LMFDB - Embedded newform 648.4.a.f.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [648,4,Mod(1,648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(648, 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("648.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 648 = 2^{3} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	648.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:	$38.2332376837$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{201}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 50 $ x^2 - x - 50
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$7.58872$ of defining polynomial
Character	$\chi$	$=$	648.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+18.1774 q^{5} -0.822553 q^{7} +O(q^{10})$	$q+18.1774 q^{5} -0.822553 q^{7} -65.5323 q^{11} +31.8226 q^{13} +124.420 q^{17} +27.8872 q^{19} -43.1774 q^{23} +205.420 q^{25} +39.8226 q^{29} +294.839 q^{31} -14.9519 q^{35} -104.177 q^{37} +307.645 q^{41} -361.081 q^{43} +397.645 q^{47} -342.323 q^{49} +107.161 q^{53} -1191.21 q^{55} +188.935 q^{59} -99.0166 q^{61} +578.453 q^{65} +425.048 q^{67} -445.532 q^{71} -499.806 q^{73} +53.9038 q^{77} +570.275 q^{79} +1308.74 q^{83} +2261.63 q^{85} +1323.00 q^{89} -26.1757 q^{91} +506.919 q^{95} +944.872 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{5} - 30 q^{7}+O(q^{10}) $ 2 * q + 8 * q^5 - 30 * q^7	$ 2 q + 8 q^{5} - 30 q^{7} - 46 q^{11} + 92 q^{13} + 22 q^{17} - 86 q^{19} - 58 q^{23} + 184 q^{25} + 108 q^{29} + 136 q^{31} + 282 q^{35} - 180 q^{37} + 672 q^{41} - 70 q^{43} + 852 q^{47} + 166 q^{49} + 668 q^{53} - 1390 q^{55} + 548 q^{59} + 284 q^{61} - 34 q^{65} + 1162 q^{67} - 806 q^{71} - 1510 q^{73} - 516 q^{77} - 22 q^{79} + 1540 q^{83} + 3304 q^{85} + 2646 q^{89} - 1782 q^{91} + 1666 q^{95} + 472 q^{97}+O(q^{100}) $ 2 * q + 8 * q^5 - 30 * q^7 - 46 * q^11 + 92 * q^13 + 22 * q^17 - 86 * q^19 - 58 * q^23 + 184 * q^25 + 108 * q^29 + 136 * q^31 + 282 * q^35 - 180 * q^37 + 672 * q^41 - 70 * q^43 + 852 * q^47 + 166 * q^49 + 668 * q^53 - 1390 * q^55 + 548 * q^59 + 284 * q^61 - 34 * q^65 + 1162 * q^67 - 806 * q^71 - 1510 * q^73 - 516 * q^77 - 22 * q^79 + 1540 * q^83 + 3304 * q^85 + 2646 * q^89 - 1782 * q^91 + 1666 * q^95 + 472 * q^97

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	0
$3$	0	0
$4$	0	0
$5$	18.1774	1.62584	0.812920	−	0.582375i	$-0.197877\pi$
$5$	18.1774	1.62584	0.812920	+	0.582375i	$0.197877\pi$
$6$	0	0
$7$	−0.822553	−0.0444137	−0.0222068	−	0.999753i	$-0.507069\pi$
$7$	−0.822553	−0.0444137	−0.0222068	+	0.999753i	$0.507069\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−65.5323	−1.79625	−0.898125	−	0.439741i	$-0.855070\pi$
$11$	−65.5323	−1.79625	−0.898125	+	0.439741i	$0.855070\pi$
$12$	0	0
$13$	31.8226	0.678922	0.339461	−	0.940620i	$-0.389755\pi$
$13$	31.8226	0.678922	0.339461	+	0.940620i	$0.389755\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	124.420	1.77507	0.887535	−	0.460741i	$-0.152416\pi$
$17$	124.420	1.77507	0.887535	+	0.460741i	$0.152416\pi$
$18$	0	0
$19$	27.8872	0.336725	0.168362	−	0.985725i	$-0.446152\pi$
$19$	27.8872	0.336725	0.168362	+	0.985725i	$0.446152\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−43.1774	−0.391440	−0.195720	−	0.980660i	$-0.562704\pi$
$23$	−43.1774	−0.391440	−0.195720	+	0.980660i	$0.562704\pi$
$24$	0	0
$25$	205.420	1.64336
$26$	0	0
$27$	0	0
$28$	0	0
$29$	39.8226	0.254995	0.127498	−	0.991839i	$-0.459305\pi$
$29$	39.8226	0.254995	0.127498	+	0.991839i	$0.459305\pi$
$30$	0	0
$31$	294.839	1.70822	0.854108	−	0.520096i	$-0.174104\pi$
$31$	294.839	1.70822	0.854108	+	0.520096i	$0.174104\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−14.9519	−0.0722096
$36$	0	0
$37$	−104.177	−0.462883	−0.231441	−	0.972849i	$-0.574344\pi$
$37$	−104.177	−0.462883	−0.231441	+	0.972849i	$0.574344\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	307.645	1.17186	0.585928	−	0.810363i	$-0.300730\pi$
$41$	307.645	1.17186	0.585928	+	0.810363i	$0.300730\pi$
$42$	0	0
$43$	−361.081	−1.28057	−0.640283	−	0.768139i	$-0.721183\pi$
$43$	−361.081	−1.28057	−0.640283	+	0.768139i	$0.721183\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	397.645	1.23410	0.617048	−	0.786926i	$-0.288328\pi$
$47$	397.645	1.23410	0.617048	+	0.786926i	$0.288328\pi$
$48$	0	0
$49$	−342.323	−0.998027
$50$	0	0
$51$	0	0
$52$	0	0
$53$	107.161	0.277730	0.138865	−	0.990311i	$-0.455655\pi$
$53$	107.161	0.277730	0.138865	+	0.990311i	$0.455655\pi$
$54$	0	0
$55$	−1191.21	−2.92041
$56$	0	0
$57$	0	0
$58$	0	0
$59$	188.935	0.416903	0.208452	−	0.978033i	$-0.433158\pi$
$59$	188.935	0.416903	0.208452	+	0.978033i	$0.433158\pi$
$60$	0	0
$61$	−99.0166	−0.207832	−0.103916	−	0.994586i	$-0.533137\pi$
