LMFDB - Embedded newform 1472.2.a.v.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1472,2,Mod(1,1472)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1472, 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("1472.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1472 = 2^{6} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1472.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:	$11.7539791775$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 736)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	1472.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.414214 q^{3} +0.585786 q^{5} +0.585786 q^{7} -2.82843 q^{9} +O(q^{10})$	$q-0.414214 q^{3} +0.585786 q^{5} +0.585786 q^{7} -2.82843 q^{9} +4.24264 q^{11} +3.82843 q^{13} -0.242641 q^{15} -2.58579 q^{17} -0.828427 q^{19} -0.242641 q^{21} +1.00000 q^{23} -4.65685 q^{25} +2.41421 q^{27} +5.00000 q^{29} +9.24264 q^{31} -1.75736 q^{33} +0.343146 q^{35} -5.07107 q^{37} -1.58579 q^{39} -1.34315 q^{41} +4.00000 q^{43} -1.65685 q^{45} +5.24264 q^{47} -6.65685 q^{49} +1.07107 q^{51} +9.31371 q^{53} +2.48528 q^{55} +0.343146 q^{57} +1.17157 q^{59} +3.65685 q^{61} -1.65685 q^{63} +2.24264 q^{65} +13.8995 q^{67} -0.414214 q^{69} -3.58579 q^{71} +7.82843 q^{73} +1.92893 q^{75} +2.48528 q^{77} +14.4853 q^{79} +7.48528 q^{81} -10.7279 q^{83} -1.51472 q^{85} -2.07107 q^{87} -8.00000 q^{89} +2.24264 q^{91} -3.82843 q^{93} -0.485281 q^{95} +9.89949 q^{97} -12.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 4 q^{5} + 4 q^{7}+O(q^{10}) $ 2 * q + 2 * q^3 + 4 * q^5 + 4 * q^7	$ 2 q + 2 q^{3} + 4 q^{5} + 4 q^{7} + 2 q^{13} + 8 q^{15} - 8 q^{17} + 4 q^{19} + 8 q^{21} + 2 q^{23} + 2 q^{25} + 2 q^{27} + 10 q^{29} + 10 q^{31} - 12 q^{33} + 12 q^{35} + 4 q^{37} - 6 q^{39} - 14 q^{41} + 8 q^{43} + 8 q^{45} + 2 q^{47} - 2 q^{49} - 12 q^{51} - 4 q^{53} - 12 q^{55} + 12 q^{57} + 8 q^{59} - 4 q^{61} + 8 q^{63} - 4 q^{65} + 8 q^{67} + 2 q^{69} - 10 q^{71} + 10 q^{73} + 18 q^{75} - 12 q^{77} + 12 q^{79} - 2 q^{81} + 4 q^{83} - 20 q^{85} + 10 q^{87} - 16 q^{89} - 4 q^{91} - 2 q^{93} + 16 q^{95} - 24 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 4 * q^5 + 4 * q^7 + 2 * q^13 + 8 * q^15 - 8 * q^17 + 4 * q^19 + 8 * q^21 + 2 * q^23 + 2 * q^25 + 2 * q^27 + 10 * q^29 + 10 * q^31 - 12 * q^33 + 12 * q^35 + 4 * q^37 - 6 * q^39 - 14 * q^41 + 8 * q^43 + 8 * q^45 + 2 * q^47 - 2 * q^49 - 12 * q^51 - 4 * q^53 - 12 * q^55 + 12 * q^57 + 8 * q^59 - 4 * q^61 + 8 * q^63 - 4 * q^65 + 8 * q^67 + 2 * q^69 - 10 * q^71 + 10 * q^73 + 18 * q^75 - 12 * q^77 + 12 * q^79 - 2 * q^81 + 4 * q^83 - 20 * q^85 + 10 * q^87 - 16 * q^89 - 4 * q^91 - 2 * q^93 + 16 * q^95 - 24 * 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
$3$	−0.414214	−0.239146	−0.119573	−	0.992825i	$-0.538153\pi$
$3$	−0.414214	−0.239146	−0.119573	+	0.992825i	$0.538153\pi$
$4$	0	0
$5$	0.585786	0.261972	0.130986	−	0.991384i	$-0.458186\pi$
$5$	0.585786	0.261972	0.130986	+	0.991384i	$0.458186\pi$
$6$	0	0
$7$	0.585786	0.221406	0.110703	−	0.993854i	$-0.464690\pi$
$7$	0.585786	0.221406	0.110703	+	0.993854i	$0.464690\pi$
$8$	0	0
$9$	−2.82843	−0.942809
$10$	0	0
$11$	4.24264	1.27920	0.639602	−	0.768706i	$-0.279099\pi$
$11$	4.24264	1.27920	0.639602	+	0.768706i	$0.279099\pi$
$12$	0	0
$13$	3.82843	1.06181	0.530907	−	0.847430i	$-0.321851\pi$
$13$	3.82843	1.06181	0.530907	+	0.847430i	$0.321851\pi$
$14$	0	0
$15$	−0.242641	−0.0626496
$16$	0	0
$17$	−2.58579	−0.627145	−0.313573	−	0.949564i	$-0.601526\pi$
$17$	−2.58579	−0.627145	−0.313573	+	0.949564i	$0.601526\pi$
$18$	0	0
$19$	−0.828427	−0.190054	−0.0950271	−	0.995475i	$-0.530294\pi$
$19$	−0.828427	−0.190054	−0.0950271	+	0.995475i	$0.530294\pi$
$20$	0	0
$21$	−0.242641	−0.0529485
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.65685	−0.931371
$26$	0	0
$27$	2.41421	0.464616
$28$	0	0
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	0	0
$31$	9.24264	1.66003	0.830014	−	0.557743i	$-0.188333\pi$
$31$	9.24264	1.66003	0.830014	+	0.557743i	$0.188333\pi$
$32$	0	0
$33$	−1.75736	−0.305917
$34$	0	0
$35$	0.343146	0.0580022
$36$	0	0
$37$	−5.07107	−0.833678	−0.416839	−	0.908980i	$-0.636862\pi$
$37$	−5.07107	−0.833678	−0.416839	+	0.908980i	$0.636862\pi$
$38$	0	0
$39$	−1.58579	−0.253929
$40$	0	0
$41$	−1.34315	−0.209764	−0.104882	−	0.994485i	$-0.533447\pi$
$41$	−1.34315	−0.209764	−0.104882	+	0.994485i	$0.533447\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	−1.65685	−0.246989
$46$	0	0
$47$	5.24264	0.764718	0.382359	−	0.924014i	$-0.375112\pi$
$47$	5.24264	0.764718	0.382359	+	0.924014i	$0.375112\pi$
$48$	0	0
$49$	−6.65685	−0.950979
$50$	0	0
$51$	1.07107	0.149979
$52$	0	0
$53$	9.31371	1.27934	0.639668	−	0.768651i	$-0.279072\pi$
$53$	9.31371	1.27934	0.639668	+	0.768651i	$0.279072\pi$
$54$	0	0
$55$	2.48528	0.335115
$56$	0	0
$57$	0.343146	0.0454508
