LMFDB - Embedded newform 441.4.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,4,Mod(1,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("441.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	441.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:	$26.0198423125$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{57}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 14 $ x^2 - x - 14
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 21)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.27492$ of defining polynomial
Character	$\chi$	$=$	441.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.27492 q^{2} -2.82475 q^{4} -4.54983 q^{5} +24.6254 q^{8} +O(q^{10})$	$q-2.27492 q^{2} -2.82475 q^{4} -4.54983 q^{5} +24.6254 q^{8} +10.3505 q^{10} +40.7492 q^{11} -53.2990 q^{13} -33.4228 q^{16} +4.54983 q^{17} -122.598 q^{19} +12.8522 q^{20} -92.7010 q^{22} -131.347 q^{23} -104.299 q^{25} +121.251 q^{26} +216.598 q^{29} +251.794 q^{31} -120.969 q^{32} -10.3505 q^{34} +11.8970 q^{37} +278.900 q^{38} -112.042 q^{40} -111.752 q^{41} +369.196 q^{43} -115.106 q^{44} +298.804 q^{46} -262.694 q^{47} +237.272 q^{50} +150.556 q^{52} +567.100 q^{53} -185.402 q^{55} -492.743 q^{58} +839.890 q^{59} +485.794 q^{61} -572.811 q^{62} +542.577 q^{64} +242.502 q^{65} -333.691 q^{67} -12.8522 q^{68} -590.248 q^{71} -490.701 q^{73} -27.0647 q^{74} +346.309 q^{76} +121.691 q^{79} +152.068 q^{80} +254.228 q^{82} +609.608 q^{83} -20.7010 q^{85} -839.890 q^{86} +1003.47 q^{88} +719.038 q^{89} +371.023 q^{92} +597.608 q^{94} +557.801 q^{95} +637.877 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{2} + 17 q^{4} + 6 q^{5} + 87 q^{8}+O(q^{10}) $ 2 * q + 3 * q^2 + 17 * q^4 + 6 * q^5 + 87 * q^8	$ 2 q + 3 q^{2} + 17 q^{4} + 6 q^{5} + 87 q^{8} + 66 q^{10} + 6 q^{11} - 16 q^{13} + 137 q^{16} - 6 q^{17} - 64 q^{19} + 222 q^{20} - 276 q^{22} - 6 q^{23} - 118 q^{25} + 318 q^{26} + 252 q^{29} - 40 q^{31} + 279 q^{32} - 66 q^{34} - 248 q^{37} + 588 q^{38} + 546 q^{40} - 450 q^{41} + 376 q^{43} - 804 q^{44} + 960 q^{46} - 12 q^{47} + 165 q^{50} + 890 q^{52} + 1104 q^{53} - 552 q^{55} - 306 q^{58} + 804 q^{59} + 428 q^{61} - 2112 q^{62} + 1289 q^{64} + 636 q^{65} + 148 q^{67} - 222 q^{68} - 954 q^{71} - 1072 q^{73} - 1398 q^{74} + 1508 q^{76} - 572 q^{79} + 1950 q^{80} - 1530 q^{82} + 1944 q^{83} - 132 q^{85} - 804 q^{86} - 1164 q^{88} + 366 q^{89} + 2856 q^{92} + 1920 q^{94} + 1176 q^{95} - 808 q^{97}+O(q^{100}) $ 2 * q + 3 * q^2 + 17 * q^4 + 6 * q^5 + 87 * q^8 + 66 * q^10 + 6 * q^11 - 16 * q^13 + 137 * q^16 - 6 * q^17 - 64 * q^19 + 222 * q^20 - 276 * q^22 - 6 * q^23 - 118 * q^25 + 318 * q^26 + 252 * q^29 - 40 * q^31 + 279 * q^32 - 66 * q^34 - 248 * q^37 + 588 * q^38 + 546 * q^40 - 450 * q^41 + 376 * q^43 - 804 * q^44 + 960 * q^46 - 12 * q^47 + 165 * q^50 + 890 * q^52 + 1104 * q^53 - 552 * q^55 - 306 * q^58 + 804 * q^59 + 428 * q^61 - 2112 * q^62 + 1289 * q^64 + 636 * q^65 + 148 * q^67 - 222 * q^68 - 954 * q^71 - 1072 * q^73 - 1398 * q^74 + 1508 * q^76 - 572 * q^79 + 1950 * q^80 - 1530 * q^82 + 1944 * q^83 - 132 * q^85 - 804 * q^86 - 1164 * q^88 + 366 * q^89 + 2856 * q^92 + 1920 * q^94 + 1176 * q^95 - 808 * 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 $

$2$	−2.27492	−0.804305	−0.402152	−	0.915573i	$-0.631738\pi$
$2$	−2.27492	−0.804305	−0.402152	+	0.915573i	$0.631738\pi$
$3$	0	0
$3$	0	0
$4$	−2.82475	−0.353094
$5$	−4.54983	−0.406950	−0.203475	−	0.979080i	$-0.565223\pi$
$5$	−4.54983	−0.406950	−0.203475	+	0.979080i	$0.565223\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	24.6254	1.08830
$9$	0	0
$10$	10.3505	0.327311
$11$	40.7492	1.11694	0.558470	−	0.829525i	$-0.311389\pi$
$11$	40.7492	1.11694	0.558470	+	0.829525i	$0.311389\pi$
$12$	0	0
$13$	−53.2990	−1.13711	−0.568557	−	0.822644i	$-0.692498\pi$
$13$	−53.2990	−1.13711	−0.568557	+	0.822644i	$0.692498\pi$
$14$	0	0
$15$	0	0
$16$	−33.4228	−0.522231
$17$	4.54983	0.0649116	0.0324558	−	0.999473i	$-0.489667\pi$
$17$	4.54983	0.0649116	0.0324558	+	0.999473i	$0.489667\pi$
$18$	0	0
$19$	−122.598	−1.48031	−0.740156	−	0.672436i	$-0.765248\pi$
$19$	−122.598	−1.48031	−0.740156	+	0.672436i	$0.765248\pi$
$20$	12.8522	0.143691
$21$	0	0
$22$	−92.7010	−0.898360
$23$	−131.347	−1.19077	−0.595387	−	0.803439i	$-0.703001\pi$
$23$	−131.347	−1.19077	−0.595387	+	0.803439i	$0.703001\pi$
$24$	0	0
$25$	−104.299	−0.834392
$26$	121.251	0.914586
$27$	0	0
$28$	0	0
$29$	216.598	1.38694	0.693470	−	0.720486i	$-0.256081\pi$
$29$	216.598	1.38694	0.693470	+	0.720486i	$0.256081\pi$
$30$	0	0
$31$	251.794	1.45882	0.729412	−	0.684075i	$-0.239794\pi$
$31$	251.794	1.45882	0.729412	+	0.684075i	$0.239794\pi$
$32$	−120.969	−0.668267
