LMFDB - Embedded newform 464.4.a.i.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [464,4,Mod(1,464)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(464, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("464.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 464 = 2^{4} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	464.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:	$27.3768862427$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.19816.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 42x - 54 $ x^3 - x^2 - 42*x - 54
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 58)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.39712$ of defining polynomial
Character	$\chi$	$=$	464.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.39712 q^{3} -3.28077 q^{5} -33.9461 q^{7} -21.2538 q^{9} +O(q^{10})$	$q-2.39712 q^{3} -3.28077 q^{5} -33.9461 q^{7} -21.2538 q^{9} -14.5759 q^{11} -86.5287 q^{13} +7.86439 q^{15} +102.550 q^{17} +105.688 q^{19} +81.3729 q^{21} +135.456 q^{23} -114.237 q^{25} +115.670 q^{27} +29.0000 q^{29} -223.883 q^{31} +34.9403 q^{33} +111.369 q^{35} -239.723 q^{37} +207.420 q^{39} +219.331 q^{41} -18.9838 q^{43} +69.7288 q^{45} -147.922 q^{47} +809.338 q^{49} -245.824 q^{51} +613.202 q^{53} +47.8202 q^{55} -253.348 q^{57} -184.206 q^{59} -13.6914 q^{61} +721.484 q^{63} +283.880 q^{65} -328.733 q^{67} -324.703 q^{69} -5.15186 q^{71} -428.481 q^{73} +273.839 q^{75} +494.796 q^{77} +392.819 q^{79} +296.578 q^{81} +454.172 q^{83} -336.442 q^{85} -69.5165 q^{87} -811.929 q^{89} +2937.31 q^{91} +536.675 q^{93} -346.738 q^{95} -11.3513 q^{97} +309.794 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{3} + 20 q^{5} - 24 q^{7} + 5 q^{9}+O(q^{10}) $ 3 * q - 2 * q^3 + 20 * q^5 - 24 * q^7 + 5 * q^9	$ 3 q - 2 q^{3} + 20 q^{5} - 24 q^{7} + 5 q^{9} - 10 q^{11} - 4 q^{13} + 130 q^{15} - 66 q^{17} + 164 q^{19} - 88 q^{21} + 204 q^{23} + 79 q^{25} + 142 q^{27} + 87 q^{29} + 86 q^{31} - 130 q^{33} - 24 q^{35} - 42 q^{37} + 394 q^{39} + 562 q^{41} - 18 q^{43} + 422 q^{45} - 654 q^{47} + 539 q^{49} - 556 q^{51} + 712 q^{53} - 142 q^{55} + 828 q^{57} - 184 q^{59} + 322 q^{61} + 784 q^{63} + 1494 q^{65} + 228 q^{67} + 684 q^{69} + 52 q^{71} - 494 q^{73} + 3048 q^{75} + 872 q^{77} + 2110 q^{79} - 1513 q^{81} + 288 q^{83} - 2704 q^{85} - 58 q^{87} + 914 q^{89} + 2984 q^{91} - 62 q^{93} + 1900 q^{95} + 218 q^{97} + 304 q^{99}+O(q^{100}) $ 3 * q - 2 * q^3 + 20 * q^5 - 24 * q^7 + 5 * q^9 - 10 * q^11 - 4 * q^13 + 130 * q^15 - 66 * q^17 + 164 * q^19 - 88 * q^21 + 204 * q^23 + 79 * q^25 + 142 * q^27 + 87 * q^29 + 86 * q^31 - 130 * q^33 - 24 * q^35 - 42 * q^37 + 394 * q^39 + 562 * q^41 - 18 * q^43 + 422 * q^45 - 654 * q^47 + 539 * q^49 - 556 * q^51 + 712 * q^53 - 142 * q^55 + 828 * q^57 - 184 * q^59 + 322 * q^61 + 784 * q^63 + 1494 * q^65 + 228 * q^67 + 684 * q^69 + 52 * q^71 - 494 * q^73 + 3048 * q^75 + 872 * q^77 + 2110 * q^79 - 1513 * q^81 + 288 * q^83 - 2704 * q^85 - 58 * q^87 + 914 * q^89 + 2984 * q^91 - 62 * q^93 + 1900 * q^95 + 218 * q^97 + 304 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−2.39712	−0.461326	−0.230663	−	0.973034i	$-0.574089\pi$
$3$	−2.39712	−0.461326	−0.230663	+	0.973034i	$0.574089\pi$
$4$	0	0
$5$	−3.28077	−0.293441	−0.146720	−	0.989178i	$-0.546872\pi$
$5$	−3.28077	−0.293441	−0.146720	+	0.989178i	$0.546872\pi$
$6$	0	0
$7$	−33.9461	−1.83292	−0.916459	−	0.400130i	$-0.868965\pi$
$7$	−33.9461	−1.83292	−0.916459	+	0.400130i	$0.868965\pi$
$8$	0	0
$9$	−21.2538	−0.787178
$10$	0	0
$11$	−14.5759	−0.399528	−0.199764	−	0.979844i	$-0.564018\pi$
$11$	−14.5759	−0.399528	−0.199764	+	0.979844i	$0.564018\pi$
$12$	0	0
$13$	−86.5287	−1.84606	−0.923029	−	0.384730i	$-0.874294\pi$
$13$	−86.5287	−1.84606	−0.923029	+	0.384730i	$0.874294\pi$
$14$	0	0
$15$	7.86439	0.135372
$16$	0	0
$17$	102.550	1.46306	0.731529	−	0.681810i	$-0.238807\pi$
$17$	102.550	1.46306	0.731529	+	0.681810i	$0.238807\pi$
$18$	0	0
$19$	105.688	1.27613	0.638067	−	0.769981i	$-0.279734\pi$
$19$	105.688	1.27613	0.638067	+	0.769981i	$0.279734\pi$
$20$	0	0
$21$	81.3729	0.845572
$22$	0	0
$23$	135.456	1.22802	0.614010	−	0.789299i	$-0.289556\pi$
$23$	135.456	1.22802	0.614010	+	0.789299i	$0.289556\pi$
$24$	0	0
$25$	−114.237	−0.913893
$26$	0	0
$27$	115.670	0.824472
$28$	0	0
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	−223.883	−1.29712	−0.648558	−	0.761165i	$-0.724628\pi$
$31$	−223.883	−1.29712	−0.648558	+	0.761165i	$0.724628\pi$
$32$	0	0
$33$	34.9403	0.184313
$34$	0	0
$35$	111.369	0.537852
$36$	0	0
$37$	−239.723	−1.06514	−0.532569	−	0.846386i	$-0.678773\pi$
$37$	−239.723	−1.06514	−0.532569	+	0.846386i	$0.678773\pi$
$38$	0	0
$39$	207.420	0.851634
$40$	0	0
$41$	219.331	0.835456	0.417728	−	0.908572i	$-0.362827\pi$
$41$	219.331	0.835456	0.417728	+	0.908572i	$0.362827\pi$
$42$	0	0
$43$	−18.9838	−0.0673255	−0.0336627	−	0.999433i	$-0.510717\pi$