$58$	0	0
$59$	1.17157	0.152526	0.0762629	−	0.997088i	$-0.475701\pi$
$59$	1.17157	0.152526	0.0762629	+	0.997088i	$0.475701\pi$
$60$	0	0
$61$	3.65685	0.468212	0.234106	−	0.972211i	$-0.424784\pi$
$61$	3.65685	0.468212	0.234106	+	0.972211i	$0.424784\pi$
$62$	0	0
$63$	−1.65685	−0.208744
$64$	0	0
$65$	2.24264	0.278165
$66$	0	0
$67$	13.8995	1.69809	0.849047	−	0.528318i	$-0.177177\pi$
$67$	13.8995	1.69809	0.849047	+	0.528318i	$0.177177\pi$
$68$	0	0
$69$	−0.414214	−0.0498655
$70$	0	0
$71$	−3.58579	−0.425555	−0.212777	−	0.977101i	$-0.568251\pi$
$71$	−3.58579	−0.425555	−0.212777	+	0.977101i	$0.568251\pi$
$72$	0	0
$73$	7.82843	0.916248	0.458124	−	0.888888i	$-0.348522\pi$
$73$	7.82843	0.916248	0.458124	+	0.888888i	$0.348522\pi$
$74$	0	0
$75$	1.92893	0.222734
$76$	0	0
$77$	2.48528	0.283224
$78$	0	0
$79$	14.4853	1.62972	0.814861	−	0.579657i	$-0.196813\pi$
$79$	14.4853	1.62972	0.814861	+	0.579657i	$0.196813\pi$
$80$	0	0
$81$	7.48528	0.831698
$82$	0	0
$83$	−10.7279	−1.17754	−0.588771	−	0.808300i	$-0.700388\pi$
$83$	−10.7279	−1.17754	−0.588771	+	0.808300i	$0.700388\pi$
$84$	0	0
$85$	−1.51472	−0.164294
$86$	0	0
$87$	−2.07107	−0.222042
$88$	0	0
$89$	−8.00000	−0.847998	−0.423999	−	0.905663i	$-0.639374\pi$
$89$	−8.00000	−0.847998	−0.423999	+	0.905663i	$0.639374\pi$
$90$	0	0
$91$	2.24264	0.235093
$92$	0	0
$93$	−3.82843	−0.396989
$94$	0	0
$95$	−0.485281	−0.0497888
$96$	0	0
$97$	9.89949	1.00514	0.502571	−	0.864536i	$-0.332388\pi$
$97$	9.89949	1.00514	0.502571	+	0.864536i	$0.332388\pi$
$98$	0	0
$99$	−12.0000	−1.20605
$100$	0	0
$101$	8.00000	0.796030	0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000	0.796030	0.398015	+	0.917379i	$0.369699\pi$
$102$	0	0
$103$	−3.89949	−0.384229	−0.192114	−	0.981373i	$-0.561534\pi$
$103$	−3.89949	−0.384229	−0.192114	+	0.981373i	$0.561534\pi$
$104$	0	0
$105$	−0.142136	−0.0138710
$106$	0	0
$107$	7.17157	0.693302	0.346651	−	0.937994i	$-0.387319\pi$
$107$	7.17157	0.693302	0.346651	+	0.937994i	$0.387319\pi$
$108$	0	0
$109$	−13.6569	−1.30809	−0.654045	−	0.756456i	$-0.726929\pi$
$109$	−13.6569	−1.30809	−0.654045	+	0.756456i	$0.726929\pi$
$110$	0	0
$111$	2.10051	0.199371
$112$	0	0
$113$	−8.24264	−0.775402	−0.387701	−	0.921785i	$-0.626731\pi$
$113$	−8.24264	−0.775402	−0.387701	+	0.921785i	$0.626731\pi$
$114$	0	0
$115$	0.585786	0.0546249
$116$	0	0
$117$	−10.8284	−1.00109
$118$	0	0
$119$	−1.51472	−0.138854
$120$	0	0
$121$	7.00000	0.636364
$122$	0	0
$123$	0.556349	0.0501643
$124$	0	0
$125$	−5.65685	−0.505964
$126$	0	0
$127$	−6.41421	−0.569169	−0.284585	−	0.958651i	$-0.591856\pi$
$127$	−6.41421	−0.569169	−0.284585	+	0.958651i	$0.591856\pi$
$128$	0	0
$129$	−1.65685	−0.145878
$130$	0	0
$131$	8.89949	0.777552	0.388776	−	0.921332i	$-0.372898\pi$
$131$	8.89949	0.777552	0.388776	+	0.921332i	$0.372898\pi$
$132$	0	0
$133$	−0.485281	−0.0420792
$134$	0	0
$135$	1.41421	0.121716
$136$	0	0
$137$	12.9706	1.10815	0.554075	−	0.832467i	$-0.313072\pi$
$137$	12.9706	1.10815	0.554075	+	0.832467i	$0.313072\pi$
$138$	0	0
$139$	4.75736	0.403514	0.201757	−	0.979436i	$-0.435335\pi$
$139$	4.75736	0.403514	0.201757	+	0.979436i	$0.435335\pi$
$140$	0	0
$141$	−2.17157	−0.182879
$142$	0	0
$143$	16.2426	1.35828
$144$	0	0
$145$	2.92893	0.243235
$146$	0	0
$147$	2.75736	0.227423
$148$	0	0
$149$	−1.31371	−0.107623	−0.0538116	−	0.998551i	$-0.517137\pi$
$149$	−1.31371	−0.107623	−0.0538116	+	0.998551i	$0.517137\pi$
$150$	0	0
$151$	4.41421	0.359224	0.179612	−	0.983738i	$-0.442516\pi$
$151$	4.41421	0.359224	0.179612	+	0.983738i	$0.442516\pi$
$152$	0	0
$153$	7.31371	0.591278
$154$	0	0
$155$	5.41421	0.434880
$156$	0	0
$157$	−2.34315	−0.187003	−0.0935017	−	0.995619i	$-0.529806\pi$
$157$	−2.34315	−0.187003	−0.0935017	+	0.995619i	$0.529806\pi$
$158$	0	0
$159$	−3.85786	−0.305949
$160$	0	0
$161$	0.585786	0.0461664
$162$	0	0
$163$	14.0711	1.10213	0.551066	−	0.834462i	$-0.314221\pi$
$163$	14.0711	1.10213	0.551066	+	0.834462i	$0.314221\pi$
$164$	0	0
$165$	−1.02944	−0.0801416
$166$	0	0
$167$	−10.1421	−0.784822	−0.392411	−	0.919790i	$-0.628359\pi$
$167$	−10.1421	−0.784822	−0.392411	+	0.919790i	$0.628359\pi$
$168$	0	0
$169$	1.65685	0.127450
$170$	0	0
$171$	2.34315	0.179185
$172$	0	0
$173$	8.48528	0.645124	0.322562	−	0.946548i	$-0.395456\pi$
$173$	8.48528	0.645124	0.322562	+	0.946548i	$0.395456\pi$
$174$	0	0
$175$	−2.72792	−0.206212
$176$	0	0
$177$	−0.485281	−0.0364760
$178$	0	0
$179$	−14.7574	−1.10302	−0.551508	−	0.834169i	$-0.685948\pi$
$179$	−14.7574	−1.10302	−0.551508	+	0.834169i	$0.685948\pi$
$180$	0	0
$181$	−15.8995	−1.18180	−0.590900	−	0.806745i	$-0.701227\pi$
$181$	−15.8995	−1.18180	−0.590900	+	0.806745i	$0.701227\pi$
$182$	0	0
$183$	−1.51472	−0.111971
$184$	0	0
$185$	−2.97056	−0.218400
$186$	0	0
$187$	−10.9706	−0.802247
$188$	0	0
$189$	1.41421	0.102869
$190$	0	0
$191$	20.2426	1.46471	0.732353	−	0.680925i	$-0.238422\pi$