$33$	0	0
$34$	−10.3505	−0.0522087
$35$	0	0
$36$	0	0
$37$	11.8970	0.0528610	0.0264305	−	0.999651i	$-0.491586\pi$
$37$	11.8970	0.0528610	0.0264305	+	0.999651i	$0.491586\pi$
$38$	278.900	1.19062
$39$	0	0
$40$	−112.042	−0.442883
$41$	−111.752	−0.425678	−0.212839	−	0.977087i	$-0.568271\pi$
$41$	−111.752	−0.425678	−0.212839	+	0.977087i	$0.568271\pi$
$42$	0	0
$43$	369.196	1.30935	0.654673	−	0.755912i	$-0.272806\pi$
$43$	369.196	1.30935	0.654673	+	0.755912i	$0.272806\pi$
$44$	−115.106	−0.394385
$45$	0	0
$46$	298.804	0.957744
$47$	−262.694	−0.815275	−0.407637	−	0.913144i	$-0.633647\pi$
$47$	−262.694	−0.815275	−0.407637	+	0.913144i	$0.633647\pi$
$48$	0	0
$49$	0	0
$50$	237.272	0.671105
$51$	0	0
$52$	150.556	0.401508
$53$	567.100	1.46976	0.734879	−	0.678199i	$-0.237239\pi$
$53$	567.100	1.46976	0.734879	+	0.678199i	$0.237239\pi$
$54$	0	0
$55$	−185.402	−0.454538
$56$	0	0
$57$	0	0
$58$	−492.743	−1.11552
$59$	839.890	1.85330	0.926648	−	0.375931i	$-0.122677\pi$
$59$	839.890	1.85330	0.926648	+	0.375931i	$0.122677\pi$
$60$	0	0
$61$	485.794	1.01966	0.509832	−	0.860274i	$-0.329707\pi$
$61$	485.794	1.01966	0.509832	+	0.860274i	$0.329707\pi$
$62$	−572.811	−1.17334
$63$	0	0
$64$	542.577	1.05972
$65$	242.502	0.462748
$66$	0	0
$67$	−333.691	−0.608460	−0.304230	−	0.952599i	$-0.598399\pi$
$67$	−333.691	−0.608460	−0.304230	+	0.952599i	$0.598399\pi$
$68$	−12.8522	−0.0229199
$69$	0	0
$70$	0	0
$71$	−590.248	−0.986613	−0.493306	−	0.869856i	$-0.664212\pi$
$71$	−590.248	−0.986613	−0.493306	+	0.869856i	$0.664212\pi$
$72$	0	0
$73$	−490.701	−0.786743	−0.393371	−	0.919380i	$-0.628691\pi$
$73$	−490.701	−0.786743	−0.393371	+	0.919380i	$0.628691\pi$
$74$	−27.0647	−0.0425164
$75$	0	0
$76$	346.309	0.522689
$77$	0	0
$78$	0	0
$79$	121.691	0.173308	0.0866539	−	0.996238i	$-0.472383\pi$
$79$	121.691	0.173308	0.0866539	+	0.996238i	$0.472383\pi$
$80$	152.068	0.212522
$81$	0	0
$82$	254.228	0.342375
$83$	609.608	0.806183	0.403091	−	0.915160i	$-0.367936\pi$
$83$	609.608	0.806183	0.403091	+	0.915160i	$0.367936\pi$
$84$	0	0
$85$	−20.7010	−0.0264157
$86$	−839.890	−1.05311
$87$	0	0
$88$	1003.47	1.21557
$89$	719.038	0.856381	0.428190	−	0.903689i	$-0.359151\pi$
$89$	719.038	0.856381	0.428190	+	0.903689i	$0.359151\pi$
$90$	0	0
$91$	0	0
$92$	371.023	0.420455
$93$	0	0
$94$	597.608	0.655729
$95$	557.801	0.602412
$96$	0	0
$97$	637.877	0.667697	0.333849	−	0.942627i	$-0.391653\pi$
$97$	637.877	0.667697	0.333849	+	0.942627i	$0.391653\pi$
$98$	0	0
$99$	0	0
$100$	294.619	0.294619
$101$	671.148	0.661205	0.330603	−	0.943770i	$-0.392748\pi$
$101$	671.148	0.661205	0.330603	+	0.943770i	$0.392748\pi$
$102$	0	0
$103$	912.412	0.872841	0.436420	−	0.899743i	$-0.356246\pi$
$103$	912.412	0.872841	0.436420	+	0.899743i	$0.356246\pi$
$104$	−1312.51	−1.23752
$105$	0	0
$106$	−1290.10	−1.18213
$107$	116.736	0.105470	0.0527350	−	0.998609i	$-0.483206\pi$
$107$	116.736	0.105470	0.0527350	+	0.998609i	$0.483206\pi$
$108$	0	0
$109$	837.176	0.735660	0.367830	−	0.929893i	$-0.380101\pi$
$109$	837.176	0.735660	0.367830	+	0.929893i	$0.380101\pi$
$110$	421.774	0.365587
$111$	0	0
$112$	0	0
$113$	1086.58	0.904572	0.452286	−	0.891873i	$-0.350609\pi$
$113$	1086.58	0.904572	0.452286	+	0.891873i	$0.350609\pi$
$114$	0	0
$115$	597.608	0.484585
$116$	−611.836	−0.489720
$117$	0	0
$118$	−1910.68	−1.49061
$119$	0	0
$120$	0	0
$121$	329.495	0.247554
$122$	−1105.14	−0.820121
$123$	0	0
$124$	−711.256	−0.515102
$125$	1043.27	0.746505
$126$	0	0
$127$	−537.113	−0.375284	−0.187642	−	0.982237i	$-0.560084\pi$
$127$	−537.113	−0.375284	−0.187642	+	0.982237i	$0.560084\pi$
$128$	−266.564	−0.184071
$129$	0	0
$130$	−551.671	−0.372190
$131$	1497.39	0.998683	0.499341	−	0.866405i	$-0.333575\pi$
$131$	1497.39	0.998683	0.499341	+	0.866405i	$0.333575\pi$
$132$	0	0
$133$	0	0
$134$	759.120	0.489388
$135$	0	0
$136$	112.042	0.0706433
$137$	1380.09	0.860650	0.430325	−	0.902674i	$-0.358399\pi$
$137$	1380.09	0.860650	0.430325	+	0.902674i	$0.358399\pi$
$138$	0	0
$139$	141.980	0.0866374	0.0433187	−	0.999061i	$-0.486207\pi$
$139$	141.980	0.0866374	0.0433187	+	0.999061i	$0.486207\pi$
$140$	0	0
$141$	0	0
$142$	1342.76	0.793537
$143$	−2171.89	−1.27009
$144$	0	0
$145$	−985.485	−0.564414
$146$	1116.30	0.632781
$147$	0	0
$148$	−33.6061	−0.0186649
$149$	1943.87	1.06878	0.534390	−	0.845238i	$-0.320542\pi$
$149$	1943.87	1.06878	0.534390	+	0.845238i	$0.320542\pi$
$150$	0	0
$151$	−2654.76	−1.43074	−0.715370	−	0.698746i	$-0.753742\pi$
$151$	−2654.76	−1.43074	−0.715370	+	0.698746i	$0.753742\pi$
$152$	−3019.03	−1.61102
$153$	0	0
$154$	0	0
$155$	−1145.62	−0.593668
$156$	0	0
$157$	−1665.22	−0.846489	−0.423244	−	0.906016i	$-0.639109\pi$
$157$	−1665.22	−0.846489	−0.423244	+	0.906016i	$0.639109\pi$
