LMFDB - Embedded newform 4592.2.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4592,2,Mod(1,4592)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4592, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4592.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4592 = 2^{4} \cdot 7 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4592.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:	$36.6673046082$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1148)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.30278$ of defining polynomial
Character	$\chi$	$=$	4592.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.302776 q^{3} +0.302776 q^{5} -1.00000 q^{7} -2.90833 q^{9} +O(q^{10})$	$q-0.302776 q^{3} +0.302776 q^{5} -1.00000 q^{7} -2.90833 q^{9} -1.00000 q^{11} -5.00000 q^{13} -0.0916731 q^{15} -8.21110 q^{17} +6.90833 q^{19} +0.302776 q^{21} -0.302776 q^{23} -4.90833 q^{25} +1.78890 q^{27} +6.69722 q^{29} -10.9083 q^{31} +0.302776 q^{33} -0.302776 q^{35} +6.60555 q^{37} +1.51388 q^{39} -1.00000 q^{41} +4.39445 q^{43} -0.880571 q^{45} +12.6056 q^{47} +1.00000 q^{49} +2.48612 q^{51} -6.30278 q^{53} -0.302776 q^{55} -2.09167 q^{57} +4.90833 q^{59} +4.21110 q^{61} +2.90833 q^{63} -1.51388 q^{65} +8.90833 q^{67} +0.0916731 q^{69} -2.21110 q^{71} +7.00000 q^{73} +1.48612 q^{75} +1.00000 q^{77} +9.39445 q^{79} +8.18335 q^{81} -10.8167 q^{83} -2.48612 q^{85} -2.02776 q^{87} +9.51388 q^{89} +5.00000 q^{91} +3.30278 q^{93} +2.09167 q^{95} -16.1194 q^{97} +2.90833 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{3} - 3 q^{5} - 2 q^{7} + 5 q^{9}+O(q^{10}) $ 2 * q + 3 * q^3 - 3 * q^5 - 2 * q^7 + 5 * q^9	$ 2 q + 3 q^{3} - 3 q^{5} - 2 q^{7} + 5 q^{9} - 2 q^{11} - 10 q^{13} - 11 q^{15} - 2 q^{17} + 3 q^{19} - 3 q^{21} + 3 q^{23} + q^{25} + 18 q^{27} + 17 q^{29} - 11 q^{31} - 3 q^{33} + 3 q^{35} + 6 q^{37} - 15 q^{39} - 2 q^{41} + 16 q^{43} - 27 q^{45} + 18 q^{47} + 2 q^{49} + 23 q^{51} - 9 q^{53} + 3 q^{55} - 15 q^{57} - q^{59} - 6 q^{61} - 5 q^{63} + 15 q^{65} + 7 q^{67} + 11 q^{69} + 10 q^{71} + 14 q^{73} + 21 q^{75} + 2 q^{77} + 26 q^{79} + 38 q^{81} - 23 q^{85} + 32 q^{87} + q^{89} + 10 q^{91} + 3 q^{93} + 15 q^{95} - 7 q^{97} - 5 q^{99}+O(q^{100}) $ 2 * q + 3 * q^3 - 3 * q^5 - 2 * q^7 + 5 * q^9 - 2 * q^11 - 10 * q^13 - 11 * q^15 - 2 * q^17 + 3 * q^19 - 3 * q^21 + 3 * q^23 + q^25 + 18 * q^27 + 17 * q^29 - 11 * q^31 - 3 * q^33 + 3 * q^35 + 6 * q^37 - 15 * q^39 - 2 * q^41 + 16 * q^43 - 27 * q^45 + 18 * q^47 + 2 * q^49 + 23 * q^51 - 9 * q^53 + 3 * q^55 - 15 * q^57 - q^59 - 6 * q^61 - 5 * q^63 + 15 * q^65 + 7 * q^67 + 11 * q^69 + 10 * q^71 + 14 * q^73 + 21 * q^75 + 2 * q^77 + 26 * q^79 + 38 * q^81 - 23 * q^85 + 32 * q^87 + q^89 + 10 * q^91 + 3 * q^93 + 15 * q^95 - 7 * q^97 - 5 * 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
$3$	−0.302776	−0.174808	−0.0874038	−	0.996173i	$-0.527857\pi$
$3$	−0.302776	−0.174808	−0.0874038	+	0.996173i	$0.527857\pi$
$4$	0	0
$5$	0.302776	0.135405	0.0677027	−	0.997706i	$-0.478433\pi$
$5$	0.302776	0.135405	0.0677027	+	0.997706i	$0.478433\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.90833	−0.969442
$10$	0	0
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	0	0
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	0	0
$15$	−0.0916731	−0.0236699
$16$	0	0
$17$	−8.21110	−1.99148	−0.995742	−	0.0921791i	$-0.970617\pi$
$17$	−8.21110	−1.99148	−0.995742	+	0.0921791i	$0.970617\pi$
$18$	0	0
$19$	6.90833	1.58488	0.792439	−	0.609951i	$-0.208811\pi$
$19$	6.90833	1.58488	0.792439	+	0.609951i	$0.208811\pi$
$20$	0	0
$21$	0.302776	0.0660711
$22$	0	0
$23$	−0.302776	−0.0631331	−0.0315665	−	0.999502i	$-0.510050\pi$
$23$	−0.302776	−0.0631331	−0.0315665	+	0.999502i	$0.510050\pi$
$24$	0	0
$25$	−4.90833	−0.981665
$26$	0	0
$27$	1.78890	0.344273
$28$	0	0
$29$	6.69722	1.24364	0.621822	−	0.783159i	$-0.286393\pi$
$29$	6.69722	1.24364	0.621822	+	0.783159i	$0.286393\pi$
$30$	0	0
$31$	−10.9083	−1.95919	−0.979597	−	0.200974i	$-0.935589\pi$
$31$	−10.9083	−1.95919	−0.979597	+	0.200974i	$0.935589\pi$
$32$	0	0
$33$	0.302776	0.0527065
$34$	0	0
$35$	−0.302776	−0.0511784
$36$	0	0
$37$	6.60555	1.08595	0.542973	−	0.839750i	$-0.317299\pi$
$37$	6.60555	1.08595	0.542973	+	0.839750i	$0.317299\pi$
$38$	0	0
$39$	1.51388	0.242415
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	4.39445	0.670147	0.335074	−	0.942192i	$-0.391239\pi$
$43$	4.39445	0.670147	0.335074	+	0.942192i	$0.391239\pi$
$44$	0	0
$45$	−0.880571	−0.131268
$46$	0	0
$47$	12.6056	1.83871	0.919354	−	0.393431i	$-0.128712\pi$
$47$	12.6056	1.83871	0.919354	+	0.393431i	$0.128712\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.48612	0.348127
$52$	0	0
$53$	−6.30278	−0.865753	−0.432876	−	0.901453i	$-0.642501\pi$
$53$	−6.30278	−0.865753	−0.432876	+	0.901453i	$0.642501\pi$