$191$	20.2426	1.46471	0.732353	+	0.680925i	$0.238422\pi$
$192$	0	0
$193$	15.4853	1.11465	0.557327	−	0.830293i	$-0.311827\pi$
$193$	15.4853	1.11465	0.557327	+	0.830293i	$0.311827\pi$
$194$	0	0
$195$	−0.928932	−0.0665222
$196$	0	0
$197$	−5.82843	−0.415258	−0.207629	−	0.978208i	$-0.566575\pi$
$197$	−5.82843	−0.415258	−0.207629	+	0.978208i	$0.566575\pi$
$198$	0	0
$199$	−2.10051	−0.148901	−0.0744504	−	0.997225i	$-0.523720\pi$
$199$	−2.10051	−0.148901	−0.0744504	+	0.997225i	$0.523720\pi$
$200$	0	0
$201$	−5.75736	−0.406093
$202$	0	0
$203$	2.92893	0.205571
$204$	0	0
$205$	−0.786797	−0.0549523
$206$	0	0
$207$	−2.82843	−0.196589
$208$	0	0
$209$	−3.51472	−0.243118
$210$	0	0
$211$	−16.4853	−1.13489	−0.567447	−	0.823410i	$-0.692069\pi$
$211$	−16.4853	−1.13489	−0.567447	+	0.823410i	$0.692069\pi$
$212$	0	0
$213$	1.48528	0.101770
$214$	0	0
$215$	2.34315	0.159801
$216$	0	0
$217$	5.41421	0.367541
$218$	0	0
$219$	−3.24264	−0.219117
$220$	0	0
$221$	−9.89949	−0.665912
$222$	0	0
$223$	−15.6569	−1.04846	−0.524230	−	0.851577i	$-0.675647\pi$
$223$	−15.6569	−1.04846	−0.524230	+	0.851577i	$0.675647\pi$
$224$	0	0
$225$	13.1716	0.878105
$226$	0	0
$227$	−25.0711	−1.66403	−0.832013	−	0.554757i	$-0.812811\pi$
$227$	−25.0711	−1.66403	−0.832013	+	0.554757i	$0.812811\pi$
$228$	0	0
$229$	18.3431	1.21215	0.606075	−	0.795408i	$-0.292743\pi$
$229$	18.3431	1.21215	0.606075	+	0.795408i	$0.292743\pi$
$230$	0	0
$231$	−1.02944	−0.0677320
$232$	0	0
$233$	−7.82843	−0.512857	−0.256429	−	0.966563i	$-0.582546\pi$
$233$	−7.82843	−0.512857	−0.256429	+	0.966563i	$0.582546\pi$
$234$	0	0
$235$	3.07107	0.200334
$236$	0	0
$237$	−6.00000	−0.389742
$238$	0	0
$239$	−29.7279	−1.92294	−0.961470	−	0.274911i	$-0.911352\pi$
$239$	−29.7279	−1.92294	−0.961470	+	0.274911i	$0.911352\pi$
$240$	0	0
$241$	12.5858	0.810722	0.405361	−	0.914157i	$-0.367146\pi$
$241$	12.5858	0.810722	0.405361	+	0.914157i	$0.367146\pi$
$242$	0	0
$243$	−10.3431	−0.663513
$244$	0	0
$245$	−3.89949	−0.249130
$246$	0	0
$247$	−3.17157	−0.201802
$248$	0	0
$249$	4.44365	0.281605
$250$	0	0
$251$	12.5858	0.794408	0.397204	−	0.917730i	$-0.369981\pi$
$251$	12.5858	0.794408	0.397204	+	0.917730i	$0.369981\pi$
$252$	0	0
$253$	4.24264	0.266733
$254$	0	0
$255$	0.627417	0.0392904
$256$	0	0
$257$	−20.7990	−1.29741	−0.648703	−	0.761042i	$-0.724688\pi$
$257$	−20.7990	−1.29741	−0.648703	+	0.761042i	$0.724688\pi$
$258$	0	0
$259$	−2.97056	−0.184582
$260$	0	0
$261$	−14.1421	−0.875376
$262$	0	0
$263$	−10.4853	−0.646550	−0.323275	−	0.946305i	$-0.604784\pi$
$263$	−10.4853	−0.646550	−0.323275	+	0.946305i	$0.604784\pi$
$264$	0	0
$265$	5.45584	0.335150
$266$	0	0
$267$	3.31371	0.202796
$268$	0	0
$269$	−1.48528	−0.0905592	−0.0452796	−	0.998974i	$-0.514418\pi$
$269$	−1.48528	−0.0905592	−0.0452796	+	0.998974i	$0.514418\pi$
$270$	0	0
$271$	10.3431	0.628301	0.314151	−	0.949373i	$-0.398280\pi$
$271$	10.3431	0.628301	0.314151	+	0.949373i	$0.398280\pi$
$272$	0	0
$273$	−0.928932	−0.0562215
$274$	0	0
$275$	−19.7574	−1.19141
$276$	0	0
$277$	−1.82843	−0.109860	−0.0549298	−	0.998490i	$-0.517493\pi$
$277$	−1.82843	−0.109860	−0.0549298	+	0.998490i	$0.517493\pi$
$278$	0	0
$279$	−26.1421	−1.56509
$280$	0	0
$281$	−30.8701	−1.84155	−0.920777	−	0.390090i	$-0.872444\pi$
$281$	−30.8701	−1.84155	−0.920777	+	0.390090i	$0.872444\pi$
$282$	0	0
$283$	−7.27208	−0.432280	−0.216140	−	0.976362i	$-0.569347\pi$
$283$	−7.27208	−0.432280	−0.216140	+	0.976362i	$0.569347\pi$
$284$	0	0
$285$	0.201010	0.0119068
$286$	0	0
$287$	−0.786797	−0.0464431
$288$	0	0
$289$	−10.3137	−0.606689
$290$	0	0
$291$	−4.10051	−0.240376
$292$	0	0
$293$	11.3137	0.660954	0.330477	−	0.943814i	$-0.392790\pi$
$293$	11.3137	0.660954	0.330477	+	0.943814i	$0.392790\pi$
$294$	0	0
$295$	0.686292	0.0399574
$296$	0	0
$297$	10.2426	0.594338
$298$	0	0
$299$	3.82843	0.221404
$300$	0	0
$301$	2.34315	0.135057
$302$	0	0
$303$	−3.31371	−0.190368
$304$	0	0
$305$	2.14214	0.122658
$306$	0	0
$307$	−31.7990	−1.81486	−0.907432	−	0.420199i	$-0.861960\pi$
$307$	−31.7990	−1.81486	−0.907432	+	0.420199i	$0.861960\pi$
$308$	0	0
$309$	1.61522	0.0918869
$310$	0	0
$311$	−27.2426	−1.54479	−0.772394	−	0.635143i	$-0.780941\pi$
$311$	−27.2426	−1.54479	−0.772394	+	0.635143i	$0.780941\pi$
$312$	0	0
$313$	−27.3137	−1.54386	−0.771931	−	0.635706i	$-0.780709\pi$
$313$	−27.3137	−1.54386	−0.771931	+	0.635706i	$0.780709\pi$
$314$	0	0
$315$	−0.970563	−0.0546850
$316$	0	0
$317$	6.34315	0.356267	0.178133	−	0.984006i	$-0.442994\pi$
$317$	6.34315	0.356267	0.178133	+	0.984006i	$0.442994\pi$
$318$	0	0
$319$	21.2132	1.18771
$320$	0	0
$321$	−2.97056	−0.165801
$322$	0	0
$323$	2.14214	0.119192
$324$	0	0
$325$	−17.8284	−0.988943
$326$	0	0
$327$	5.65685	0.312825
$328$	0	0
$329$	3.07107	0.169313
$330$	0	0
$331$	33.7279	1.85385	0.926927	−	0.375241i	$-0.122440\pi$