$158$	−276.837	−0.139392
$159$	0	0
$160$	550.390	0.271951
$161$	0	0
$162$	0	0
$163$	−33.0732	−0.0158926	−0.00794629	−	0.999968i	$-0.502529\pi$
$163$	−33.0732	−0.0158926	−0.00794629	+	0.999968i	$0.502529\pi$
$164$	315.673	0.150304
$165$	0	0
$166$	−1386.81	−0.648417
$167$	1654.48	0.766630	0.383315	−	0.923618i	$-0.374782\pi$
$167$	1654.48	0.766630	0.383315	+	0.923618i	$0.374782\pi$
$168$	0	0
$169$	643.784	0.293029
$170$	47.0930	0.0212463
$171$	0	0
$172$	−1042.89	−0.462322
$173$	64.1909	0.0282101	0.0141050	−	0.999901i	$-0.495510\pi$
$173$	64.1909	0.0282101	0.0141050	+	0.999901i	$0.495510\pi$
$174$	0	0
$175$	0	0
$176$	−1361.95	−0.583300
$177$	0	0
$178$	−1635.75	−0.688791
$179$	−3914.68	−1.63462	−0.817309	−	0.576200i	$-0.804535\pi$
$179$	−3914.68	−1.63462	−0.817309	+	0.576200i	$0.804535\pi$
$180$	0	0
$181$	2058.04	0.845156	0.422578	−	0.906327i	$-0.361125\pi$
$181$	2058.04	0.845156	0.422578	+	0.906327i	$0.361125\pi$
$182$	0	0
$183$	0	0
$184$	−3234.48	−1.29592
$185$	−54.1295	−0.0215118
$186$	0	0
$187$	185.402	0.0725023
$188$	742.046	0.287869
$189$	0	0
$190$	−1268.95	−0.484523
$191$	−428.048	−0.162160	−0.0810798	−	0.996708i	$-0.525837\pi$
$191$	−428.048	−0.162160	−0.0810798	+	0.996708i	$0.525837\pi$
$192$	0	0
$193$	1604.93	0.598576	0.299288	−	0.954163i	$-0.403251\pi$
$193$	1604.93	0.598576	0.299288	+	0.954163i	$0.403251\pi$
$194$	−1451.12	−0.537032
$195$	0	0
$196$	0	0
$197$	−3738.83	−1.35218	−0.676092	−	0.736817i	$-0.736328\pi$
$197$	−3738.83	−1.35218	−0.676092	+	0.736817i	$0.736328\pi$
$198$	0	0
$199$	349.030	0.124332	0.0621660	−	0.998066i	$-0.480199\pi$
$199$	349.030	0.124332	0.0621660	+	0.998066i	$0.480199\pi$
$200$	−2568.41	−0.908069
$201$	0	0
$202$	−1526.81	−0.531810
$203$	0	0
$204$	0	0
$205$	508.455	0.173230
$206$	−2075.66	−0.702030
$207$	0	0
$208$	1781.40	0.593836
$209$	−4995.77	−1.65342
$210$	0	0
$211$	2588.58	0.844574	0.422287	−	0.906462i	$-0.361227\pi$
$211$	2588.58	0.844574	0.422287	+	0.906462i	$0.361227\pi$
$212$	−1601.92	−0.518962
$213$	0	0
$214$	−265.565	−0.0848300
$215$	−1679.78	−0.532838
$216$	0	0
$217$	0	0
$218$	−1904.51	−0.591695
$219$	0	0
$220$	523.715	0.160495
$221$	−242.502	−0.0738119
$222$	0	0
$223$	3236.21	0.971804	0.485902	−	0.874013i	$-0.338491\pi$
$223$	3236.21	0.971804	0.485902	+	0.874013i	$0.338491\pi$
$224$	0	0
$225$	0	0
$226$	−2471.88	−0.727552
$227$	−5631.62	−1.64662	−0.823312	−	0.567589i	$-0.807876\pi$
$227$	−5631.62	−1.64662	−0.823312	+	0.567589i	$0.807876\pi$
$228$	0	0
$229$	−3770.25	−1.08797	−0.543985	−	0.839095i	$-0.683085\pi$
$229$	−3770.25	−1.08797	−0.543985	+	0.839095i	$0.683085\pi$
$230$	−1359.51	−0.389754
$231$	0	0
$232$	5333.82	1.50941
$233$	6560.90	1.84472	0.922358	−	0.386336i	$-0.126259\pi$
$233$	6560.90	1.84472	0.922358	+	0.386336i	$0.126259\pi$
$234$	0	0
$235$	1195.22	0.331776
$236$	−2372.48	−0.654387
$237$	0	0
$238$	0	0
$239$	771.444	0.208789	0.104394	−	0.994536i	$-0.466710\pi$
$239$	771.444	0.208789	0.104394	+	0.994536i	$0.466710\pi$
$240$	0	0
$241$	−1252.10	−0.334668	−0.167334	−	0.985900i	$-0.553516\pi$
$241$	−1252.10	−0.334668	−0.167334	+	0.985900i	$0.553516\pi$
$242$	−749.574	−0.199109
$243$	0	0
$244$	−1372.25	−0.360037
$245$	0	0
$246$	0	0
$247$	6534.35	1.68328
$248$	6200.53	1.58764
$249$	0	0
$250$	−2373.36	−0.600418
$251$	5166.27	1.29917	0.649586	−	0.760288i	$-0.274942\pi$
$251$	5166.27	1.29917	0.649586	+	0.760288i	$0.274942\pi$
$252$	0	0
$253$	−5352.29	−1.33002
$254$	1221.89	0.301843
$255$	0	0
$256$	−3734.21	−0.911672
$257$	2767.45	0.671707	0.335854	−	0.941914i	$-0.390975\pi$
$257$	2767.45	0.671707	0.335854	+	0.941914i	$0.390975\pi$
$258$	0	0
$259$	0	0
$260$	−685.007	−0.163394
$261$	0	0
$262$	−3406.44	−0.803245
$263$	4101.78	0.961699	0.480849	−	0.876803i	$-0.340328\pi$
$263$	4101.78	0.961699	0.480849	+	0.876803i	$0.340328\pi$
$264$	0	0
$265$	−2580.21	−0.598117
$266$	0	0
$267$	0	0
$268$	942.594	0.214844
$269$	−6950.84	−1.57546	−0.787732	−	0.616018i	$-0.788745\pi$
$269$	−6950.84	−1.57546	−0.787732	+	0.616018i	$0.788745\pi$
$270$	0	0
$271$	7140.29	1.60052	0.800262	−	0.599651i	$-0.204694\pi$
$271$	7140.29	1.60052	0.800262	+	0.599651i	$0.204694\pi$
$272$	−152.068	−0.0338988
$273$	0	0
$274$	−3139.59	−0.692225
$275$	−4250.10	−0.931966
$276$	0	0
$277$	1320.51	0.286433	0.143217	−	0.989691i	$-0.454255\pi$
$277$	1320.51	0.286433	0.143217	+	0.989691i	$0.454255\pi$
$278$	−322.993	−0.0696829
$279$	0	0
$280$	0	0
$281$	204.309	0.0433738	0.0216869	−	0.999765i	$-0.493096\pi$
$281$	204.309	0.0433738	0.0216869	+	0.999765i	$0.493096\pi$
$282$	0	0
$283$	−975.794	−0.204964	−0.102482	−	0.994735i	$-0.532678\pi$
$283$	−975.794	−0.204964	−0.102482	+	0.994735i	$0.532678\pi$
$284$	1667.30	0.348367
$285$	0	0
