.0743592
$688$	0	0
$689$	−15.5713	−0.593221
$690$	0	0
$691$	30.8579	1.17389	0.586945	−	0.809627i	$-0.300331\pi$
$691$	30.8579	1.17389	0.586945	+	0.809627i	$0.300331\pi$
$692$	0	0
$693$	−2.01511	−0.0765478
$694$	0	0
$695$	−0.460332	−0.0174614
$696$	0	0
$697$	−14.5984	−0.552952
$698$	0	0
$699$	−0.881820	−0.0333535
$700$	0	0
$701$	−49.9348	−1.88601	−0.943006	−	0.332775i	$-0.892015\pi$
$701$	−49.9348	−1.88601	−0.943006	+	0.332775i	$0.892015\pi$
$702$	0	0
$703$	−2.06794	−0.0779940
$704$	0	0
$705$	−1.48920	−0.0560867
$706$	0	0
$707$	−0.354276	−0.0133239
$708$	0	0
$709$	30.1773	1.13333	0.566665	−	0.823948i	$-0.308233\pi$
$709$	30.1773	1.13333	0.566665	+	0.823948i	$0.308233\pi$
$710$	0	0
$711$	−21.2950	−0.798625
$712$	0	0
$713$	10.4852	0.392674
$714$	0	0
$715$	−7.84994	−0.293571
$716$	0	0
$717$	0.175206	0.00654318
$718$	0	0
$719$	25.5563	0.953089	0.476544	−	0.879150i	$-0.341889\pi$
$719$	25.5563	0.953089	0.476544	+	0.879150i	$0.341889\pi$
$720$	0	0
$721$	16.5750	0.617284
$722$	0	0
$723$	−1.78885	−0.0665279
$724$	0	0
$725$	0.0236321	0.000877674 0
$726$	0	0
$727$	11.4845	0.425937	0.212969	−	0.977059i	$-0.431687\pi$
$727$	11.4845	0.425937	0.212969	+	0.977059i	$0.431687\pi$
$728$	0	0
$729$	−26.4468	−0.979510
$730$	0	0
$731$	14.0000	0.517807
$732$	0	0
$733$	−52.0835	−1.92375	−0.961873	−	0.273495i	$-0.911820\pi$
$733$	−52.0835	−1.92375	−0.961873	+	0.273495i	$0.911820\pi$
$734$	0	0
$735$	0.222631	0.00821185
$736$	0	0
$737$	9.38977	0.345877
$738$	0	0
$739$	−25.2269	−0.927986	−0.463993	−	0.885839i	$-0.653584\pi$
$739$	−25.2269	−0.927986	−0.463993	+	0.885839i	$0.653584\pi$
$740$	0	0
$741$	−0.534889	−0.0196496
$742$	0	0
$743$	23.8950	0.876623	0.438311	−	0.898823i	$-0.355577\pi$
$743$	23.8950	0.876623	0.438311	+	0.898823i	$0.355577\pi$
$744$	0	0
$745$	23.2588	0.852136
$746$	0	0
$747$	−45.3939	−1.66087
$748$	0	0
$749$	17.7782	0.649602
$750$	0	0
$751$	29.7013	1.08381	0.541907	−	0.840438i	$-0.317702\pi$
$751$	29.7013	1.08381	0.541907	+	0.840438i	$0.317702\pi$
$752$	0	0
$753$	1.48088	0.0539664
$754$	0	0
$755$	−25.2190	−0.917815
$756$	0	0
$757$	47.1188	1.71256	0.856280	−	0.516511i	$-0.172770\pi$
$757$	47.1188	1.71256	0.856280	+	0.516511i	$0.172770\pi$
$758$	0	0
$759$	0.229955	0.00834685
$760$	0	0
$761$	11.5153	0.417430	0.208715	−	0.977977i	$-0.433072\pi$
$761$	11.5153	0.417430	0.208715	+	0.977977i	$0.433072\pi$
$762$	0	0
$763$	10.1967	0.369145
$764$	0	0
$765$	25.4472	0.920046
$766$	0	0
$767$	36.6983	1.32510
$768$	0	0
$769$	27.5195	0.992378	0.496189	−	0.868215i	$-0.334732\pi$
$769$	27.5195	0.992378	0.496189	+	0.868215i	$0.334732\pi$
$770$	0	0
$771$	2.45223	0.0883150
$772$	0	0
$773$	13.6192	0.489850	0.244925	−	0.969542i	$-0.421237\pi$
$773$	13.6192	0.489850	0.244925	+	0.969542i	$0.421237\pi$
$774$	0	0
$775$	0.547727	0.0196749
$776$	0	0
$777$	0.210624	0.00755611
$778$	0	0
$779$	−3.74895	−0.134320
$780$	0	0