$54$	0	0
$55$	−0.302776	−0.0408263
$56$	0	0
$57$	−2.09167	−0.277049
$58$	0	0
$59$	4.90833	0.639010	0.319505	−	0.947585i	$-0.396483\pi$
$59$	4.90833	0.639010	0.319505	+	0.947585i	$0.396483\pi$
$60$	0	0
$61$	4.21110	0.539176	0.269588	−	0.962976i	$-0.413112\pi$
$61$	4.21110	0.539176	0.269588	+	0.962976i	$0.413112\pi$
$62$	0	0
$63$	2.90833	0.366415
$64$	0	0
$65$	−1.51388	−0.187773
$66$	0	0
$67$	8.90833	1.08833	0.544163	−	0.838980i	$-0.316847\pi$
$67$	8.90833	1.08833	0.544163	+	0.838980i	$0.316847\pi$
$68$	0	0
$69$	0.0916731	0.0110361
$70$	0	0
$71$	−2.21110	−0.262410	−0.131205	−	0.991355i	$-0.541885\pi$
$71$	−2.21110	−0.262410	−0.131205	+	0.991355i	$0.541885\pi$
$72$	0	0
$73$	7.00000	0.819288	0.409644	−	0.912245i	$-0.365653\pi$
$73$	7.00000	0.819288	0.409644	+	0.912245i	$0.365653\pi$
$74$	0	0
$75$	1.48612	0.171603
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	9.39445	1.05696	0.528479	−	0.848946i	$-0.322763\pi$
$79$	9.39445	1.05696	0.528479	+	0.848946i	$0.322763\pi$
$80$	0	0
$81$	8.18335	0.909261
$82$	0	0
$83$	−10.8167	−1.18728	−0.593641	−	0.804730i	$-0.702310\pi$
$83$	−10.8167	−1.18728	−0.593641	+	0.804730i	$0.702310\pi$
$84$	0	0
$85$	−2.48612	−0.269658
$86$	0	0
$87$	−2.02776	−0.217398
$88$	0	0
$89$	9.51388	1.00847	0.504235	−	0.863567i	$-0.331775\pi$
$89$	9.51388	1.00847	0.504235	+	0.863567i	$0.331775\pi$
$90$	0	0
$91$	5.00000	0.524142
$92$	0	0
$93$	3.30278	0.342482
$94$	0	0
$95$	2.09167	0.214601
$96$	0	0
$97$	−16.1194	−1.63668	−0.818340	−	0.574734i	$-0.805105\pi$
$97$	−16.1194	−1.63668	−0.818340	+	0.574734i	$0.805105\pi$
$98$	0	0
$99$	2.90833	0.292298
$100$	0	0
$101$	−1.60555	−0.159758	−0.0798792	−	0.996805i	$-0.525453\pi$
$101$	−1.60555	−0.159758	−0.0798792	+	0.996805i	$0.525453\pi$
$102$	0	0
$103$	11.3028	1.11370	0.556848	−	0.830615i	$-0.312011\pi$
$103$	11.3028	1.11370	0.556848	+	0.830615i	$0.312011\pi$
$104$	0	0
$105$	0.0916731	0.00894638
$106$	0	0
$107$	5.81665	0.562317	0.281159	−	0.959661i	$-0.409281\pi$
$107$	5.81665	0.562317	0.281159	+	0.959661i	$0.409281\pi$
$108$	0	0
$109$	−3.21110	−0.307568	−0.153784	−	0.988105i	$-0.549146\pi$
$109$	−3.21110	−0.307568	−0.153784	+	0.988105i	$0.549146\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	20.2111	1.90130	0.950650	−	0.310264i	$-0.100418\pi$
$113$	20.2111	1.90130	0.950650	+	0.310264i	$0.100418\pi$
$114$	0	0
$115$	−0.0916731	−0.00854856
$116$	0	0
$117$	14.5416	1.34437
$118$	0	0
$119$	8.21110	0.752711
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	0.302776	0.0273004
$124$	0	0
$125$	−3.00000	−0.268328
$126$	0	0
$127$	−7.00000	−0.621150	−0.310575	−	0.950549i	$-0.600522\pi$
$127$	−7.00000	−0.621150	−0.310575	+	0.950549i	$0.600522\pi$
$128$	0	0
$129$	−1.33053	−0.117147
$130$	0	0
$131$	−11.0917	−0.969084	−0.484542	−	0.874768i	$-0.661014\pi$
$131$	−11.0917	−0.969084	−0.484542	+	0.874768i	$0.661014\pi$
$132$	0	0
$133$	−6.90833	−0.599028
$134$	0	0
$135$	0.541635	0.0466165
$136$	0	0
$137$	12.9083	1.10283	0.551416	−	0.834230i	$-0.314088\pi$
$137$	12.9083	1.10283	0.551416	+	0.834230i	$0.314088\pi$
$138$	0	0
$139$	8.39445	0.712008	0.356004	−	0.934484i	$-0.384139\pi$
$139$	8.39445	0.712008	0.356004	+	0.934484i	$0.384139\pi$
$140$	0	0
$141$	−3.81665	−0.321420
$142$	0	0
$143$	5.00000	0.418121
$144$	0	0
$145$	2.02776	0.168396
$146$	0	0
$147$	−0.302776	−0.0249725
$148$	0	0
$149$	−15.2111	−1.24614	−0.623071	−	0.782165i	$-0.714115\pi$
$149$	−15.2111	−1.24614	−0.623071	+	0.782165i	$0.714115\pi$
$150$	0	0
$151$	5.00000	0.406894	0.203447	−	0.979086i	$-0.434786\pi$
$151$	5.00000	0.406894	0.203447	+	0.979086i	$0.434786\pi$
$152$	0	0
$153$	23.8806	1.93063
$154$	0	0
$155$	−3.30278	−0.265285
$156$	0	0
$157$	10.4222	0.831783	0.415891	−	0.909414i	$-0.363470\pi$
$157$	10.4222	0.831783	0.415891	+	0.909414i	$0.363470\pi$
$158$	0	0
$159$	1.90833	0.151340
$160$	0	0
$161$	0.302776	0.0238621
$162$	0	0
$163$	2.78890	0.218443	0.109222	−	0.994017i	$-0.465164\pi$
$163$	2.78890	0.218443	0.109222	+	0.994017i	$0.465164\pi$
$164$	0	0
$165$	0.0916731	0.00713674
$166$	0	0
$167$	10.2111	0.790159	0.395079	−	0.918647i	$-0.370717\pi$
$167$	10.2111	0.790159	0.395079	+	0.918647i	$0.370717\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	−20.0917	−1.53645
$172$	0	0
$173$	−10.8167	−0.822375	−0.411187	−	0.911551i	$-0.634886\pi$
$173$	−10.8167	−0.822375	−0.411187	+	0.911551i	$0.634886\pi$
$174$	0	0
$175$	4.90833	0.371035
$176$	0	0
$177$	−1.48612	−0.111704
$178$	0	0
$179$	−16.1194	−1.20482	−0.602411	−	0.798186i	$-0.705793\pi$
$179$	−16.1194	−1.20482	−0.602411	+	0.798186i	$0.705793\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	−1.27502	−0.0942521
$184$	0	0
$185$	2.00000	0.147043
$186$	0	0
$187$	8.21110	0.600455
$188$	0	0
$189$	−1.78890	−0.130123
$190$	0	0
$191$	−9.21110	−0.666492	−0.333246	−	0.942840i	$-0.608144\pi$