$286$	4940.87	1.02154
$287$	0	0
$288$	0	0
$289$	−4892.30	−0.995786
$290$	2241.90	0.453961
$291$	0	0
$292$	1386.11	0.277794
$293$	607.919	0.121212	0.0606058	−	0.998162i	$-0.480697\pi$
$293$	607.919	0.121212	0.0606058	+	0.998162i	$0.480697\pi$
$294$	0	0
$295$	−3821.36	−0.754198
$296$	292.969	0.0575286
$297$	0	0
$298$	−4422.14	−0.859624
$299$	7000.67	1.35405
$300$	0	0
$301$	0	0
$302$	6039.37	1.15075
$303$	0	0
$304$	4097.56	0.773064
$305$	−2210.28	−0.414952
$306$	0	0
$307$	−8037.08	−1.49414	−0.747069	−	0.664747i	$-0.768539\pi$
$307$	−8037.08	−1.49414	−0.747069	+	0.664747i	$0.768539\pi$
$308$	0	0
$309$	0	0
$310$	2606.19	0.477490
$311$	5311.60	0.968468	0.484234	−	0.874939i	$-0.339098\pi$
$311$	5311.60	0.968468	0.484234	+	0.874939i	$0.339098\pi$
$312$	0	0
$313$	1531.61	0.276587	0.138293	−	0.990391i	$-0.455838\pi$
$313$	1531.61	0.276587	0.138293	+	0.990391i	$0.455838\pi$
$314$	3788.23	0.680835
$315$	0	0
$316$	−343.747	−0.0611939
$317$	−4219.19	−0.747549	−0.373775	−	0.927520i	$-0.621937\pi$
$317$	−4219.19	−0.747549	−0.373775	+	0.927520i	$0.621937\pi$
$318$	0	0
$319$	8826.19	1.54913
$320$	−2468.64	−0.431253
$321$	0	0
$322$	0	0
$323$	−557.801	−0.0960893
$324$	0	0
$325$	5559.03	0.948799
$326$	75.2387	0.0127825
$327$	0	0
$328$	−2751.95	−0.463265
$329$	0	0
$330$	0	0
$331$	8298.19	1.37797	0.688987	−	0.724773i	$-0.258056\pi$
$331$	8298.19	1.37797	0.688987	+	0.724773i	$0.258056\pi$
$332$	−1721.99	−0.284658
$333$	0	0
$334$	−3763.79	−0.616604
$335$	1518.24	0.247613
$336$	0	0
$337$	−4348.44	−0.702892	−0.351446	−	0.936208i	$-0.614310\pi$
$337$	−4348.44	−0.702892	−0.351446	+	0.936208i	$0.614310\pi$
$338$	−1464.56	−0.235684
$339$	0	0
$340$	58.4752	0.00932724
$341$	10260.4	1.62942
$342$	0	0
$343$	0	0
$344$	9091.60	1.42496
$345$	0	0
$346$	−146.029	−0.0226895
$347$	8345.54	1.29110	0.645550	−	0.763718i	$-0.276628\pi$
$347$	8345.54	1.29110	0.645550	+	0.763718i	$0.276628\pi$
$348$	0	0
$349$	9982.54	1.53110	0.765549	−	0.643378i	$-0.222468\pi$
$349$	9982.54	1.53110	0.765549	+	0.643378i	$0.222468\pi$
$350$	0	0
$351$	0	0
$352$	−4929.40	−0.746414
$353$	8801.59	1.32709	0.663543	−	0.748138i	$-0.269052\pi$
$353$	8801.59	1.32709	0.663543	+	0.748138i	$0.269052\pi$
$354$	0	0
$355$	2685.53	0.401502
$356$	−2031.10	−0.302383
$357$	0	0
$358$	8905.56	1.31473
$359$	524.039	0.0770409	0.0385205	−	0.999258i	$-0.487736\pi$
$359$	524.039	0.0770409	0.0385205	+	0.999258i	$0.487736\pi$
$360$	0	0
$361$	8171.27	1.19132
$362$	−4681.88	−0.679763
$363$	0	0
$364$	0	0
$365$	2232.61	0.320165
$366$	0	0
$367$	6362.72	0.904991	0.452495	−	0.891767i	$-0.350534\pi$
$367$	6362.72	0.904991	0.452495	+	0.891767i	$0.350534\pi$
$368$	4389.99	0.621858
$369$	0	0
$370$	123.140	0.0173020
$371$	0	0
$372$	0	0
$373$	−11265.8	−1.56387	−0.781935	−	0.623361i	$-0.785767\pi$
$373$	−11265.8	−1.56387	−0.781935	+	0.623361i	$0.785767\pi$
$374$	−421.774	−0.0583140
$375$	0	0
$376$	−6468.96	−0.887263
$377$	−11544.5	−1.57711
$378$	0	0
$379$	−1151.71	−0.156094	−0.0780470	−	0.996950i	$-0.524868\pi$
$379$	−1151.71	−0.156094	−0.0780470	+	0.996950i	$0.524868\pi$
$380$	−1575.65	−0.212708
$381$	0	0
$382$	973.774	0.130426
$383$	−151.554	−0.0202195	−0.0101097	−	0.999949i	$-0.503218\pi$
$383$	−151.554	−0.0202195	−0.0101097	+	0.999949i	$0.503218\pi$
$384$	0	0
$385$	0	0
$386$	−3651.08	−0.481437
$387$	0	0
$388$	−1801.84	−0.235760
$389$	−4794.18	−0.624870	−0.312435	−	0.949939i	$-0.601145\pi$
$389$	−4794.18	−0.624870	−0.312435	+	0.949939i	$0.601145\pi$
$390$	0	0
$391$	−597.608	−0.0772950
$392$	0	0
$393$	0	0
$394$	8505.52	1.08757
$395$	−553.674	−0.0705275
$396$	0	0
$397$	4623.94	0.584556	0.292278	−	0.956333i	$-0.405587\pi$
$397$	4623.94	0.584556	0.292278	+	0.956333i	$0.405587\pi$
$398$	−794.014	−0.100001
$399$	0	0
$400$	3485.96	0.435745
$401$	3610.63	0.449642	0.224821	−	0.974400i	$-0.427820\pi$
$401$	3610.63	0.449642	0.224821	+	0.974400i	$0.427820\pi$
$402$	0	0
$403$	−13420.4	−1.65885
$404$	−1895.83	−0.233467
$405$	0	0
$406$	0	0
$407$	484.794	0.0590426
$408$	0	0
$409$	−8959.57	−1.08318	−0.541592	−	0.840641i	$-0.682178\pi$
$409$	−8959.57	−1.08318	−0.541592	+	0.840641i	$0.682178\pi$
$410$	−1156.69	−0.139329
$411$	0	0
$412$	−2577.34	−0.308195
$413$	0	0
$414$	0	0
$415$	−2773.62	−0.328076
$416$	6447.54	0.759896
$417$	0	0
$418$	11365.0	1.32985
$419$	−7078.28	−0.825290	−0.412645	−	0.910892i	$-0.635395\pi$
$419$	−7078.28	−0.825290	−0.412645	+	0.910892i	$0.635395\pi$
$420$	0	0
$421$	11551.5	1.33725	0.668626	−	0.743599i	$-0.266883\pi$
$421$	11551.5	1.33725	0.668626	+	0.743599i	$0.266883\pi$
$422$	−5888.80	−0.679295
$423$	0	0
$424$	13965.1	1.59954
$425$	−474.543	−0.0541617
$426$	0	0
$427$	0	0
$428$	−329.750	−0.0372408
$429$	0	0
$430$	3821.36	0.428564