$331$	33.7279	1.85385	0.926927	+	0.375241i	$0.122440\pi$
$332$	0	0
$333$	14.3431	0.786000
$334$	0	0
$335$	8.14214	0.444852
$336$	0	0
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	0	0
$339$	3.41421	0.185435
$340$	0	0
$341$	39.2132	2.12351
$342$	0	0
$343$	−8.00000	−0.431959
$344$	0	0
$345$	−0.242641	−0.0130633
$346$	0	0
$347$	−2.00000	−0.107366	−0.0536828	−	0.998558i	$-0.517096\pi$
$347$	−2.00000	−0.107366	−0.0536828	+	0.998558i	$0.517096\pi$
$348$	0	0
$349$	28.6569	1.53397	0.766983	−	0.641667i	$-0.221757\pi$
$349$	28.6569	1.53397	0.766983	+	0.641667i	$0.221757\pi$
$350$	0	0
$351$	9.24264	0.493336
$352$	0	0
$353$	−17.8284	−0.948911	−0.474456	−	0.880279i	$-0.657355\pi$
$353$	−17.8284	−0.948911	−0.474456	+	0.880279i	$0.657355\pi$
$354$	0	0
$355$	−2.10051	−0.111483
$356$	0	0
$357$	0.627417	0.0332064
$358$	0	0
$359$	−8.14214	−0.429725	−0.214863	−	0.976644i	$-0.568930\pi$
$359$	−8.14214	−0.429725	−0.214863	+	0.976644i	$0.568930\pi$
$360$	0	0
$361$	−18.3137	−0.963879
$362$	0	0
$363$	−2.89949	−0.152184
$364$	0	0
$365$	4.58579	0.240031
$366$	0	0
$367$	21.5563	1.12523	0.562616	−	0.826718i	$-0.309795\pi$
$367$	21.5563	1.12523	0.562616	+	0.826718i	$0.309795\pi$
$368$	0	0
$369$	3.79899	0.197768
$370$	0	0
$371$	5.45584	0.283253
$372$	0	0
$373$	1.07107	0.0554578	0.0277289	−	0.999615i	$-0.491172\pi$
$373$	1.07107	0.0554578	0.0277289	+	0.999615i	$0.491172\pi$
$374$	0	0
$375$	2.34315	0.121000
$376$	0	0
$377$	19.1421	0.985870
$378$	0	0
$379$	−9.65685	−0.496039	−0.248020	−	0.968755i	$-0.579780\pi$
$379$	−9.65685	−0.496039	−0.248020	+	0.968755i	$0.579780\pi$
$380$	0	0
$381$	2.65685	0.136115
$382$	0	0
$383$	−3.31371	−0.169323	−0.0846613	−	0.996410i	$-0.526981\pi$
$383$	−3.31371	−0.169323	−0.0846613	+	0.996410i	$0.526981\pi$
$384$	0	0
$385$	1.45584	0.0741967
$386$	0	0
$387$	−11.3137	−0.575108
$388$	0	0
$389$	3.31371	0.168012	0.0840058	−	0.996465i	$-0.473229\pi$
$389$	3.31371	0.168012	0.0840058	+	0.996465i	$0.473229\pi$
$390$	0	0
$391$	−2.58579	−0.130769
$392$	0	0
$393$	−3.68629	−0.185949
$394$	0	0
$395$	8.48528	0.426941
$396$	0	0
$397$	−15.9706	−0.801540	−0.400770	−	0.916179i	$-0.631257\pi$
$397$	−15.9706	−0.801540	−0.400770	+	0.916179i	$0.631257\pi$
$398$	0	0
$399$	0.201010	0.0100631
$400$	0	0
$401$	−2.92893	−0.146264	−0.0731319	−	0.997322i	$-0.523299\pi$
$401$	−2.92893	−0.146264	−0.0731319	+	0.997322i	$0.523299\pi$
$402$	0	0
$403$	35.3848	1.76264
$404$	0	0
$405$	4.38478	0.217881
$406$	0	0
$407$	−21.5147	−1.06645
$408$	0	0
$409$	3.14214	0.155369	0.0776843	−	0.996978i	$-0.475247\pi$
$409$	3.14214	0.155369	0.0776843	+	0.996978i	$0.475247\pi$
$410$	0	0
$411$	−5.37258	−0.265010
$412$	0	0
$413$	0.686292	0.0337702
$414$	0	0
$415$	−6.28427	−0.308483
$416$	0	0
$417$	−1.97056	−0.0964989
$418$	0	0
$419$	−35.3137	−1.72519	−0.862594	−	0.505897i	$-0.831162\pi$
$419$	−35.3137	−1.72519	−0.862594	+	0.505897i	$0.831162\pi$
$420$	0	0
$421$	1.89949	0.0925757	0.0462879	−	0.998928i	$-0.485261\pi$
$421$	1.89949	0.0925757	0.0462879	+	0.998928i	$0.485261\pi$
$422$	0	0
$423$	−14.8284	−0.720983
$424$	0	0
$425$	12.0416	0.584105
$426$	0	0
$427$	2.14214	0.103665
$428$	0	0
$429$	−6.72792	−0.324827
$430$	0	0
$431$	−27.3137	−1.31566	−0.657828	−	0.753169i	$-0.728524\pi$
$431$	−27.3137	−1.31566	−0.657828	+	0.753169i	$0.728524\pi$
$432$	0	0
$433$	9.07107	0.435928	0.217964	−	0.975957i	$-0.430059\pi$
$433$	9.07107	0.435928	0.217964	+	0.975957i	$0.430059\pi$
$434$	0	0
$435$	−1.21320	−0.0581687
$436$	0	0
$437$	−0.828427	−0.0396290
$438$	0	0
$439$	−31.5269	−1.50470	−0.752349	−	0.658765i	$-0.771079\pi$
$439$	−31.5269	−1.50470	−0.752349	+	0.658765i	$0.771079\pi$
$440$	0	0
$441$	18.8284	0.896592
$442$	0	0
$443$	−18.5563	−0.881639	−0.440819	−	0.897596i	$-0.645312\pi$
$443$	−18.5563	−0.881639	−0.440819	+	0.897596i	$0.645312\pi$
$444$	0	0
$445$	−4.68629	−0.222152
$446$	0	0
$447$	0.544156	0.0257377
$448$	0	0
$449$	−39.9411	−1.88494	−0.942469	−	0.334293i	$-0.891502\pi$
$449$	−39.9411	−1.88494	−0.942469	+	0.334293i	$0.891502\pi$
$450$	0	0
$451$	−5.69848	−0.268331
$452$	0	0
$453$	−1.82843	−0.0859070
$454$	0	0
$455$	1.31371	0.0615876
$456$	0	0
$457$	−6.38478	−0.298667	−0.149334	−	0.988787i	$-0.547713\pi$
$457$	−6.38478	−0.298667	−0.149334	+	0.988787i	$0.547713\pi$
$458$	0	0
$459$	−6.24264	−0.291382
$460$	0	0
$461$	39.9706	1.86161	0.930807	−	0.365510i	$-0.119105\pi$
$461$	39.9706	1.86161	0.930807	+	0.365510i	$0.119105\pi$
$462$	0	0
$463$	30.9706	1.43932	0.719662	−	0.694325i	$-0.244297\pi$
$463$	30.9706	1.43932	0.719662	+	0.694325i	$0.244297\pi$
$464$	0	0
$465$	−2.24264	−0.104000
$466$	0	0
$467$	−21.5147	−0.995582	−0.497791	−	0.867297i	$-0.665855\pi$
$467$	−21.5147	−0.995582	−0.497791	+	0.867297i	$0.665855\pi$
$468$	0	0
$469$	8.14214	0.375969
$470$	0	0
$471$	0.970563	0.0447212
$472$	0	0
$473$	16.9706	0.780307
$474$	0	0
$475$	3.85786	0.177011
$476$	0	0