$61$	−99.0166	−0.207832	−0.103916	+	0.994586i	$0.533137\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	578.453	1.10382
$66$	0	0
$67$	425.048	0.775043	0.387522	−	0.921861i	$-0.373331\pi$
$67$	425.048	0.775043	0.387522	+	0.921861i	$0.373331\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−445.532	−0.744718	−0.372359	−	0.928089i	$-0.621451\pi$
$71$	−445.532	−0.744718	−0.372359	+	0.928089i	$0.621451\pi$
$72$	0	0
$73$	−499.806	−0.801341	−0.400670	−	0.916222i	$-0.631223\pi$
$73$	−499.806	−0.801341	−0.400670	+	0.916222i	$0.631223\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	53.9038	0.0797781
$78$	0	0
$79$	570.275	0.812164	0.406082	−	0.913837i	$-0.366895\pi$
$79$	570.275	0.812164	0.406082	+	0.913837i	$0.366895\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1308.74	1.73076	0.865381	−	0.501115i	$-0.167077\pi$
$83$	1308.74	1.73076	0.865381	+	0.501115i	$0.167077\pi$
$84$	0	0
$85$	2261.63	2.88598
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1323.00	1.57570	0.787852	−	0.615864i	$-0.211193\pi$
$89$	1323.00	1.57570	0.787852	+	0.615864i	$0.211193\pi$
$90$	0	0
$91$	−26.1757	−0.0301534
$92$	0	0
$93$	0	0
$94$	0	0
$95$	506.919	0.547461
$96$	0	0
$97$	944.872	0.989044	0.494522	−	0.869165i	$-0.335343\pi$
$97$	944.872	0.989044	0.494522	+	0.869165i	$0.335343\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1398.10	−1.37739	−0.688694	−	0.725052i	$-0.741815\pi$
$101$	−1398.10	−1.37739	−0.688694	+	0.725052i	$0.741815\pi$
$102$	0	0
$103$	−960.551	−0.918892	−0.459446	−	0.888206i	$-0.651952\pi$
$103$	−960.551	−0.918892	−0.459446	+	0.888206i	$0.651952\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1935.87	1.74905	0.874523	−	0.484983i	$-0.161174\pi$
$107$	1935.87	1.74905	0.874523	+	0.484983i	$0.161174\pi$
$108$	0	0
$109$	−1305.40	−1.14711	−0.573555	−	0.819167i	$-0.694436\pi$
$109$	−1305.40	−1.14711	−0.573555	+	0.819167i	$0.694436\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1094.03	0.910777	0.455389	−	0.890293i	$-0.349500\pi$
$113$	1094.03	0.910777	0.455389	+	0.890293i	$0.349500\pi$
$114$	0	0
$115$	−784.856	−0.636419
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−102.342	−0.0788374
$120$	0	0
$121$	2963.49	2.22651
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1461.82	1.04600
$126$	0	0
$127$	−1301.05	−0.909050	−0.454525	−	0.890734i	$-0.650191\pi$
$127$	−1301.05	−0.909050	−0.454525	+	0.890734i	$0.650191\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1957.67	1.30566	0.652832	−	0.757503i	$-0.273581\pi$
$131$	1957.67	1.30566	0.652832	+	0.757503i	$0.273581\pi$
$132$	0	0
$133$	−22.9387	−0.0149552
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2553.97	−1.59270	−0.796351	−	0.604834i	$-0.793239\pi$
$137$	−2553.97	−1.59270	−0.796351	+	0.604834i	$0.793239\pi$
$138$	0	0
$139$	205.260	0.125252	0.0626258	−	0.998037i	$-0.480053\pi$
$139$	205.260	0.125252	0.0626258	+	0.998037i	$0.480053\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2085.41	−1.21951
$144$	0	0
$145$	723.872	0.414582
$146$	0	0
$147$	0	0
$148$	0	0
$149$	164.240	0.0903027	0.0451513	−	0.998980i	$-0.485623\pi$
$149$	164.240	0.0903027	0.0451513	+	0.998980i	$0.485623\pi$
$150$	0	0
$151$	3427.71	1.84731	0.923654	−	0.383228i	$-0.125188\pi$
$151$	3427.71	1.84731	0.923654	+	0.383228i	$0.125188\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5359.42	2.77729
$156$	0	0
$157$	−988.085	−0.502279	−0.251139	−	0.967951i	$-0.580805\pi$
$157$	−988.085	−0.502279	−0.251139	+	0.967951i	$0.580805\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	35.5157	0.0173853
$162$	0	0
$163$	−2845.45	−1.36732	−0.683660	−	0.729801i	$-0.739613\pi$
$163$	−2845.45	−1.36732	−0.683660	+	0.729801i	$0.739613\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3205.89	−1.48550	−0.742752	−	0.669566i	$-0.766480\pi$
$167$	−3205.89	−1.48550	−0.742752	+	0.669566i	$0.766480\pi$
$168$	0	0
$169$	−1184.33	−0.539065
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1785.63	0.784735	0.392368	−	0.919808i	$-0.371656\pi$
$173$	1785.63	0.784735	0.392368	+	0.919808i	$0.371656\pi$
$174$	0	0
$175$	−168.969	−0.0729875
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1318.84	−0.550696	−0.275348	−	0.961345i	$-0.588793\pi$
$179$	−1318.84	−0.550696	−0.275348	+	0.961345i	$0.588793\pi$
$180$	0	0
$181$	3462.97	1.42210	0.711051	−	0.703141i	$-0.248220\pi$
$181$	3462.97	1.42210	0.711051	+	0.703141i	$0.248220\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1893.68	−0.752574
$186$	0	0
$187$	−8153.51	−3.18847
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1591.31	0.602844	0.301422	−	0.953491i	$-0.402539\pi$
$191$	1591.31	0.602844	0.301422	+	0.953491i	$0.402539\pi$