$43$	−18.9838	−0.0673255	−0.0336627	+	0.999433i	$0.510717\pi$
$44$	0	0
$45$	69.7288	0.230990
$46$	0	0
$47$	−147.922	−0.459077	−0.229539	−	0.973300i	$-0.573722\pi$
$47$	−147.922	−0.459077	−0.229539	+	0.973300i	$0.573722\pi$
$48$	0	0
$49$	809.338	2.35959
$50$	0	0
$51$	−245.824	−0.674947
$52$	0	0
$53$	613.202	1.58924	0.794621	−	0.607106i	$-0.207670\pi$
$53$	613.202	1.58924	0.794621	+	0.607106i	$0.207670\pi$
$54$	0	0
$55$	47.8202	0.117238
$56$	0	0
$57$	−253.348	−0.588714
$58$	0	0
$59$	−184.206	−0.406468	−0.203234	−	0.979130i	$-0.565145\pi$
$59$	−184.206	−0.406468	−0.203234	+	0.979130i	$0.565145\pi$
$60$	0	0
$61$	−13.6914	−0.0287377	−0.0143689	−	0.999897i	$-0.504574\pi$
$61$	−13.6914	−0.0287377	−0.0143689	+	0.999897i	$0.504574\pi$
$62$	0	0
$63$	721.484	1.44283
$64$	0	0
$65$	283.880	0.541708
$66$	0	0
$67$	−328.733	−0.599421	−0.299710	−	0.954030i	$-0.596890\pi$
$67$	−328.733	−0.599421	−0.299710	+	0.954030i	$0.596890\pi$
$68$	0	0
$69$	−324.703	−0.566517
$70$	0	0
$71$	−5.15186	−0.00861145	−0.00430573	−	0.999991i	$-0.501371\pi$
$71$	−5.15186	−0.00861145	−0.00430573	+	0.999991i	$0.501371\pi$
$72$	0	0
$73$	−428.481	−0.686984	−0.343492	−	0.939156i	$-0.611610\pi$
$73$	−428.481	−0.686984	−0.343492	+	0.939156i	$0.611610\pi$
$74$	0	0
$75$	273.839	0.421602
$76$	0	0
$77$	494.796	0.732302
$78$	0	0
$79$	392.819	0.559437	0.279719	−	0.960082i	$-0.409759\pi$
$79$	392.819	0.559437	0.279719	+	0.960082i	$0.409759\pi$
$80$	0	0
$81$	296.578	0.406828
$82$	0	0
$83$	454.172	0.600624	0.300312	−	0.953841i	$-0.402909\pi$
$83$	454.172	0.600624	0.300312	+	0.953841i	$0.402909\pi$
$84$	0	0
$85$	−336.442	−0.429321
$86$	0	0
$87$	−69.5165	−0.0856661
$88$	0	0
$89$	−811.929	−0.967015	−0.483507	−	0.875340i	$-0.660637\pi$
$89$	−811.929	−0.967015	−0.483507	+	0.875340i	$0.660637\pi$
$90$	0	0
$91$	2937.31	3.38367
$92$	0	0
$93$	536.675	0.598394
$94$	0	0
$95$	−346.738	−0.374470
$96$	0	0
$97$	−11.3513	−0.0118820	−0.00594098	−	0.999982i	$-0.501891\pi$
$97$	−11.3513	−0.0118820	−0.00594098	+	0.999982i	$0.501891\pi$
$98$	0	0
$99$	309.794	0.314500
$100$	0	0
$101$	1548.46	1.52552	0.762762	−	0.646679i	$-0.223843\pi$
$101$	1548.46	1.52552	0.762762	+	0.646679i	$0.223843\pi$
$102$	0	0
$103$	−861.888	−0.824508	−0.412254	−	0.911069i	$-0.635258\pi$
$103$	−861.888	−0.824508	−0.412254	+	0.911069i	$0.635258\pi$
$104$	0	0
$105$	−266.965	−0.248125
$106$	0	0
$107$	1374.40	1.24176	0.620880	−	0.783906i	$-0.286775\pi$
$107$	1374.40	1.24176	0.620880	+	0.783906i	$0.286775\pi$
$108$	0	0
$109$	−467.521	−0.410829	−0.205415	−	0.978675i	$-0.565854\pi$
$109$	−467.521	−0.410829	−0.205415	+	0.978675i	$0.565854\pi$
$110$	0	0
$111$	574.644	0.491376
$112$	0	0
$113$	655.223	0.545471	0.272736	−	0.962089i	$-0.412072\pi$
$113$	655.223	0.545471	0.272736	+	0.962089i	$0.412072\pi$
$114$	0	0
$115$	−444.398	−0.360351
$116$	0	0
$117$	1839.07	1.45318
$118$	0	0
$119$	−3481.17	−2.68166
$120$	0	0
$121$	−1118.54	−0.840377
$122$	0	0
$123$	−525.762	−0.385418
$124$	0	0
$125$	784.879	0.561614
$126$	0	0
$127$	−830.607	−0.580350	−0.290175	−	0.956974i	$-0.593714\pi$
$127$	−830.607	−0.580350	−0.290175	+	0.956974i	$0.593714\pi$
$128$	0	0
$129$	45.5063	0.0310590
$130$	0	0
$131$	−783.195	−0.522351	−0.261176	−	0.965291i	$-0.584110\pi$
$131$	−783.195	−0.522351	−0.261176	+	0.965291i	$0.584110\pi$
$132$	0	0
$133$	−3587.71	−2.33905
$134$	0	0
$135$	−379.487	−0.241933
$136$	0	0
$137$	−1362.28	−0.849545	−0.424773	−	0.905300i	$-0.639646\pi$
$137$	−1362.28	−0.849545	−0.424773	+	0.905300i	$0.639646\pi$
$138$	0	0
$139$	874.798	0.533809	0.266904	−	0.963723i	$-0.413999\pi$
$139$	874.798	0.533809	0.266904	+	0.963723i	$0.413999\pi$
$140$	0	0
$141$	354.587	0.211784
$142$	0	0
$143$	1261.24	0.737552
$144$	0	0
$145$	−95.1422	−0.0544905
$146$	0	0
$147$	−1940.08	−1.08854
$148$	0	0
$149$	1938.20	1.06566	0.532830	−	0.846222i	$-0.321129\pi$
$149$	1938.20	1.06566	0.532830	+	0.846222i	$0.321129\pi$
$150$	0	0
$151$	1460.33	0.787018	0.393509	−	0.919321i	$-0.371261\pi$
$151$	1460.33	0.787018	0.393509	+	0.919321i	$0.371261\pi$
$152$	0	0
$153$	−2179.58	−1.15169
$154$	0	0
$155$	734.508	0.380627
$156$	0	0
$157$	421.035	0.214027	0.107014	−	0.994258i	$-0.465871\pi$
$157$	421.035	0.214027	0.107014	+	0.994258i	$0.465871\pi$
$158$	0	0
$159$	−1469.92	−0.733158
$160$	0	0
$161$	−4598.19	−2.25086
$162$	0	0
$163$	796.182	0.382588	0.191294	−	0.981533i	$-0.438732\pi$
$163$	796.182	0.382588	0.191294	+	0.981533i	$0.438732\pi$
$164$	0	0
$165$	−114.631	−0.0540848
$166$	0	0
$167$	−3625.72	−1.68004	−0.840020	−	0.542556i	$-0.817457\pi$
$167$	−3625.72	−1.68004	−0.840020	+	0.542556i	$0.817457\pi$
$168$	0	0
$169$	5290.22	2.40793
$170$	0	0
$171$	−2246.28	−1.00455
$172$	0	0
$173$	587.037	0.257986	0.128993	−	0.991645i	$-0.458825\pi$
$173$	587.037	0.257986	0.128993	+	0.991645i	$0.458825\pi$
$174$	0	0
$175$	3877.89	1.67509
$176$	0	0
$177$	441.564	0.187514
$178$	0	0
$179$	908.839	0.379496	0.189748	−	0.981833i	$-0.439233\pi$