$431$	4064.38	0.454232	0.227116	−	0.973868i	$-0.427070\pi$
$431$	4064.38	0.454232	0.227116	+	0.973868i	$0.427070\pi$
$432$	0	0
$433$	−17456.3	−1.93740	−0.968701	−	0.248229i	$-0.920151\pi$
$433$	−17456.3	−1.93740	−0.968701	+	0.248229i	$0.920151\pi$
$434$	0	0
$435$	0	0
$436$	−2364.81	−0.259757
$437$	16102.9	1.76271
$438$	0	0
$439$	−4595.39	−0.499604	−0.249802	−	0.968297i	$-0.580365\pi$
$439$	−4595.39	−0.499604	−0.249802	+	0.968297i	$0.580365\pi$
$440$	−4565.60	−0.494674
$441$	0	0
$442$	551.671	0.0593672
$443$	306.214	0.0328412	0.0164206	−	0.999865i	$-0.494773\pi$
$443$	306.214	0.0328412	0.0164206	+	0.999865i	$0.494773\pi$
$444$	0	0
$445$	−3271.50	−0.348504
$446$	−7362.10	−0.781627
$447$	0	0
$448$	0	0
$449$	−9229.22	−0.970053	−0.485026	−	0.874500i	$-0.661190\pi$
$449$	−9229.22	−0.970053	−0.485026	+	0.874500i	$0.661190\pi$
$450$	0	0
$451$	−4553.82	−0.475457
$452$	−3069.31	−0.319399
$453$	0	0
$454$	12811.5	1.32439
$455$	0	0
$456$	0	0
$457$	−10992.2	−1.12515	−0.562577	−	0.826745i	$-0.690190\pi$
$457$	−10992.2	−1.12515	−0.562577	+	0.826745i	$0.690190\pi$
$458$	8577.01	0.875059
$459$	0	0
$460$	−1688.09	−0.171104
$461$	7387.88	0.746394	0.373197	−	0.927752i	$-0.378261\pi$
$461$	7387.88	0.746394	0.373197	+	0.927752i	$0.378261\pi$
$462$	0	0
$463$	10163.8	1.02020	0.510101	−	0.860114i	$-0.329608\pi$
$463$	10163.8	1.02020	0.510101	+	0.860114i	$0.329608\pi$
$464$	−7239.30	−0.724302
$465$	0	0
$466$	−14925.5	−1.48371
$467$	−15814.6	−1.56705	−0.783524	−	0.621362i	$-0.786580\pi$
$467$	−15814.6	−1.56705	−0.783524	+	0.621362i	$0.786580\pi$
$468$	0	0
$469$	0	0
$470$	−2719.02	−0.266849
$471$	0	0
$472$	20682.6	2.01694
$473$	15044.4	1.46246
$474$	0	0
$475$	12786.9	1.23516
$476$	0	0
$477$	0	0
$478$	−1754.97	−0.167930
$479$	−1444.85	−0.137823	−0.0689113	−	0.997623i	$-0.521953\pi$
$479$	−1444.85	−0.137823	−0.0689113	+	0.997623i	$0.521953\pi$
$480$	0	0
$481$	−634.099	−0.0601090
$482$	2848.43	0.269175
$483$	0	0
$484$	−930.742	−0.0874100
$485$	−2902.24	−0.271719
$486$	0	0
$487$	−489.402	−0.0455378	−0.0227689	−	0.999741i	$-0.507248\pi$
$487$	−489.402	−0.0455378	−0.0227689	+	0.999741i	$0.507248\pi$
$488$	11962.9	1.10970
$489$	0	0
$490$	0	0
$491$	3941.30	0.362257	0.181129	−	0.983459i	$-0.442025\pi$
$491$	3941.30	0.362257	0.181129	+	0.983459i	$0.442025\pi$
$492$	0	0
$493$	985.485	0.0900284
$494$	−14865.1	−1.35387
$495$	0	0
$496$	−8415.65	−0.761843
$497$	0	0
$498$	0	0
$499$	11.0894	0.000994850 0	0.000497425	−	1.00000i	$-0.499842\pi$
$499$	11.0894	0.000994850 0	0.000497425		1.00000i	$0.499842\pi$
$500$	−2946.99	−0.263586
$501$	0	0
$502$	−11752.8	−1.04493
$503$	7088.41	0.628343	0.314172	−	0.949366i	$-0.398273\pi$
$503$	7088.41	0.628343	0.314172	+	0.949366i	$0.398273\pi$
$504$	0	0
$505$	−3053.61	−0.269077
$506$	12176.0	1.06974
$507$	0	0
$508$	1517.21	0.132511
$509$	17588.4	1.53162	0.765810	−	0.643067i	$-0.222338\pi$
$509$	17588.4	1.53162	0.765810	+	0.643067i	$0.222338\pi$
$510$	0	0
$511$	0	0
$512$	10627.5	0.917333
$513$	0	0
$514$	−6295.72	−0.540257
$515$	−4151.32	−0.355202
$516$	0	0
$517$	−10704.6	−0.910613
$518$	0	0
$519$	0	0
$520$	5971.70	0.503609
$521$	−11646.6	−0.979360	−0.489680	−	0.871902i	$-0.662886\pi$
$521$	−11646.6	−0.979360	−0.489680	+	0.871902i	$0.662886\pi$
$522$	0	0
$523$	−8965.82	−0.749614	−0.374807	−	0.927103i	$-0.622291\pi$
$523$	−8965.82	−0.749614	−0.374807	+	0.927103i	$0.622291\pi$
$524$	−4229.75	−0.352629
$525$	0	0
$526$	−9331.22	−0.773499
$527$	1145.62	0.0946946
$528$	0	0
$529$	5085.08	0.417941
$530$	5869.76	0.481068
$531$	0	0
$532$	0	0
$533$	5956.30	0.484045
$534$	0	0
$535$	−531.129	−0.0429210
$536$	−8217.28	−0.662187
$537$	0	0
$538$	15812.6	1.26715
$539$	0	0
$540$	0	0
$541$	−195.272	−0.0155183	−0.00775914	−	0.999970i	$-0.502470\pi$
$541$	−195.272	−0.0155183	−0.00775914	+	0.999970i	$0.502470\pi$
$542$	−16243.6	−1.28731
$543$	0	0
$544$	−550.390	−0.0433783
$545$	−3809.01	−0.299376
$546$	0	0
$547$	−1399.26	−0.109375	−0.0546874	−	0.998504i	$-0.517416\pi$
$547$	−1399.26	−0.109375	−0.0546874	+	0.998504i	$0.517416\pi$
$548$	−3898.41	−0.303890
$549$	0	0
$550$	9668.62	0.749584
$551$	−26554.5	−2.05310
$552$	0	0
$553$	0	0
$554$	−3004.06	−0.230380
$555$	0	0
$556$	−401.059	−0.0305911
$557$	−43.0467	−0.00327459	−0.00163730	−	0.999999i	$-0.500521\pi$
$557$	−43.0467	−0.00327459	−0.00163730	+	0.999999i	$0.500521\pi$
$558$	0	0
$559$	−19677.8	−1.48888
$560$	0	0
$561$	0	0
$562$	−464.786	−0.0348858
$563$	−19232.9	−1.43973	−0.719865	−	0.694114i	$-0.755797\pi$
$563$	−19232.9	−1.43973	−0.719865	+	0.694114i	$0.755797\pi$
$564$	0	0
$565$	−4943.75	−0.368115
$566$	2219.85	0.164854
$567$	0	0
$568$	−14535.1	−1.07373
$569$	−5163.98	−0.380466	−0.190233	−	0.981739i	$-0.560924\pi$