$477$	−26.3431	−1.20617
$478$	0	0
$479$	4.38478	0.200346	0.100173	−	0.994970i	$-0.468060\pi$
$479$	4.38478	0.200346	0.100173	+	0.994970i	$0.468060\pi$
$480$	0	0
$481$	−19.4142	−0.885212
$482$	0	0
$483$	−0.242641	−0.0110405
$484$	0	0
$485$	5.79899	0.263319
$486$	0	0
$487$	25.7279	1.16584	0.582922	−	0.812528i	$-0.301909\pi$
$487$	25.7279	1.16584	0.582922	+	0.812528i	$0.301909\pi$
$488$	0	0
$489$	−5.82843	−0.263571
$490$	0	0
$491$	−17.8701	−0.806464	−0.403232	−	0.915098i	$-0.632113\pi$
$491$	−17.8701	−0.806464	−0.403232	+	0.915098i	$0.632113\pi$
$492$	0	0
$493$	−12.9289	−0.582290
$494$	0	0
$495$	−7.02944	−0.315950
$496$	0	0
$497$	−2.10051	−0.0942205
$498$	0	0
$499$	30.0711	1.34617	0.673083	−	0.739567i	$-0.264970\pi$
$499$	30.0711	1.34617	0.673083	+	0.739567i	$0.264970\pi$
$500$	0	0
$501$	4.20101	0.187687
$502$	0	0
$503$	31.4558	1.40255	0.701273	−	0.712892i	$-0.252615\pi$
$503$	31.4558	1.40255	0.701273	+	0.712892i	$0.252615\pi$
$504$	0	0
$505$	4.68629	0.208537
$506$	0	0
$507$	−0.686292	−0.0304793
$508$	0	0
$509$	26.4558	1.17263	0.586317	−	0.810081i	$-0.300577\pi$
$509$	26.4558	1.17263	0.586317	+	0.810081i	$0.300577\pi$
$510$	0	0
$511$	4.58579	0.202863
$512$	0	0
$513$	−2.00000	−0.0883022
$514$	0	0
$515$	−2.28427	−0.100657
$516$	0	0
$517$	22.2426	0.978230
$518$	0	0
$519$	−3.51472	−0.154279
$520$	0	0
$521$	−26.6274	−1.16657	−0.583284	−	0.812268i	$-0.698233\pi$
$521$	−26.6274	−1.16657	−0.583284	+	0.812268i	$0.698233\pi$
$522$	0	0
$523$	38.7279	1.69345	0.846727	−	0.532028i	$-0.178570\pi$
$523$	38.7279	1.69345	0.846727	+	0.532028i	$0.178570\pi$
$524$	0	0
$525$	1.12994	0.0493147
$526$	0	0
$527$	−23.8995	−1.04108
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−3.31371	−0.143803
$532$	0	0
$533$	−5.14214	−0.222731
$534$	0	0
$535$	4.20101	0.181626
$536$	0	0
$537$	6.11270	0.263782
$538$	0	0
$539$	−28.2426	−1.21650
$540$	0	0
$541$	−27.6274	−1.18780	−0.593898	−	0.804541i	$-0.702412\pi$
$541$	−27.6274	−1.18780	−0.593898	+	0.804541i	$0.702412\pi$
$542$	0	0
$543$	6.58579	0.282623
$544$	0	0
$545$	−8.00000	−0.342682
$546$	0	0
$547$	30.4142	1.30042	0.650209	−	0.759755i	$-0.274681\pi$
$547$	30.4142	1.30042	0.650209	+	0.759755i	$0.274681\pi$
$548$	0	0
$549$	−10.3431	−0.441435
$550$	0	0
$551$	−4.14214	−0.176461
$552$	0	0
$553$	8.48528	0.360831
$554$	0	0
$555$	1.23045	0.0522296
$556$	0	0
$557$	−37.6569	−1.59557	−0.797786	−	0.602941i	$-0.793996\pi$
$557$	−37.6569	−1.59557	−0.797786	+	0.602941i	$0.793996\pi$
$558$	0	0
$559$	15.3137	0.647701
$560$	0	0
$561$	4.54416	0.191854
$562$	0	0
$563$	32.9706	1.38954	0.694772	−	0.719230i	$-0.255505\pi$
$563$	32.9706	1.38954	0.694772	+	0.719230i	$0.255505\pi$
$564$	0	0
$565$	−4.82843	−0.203133
$566$	0	0
$567$	4.38478	0.184143
$568$	0	0
$569$	−22.9289	−0.961231	−0.480615	−	0.876931i	$-0.659587\pi$
$569$	−22.9289	−0.961231	−0.480615	+	0.876931i	$0.659587\pi$
$570$	0	0
$571$	−24.5858	−1.02888	−0.514442	−	0.857525i	$-0.672001\pi$
$571$	−24.5858	−1.02888	−0.514442	+	0.857525i	$0.672001\pi$
$572$	0	0
$573$	−8.38478	−0.350279
$574$	0	0
$575$	−4.65685	−0.194204
$576$	0	0
$577$	41.8284	1.74134	0.870670	−	0.491867i	$-0.163686\pi$
$577$	41.8284	1.74134	0.870670	+	0.491867i	$0.163686\pi$
$578$	0	0
$579$	−6.41421	−0.266566
$580$	0	0
$581$	−6.28427	−0.260716
$582$	0	0
$583$	39.5147	1.63653
$584$	0	0
$585$	−6.34315	−0.262257
$586$	0	0
$587$	38.0122	1.56893	0.784466	−	0.620172i	$-0.212937\pi$
$587$	38.0122	1.56893	0.784466	+	0.620172i	$0.212937\pi$
$588$	0	0
$589$	−7.65685	−0.315495
$590$	0	0
$591$	2.41421	0.0993075
$592$	0	0
$593$	20.4853	0.841230	0.420615	−	0.907239i	$-0.361814\pi$
$593$	20.4853	0.841230	0.420615	+	0.907239i	$0.361814\pi$
$594$	0	0
$595$	−0.887302	−0.0363758
$596$	0	0
$597$	0.870058	0.0356091
$598$	0	0
$599$	−11.3137	−0.462266	−0.231133	−	0.972922i	$-0.574243\pi$
$599$	−11.3137	−0.462266	−0.231133	+	0.972922i	$0.574243\pi$
$600$	0	0
$601$	24.3137	0.991777	0.495888	−	0.868386i	$-0.334842\pi$
$601$	24.3137	0.991777	0.495888	+	0.868386i	$0.334842\pi$
$602$	0	0
$603$	−39.3137	−1.60098
$604$	0	0
$605$	4.10051	0.166709
$606$	0	0
$607$	−25.4558	−1.03322	−0.516610	−	0.856221i	$-0.672806\pi$
$607$	−25.4558	−1.03322	−0.516610	+	0.856221i	$0.672806\pi$
$608$	0	0
$609$	−1.21320	−0.0491615
$610$	0	0
$611$	20.0711	0.811988
$612$	0	0
$613$	−4.44365	−0.179477	−0.0897387	−	0.995965i	$-0.528603\pi$
$613$	−4.44365	−0.179477	−0.0897387	+	0.995965i	$0.528603\pi$
$614$	0	0
$615$	0.325902	0.0131416
$616$	0	0
$617$	35.6569	1.43549	0.717745	−	0.696306i	$-0.245174\pi$
$617$	35.6569	1.43549	0.717745	+	0.696306i	$0.245174\pi$
$618$	0	0
$619$	28.9706	1.16443	0.582213	−	0.813037i	$-0.302187\pi$
$619$	28.9706	1.16443	0.582213	+	0.813037i	$0.302187\pi$
$620$	0	0
$621$	2.41421	0.0968791
$622$	0	0
$623$	−4.68629	−0.187752
$624$	0	0
$625$	19.9706	0.798823
$626$	0	0
$627$	1.45584	0.0581408