$191$	−9.21110	−0.666492	−0.333246	+	0.942840i	$0.608144\pi$
$192$	0	0
$193$	15.4222	1.11011	0.555057	−	0.831812i	$-0.312696\pi$
$193$	15.4222	1.11011	0.555057	+	0.831812i	$0.312696\pi$
$194$	0	0
$195$	0.458365	0.0328242
$196$	0	0
$197$	8.90833	0.634692	0.317346	−	0.948310i	$-0.397208\pi$
$197$	8.90833	0.634692	0.317346	+	0.948310i	$0.397208\pi$
$198$	0	0
$199$	−14.2111	−1.00740	−0.503699	−	0.863879i	$-0.668028\pi$
$199$	−14.2111	−1.00740	−0.503699	+	0.863879i	$0.668028\pi$
$200$	0	0
$201$	−2.69722	−0.190248
$202$	0	0
$203$	−6.69722	−0.470053
$204$	0	0
$205$	−0.302776	−0.0211468
$206$	0	0
$207$	0.880571	0.0612039
$208$	0	0
$209$	−6.90833	−0.477859
$210$	0	0
$211$	−9.11943	−0.627807	−0.313904	−	0.949455i	$-0.601637\pi$
$211$	−9.11943	−0.627807	−0.313904	+	0.949455i	$0.601637\pi$
$212$	0	0
$213$	0.669468	0.0458712
$214$	0	0
$215$	1.33053	0.0907415
$216$	0	0
$217$	10.9083	0.740505
$218$	0	0
$219$	−2.11943	−0.143218
$220$	0	0
$221$	41.0555	2.76169
$222$	0	0
$223$	8.30278	0.555995	0.277997	−	0.960582i	$-0.410329\pi$
$223$	8.30278	0.555995	0.277997	+	0.960582i	$0.410329\pi$
$224$	0	0
$225$	14.2750	0.951668
$226$	0	0
$227$	16.8167	1.11616	0.558080	−	0.829787i	$-0.311538\pi$
$227$	16.8167	1.11616	0.558080	+	0.829787i	$0.311538\pi$
$228$	0	0
$229$	−21.3944	−1.41378	−0.706892	−	0.707321i	$-0.749904\pi$
$229$	−21.3944	−1.41378	−0.706892	+	0.707321i	$0.749904\pi$
$230$	0	0
$231$	−0.302776	−0.0199212
$232$	0	0
$233$	−10.0917	−0.661127	−0.330564	−	0.943784i	$-0.607239\pi$
$233$	−10.0917	−0.661127	−0.330564	+	0.943784i	$0.607239\pi$
$234$	0	0
$235$	3.81665	0.248971
$236$	0	0
$237$	−2.84441	−0.184764
$238$	0	0
$239$	3.48612	0.225498	0.112749	−	0.993623i	$-0.464034\pi$
$239$	3.48612	0.225498	0.112749	+	0.993623i	$0.464034\pi$
$240$	0	0
$241$	−9.81665	−0.632346	−0.316173	−	0.948702i	$-0.602398\pi$
$241$	−9.81665	−0.632346	−0.316173	+	0.948702i	$0.602398\pi$
$242$	0	0
$243$	−7.84441	−0.503219
$244$	0	0
$245$	0.302776	0.0193436
$246$	0	0
$247$	−34.5416	−2.19783
$248$	0	0
$249$	3.27502	0.207546
$250$	0	0
$251$	−16.8167	−1.06146	−0.530729	−	0.847542i	$-0.678082\pi$
$251$	−16.8167	−1.06146	−0.530729	+	0.847542i	$0.678082\pi$
$252$	0	0
$253$	0.302776	0.0190353
$254$	0	0
$255$	0.752737	0.0471382
$256$	0	0
$257$	−19.0000	−1.18519	−0.592594	−	0.805502i	$-0.701896\pi$
$257$	−19.0000	−1.18519	−0.592594	+	0.805502i	$0.701896\pi$
$258$	0	0
$259$	−6.60555	−0.410449
$260$	0	0
$261$	−19.4777	−1.20564
$262$	0	0
$263$	−15.6056	−0.962280	−0.481140	−	0.876644i	$-0.659777\pi$
$263$	−15.6056	−0.962280	−0.481140	+	0.876644i	$0.659777\pi$
$264$	0	0
$265$	−1.90833	−0.117228
$266$	0	0
$267$	−2.88057	−0.176288
$268$	0	0
$269$	18.6972	1.13999	0.569995	−	0.821648i	$-0.306945\pi$
$269$	18.6972	1.13999	0.569995	+	0.821648i	$0.306945\pi$
$270$	0	0
$271$	19.0000	1.15417	0.577084	−	0.816685i	$-0.304191\pi$
$271$	19.0000	1.15417	0.577084	+	0.816685i	$0.304191\pi$
$272$	0	0
$273$	−1.51388	−0.0916241
$274$	0	0
$275$	4.90833	0.295983
$276$	0	0
$277$	−4.39445	−0.264037	−0.132018	−	0.991247i	$-0.542146\pi$
$277$	−4.39445	−0.264037	−0.132018	+	0.991247i	$0.542146\pi$
$278$	0	0
$279$	31.7250	1.89932
$280$	0	0
$281$	−5.00000	−0.298275	−0.149137	−	0.988816i	$-0.547650\pi$
$281$	−5.00000	−0.298275	−0.149137	+	0.988816i	$0.547650\pi$
$282$	0	0
$283$	31.6333	1.88040	0.940202	−	0.340616i	$-0.110636\pi$
$283$	31.6333	1.88040	0.940202	+	0.340616i	$0.110636\pi$
$284$	0	0
$285$	−0.633308	−0.0375139
$286$	0	0
$287$	1.00000	0.0590281
$288$	0	0
$289$	50.4222	2.96601
$290$	0	0
$291$	4.88057	0.286104
$292$	0	0
$293$	29.8444	1.74353	0.871765	−	0.489925i	$-0.162976\pi$
$293$	29.8444	1.74353	0.871765	+	0.489925i	$0.162976\pi$
$294$	0	0
$295$	1.48612	0.0865254
$296$	0	0
$297$	−1.78890	−0.103802
$298$	0	0
$299$	1.51388	0.0875498
$300$	0	0
$301$	−4.39445	−0.253292
$302$	0	0
$303$	0.486122	0.0279270
$304$	0	0
$305$	1.27502	0.0730074
$306$	0	0
$307$	11.2111	0.639851	0.319926	−	0.947443i	$-0.396342\pi$
$307$	11.2111	0.639851	0.319926	+	0.947443i	$0.396342\pi$
$308$	0	0
$309$	−3.42221	−0.194682
$310$	0	0
$311$	−5.90833	−0.335030	−0.167515	−	0.985869i	$-0.553574\pi$
$311$	−5.90833	−0.335030	−0.167515	+	0.985869i	$0.553574\pi$
$312$	0	0
$313$	15.6056	0.882078	0.441039	−	0.897488i	$-0.354610\pi$
$313$	15.6056	0.882078	0.441039	+	0.897488i	$0.354610\pi$
$314$	0	0
$315$	0.880571	0.0496145
$316$	0	0
$317$	16.2111	0.910506	0.455253	−	0.890362i	$-0.349549\pi$
$317$	16.2111	0.910506	0.455253	+	0.890362i	$0.349549\pi$
$318$	0	0
$319$	−6.69722	−0.374973
$320$	0	0
$321$	−1.76114	−0.0982973
$322$	0	0
$323$	−56.7250	−3.15626
$324$	0	0
$325$	24.5416	1.36132
$326$	0	0
$327$	0.972244	0.0537652
$328$	0	0
$329$	−12.6056	−0.694967
$330$	0	0
$331$	27.6333	1.51886	0.759432	−	0.650587i	$-0.225477\pi$
$331$	27.6333	1.51886	0.759432	+	0.650587i	$0.225477\pi$
$332$	0	0