$192$	0	0
$193$	−764.456	−0.285113	−0.142556	−	0.989787i	$-0.545532\pi$
$193$	−764.456	−0.285113	−0.142556	+	0.989787i	$0.545532\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1191.63	0.430965	0.215483	−	0.976508i	$-0.430868\pi$
$197$	1191.63	0.430965	0.215483	+	0.976508i	$0.430868\pi$
$198$	0	0
$199$	−2708.20	−0.964719	−0.482360	−	0.875973i	$-0.660220\pi$
$199$	−2708.20	−0.964719	−0.482360	+	0.875973i	$0.660220\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−32.7562	−0.0113253
$204$	0	0
$205$	5592.20	1.90525
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1827.52	−0.604842
$210$	0	0
$211$	−3298.95	−1.07635	−0.538173	−	0.842834i	$-0.680885\pi$
$211$	−3298.95	−1.07635	−0.538173	+	0.842834i	$0.680885\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−6563.54	−2.08200
$216$	0	0
$217$	−242.521	−0.0758682
$218$	0	0
$219$	0	0
$220$	0	0
$221$	3959.35	1.20513
$222$	0	0
$223$	−2521.67	−0.757234	−0.378617	−	0.925553i	$-0.623600\pi$
$223$	−2521.67	−0.757234	−0.378617	+	0.925553i	$0.623600\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3314.67	−0.969173	−0.484586	−	0.874743i	$-0.661030\pi$
$227$	−3314.67	−0.969173	−0.484586	+	0.874743i	$0.661030\pi$
$228$	0	0
$229$	−4129.66	−1.19168	−0.595842	−	0.803101i	$-0.703182\pi$
$229$	−4129.66	−1.19168	−0.595842	+	0.803101i	$0.703182\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	986.257	0.277304	0.138652	−	0.990341i	$-0.455723\pi$
$233$	986.257	0.277304	0.138652	+	0.990341i	$0.455723\pi$
$234$	0	0
$235$	7228.17	2.00644
$236$	0	0
$237$	0	0
$238$	0	0
$239$	358.998	0.0971618	0.0485809	−	0.998819i	$-0.484530\pi$
$239$	358.998	0.0971618	0.0485809	+	0.998819i	$0.484530\pi$
$240$	0	0
$241$	2042.03	0.545804	0.272902	−	0.962042i	$-0.412016\pi$
$241$	2042.03	0.545804	0.272902	+	0.962042i	$0.412016\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−6222.57	−1.62263
$246$	0	0
$247$	887.443	0.228610
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−2609.69	−0.656264	−0.328132	−	0.944632i	$-0.606419\pi$
$251$	−2609.69	−0.656264	−0.328132	+	0.944632i	$0.606419\pi$
$252$	0	0
$253$	2829.52	0.703124
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−878.837	−0.213309	−0.106654	−	0.994296i	$-0.534014\pi$
$257$	−878.837	−0.213309	−0.106654	+	0.994296i	$0.534014\pi$
$258$	0	0
$259$	85.6915	0.0205583
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−3217.33	−0.754332	−0.377166	−	0.926146i	$-0.623101\pi$
$263$	−3217.33	−0.754332	−0.377166	+	0.926146i	$0.623101\pi$
$264$	0	0
$265$	1947.91	0.451544
$266$	0	0
$267$	0	0
$268$	0	0
$269$	7978.38	1.80837	0.904183	−	0.427146i	$-0.140481\pi$
$269$	7978.38	1.80837	0.904183	+	0.427146i	$0.140481\pi$
$270$	0	0
$271$	4248.21	0.952251	0.476126	−	0.879377i	$-0.342041\pi$
$271$	4248.21	0.952251	0.476126	+	0.879377i	$0.342041\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−13461.6	−2.95188
$276$	0	0
$277$	831.035	0.180260	0.0901301	−	0.995930i	$-0.471272\pi$
$277$	831.035	0.180260	0.0901301	+	0.995930i	$0.471272\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−2962.00	−0.628819	−0.314410	−	0.949287i	$-0.601807\pi$
$281$	−2962.00	−0.628819	−0.314410	+	0.949287i	$0.601807\pi$
$282$	0	0
$283$	−4430.84	−0.930693	−0.465346	−	0.885129i	$-0.654070\pi$
$283$	−4430.84	−0.930693	−0.465346	+	0.885129i	$0.654070\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−253.054	−0.0520465
$288$	0	0
$289$	10567.2	2.15087
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−3422.86	−0.682477	−0.341239	−	0.939977i	$-0.610846\pi$
$293$	−3422.86	−0.682477	−0.341239	+	0.939977i	$0.610846\pi$
$294$	0	0
$295$	3434.36	0.677818
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1374.02	−0.265757
$300$	0	0
$301$	297.009	0.0568747
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1799.87	−0.337902
$306$	0	0
$307$	139.734	0.0259774	0.0129887	−	0.999916i	$-0.495865\pi$
$307$	139.734	0.0259774	0.0129887	+	0.999916i	$0.495865\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	7599.88	1.38569	0.692845	−	0.721086i	$-0.256357\pi$
$311$	7599.88	1.38569	0.692845	+	0.721086i	$0.256357\pi$
$312$	0	0
$313$	−3017.52	−0.544921	−0.272460	−	0.962167i	$-0.587837\pi$
$313$	−3017.52	−0.544921	−0.272460	+	0.962167i	$0.587837\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1921.21	−0.340396	−0.170198	−	0.985410i	$-0.554441\pi$
$317$	−1921.21	−0.340396	−0.170198	+	0.985410i	$0.554441\pi$
$318$	0	0
$319$	−2609.67	−0.458035
$320$	0	0
$321$	0	0
$322$	0	0
$323$	3469.72	0.597710
$324$	0	0
$325$	6536.98	1.11571
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−327.084	−0.0548108
$330$	0	0
$331$	912.412	0.151513	0.0757563	−	0.997126i	$-0.475863\pi$
