LMFDB - Embedded newform 920.2.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [920,2,Mod(1,920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(920, 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("920.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 920 = 2^{3} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	920.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:	$7.34623698596$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.621.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.523976$ of defining polynomial
Character	$\chi$	$=$	920.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.523976 q^{3} -1.00000 q^{5} +0.476024 q^{7} -2.72545 q^{9} +O(q^{10})$	$q-0.523976 q^{3} -1.00000 q^{5} +0.476024 q^{7} -2.72545 q^{9} -1.67750 q^{11} +2.67750 q^{13} +0.523976 q^{15} +1.67750 q^{17} +7.92692 q^{19} -0.249425 q^{21} +1.00000 q^{23} +1.00000 q^{25} +3.00000 q^{27} -2.20147 q^{29} +6.77340 q^{31} +0.878968 q^{33} -0.476024 q^{35} +4.00000 q^{37} -1.40294 q^{39} +5.97487 q^{41} -0.402945 q^{43} +2.72545 q^{45} +1.79853 q^{47} -6.77340 q^{49} -0.878968 q^{51} -10.8059 q^{53} +1.67750 q^{55} -4.15352 q^{57} +9.45090 q^{59} +6.32250 q^{61} -1.29738 q^{63} -2.67750 q^{65} +14.7100 q^{67} -0.523976 q^{69} +9.97487 q^{71} -6.10557 q^{73} -0.523976 q^{75} -0.798528 q^{77} +6.60442 q^{81} +4.49885 q^{83} -1.67750 q^{85} +1.15352 q^{87} -3.59706 q^{89} +1.27455 q^{91} -3.54910 q^{93} -7.92692 q^{95} -6.08044 q^{97} +4.57193 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^5 + 3 * q^7 + 3 * q^9	$ 3 q - 3 q^{5} + 3 q^{7} + 3 q^{9} + 3 q^{11} - 3 q^{17} + 3 q^{19} + 12 q^{21} + 3 q^{23} + 3 q^{25} + 9 q^{27} + 3 q^{29} + 6 q^{31} - 15 q^{33} - 3 q^{35} + 12 q^{37} + 15 q^{39} - 6 q^{41} + 18 q^{43} - 3 q^{45} + 15 q^{47} - 6 q^{49} + 15 q^{51} + 6 q^{53} - 3 q^{55} - 6 q^{57} + 6 q^{59} + 27 q^{61} + 12 q^{63} + 12 q^{67} + 6 q^{71} - 15 q^{73} - 12 q^{77} - 9 q^{81} - 12 q^{83} + 3 q^{85} - 3 q^{87} - 30 q^{89} + 15 q^{91} - 33 q^{93} - 3 q^{95} + 9 q^{97} + 9 q^{99}+O(q^{100}) $ 3 * q - 3 * q^5 + 3 * q^7 + 3 * q^9 + 3 * q^11 - 3 * q^17 + 3 * q^19 + 12 * q^21 + 3 * q^23 + 3 * q^25 + 9 * q^27 + 3 * q^29 + 6 * q^31 - 15 * q^33 - 3 * q^35 + 12 * q^37 + 15 * q^39 - 6 * q^41 + 18 * q^43 - 3 * q^45 + 15 * q^47 - 6 * q^49 + 15 * q^51 + 6 * q^53 - 3 * q^55 - 6 * q^57 + 6 * q^59 + 27 * q^61 + 12 * q^63 + 12 * q^67 + 6 * q^71 - 15 * q^73 - 12 * q^77 - 9 * q^81 - 12 * q^83 + 3 * q^85 - 3 * q^87 - 30 * q^89 + 15 * q^91 - 33 * q^93 - 3 * q^95 + 9 * q^97 + 9 * 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
$3$	−0.523976	−0.302518	−0.151259	−	0.988494i	$-0.548333\pi$
$3$	−0.523976	−0.302518	−0.151259	+	0.988494i	$0.548333\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0.476024	0.179920	0.0899600	−	0.995945i	$-0.471326\pi$
$7$	0.476024	0.179920	0.0899600	+	0.995945i	$0.471326\pi$
$8$	0	0
$9$	−2.72545	−0.908483
$10$	0	0
$11$	−1.67750	−0.505784	−0.252892	−	0.967495i	$-0.581382\pi$
$11$	−1.67750	−0.505784	−0.252892	+	0.967495i	$0.581382\pi$
$12$	0	0
$13$	2.67750	0.742604	0.371302	−	0.928512i	$-0.378911\pi$
$13$	2.67750	0.742604	0.371302	+	0.928512i	$0.378911\pi$
$14$	0	0
$15$	0.523976	0.135290
$16$	0	0
$17$	1.67750	0.406853	0.203426	−	0.979090i	$-0.434792\pi$
$17$	1.67750	0.406853	0.203426	+	0.979090i	$0.434792\pi$
$18$	0	0
$19$	7.92692	1.81856	0.909280	−	0.416184i	$-0.136633\pi$
$19$	7.92692	1.81856	0.909280	+	0.416184i	$0.136633\pi$
$20$	0	0
$21$	−0.249425	−0.0544290
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	3.00000	0.577350
$28$	0	0
$29$	−2.20147	−0.408803	−0.204402	−	0.978887i	$-0.565525\pi$
$29$	−2.20147	−0.408803	−0.204402	+	0.978887i	$0.565525\pi$
$30$	0	0
$31$	6.77340	1.21654	0.608269	−	0.793731i	$-0.291864\pi$
$31$	6.77340	1.21654	0.608269	+	0.793731i	$0.291864\pi$
$32$	0	0
$33$	0.878968	0.153009
$34$	0	0
$35$	−0.476024	−0.0804627
$36$	0	0
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	0	0
$39$	−1.40294	−0.224651
$40$	0	0
$41$	5.97487	0.933119	0.466559	−	0.884490i	$-0.345493\pi$
$41$	5.97487	0.933119	0.466559	+	0.884490i	$0.345493\pi$
$42$	0	0
$43$	−0.402945	−0.0614485	−0.0307242	−	0.999528i	$-0.509781\pi$
$43$	−0.402945	−0.0614485	−0.0307242	+	0.999528i	$0.509781\pi$
$44$	0	0
$45$	2.72545	0.406286
$46$	0	0
$47$	1.79853	0.262342	0.131171	−	0.991360i	$-0.458126\pi$
$47$	1.79853	0.262342	0.131171	+	0.991360i	$0.458126\pi$
$48$	0	0
$49$	−6.77340	−0.967629
$50$	0	0
$51$	−0.878968	−0.123080
$52$	0	0
$53$	−10.8059	−1.48430	−0.742152	−	0.670232i	$-0.766195\pi$
$53$	−10.8059	−1.48430	−0.742152	+	0.670232i	$0.766195\pi$
$54$	0	0
$55$	1.67750	0.226194
$56$	0	0
$57$	−4.15352	−0.550147
$58$	0	0
$59$	9.45090	1.23040	0.615201	−	0.788370i	$-0.289075\pi$
$59$	9.45090	1.23040	0.615201	+	0.788370i	$0.289075\pi$