$333$	−19.2111	−1.05276
$334$	0	0
$335$	2.69722	0.147365
$336$	0	0
$337$	15.2111	0.828602	0.414301	−	0.910140i	$-0.364026\pi$
$337$	15.2111	0.828602	0.414301	+	0.910140i	$0.364026\pi$
$338$	0	0
$339$	−6.11943	−0.332362
$340$	0	0
$341$	10.9083	0.590719
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0.0277564	0.00149435
$346$	0	0
$347$	26.6056	1.42826	0.714130	−	0.700013i	$-0.246822\pi$
$347$	26.6056	1.42826	0.714130	+	0.700013i	$0.246822\pi$
$348$	0	0
$349$	−12.0278	−0.643831	−0.321916	−	0.946768i	$-0.604327\pi$
$349$	−12.0278	−0.643831	−0.321916	+	0.946768i	$0.604327\pi$
$350$	0	0
$351$	−8.94449	−0.477421
$352$	0	0
$353$	26.8444	1.42878	0.714392	−	0.699746i	$-0.246703\pi$
$353$	26.8444	1.42878	0.714392	+	0.699746i	$0.246703\pi$
$354$	0	0
$355$	−0.669468	−0.0355317
$356$	0	0
$357$	−2.48612	−0.131580
$358$	0	0
$359$	−1.39445	−0.0735962	−0.0367981	−	0.999323i	$-0.511716\pi$
$359$	−1.39445	−0.0735962	−0.0367981	+	0.999323i	$0.511716\pi$
$360$	0	0
$361$	28.7250	1.51184
$362$	0	0
$363$	3.02776	0.158916
$364$	0	0
$365$	2.11943	0.110936
$366$	0	0
$367$	0.513878	0.0268242	0.0134121	−	0.999910i	$-0.495731\pi$
$367$	0.513878	0.0268242	0.0134121	+	0.999910i	$0.495731\pi$
$368$	0	0
$369$	2.90833	0.151401
$370$	0	0
$371$	6.30278	0.327224
$372$	0	0
$373$	−18.3028	−0.947682	−0.473841	−	0.880610i	$-0.657133\pi$
$373$	−18.3028	−0.947682	−0.473841	+	0.880610i	$0.657133\pi$
$374$	0	0
$375$	0.908327	0.0469058
$376$	0	0
$377$	−33.4861	−1.72462
$378$	0	0
$379$	−5.81665	−0.298781	−0.149391	−	0.988778i	$-0.547731\pi$
$379$	−5.81665	−0.298781	−0.149391	+	0.988778i	$0.547731\pi$
$380$	0	0
$381$	2.11943	0.108582
$382$	0	0
$383$	−6.39445	−0.326741	−0.163371	−	0.986565i	$-0.552237\pi$
$383$	−6.39445	−0.326741	−0.163371	+	0.986565i	$0.552237\pi$
$384$	0	0
$385$	0.302776	0.0154309
$386$	0	0
$387$	−12.7805	−0.649669
$388$	0	0
$389$	−3.90833	−0.198160	−0.0990800	−	0.995079i	$-0.531590\pi$
$389$	−3.90833	−0.198160	−0.0990800	+	0.995079i	$0.531590\pi$
$390$	0	0
$391$	2.48612	0.125729
$392$	0	0
$393$	3.35829	0.169403
$394$	0	0
$395$	2.84441	0.143118
$396$	0	0
$397$	−18.7250	−0.939780	−0.469890	−	0.882725i	$-0.655706\pi$
$397$	−18.7250	−0.939780	−0.469890	+	0.882725i	$0.655706\pi$
$398$	0	0
$399$	2.09167	0.104715
$400$	0	0
$401$	−23.8167	−1.18935	−0.594673	−	0.803967i	$-0.702719\pi$
$401$	−23.8167	−1.18935	−0.594673	+	0.803967i	$0.702719\pi$
$402$	0	0
$403$	54.5416	2.71691
$404$	0	0
$405$	2.47772	0.123119
$406$	0	0
$407$	−6.60555	−0.327425
$408$	0	0
$409$	−3.60555	−0.178283	−0.0891415	−	0.996019i	$-0.528412\pi$
$409$	−3.60555	−0.178283	−0.0891415	+	0.996019i	$0.528412\pi$
$410$	0	0
$411$	−3.90833	−0.192784
$412$	0	0
$413$	−4.90833	−0.241523
$414$	0	0
$415$	−3.27502	−0.160764
$416$	0	0
$417$	−2.54163	−0.124464
$418$	0	0
$419$	−11.7250	−0.572803	−0.286401	−	0.958110i	$-0.592459\pi$
$419$	−11.7250	−0.572803	−0.286401	+	0.958110i	$0.592459\pi$
$420$	0	0
$421$	18.0000	0.877266	0.438633	−	0.898666i	$-0.355463\pi$
$421$	18.0000	0.877266	0.438633	+	0.898666i	$0.355463\pi$
$422$	0	0
$423$	−36.6611	−1.78252
$424$	0	0
$425$	40.3028	1.95497
$426$	0	0
$427$	−4.21110	−0.203790
$428$	0	0
$429$	−1.51388	−0.0730907
$430$	0	0
$431$	7.60555	0.366347	0.183173	−	0.983081i	$-0.441363\pi$
$431$	7.60555	0.366347	0.183173	+	0.983081i	$0.441363\pi$
$432$	0	0
$433$	−22.9083	−1.10090	−0.550452	−	0.834867i	$-0.685545\pi$
$433$	−22.9083	−1.10090	−0.550452	+	0.834867i	$0.685545\pi$
$434$	0	0
$435$	−0.613955	−0.0294369
$436$	0	0
$437$	−2.09167	−0.100058
$438$	0	0
$439$	22.6333	1.08023	0.540114	−	0.841592i	$-0.318381\pi$
$439$	22.6333	1.08023	0.540114	+	0.841592i	$0.318381\pi$
$440$	0	0
$441$	−2.90833	−0.138492
$442$	0	0
$443$	34.5416	1.64112	0.820561	−	0.571559i	$-0.193661\pi$
$443$	34.5416	1.64112	0.820561	+	0.571559i	$0.193661\pi$
$444$	0	0
$445$	2.88057	0.136552
$446$	0	0
$447$	4.60555	0.217835
$448$	0	0
$449$	−8.78890	−0.414774	−0.207387	−	0.978259i	$-0.566496\pi$
$449$	−8.78890	−0.414774	−0.207387	+	0.978259i	$0.566496\pi$
$450$	0	0
$451$	1.00000	0.0470882
$452$	0	0
$453$	−1.51388	−0.0711282
$454$	0	0
$455$	1.51388	0.0709717
$456$	0	0
$457$	30.8167	1.44154	0.720771	−	0.693173i	$-0.243788\pi$
$457$	30.8167	1.44154	0.720771	+	0.693173i	$0.243788\pi$
$458$	0	0
$459$	−14.6888	−0.685615
$460$	0	0
$461$	18.8167	0.876379	0.438189	−	0.898883i	$-0.355620\pi$
$461$	18.8167	0.876379	0.438189	+	0.898883i	$0.355620\pi$
$462$	0	0
$463$	−18.9361	−0.880034	−0.440017	−	0.897989i	$-0.645028\pi$
$463$	−18.9361	−0.880034	−0.440017	+	0.897989i	$0.645028\pi$
$464$	0	0
$465$	1.00000	0.0463739
$466$	0	0
$467$	−8.21110	−0.379965	−0.189982	−	0.981788i	$-0.560843\pi$
$467$	−8.21110	−0.379965	−0.189982	+	0.981788i	$0.560843\pi$
$468$	0	0
$469$	−8.90833	−0.411348
$470$	0	0
$471$	−3.15559	−0.145402
$472$	0	0
$473$	−4.39445	−0.202057