$179$	908.839	0.379496	0.189748	+	0.981833i	$0.439233\pi$
$180$	0	0
$181$	2139.11	0.878447	0.439223	−	0.898378i	$-0.355254\pi$
$181$	2139.11	0.878447	0.439223	+	0.898378i	$0.355254\pi$
$182$	0	0
$183$	32.8199	0.0132575
$184$	0	0
$185$	786.473	0.312555
$186$	0	0
$187$	−1494.76	−0.584533
$188$	0	0
$189$	−3926.55	−1.51119
$190$	0	0
$191$	1715.31	0.649819	0.324910	−	0.945745i	$-0.394666\pi$
$191$	1715.31	0.649819	0.324910	+	0.945745i	$0.394666\pi$
$192$	0	0
$193$	−2509.88	−0.936090	−0.468045	−	0.883705i	$-0.655042\pi$
$193$	−2509.88	−0.936090	−0.468045	+	0.883705i	$0.655042\pi$
$194$	0	0
$195$	−680.496	−0.249904
$196$	0	0
$197$	−688.689	−0.249071	−0.124536	−	0.992215i	$-0.539744\pi$
$197$	−688.689	−0.249071	−0.124536	+	0.992215i	$0.539744\pi$
$198$	0	0
$199$	−1106.28	−0.394079	−0.197040	−	0.980396i	$-0.563133\pi$
$199$	−1106.28	−0.394079	−0.197040	+	0.980396i	$0.563133\pi$
$200$	0	0
$201$	788.014	0.276528
$202$	0	0
$203$	−984.437	−0.340364
$204$	0	0
$205$	−719.572	−0.245157
$206$	0	0
$207$	−2878.95	−0.966670
$208$	0	0
$209$	−1540.51	−0.509851
$210$	0	0
$211$	2094.08	0.683234	0.341617	−	0.939839i	$-0.389025\pi$
$211$	2094.08	0.683234	0.341617	+	0.939839i	$0.389025\pi$
$212$	0	0
$213$	12.3496	0.00397269
$214$	0	0
$215$	62.2812	0.0197560
$216$	0	0
$217$	7599.96	2.37751
$218$	0	0
$219$	1027.12	0.316924
$220$	0	0
$221$	−8873.51	−2.70089
$222$	0	0
$223$	−6033.11	−1.81169	−0.905845	−	0.423610i	$-0.860763\pi$
$223$	−6033.11	−1.81169	−0.905845	+	0.423610i	$0.860763\pi$
$224$	0	0
$225$	2427.96	0.719396
$226$	0	0
$227$	−189.492	−0.0554053	−0.0277027	−	0.999616i	$-0.508819\pi$
$227$	−189.492	−0.0554053	−0.0277027	+	0.999616i	$0.508819\pi$
$228$	0	0
$229$	−2347.80	−0.677498	−0.338749	−	0.940877i	$-0.610004\pi$
$229$	−2347.80	−0.677498	−0.338749	+	0.940877i	$0.610004\pi$
$230$	0	0
$231$	−1186.09	−0.337830
$232$	0	0
$233$	551.777	0.155142	0.0775710	−	0.996987i	$-0.475284\pi$
$233$	551.777	0.155142	0.0775710	+	0.996987i	$0.475284\pi$
$234$	0	0
$235$	485.297	0.134712
$236$	0	0
$237$	−941.633	−0.258083
$238$	0	0
$239$	4533.45	1.22696	0.613482	−	0.789708i	$-0.289768\pi$
$239$	4533.45	1.22696	0.613482	+	0.789708i	$0.289768\pi$
$240$	0	0
$241$	1148.28	0.306917	0.153458	−	0.988155i	$-0.450959\pi$
$241$	1148.28	0.306917	0.153458	+	0.988155i	$0.450959\pi$
$242$	0	0
$243$	−3834.03	−1.01215
$244$	0	0
$245$	−2655.25	−0.692398
$246$	0	0
$247$	−9145.07	−2.35582
$248$	0	0
$249$	−1088.70	−0.277084
$250$	0	0
$251$	7236.42	1.81976	0.909878	−	0.414876i	$-0.136175\pi$
$251$	7236.42	1.81976	0.909878	+	0.414876i	$0.136175\pi$
$252$	0	0
$253$	−1974.39	−0.490628
$254$	0	0
$255$	806.492	0.198057
$256$	0	0
$257$	5051.89	1.22618	0.613089	−	0.790013i	$-0.289927\pi$
$257$	5051.89	1.22618	0.613089	+	0.790013i	$0.289927\pi$
$258$	0	0
$259$	8137.64	1.95231
$260$	0	0
$261$	−616.361	−0.146175
$262$	0	0
$263$	−6368.81	−1.49322	−0.746611	−	0.665260i	$-0.768320\pi$
$263$	−6368.81	−1.49322	−0.746611	+	0.665260i	$0.768320\pi$
$264$	0	0
$265$	−2011.77	−0.466348
$266$	0	0
$267$	1946.29	0.446109
$268$	0	0
$269$	3299.89	0.747948	0.373974	−	0.927439i	$-0.377995\pi$
$269$	3299.89	0.747948	0.373974	+	0.927439i	$0.377995\pi$
$270$	0	0
$271$	2235.21	0.501031	0.250515	−	0.968113i	$-0.419400\pi$
$271$	2235.21	0.501031	0.250515	+	0.968113i	$0.419400\pi$
$272$	0	0
$273$	−7041.09	−1.56098
$274$	0	0
$275$	1665.10	0.365126
$276$	0	0
$277$	−2354.61	−0.510739	−0.255369	−	0.966844i	$-0.582197\pi$
$277$	−2354.61	−0.510739	−0.255369	+	0.966844i	$0.582197\pi$
$278$	0	0
$279$	4758.37	1.02106
$280$	0	0
$281$	−6858.62	−1.45605	−0.728027	−	0.685549i	$-0.759562\pi$
$281$	−6858.62	−1.45605	−0.728027	+	0.685549i	$0.759562\pi$
$282$	0	0
$283$	3031.29	0.636719	0.318360	−	0.947970i	$-0.396868\pi$
$283$	3031.29	0.636719	0.318360	+	0.947970i	$0.396868\pi$
$284$	0	0
$285$	831.174	0.172753
$286$	0	0
$287$	−7445.42	−1.53132
$288$	0	0
$289$	5603.47	1.14054
$290$	0	0
$291$	27.2104	0.00548145
$292$	0	0
$293$	1921.35	0.383094	0.191547	−	0.981483i	$-0.438650\pi$
$293$	1921.35	0.383094	0.191547	+	0.981483i	$0.438650\pi$
$294$	0	0
$295$	604.337	0.119274
$296$	0	0
$297$	−1686.00	−0.329400
$298$	0	0
$299$	−11720.8	−2.26699
$300$	0	0
$301$	644.424	0.123402
$302$	0	0
$303$	−3711.86	−0.703764
$304$	0	0
$305$	44.9182	0.00843282
$306$	0	0
$307$	8734.92	1.62387	0.811935	−	0.583747i	$-0.198414\pi$
$307$	8734.92	1.62387	0.811935	+	0.583747i	$0.198414\pi$
$308$	0	0
$309$	2066.05	0.380367
$310$	0	0
$311$	198.321	0.0361599	0.0180800	−	0.999837i	$-0.494245\pi$
$311$	198.321	0.0361599	0.0180800	+	0.999837i	$0.494245\pi$
$312$	0	0
$313$	−3759.23	−0.678864	−0.339432	−	0.940631i	$-0.610235\pi$
$313$	−3759.23	−0.678864	−0.339432	+	0.940631i	$0.610235\pi$
$314$	0	0
$315$	−2367.02	−0.423386
$316$	0	0
$317$	−5543.84	−0.982249	−0.491124	−	0.871090i	$-0.663414\pi$
$317$	−5543.84	−0.982249	−0.491124	+	0.871090i	$0.663414\pi$
$318$	0	0
$319$	−422.702	−0.0741905
$320$	0	0
$321$	−3294.60	−0.572856