$60$	0	0
$61$	6.32250	0.809514	0.404757	−	0.914424i	$-0.367356\pi$
$61$	6.32250	0.809514	0.404757	+	0.914424i	$0.367356\pi$
$62$	0	0
$63$	−1.29738	−0.163454
$64$	0	0
$65$	−2.67750	−0.332102
$66$	0	0
$67$	14.7100	1.79711	0.898555	−	0.438860i	$-0.144618\pi$
$67$	14.7100	1.79711	0.898555	+	0.438860i	$0.144618\pi$
$68$	0	0
$69$	−0.523976	−0.0630793
$70$	0	0
$71$	9.97487	1.18380	0.591900	−	0.806012i	$-0.298378\pi$
$71$	9.97487	1.18380	0.591900	+	0.806012i	$0.298378\pi$
$72$	0	0
$73$	−6.10557	−0.714603	−0.357301	−	0.933989i	$-0.616303\pi$
$73$	−6.10557	−0.714603	−0.357301	+	0.933989i	$0.616303\pi$
$74$	0	0
$75$	−0.523976	−0.0605036
$76$	0	0
$77$	−0.798528	−0.0910007
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	6.60442	0.733824
$82$	0	0
$83$	4.49885	0.493813	0.246906	−	0.969039i	$-0.420586\pi$
$83$	4.49885	0.493813	0.246906	+	0.969039i	$0.420586\pi$
$84$	0	0
$85$	−1.67750	−0.181950
$86$	0	0
$87$	1.15352	0.123670
$88$	0	0
$89$	−3.59706	−0.381287	−0.190644	−	0.981659i	$-0.561057\pi$
$89$	−3.59706	−0.381287	−0.190644	+	0.981659i	$0.561057\pi$
$90$	0	0
$91$	1.27455	0.133609
$92$	0	0
$93$	−3.54910	−0.368025
$94$	0	0
$95$	−7.92692	−0.813285
$96$	0	0
$97$	−6.08044	−0.617375	−0.308688	−	0.951163i	$-0.599890\pi$
$97$	−6.08044	−0.617375	−0.308688	+	0.951163i	$0.599890\pi$
$98$	0	0
$99$	4.57193	0.459496
$100$	0	0
$101$	11.4509	1.13941	0.569703	−	0.821850i	$-0.307058\pi$
$101$	11.4509	1.13941	0.569703	+	0.821850i	$0.307058\pi$
$102$	0	0
$103$	−2.32250	−0.228843	−0.114422	−	0.993432i	$-0.536501\pi$
$103$	−2.32250	−0.228843	−0.114422	+	0.993432i	$0.536501\pi$
$104$	0	0
$105$	0.249425	0.0243414
$106$	0	0
$107$	−7.85384	−0.759260	−0.379630	−	0.925138i	$-0.623949\pi$
$107$	−7.85384	−0.759260	−0.379630	+	0.925138i	$0.623949\pi$
$108$	0	0
$109$	13.9269	1.33396	0.666979	−	0.745077i	$-0.267587\pi$
$109$	13.9269	1.33396	0.666979	+	0.745077i	$0.267587\pi$
$110$	0	0
$111$	−2.09591	−0.198935
$112$	0	0
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	−7.29738	−0.674643
$118$	0	0
$119$	0.798528	0.0732009
$120$	0	0
$121$	−8.18601	−0.744182
$122$	0	0
$123$	−3.13069	−0.282285
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−11.1033	−0.985256	−0.492628	−	0.870240i	$-0.663964\pi$
$127$	−11.1033	−0.985256	−0.492628	+	0.870240i	$0.663964\pi$
$128$	0	0
$129$	0.211133	0.0185893
$130$	0	0
$131$	7.65237	0.668591	0.334295	−	0.942468i	$-0.391502\pi$
$131$	7.65237	0.668591	0.334295	+	0.942468i	$0.391502\pi$
$132$	0	0
$133$	3.77340	0.327195
$134$	0	0
$135$	−3.00000	−0.258199
$136$	0	0
$137$	−21.3778	−1.82643	−0.913215	−	0.407478i	$-0.866408\pi$
$137$	−21.3778	−1.82643	−0.913215	+	0.407478i	$0.866408\pi$
$138$	0	0
$139$	−8.60442	−0.729817	−0.364909	−	0.931043i	$-0.618900\pi$
$139$	−8.60442	−0.729817	−0.364909	+	0.931043i	$0.618900\pi$
$140$	0	0
$141$	−0.942386	−0.0793632
$142$	0	0
$143$	−4.49149	−0.375597
$144$	0	0
$145$	2.20147	0.182822
$146$	0	0
$147$	3.54910	0.292725
$148$	0	0
$149$	4.97487	0.407558	0.203779	−	0.979017i	$-0.434678\pi$
$149$	4.97487	0.407558	0.203779	+	0.979017i	$0.434678\pi$
$150$	0	0
$151$	−0.926921	−0.0754318	−0.0377159	−	0.999289i	$-0.512008\pi$
$151$	−0.926921	−0.0754318	−0.0377159	+	0.999289i	$0.512008\pi$
$152$	0	0
$153$	−4.57193	−0.369619
$154$	0	0
$155$	−6.77340	−0.544053
$156$	0	0
$157$	22.3527	1.78394	0.891970	−	0.452096i	$-0.149323\pi$
$157$	22.3527	1.78394	0.891970	+	0.452096i	$0.149323\pi$
$158$	0	0
$159$	5.66203	0.449028
$160$	0	0
$161$	0.476024	0.0375159
$162$	0	0
$163$	−7.93658	−0.621641	−0.310821	−	0.950469i	$-0.600604\pi$
$163$	−7.93658	−0.621641	−0.310821	+	0.950469i	$0.600604\pi$
$164$	0	0
$165$	−0.878968	−0.0684276
$166$	0	0
$167$	12.8059	0.990949	0.495475	−	0.868622i	$-0.334994\pi$
$167$	12.8059	0.990949	0.495475	+	0.868622i	$0.334994\pi$
$168$	0	0
$169$	−5.83102	−0.448540
$170$	0	0
$171$	−21.6044	−1.65213
$172$	0	0
$173$	−24.6752	−1.87602	−0.938010	−	0.346608i	$-0.887334\pi$
$173$	−24.6752	−1.87602	−0.938010	+	0.346608i	$0.887334\pi$
$174$	0	0
$175$	0.476024	0.0359840
$176$	0	0
$177$	−4.95205	−0.372219
$178$	0	0
$179$	−13.1033	−0.979384	−0.489692	−	0.871895i	$-0.662891\pi$
$179$	−13.1033	−0.979384	−0.489692	+	0.871895i	$0.662891\pi$
$180$	0	0
$181$	9.52398	0.707912	0.353956	−	0.935262i	$-0.384836\pi$
$181$	9.52398	0.707912	0.353956	+	0.935262i	$0.384836\pi$
$182$	0	0
$183$	−3.31284	−0.244892
$184$	0	0
$185$	−4.00000	−0.294086
$186$	0	0
$187$	−2.81399	−0.205780
$188$	0	0
$189$	1.42807	0.103877
$190$	0	0
$191$	5.85384	0.423569	0.211785	−	0.977316i	$-0.432072\pi$
$191$	5.85384	0.423569	0.211785	+	0.977316i	$0.432072\pi$
$192$	0	0
$193$	4.41261	0.317626	0.158813	−	0.987309i	$-0.449233\pi$
$193$	4.41261	0.317626	0.158813	+	0.987309i	$0.449233\pi$
$194$	0	0
$195$	1.40294	0.100467
$196$	0	0
$197$	11.4258	0.814053	0.407026	−	0.913416i	$-0.366566\pi$