$331$	912.412	0.151513	0.0757563	+	0.997126i	$0.475863\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	7726.29	1.26010
$336$	0	0
$337$	−6014.62	−0.972218	−0.486109	−	0.873898i	$-0.661584\pi$
$337$	−6014.62	−0.972218	−0.486109	+	0.873898i	$0.661584\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−19321.5	−3.06838
$342$	0	0
$343$	563.715	0.0887398
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2739.83	0.423867	0.211934	−	0.977284i	$-0.432024\pi$
$347$	2739.83	0.423867	0.211934	+	0.977284i	$0.432024\pi$
$348$	0	0
$349$	−7608.69	−1.16700	−0.583501	−	0.812112i	$-0.698318\pi$
$349$	−7608.69	−1.16700	−0.583501	+	0.812112i	$0.698318\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−8725.11	−1.31555	−0.657777	−	0.753212i	$-0.728503\pi$
$353$	−8725.11	−1.31555	−0.657777	+	0.753212i	$0.728503\pi$
$354$	0	0
$355$	−8098.64	−1.21079
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−10470.0	−1.53923	−0.769614	−	0.638509i	$-0.779551\pi$
$359$	−10470.0	−1.53923	−0.769614	+	0.638509i	$0.779551\pi$
$360$	0	0
$361$	−6081.30	−0.886616
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−9085.20	−1.30285
$366$	0	0
$367$	−7894.14	−1.12281	−0.561405	−	0.827542i	$-0.689739\pi$
$367$	−7894.14	−1.12281	−0.561405	+	0.827542i	$0.689739\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−88.1455	−0.0123350
$372$	0	0
$373$	−269.940	−0.0374718	−0.0187359	−	0.999824i	$-0.505964\pi$
$373$	−269.940	−0.0374718	−0.0187359	+	0.999824i	$0.505964\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1267.26	0.173122
$378$	0	0
$379$	−8580.72	−1.16296	−0.581480	−	0.813561i	$-0.697526\pi$
$379$	−8580.72	−1.16296	−0.581480	+	0.813561i	$0.697526\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	3414.93	0.455599	0.227800	−	0.973708i	$-0.426847\pi$
$383$	3414.93	0.455599	0.227800	+	0.973708i	$0.426847\pi$
$384$	0	0
$385$	979.834	0.129706
$386$	0	0
$387$	0	0
$388$	0	0
$389$	3304.00	0.430642	0.215321	−	0.976543i	$-0.430920\pi$
$389$	3304.00	0.430642	0.215321	+	0.976543i	$0.430920\pi$
$390$	0	0
$391$	−5372.12	−0.694833
$392$	0	0
$393$	0	0
$394$	0	0
$395$	10366.1	1.32045
$396$	0	0
$397$	−12756.1	−1.61262	−0.806311	−	0.591492i	$-0.798539\pi$
$397$	−12756.1	−1.61262	−0.806311	+	0.591492i	$0.798539\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−4628.52	−0.576402	−0.288201	−	0.957570i	$-0.593057\pi$
$401$	−4628.52	−0.576402	−0.288201	+	0.957570i	$0.593057\pi$
$402$	0	0
$403$	9382.53	1.15975
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6826.99	0.831453
$408$	0	0
$409$	−2965.55	−0.358526	−0.179263	−	0.983801i	$-0.557371\pi$
$409$	−2965.55	−0.358526	−0.179263	+	0.983801i	$0.557371\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−155.409	−0.0185162
$414$	0	0
$415$	23789.6	2.81394
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1870.26	−0.218063	−0.109031	−	0.994038i	$-0.534775\pi$
$419$	−1870.26	−0.218063	−0.109031	+	0.994038i	$0.534775\pi$
$420$	0	0
$421$	−12650.6	−1.46450	−0.732248	−	0.681038i	$-0.761529\pi$
$421$	−12650.6	−1.46450	−0.732248	+	0.681038i	$0.761529\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	25558.2	2.91707
$426$	0	0
$427$	81.4464	0.00923060
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3266.47	0.365059	0.182530	−	0.983200i	$-0.441571\pi$
$431$	3266.47	0.365059	0.182530	+	0.983200i	$0.441571\pi$
$432$	0	0
$433$	6788.95	0.753478	0.376739	−	0.926319i	$-0.377045\pi$
$433$	6788.95	0.753478	0.376739	+	0.926319i	$0.377045\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1204.10	−0.131808
$438$	0	0
$439$	4918.55	0.534737	0.267368	−	0.963594i	$-0.413846\pi$
$439$	4918.55	0.534737	0.267368	+	0.963594i	$0.413846\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−14033.5	−1.50508	−0.752540	−	0.658546i	$-0.771172\pi$
$443$	−14033.5	−1.50508	−0.752540	+	0.658546i	$0.771172\pi$
$444$	0	0
$445$	24048.8	2.56184
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−5374.04	−0.564848	−0.282424	−	0.959290i	$-0.591138\pi$
$449$	−5374.04	−0.564848	−0.282424	+	0.959290i	$0.591138\pi$
$450$	0	0
$451$	−20160.7	−2.10495
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−475.808	−0.0490247
$456$	0	0
$457$	12169.2	1.24563	0.622815	−	0.782369i	$-0.285989\pi$
$457$	12169.2	1.24563	0.622815	+	0.782369i	$0.285989\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−6186.08	−0.624977	−0.312489	−	0.949922i	$-0.601163\pi$
$461$	−6186.08	−0.624977	−0.312489	+	0.949922i	$0.601163\pi$
$462$	0	0
$463$	15197.8	1.52549	0.762743	−	0.646701i	$-0.223852\pi$
$463$	15197.8	1.52549	0.762743	+	0.646701i	$0.223852\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1981.82	0.196377	0.0981883	−	0.995168i	$-0.468695\pi$