$781$	−7.17551	−0.256760
$782$	0	0
$783$	−0.0816005	−0.00291616
$784$	0	0
$785$	−21.9246	−0.782523
$786$	0	0
$787$	−34.6258	−1.23428	−0.617138	−	0.786855i	$-0.711708\pi$
$787$	−34.6258	−1.23428	−0.617138	+	0.786855i	$0.711708\pi$
$788$	0	0
$789$	2.41400	0.0859408
$790$	0	0
$791$	9.69262	0.344630
$792$	0	0
$793$	−22.8151	−0.810189
$794$	0	0
$795$	−0.653768	−0.0231868
$796$	0	0
$797$	−35.0863	−1.24282	−0.621410	−	0.783486i	$-0.713440\pi$
$797$	−35.0863	−1.24282	−0.621410	+	0.783486i	$0.713440\pi$
$798$	0	0
$799$	−25.9219	−0.917051
$800$	0	0
$801$	−2.56138	−0.0905017
$802$	0	0
$803$	5.81247	0.205118
$804$	0	0
$805$	7.39287	0.260565
$806$	0	0
$807$	−0.0308623	−0.00108641
$808$	0	0
$809$	−28.3561	−0.996947	−0.498473	−	0.866905i	$-0.666106\pi$
$809$	−28.3561	−0.996947	−0.498473	+	0.866905i	$0.666106\pi$
$810$	0	0
$811$	2.29432	0.0805646	0.0402823	−	0.999188i	$-0.487174\pi$
$811$	2.29432	0.0805646	0.0402823	+	0.999188i	$0.487174\pi$
$812$	0	0
$813$	1.74360	0.0611508
$814$	0	0
$815$	8.27732	0.289942
$816$	0	0
$817$	3.59528	0.125783
$818$	0	0
$819$	−15.8533	−0.553958
$820$	0	0
$821$	41.7766	1.45801	0.729007	−	0.684506i	$-0.239982\pi$
$821$	41.7766	1.45801	0.729007	+	0.684506i	$0.239982\pi$
$822$	0	0
$823$	45.2110	1.57596	0.787979	−	0.615702i	$-0.211128\pi$
$823$	45.2110	1.57596	0.787979	+	0.615702i	$0.211128\pi$
$824$	0	0
$825$	0.0120124	0.000418219 0
$826$	0	0
$827$	5.06276	0.176049	0.0880246	−	0.996118i	$-0.471945\pi$
$827$	5.06276	0.176049	0.0880246	+	0.996118i	$0.471945\pi$
$828$	0	0
$829$	−51.1701	−1.77721	−0.888606	−	0.458671i	$-0.848326\pi$
$829$	−51.1701	−1.77721	−0.888606	+	0.458671i	$0.848326\pi$
$830$	0	0
$831$	−2.14369	−0.0743637
$832$	0	0
$833$	3.87523	0.134269
$834$	0	0
$835$	−18.5791	−0.642956
$836$	0	0
$837$	−1.89128	−0.0653721
$838$	0	0
$839$	−40.5224	−1.39899	−0.699495	−	0.714638i	$-0.746592\pi$
$839$	−40.5224	−1.39899	−0.699495	+	0.714638i	$0.746592\pi$
$840$	0	0
$841$	−28.9819	−0.999377
$842$	0	0
$843$	−1.12485	−0.0387420
$844$	0	0
$845$	−33.2037	−1.14224
$846$	0	0
$847$	−10.5457	−0.362355
$848$	0	0
$849$	−2.41291	−0.0828109
$850$	0	0
$851$	6.99419	0.239758
$852$	0	0
$853$	8.02166	0.274656	0.137328	−	0.990526i	$-0.456148\pi$
$853$	8.02166	0.274656	0.137328	+	0.990526i	$0.456148\pi$
$854$	0	0
$855$	6.53500	0.223492
$856$	0	0
$857$	−0.785744	−0.0268405	−0.0134203	−	0.999910i	$-0.504272\pi$
$857$	−0.785744	−0.0268405	−0.0134203	+	0.999910i	$0.504272\pi$
$858$	0	0
$859$	−17.1625	−0.585577	−0.292789	−	0.956177i	$-0.594583\pi$
$859$	−17.1625	−0.585577	−0.292789	+	0.956177i	$0.594583\pi$
$860$	0	0
$861$	0.381839	0.0130130
$862$	0	0
$863$	8.01096	0.272696	0.136348	−	0.990661i	$-0.456463\pi$
$863$	8.01096	0.272696	0.136348	+	0.990661i	$0.456463\pi$
$864$	0	0
$865$	3.10896	0.105708
$866$	0	0
$867$	0.200959	0.00682494
$868$	0	0
$869$	4.80081	0.162856
$870$	0	0
$871$	73.8710	2.50302
$872$	0	0