$197$	11.4258	0.814053	0.407026	+	0.913416i	$0.366566\pi$
$198$	0	0
$199$	−0.307039	−0.0217654	−0.0108827	−	0.999941i	$-0.503464\pi$
$199$	−0.307039	−0.0217654	−0.0108827	+	0.999941i	$0.503464\pi$
$200$	0	0
$201$	−7.70768	−0.543658
$202$	0	0
$203$	−1.04795	−0.0735519
$204$	0	0
$205$	−5.97487	−0.417303
$206$	0	0
$207$	−2.72545	−0.189432
$208$	0	0
$209$	−13.2974	−0.919799
$210$	0	0
$211$	17.3047	1.19131	0.595654	−	0.803241i	$-0.296893\pi$
$211$	17.3047	1.19131	0.595654	+	0.803241i	$0.296893\pi$
$212$	0	0
$213$	−5.22660	−0.358121
$214$	0	0
$215$	0.402945	0.0274806
$216$	0	0
$217$	3.22430	0.218880
$218$	0	0
$219$	3.19917	0.216180
$220$	0	0
$221$	4.49149	0.302130
$222$	0	0
$223$	−1.90409	−0.127508	−0.0637538	−	0.997966i	$-0.520307\pi$
$223$	−1.90409	−0.127508	−0.0637538	+	0.997966i	$0.520307\pi$
$224$	0	0
$225$	−2.72545	−0.181697
$226$	0	0
$227$	17.3550	1.15189	0.575946	−	0.817488i	$-0.304634\pi$
$227$	17.3550	1.15189	0.575946	+	0.817488i	$0.304634\pi$
$228$	0	0
$229$	−6.19181	−0.409166	−0.204583	−	0.978849i	$-0.565584\pi$
$229$	−6.19181	−0.409166	−0.204583	+	0.978849i	$0.565584\pi$
$230$	0	0
$231$	0.418410	0.0275293
$232$	0	0
$233$	−15.7985	−1.03500	−0.517498	−	0.855684i	$-0.673137\pi$
$233$	−15.7985	−1.03500	−0.517498	+	0.855684i	$0.673137\pi$
$234$	0	0
$235$	−1.79853	−0.117323
$236$	0	0
$237$	0	0
$238$	0	0
$239$	6.34763	0.410594	0.205297	−	0.978700i	$-0.434184\pi$
$239$	6.34763	0.410594	0.205297	+	0.978700i	$0.434184\pi$
$240$	0	0
$241$	5.69296	0.366716	0.183358	−	0.983046i	$-0.441303\pi$
$241$	5.69296	0.366716	0.183358	+	0.983046i	$0.441303\pi$
$242$	0	0
$243$	−12.4606	−0.799345
$244$	0	0
$245$	6.77340	0.432737
$246$	0	0
$247$	21.2243	1.35047
$248$	0	0
$249$	−2.35729	−0.149387
$250$	0	0
$251$	−28.9246	−1.82571	−0.912853	−	0.408288i	$-0.866126\pi$
$251$	−28.9246	−1.82571	−0.912853	+	0.408288i	$0.866126\pi$
$252$	0	0
$253$	−1.67750	−0.105463
$254$	0	0
$255$	0.878968	0.0550431
$256$	0	0
$257$	−14.5085	−0.905016	−0.452508	−	0.891760i	$-0.649471\pi$
$257$	−14.5085	−0.905016	−0.452508	+	0.891760i	$0.649471\pi$
$258$	0	0
$259$	1.90409	0.118315
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	−15.7734	−0.972630	−0.486315	−	0.873784i	$-0.661659\pi$
$263$	−15.7734	−0.972630	−0.486315	+	0.873784i	$0.661659\pi$
$264$	0	0
$265$	10.8059	0.663801
$266$	0	0
$267$	1.88477	0.115346
$268$	0	0
$269$	−6.20147	−0.378110	−0.189055	−	0.981966i	$-0.560543\pi$
$269$	−6.20147	−0.378110	−0.189055	+	0.981966i	$0.560543\pi$
$270$	0	0
$271$	−25.5696	−1.55324	−0.776622	−	0.629967i	$-0.783069\pi$
$271$	−25.5696	−1.55324	−0.776622	+	0.629967i	$0.783069\pi$
$272$	0	0
$273$	−0.667835	−0.0404192
$274$	0	0
$275$	−1.67750	−0.101157
$276$	0	0
$277$	7.03829	0.422890	0.211445	−	0.977390i	$-0.432183\pi$
$277$	7.03829	0.422890	0.211445	+	0.977390i	$0.432183\pi$
$278$	0	0
$279$	−18.4606	−1.10520
$280$	0	0
$281$	29.0627	1.73373	0.866867	−	0.498540i	$-0.166130\pi$
$281$	29.0627	1.73373	0.866867	+	0.498540i	$0.166130\pi$
$282$	0	0
$283$	−18.6141	−1.10649	−0.553246	−	0.833018i	$-0.686611\pi$
$283$	−18.6141	−1.10649	−0.553246	+	0.833018i	$0.686611\pi$
$284$	0	0
$285$	4.15352	0.246033
$286$	0	0
$287$	2.84418	0.167887
$288$	0	0
$289$	−14.1860	−0.834471
$290$	0	0
$291$	3.18601	0.186767
$292$	0	0
$293$	23.4006	1.36708	0.683540	−	0.729913i	$-0.260439\pi$
$293$	23.4006	1.36708	0.683540	+	0.729913i	$0.260439\pi$
$294$	0	0
$295$	−9.45090	−0.550253
$296$	0	0
$297$	−5.03249	−0.292015
$298$	0	0
$299$	2.67750	0.154844
$300$	0	0
$301$	−0.191811	−0.0110558
$302$	0	0
$303$	−6.00000	−0.344691
$304$	0	0
$305$	−6.32250	−0.362026
$306$	0	0
$307$	−1.53134	−0.0873981	−0.0436990	−	0.999045i	$-0.513914\pi$
$307$	−1.53134	−0.0873981	−0.0436990	+	0.999045i	$0.513914\pi$
$308$	0	0
$309$	1.21694	0.0692291
$310$	0	0
$311$	−7.55646	−0.428488	−0.214244	−	0.976780i	$-0.568729\pi$
$311$	−7.55646	−0.428488	−0.214244	+	0.976780i	$0.568729\pi$
$312$	0	0
$313$	−2.32987	−0.131692	−0.0658459	−	0.997830i	$-0.520975\pi$
$313$	−2.32987	−0.131692	−0.0658459	+	0.997830i	$0.520975\pi$
$314$	0	0
$315$	1.29738	0.0730990
$316$	0	0
$317$	−15.3778	−0.863704	−0.431852	−	0.901944i	$-0.642140\pi$
$317$	−15.3778	−0.863704	−0.431852	+	0.901944i	$0.642140\pi$
$318$	0	0
$319$	3.69296	0.206766
$320$	0	0
$321$	4.11523	0.229690
$322$	0	0
$323$	13.2974	0.739886
$324$	0	0
$325$	2.67750	0.148521
$326$	0	0
$327$	−7.29738	−0.403546
$328$	0	0
$329$	0.856142	0.0472006
$330$	0	0
$331$	22.9115	1.25933	0.629664	−	0.776868i	$-0.283193\pi$
$331$	22.9115	1.25933	0.629664	+	0.776868i	$0.283193\pi$
$332$	0	0
$333$	−10.9018	−0.597415
$334$	0	0
$335$	−14.7100	−0.803692
$336$	0	0
$337$	26.5410	1.44578	0.722890	−	0.690963i	$-0.242813\pi$
$337$	26.5410	1.44578	0.722890	+	0.690963i	$0.242813\pi$
$338$	0	0
$339$	5.23976	0.284585
$340$	0	0