$474$	0	0
$475$	−33.9083	−1.55582
$476$	0	0
$477$	18.3305	0.839297
$478$	0	0
$479$	−40.1194	−1.83310	−0.916552	−	0.399916i	$-0.869039\pi$
$479$	−40.1194	−1.83310	−0.916552	+	0.399916i	$0.869039\pi$
$480$	0	0
$481$	−33.0278	−1.50594
$482$	0	0
$483$	−0.0916731	−0.00417127
$484$	0	0
$485$	−4.88057	−0.221615
$486$	0	0
$487$	17.3944	0.788218	0.394109	−	0.919064i	$-0.371053\pi$
$487$	17.3944	0.788218	0.394109	+	0.919064i	$0.371053\pi$
$488$	0	0
$489$	−0.844410	−0.0381855
$490$	0	0
$491$	7.72498	0.348624	0.174312	−	0.984691i	$-0.444230\pi$
$491$	7.72498	0.348624	0.174312	+	0.984691i	$0.444230\pi$
$492$	0	0
$493$	−54.9916	−2.47670
$494$	0	0
$495$	0.880571	0.0395787
$496$	0	0
$497$	2.21110	0.0991815
$498$	0	0
$499$	30.2389	1.35368	0.676839	−	0.736131i	$-0.263349\pi$
$499$	30.2389	1.35368	0.676839	+	0.736131i	$0.263349\pi$
$500$	0	0
$501$	−3.09167	−0.138126
$502$	0	0
$503$	12.6972	0.566141	0.283071	−	0.959099i	$-0.408647\pi$
$503$	12.6972	0.566141	0.283071	+	0.959099i	$0.408647\pi$
$504$	0	0
$505$	−0.486122	−0.0216321
$506$	0	0
$507$	−3.63331	−0.161361
$508$	0	0
$509$	−24.0917	−1.06784	−0.533922	−	0.845534i	$-0.679283\pi$
$509$	−24.0917	−1.06784	−0.533922	+	0.845534i	$0.679283\pi$
$510$	0	0
$511$	−7.00000	−0.309662
$512$	0	0
$513$	12.3583	0.545632
$514$	0	0
$515$	3.42221	0.150800
$516$	0	0
$517$	−12.6056	−0.554392
$518$	0	0
$519$	3.27502	0.143757
$520$	0	0
$521$	10.3028	0.451373	0.225686	−	0.974200i	$-0.427538\pi$
$521$	10.3028	0.451373	0.225686	+	0.974200i	$0.427538\pi$
$522$	0	0
$523$	29.6056	1.29456	0.647280	−	0.762252i	$-0.275906\pi$
$523$	29.6056	1.29456	0.647280	+	0.762252i	$0.275906\pi$
$524$	0	0
$525$	−1.48612	−0.0648597
$526$	0	0
$527$	89.5694	3.90170
$528$	0	0
$529$	−22.9083	−0.996014
$530$	0	0
$531$	−14.2750	−0.619483
$532$	0	0
$533$	5.00000	0.216574
$534$	0	0
$535$	1.76114	0.0761408
$536$	0	0
$537$	4.88057	0.210612
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−40.3583	−1.73514	−0.867569	−	0.497317i	$-0.834319\pi$
$541$	−40.3583	−1.73514	−0.867569	+	0.497317i	$0.834319\pi$
$542$	0	0
$543$	3.02776	0.129933
$544$	0	0
$545$	−0.972244	−0.0416463
$546$	0	0
$547$	2.57779	0.110219	0.0551093	−	0.998480i	$-0.482449\pi$
$547$	2.57779	0.110219	0.0551093	+	0.998480i	$0.482449\pi$
$548$	0	0
$549$	−12.2473	−0.522700
$550$	0	0
$551$	46.2666	1.97102
$552$	0	0
$553$	−9.39445	−0.399493
$554$	0	0
$555$	−0.605551	−0.0257042
$556$	0	0
$557$	13.5416	0.573777	0.286889	−	0.957964i	$-0.407379\pi$
$557$	13.5416	0.573777	0.286889	+	0.957964i	$0.407379\pi$
$558$	0	0
$559$	−21.9722	−0.929327
$560$	0	0
$561$	−2.48612	−0.104964
$562$	0	0
$563$	34.6333	1.45962	0.729810	−	0.683650i	$-0.239609\pi$
$563$	34.6333	1.45962	0.729810	+	0.683650i	$0.239609\pi$
$564$	0	0
$565$	6.11943	0.257446
$566$	0	0
$567$	−8.18335	−0.343668
$568$	0	0
$569$	22.5416	0.944994	0.472497	−	0.881332i	$-0.343353\pi$
$569$	22.5416	0.944994	0.472497	+	0.881332i	$0.343353\pi$
$570$	0	0
$571$	−45.6333	−1.90969	−0.954847	−	0.297097i	$-0.903981\pi$
$571$	−45.6333	−1.90969	−0.954847	+	0.297097i	$0.903981\pi$
$572$	0	0
$573$	2.78890	0.116508
$574$	0	0
$575$	1.48612	0.0619756
$576$	0	0
$577$	7.42221	0.308990	0.154495	−	0.987994i	$-0.450625\pi$
$577$	7.42221	0.308990	0.154495	+	0.987994i	$0.450625\pi$
$578$	0	0
$579$	−4.66947	−0.194056
$580$	0	0
$581$	10.8167	0.448750
$582$	0	0
$583$	6.30278	0.261034
$584$	0	0
$585$	4.40285	0.182036
$586$	0	0
$587$	33.2389	1.37191	0.685957	−	0.727642i	$-0.259384\pi$
$587$	33.2389	1.37191	0.685957	+	0.727642i	$0.259384\pi$
$588$	0	0
$589$	−75.3583	−3.10508
$590$	0	0
$591$	−2.69722	−0.110949
$592$	0	0
$593$	−24.8444	−1.02024	−0.510119	−	0.860104i	$-0.670399\pi$
$593$	−24.8444	−1.02024	−0.510119	+	0.860104i	$0.670399\pi$
$594$	0	0
$595$	2.48612	0.101921
$596$	0	0
$597$	4.30278	0.176101
$598$	0	0
$599$	1.81665	0.0742265	0.0371132	−	0.999311i	$-0.488184\pi$
$599$	1.81665	0.0742265	0.0371132	+	0.999311i	$0.488184\pi$
$600$	0	0
$601$	9.30278	0.379468	0.189734	−	0.981836i	$-0.439237\pi$
$601$	9.30278	0.379468	0.189734	+	0.981836i	$0.439237\pi$
$602$	0	0
$603$	−25.9083	−1.05507
$604$	0	0
$605$	−3.02776	−0.123096
$606$	0	0
$607$	−12.0278	−0.488192	−0.244096	−	0.969751i	$-0.578491\pi$
$607$	−12.0278	−0.488192	−0.244096	+	0.969751i	$0.578491\pi$
$608$	0	0
$609$	2.02776	0.0821688
$610$	0	0
$611$	−63.0278	−2.54983
$612$	0	0
$613$	31.1472	1.25802	0.629011	−	0.777396i	$-0.283460\pi$
$613$	31.1472	1.25802	0.629011	+	0.777396i	$0.283460\pi$
$614$	0	0
$615$	0.0916731	0.00369662
$616$	0	0
$617$	7.97224	0.320950	0.160475	−	0.987040i	$-0.448697\pi$
$617$	7.97224	0.320950	0.160475	+	0.987040i	$0.448697\pi$
$618$	0	0
$619$	32.5139	1.30684	0.653422	−	0.756994i	$-0.273333\pi$
$619$	32.5139	1.30684	0.653422	+	0.756994i	$0.273333\pi$
$620$	0	0
$621$	−0.541635	−0.0217350
$622$	0	0
$623$	−9.51388	−0.381165