$628$	0	0
$629$	13.1127	0.522838
$630$	0	0
$631$	17.6569	0.702908	0.351454	−	0.936205i	$-0.385687\pi$
$631$	17.6569	0.702908	0.351454	+	0.936205i	$0.385687\pi$
$632$	0	0
$633$	6.82843	0.271406
$634$	0	0
$635$	−3.75736	−0.149106
$636$	0	0
$637$	−25.4853	−1.00976
$638$	0	0
$639$	10.1421	0.401217
$640$	0	0
$641$	5.65685	0.223432	0.111716	−	0.993740i	$-0.464365\pi$
$641$	5.65685	0.223432	0.111716	+	0.993740i	$0.464365\pi$
$642$	0	0
$643$	24.0416	0.948109	0.474055	−	0.880495i	$-0.342790\pi$
$643$	24.0416	0.948109	0.474055	+	0.880495i	$0.342790\pi$
$644$	0	0
$645$	−0.970563	−0.0382159
$646$	0	0
$647$	7.24264	0.284738	0.142369	−	0.989814i	$-0.454528\pi$
$647$	7.24264	0.284738	0.142369	+	0.989814i	$0.454528\pi$
$648$	0	0
$649$	4.97056	0.195112
$650$	0	0
$651$	−2.24264	−0.0878960
$652$	0	0
$653$	17.8284	0.697680	0.348840	−	0.937182i	$-0.386576\pi$
$653$	17.8284	0.697680	0.348840	+	0.937182i	$0.386576\pi$
$654$	0	0
$655$	5.21320	0.203697
$656$	0	0
$657$	−22.1421	−0.863847
$658$	0	0
$659$	−31.8995	−1.24263	−0.621314	−	0.783562i	$-0.713401\pi$
$659$	−31.8995	−1.24263	−0.621314	+	0.783562i	$0.713401\pi$
$660$	0	0
$661$	−16.6274	−0.646732	−0.323366	−	0.946274i	$-0.604814\pi$
$661$	−16.6274	−0.646732	−0.323366	+	0.946274i	$0.604814\pi$
$662$	0	0
$663$	4.10051	0.159250
$664$	0	0
$665$	−0.284271	−0.0110236
$666$	0	0
$667$	5.00000	0.193601
$668$	0	0
$669$	6.48528	0.250735
$670$	0	0
$671$	15.5147	0.598939
$672$	0	0
$673$	−21.9706	−0.846903	−0.423451	−	0.905919i	$-0.639182\pi$
$673$	−21.9706	−0.846903	−0.423451	+	0.905919i	$0.639182\pi$
$674$	0	0
$675$	−11.2426	−0.432729
$676$	0	0
$677$	−18.2843	−0.702722	−0.351361	−	0.936240i	$-0.614281\pi$
$677$	−18.2843	−0.702722	−0.351361	+	0.936240i	$0.614281\pi$
$678$	0	0
$679$	5.79899	0.222545
$680$	0	0
$681$	10.3848	0.397945
$682$	0	0
$683$	41.2426	1.57811	0.789053	−	0.614325i	$-0.210572\pi$
$683$	41.2426	1.57811	0.789053	+	0.614325i	$0.210572\pi$
$684$	0	0
$685$	7.59798	0.290304
$686$	0	0
$687$	−7.59798	−0.289881
$688$	0	0
$689$	35.6569	1.35842
$690$	0	0
$691$	−18.6274	−0.708620	−0.354310	−	0.935128i	$-0.615284\pi$
$691$	−18.6274	−0.708620	−0.354310	+	0.935128i	$0.615284\pi$
$692$	0	0
$693$	−7.02944	−0.267026
$694$	0	0
$695$	2.78680	0.105709
$696$	0	0
$697$	3.47309	0.131553
$698$	0	0
$699$	3.24264	0.122648
$700$	0	0
$701$	−4.92893	−0.186163	−0.0930816	−	0.995658i	$-0.529672\pi$
$701$	−4.92893	−0.186163	−0.0930816	+	0.995658i	$0.529672\pi$
$702$	0	0
$703$	4.20101	0.158444
$704$	0	0
$705$	−1.27208	−0.0479092
$706$	0	0
$707$	4.68629	0.176246
$708$	0	0
$709$	15.7574	0.591780	0.295890	−	0.955222i	$-0.404384\pi$
$709$	15.7574	0.591780	0.295890	+	0.955222i	$0.404384\pi$
$710$	0	0
$711$	−40.9706	−1.53652
$712$	0	0
$713$	9.24264	0.346140
$714$	0	0
$715$	9.51472	0.355830
$716$	0	0
$717$	12.3137	0.459864
$718$	0	0
$719$	27.3137	1.01863	0.509315	−	0.860580i	$-0.329899\pi$
$719$	27.3137	1.01863	0.509315	+	0.860580i	$0.329899\pi$
$720$	0	0
$721$	−2.28427	−0.0850707
$722$	0	0
$723$	−5.21320	−0.193881
$724$	0	0
$725$	−23.2843	−0.864756
$726$	0	0
$727$	51.5563	1.91212	0.956060	−	0.293172i	$-0.0947110\pi$
$727$	51.5563	1.91212	0.956060	+	0.293172i	$0.0947110\pi$
$728$	0	0
$729$	−18.1716	−0.673021
$730$	0	0
$731$	−10.3431	−0.382555
$732$	0	0
$733$	9.21320	0.340297	0.170149	−	0.985418i	$-0.445575\pi$
$733$	9.21320	0.340297	0.170149	+	0.985418i	$0.445575\pi$
$734$	0	0
$735$	1.61522	0.0595784
$736$	0	0
$737$	58.9706	2.17221
$738$	0	0
$739$	16.8995	0.621658	0.310829	−	0.950466i	$-0.399393\pi$
$739$	16.8995	0.621658	0.310829	+	0.950466i	$0.399393\pi$
$740$	0	0
$741$	1.31371	0.0482603
$742$	0	0
$743$	42.0416	1.54236	0.771179	−	0.636618i	$-0.219667\pi$
$743$	42.0416	1.54236	0.771179	+	0.636618i	$0.219667\pi$
$744$	0	0
$745$	−0.769553	−0.0281942
$746$	0	0
$747$	30.3431	1.11020
$748$	0	0
$749$	4.20101	0.153502
$750$	0	0
$751$	−26.6274	−0.971648	−0.485824	−	0.874057i	$-0.661480\pi$
$751$	−26.6274	−0.971648	−0.485824	+	0.874057i	$0.661480\pi$
$752$	0	0
$753$	−5.21320	−0.189980
$754$	0	0
$755$	2.58579	0.0941064
$756$	0	0
$757$	−38.0416	−1.38265	−0.691323	−	0.722546i	$-0.742972\pi$
$757$	−38.0416	−1.38265	−0.691323	+	0.722546i	$0.742972\pi$
$758$	0	0
$759$	−1.75736	−0.0637881
$760$	0	0
$761$	−43.6274	−1.58149	−0.790746	−	0.612144i	$-0.790307\pi$
$761$	−43.6274	−1.58149	−0.790746	+	0.612144i	$0.790307\pi$
$762$	0	0
$763$	−8.00000	−0.289619
$764$	0	0
$765$	4.28427	0.154898
$766$	0	0
$767$	4.48528	0.161954
$768$	0	0
$769$	−38.0416	−1.37182	−0.685908	−	0.727688i	$-0.740595\pi$
$769$	−38.0416	−1.37182	−0.685908	+	0.727688i	$0.740595\pi$
$770$	0	0
$771$	8.61522	0.310270
$772$	0	0
$773$	−44.9706	−1.61748	−0.808739	−	0.588167i	$-0.799850\pi$
$773$	−44.9706	−1.61748	−0.808739	+	0.588167i	$0.799850\pi$
$774$	0	0
$775$	−43.0416	−1.54610
$776$	0	0
$777$	1.23045	0.0441421