$341$	−11.3624	−0.615306
$342$	0	0
$343$	−6.55646	−0.354016
$344$	0	0
$345$	0.523976	0.0282099
$346$	0	0
$347$	−3.27455	−0.175787	−0.0878936	−	0.996130i	$-0.528014\pi$
$347$	−3.27455	−0.175787	−0.0878936	+	0.996130i	$0.528014\pi$
$348$	0	0
$349$	32.0553	1.71588	0.857941	−	0.513749i	$-0.171744\pi$
$349$	32.0553	1.71588	0.857941	+	0.513749i	$0.171744\pi$
$350$	0	0
$351$	8.03249	0.428742
$352$	0	0
$353$	11.1535	0.593642	0.296821	−	0.954933i	$-0.404074\pi$
$353$	11.1535	0.593642	0.296821	+	0.954933i	$0.404074\pi$
$354$	0	0
$355$	−9.97487	−0.529411
$356$	0	0
$357$	−0.418410	−0.0221446
$358$	0	0
$359$	22.4029	1.18238	0.591191	−	0.806532i	$-0.298658\pi$
$359$	22.4029	1.18238	0.591191	+	0.806532i	$0.298658\pi$
$360$	0	0
$361$	43.8361	2.30716
$362$	0	0
$363$	4.28927	0.225129
$364$	0	0
$365$	6.10557	0.319580
$366$	0	0
$367$	14.2877	0.745813	0.372906	−	0.927869i	$-0.378361\pi$
$367$	14.2877	0.745813	0.372906	+	0.927869i	$0.378361\pi$
$368$	0	0
$369$	−16.2842	−0.847722
$370$	0	0
$371$	−5.14386	−0.267056
$372$	0	0
$373$	−1.23976	−0.0641925	−0.0320963	−	0.999485i	$-0.510218\pi$
$373$	−1.23976	−0.0641925	−0.0320963	+	0.999485i	$0.510218\pi$
$374$	0	0
$375$	0.523976	0.0270580
$376$	0	0
$377$	−5.89443	−0.303579
$378$	0	0
$379$	4.38748	0.225370	0.112685	−	0.993631i	$-0.464055\pi$
$379$	4.38748	0.225370	0.112685	+	0.993631i	$0.464055\pi$
$380$	0	0
$381$	5.81785	0.298057
$382$	0	0
$383$	18.4989	0.945247	0.472624	−	0.881264i	$-0.343307\pi$
$383$	18.4989	0.945247	0.472624	+	0.881264i	$0.343307\pi$
$384$	0	0
$385$	0.798528	0.0406967
$386$	0	0
$387$	1.09821	0.0558249
$388$	0	0
$389$	21.6775	1.09909	0.549546	−	0.835463i	$-0.314801\pi$
$389$	21.6775	1.09909	0.549546	+	0.835463i	$0.314801\pi$
$390$	0	0
$391$	1.67750	0.0848346
$392$	0	0
$393$	−4.00966	−0.202261
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−20.6849	−1.03814	−0.519072	−	0.854731i	$-0.673722\pi$
$397$	−20.6849	−1.03814	−0.519072	+	0.854731i	$0.673722\pi$
$398$	0	0
$399$	−1.97717	−0.0989825
$400$	0	0
$401$	−28.5638	−1.42641	−0.713205	−	0.700956i	$-0.752757\pi$
$401$	−28.5638	−1.42641	−0.713205	+	0.700956i	$0.752757\pi$
$402$	0	0
$403$	18.1358	0.903406
$404$	0	0
$405$	−6.60442	−0.328176
$406$	0	0
$407$	−6.70998	−0.332602
$408$	0	0
$409$	16.7734	0.829391	0.414696	−	0.909960i	$-0.363888\pi$
$409$	16.7734	0.829391	0.414696	+	0.909960i	$0.363888\pi$
$410$	0	0
$411$	11.2015	0.552528
$412$	0	0
$413$	4.49885	0.221374
$414$	0	0
$415$	−4.49885	−0.220840
$416$	0	0
$417$	4.50851	0.220783
$418$	0	0
$419$	−4.59476	−0.224469	−0.112234	−	0.993682i	$-0.535801\pi$
$419$	−4.59476	−0.224469	−0.112234	+	0.993682i	$0.535801\pi$
$420$	0	0
$421$	31.8310	1.55135	0.775674	−	0.631133i	$-0.217410\pi$
$421$	31.8310	1.55135	0.775674	+	0.631133i	$0.217410\pi$
$422$	0	0
$423$	−4.90179	−0.238333
$424$	0	0
$425$	1.67750	0.0813705
$426$	0	0
$427$	3.00966	0.145648
$428$	0	0
$429$	2.35343	0.113625
$430$	0	0
$431$	−16.0650	−0.773823	−0.386911	−	0.922117i	$-0.626458\pi$
$431$	−16.0650	−0.773823	−0.386911	+	0.922117i	$0.626458\pi$
$432$	0	0
$433$	−7.68486	−0.369311	−0.184655	−	0.982803i	$-0.559117\pi$
$433$	−7.68486	−0.369311	−0.184655	+	0.982803i	$0.559117\pi$
$434$	0	0
$435$	−1.15352	−0.0553070
$436$	0	0
$437$	7.92692	0.379196
$438$	0	0
$439$	15.9822	0.762790	0.381395	−	0.924412i	$-0.375444\pi$
$439$	15.9822	0.762790	0.381395	+	0.924412i	$0.375444\pi$
$440$	0	0
$441$	18.4606	0.879074
$442$	0	0
$443$	−3.57929	−0.170057	−0.0850286	−	0.996379i	$-0.527098\pi$
$443$	−3.57929	−0.170057	−0.0850286	+	0.996379i	$0.527098\pi$
$444$	0	0
$445$	3.59706	0.170517
$446$	0	0
$447$	−2.60672	−0.123293
$448$	0	0
$449$	−28.5217	−1.34602	−0.673011	−	0.739633i	$-0.734999\pi$
$449$	−28.5217	−1.34602	−0.673011	+	0.739633i	$0.734999\pi$
$450$	0	0
$451$	−10.0228	−0.471956
$452$	0	0
$453$	0.485685	0.0228195
$454$	0	0
$455$	−1.27455	−0.0597519
$456$	0	0
$457$	−4.49885	−0.210447	−0.105224	−	0.994449i	$-0.533556\pi$
$457$	−4.49885	−0.210447	−0.105224	+	0.994449i	$0.533556\pi$
$458$	0	0
$459$	5.03249	0.234896
$460$	0	0
$461$	2.29738	0.107000	0.0534998	−	0.998568i	$-0.482962\pi$
$461$	2.29738	0.107000	0.0534998	+	0.998568i	$0.482962\pi$
$462$	0	0
$463$	−33.4966	−1.55672	−0.778358	−	0.627820i	$-0.783947\pi$
$463$	−33.4966	−1.55672	−0.778358	+	0.627820i	$0.783947\pi$
$464$	0	0
$465$	3.54910	0.164586
$466$	0	0
$467$	−5.23976	−0.242467	−0.121234	−	0.992624i	$-0.538685\pi$
$467$	−5.23976	−0.242467	−0.121234	+	0.992624i	$0.538685\pi$
$468$	0	0
$469$	7.00230	0.323336
$470$	0	0
$471$	−11.7123	−0.539674
$472$	0	0
$473$	0.675938	0.0310797
$474$	0	0
$475$	7.92692	0.363712
$476$	0	0
$477$	29.4509	1.34846
$478$	0	0
$479$	−2.40294	−0.109793	−0.0548967	−	0.998492i	$-0.517483\pi$
$479$	−2.40294	−0.109793	−0.0548967	+	0.998492i	$0.517483\pi$
$480$	0	0
$481$	10.7100	0.488333
$482$	0	0