$624$	0	0
$625$	23.6333	0.945332
$626$	0	0
$627$	2.09167	0.0835334
$628$	0	0
$629$	−54.2389	−2.16264
$630$	0	0
$631$	35.8444	1.42694	0.713472	−	0.700684i	$-0.247122\pi$
$631$	35.8444	1.42694	0.713472	+	0.700684i	$0.247122\pi$
$632$	0	0
$633$	2.76114	0.109746
$634$	0	0
$635$	−2.11943	−0.0841070
$636$	0	0
$637$	−5.00000	−0.198107
$638$	0	0
$639$	6.43061	0.254391
$640$	0	0
$641$	42.6333	1.68391	0.841957	−	0.539544i	$-0.181403\pi$
$641$	42.6333	1.68391	0.841957	+	0.539544i	$0.181403\pi$
$642$	0	0
$643$	−41.9638	−1.65489	−0.827446	−	0.561545i	$-0.810207\pi$
$643$	−41.9638	−1.65489	−0.827446	+	0.561545i	$0.810207\pi$
$644$	0	0
$645$	−0.402853	−0.0158623
$646$	0	0
$647$	−30.0278	−1.18051	−0.590256	−	0.807216i	$-0.700973\pi$
$647$	−30.0278	−1.18051	−0.590256	+	0.807216i	$0.700973\pi$
$648$	0	0
$649$	−4.90833	−0.192669
$650$	0	0
$651$	−3.30278	−0.129446
$652$	0	0
$653$	2.72498	0.106637	0.0533184	−	0.998578i	$-0.483020\pi$
$653$	2.72498	0.106637	0.0533184	+	0.998578i	$0.483020\pi$
$654$	0	0
$655$	−3.35829	−0.131219
$656$	0	0
$657$	−20.3583	−0.794252
$658$	0	0
$659$	−19.6056	−0.763724	−0.381862	−	0.924219i	$-0.624717\pi$
$659$	−19.6056	−0.763724	−0.381862	+	0.924219i	$0.624717\pi$
$660$	0	0
$661$	21.3944	0.832148	0.416074	−	0.909331i	$-0.363406\pi$
$661$	21.3944	0.832148	0.416074	+	0.909331i	$0.363406\pi$
$662$	0	0
$663$	−12.4306	−0.482765
$664$	0	0
$665$	−2.09167	−0.0811116
$666$	0	0
$667$	−2.02776	−0.0785150
$668$	0	0
$669$	−2.51388	−0.0971921
$670$	0	0
$671$	−4.21110	−0.162568
$672$	0	0
$673$	17.1833	0.662369	0.331185	−	0.943566i	$-0.392552\pi$
$673$	17.1833	0.662369	0.331185	+	0.943566i	$0.392552\pi$
$674$	0	0
$675$	−8.78049	−0.337961
$676$	0	0
$677$	−30.3583	−1.16676	−0.583382	−	0.812198i	$-0.698271\pi$
$677$	−30.3583	−1.16676	−0.583382	+	0.812198i	$0.698271\pi$
$678$	0	0
$679$	16.1194	0.618607
$680$	0	0
$681$	−5.09167	−0.195113
$682$	0	0
$683$	38.6333	1.47826	0.739131	−	0.673561i	$-0.235236\pi$
$683$	38.6333	1.47826	0.739131	+	0.673561i	$0.235236\pi$
$684$	0	0
$685$	3.90833	0.149329
$686$	0	0
$687$	6.47772	0.247140
$688$	0	0
$689$	31.5139	1.20058
$690$	0	0
$691$	25.3028	0.962563	0.481281	−	0.876566i	$-0.340171\pi$
$691$	25.3028	0.962563	0.481281	+	0.876566i	$0.340171\pi$
$692$	0	0
$693$	−2.90833	−0.110478
$694$	0	0
$695$	2.54163	0.0964097
$696$	0	0
$697$	8.21110	0.311018
$698$	0	0
$699$	3.05551	0.115570
$700$	0	0
$701$	−22.9083	−0.865236	−0.432618	−	0.901577i	$-0.642410\pi$
$701$	−22.9083	−0.865236	−0.432618	+	0.901577i	$0.642410\pi$
$702$	0	0
$703$	45.6333	1.72109
$704$	0	0
$705$	−1.15559	−0.0435220
$706$	0	0
$707$	1.60555	0.0603830
$708$	0	0
$709$	−6.60555	−0.248077	−0.124038	−	0.992277i	$-0.539585\pi$
$709$	−6.60555	−0.248077	−0.124038	+	0.992277i	$0.539585\pi$
$710$	0	0
$711$	−27.3221	−1.02466
$712$	0	0
$713$	3.30278	0.123690
$714$	0	0
$715$	1.51388	0.0566158
$716$	0	0
$717$	−1.05551	−0.0394188
$718$	0	0
$719$	−37.2389	−1.38878	−0.694388	−	0.719601i	$-0.744325\pi$
$719$	−37.2389	−1.38878	−0.694388	+	0.719601i	$0.744325\pi$
$720$	0	0
$721$	−11.3028	−0.420937
$722$	0	0
$723$	2.97224	0.110539
$724$	0	0
$725$	−32.8722	−1.22084
$726$	0	0
$727$	−5.66947	−0.210269	−0.105134	−	0.994458i	$-0.533527\pi$
$727$	−5.66947	−0.210269	−0.105134	+	0.994458i	$0.533527\pi$
$728$	0	0
$729$	−22.1749	−0.821294
$730$	0	0
$731$	−36.0833	−1.33459
$732$	0	0
$733$	−37.4222	−1.38222	−0.691110	−	0.722749i	$-0.742878\pi$
$733$	−37.4222	−1.38222	−0.691110	+	0.722749i	$0.742878\pi$
$734$	0	0
$735$	−0.0916731	−0.00338141
$736$	0	0
$737$	−8.90833	−0.328142
$738$	0	0
$739$	−18.5139	−0.681044	−0.340522	−	0.940237i	$-0.610604\pi$
$739$	−18.5139	−0.681044	−0.340522	+	0.940237i	$0.610604\pi$
$740$	0	0
$741$	10.4584	0.384198
$742$	0	0
$743$	38.5694	1.41497	0.707487	−	0.706726i	$-0.249829\pi$
$743$	38.5694	1.41497	0.707487	+	0.706726i	$0.249829\pi$
$744$	0	0
$745$	−4.60555	−0.168734
$746$	0	0
$747$	31.4584	1.15100
$748$	0	0
$749$	−5.81665	−0.212536
$750$	0	0
$751$	31.2389	1.13992	0.569961	−	0.821672i	$-0.306958\pi$
$751$	31.2389	1.13992	0.569961	+	0.821672i	$0.306958\pi$
$752$	0	0
$753$	5.09167	0.185551
$754$	0	0
$755$	1.51388	0.0550957
$756$	0	0
$757$	29.0278	1.05503	0.527516	−	0.849545i	$-0.323124\pi$
$757$	29.0278	1.05503	0.527516	+	0.849545i	$0.323124\pi$
$758$	0	0
$759$	−0.0916731	−0.00332752
$760$	0	0
$761$	32.0000	1.16000	0.580000	−	0.814617i	$-0.303053\pi$
$761$	32.0000	1.16000	0.580000	+	0.814617i	$0.303053\pi$
$762$	0	0
$763$	3.21110	0.116250
$764$	0	0
$765$	7.23045	0.261418
$766$	0	0
$767$	−24.5416	−0.886147
$768$	0	0
$769$	−2.97224	−0.107182	−0.0535909	−	0.998563i	$-0.517067\pi$
$769$	−2.97224	−0.107182	−0.0535909	+	0.998563i	$0.517067\pi$
$770$	0	0
$771$	5.75274	0.207180
$772$	0	0
$773$	−21.8444	−0.785689	−0.392844	−	0.919605i	$-0.628509\pi$
$773$	−21.8444	−0.785689	−0.392844	+	0.919605i	$0.628509\pi$