$467$	1981.82	0.196377	0.0981883	+	0.995168i	$0.468695\pi$
$468$	0	0
$469$	−349.625	−0.0344225
$470$	0	0
$471$	0	0
$472$	0	0
$473$	23662.5	2.30022
$474$	0	0
$475$	5728.58	0.553359
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−9779.52	−0.932855	−0.466427	−	0.884559i	$-0.654459\pi$
$479$	−9779.52	−0.932855	−0.466427	+	0.884559i	$0.654459\pi$
$480$	0	0
$481$	−3315.19	−0.314261
$482$	0	0
$483$	0	0
$484$	0	0
$485$	17175.4	1.60803
$486$	0	0
$487$	18668.9	1.73710	0.868549	−	0.495603i	$-0.165053\pi$
$487$	18668.9	1.73710	0.868549	+	0.495603i	$0.165053\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−16006.2	−1.47118	−0.735591	−	0.677426i	$-0.763095\pi$
$491$	−16006.2	−1.47118	−0.735591	+	0.677426i	$0.763095\pi$
$492$	0	0
$493$	4954.71	0.452634
$494$	0	0
$495$	0	0
$496$	0	0
$497$	366.474	0.0330757
$498$	0	0
$499$	9860.57	0.884609	0.442304	−	0.896865i	$-0.354161\pi$
$499$	9860.57	0.884609	0.442304	+	0.896865i	$0.354161\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	10512.0	0.931825	0.465913	−	0.884831i	$-0.345726\pi$
$503$	10512.0	0.931825	0.465913	+	0.884831i	$0.345726\pi$
$504$	0	0
$505$	−25413.9	−2.23941
$506$	0	0
$507$	0	0
$508$	0	0
$509$	16667.3	1.45140	0.725700	−	0.688011i	$-0.241516\pi$
$509$	16667.3	1.45140	0.725700	+	0.688011i	$0.241516\pi$
$510$	0	0
$511$	411.117	0.0355905
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−17460.4	−1.49397
$516$	0	0
$517$	−26058.6	−2.21674
$518$	0	0
$519$	0	0
$520$	0	0
$521$	10923.9	0.918586	0.459293	−	0.888285i	$-0.348103\pi$
$521$	10923.9	0.918586	0.459293	+	0.888285i	$0.348103\pi$
$522$	0	0
$523$	−17488.0	−1.46213	−0.731067	−	0.682305i	$-0.760978\pi$
$523$	−17488.0	−1.46213	−0.731067	+	0.682305i	$0.760978\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	36683.8	3.03220
$528$	0	0
$529$	−10302.7	−0.846775
$530$	0	0
$531$	0	0
$532$	0	0
$533$	9790.05	0.795599
$534$	0	0
$535$	35189.2	2.84367
$536$	0	0
$537$	0	0
$538$	0	0
$539$	22433.3	1.79271
$540$	0	0
$541$	−8724.14	−0.693309	−0.346655	−	0.937993i	$-0.612682\pi$
$541$	−8724.14	−0.693309	−0.346655	+	0.937993i	$0.612682\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−23728.9	−1.86502
$546$	0	0
$547$	−959.834	−0.0750266	−0.0375133	−	0.999296i	$-0.511944\pi$
$547$	−959.834	−0.0750266	−0.0375133	+	0.999296i	$0.511944\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1110.54	0.0858632
$552$	0	0
$553$	−469.082	−0.0360712
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−14212.0	−1.08112	−0.540558	−	0.841307i	$-0.681787\pi$
$557$	−14212.0	−1.08112	−0.540558	+	0.841307i	$0.681787\pi$
$558$	0	0
$559$	−11490.5	−0.869405
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−7110.56	−0.532281	−0.266141	−	0.963934i	$-0.585749\pi$
$563$	−7110.56	−0.532281	−0.266141	+	0.963934i	$0.585749\pi$
$564$	0	0
$565$	19886.7	1.48078
$566$	0	0
$567$	0	0
$568$	0	0
$569$	4859.18	0.358010	0.179005	−	0.983848i	$-0.442712\pi$
$569$	4859.18	0.358010	0.179005	+	0.983848i	$0.442712\pi$
$570$	0	0
$571$	8580.29	0.628851	0.314425	−	0.949282i	$-0.398188\pi$
$571$	8580.29	0.628851	0.314425	+	0.949282i	$0.398188\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−8869.49	−0.643276
$576$	0	0
$577$	−1029.99	−0.0743136	−0.0371568	−	0.999309i	$-0.511830\pi$
$577$	−1029.99	−0.0743136	−0.0371568	+	0.999309i	$0.511830\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1076.51	−0.0768695
$582$	0	0
$583$	−7022.50	−0.498872
$584$	0	0
$585$	0	0
$586$	0	0
$587$	15541.4	1.09278	0.546391	−	0.837530i	$-0.316001\pi$
$587$	15541.4	1.09278	0.546391	+	0.837530i	$0.316001\pi$
$588$	0	0
$589$	8222.25	0.575198
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−12643.5	−0.875562	−0.437781	−	0.899082i	$-0.644235\pi$
$593$	−12643.5	−0.875562	−0.437781	+	0.899082i	$0.644235\pi$
$594$	0	0
$595$	−1860.31	−0.128177
$596$	0	0
$597$	0	0
$598$	0	0
$599$	17162.9	1.17071	0.585357	−	0.810776i	$-0.300954\pi$
$599$	17162.9	1.17071	0.585357	+	0.810776i	$0.300954\pi$
$600$	0	0
$601$	−4010.19	−0.272178	−0.136089	−	0.990697i	$-0.543453\pi$
$601$	−4010.19	−0.272178	−0.136089	+	0.990697i	$0.543453\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	53868.6	3.61995
$606$	0	0
$607$	17321.2	1.15823	0.579114	−	0.815247i	$-0.303399\pi$
$607$	17321.2	1.15823	0.579114	+	0.815247i	$0.303399\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	12654.1	0.837855
$612$	0	0
$613$	2474.79	0.163060	0.0815300	−	0.996671i	$-0.474019\pi$
$613$	2474.79	0.163060	0.0815300	+	0.996671i	$0.474019\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	11844.4	0.772831	0.386416	−	0.922325i	$-0.373713\pi$
$617$	11844.4	0.772831	0.386416	+	0.922325i	$0.373713\pi$
$618$	0	0