$569$	−5163.98	−0.380466	−0.190233	+	0.981739i	$0.560924\pi$
$570$	0	0
$571$	−10231.9	−0.749899	−0.374950	−	0.927045i	$-0.622340\pi$
$571$	−10231.9	−0.749899	−0.374950	+	0.927045i	$0.622340\pi$
$572$	6135.05	0.448460
$573$	0	0
$574$	0	0
$575$	13699.4	0.993572
$576$	0	0
$577$	−16563.7	−1.19507	−0.597537	−	0.801842i	$-0.703854\pi$
$577$	−16563.7	−1.19507	−0.597537	+	0.801842i	$0.703854\pi$
$578$	11129.6	0.800916
$579$	0	0
$580$	2783.75	0.199291
$581$	0	0
$582$	0	0
$583$	23108.8	1.64163
$584$	−12083.7	−0.856212
$585$	0	0
$586$	−1382.96	−0.0974910
$587$	16020.6	1.12648	0.563239	−	0.826294i	$-0.309555\pi$
$587$	16020.6	1.12648	0.563239	+	0.826294i	$0.309555\pi$
$588$	0	0
$589$	−30869.4	−2.15951
$590$	8693.28	0.606605
$591$	0	0
$592$	−397.631	−0.0276057
$593$	−6771.14	−0.468900	−0.234450	−	0.972128i	$-0.575329\pi$
$593$	−6771.14	−0.468900	−0.234450	+	0.972128i	$0.575329\pi$
$594$	0	0
$595$	0	0
$596$	−5490.95	−0.377379
$597$	0	0
$598$	−15926.0	−1.08906
$599$	−11070.2	−0.755120	−0.377560	−	0.925985i	$-0.623237\pi$
$599$	−11070.2	−0.755120	−0.377560	+	0.925985i	$0.623237\pi$
$600$	0	0
$601$	24187.7	1.64166	0.820830	−	0.571173i	$-0.193511\pi$
$601$	24187.7	1.64166	0.820830	+	0.571173i	$0.193511\pi$
$602$	0	0
$603$	0	0
$604$	7499.05	0.505185
$605$	−1499.15	−0.100742
$606$	0	0
$607$	10074.1	0.673631	0.336816	−	0.941571i	$-0.390650\pi$
$607$	10074.1	0.673631	0.336816	+	0.941571i	$0.390650\pi$
$608$	14830.6	0.989243
$609$	0	0
$610$	5028.21	0.333748
$611$	14001.3	0.927060
$612$	0	0
$613$	−11114.6	−0.732323	−0.366161	−	0.930551i	$-0.619328\pi$
$613$	−11114.6	−0.732323	−0.366161	+	0.930551i	$0.619328\pi$
$614$	18283.7	1.20174
$615$	0	0
$616$	0	0
$617$	−20496.4	−1.33737	−0.668683	−	0.743548i	$-0.733142\pi$
$617$	−20496.4	−1.33737	−0.668683	+	0.743548i	$0.733142\pi$
$618$	0	0
$619$	16714.4	1.08532	0.542658	−	0.839954i	$-0.317418\pi$
$619$	16714.4	1.08532	0.542658	+	0.839954i	$0.317418\pi$
$620$	3236.10	0.209621
$621$	0	0
$622$	−12083.5	−0.778943
$623$	0	0
$624$	0	0
$625$	8290.66	0.530602
$626$	−3484.28	−0.222460
$627$	0	0
$628$	4703.82	0.298890
$629$	54.1295	0.00343129
$630$	0	0
$631$	9168.53	0.578437	0.289218	−	0.957263i	$-0.406605\pi$
$631$	9168.53	0.578437	0.289218	+	0.957263i	$0.406605\pi$
$632$	2996.69	0.188611
$633$	0	0
$634$	9598.30	0.601257
$635$	2443.77	0.152722
$636$	0	0
$637$	0	0
$638$	−20078.9	−1.24597
$639$	0	0
$640$	1212.82	0.0749078
$641$	4273.37	0.263319	0.131660	−	0.991295i	$-0.457969\pi$
$641$	4273.37	0.263319	0.131660	+	0.991295i	$0.457969\pi$
$642$	0	0
$643$	−2955.75	−0.181281	−0.0906404	−	0.995884i	$-0.528891\pi$
$643$	−2955.75	−0.181281	−0.0906404	+	0.995884i	$0.528891\pi$
$644$	0	0
$645$	0	0
$646$	1268.95	0.0772851
$647$	22701.2	1.37941	0.689704	−	0.724091i	$-0.257741\pi$
$647$	22701.2	1.37941	0.689704	+	0.724091i	$0.257741\pi$
$648$	0	0
$649$	34224.8	2.07002
$650$	−12646.3	−0.763124
$651$	0	0
$652$	93.4235	0.00561158
$653$	−1537.81	−0.0921582	−0.0460791	−	0.998938i	$-0.514673\pi$
$653$	−1537.81	−0.0921582	−0.0460791	+	0.998938i	$0.514673\pi$
$654$	0	0
$655$	−6812.87	−0.406414
$656$	3735.08	0.222302
$657$	0	0
$658$	0	0
$659$	−12338.1	−0.729323	−0.364661	−	0.931140i	$-0.618815\pi$
$659$	−12338.1	−0.729323	−0.364661	+	0.931140i	$0.618815\pi$
$660$	0	0
$661$	−1845.10	−0.108572	−0.0542859	−	0.998525i	$-0.517288\pi$
$661$	−1845.10	−0.108572	−0.0542859	+	0.998525i	$0.517288\pi$
$662$	−18877.7	−1.10831
$663$	0	0
$664$	15011.8	0.877369
$665$	0	0
$666$	0	0
$667$	−28449.5	−1.65153
$668$	−4673.48	−0.270692
$669$	0	0
$670$	−3453.87	−0.199156
$671$	19795.7	1.13890
$672$	0	0
$673$	23955.4	1.37208	0.686041	−	0.727563i	$-0.259347\pi$
$673$	23955.4	1.37208	0.686041	+	0.727563i	$0.259347\pi$
$674$	9892.34	0.565339
$675$	0	0
$676$	−1818.53	−0.103467
$677$	−3678.26	−0.208814	−0.104407	−	0.994535i	$-0.533294\pi$
$677$	−3678.26	−0.208814	−0.104407	+	0.994535i	$0.533294\pi$
$678$	0	0
$679$	0	0
$680$	−509.771	−0.0287482
$681$	0	0
$682$	−23341.6	−1.31055
$683$	−4390.87	−0.245991	−0.122996	−	0.992407i	$-0.539250\pi$
$683$	−4390.87	−0.245991	−0.122996	+	0.992407i	$0.539250\pi$
$684$	0	0
$685$	−6279.18	−0.350241
$686$	0	0
$687$	0	0
$688$	−12339.6	−0.683781
$689$	−30225.8	−1.67128
$690$	0	0
$691$	−10371.7	−0.570994	−0.285497	−	0.958380i	$-0.592159\pi$
$691$	−10371.7	−0.570994	−0.285497	+	0.958380i	$0.592159\pi$
$692$	−181.323	−0.00996081
$693$	0	0
$694$	−18985.4	−1.03844
$695$	−645.986	−0.0352570
$696$	0	0
$697$	−508.455	−0.0276314
$698$	−22709.4	−1.23147
$699$	0	0
$700$	0	0
$701$	−109.675	−0.00590922	−0.00295461	−	0.999996i	$-0.500940\pi$
$701$	−109.675	−0.00590922	−0.00295461	+	0.999996i	$0.500940\pi$
$702$	0	0
$703$	−1458.55	−0.0782508
$704$	22109.6	1.18364