$322$	0	0
$323$	10838.3	1.86706
$324$	0	0
$325$	9884.75	1.68710
$326$	0	0
$327$	1120.70	0.189526
$328$	0	0
$329$	5021.37	0.841451
$330$	0	0
$331$	2875.96	0.477574	0.238787	−	0.971072i	$-0.423250\pi$
$331$	2875.96	0.477574	0.238787	+	0.971072i	$0.423250\pi$
$332$	0	0
$333$	5095.02	0.838454
$334$	0	0
$335$	1078.50	0.175894
$336$	0	0
$337$	8141.60	1.31603	0.658014	−	0.753006i	$-0.271397\pi$
$337$	8141.60	1.31603	0.658014	+	0.753006i	$0.271397\pi$
$338$	0	0
$339$	−1570.65	−0.251640
$340$	0	0
$341$	3263.31	0.518234
$342$	0	0
$343$	−15830.3	−2.49201
$344$	0	0
$345$	1065.28	0.166239
$346$	0	0
$347$	8941.88	1.38336	0.691679	−	0.722205i	$-0.256871\pi$
$347$	8941.88	1.38336	0.691679	+	0.722205i	$0.256871\pi$
$348$	0	0
$349$	−3061.39	−0.469548	−0.234774	−	0.972050i	$-0.575435\pi$
$349$	−3061.39	−0.469548	−0.234774	+	0.972050i	$0.575435\pi$
$350$	0	0
$351$	−10008.8	−1.52202
$352$	0	0
$353$	2871.64	0.432979	0.216490	−	0.976285i	$-0.430539\pi$
$353$	2871.64	0.432979	0.216490	+	0.976285i	$0.430539\pi$
$354$	0	0
$355$	16.9020	0.00252695
$356$	0	0
$357$	8344.78	1.23712
$358$	0	0
$359$	9773.24	1.43680	0.718401	−	0.695629i	$-0.244874\pi$
$359$	9773.24	1.43680	0.718401	+	0.695629i	$0.244874\pi$
$360$	0	0
$361$	4311.01	0.628519
$362$	0	0
$363$	2681.28	0.387688
$364$	0	0
$365$	1405.74	0.201589
$366$	0	0
$367$	6064.61	0.862588	0.431294	−	0.902211i	$-0.358057\pi$
$367$	6064.61	0.862588	0.431294	+	0.902211i	$0.358057\pi$
$368$	0	0
$369$	−4661.61	−0.657653
$370$	0	0
$371$	−20815.8	−2.91295
$372$	0	0
$373$	2684.28	0.372618	0.186309	−	0.982491i	$-0.440347\pi$
$373$	2684.28	0.372618	0.186309	+	0.982491i	$0.440347\pi$
$374$	0	0
$375$	−1881.45	−0.259087
$376$	0	0
$377$	−2509.33	−0.342804
$378$	0	0
$379$	−2069.54	−0.280489	−0.140244	−	0.990117i	$-0.544789\pi$
$379$	−2069.54	−0.280489	−0.140244	+	0.990117i	$0.544789\pi$
$380$	0	0
$381$	1991.06	0.267731
$382$	0	0
$383$	−8550.46	−1.14075	−0.570376	−	0.821384i	$-0.693203\pi$
$383$	−8550.46	−1.14075	−0.570376	+	0.821384i	$0.693203\pi$
$384$	0	0
$385$	−1623.31	−0.214887
$386$	0	0
$387$	403.477	0.0529971
$388$	0	0
$389$	−10974.4	−1.43040	−0.715199	−	0.698921i	$-0.753664\pi$
$389$	−10974.4	−1.43040	−0.715199	+	0.698921i	$0.753664\pi$
$390$	0	0
$391$	13890.9	1.79666
$392$	0	0
$393$	1877.41	0.240974
$394$	0	0
$395$	−1288.75	−0.164162
$396$	0	0
$397$	973.767	0.123103	0.0615516	−	0.998104i	$-0.480395\pi$
$397$	973.767	0.123103	0.0615516	+	0.998104i	$0.480395\pi$
$398$	0	0
$399$	8600.16	1.07906
$400$	0	0
$401$	2196.32	0.273514	0.136757	−	0.990605i	$-0.456332\pi$
$401$	2196.32	0.273514	0.136757	+	0.990605i	$0.456332\pi$
$402$	0	0
$403$	19372.3	2.39455
$404$	0	0
$405$	−973.002	−0.119380
$406$	0	0
$407$	3494.18	0.425553
$408$	0	0
$409$	7847.40	0.948726	0.474363	−	0.880329i	$-0.342678\pi$
$409$	7847.40	0.948726	0.474363	+	0.880329i	$0.342678\pi$
$410$	0	0
$411$	3265.56	0.391917
$412$	0	0
$413$	6253.08	0.745022
$414$	0	0
$415$	−1490.03	−0.176248
$416$	0	0
$417$	−2097.00	−0.246260
$418$	0	0
$419$	6943.21	0.809541	0.404771	−	0.914418i	$-0.367351\pi$
$419$	6943.21	0.809541	0.404771	+	0.914418i	$0.367351\pi$
$420$	0	0
$421$	−10527.3	−1.21869	−0.609343	−	0.792907i	$-0.708567\pi$
$421$	−10527.3	−1.21869	−0.609343	+	0.792907i	$0.708567\pi$
$422$	0	0
$423$	3143.91	0.361376
$424$	0	0
$425$	−11714.9	−1.33708
$426$	0	0
$427$	464.769	0.0526739
$428$	0	0
$429$	−3023.34	−0.340252
$430$	0	0
$431$	1735.58	0.193967	0.0969836	−	0.995286i	$-0.469081\pi$
$431$	1735.58	0.193967	0.0969836	+	0.995286i	$0.469081\pi$
$432$	0	0
$433$	−9380.15	−1.04107	−0.520533	−	0.853842i	$-0.674267\pi$
$433$	−9380.15	−1.04107	−0.520533	+	0.853842i	$0.674267\pi$
$434$	0	0
$435$	228.067	0.0251379
$436$	0	0
$437$	14316.1	1.56712
$438$	0	0
$439$	4099.58	0.445700	0.222850	−	0.974853i	$-0.428464\pi$
$439$	4099.58	0.445700	0.222850	+	0.974853i	$0.428464\pi$
$440$	0	0
$441$	−17201.5	−1.85741
$442$	0	0
$443$	−5718.20	−0.613273	−0.306636	−	0.951827i	$-0.599204\pi$
$443$	−5718.20	−0.613273	−0.306636	+	0.951827i	$0.599204\pi$
$444$	0	0
$445$	2663.75	0.283761
$446$	0	0
$447$	−4646.09	−0.491616
$448$	0	0
$449$	2474.77	0.260115	0.130057	−	0.991506i	$-0.458484\pi$
$449$	2474.77	0.260115	0.130057	+	0.991506i	$0.458484\pi$
$450$	0	0
$451$	−3196.95	−0.333788
$452$	0	0
$453$	−3500.58	−0.363072
$454$	0	0
$455$	−9636.63	−0.992906
$456$	0	0
$457$	10932.4	1.11903	0.559513	−	0.828822i	$-0.310988\pi$
$457$	10932.4	1.11903	0.559513	+	0.828822i	$0.310988\pi$
$458$	0	0
$459$	11862.0	1.20625
$460$	0	0
$461$	16589.3	1.67602	0.838008	−	0.545659i	$-0.183720\pi$
$461$	16589.3	1.67602	0.838008	+	0.545659i	$0.183720\pi$
$462$	0	0
$463$	−894.244	−0.0897604	−0.0448802	−	0.998992i	$-0.514291\pi$
$463$	−894.244	−0.0897604	−0.0448802	+	0.998992i	$0.514291\pi$
$464$	0	0
$465$	−1760.70	−0.175593
$466$	0	0
$467$	8011.09	0.793809	0.396905	−	0.917860i	$-0.370084\pi$
$467$	8011.09	0.793809	0.396905	+	0.917860i	$0.370084\pi$