$483$	−0.249425	−0.0113492
$484$	0	0
$485$	6.08044	0.276099
$486$	0	0
$487$	25.8944	1.17339	0.586694	−	0.809808i	$-0.300429\pi$
$487$	25.8944	1.17339	0.586694	+	0.809808i	$0.300429\pi$
$488$	0	0
$489$	4.15858	0.188058
$490$	0	0
$491$	−0.489189	−0.0220768	−0.0110384	−	0.999939i	$-0.503514\pi$
$491$	−0.489189	−0.0220768	−0.0110384	+	0.999939i	$0.503514\pi$
$492$	0	0
$493$	−3.69296	−0.166323
$494$	0	0
$495$	−4.57193	−0.205493
$496$	0	0
$497$	4.74828	0.212989
$498$	0	0
$499$	−38.4583	−1.72163	−0.860814	−	0.508920i	$-0.830045\pi$
$499$	−38.4583	−1.72163	−0.860814	+	0.508920i	$0.830045\pi$
$500$	0	0
$501$	−6.70998	−0.299780
$502$	0	0
$503$	37.8960	1.68970	0.844849	−	0.535004i	$-0.179690\pi$
$503$	37.8960	1.68970	0.844849	+	0.535004i	$0.179690\pi$
$504$	0	0
$505$	−11.4509	−0.509558
$506$	0	0
$507$	3.05531	0.135691
$508$	0	0
$509$	8.70032	0.385635	0.192818	−	0.981235i	$-0.438237\pi$
$509$	8.70032	0.385635	0.192818	+	0.981235i	$0.438237\pi$
$510$	0	0
$511$	−2.90639	−0.128571
$512$	0	0
$513$	23.7808	1.04995
$514$	0	0
$515$	2.32250	0.102342
$516$	0	0
$517$	−3.01702	−0.132689
$518$	0	0
$519$	12.9292	0.567530
$520$	0	0
$521$	25.1129	1.10022	0.550109	−	0.835093i	$-0.314586\pi$
$521$	25.1129	1.10022	0.550109	+	0.835093i	$0.314586\pi$
$522$	0	0
$523$	12.7100	0.555769	0.277884	−	0.960615i	$-0.410367\pi$
$523$	12.7100	0.555769	0.277884	+	0.960615i	$0.410367\pi$
$524$	0	0
$525$	−0.249425	−0.0108858
$526$	0	0
$527$	11.3624	0.494952
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−25.7579	−1.11780
$532$	0	0
$533$	15.9977	0.692937
$534$	0	0
$535$	7.85384	0.339551
$536$	0	0
$537$	6.86580	0.296281
$538$	0	0
$539$	11.3624	0.489411
$540$	0	0
$541$	−0.251725	−0.0108225	−0.00541124	−	0.999985i	$-0.501722\pi$
$541$	−0.251725	−0.0108225	−0.00541124	+	0.999985i	$0.501722\pi$
$542$	0	0
$543$	−4.99034	−0.214156
$544$	0	0
$545$	−13.9269	−0.596564
$546$	0	0
$547$	24.3395	1.04068	0.520342	−	0.853958i	$-0.325805\pi$
$547$	24.3395	1.04068	0.520342	+	0.853958i	$0.325805\pi$
$548$	0	0
$549$	−17.2317	−0.735429
$550$	0	0
$551$	−17.4509	−0.743433
$552$	0	0
$553$	0	0
$554$	0	0
$555$	2.09591	0.0889662
$556$	0	0
$557$	−23.4966	−0.995581	−0.497790	−	0.867297i	$-0.665855\pi$
$557$	−23.4966	−0.995581	−0.497790	+	0.867297i	$0.665855\pi$
$558$	0	0
$559$	−1.07888	−0.0456319
$560$	0	0
$561$	1.47447	0.0622520
$562$	0	0
$563$	−19.5159	−0.822496	−0.411248	−	0.911523i	$-0.634907\pi$
$563$	−19.5159	−0.822496	−0.411248	+	0.911523i	$0.634907\pi$
$564$	0	0
$565$	10.0000	0.420703
$566$	0	0
$567$	3.14386	0.132030
$568$	0	0
$569$	−8.51817	−0.357100	−0.178550	−	0.983931i	$-0.557141\pi$
$569$	−8.51817	−0.357100	−0.178550	+	0.983931i	$0.557141\pi$
$570$	0	0
$571$	−27.4811	−1.15005	−0.575024	−	0.818137i	$-0.695007\pi$
$571$	−27.4811	−1.15005	−0.575024	+	0.818137i	$0.695007\pi$
$572$	0	0
$573$	−3.06728	−0.128137
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	−26.1512	−1.08869	−0.544345	−	0.838862i	$-0.683222\pi$
$577$	−26.1512	−1.08869	−0.544345	+	0.838862i	$0.683222\pi$
$578$	0	0
$579$	−2.31210	−0.0960877
$580$	0	0
$581$	2.14156	0.0888468
$582$	0	0
$583$	18.1268	0.750737
$584$	0	0
$585$	7.29738	0.301709
$586$	0	0
$587$	−34.3276	−1.41685	−0.708425	−	0.705786i	$-0.750594\pi$
$587$	−34.3276	−1.41685	−0.708425	+	0.705786i	$0.750594\pi$
$588$	0	0
$589$	53.6922	2.21235
$590$	0	0
$591$	−5.98683	−0.246265
$592$	0	0
$593$	15.9497	0.654978	0.327489	−	0.944855i	$-0.393798\pi$
$593$	15.9497	0.654978	0.327489	+	0.944855i	$0.393798\pi$
$594$	0	0
$595$	−0.798528	−0.0327364
$596$	0	0
$597$	0.160881	0.00658443
$598$	0	0
$599$	40.5940	1.65863	0.829313	−	0.558784i	$-0.188732\pi$
$599$	40.5940	1.65863	0.829313	+	0.558784i	$0.188732\pi$
$600$	0	0
$601$	−29.0302	−1.18417	−0.592083	−	0.805877i	$-0.701694\pi$
$601$	−29.0302	−1.18417	−0.592083	+	0.805877i	$0.701694\pi$
$602$	0	0
$603$	−40.0913	−1.63264
$604$	0	0
$605$	8.18601	0.332809
$606$	0	0
$607$	16.9977	0.689915	0.344958	−	0.938618i	$-0.387893\pi$
$607$	16.9977	0.689915	0.344958	+	0.938618i	$0.387893\pi$
$608$	0	0
$609$	0.549103	0.0222508
$610$	0	0
$611$	4.81555	0.194816
$612$	0	0
$613$	30.3024	1.22390	0.611952	−	0.790895i	$-0.290385\pi$
$613$	30.3024	1.22390	0.611952	+	0.790895i	$0.290385\pi$
$614$	0	0
$615$	3.13069	0.126242
$616$	0	0
$617$	−6.13069	−0.246812	−0.123406	−	0.992356i	$-0.539382\pi$
$617$	−6.13069	−0.246812	−0.123406	+	0.992356i	$0.539382\pi$
$618$	0	0
$619$	−33.7305	−1.35574	−0.677872	−	0.735180i	$-0.737098\pi$
$619$	−33.7305	−1.35574	−0.677872	+	0.735180i	$0.737098\pi$
$620$	0	0
$621$	3.00000	0.120386
$622$	0	0
$623$	−1.71228	−0.0686012
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	6.96751	0.278256
$628$	0	0
$629$	6.70998	0.267545
$630$	0	0
$631$	4.85614	0.193320	0.0966600	−	0.995317i	$-0.469184\pi$
$631$	4.85614	0.193320	0.0966600	+	0.995317i	$0.469184\pi$