$705$	0	0
$706$	−20022.9	−1.06738
$707$	0	0
$708$	0	0
$709$	26918.8	1.42589	0.712944	−	0.701221i	$-0.247361\pi$
$709$	26918.8	1.42589	0.712944	+	0.701221i	$0.247361\pi$
$710$	−6109.35	−0.322930
$711$	0	0
$712$	17706.6	0.931999
$713$	−33072.4	−1.73713
$714$	0	0
$715$	9881.74	0.516862
$716$	11058.0	0.577174
$717$	0	0
$718$	−1192.14	−0.0619644
$719$	15170.8	0.786889	0.393445	−	0.919348i	$-0.371283\pi$
$719$	15170.8	0.786889	0.393445	+	0.919348i	$0.371283\pi$
$720$	0	0
$721$	0	0
$722$	−18589.0	−0.958185
$723$	0	0
$724$	−5813.46	−0.298419
$725$	−22591.0	−1.15725
$726$	0	0
$727$	33286.9	1.69813	0.849066	−	0.528288i	$-0.177166\pi$
$727$	33286.9	1.69813	0.849066	+	0.528288i	$0.177166\pi$
$728$	0	0
$729$	0	0
$730$	−5079.00	−0.257510
$731$	1679.78	0.0849917
$732$	0	0
$733$	−20544.0	−1.03521	−0.517607	−	0.855619i	$-0.673177\pi$
$733$	−20544.0	−1.03521	−0.517607	+	0.855619i	$0.673177\pi$
$734$	−14474.7	−0.727888
$735$	0	0
$736$	15889.0	0.795755
$737$	−13597.6	−0.679614
$738$	0	0
$739$	34357.2	1.71022	0.855109	−	0.518449i	$-0.173490\pi$
$739$	34357.2	1.71022	0.855109	+	0.518449i	$0.173490\pi$
$740$	152.902	0.00759568
$741$	0	0
$742$	0	0
$743$	−8166.99	−0.403254	−0.201627	−	0.979462i	$-0.564623\pi$
$743$	−8166.99	−0.403254	−0.201627	+	0.979462i	$0.564623\pi$
$744$	0	0
$745$	−8844.29	−0.434939
$746$	25628.9	1.25783
$747$	0	0
$748$	−523.715	−0.0256001
$749$	0	0
$750$	0	0
$751$	17080.1	0.829909	0.414954	−	0.909842i	$-0.363798\pi$
$751$	17080.1	0.829909	0.414954	+	0.909842i	$0.363798\pi$
$752$	8779.97	0.425761
$753$	0	0
$754$	26262.7	1.26848
$755$	12078.7	0.582239
$756$	0	0
$757$	−16324.0	−0.783758	−0.391879	−	0.920017i	$-0.628175\pi$
$757$	−16324.0	−0.783758	−0.391879	+	0.920017i	$0.628175\pi$
$758$	2620.06	0.125547
$759$	0	0
$760$	13736.1	0.655605
$761$	−32366.2	−1.54175	−0.770875	−	0.636986i	$-0.780181\pi$
$761$	−32366.2	−1.54175	−0.770875	+	0.636986i	$0.780181\pi$
$762$	0	0
$763$	0	0
$764$	1209.13	0.0572576
$765$	0	0
$766$	344.774	0.0162626
$767$	−44765.3	−2.10741
$768$	0	0
$769$	7948.44	0.372728	0.186364	−	0.982481i	$-0.440330\pi$
$769$	7948.44	0.372728	0.186364	+	0.982481i	$0.440330\pi$
$770$	0	0
$771$	0	0
$772$	−4533.52	−0.211354
$773$	17819.3	0.829127	0.414564	−	0.910020i	$-0.363934\pi$
$773$	17819.3	0.829127	0.414564	+	0.910020i	$0.363934\pi$
$774$	0	0
$775$	−26261.9	−1.21723
$776$	15708.0	0.726655
$777$	0	0
$778$	10906.4	0.502586
$779$	13700.6	0.630136
$780$	0	0
$781$	−24052.1	−1.10199
$782$	1359.51	0.0621687
$783$	0	0
$784$	0	0
$785$	7576.46	0.344478
$786$	0	0
$787$	−2912.38	−0.131912	−0.0659562	−	0.997823i	$-0.521010\pi$
$787$	−2912.38	−0.131912	−0.0659562	+	0.997823i	$0.521010\pi$
$788$	10561.3	0.477448
$789$	0	0
$790$	1259.56	0.0567256
$791$	0	0
$792$	0	0
$793$	−25892.3	−1.15948
$794$	−10519.1	−0.470162
$795$	0	0
$796$	−985.923	−0.0439009
$797$	−33789.1	−1.50172	−0.750861	−	0.660460i	$-0.770361\pi$
$797$	−33789.1	−1.50172	−0.750861	+	0.660460i	$0.770361\pi$
$798$	0	0
$799$	−1195.22	−0.0529208
$800$	12617.0	0.557597
$801$	0	0
$802$	−8213.90	−0.361649
$803$	−19995.7	−0.878744
$804$	0	0
$805$	0	0
$806$	30530.2	1.33422
$807$	0	0
$808$	16527.3	0.719589
$809$	−1252.13	−0.0544159	−0.0272079	−	0.999630i	$-0.508662\pi$
$809$	−1252.13	−0.0544159	−0.0272079	+	0.999630i	$0.508662\pi$
$810$	0	0
$811$	31913.1	1.38178	0.690889	−	0.722961i	$-0.257219\pi$
$811$	31913.1	1.38178	0.690889	+	0.722961i	$0.257219\pi$
$812$	0	0
$813$	0	0
$814$	−1102.87	−0.0474882
$815$	150.477	0.00646748
$816$	0	0
$817$	−45262.7	−1.93824
$818$	20382.3	0.871210
$819$	0	0
$820$	−1436.26	−0.0611663
$821$	−30742.4	−1.30684	−0.653421	−	0.756995i	$-0.726667\pi$
$821$	−30742.4	−1.30684	−0.653421	+	0.756995i	$0.726667\pi$
$822$	0	0
$823$	−13822.6	−0.585449	−0.292724	−	0.956197i	$-0.594562\pi$
$823$	−13822.6	−0.585449	−0.292724	+	0.956197i	$0.594562\pi$
$824$	22468.5	0.949913
$825$	0	0
$826$	0	0
$827$	42107.1	1.77051	0.885253	−	0.465110i	$-0.153985\pi$
$827$	42107.1	1.77051	0.885253	+	0.465110i	$0.153985\pi$
$828$	0	0
$829$	38763.8	1.62403	0.812015	−	0.583636i	$-0.198371\pi$
$829$	38763.8	1.62403	0.812015	+	0.583636i	$0.198371\pi$
$830$	6309.75	0.263873
$831$	0	0
$832$	−28918.8	−1.20502
$833$	0	0
$834$	0	0
$835$	−7527.59	−0.311980
$836$	14111.8	0.583812
$837$	0	0
$838$	16102.5	0.663784
$839$	16896.3	0.695262	0.347631	−	0.937631i	$-0.386986\pi$
$839$	16896.3	0.695262	0.347631	+	0.937631i	$0.386986\pi$
$840$	0	0
$841$	22525.7	0.923601
$842$	−26278.6	−1.07556
$843$	0	0
$844$	−7312.09	−0.298214
$845$	−2929.11	−0.119248
$846$	0	0
$847$	0	0
$848$	−18954.0	−0.767552
$849$	0	0
$850$	1079.55	0.0435625
$851$	−1562.64	−0.0629455
$852$	0	0
$853$	−46429.3	−1.86367	−0.931833	−	0.362887i	$-0.881791\pi$