$778$	0	0
$779$	1.11270	0.0398666
$780$	0	0
$781$	−15.2132	−0.544371
$782$	0	0
$783$	12.0711	0.431385
$784$	0	0
$785$	−1.37258	−0.0489896
$786$	0	0
$787$	36.0000	1.28326	0.641631	−	0.767014i	$-0.278258\pi$
$787$	36.0000	1.28326	0.641631	+	0.767014i	$0.278258\pi$
$788$	0	0
$789$	4.34315	0.154620
$790$	0	0
$791$	−4.82843	−0.171679
$792$	0	0
$793$	14.0000	0.497155
$794$	0	0
$795$	−2.25988	−0.0801498
$796$	0	0
$797$	−38.9706	−1.38041	−0.690204	−	0.723615i	$-0.742479\pi$
$797$	−38.9706	−1.38041	−0.690204	+	0.723615i	$0.742479\pi$
$798$	0	0
$799$	−13.5563	−0.479589
$800$	0	0
$801$	22.6274	0.799500
$802$	0	0
$803$	33.2132	1.17207
$804$	0	0
$805$	0.343146	0.0120943
$806$	0	0
$807$	0.615224	0.0216569
$808$	0	0
$809$	18.1421	0.637844	0.318922	−	0.947781i	$-0.396679\pi$
$809$	18.1421	0.637844	0.318922	+	0.947781i	$0.396679\pi$
$810$	0	0
$811$	12.2721	0.430931	0.215465	−	0.976511i	$-0.430873\pi$
$811$	12.2721	0.430931	0.215465	+	0.976511i	$0.430873\pi$
$812$	0	0
$813$	−4.28427	−0.150256
$814$	0	0
$815$	8.24264	0.288727
$816$	0	0
$817$	−3.31371	−0.115932
$818$	0	0
$819$	−6.34315	−0.221647
$820$	0	0
$821$	−51.7990	−1.80780	−0.903899	−	0.427747i	$-0.859307\pi$
$821$	−51.7990	−1.80780	−0.903899	+	0.427747i	$0.859307\pi$
$822$	0	0
$823$	27.0416	0.942612	0.471306	−	0.881970i	$-0.343783\pi$
$823$	27.0416	0.942612	0.471306	+	0.881970i	$0.343783\pi$
$824$	0	0
$825$	8.18377	0.284922
$826$	0	0
$827$	−20.9706	−0.729218	−0.364609	−	0.931161i	$-0.618797\pi$
$827$	−20.9706	−0.729218	−0.364609	+	0.931161i	$0.618797\pi$
$828$	0	0
$829$	9.65685	0.335396	0.167698	−	0.985838i	$-0.446367\pi$
$829$	9.65685	0.335396	0.167698	+	0.985838i	$0.446367\pi$
$830$	0	0
$831$	0.757359	0.0262725
$832$	0	0
$833$	17.2132	0.596402
$834$	0	0
$835$	−5.94113	−0.205601
$836$	0	0
$837$	22.3137	0.771275
$838$	0	0
$839$	−49.3553	−1.70394	−0.851968	−	0.523594i	$-0.824591\pi$
$839$	−49.3553	−1.70394	−0.851968	+	0.523594i	$0.824591\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	0	0
$843$	12.7868	0.440401
$844$	0	0
$845$	0.970563	0.0333884
$846$	0	0
$847$	4.10051	0.140895
$848$	0	0
$849$	3.01219	0.103378
$850$	0	0
$851$	−5.07107	−0.173834
$852$	0	0
$853$	14.3431	0.491100	0.245550	−	0.969384i	$-0.421031\pi$
$853$	14.3431	0.491100	0.245550	+	0.969384i	$0.421031\pi$
$854$	0	0
$855$	1.37258	0.0469413
$856$	0	0
$857$	−6.45584	−0.220527	−0.110264	−	0.993902i	$-0.535170\pi$
$857$	−6.45584	−0.220527	−0.110264	+	0.993902i	$0.535170\pi$
$858$	0	0
$859$	−44.8995	−1.53195	−0.765975	−	0.642870i	$-0.777744\pi$
$859$	−44.8995	−1.53195	−0.765975	+	0.642870i	$0.777744\pi$
$860$	0	0
$861$	0.325902	0.0111067
$862$	0	0
$863$	6.41421	0.218342	0.109171	−	0.994023i	$-0.465180\pi$
$863$	6.41421	0.218342	0.109171	+	0.994023i	$0.465180\pi$
$864$	0	0
$865$	4.97056	0.169004
$866$	0	0
$867$	4.27208	0.145087
$868$	0	0
$869$	61.4558	2.08475
$870$	0	0
$871$	53.2132	1.80306
$872$	0	0
$873$	−28.0000	−0.947656
$874$	0	0
$875$	−3.31371	−0.112024
$876$	0	0
$877$	24.0000	0.810422	0.405211	−	0.914223i	$-0.367198\pi$
$877$	24.0000	0.810422	0.405211	+	0.914223i	$0.367198\pi$
$878$	0	0
$879$	−4.68629	−0.158065
$880$	0	0
$881$	−24.2426	−0.816755	−0.408378	−	0.912813i	$-0.633905\pi$
$881$	−24.2426	−0.816755	−0.408378	+	0.912813i	$0.633905\pi$
$882$	0	0
$883$	24.6863	0.830760	0.415380	−	0.909648i	$-0.363649\pi$
$883$	24.6863	0.830760	0.415380	+	0.909648i	$0.363649\pi$
$884$	0	0
$885$	−0.284271	−0.00955567
$886$	0	0
$887$	40.0122	1.34348	0.671739	−	0.740788i	$-0.265548\pi$
$887$	40.0122	1.34348	0.671739	+	0.740788i	$0.265548\pi$
$888$	0	0
$889$	−3.75736	−0.126018
$890$	0	0
$891$	31.7574	1.06391
$892$	0	0
$893$	−4.34315	−0.145338
$894$	0	0
$895$	−8.64466	−0.288959
$896$	0	0
$897$	−1.58579	−0.0529479
$898$	0	0
$899$	46.2132	1.54130
$900$	0	0
$901$	−24.0833	−0.802330
$902$	0	0
$903$	−0.970563	−0.0322983
$904$	0	0
$905$	−9.31371	−0.309598
$906$	0	0
$907$	−36.1421	−1.20008	−0.600040	−	0.799970i	$-0.704849\pi$
$907$	−36.1421	−1.20008	−0.600040	+	0.799970i	$0.704849\pi$
$908$	0	0
$909$	−22.6274	−0.750504
$910$	0	0
$911$	0.142136	0.00470916	0.00235458	−	0.999997i	$-0.499251\pi$
$911$	0.142136	0.00470916	0.00235458	+	0.999997i	$0.499251\pi$
$912$	0	0
$913$	−45.5147	−1.50632
$914$	0	0
$915$	−0.887302	−0.0293333
$916$	0	0
$917$	5.21320	0.172155
$918$	0	0
$919$	47.6985	1.57343	0.786714	−	0.617318i	$-0.211781\pi$
$919$	47.6985	1.57343	0.786714	+	0.617318i	$0.211781\pi$
$920$	0	0
$921$	13.1716	0.434018
$922$	0	0
$923$	−13.7279	−0.451860
$924$	0	0
$925$	23.6152	0.776464
$926$	0	0
$927$	11.0294	0.362254
$928$	0	0
$929$	17.4853	0.573673	0.286837	−	0.957979i	$-0.407396\pi$
$929$	17.4853	0.573673	0.286837	+	0.957979i	$0.407396\pi$
$930$	0	0
$931$	5.51472	0.180738
$932$	0	0
$933$	11.2843	0.369430
$934$	0	0
$935$	−6.42641	−0.210166
$936$	0	0
$937$	−32.8701	−1.07382	−0.536909	−	0.843640i	$-0.680408\pi$