$632$	0	0
$633$	−9.06728	−0.360392
$634$	0	0
$635$	11.1033	0.440620
$636$	0	0
$637$	−18.1358	−0.718565
$638$	0	0
$639$	−27.1860	−1.07546
$640$	0	0
$641$	7.78887	0.307642	0.153821	−	0.988099i	$-0.450842\pi$
$641$	7.78887	0.307642	0.153821	+	0.988099i	$0.450842\pi$
$642$	0	0
$643$	43.6427	1.72110	0.860550	−	0.509366i	$-0.170120\pi$
$643$	43.6427	1.72110	0.860550	+	0.509366i	$0.170120\pi$
$644$	0	0
$645$	−0.211133	−0.00831337
$646$	0	0
$647$	−15.4606	−0.607817	−0.303909	−	0.952701i	$-0.598292\pi$
$647$	−15.4606	−0.607817	−0.303909	+	0.952701i	$0.598292\pi$
$648$	0	0
$649$	−15.8538	−0.622318
$650$	0	0
$651$	−1.68946	−0.0662150
$652$	0	0
$653$	−9.70613	−0.379830	−0.189915	−	0.981801i	$-0.560821\pi$
$653$	−9.70613	−0.379830	−0.189915	+	0.981801i	$0.560821\pi$
$654$	0	0
$655$	−7.65237	−0.299003
$656$	0	0
$657$	16.6404	0.649204
$658$	0	0
$659$	50.7100	1.97538	0.987690	−	0.156422i	$-0.0499961\pi$
$659$	50.7100	1.97538	0.987690	+	0.156422i	$0.0499961\pi$
$660$	0	0
$661$	33.0132	1.28406	0.642032	−	0.766678i	$-0.278092\pi$
$661$	33.0132	1.28406	0.642032	+	0.766678i	$0.278092\pi$
$662$	0	0
$663$	−2.35343	−0.0913998
$664$	0	0
$665$	−3.77340	−0.146326
$666$	0	0
$667$	−2.20147	−0.0852413
$668$	0	0
$669$	0.997701	0.0385733
$670$	0	0
$671$	−10.6060	−0.409439
$672$	0	0
$673$	19.2494	0.742011	0.371005	−	0.928631i	$-0.379013\pi$
$673$	19.2494	0.742011	0.371005	+	0.928631i	$0.379013\pi$
$674$	0	0
$675$	3.00000	0.115470
$676$	0	0
$677$	−25.1941	−0.968288	−0.484144	−	0.874988i	$-0.660869\pi$
$677$	−25.1941	−0.968288	−0.484144	+	0.874988i	$0.660869\pi$
$678$	0	0
$679$	−2.89443	−0.111078
$680$	0	0
$681$	−9.09361	−0.348468
$682$	0	0
$683$	9.48339	0.362872	0.181436	−	0.983403i	$-0.441926\pi$
$683$	9.48339	0.362872	0.181436	+	0.983403i	$0.441926\pi$
$684$	0	0
$685$	21.3778	0.816804
$686$	0	0
$687$	3.24436	0.123780
$688$	0	0
$689$	−28.9327	−1.10225
$690$	0	0
$691$	−23.8538	−0.907443	−0.453721	−	0.891144i	$-0.649904\pi$
$691$	−23.8538	−0.907443	−0.453721	+	0.891144i	$0.649904\pi$
$692$	0	0
$693$	2.17635	0.0826726
$694$	0	0
$695$	8.60442	0.326384
$696$	0	0
$697$	10.0228	0.379642
$698$	0	0
$699$	8.27806	0.313105
$700$	0	0
$701$	19.8767	0.750731	0.375366	−	0.926877i	$-0.377517\pi$
$701$	19.8767	0.750731	0.375366	+	0.926877i	$0.377517\pi$
$702$	0	0
$703$	31.7077	1.19588
$704$	0	0
$705$	0.942386	0.0354923
$706$	0	0
$707$	5.45090	0.205002
$708$	0	0
$709$	−6.72545	−0.252580	−0.126290	−	0.991993i	$-0.540307\pi$
$709$	−6.72545	−0.252580	−0.126290	+	0.991993i	$0.540307\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	6.77340	0.253666
$714$	0	0
$715$	4.49149	0.167972
$716$	0	0
$717$	−3.32601	−0.124212
$718$	0	0
$719$	−38.2294	−1.42571	−0.712857	−	0.701309i	$-0.752599\pi$
$719$	−38.2294	−1.42571	−0.712857	+	0.701309i	$0.752599\pi$
$720$	0	0
$721$	−1.10557	−0.0411735
$722$	0	0
$723$	−2.98298	−0.110938
$724$	0	0
$725$	−2.20147	−0.0817606
$726$	0	0
$727$	−3.72315	−0.138084	−0.0690420	−	0.997614i	$-0.521994\pi$
$727$	−3.72315	−0.138084	−0.0690420	+	0.997614i	$0.521994\pi$
$728$	0	0
$729$	−13.2842	−0.492008
$730$	0	0
$731$	−0.675938	−0.0250005
$732$	0	0
$733$	−8.49885	−0.313912	−0.156956	−	0.987606i	$-0.550168\pi$
$733$	−8.49885	−0.313912	−0.156956	+	0.987606i	$0.550168\pi$
$734$	0	0
$735$	−3.54910	−0.130911
$736$	0	0
$737$	−24.6759	−0.908950
$738$	0	0
$739$	−0.251725	−0.00925984	−0.00462992	−	0.999989i	$-0.501474\pi$
$739$	−0.251725	−0.00925984	−0.00462992	+	0.999989i	$0.501474\pi$
$740$	0	0
$741$	−11.1210	−0.408541
$742$	0	0
$743$	−29.7305	−1.09071	−0.545353	−	0.838206i	$-0.683605\pi$
$743$	−29.7305	−1.09071	−0.545353	+	0.838206i	$0.683605\pi$
$744$	0	0
$745$	−4.97487	−0.182265
$746$	0	0
$747$	−12.2614	−0.448621
$748$	0	0
$749$	−3.73861	−0.136606
$750$	0	0
$751$	10.6597	0.388979	0.194490	−	0.980905i	$-0.437695\pi$
$751$	10.6597	0.388979	0.194490	+	0.980905i	$0.437695\pi$
$752$	0	0
$753$	15.1558	0.552309
$754$	0	0
$755$	0.926921	0.0337341
$756$	0	0
$757$	−16.1918	−0.588501	−0.294251	−	0.955728i	$-0.595070\pi$
$757$	−16.1918	−0.588501	−0.294251	+	0.955728i	$0.595070\pi$
$758$	0	0
$759$	0.878968	0.0319045
$760$	0	0
$761$	−46.9343	−1.70137	−0.850683	−	0.525679i	$-0.823811\pi$
$761$	−46.9343	−1.70137	−0.850683	+	0.525679i	$0.823811\pi$
$762$	0	0
$763$	6.62954	0.240006
$764$	0	0
$765$	4.57193	0.165298
$766$	0	0
$767$	25.3047	0.913701
$768$	0	0
$769$	−35.9954	−1.29803	−0.649014	−	0.760777i	$-0.724818\pi$
$769$	−35.9954	−1.29803	−0.649014	+	0.760777i	$0.724818\pi$
$770$	0	0
$771$	7.60212	0.273784
$772$	0	0
$773$	−21.8229	−0.784916	−0.392458	−	0.919770i	$-0.628375\pi$
$773$	−21.8229	−0.784916	−0.392458	+	0.919770i	$0.628375\pi$
$774$	0	0
$775$	6.77340	0.243308
$776$	0	0
$777$	−0.997701	−0.0357923
$778$	0	0
$779$	47.3624	1.69693
$780$	0	0
$781$	−16.7328	−0.598747
$782$	0	0