$468$	0	0
$469$	11159.2	1.09869
$470$	0	0
$471$	−1009.27	−0.0987363
$472$	0	0
$473$	276.706	0.0268984
$474$	0	0
$475$	−12073.5	−1.16625
$476$	0	0
$477$	−13032.9	−1.25102
$478$	0	0
$479$	14558.7	1.38873	0.694366	−	0.719622i	$-0.255685\pi$
$479$	14558.7	1.38873	0.694366	+	0.719622i	$0.255685\pi$
$480$	0	0
$481$	20742.9	1.96631
$482$	0	0
$483$	11022.4	1.03838
$484$	0	0
$485$	37.2409	0.00348665
$486$	0	0
$487$	17666.8	1.64386	0.821928	−	0.569591i	$-0.192898\pi$
$487$	17666.8	1.64386	0.821928	+	0.569591i	$0.192898\pi$
$488$	0	0
$489$	−1908.54	−0.176498
$490$	0	0
$491$	4904.36	0.450775	0.225388	−	0.974269i	$-0.427635\pi$
$491$	4904.36	0.450775	0.225388	+	0.974269i	$0.427635\pi$
$492$	0	0
$493$	2973.95	0.271683
$494$	0	0
$495$	−1016.36	−0.0922870
$496$	0	0
$497$	174.886	0.0157841
$498$	0	0
$499$	−2496.99	−0.224009	−0.112005	−	0.993708i	$-0.535727\pi$
$499$	−2496.99	−0.224009	−0.112005	+	0.993708i	$0.535727\pi$
$500$	0	0
$501$	8691.29	0.775046
$502$	0	0
$503$	968.191	0.0858241	0.0429120	−	0.999079i	$-0.486336\pi$
$503$	968.191	0.0858241	0.0429120	+	0.999079i	$0.486336\pi$
$504$	0	0
$505$	−5080.15	−0.447651
$506$	0	0
$507$	−12681.3	−1.11084
$508$	0	0
$509$	3379.07	0.294253	0.147126	−	0.989118i	$-0.452998\pi$
$509$	3379.07	0.294253	0.147126	+	0.989118i	$0.452998\pi$
$510$	0	0
$511$	14545.2	1.25919
$512$	0	0
$513$	12225.0	1.05214
$514$	0	0
$515$	2827.65	0.241944
$516$	0	0
$517$	2156.10	0.183414
$518$	0	0
$519$	−1407.20	−0.119016
$520$	0	0
$521$	10941.9	0.920098	0.460049	−	0.887893i	$-0.347832\pi$
$521$	10941.9	0.920098	0.460049	+	0.887893i	$0.347832\pi$
$522$	0	0
$523$	−8947.20	−0.748057	−0.374029	−	0.927417i	$-0.622024\pi$
$523$	−8947.20	−0.748057	−0.374029	+	0.927417i	$0.622024\pi$
$524$	0	0
$525$	−9295.76	−0.772762
$526$	0	0
$527$	−22959.2	−1.89776
$528$	0	0
$529$	6181.21	0.508031
$530$	0	0
$531$	3915.09	0.319963
$532$	0	0
$533$	−18978.4	−1.54230
$534$	0	0
$535$	−4509.09	−0.364383
$536$	0	0
$537$	−2178.60	−0.175071
$538$	0	0
$539$	−11796.8	−0.942720
$540$	0	0
$541$	9008.10	0.715875	0.357938	−	0.933746i	$-0.383480\pi$
$541$	9008.10	0.715875	0.357938	+	0.933746i	$0.383480\pi$
$542$	0	0
$543$	−5127.71	−0.405250
$544$	0	0
$545$	1533.83	0.120554
$546$	0	0
$547$	−12980.3	−1.01462	−0.507312	−	0.861763i	$-0.669361\pi$
$547$	−12980.3	−1.01462	−0.507312	+	0.861763i	$0.669361\pi$
$548$	0	0
$549$	290.994	0.0226217
$550$	0	0
$551$	3064.96	0.236972
$552$	0	0
$553$	−13334.7	−1.02540
$554$	0	0
$555$	−1885.27	−0.144190
$556$	0	0
$557$	−8777.82	−0.667734	−0.333867	−	0.942620i	$-0.608354\pi$
$557$	−8777.82	−0.667734	−0.333867	+	0.942620i	$0.608354\pi$
$558$	0	0
$559$	1642.64	0.124287
$560$	0	0
$561$	3583.12	0.269660
$562$	0	0
$563$	−3734.91	−0.279588	−0.139794	−	0.990181i	$-0.544644\pi$
$563$	−3734.91	−0.279588	−0.139794	+	0.990181i	$0.544644\pi$
$564$	0	0
$565$	−2149.63	−0.160063
$566$	0	0
$567$	−10067.7	−0.745682
$568$	0	0
$569$	−14373.7	−1.05901	−0.529504	−	0.848307i	$-0.677622\pi$
$569$	−14373.7	−1.05901	−0.529504	+	0.848307i	$0.677622\pi$
$570$	0	0
$571$	−8147.54	−0.597135	−0.298567	−	0.954389i	$-0.596509\pi$
$571$	−8147.54	−0.597135	−0.298567	+	0.954389i	$0.596509\pi$
$572$	0	0
$573$	−4111.81	−0.299779
$574$	0	0
$575$	−15474.0	−1.12228
$576$	0	0
$577$	9438.43	0.680982	0.340491	−	0.940248i	$-0.389407\pi$
$577$	9438.43	0.680982	0.340491	+	0.940248i	$0.389407\pi$
$578$	0	0
$579$	6016.49	0.431843
$580$	0	0
$581$	−15417.4	−1.10089
$582$	0	0
$583$	−8937.99	−0.634946
$584$	0	0
$585$	−6033.54	−0.426421
$586$	0	0
$587$	3498.97	0.246027	0.123014	−	0.992405i	$-0.460744\pi$
$587$	3498.97	0.246027	0.123014	+	0.992405i	$0.460744\pi$
$588$	0	0
$589$	−23661.8	−1.65530
$590$	0	0
$591$	1650.87	0.114903
$592$	0	0
$593$	7383.92	0.511335	0.255667	−	0.966765i	$-0.417705\pi$
$593$	7383.92	0.511335	0.255667	+	0.966765i	$0.417705\pi$
$594$	0	0
$595$	11420.9	0.786909
$596$	0	0
$597$	2651.87	0.181799
$598$	0	0
$599$	−14673.8	−1.00093	−0.500463	−	0.865758i	$-0.666837\pi$
$599$	−14673.8	−1.00093	−0.500463	+	0.865758i	$0.666837\pi$
$600$	0	0
$601$	−10950.2	−0.743209	−0.371605	−	0.928391i	$-0.621192\pi$
$601$	−10950.2	−0.743209	−0.371605	+	0.928391i	$0.621192\pi$
$602$	0	0
$603$	6986.84	0.471851
$604$	0	0
$605$	3669.67	0.246601
$606$	0	0
$607$	−17469.1	−1.16812	−0.584059	−	0.811712i	$-0.698536\pi$
$607$	−17469.1	−1.16812	−0.584059	+	0.811712i	$0.698536\pi$
$608$	0	0
$609$	2359.81	0.157019
$610$	0	0
$611$	12799.5	0.847484
$612$	0	0
$613$	5866.70	0.386548	0.193274	−	0.981145i	$-0.438089\pi$
$613$	5866.70	0.386548	0.193274	+	0.981145i	$0.438089\pi$
$614$	0	0
$615$	1724.90	0.113097
$616$	0	0
$617$	−14383.8	−0.938528	−0.469264	−	0.883058i	$-0.655481\pi$
$617$	−14383.8	−0.938528	−0.469264	+	0.883058i	$0.655481\pi$
$618$	0	0
$619$	−28631.0	−1.85909	−0.929545	−	0.368710i	$-0.879800\pi$
$619$	−28631.0	−1.85909	−0.929545	+	0.368710i	$0.879800\pi$
$620$	0	0
$621$	15668.2	1.01247
$622$	0	0
$623$	27561.8	1.77246