$619$	−7848.64	−0.509634	−0.254817	−	0.966989i	$-0.582015\pi$
$619$	−7848.64	−0.509634	−0.254817	+	0.966989i	$0.582015\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−1088.24	−0.0699829
$624$	0	0
$625$	894.755	0.0572643
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−12961.7	−0.821649
$630$	0	0
$631$	−11346.0	−0.715811	−0.357906	−	0.933758i	$-0.616509\pi$
$631$	−11346.0	−0.715811	−0.357906	+	0.933758i	$0.616509\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−23649.7	−1.47797
$636$	0	0
$637$	−10893.6	−0.677583
$638$	0	0
$639$	0	0
$640$	0	0
$641$	17585.2	1.08358	0.541791	−	0.840513i	$-0.317747\pi$
$641$	17585.2	1.08358	0.541791	+	0.840513i	$0.317747\pi$
$642$	0	0
$643$	−10644.9	−0.652866	−0.326433	−	0.945220i	$-0.605847\pi$
$643$	−10644.9	−0.652866	−0.326433	+	0.945220i	$0.605847\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	7872.89	0.478385	0.239193	−	0.970972i	$-0.423117\pi$
$647$	7872.89	0.478385	0.239193	+	0.970972i	$0.423117\pi$
$648$	0	0
$649$	−12381.4	−0.748862
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−2721.08	−0.163069	−0.0815347	−	0.996671i	$-0.525982\pi$
$653$	−2721.08	−0.163069	−0.0815347	+	0.996671i	$0.525982\pi$
$654$	0	0
$655$	35585.4	2.12280
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−2000.76	−0.118268	−0.0591338	−	0.998250i	$-0.518834\pi$
$659$	−2000.76	−0.118268	−0.0591338	+	0.998250i	$0.518834\pi$
$660$	0	0
$661$	−27738.0	−1.63220	−0.816100	−	0.577911i	$-0.803868\pi$
$661$	−27738.0	−1.63220	−0.816100	+	0.577911i	$0.803868\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−416.968	−0.0243148
$666$	0	0
$667$	−1719.44	−0.0998153
$668$	0	0
$669$	0	0
$670$	0	0
$671$	6488.79	0.373319
$672$	0	0
$673$	−24693.0	−1.41433	−0.707166	−	0.707047i	$-0.750027\pi$
$673$	−24693.0	−1.41433	−0.707166	+	0.707047i	$0.750027\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−14764.7	−0.838189	−0.419095	−	0.907943i	$-0.637652\pi$
$677$	−14764.7	−0.838189	−0.419095	+	0.907943i	$0.637652\pi$
$678$	0	0
$679$	−777.208	−0.0439271
$680$	0	0
$681$	0	0
$682$	0	0
$683$	571.548	0.0320200	0.0160100	−	0.999872i	$-0.494904\pi$
$683$	571.548	0.0320200	0.0160100	+	0.999872i	$0.494904\pi$
$684$	0	0
$685$	−46424.6	−2.58948
$686$	0	0
$687$	0	0
$688$	0	0
$689$	3410.13	0.188557
$690$	0	0
$691$	−3507.58	−0.193104	−0.0965519	−	0.995328i	$-0.530781\pi$
$691$	−3507.58	−0.193104	−0.0965519	+	0.995328i	$0.530781\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3731.11	0.203639
$696$	0	0
$697$	38277.1	2.08013
$698$	0	0
$699$	0	0
$700$	0	0
$701$	21145.2	1.13929	0.569644	−	0.821891i	$-0.307081\pi$
$701$	21145.2	1.13929	0.569644	+	0.821891i	$0.307081\pi$
$702$	0	0
$703$	−2905.22	−0.155864
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1150.01	0.0611749
$708$	0	0
$709$	−15815.2	−0.837734	−0.418867	−	0.908048i	$-0.637573\pi$
$709$	−15815.2	−0.837734	−0.418867	+	0.908048i	$0.637573\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−12730.4	−0.668664
$714$	0	0
$715$	−37907.4	−1.98273
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−33309.8	−1.72774	−0.863871	−	0.503713i	$-0.831967\pi$
$719$	−33309.8	−1.72774	−0.863871	+	0.503713i	$0.831967\pi$
$720$	0	0
$721$	790.104	0.0408114
$722$	0	0
$723$	0	0
$724$	0	0
$725$	8180.33	0.419048
$726$	0	0
$727$	7037.29	0.359008	0.179504	−	0.983757i	$-0.442551\pi$
$727$	7037.29	0.359008	0.179504	+	0.983757i	$0.442551\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−44925.6	−2.27310
$732$	0	0
$733$	12661.3	0.638000	0.319000	−	0.947755i	$-0.396653\pi$
$733$	12661.3	0.638000	0.319000	+	0.947755i	$0.396653\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−27854.4	−1.39217
$738$	0	0
$739$	32709.4	1.62819	0.814096	−	0.580730i	$-0.197233\pi$
$739$	32709.4	1.62819	0.814096	+	0.580730i	$0.197233\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	29381.2	1.45073	0.725364	−	0.688365i	$-0.241671\pi$
$743$	29381.2	1.45073	0.725364	+	0.688365i	$0.241671\pi$
$744$	0	0
$745$	2985.47	0.146818
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1592.36	−0.0776816
$750$	0	0
$751$	2225.64	0.108142	0.0540712	−	0.998537i	$-0.482780\pi$
$751$	2225.64	0.108142	0.0540712	+	0.998537i	$0.482780\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	62307.1	3.00343
$756$	0	0
$757$	32628.1	1.56656	0.783281	−	0.621668i	$-0.213545\pi$
$757$	32628.1	1.56656	0.783281	+	0.621668i	$0.213545\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−18690.6	−0.890322	−0.445161	−	0.895451i	$-0.646854\pi$
$761$	−18690.6	−0.890322	−0.445161	+	0.895451i	$0.646854\pi$
$762$	0	0
$763$	1073.76	0.0509474
$764$	0	0
$765$	0	0
$766$	0	0
$767$	6012.40	0.283045
$768$	0	0
$769$	6845.67	0.321016	0.160508	−	0.987035i	$-0.448687\pi$