$783$	−6.60442	−0.236023
$784$	0	0
$785$	−22.3527	−0.797802
$786$	0	0
$787$	−7.16318	−0.255340	−0.127670	−	0.991817i	$-0.540750\pi$
$787$	−7.16318	−0.255340	−0.127670	+	0.991817i	$0.540750\pi$
$788$	0	0
$789$	8.26489	0.294238
$790$	0	0
$791$	−4.76024	−0.169255
$792$	0	0
$793$	16.9285	0.601148
$794$	0	0
$795$	−5.66203	−0.200812
$796$	0	0
$797$	−48.0604	−1.70239	−0.851193	−	0.524853i	$-0.824120\pi$
$797$	−48.0604	−1.70239	−0.851193	+	0.524853i	$0.824120\pi$
$798$	0	0
$799$	3.01702	0.106735
$800$	0	0
$801$	9.80359	0.346393
$802$	0	0
$803$	10.2421	0.361435
$804$	0	0
$805$	−0.476024	−0.0167776
$806$	0	0
$807$	3.24943	0.114385
$808$	0	0
$809$	0.0611183	0.00214880	0.00107440	−	0.999999i	$-0.499658\pi$
$809$	0.0611183	0.00214880	0.00107440	+	0.999999i	$0.499658\pi$
$810$	0	0
$811$	25.9092	0.909794	0.454897	−	0.890544i	$-0.349676\pi$
$811$	25.9092	0.909794	0.454897	+	0.890544i	$0.349676\pi$
$812$	0	0
$813$	13.3979	0.469884
$814$	0	0
$815$	7.93658	0.278006
$816$	0	0
$817$	−3.19411	−0.111748
$818$	0	0
$819$	−3.47372	−0.121382
$820$	0	0
$821$	−21.0936	−0.736172	−0.368086	−	0.929792i	$-0.619987\pi$
$821$	−21.0936	−0.736172	−0.368086	+	0.929792i	$0.619987\pi$
$822$	0	0
$823$	−37.2641	−1.29895	−0.649473	−	0.760384i	$-0.725011\pi$
$823$	−37.2641	−1.29895	−0.649473	+	0.760384i	$0.725011\pi$
$824$	0	0
$825$	0.878968	0.0306017
$826$	0	0
$827$	−4.45320	−0.154853	−0.0774264	−	0.996998i	$-0.524670\pi$
$827$	−4.45320	−0.154853	−0.0774264	+	0.996998i	$0.524670\pi$
$828$	0	0
$829$	−38.1300	−1.32431	−0.662154	−	0.749368i	$-0.730358\pi$
$829$	−38.1300	−1.32431	−0.662154	+	0.749368i	$0.730358\pi$
$830$	0	0
$831$	−3.68790	−0.127932
$832$	0	0
$833$	−11.3624	−0.393682
$834$	0	0
$835$	−12.8059	−0.443166
$836$	0	0
$837$	20.3202	0.702369
$838$	0	0
$839$	−7.15858	−0.247142	−0.123571	−	0.992336i	$-0.539435\pi$
$839$	−7.15858	−0.247142	−0.123571	+	0.992336i	$0.539435\pi$
$840$	0	0
$841$	−24.1535	−0.832880
$842$	0	0
$843$	−15.2282	−0.524486
$844$	0	0
$845$	5.83102	0.200593
$846$	0	0
$847$	−3.89673	−0.133893
$848$	0	0
$849$	9.75334	0.334734
$850$	0	0
$851$	4.00000	0.137118
$852$	0	0
$853$	−49.6199	−1.69895	−0.849476	−	0.527627i	$-0.823082\pi$
$853$	−49.6199	−1.69895	−0.849476	+	0.527627i	$0.823082\pi$
$854$	0	0
$855$	21.6044	0.738855
$856$	0	0
$857$	−44.0360	−1.50424	−0.752120	−	0.659026i	$-0.770969\pi$
$857$	−44.0360	−1.50424	−0.752120	+	0.659026i	$0.770969\pi$
$858$	0	0
$859$	−49.5062	−1.68913	−0.844565	−	0.535453i	$-0.820141\pi$
$859$	−49.5062	−1.68913	−0.844565	+	0.535453i	$0.820141\pi$
$860$	0	0
$861$	−1.49028	−0.0507887
$862$	0	0
$863$	20.7962	0.707912	0.353956	−	0.935262i	$-0.384836\pi$
$863$	20.7962	0.707912	0.353956	+	0.935262i	$0.384836\pi$
$864$	0	0
$865$	24.6752	0.838982
$866$	0	0
$867$	7.43313	0.252442
$868$	0	0
$869$	0	0
$870$	0	0
$871$	39.3859	1.33454
$872$	0	0
$873$	16.5719	0.560875
$874$	0	0
$875$	−0.476024	−0.0160925
$876$	0	0
$877$	−5.42071	−0.183044	−0.0915222	−	0.995803i	$-0.529173\pi$
$877$	−5.42071	−0.183044	−0.0915222	+	0.995803i	$0.529173\pi$
$878$	0	0
$879$	−12.2614	−0.413566
$880$	0	0
$881$	−45.8036	−1.54316	−0.771581	−	0.636131i	$-0.780534\pi$
$881$	−45.8036	−1.54316	−0.771581	+	0.636131i	$0.780534\pi$
$882$	0	0
$883$	57.1860	1.92446	0.962231	−	0.272234i	$-0.0877624\pi$
$883$	57.1860	1.92446	0.962231	+	0.272234i	$0.0877624\pi$
$884$	0	0
$885$	4.95205	0.166461
$886$	0	0
$887$	−15.5371	−0.521686	−0.260843	−	0.965381i	$-0.584001\pi$
$887$	−15.5371	−0.521686	−0.260843	+	0.965381i	$0.584001\pi$
$888$	0	0
$889$	−5.28542	−0.177267
$890$	0	0
$891$	−11.0789	−0.371157
$892$	0	0
$893$	14.2568	0.477085
$894$	0	0
$895$	13.1033	0.437994
$896$	0	0
$897$	−1.40294	−0.0468430
$898$	0	0
$899$	−14.9115	−0.497325
$900$	0	0
$901$	−18.1268	−0.603892
$902$	0	0
$903$	0.100505	0.00334458
$904$	0	0
$905$	−9.52398	−0.316588
$906$	0	0
$907$	−32.8515	−1.09082	−0.545409	−	0.838170i	$-0.683626\pi$
$907$	−32.8515	−1.09082	−0.545409	+	0.838170i	$0.683626\pi$
$908$	0	0
$909$	−31.2088	−1.03513
$910$	0	0
$911$	8.95205	0.296595	0.148297	−	0.988943i	$-0.452621\pi$
$911$	8.95205	0.296595	0.148297	+	0.988943i	$0.452621\pi$
$912$	0	0
$913$	−7.54680	−0.249763
$914$	0	0
$915$	3.31284	0.109519
$916$	0	0
$917$	3.64271	0.120293
$918$	0	0
$919$	−49.8229	−1.64351	−0.821753	−	0.569844i	$-0.807004\pi$
$919$	−49.8229	−1.64351	−0.821753	+	0.569844i	$0.807004\pi$
$920$	0	0
$921$	0.802385	0.0264395
$922$	0	0
$923$	26.7077	0.879094
$924$	0	0
$925$	4.00000	0.131519
$926$	0	0
$927$	6.32987	0.207900
$928$	0	0
$929$	25.0723	0.822597	0.411298	−	0.911501i	$-0.365075\pi$
$929$	25.0723	0.822597	0.411298	+	0.911501i	$0.365075\pi$
$930$	0	0
$931$	−53.6922	−1.75969
$932$	0	0
$933$	3.95941	0.129625
$934$	0	0
$935$	2.81399	0.0920274
$936$	0	0
$937$	21.3322	0.696891	0.348446	−	0.937329i	$-0.386710\pi$