$937$	−32.8701	−1.07382	−0.536909	+	0.843640i	$0.680408\pi$
$938$	0	0
$939$	11.3137	0.369209
$940$	0	0
$941$	12.0416	0.392546	0.196273	−	0.980549i	$-0.437116\pi$
$941$	12.0416	0.392546	0.196273	+	0.980549i	$0.437116\pi$
$942$	0	0
$943$	−1.34315	−0.0437388
$944$	0	0
$945$	0.828427	0.0269487
$946$	0	0
$947$	4.69848	0.152680	0.0763401	−	0.997082i	$-0.475677\pi$
$947$	4.69848	0.152680	0.0763401	+	0.997082i	$0.475677\pi$
$948$	0	0
$949$	29.9706	0.972886
$950$	0	0
$951$	−2.62742	−0.0851998
$952$	0	0
$953$	−14.9706	−0.484944	−0.242472	−	0.970158i	$-0.577958\pi$
$953$	−14.9706	−0.484944	−0.242472	+	0.970158i	$0.577958\pi$
$954$	0	0
$955$	11.8579	0.383711
$956$	0	0
$957$	−8.78680	−0.284037
$958$	0	0
$959$	7.59798	0.245352
$960$	0	0
$961$	54.4264	1.75569
$962$	0	0
$963$	−20.2843	−0.653652
$964$	0	0
$965$	9.07107	0.292008
$966$	0	0
$967$	4.21320	0.135487	0.0677437	−	0.997703i	$-0.478420\pi$
$967$	4.21320	0.135487	0.0677437	+	0.997703i	$0.478420\pi$
$968$	0	0
$969$	−0.887302	−0.0285042
$970$	0	0
$971$	−13.1127	−0.420807	−0.210403	−	0.977615i	$-0.567478\pi$
$971$	−13.1127	−0.420807	−0.210403	+	0.977615i	$0.567478\pi$
$972$	0	0
$973$	2.78680	0.0893406
$974$	0	0
$975$	7.38478	0.236502
$976$	0	0
$977$	22.9289	0.733562	0.366781	−	0.930307i	$-0.380460\pi$
$977$	22.9289	0.733562	0.366781	+	0.930307i	$0.380460\pi$
$978$	0	0
$979$	−33.9411	−1.08476
$980$	0	0
$981$	38.6274	1.23328
$982$	0	0
$983$	−20.1421	−0.642434	−0.321217	−	0.947006i	$-0.604092\pi$
$983$	−20.1421	−0.642434	−0.321217	+	0.947006i	$0.604092\pi$
$984$	0	0
$985$	−3.41421	−0.108786
$986$	0	0
$987$	−1.27208	−0.0404907
$988$	0	0
$989$	4.00000	0.127193
$990$	0	0
$991$	2.34315	0.0744325	0.0372162	−	0.999307i	$-0.488151\pi$
$991$	2.34315	0.0744325	0.0372162	+	0.999307i	$0.488151\pi$
$992$	0	0
$993$	−13.9706	−0.443342
$994$	0	0
$995$	−1.23045	−0.0390078
$996$	0	0
$997$	−42.4264	−1.34366	−0.671829	−	0.740706i	$-0.734491\pi$
$997$	−42.4264	−1.34366	−0.671829	+	0.740706i	$0.734491\pi$
$998$	0	0
$999$	−12.2426	−0.387340

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1472.2.a.v.1.1		2
4.3	odd	2		1472.2.a.o.1.2		2
8.3	odd	2		736.2.a.d.1.1	yes	2
8.5	even	2		736.2.a.a.1.2	✓	2
24.5	odd	2		6624.2.a.t.1.1		2
24.11	even	2		6624.2.a.s.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
736.2.a.a.1.2	✓	2	8.5	even	2
736.2.a.d.1.1	yes	2	8.3	odd	2
1472.2.a.o.1.2		2	4.3	odd	2
1472.2.a.v.1.1		2	1.1	even	1	trivial
6624.2.a.s.1.1		2	24.11	even	2
6624.2.a.t.1.1		2	24.5	odd	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:	\( N \)	\(=\)	\( 1472 = 2^{6} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1472.a (trivial)

\(f(q)\)	\(=\)	\(q-0.414214 q^{3} +0.585786 q^{5} +0.585786 q^{7} -2.82843 q^{9} +O(q^{10})\)	\(q-0.414214 q^{3} +0.585786 q^{5} +0.585786 q^{7} -2.82843 q^{9} +4.24264 q^{11} +3.82843 q^{13} -0.242641 q^{15} -2.58579 q^{17} -0.828427 q^{19} -0.242641 q^{21} +1.00000 q^{23} -4.65685 q^{25} +2.41421 q^{27} +5.00000 q^{29} +9.24264 q^{31} -1.75736 q^{33} +0.343146 q^{35} -5.07107 q^{37} -1.58579 q^{39} -1.34315 q^{41} +4.00000 q^{43} -1.65685 q^{45} +5.24264 q^{47} -6.65685 q^{49} +1.07107 q^{51} +9.31371 q^{53} +2.48528 q^{55} +0.343146 q^{57} +1.17157 q^{59} +3.65685 q^{61} -1.65685 q^{63} +2.24264 q^{65} +13.8995 q^{67} -0.414214 q^{69} -3.58579 q^{71} +7.82843 q^{73} +1.92893 q^{75} +2.48528 q^{77} +14.4853 q^{79} +7.48528 q^{81} -10.7279 q^{83} -1.51472 q^{85} -2.07107 q^{87} -8.00000 q^{89} +2.24264 q^{91} -3.82843 q^{93} -0.485281 q^{95} +9.89949 q^{97} -12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 4 q^{5} + 4 q^{7}+O(q^{10}) \) 2 * q + 2 * q^3 + 4 * q^5 + 4 * q^7	\( 2 q + 2 q^{3} + 4 q^{5} + 4 q^{7} + 2 q^{13} + 8 q^{15} - 8 q^{17} + 4 q^{19} + 8 q^{21} + 2 q^{23} + 2 q^{25} + 2 q^{27} + 10 q^{29} + 10 q^{31} - 12 q^{33} + 12 q^{35} + 4 q^{37} - 6 q^{39} - 14 q^{41} + 8 q^{43} + 8 q^{45} + 2 q^{47} - 2 q^{49} - 12 q^{51} - 4 q^{53} - 12 q^{55} + 12 q^{57} + 8 q^{59} - 4 q^{61} + 8 q^{63} - 4 q^{65} + 8 q^{67} + 2 q^{69} - 10 q^{71} + 10 q^{73} + 18 q^{75} - 12 q^{77} + 12 q^{79} - 2 q^{81} + 4 q^{83} - 20 q^{85} + 10 q^{87} - 16 q^{89} - 4 q^{91} - 2 q^{93} + 16 q^{95} - 24 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 4 * q^5 + 4 * q^7 + 2 * q^13 + 8 * q^15 - 8 * q^17 + 4 * q^19 + 8 * q^21 + 2 * q^23 + 2 * q^25 + 2 * q^27 + 10 * q^29 + 10 * q^31 - 12 * q^33 + 12 * q^35 + 4 * q^37 - 6 * q^39 - 14 * q^41 + 8 * q^43 + 8 * q^45 + 2 * q^47 - 2 * q^49 - 12 * q^51 - 4 * q^53 - 12 * q^55 + 12 * q^57 + 8 * q^59 - 4 * q^61 + 8 * q^63 - 4 * q^65 + 8 * q^67 + 2 * q^69 - 10 * q^71 + 10 * q^73 + 18 * q^75 - 12 * q^77 + 12 * q^79 - 2 * q^81 + 4 * q^83 - 20 * q^85 + 10 * q^87 - 16 * q^89 - 4 * q^91 - 2 * q^93 + 16 * q^95 - 24 * q^99

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