$769$	6845.67	0.321016	0.160508	+	0.987035i	$0.448687\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	161.873	0.00753190	0.00376595	−	0.999993i	$-0.498801\pi$
$773$	161.873	0.00753190	0.00376595	+	0.999993i	$0.498801\pi$
$774$	0	0
$775$	60565.7	2.80721
$776$	0	0
$777$	0	0
$778$	0	0
$779$	8579.37	0.394593
$780$	0	0
$781$	29196.8	1.33770
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−17960.9	−0.816625
$786$	0	0
$787$	9959.12	0.451085	0.225543	−	0.974233i	$-0.427584\pi$
$787$	9959.12	0.451085	0.225543	+	0.974233i	$0.427584\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−899.899	−0.0404510
$792$	0	0
$793$	−3150.96	−0.141102
$794$	0	0
$795$	0	0
$796$	0	0
$797$	4656.36	0.206947	0.103473	−	0.994632i	$-0.467004\pi$
$797$	4656.36	0.206947	0.103473	+	0.994632i	$0.467004\pi$
$798$	0	0
$799$	49474.8	2.19061
$800$	0	0
$801$	0	0
$802$	0	0
$803$	32753.5	1.43941
$804$	0	0
$805$	645.586	0.0282657
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−173.564	−0.00754289	−0.00377144	−	0.999993i	$-0.501200\pi$
$809$	−173.564	−0.00754289	−0.00377144	+	0.999993i	$0.501200\pi$
$810$	0	0
$811$	−22674.1	−0.981744	−0.490872	−	0.871232i	$-0.663322\pi$
$811$	−22674.1	−0.981744	−0.490872	+	0.871232i	$0.663322\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−51723.1	−2.22304
$816$	0	0
$817$	−10069.6	−0.431199
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−40832.2	−1.73575	−0.867877	−	0.496779i	$-0.834516\pi$
$821$	−40832.2	−1.73575	−0.867877	+	0.496779i	$0.834516\pi$
$822$	0	0
$823$	−4881.01	−0.206733	−0.103367	−	0.994643i	$-0.532961\pi$
$823$	−4881.01	−0.206733	−0.103367	+	0.994643i	$0.532961\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	22213.4	0.934022	0.467011	−	0.884251i	$-0.345331\pi$
$827$	22213.4	0.934022	0.467011	+	0.884251i	$0.345331\pi$
$828$	0	0
$829$	35477.2	1.48634	0.743168	−	0.669104i	$-0.233322\pi$
$829$	35477.2	1.48634	0.743168	+	0.669104i	$0.233322\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−42591.7	−1.77157
$834$	0	0
$835$	−58274.9	−2.41519
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−23505.3	−0.967213	−0.483607	−	0.875285i	$-0.660673\pi$
$839$	−23505.3	−0.967213	−0.483607	+	0.875285i	$0.660673\pi$
$840$	0	0
$841$	−22803.2	−0.934977
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−21528.0	−0.876433
$846$	0	0
$847$	−2437.63	−0.0988876
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4498.12	0.181191
$852$	0	0
$853$	−12434.1	−0.499103	−0.249551	−	0.968362i	$-0.580283\pi$
$853$	−12434.1	−0.499103	−0.249551	+	0.968362i	$0.580283\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−795.866	−0.0317226	−0.0158613	−	0.999874i	$-0.505049\pi$
$857$	−795.866	−0.0317226	−0.0158613	+	0.999874i	$0.505049\pi$
$858$	0	0
$859$	−8583.95	−0.340955	−0.170478	−	0.985362i	$-0.554531\pi$
$859$	−8583.95	−0.340955	−0.170478	+	0.985362i	$0.554531\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−37865.4	−1.49357	−0.746787	−	0.665063i	$-0.768405\pi$
$863$	−37865.4	−1.49357	−0.746787	+	0.665063i	$0.768405\pi$
$864$	0	0
$865$	32458.3	1.27585
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−37371.5	−1.45885
$870$	0	0
$871$	13526.1	0.526194
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1202.43	−0.0464565
$876$	0	0
$877$	4604.56	0.177292	0.0886459	−	0.996063i	$-0.471746\pi$
$877$	4604.56	0.177292	0.0886459	+	0.996063i	$0.471746\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	13021.9	0.497979	0.248990	−	0.968506i	$-0.419901\pi$
$881$	13021.9	0.497979	0.248990	+	0.968506i	$0.419901\pi$
$882$	0	0
$883$	−4301.10	−0.163923	−0.0819613	−	0.996636i	$-0.526118\pi$
$883$	−4301.10	−0.163923	−0.0819613	+	0.996636i	$0.526118\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	15195.8	0.575225	0.287613	−	0.957747i	$-0.407138\pi$
$887$	15195.8	0.575225	0.287613	+	0.957747i	$0.407138\pi$
$888$	0	0
$889$	1070.18	0.0403743
$890$	0	0
$891$	0	0
$892$	0	0
$893$	11089.2	0.415551
$894$	0	0
$895$	−23973.1	−0.895344
$896$	0	0
$897$	0	0
$898$	0	0
$899$	11741.2	0.435587
$900$	0	0
$901$	13332.9	0.492990
$902$	0	0
$903$	0	0
$904$	0	0
$905$	62947.9	2.31211
$906$	0	0
$907$	−4657.99	−0.170525	−0.0852624	−	0.996359i	$-0.527173\pi$
$907$	−4657.99	−0.170525	−0.0852624	+	0.996359i	$0.527173\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−12892.8	−0.468889	−0.234444	−	0.972130i	$-0.575327\pi$
$911$	−12892.8	−0.468889	−0.234444	+	0.972130i	$0.575327\pi$
$912$	0	0
$913$	−85765.0	−3.10888
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1610.28	−0.0579894
$918$	0	0
$919$	−36405.3	−1.30675	−0.653373	−	0.757036i	$-0.726647\pi$
$919$	−36405.3	−1.30675	−0.653373	+	0.757036i	$0.726647\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−14178.0	−0.505605