$624$	0	0
$625$	11704.6	0.749092
$626$	0	0
$627$	3692.78	0.235208
$628$	0	0
$629$	−24583.5	−1.55836
$630$	0	0
$631$	2860.48	0.180466	0.0902329	−	0.995921i	$-0.471239\pi$
$631$	2860.48	0.180466	0.0902329	+	0.995921i	$0.471239\pi$
$632$	0	0
$633$	−5019.76	−0.315194
$634$	0	0
$635$	2725.03	0.170298
$636$	0	0
$637$	−70031.0	−4.35593
$638$	0	0
$639$	109.497	0.00677875
$640$	0	0
$641$	6003.37	0.369920	0.184960	−	0.982746i	$-0.440784\pi$
$641$	6003.37	0.369920	0.184960	+	0.982746i	$0.440784\pi$
$642$	0	0
$643$	−6690.61	−0.410345	−0.205173	−	0.978726i	$-0.565776\pi$
$643$	−6690.61	−0.410345	−0.205173	+	0.978726i	$0.565776\pi$
$644$	0	0
$645$	−149.296	−0.00911397
$646$	0	0
$647$	8697.55	0.528495	0.264247	−	0.964455i	$-0.414876\pi$
$647$	8697.55	0.528495	0.264247	+	0.964455i	$0.414876\pi$
$648$	0	0
$649$	2684.98	0.162395
$650$	0	0
$651$	−18218.0	−1.09681
$652$	0	0
$653$	21260.9	1.27413	0.637063	−	0.770812i	$-0.280149\pi$
$653$	21260.9	1.27413	0.637063	+	0.770812i	$0.280149\pi$
$654$	0	0
$655$	2569.48	0.153279
$656$	0	0
$657$	9106.85	0.540779
$658$	0	0
$659$	−11717.3	−0.692625	−0.346313	−	0.938119i	$-0.612566\pi$
$659$	−11717.3	−0.692625	−0.346313	+	0.938119i	$0.612566\pi$
$660$	0	0
$661$	27523.1	1.61955	0.809775	−	0.586740i	$-0.199589\pi$
$661$	27523.1	1.61955	0.809775	+	0.586740i	$0.199589\pi$
$662$	0	0
$663$	21270.9	1.24599
$664$	0	0
$665$	11770.4	0.686372
$666$	0	0
$667$	3928.21	0.228037
$668$	0	0
$669$	14462.1	0.835780
$670$	0	0
$671$	199.565	0.0114815
$672$	0	0
$673$	−30200.5	−1.72978	−0.864891	−	0.501960i	$-0.832612\pi$
$673$	−30200.5	−1.72978	−0.864891	+	0.501960i	$0.832612\pi$
$674$	0	0
$675$	−13213.8	−0.753479
$676$	0	0
$677$	19986.2	1.13461	0.567306	−	0.823507i	$-0.307986\pi$
$677$	19986.2	1.13461	0.567306	+	0.823507i	$0.307986\pi$
$678$	0	0
$679$	385.332	0.0217786
$680$	0	0
$681$	454.234	0.0255599
$682$	0	0
$683$	6037.11	0.338219	0.169110	−	0.985597i	$-0.445911\pi$
$683$	6037.11	0.338219	0.169110	+	0.985597i	$0.445911\pi$
$684$	0	0
$685$	4469.33	0.249291
$686$	0	0
$687$	5627.96	0.312547
$688$	0	0
$689$	−53059.6	−2.93383
$690$	0	0
$691$	28078.2	1.54579	0.772897	−	0.634531i	$-0.218807\pi$
$691$	28078.2	1.54579	0.772897	+	0.634531i	$0.218807\pi$
$692$	0	0
$693$	−10516.3	−0.576452
$694$	0	0
$695$	−2870.01	−0.156641
$696$	0	0
$697$	22492.3	1.22232
$698$	0	0
$699$	−1322.68	−0.0715711
$700$	0	0
$701$	32839.5	1.76938	0.884688	−	0.466184i	$-0.154372\pi$
$701$	32839.5	1.76938	0.884688	+	0.466184i	$0.154372\pi$
$702$	0	0
$703$	−25335.9	−1.35926
$704$	0	0
$705$	−1163.32	−0.0621461
$706$	0	0
$707$	−52564.3	−2.79616
$708$	0	0
$709$	30391.9	1.60986	0.804932	−	0.593367i	$-0.202202\pi$
$709$	30391.9	1.60986	0.804932	+	0.593367i	$0.202202\pi$
$710$	0	0
$711$	−8348.89	−0.440377
$712$	0	0
$713$	−30326.2	−1.59288
$714$	0	0
$715$	−4137.82	−0.216428
$716$	0	0
$717$	−10867.2	−0.566031
$718$	0	0
$719$	28162.9	1.46078	0.730390	−	0.683031i	$-0.239338\pi$
$719$	28162.9	1.46078	0.730390	+	0.683031i	$0.239338\pi$
$720$	0	0
$721$	29257.7	1.51125
$722$	0	0
$723$	−2752.56	−0.141589
$724$	0	0
$725$	−3312.86	−0.169706
$726$	0	0
$727$	36419.3	1.85794	0.928968	−	0.370161i	$-0.120698\pi$
$727$	36419.3	1.85794	0.928968	+	0.370161i	$0.120698\pi$
$728$	0	0
$729$	1183.03	0.0601040
$730$	0	0
$731$	−1946.78	−0.0985011
$732$	0	0
$733$	8525.28	0.429589	0.214794	−	0.976659i	$-0.431092\pi$
$733$	8525.28	0.429589	0.214794	+	0.976659i	$0.431092\pi$
$734$	0	0
$735$	6364.95	0.319421
$736$	0	0
$737$	4791.60	0.239485
$738$	0	0
$739$	−18309.2	−0.911389	−0.455695	−	0.890136i	$-0.650609\pi$
$739$	−18309.2	−0.911389	−0.455695	+	0.890136i	$0.650609\pi$
$740$	0	0
$741$	21921.8	1.08680
$742$	0	0
$743$	−12992.8	−0.641534	−0.320767	−	0.947158i	$-0.603941\pi$
$743$	−12992.8	−0.641534	−0.320767	+	0.947158i	$0.603941\pi$
$744$	0	0
$745$	−6358.77	−0.312708
$746$	0	0
$747$	−9652.88	−0.472799
$748$	0	0
$749$	−46655.5	−2.27604
$750$	0	0
$751$	3443.18	0.167302	0.0836508	−	0.996495i	$-0.473342\pi$
$751$	3443.18	0.167302	0.0836508	+	0.996495i	$0.473342\pi$
$752$	0	0
$753$	−17346.6	−0.839501
$754$	0	0
$755$	−4790.99	−0.230943
$756$	0	0
$757$	3498.73	0.167983	0.0839917	−	0.996466i	$-0.473233\pi$
$757$	3498.73	0.167983	0.0839917	+	0.996466i	$0.473233\pi$
$758$	0	0
$759$	4732.85	0.226339
$760$	0	0
$761$	−21460.2	−1.02225	−0.511124	−	0.859507i	$-0.670771\pi$
$761$	−21460.2	−1.02225	−0.511124	+	0.859507i	$0.670771\pi$
$762$	0	0
$763$	15870.5	0.753016
$764$	0	0
$765$	7150.67	0.337952
$766$	0	0
$767$	15939.1	0.750363
$768$	0	0
$769$	−8766.41	−0.411086	−0.205543	−	0.978648i	$-0.565896\pi$
$769$	−8766.41	−0.411086	−0.205543	+	0.978648i	$0.565896\pi$
$770$	0	0
$771$	−12110.0	−0.565668
$772$	0	0
$773$	−17969.5	−0.836116	−0.418058	−	0.908420i	$-0.637289\pi$
$773$	−17969.5	−0.836116	−0.418058	+	0.908420i	$0.637289\pi$
$774$	0	0
$775$	25575.7	1.18543
$776$	0	0
$777$	−19506.9	−0.900652
$778$	0	0
$779$	23180.7	1.06615
$780$	0	0
$781$	75.0931	0.00344052