$774$	0	0
$775$	53.5416	1.92327
$776$	0	0
$777$	2.00000	0.0717496
$778$	0	0
$779$	−6.90833	−0.247516
$780$	0	0
$781$	2.21110	0.0791195
$782$	0	0
$783$	11.9806	0.428153
$784$	0	0
$785$	3.15559	0.112628
$786$	0	0
$787$	10.6056	0.378047	0.189024	−	0.981973i	$-0.439468\pi$
$787$	10.6056	0.378047	0.189024	+	0.981973i	$0.439468\pi$
$788$	0	0
$789$	4.72498	0.168214
$790$	0	0
$791$	−20.2111	−0.718624
$792$	0	0
$793$	−21.0555	−0.747703
$794$	0	0
$795$	0.577795	0.0204923
$796$	0	0
$797$	−28.6333	−1.01424	−0.507122	−	0.861874i	$-0.669291\pi$
$797$	−28.6333	−1.01424	−0.507122	+	0.861874i	$0.669291\pi$
$798$	0	0
$799$	−103.505	−3.66176
$800$	0	0
$801$	−27.6695	−0.977653
$802$	0	0
$803$	−7.00000	−0.247025
$804$	0	0
$805$	0.0916731	0.00323105
$806$	0	0
$807$	−5.66106	−0.199279
$808$	0	0
$809$	−45.3305	−1.59374	−0.796868	−	0.604153i	$-0.793512\pi$
$809$	−45.3305	−1.59374	−0.796868	+	0.604153i	$0.793512\pi$
$810$	0	0
$811$	3.00000	0.105344	0.0526721	−	0.998612i	$-0.483226\pi$
$811$	3.00000	0.105344	0.0526721	+	0.998612i	$0.483226\pi$
$812$	0	0
$813$	−5.75274	−0.201757
$814$	0	0
$815$	0.844410	0.0295784
$816$	0	0
$817$	30.3583	1.06210
$818$	0	0
$819$	−14.5416	−0.508126
$820$	0	0
$821$	39.8444	1.39058	0.695290	−	0.718730i	$-0.255276\pi$
$821$	39.8444	1.39058	0.695290	+	0.718730i	$0.255276\pi$
$822$	0	0
$823$	31.4861	1.09754	0.548769	−	0.835974i	$-0.315097\pi$
$823$	31.4861	1.09754	0.548769	+	0.835974i	$0.315097\pi$
$824$	0	0
$825$	−1.48612	−0.0517401
$826$	0	0
$827$	−9.78890	−0.340393	−0.170197	−	0.985410i	$-0.554440\pi$
$827$	−9.78890	−0.340393	−0.170197	+	0.985410i	$0.554440\pi$
$828$	0	0
$829$	−42.6333	−1.48072	−0.740358	−	0.672213i	$-0.765344\pi$
$829$	−42.6333	−1.48072	−0.740358	+	0.672213i	$0.765344\pi$
$830$	0	0
$831$	1.33053	0.0461556
$832$	0	0
$833$	−8.21110	−0.284498
$834$	0	0
$835$	3.09167	0.106992
$836$	0	0
$837$	−19.5139	−0.674498
$838$	0	0
$839$	19.3944	0.669571	0.334785	−	0.942294i	$-0.391336\pi$
$839$	19.3944	0.669571	0.334785	+	0.942294i	$0.391336\pi$
$840$	0	0
$841$	15.8528	0.546649
$842$	0	0
$843$	1.51388	0.0521407
$844$	0	0
$845$	3.63331	0.124990
$846$	0	0
$847$	10.0000	0.343604
$848$	0	0
$849$	−9.57779	−0.328709
$850$	0	0
$851$	−2.00000	−0.0685591
$852$	0	0
$853$	−53.6333	−1.83637	−0.918185	−	0.396152i	$-0.870345\pi$
$853$	−53.6333	−1.83637	−0.918185	+	0.396152i	$0.870345\pi$
$854$	0	0
$855$	−6.08327	−0.208043
$856$	0	0
$857$	−27.4861	−0.938908	−0.469454	−	0.882957i	$-0.655549\pi$
$857$	−27.4861	−0.938908	−0.469454	+	0.882957i	$0.655549\pi$
$858$	0	0
$859$	−11.5139	−0.392848	−0.196424	−	0.980519i	$-0.562933\pi$
$859$	−11.5139	−0.392848	−0.196424	+	0.980519i	$0.562933\pi$
$860$	0	0
$861$	−0.302776	−0.0103186
$862$	0	0
$863$	−41.8444	−1.42440	−0.712200	−	0.701976i	$-0.752301\pi$
$863$	−41.8444	−1.42440	−0.712200	+	0.701976i	$0.752301\pi$
$864$	0	0
$865$	−3.27502	−0.111354
$866$	0	0
$867$	−15.2666	−0.518481
$868$	0	0
$869$	−9.39445	−0.318685
$870$	0	0
$871$	−44.5416	−1.50924
$872$	0	0
$873$	46.8806	1.58667
$874$	0	0
$875$	3.00000	0.101419
$876$	0	0
$877$	−27.9638	−0.944272	−0.472136	−	0.881526i	$-0.656517\pi$
$877$	−27.9638	−0.944272	−0.472136	+	0.881526i	$0.656517\pi$
$878$	0	0
$879$	−9.03616	−0.304782
$880$	0	0
$881$	−8.78890	−0.296105	−0.148053	−	0.988979i	$-0.547301\pi$
$881$	−8.78890	−0.296105	−0.148053	+	0.988979i	$0.547301\pi$
$882$	0	0
$883$	43.5694	1.46623	0.733113	−	0.680106i	$-0.238066\pi$
$883$	43.5694	1.46623	0.733113	+	0.680106i	$0.238066\pi$
$884$	0	0
$885$	−0.449961	−0.0151253
$886$	0	0
$887$	−40.8444	−1.37142	−0.685711	−	0.727874i	$-0.740508\pi$
$887$	−40.8444	−1.37142	−0.685711	+	0.727874i	$0.740508\pi$
$888$	0	0
$889$	7.00000	0.234772
$890$	0	0
$891$	−8.18335	−0.274152
$892$	0	0
$893$	87.0833	2.91413
$894$	0	0
$895$	−4.88057	−0.163139
$896$	0	0
$897$	−0.458365	−0.0153044
$898$	0	0
$899$	−73.0555	−2.43654
$900$	0	0
$901$	51.7527	1.72413
$902$	0	0
$903$	1.33053	0.0442773
$904$	0	0
$905$	−3.02776	−0.100646
$906$	0	0
$907$	−4.88057	−0.162057	−0.0810283	−	0.996712i	$-0.525820\pi$
$907$	−4.88057	−0.162057	−0.0810283	+	0.996712i	$0.525820\pi$
$908$	0	0
$909$	4.66947	0.154876
$910$	0	0
$911$	−28.8444	−0.955658	−0.477829	−	0.878453i	$-0.658576\pi$
$911$	−28.8444	−0.955658	−0.477829	+	0.878453i	$0.658576\pi$
$912$	0	0
$913$	10.8167	0.357979
$914$	0	0
$915$	−0.386045	−0.0127622
$916$	0	0
$917$	11.0917	0.366279
$918$	0	0
$919$	16.5139	0.544743	0.272371	−	0.962192i	$-0.412192\pi$
$919$	16.5139	0.544743	0.272371	+	0.962192i	$0.412192\pi$
$920$	0	0
$921$	−3.39445	−0.111851
$922$	0	0
$923$	11.0555	0.363897
$924$	0	0
$925$	−32.4222	−1.06604
$926$	0	0
$927$	−32.8722	−1.07966
$928$	0	0
$929$	21.5416	0.706758	0.353379	−	0.935480i	$-0.385033\pi$
$929$	21.5416	0.706758	0.353379	+	0.935480i	$0.385033\pi$
$930$	0	0
$931$	6.90833	0.226411