$937$	21.3322	0.696891	0.348446	+	0.937329i	$0.386710\pi$
$938$	0	0
$939$	1.22079	0.0398391
$940$	0	0
$941$	3.93428	0.128254	0.0641270	−	0.997942i	$-0.479574\pi$
$941$	3.93428	0.128254	0.0641270	+	0.997942i	$0.479574\pi$
$942$	0	0
$943$	5.97487	0.194569
$944$	0	0
$945$	−1.42807	−0.0464551
$946$	0	0
$947$	48.6420	1.58065	0.790326	−	0.612687i	$-0.209911\pi$
$947$	48.6420	1.58065	0.790326	+	0.612687i	$0.209911\pi$
$948$	0	0
$949$	−16.3476	−0.530667
$950$	0	0
$951$	8.05761	0.261286
$952$	0	0
$953$	−15.5623	−0.504111	−0.252056	−	0.967713i	$-0.581107\pi$
$953$	−15.5623	−0.504111	−0.252056	+	0.967713i	$0.581107\pi$
$954$	0	0
$955$	−5.85384	−0.189426
$956$	0	0
$957$	−1.93502	−0.0625505
$958$	0	0
$959$	−10.1763	−0.328611
$960$	0	0
$961$	14.8790	0.479967
$962$	0	0
$963$	21.4052	0.689774
$964$	0	0
$965$	−4.41261	−0.142047
$966$	0	0
$967$	6.80083	0.218700	0.109350	−	0.994003i	$-0.465123\pi$
$967$	6.80083	0.218700	0.109350	+	0.994003i	$0.465123\pi$
$968$	0	0
$969$	−6.96751	−0.223829
$970$	0	0
$971$	6.46406	0.207442	0.103721	−	0.994606i	$-0.466925\pi$
$971$	6.46406	0.207442	0.103721	+	0.994606i	$0.466925\pi$
$972$	0	0
$973$	−4.09591	−0.131309
$974$	0	0
$975$	−1.40294	−0.0449302
$976$	0	0
$977$	9.18601	0.293886	0.146943	−	0.989145i	$-0.453057\pi$
$977$	9.18601	0.293886	0.146943	+	0.989145i	$0.453057\pi$
$978$	0	0
$979$	6.03405	0.192849
$980$	0	0
$981$	−37.9571	−1.21188
$982$	0	0
$983$	15.3705	0.490241	0.245121	−	0.969493i	$-0.421172\pi$
$983$	15.3705	0.490241	0.245121	+	0.969493i	$0.421172\pi$
$984$	0	0
$985$	−11.4258	−0.364055
$986$	0	0
$987$	−0.448598	−0.0142790
$988$	0	0
$989$	−0.402945	−0.0128129
$990$	0	0
$991$	−52.0109	−1.65218	−0.826090	−	0.563538i	$-0.809440\pi$
$991$	−52.0109	−1.65218	−0.826090	+	0.563538i	$0.809440\pi$
$992$	0	0
$993$	−12.0051	−0.380969
$994$	0	0
$995$	0.307039	0.00973379
$996$	0	0
$997$	15.7123	0.497613	0.248807	−	0.968553i	$-0.419962\pi$
$997$	15.7123	0.497613	0.248807	+	0.968553i	$0.419962\pi$
$998$	0	0
$999$	12.0000	0.379663

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	920.2.a.g.1.2	✓	3
3.2	odd	2		8280.2.a.bo.1.2		3
4.3	odd	2		1840.2.a.t.1.2		3
5.2	odd	4		4600.2.e.r.4049.4		6
5.3	odd	4		4600.2.e.r.4049.3		6
5.4	even	2		4600.2.a.y.1.2		3
8.3	odd	2		7360.2.a.ca.1.2		3
8.5	even	2		7360.2.a.cb.1.2		3
20.19	odd	2		9200.2.a.cd.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.g.1.2	✓	3	1.1	even	1	trivial
1840.2.a.t.1.2		3	4.3	odd	2
4600.2.a.y.1.2		3	5.4	even	2
4600.2.e.r.4049.3		6	5.3	odd	4
4600.2.e.r.4049.4		6	5.2	odd	4
7360.2.a.ca.1.2		3	8.3	odd	2
7360.2.a.cb.1.2		3	8.5	even	2
8280.2.a.bo.1.2		3	3.2	odd	2
9200.2.a.cd.1.2		3	20.19	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:	\( N \)	\(=\)	\( 920 = 2^{3} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	920.a (trivial)

\(f(q)\)	\(=\)	\(q-0.523976 q^{3} -1.00000 q^{5} +0.476024 q^{7} -2.72545 q^{9} +O(q^{10})\)	\(q-0.523976 q^{3} -1.00000 q^{5} +0.476024 q^{7} -2.72545 q^{9} -1.67750 q^{11} +2.67750 q^{13} +0.523976 q^{15} +1.67750 q^{17} +7.92692 q^{19} -0.249425 q^{21} +1.00000 q^{23} +1.00000 q^{25} +3.00000 q^{27} -2.20147 q^{29} +6.77340 q^{31} +0.878968 q^{33} -0.476024 q^{35} +4.00000 q^{37} -1.40294 q^{39} +5.97487 q^{41} -0.402945 q^{43} +2.72545 q^{45} +1.79853 q^{47} -6.77340 q^{49} -0.878968 q^{51} -10.8059 q^{53} +1.67750 q^{55} -4.15352 q^{57} +9.45090 q^{59} +6.32250 q^{61} -1.29738 q^{63} -2.67750 q^{65} +14.7100 q^{67} -0.523976 q^{69} +9.97487 q^{71} -6.10557 q^{73} -0.523976 q^{75} -0.798528 q^{77} +6.60442 q^{81} +4.49885 q^{83} -1.67750 q^{85} +1.15352 q^{87} -3.59706 q^{89} +1.27455 q^{91} -3.54910 q^{93} -7.92692 q^{95} -6.08044 q^{97} +4.57193 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^5 + 3 * q^7 + 3 * q^9	\( 3 q - 3 q^{5} + 3 q^{7} + 3 q^{9} + 3 q^{11} - 3 q^{17} + 3 q^{19} + 12 q^{21} + 3 q^{23} + 3 q^{25} + 9 q^{27} + 3 q^{29} + 6 q^{31} - 15 q^{33} - 3 q^{35} + 12 q^{37} + 15 q^{39} - 6 q^{41} + 18 q^{43} - 3 q^{45} + 15 q^{47} - 6 q^{49} + 15 q^{51} + 6 q^{53} - 3 q^{55} - 6 q^{57} + 6 q^{59} + 27 q^{61} + 12 q^{63} + 12 q^{67} + 6 q^{71} - 15 q^{73} - 12 q^{77} - 9 q^{81} - 12 q^{83} + 3 q^{85} - 3 q^{87} - 30 q^{89} + 15 q^{91} - 33 q^{93} - 3 q^{95} + 9 q^{97} + 9 q^{99}+O(q^{100}) \) 3 * q - 3 * q^5 + 3 * q^7 + 3 * q^9 + 3 * q^11 - 3 * q^17 + 3 * q^19 + 12 * q^21 + 3 * q^23 + 3 * q^25 + 9 * q^27 + 3 * q^29 + 6 * q^31 - 15 * q^33 - 3 * q^35 + 12 * q^37 + 15 * q^39 - 6 * q^41 + 18 * q^43 - 3 * q^45 + 15 * q^47 - 6 * q^49 + 15 * q^51 + 6 * q^53 - 3 * q^55 - 6 * q^57 + 6 * q^59 + 27 * q^61 + 12 * q^63 + 12 * q^67 + 6 * q^71 - 15 * q^73 - 12 * q^77 - 9 * q^81 - 12 * q^83 + 3 * q^85 - 3 * q^87 - 30 * q^89 + 15 * q^91 - 33 * q^93 - 3 * q^95 + 9 * q^97 + 9 * q^99

Introduction

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