$782$	0	0
$783$	3354.44	0.153101
$784$	0	0
$785$	−1381.32	−0.0628042
$786$	0	0
$787$	2799.53	0.126801	0.0634005	−	0.997988i	$-0.479805\pi$
$787$	2799.53	0.126801	0.0634005	+	0.997988i	$0.479805\pi$
$788$	0	0
$789$	15266.8	0.688862
$790$	0	0
$791$	−22242.3	−0.999803
$792$	0	0
$793$	1184.70	0.0530515
$794$	0	0
$795$	4822.46	0.215138
$796$	0	0
$797$	−21771.2	−0.967597	−0.483799	−	0.875179i	$-0.660743\pi$
$797$	−21771.2	−0.967597	−0.483799	+	0.875179i	$0.660743\pi$
$798$	0	0
$799$	−15169.4	−0.671657
$800$	0	0
$801$	17256.6	0.761213
$802$	0	0
$803$	6245.50	0.274469
$804$	0	0
$805$	15085.6	0.660493
$806$	0	0
$807$	−7910.24	−0.345048
$808$	0	0
$809$	−28676.9	−1.24626	−0.623130	−	0.782118i	$-0.714139\pi$
$809$	−28676.9	−1.24626	−0.623130	+	0.782118i	$0.714139\pi$
$810$	0	0
$811$	−12021.0	−0.520486	−0.260243	−	0.965543i	$-0.583803\pi$
$811$	−12021.0	−0.520486	−0.260243	+	0.965543i	$0.583803\pi$
$812$	0	0
$813$	−5358.07	−0.231139
$814$	0	0
$815$	−2612.09	−0.112267
$816$	0	0
$817$	−2006.36	−0.0859164
$818$	0	0
$819$	−62429.1	−2.66355
$820$	0	0
$821$	−14455.7	−0.614504	−0.307252	−	0.951628i	$-0.599409\pi$
$821$	−14455.7	−0.614504	−0.307252	+	0.951628i	$0.599409\pi$
$822$	0	0
$823$	43192.9	1.82941	0.914707	−	0.404117i	$-0.132421\pi$
$823$	43192.9	1.82941	0.914707	+	0.404117i	$0.132421\pi$
$824$	0	0
$825$	−3991.46	−0.168442
$826$	0	0
$827$	−10474.2	−0.440415	−0.220207	−	0.975453i	$-0.570673\pi$
$827$	−10474.2	−0.440415	−0.220207	+	0.975453i	$0.570673\pi$
$828$	0	0
$829$	−13114.2	−0.549425	−0.274713	−	0.961526i	$-0.588583\pi$
$829$	−13114.2	−0.549425	−0.274713	+	0.961526i	$0.588583\pi$
$830$	0	0
$831$	5644.28	0.235617
$832$	0	0
$833$	82997.5	3.45221
$834$	0	0
$835$	11895.1	0.492992
$836$	0	0
$837$	−25896.6	−1.06944
$838$	0	0
$839$	−4806.78	−0.197793	−0.0988965	−	0.995098i	$-0.531531\pi$
$839$	−4806.78	−0.197793	−0.0988965	+	0.995098i	$0.531531\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	16440.9	0.671715
$844$	0	0
$845$	−17356.0	−0.706584
$846$	0	0
$847$	37970.1	1.54034
$848$	0	0
$849$	−7266.37	−0.293735
$850$	0	0
$851$	−32471.8	−1.30801
$852$	0	0
$853$	49150.9	1.97291	0.986456	−	0.164026i	$-0.0524481\pi$
$853$	49150.9	1.97291	0.986456	+	0.164026i	$0.0524481\pi$
$854$	0	0
$855$	7369.52	0.294774
$856$	0	0
$857$	44886.9	1.78916	0.894578	−	0.446911i	$-0.147476\pi$
$857$	44886.9	1.78916	0.894578	+	0.446911i	$0.147476\pi$
$858$	0	0
$859$	28327.3	1.12516	0.562582	−	0.826742i	$-0.309808\pi$
$859$	28327.3	1.12516	0.562582	+	0.826742i	$0.309808\pi$
$860$	0	0
$861$	17847.6	0.706438
$862$	0	0
$863$	−28693.4	−1.13179	−0.565895	−	0.824477i	$-0.691469\pi$
$863$	−28693.4	−1.13179	−0.565895	+	0.824477i	$0.691469\pi$
$864$	0	0
$865$	−1925.93	−0.0757036
$866$	0	0
$867$	−13432.2	−0.526160
$868$	0	0
$869$	−5725.70	−0.223511
$870$	0	0
$871$	28444.9	1.10657
$872$	0	0
$873$	241.258	0.00935322
$874$	0	0
$875$	−26643.6	−1.02939
$876$	0	0
$877$	−29417.1	−1.13266	−0.566332	−	0.824177i	$-0.691638\pi$
$877$	−29417.1	−1.13266	−0.566332	+	0.824177i	$0.691638\pi$
$878$	0	0
$879$	−4605.71	−0.176731
$880$	0	0
$881$	−6154.67	−0.235365	−0.117682	−	0.993051i	$-0.537546\pi$
$881$	−6154.67	−0.235365	−0.117682	+	0.993051i	$0.537546\pi$
$882$	0	0
$883$	−28371.7	−1.08130	−0.540648	−	0.841249i	$-0.681821\pi$
$883$	−28371.7	−1.08130	−0.540648	+	0.841249i	$0.681821\pi$
$884$	0	0
$885$	−1448.67	−0.0550243
$886$	0	0
$887$	−36685.3	−1.38869	−0.694346	−	0.719641i	$-0.744306\pi$
$887$	−36685.3	−1.38869	−0.694346	+	0.719641i	$0.744306\pi$
$888$	0	0
$889$	28195.9	1.06373
$890$	0	0
$891$	−4322.90	−0.162539
$892$	0	0
$893$	−15633.6	−0.585845
$894$	0	0
$895$	−2981.69	−0.111360
$896$	0	0
$897$	28096.2	1.04582
$898$	0	0
$899$	−6492.61	−0.240868
$900$	0	0
$901$	62883.8	2.32515
$902$	0	0
$903$	−1544.76	−0.0569285
$904$	0	0
$905$	−7017.92	−0.257772
$906$	0	0
$907$	38189.3	1.39808	0.699039	−	0.715084i	$-0.253612\pi$
$907$	38189.3	1.39808	0.699039	+	0.715084i	$0.253612\pi$
$908$	0	0
$909$	−32910.8	−1.20086
$910$	0	0
$911$	−6696.40	−0.243537	−0.121768	−	0.992559i	$-0.538856\pi$
$911$	−6696.40	−0.243537	−0.121768	+	0.992559i	$0.538856\pi$
$912$	0	0
$913$	−6619.98	−0.239966
$914$	0	0
$915$	−107.674	−0.00389028
$916$	0	0
$917$	26586.4	0.957427
$918$	0	0
$919$	−48279.1	−1.73295	−0.866474	−	0.499222i	$-0.833619\pi$
$919$	−48279.1	−1.73295	−0.866474	+	0.499222i	$0.833619\pi$
$920$	0	0
$921$	−20938.7	−0.749134
$922$	0	0
$923$	445.784	0.0158972
$924$	0	0
$925$	27385.1	0.973423
$926$	0	0
$927$	18318.4	0.649035
$928$	0	0
$929$	22375.0	0.790206	0.395103	−	0.918637i	$-0.370709\pi$
$929$	22375.0	0.790206	0.395103	+	0.918637i	$0.370709\pi$
$930$	0	0
$931$	85537.5	3.01115
$932$	0	0
$933$	−475.399	−0.0166815
$934$	0	0
$935$	4903.95	0.171526
$936$	0	0
$937$	−27929.2	−0.973754	−0.486877	−	0.873470i	$-0.661864\pi$
$937$	−27929.2	−0.973754	−0.486877	+	0.873470i	$0.661864\pi$
$938$	0	0
$939$	9011.33	0.313178
$940$	0	0