$873$	32.9243	1.11432
$874$	0	0
$875$	11.3682	0.384315
$876$	0	0
$877$	−7.20234	−0.243206	−0.121603	−	0.992579i	$-0.538803\pi$
$877$	−7.20234	−0.243206	−0.121603	+	0.992579i	$0.538803\pi$
$878$	0	0
$879$	1.34520	0.0453724
$880$	0	0
$881$	28.3868	0.956377	0.478188	−	0.878257i	$-0.341294\pi$
$881$	28.3868	0.956377	0.478188	+	0.878257i	$0.341294\pi$
$882$	0	0
$883$	27.6390	0.930126	0.465063	−	0.885278i	$-0.346032\pi$
$883$	27.6390	0.930126	0.465063	+	0.885278i	$0.346032\pi$
$884$	0	0
$885$	1.54079	0.0517931
$886$	0	0
$887$	−6.99105	−0.234737	−0.117368	−	0.993088i	$-0.537446\pi$
$887$	−6.99105	−0.234737	−0.117368	+	0.993088i	$0.537446\pi$
$888$	0	0
$889$	20.2674	0.679746
$890$	0	0
$891$	6.00386	0.201137
$892$	0	0
$893$	−6.65691	−0.222765
$894$	0	0
$895$	37.8766	1.26607
$896$	0	0
$897$	1.80910	0.0604040
$898$	0	0
$899$	0.418685	0.0139639
$900$	0	0
$901$	−11.3798	−0.379118
$902$	0	0
$903$	−0.366187	−0.0121859
$904$	0	0
$905$	−10.4923	−0.348776
$906$	0	0
$907$	8.75000	0.290539	0.145270	−	0.989392i	$-0.453595\pi$
$907$	8.75000	0.290539	0.145270	+	0.989392i	$0.453595\pi$
$908$	0	0
$909$	1.05919	0.0351311
$910$	0	0
$911$	−23.5428	−0.780009	−0.390005	−	0.920813i	$-0.627527\pi$
$911$	−23.5428	−0.780009	−0.390005	+	0.920813i	$0.627527\pi$
$912$	0	0
$913$	10.2337	0.338687
$914$	0	0
$915$	−0.957901	−0.0316672
$916$	0	0
$917$	17.8643	0.589932
$918$	0	0
$919$	−53.5349	−1.76595	−0.882976	−	0.469417i	$-0.844464\pi$
$919$	−53.5349	−1.76595	−0.882976	+	0.469417i	$0.844464\pi$
$920$	0	0
$921$	−1.46290	−0.0482041
$922$	0	0
$923$	−56.4510	−1.85811
$924$	0	0
$925$	0.365363	0.0120131
$926$	0	0
$927$	−49.5546	−1.62759
$928$	0	0
$929$	43.7405	1.43508	0.717540	−	0.696518i	$-0.245268\pi$
$929$	43.7405	1.43508	0.717540	+	0.696518i	$0.245268\pi$
$930$	0	0
$931$	0.995183	0.0326158
$932$	0	0
$933$	2.27358	0.0744337
$934$	0	0
$935$	−5.73689	−0.187617
$936$	0	0
$937$	−8.90798	−0.291011	−0.145506	−	0.989357i	$-0.546481\pi$
$937$	−8.90798	−0.291011	−0.145506	+	0.989357i	$0.546481\pi$
$938$	0	0
$939$	2.49360	0.0813755
$940$	0	0
$941$	24.1765	0.788133	0.394066	−	0.919082i	$-0.371068\pi$
$941$	24.1765	0.788133	0.394066	+	0.919082i	$0.371068\pi$
$942$	0	0
$943$	12.6797	0.412907
$944$	0	0
$945$	−1.33350	−0.0433786
$946$	0	0
$947$	−8.55120	−0.277877	−0.138938	−	0.990301i	$-0.544369\pi$
$947$	−8.55120	−0.277877	−0.138938	+	0.990301i	$0.544369\pi$
$948$	0	0
$949$	45.7277	1.48438
$950$	0	0
$951$	−3.13732	−0.101735
$952$	0	0
$953$	−23.7559	−0.769529	−0.384765	−	0.923015i	$-0.625717\pi$
$953$	−23.7559	−0.769529	−0.384765	+	0.923015i	$0.625717\pi$
$954$	0	0
$955$	19.5512	0.632662
$956$	0	0
$957$	0.00918236	0.000296823 0
$958$	0	0
$959$	11.4894	0.371012
$960$	0	0
$961$	−21.2960	−0.686968
$962$	0	0
$963$	−53.1520	−1.71280
$964$	0	0
$965$	26.3370	0.847817
$966$	0	0
$967$	−56.9403	−1.83108	−0.915538	−	0.402231i	$-0.868235\pi$