$924$	0	0
$925$	−21400.1	−0.760682
$926$	0	0
$927$	0	0
$928$	0	0
$929$	4279.70	0.151143	0.0755717	−	0.997140i	$-0.475922\pi$
$929$	4279.70	0.151143	0.0755717	+	0.997140i	$0.475922\pi$
$930$	0	0
$931$	−9546.45	−0.336061
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−148210.	−5.18394
$936$	0	0
$937$	21006.6	0.732396	0.366198	−	0.930537i	$-0.380659\pi$
$937$	21006.6	0.732396	0.366198	+	0.930537i	$0.380659\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−13529.1	−0.468688	−0.234344	−	0.972154i	$-0.575294\pi$
$941$	−13529.1	−0.468688	−0.234344	+	0.972154i	$0.575294\pi$
$942$	0	0
$943$	−13283.3	−0.458711
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4153.14	−0.142512	−0.0712561	−	0.997458i	$-0.522701\pi$
$947$	−4153.14	−0.142512	−0.0712561	+	0.997458i	$0.522701\pi$
$948$	0	0
$949$	−15905.1	−0.544048
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−18387.5	−0.625005	−0.312503	−	0.949917i	$-0.601167\pi$
$953$	−18387.5	−0.625005	−0.312503	+	0.949917i	$0.601167\pi$
$954$	0	0
$955$	28926.0	0.980128
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2100.77	0.0707378
$960$	0	0
$961$	57139.1	1.91800
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−13895.9	−0.463548
$966$	0	0
$967$	−34454.1	−1.14578	−0.572889	−	0.819633i	$-0.694178\pi$
$967$	−34454.1	−1.14578	−0.572889	+	0.819633i	$0.694178\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−2689.17	−0.0888770	−0.0444385	−	0.999012i	$-0.514150\pi$
$971$	−2689.17	−0.0888770	−0.0444385	+	0.999012i	$0.514150\pi$
$972$	0	0
$973$	−168.838	−0.00556288
$974$	0	0
$975$	0	0
$976$	0	0
$977$	148.959	0.00487782	0.00243891	−	0.999997i	$-0.499224\pi$
$977$	148.959	0.00487782	0.00243891	+	0.999997i	$0.499224\pi$
$978$	0	0
$979$	−86699.3	−2.83036
$980$	0	0
$981$	0	0
$982$	0	0
$983$	32221.8	1.04549	0.522744	−	0.852490i	$-0.324908\pi$
$983$	32221.8	1.04549	0.522744	+	0.852490i	$0.324908\pi$
$984$	0	0
$985$	21660.8	0.700681
$986$	0	0
$987$	0	0
$988$	0	0
$989$	15590.6	0.501265
$990$	0	0
$991$	2979.37	0.0955023	0.0477512	−	0.998859i	$-0.484795\pi$
$991$	2979.37	0.0955023	0.0477512	+	0.998859i	$0.484795\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−49228.1	−1.56848
$996$	0	0
$997$	−32093.8	−1.01948	−0.509740	−	0.860328i	$-0.670258\pi$
$997$	−32093.8	−1.01948	−0.509740	+	0.860328i	$0.670258\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.4.a.f.1.2	yes	2
3.2	odd	2		648.4.a.c.1.1	✓	2
4.3	odd	2		1296.4.a.q.1.2		2
9.2	odd	6		648.4.i.t.433.2		4
9.4	even	3		648.4.i.n.217.1		4
9.5	odd	6		648.4.i.t.217.2		4
9.7	even	3		648.4.i.n.433.1		4
12.11	even	2		1296.4.a.m.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.4.a.c.1.1	✓	2	3.2	odd	2
648.4.a.f.1.2	yes	2	1.1	even	1	trivial
648.4.i.n.217.1		4	9.4	even	3
648.4.i.n.433.1		4	9.7	even	3
648.4.i.t.217.2		4	9.5	odd	6
648.4.i.t.433.2		4	9.2	odd	6
1296.4.a.m.1.1		2	12.11	even	2
1296.4.a.q.1.2		2	4.3	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:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	648.a (trivial)

\(f(q)\)	\(=\)	\(q+18.1774 q^{5} -0.822553 q^{7} +O(q^{10})\)	\(q+18.1774 q^{5} -0.822553 q^{7} -65.5323 q^{11} +31.8226 q^{13} +124.420 q^{17} +27.8872 q^{19} -43.1774 q^{23} +205.420 q^{25} +39.8226 q^{29} +294.839 q^{31} -14.9519 q^{35} -104.177 q^{37} +307.645 q^{41} -361.081 q^{43} +397.645 q^{47} -342.323 q^{49} +107.161 q^{53} -1191.21 q^{55} +188.935 q^{59} -99.0166 q^{61} +578.453 q^{65} +425.048 q^{67} -445.532 q^{71} -499.806 q^{73} +53.9038 q^{77} +570.275 q^{79} +1308.74 q^{83} +2261.63 q^{85} +1323.00 q^{89} -26.1757 q^{91} +506.919 q^{95} +944.872 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{5} - 30 q^{7}+O(q^{10}) \) 2 * q + 8 * q^5 - 30 * q^7	\( 2 q + 8 q^{5} - 30 q^{7} - 46 q^{11} + 92 q^{13} + 22 q^{17} - 86 q^{19} - 58 q^{23} + 184 q^{25} + 108 q^{29} + 136 q^{31} + 282 q^{35} - 180 q^{37} + 672 q^{41} - 70 q^{43} + 852 q^{47} + 166 q^{49} + 668 q^{53} - 1390 q^{55} + 548 q^{59} + 284 q^{61} - 34 q^{65} + 1162 q^{67} - 806 q^{71} - 1510 q^{73} - 516 q^{77} - 22 q^{79} + 1540 q^{83} + 3304 q^{85} + 2646 q^{89} - 1782 q^{91} + 1666 q^{95} + 472 q^{97}+O(q^{100}) \) 2 * q + 8 * q^5 - 30 * q^7 - 46 * q^11 + 92 * q^13 + 22 * q^17 - 86 * q^19 - 58 * q^23 + 184 * q^25 + 108 * q^29 + 136 * q^31 + 282 * q^35 - 180 * q^37 + 672 * q^41 - 70 * q^43 + 852 * q^47 + 166 * q^49 + 668 * q^53 - 1390 * q^55 + 548 * q^59 + 284 * q^61 - 34 * q^65 + 1162 * q^67 - 806 * q^71 - 1510 * q^73 - 516 * q^77 - 22 * q^79 + 1540 * q^83 + 3304 * q^85 + 2646 * q^89 - 1782 * q^91 + 1666 * q^95 + 472 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more