$941$	−11456.3	−0.396881	−0.198440	−	0.980113i	$-0.563588\pi$
$941$	−11456.3	−0.396881	−0.198440	+	0.980113i	$0.563588\pi$
$942$	0	0
$943$	29709.6	1.02596
$944$	0	0
$945$	12882.1	0.443444
$946$	0	0
$947$	38730.6	1.32901	0.664506	−	0.747283i	$-0.268642\pi$
$947$	38730.6	1.32901	0.664506	+	0.747283i	$0.268642\pi$
$948$	0	0
$949$	37075.9	1.26821
$950$	0	0
$951$	13289.2	0.453137
$952$	0	0
$953$	56157.0	1.90882	0.954408	−	0.298504i	$-0.0964877\pi$
$953$	56157.0	1.90882	0.954408	+	0.298504i	$0.0964877\pi$
$954$	0	0
$955$	−5627.53	−0.190683
$956$	0	0
$957$	1013.27	0.0342260
$958$	0	0
$959$	46244.2	1.55715
$960$	0	0
$961$	20332.7	0.682511
$962$	0	0
$963$	−29211.3	−0.977487
$964$	0	0
$965$	8234.34	0.274687
$966$	0	0
$967$	−7078.24	−0.235389	−0.117694	−	0.993050i	$-0.537550\pi$
$967$	−7078.24	−0.235389	−0.117694	+	0.993050i	$0.537550\pi$
$968$	0	0
$969$	−25980.7	−0.861323
$970$	0	0
$971$	49719.3	1.64322	0.821610	−	0.570049i	$-0.193076\pi$
$971$	49719.3	1.64322	0.821610	+	0.570049i	$0.193076\pi$
$972$	0	0
$973$	−29696.0	−0.978427
$974$	0	0
$975$	−23694.9	−0.778302
$976$	0	0
$977$	41123.3	1.34662	0.673312	−	0.739358i	$-0.264871\pi$
$977$	41123.3	1.34662	0.673312	+	0.739358i	$0.264871\pi$
$978$	0	0
$979$	11834.6	0.386349
$980$	0	0
$981$	9936.60	0.323396
$982$	0	0
$983$	13824.2	0.448549	0.224275	−	0.974526i	$-0.427999\pi$
$983$	13824.2	0.448549	0.224275	+	0.974526i	$0.427999\pi$
$984$	0	0
$985$	2259.43	0.0730876
$986$	0	0
$987$	−12036.8	−0.388183
$988$	0	0
$989$	−2571.46	−0.0826770
$990$	0	0
$991$	13606.6	0.436154	0.218077	−	0.975932i	$-0.430022\pi$
$991$	13606.6	0.436154	0.218077	+	0.975932i	$0.430022\pi$
$992$	0	0
$993$	−6894.02	−0.220317
$994$	0	0
$995$	3629.43	0.115639
$996$	0	0
$997$	−41867.4	−1.32995	−0.664973	−	0.746868i	$-0.731557\pi$
$997$	−41867.4	−1.32995	−0.664973	+	0.746868i	$0.731557\pi$
$998$	0	0
$999$	−27728.7	−0.878177

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	464.4.a.i.1.2		3
4.3	odd	2		58.4.a.d.1.2	✓	3
8.3	odd	2		1856.4.a.r.1.2		3
8.5	even	2		1856.4.a.s.1.2		3
12.11	even	2		522.4.a.k.1.3		3
20.19	odd	2		1450.4.a.h.1.2		3
116.115	odd	2		1682.4.a.d.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
58.4.a.d.1.2	✓	3	4.3	odd	2
464.4.a.i.1.2		3	1.1	even	1	trivial
522.4.a.k.1.3		3	12.11	even	2
1450.4.a.h.1.2		3	20.19	odd	2
1682.4.a.d.1.2		3	116.115	odd	2
1856.4.a.r.1.2		3	8.3	odd	2
1856.4.a.s.1.2		3	8.5	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:	\( N \)	\(=\)	\( 464 = 2^{4} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	464.a (trivial)

\(f(q)\)	\(=\)	\(q-2.39712 q^{3} -3.28077 q^{5} -33.9461 q^{7} -21.2538 q^{9} +O(q^{10})\)	\(q-2.39712 q^{3} -3.28077 q^{5} -33.9461 q^{7} -21.2538 q^{9} -14.5759 q^{11} -86.5287 q^{13} +7.86439 q^{15} +102.550 q^{17} +105.688 q^{19} +81.3729 q^{21} +135.456 q^{23} -114.237 q^{25} +115.670 q^{27} +29.0000 q^{29} -223.883 q^{31} +34.9403 q^{33} +111.369 q^{35} -239.723 q^{37} +207.420 q^{39} +219.331 q^{41} -18.9838 q^{43} +69.7288 q^{45} -147.922 q^{47} +809.338 q^{49} -245.824 q^{51} +613.202 q^{53} +47.8202 q^{55} -253.348 q^{57} -184.206 q^{59} -13.6914 q^{61} +721.484 q^{63} +283.880 q^{65} -328.733 q^{67} -324.703 q^{69} -5.15186 q^{71} -428.481 q^{73} +273.839 q^{75} +494.796 q^{77} +392.819 q^{79} +296.578 q^{81} +454.172 q^{83} -336.442 q^{85} -69.5165 q^{87} -811.929 q^{89} +2937.31 q^{91} +536.675 q^{93} -346.738 q^{95} -11.3513 q^{97} +309.794 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{3} + 20 q^{5} - 24 q^{7} + 5 q^{9}+O(q^{10}) \) 3 * q - 2 * q^3 + 20 * q^5 - 24 * q^7 + 5 * q^9	\( 3 q - 2 q^{3} + 20 q^{5} - 24 q^{7} + 5 q^{9} - 10 q^{11} - 4 q^{13} + 130 q^{15} - 66 q^{17} + 164 q^{19} - 88 q^{21} + 204 q^{23} + 79 q^{25} + 142 q^{27} + 87 q^{29} + 86 q^{31} - 130 q^{33} - 24 q^{35} - 42 q^{37} + 394 q^{39} + 562 q^{41} - 18 q^{43} + 422 q^{45} - 654 q^{47} + 539 q^{49} - 556 q^{51} + 712 q^{53} - 142 q^{55} + 828 q^{57} - 184 q^{59} + 322 q^{61} + 784 q^{63} + 1494 q^{65} + 228 q^{67} + 684 q^{69} + 52 q^{71} - 494 q^{73} + 3048 q^{75} + 872 q^{77} + 2110 q^{79} - 1513 q^{81} + 288 q^{83} - 2704 q^{85} - 58 q^{87} + 914 q^{89} + 2984 q^{91} - 62 q^{93} + 1900 q^{95} + 218 q^{97} + 304 q^{99}+O(q^{100}) \) 3 * q - 2 * q^3 + 20 * q^5 - 24 * q^7 + 5 * q^9 - 10 * q^11 - 4 * q^13 + 130 * q^15 - 66 * q^17 + 164 * q^19 - 88 * q^21 + 204 * q^23 + 79 * q^25 + 142 * q^27 + 87 * q^29 + 86 * q^31 - 130 * q^33 - 24 * q^35 - 42 * q^37 + 394 * q^39 + 562 * q^41 - 18 * q^43 + 422 * q^45 - 654 * q^47 + 539 * q^49 - 556 * q^51 + 712 * q^53 - 142 * q^55 + 828 * q^57 - 184 * q^59 + 322 * q^61 + 784 * q^63 + 1494 * q^65 + 228 * q^67 + 684 * q^69 + 52 * q^71 - 494 * q^73 + 3048 * q^75 + 872 * q^77 + 2110 * q^79 - 1513 * q^81 + 288 * q^83 - 2704 * q^85 - 58 * q^87 + 914 * q^89 + 2984 * q^91 - 62 * q^93 + 1900 * q^95 + 218 * q^97 + 304 * q^99

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