$967$	−56.9403	−1.83108	−0.915538	+	0.402231i	$0.868235\pi$
$968$	0	0
$969$	−0.390907	−0.0125578
$970$	0	0
$971$	−25.9137	−0.831610	−0.415805	−	0.909454i	$-0.636500\pi$
$971$	−25.9137	−0.831610	−0.415805	+	0.909454i	$0.636500\pi$
$972$	0	0
$973$	0.209585	0.00671898
$974$	0	0
$975$	0.0945039	0.00302655
$976$	0	0
$977$	13.2222	0.423014	0.211507	−	0.977376i	$-0.432163\pi$
$977$	13.2222	0.423014	0.211507	+	0.977376i	$0.432163\pi$
$978$	0	0
$979$	0.577444	0.0184552
$980$	0	0
$981$	−30.4854	−0.973322
$982$	0	0
$983$	−39.0935	−1.24689	−0.623445	−	0.781867i	$-0.714268\pi$
$983$	−39.0935	−1.24689	−0.623445	+	0.781867i	$0.714268\pi$
$984$	0	0
$985$	40.6508	1.29524
$986$	0	0
$987$	0.678020	0.0215816
$988$	0	0
$989$	−12.1599	−0.386664
$990$	0	0
$991$	−21.0291	−0.668012	−0.334006	−	0.942571i	$-0.608401\pi$
$991$	−21.0291	−0.668012	−0.334006	+	0.942571i	$0.608401\pi$
$992$	0	0
$993$	−3.38849	−0.107530
$994$	0	0
$995$	4.68387	0.148489
$996$	0	0
$997$	26.9937	0.854899	0.427449	−	0.904039i	$-0.359412\pi$
$997$	26.9937	0.854899	0.427449	+	0.904039i	$0.359412\pi$
$998$	0	0
$999$	−1.26158	−0.0399147

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3584.2.a.l.1.3	yes	6
4.3	odd	2		3584.2.a.e.1.4	✓	6
8.3	odd	2		3584.2.a.k.1.3	yes	6
8.5	even	2		3584.2.a.f.1.4	yes	6
16.3	odd	4		3584.2.b.k.1793.7		12
16.5	even	4		3584.2.b.i.1793.7		12
16.11	odd	4		3584.2.b.k.1793.6		12
16.13	even	4		3584.2.b.i.1793.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3584.2.a.e.1.4	✓	6	4.3	odd	2
3584.2.a.f.1.4	yes	6	8.5	even	2
3584.2.a.k.1.3	yes	6	8.3	odd	2
3584.2.a.l.1.3	yes	6	1.1	even	1	trivial
3584.2.b.i.1793.6		12	16.13	even	4
3584.2.b.i.1793.7		12	16.5	even	4
3584.2.b.k.1793.6		12	16.11	odd	4
3584.2.b.k.1793.7		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)

\(f(q)\)	\(=\)	\(q-0.101362 q^{3} -2.19640 q^{5} +1.00000 q^{7} -2.98973 q^{9} +O(q^{10})\)	\(q-0.101362 q^{3} -2.19640 q^{5} +1.00000 q^{7} -2.98973 q^{9} +0.674013 q^{11} +5.30258 q^{13} +0.222631 q^{15} +3.87523 q^{17} +0.995183 q^{19} -0.101362 q^{21} -3.36590 q^{23} -0.175829 q^{25} +0.607128 q^{27} -0.134404 q^{29} -3.11512 q^{31} -0.0683190 q^{33} -2.19640 q^{35} -2.07795 q^{37} -0.537478 q^{39} -3.76710 q^{41} +3.61268 q^{43} +6.56663 q^{45} -6.68913 q^{47} +1.00000 q^{49} -0.392799 q^{51} -2.93656 q^{53} -1.48040 q^{55} -0.100873 q^{57} +6.92085 q^{59} -4.30265 q^{61} -2.98973 q^{63} -11.6466 q^{65} +13.9311 q^{67} +0.341173 q^{69} -10.6460 q^{71} +8.62368 q^{73} +0.0178223 q^{75} +0.674013 q^{77} +7.12273 q^{79} +8.90764 q^{81} +15.1833 q^{83} -8.51155 q^{85} +0.0136234 q^{87} +0.856726 q^{89} +5.30258 q^{91} +0.315754 q^{93} -2.18582 q^{95} -11.0125 q^{97} -2.01511 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{3} + 6 q^{7} + 10 q^{9}+O(q^{10}) \) 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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