$932$	0	0
$933$	1.78890	0.0585659
$934$	0	0
$935$	2.48612	0.0813049
$936$	0	0
$937$	−1.97224	−0.0644304	−0.0322152	−	0.999481i	$-0.510256\pi$
$937$	−1.97224	−0.0644304	−0.0322152	+	0.999481i	$0.510256\pi$
$938$	0	0
$939$	−4.72498	−0.154194
$940$	0	0
$941$	−1.69722	−0.0553279	−0.0276640	−	0.999617i	$-0.508807\pi$
$941$	−1.69722	−0.0553279	−0.0276640	+	0.999617i	$0.508807\pi$
$942$	0	0
$943$	0.302776	0.00985973
$944$	0	0
$945$	−0.541635	−0.0176194
$946$	0	0
$947$	−44.0917	−1.43279	−0.716393	−	0.697697i	$-0.754208\pi$
$947$	−44.0917	−1.43279	−0.716393	+	0.697697i	$0.754208\pi$
$948$	0	0
$949$	−35.0000	−1.13615
$950$	0	0
$951$	−4.90833	−0.159163
$952$	0	0
$953$	−10.1472	−0.328700	−0.164350	−	0.986402i	$-0.552553\pi$
$953$	−10.1472	−0.328700	−0.164350	+	0.986402i	$0.552553\pi$
$954$	0	0
$955$	−2.78890	−0.0902466
$956$	0	0
$957$	2.02776	0.0655481
$958$	0	0
$959$	−12.9083	−0.416832
$960$	0	0
$961$	87.9916	2.83844
$962$	0	0
$963$	−16.9167	−0.545134
$964$	0	0
$965$	4.66947	0.150315
$966$	0	0
$967$	9.57779	0.308001	0.154001	−	0.988071i	$-0.450784\pi$
$967$	9.57779	0.308001	0.154001	+	0.988071i	$0.450784\pi$
$968$	0	0
$969$	17.1749	0.551739
$970$	0	0
$971$	15.9722	0.512574	0.256287	−	0.966601i	$-0.417501\pi$
$971$	15.9722	0.512574	0.256287	+	0.966601i	$0.417501\pi$
$972$	0	0
$973$	−8.39445	−0.269114
$974$	0	0
$975$	−7.43061	−0.237970
$976$	0	0
$977$	−23.3028	−0.745522	−0.372761	−	0.927927i	$-0.621589\pi$
$977$	−23.3028	−0.745522	−0.372761	+	0.927927i	$0.621589\pi$
$978$	0	0
$979$	−9.51388	−0.304065
$980$	0	0
$981$	9.33894	0.298169
$982$	0	0
$983$	−32.0278	−1.02153	−0.510763	−	0.859721i	$-0.670637\pi$
$983$	−32.0278	−1.02153	−0.510763	+	0.859721i	$0.670637\pi$
$984$	0	0
$985$	2.69722	0.0859407
$986$	0	0
$987$	3.81665	0.121485
$988$	0	0
$989$	−1.33053	−0.0423085
$990$	0	0
$991$	−15.1833	−0.482315	−0.241157	−	0.970486i	$-0.577527\pi$
$991$	−15.1833	−0.482315	−0.241157	+	0.970486i	$0.577527\pi$
$992$	0	0
$993$	−8.36669	−0.265509
$994$	0	0
$995$	−4.30278	−0.136407
$996$	0	0
$997$	−37.7250	−1.19476	−0.597381	−	0.801958i	$-0.703792\pi$
$997$	−37.7250	−1.19476	−0.597381	+	0.801958i	$0.703792\pi$
$998$	0	0
$999$	11.8167	0.373862

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4592.2.a.o.1.1		2
4.3	odd	2		1148.2.a.a.1.2	✓	2
28.27	even	2		8036.2.a.g.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.a.a.1.2	✓	2	4.3	odd	2
4592.2.a.o.1.1		2	1.1	even	1	trivial
8036.2.a.g.1.1		2	28.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4592 = 2^{4} \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4592.a (trivial)

\(f(q)\)	\(=\)	\(q-0.302776 q^{3} +0.302776 q^{5} -1.00000 q^{7} -2.90833 q^{9} +O(q^{10})\)	\(q-0.302776 q^{3} +0.302776 q^{5} -1.00000 q^{7} -2.90833 q^{9} -1.00000 q^{11} -5.00000 q^{13} -0.0916731 q^{15} -8.21110 q^{17} +6.90833 q^{19} +0.302776 q^{21} -0.302776 q^{23} -4.90833 q^{25} +1.78890 q^{27} +6.69722 q^{29} -10.9083 q^{31} +0.302776 q^{33} -0.302776 q^{35} +6.60555 q^{37} +1.51388 q^{39} -1.00000 q^{41} +4.39445 q^{43} -0.880571 q^{45} +12.6056 q^{47} +1.00000 q^{49} +2.48612 q^{51} -6.30278 q^{53} -0.302776 q^{55} -2.09167 q^{57} +4.90833 q^{59} +4.21110 q^{61} +2.90833 q^{63} -1.51388 q^{65} +8.90833 q^{67} +0.0916731 q^{69} -2.21110 q^{71} +7.00000 q^{73} +1.48612 q^{75} +1.00000 q^{77} +9.39445 q^{79} +8.18335 q^{81} -10.8167 q^{83} -2.48612 q^{85} -2.02776 q^{87} +9.51388 q^{89} +5.00000 q^{91} +3.30278 q^{93} +2.09167 q^{95} -16.1194 q^{97} +2.90833 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{3} - 3 q^{5} - 2 q^{7} + 5 q^{9}+O(q^{10}) \) 2 * q + 3 * q^3 - 3 * q^5 - 2 * q^7 + 5 * q^9	\( 2 q + 3 q^{3} - 3 q^{5} - 2 q^{7} + 5 q^{9} - 2 q^{11} - 10 q^{13} - 11 q^{15} - 2 q^{17} + 3 q^{19} - 3 q^{21} + 3 q^{23} + q^{25} + 18 q^{27} + 17 q^{29} - 11 q^{31} - 3 q^{33} + 3 q^{35} + 6 q^{37} - 15 q^{39} - 2 q^{41} + 16 q^{43} - 27 q^{45} + 18 q^{47} + 2 q^{49} + 23 q^{51} - 9 q^{53} + 3 q^{55} - 15 q^{57} - q^{59} - 6 q^{61} - 5 q^{63} + 15 q^{65} + 7 q^{67} + 11 q^{69} + 10 q^{71} + 14 q^{73} + 21 q^{75} + 2 q^{77} + 26 q^{79} + 38 q^{81} - 23 q^{85} + 32 q^{87} + q^{89} + 10 q^{91} + 3 q^{93} + 15 q^{95} - 7 q^{97} - 5 q^{99}+O(q^{100}) \) 2 * q + 3 * q^3 - 3 * q^5 - 2 * q^7 + 5 * q^9 - 2 * q^11 - 10 * q^13 - 11 * q^15 - 2 * q^17 + 3 * q^19 - 3 * q^21 + 3 * q^23 + q^25 + 18 * q^27 + 17 * q^29 - 11 * q^31 - 3 * q^33 + 3 * q^35 + 6 * q^37 - 15 * q^39 - 2 * q^41 + 16 * q^43 - 27 * q^45 + 18 * q^47 + 2 * q^49 + 23 * q^51 - 9 * q^53 + 3 * q^55 - 15 * q^57 - q^59 - 6 * q^61 - 5 * q^63 + 15 * q^65 + 7 * q^67 + 11 * q^69 + 10 * q^71 + 14 * q^73 + 21 * q^75 + 2 * q^77 + 26 * q^79 + 38 * q^81 - 23 * q^85 + 32 * q^87 + q^89 + 10 * q^91 + 3 * q^93 + 15 * q^95 - 7 * q^97 - 5 * q^99

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