$853$	−46429.3	−1.86367	−0.931833	+	0.362887i	$0.881791\pi$
$854$	0	0
$855$	0	0
$856$	2874.67	0.114783
$857$	21206.4	0.845272	0.422636	−	0.906300i	$-0.361105\pi$
$857$	21206.4	0.845272	0.422636	+	0.906300i	$0.361105\pi$
$858$	0	0
$859$	−13876.2	−0.551163	−0.275581	−	0.961278i	$-0.588870\pi$
$859$	−13876.2	−0.551163	−0.275581	+	0.961278i	$0.588870\pi$
$860$	4744.96	0.188142
$861$	0	0
$862$	−9246.12	−0.365341
$863$	−14337.1	−0.565515	−0.282757	−	0.959191i	$-0.591249\pi$
$863$	−14337.1	−0.565515	−0.282757	+	0.959191i	$0.591249\pi$
$864$	0	0
$865$	−292.058	−0.0114801
$866$	39711.6	1.55826
$867$	0	0
$868$	0	0
$869$	4958.81	0.193574
$870$	0	0
$871$	17785.4	0.691889
$872$	20615.8	0.800619
$873$	0	0
$874$	−36632.8	−1.41776
$875$	0	0
$876$	0	0
$877$	−24369.3	−0.938304	−0.469152	−	0.883118i	$-0.655440\pi$
$877$	−24369.3	−0.938304	−0.469152	+	0.883118i	$0.655440\pi$
$878$	10454.1	0.401834
$879$	0	0
$880$	6196.65	0.237374
$881$	−26127.0	−0.999140	−0.499570	−	0.866273i	$-0.666509\pi$
$881$	−26127.0	−0.999140	−0.499570	+	0.866273i	$0.666509\pi$
$882$	0	0
$883$	−15713.1	−0.598855	−0.299428	−	0.954119i	$-0.596796\pi$
$883$	−15713.1	−0.598855	−0.299428	+	0.954119i	$0.596796\pi$
$884$	685.007	0.0260625
$885$	0	0
$886$	−696.611	−0.0264143
$887$	13139.5	0.497385	0.248692	−	0.968583i	$-0.419999\pi$
$887$	13139.5	0.497385	0.248692	+	0.968583i	$0.419999\pi$
$888$	0	0
$889$	0	0
$890$	7442.40	0.280303
$891$	0	0
$892$	−9141.48	−0.343138
$893$	32205.8	1.20686
$894$	0	0
$895$	17811.1	0.665207
$896$	0	0
$897$	0	0
$898$	20995.7	0.780218
$899$	54538.1	2.02330
$900$	0	0
$901$	2580.21	0.0954043
$902$	10359.6	0.382412
$903$	0	0
$904$	26757.4	0.984446
$905$	−9363.76	−0.343936
$906$	0	0
$907$	−3799.71	−0.139104	−0.0695519	−	0.997578i	$-0.522157\pi$
$907$	−3799.71	−0.139104	−0.0695519	+	0.997578i	$0.522157\pi$
$908$	15907.9	0.581413
$909$	0	0
$910$	0	0
$911$	51528.4	1.87400	0.936998	−	0.349334i	$-0.113592\pi$
$911$	51528.4	1.87400	0.936998	+	0.349334i	$0.113592\pi$
$912$	0	0
$913$	24841.0	0.900458
$914$	25006.4	0.904966
$915$	0	0
$916$	10650.0	0.384156
$917$	0	0
$918$	0	0
$919$	−16984.7	−0.609657	−0.304828	−	0.952407i	$-0.598599\pi$
$919$	−16984.7	−0.609657	−0.304828	+	0.952407i	$0.598599\pi$
$920$	14716.3	0.527373
$921$	0	0
$922$	−16806.8	−0.600329
$923$	31459.6	1.12189
$924$	0	0
$925$	−1240.85	−0.0441068
$926$	−23121.9	−0.820553
$927$	0	0
$928$	−26201.7	−0.926846
$929$	−5451.85	−0.192540	−0.0962699	−	0.995355i	$-0.530691\pi$
$929$	−5451.85	−0.192540	−0.0962699	+	0.995355i	$0.530691\pi$
$930$	0	0
$931$	0	0
$932$	−18532.9	−0.651358
$933$	0	0
$934$	35976.8	1.26038
$935$	−843.548	−0.0295048
$936$	0	0
$937$	−42429.4	−1.47930	−0.739652	−	0.672989i	$-0.765010\pi$
$937$	−42429.4	−1.47930	−0.739652	+	0.672989i	$0.765010\pi$
$938$	0	0
$939$	0	0
$940$	−3376.19	−0.117148
$941$	−32977.9	−1.14245	−0.571226	−	0.820793i	$-0.693532\pi$
$941$	−32977.9	−1.14245	−0.571226	+	0.820793i	$0.693532\pi$
$942$	0	0
$943$	14678.4	0.506886
$944$	−28071.5	−0.967848
$945$	0	0
$946$	−34224.8	−1.17626
$947$	23753.4	0.815082	0.407541	−	0.913187i	$-0.366386\pi$
$947$	23753.4	0.815082	0.407541	+	0.913187i	$0.366386\pi$
$948$	0	0
$949$	26153.9	0.894616
$950$	−29089.0	−0.993445
$951$	0	0
$952$	0	0
$953$	28074.3	0.954267	0.477134	−	0.878831i	$-0.341676\pi$
$953$	28074.3	0.954267	0.477134	+	0.878831i	$0.341676\pi$
$954$	0	0
$955$	1947.55	0.0659908
$956$	−2179.14	−0.0737221
$957$	0	0
$958$	3286.92	0.110851
$959$	0	0
$960$	0	0
$961$	33609.2	1.12817
$962$	1442.52	0.0483460
$963$	0	0
$964$	3536.88	0.118169
$965$	−7302.15	−0.243590
$966$	0	0
$967$	−11150.3	−0.370806	−0.185403	−	0.982663i	$-0.559359\pi$
$967$	−11150.3	−0.370806	−0.185403	+	0.982663i	$0.559359\pi$
$968$	8113.95	0.269414
$969$	0	0
$970$	6602.35	0.218545
$971$	6059.04	0.200251	0.100126	−	0.994975i	$-0.468076\pi$
$971$	6059.04	0.200251	0.100126	+	0.994975i	$0.468076\pi$
$972$	0	0
$973$	0	0
$974$	1113.35	0.0366263
$975$	0	0
$976$	−16236.6	−0.532500
$977$	−5700.49	−0.186668	−0.0933341	−	0.995635i	$-0.529752\pi$
$977$	−5700.49	−0.186668	−0.0933341	+	0.995635i	$0.529752\pi$
$978$	0	0
$979$	29300.2	0.956526
$980$	0	0
$981$	0	0
$982$	−8966.12	−0.291365
$983$	197.480	0.00640757	0.00320378	−	0.999995i	$-0.498980\pi$
$983$	197.480	0.00640757	0.00320378	+	0.999995i	$0.498980\pi$
$984$	0	0
$985$	17011.0	0.550271
$986$	−2241.90	−0.0724103
$987$	0	0
$988$	−18457.9	−0.594357
$989$	−48492.9	−1.55913
$990$	0	0
$991$	20620.8	0.660990	0.330495	−	0.943808i	$-0.392784\pi$
$991$	20620.8	0.660990	0.330495	+	0.943808i	$0.392784\pi$
$992$	−30459.3	−0.974884
$993$	0	0
$994$	0	0
$995$	−1588.03	−0.0505969
$996$	0	0
$997$	−19326.8	−0.613928	−0.306964	−	0.951721i	$-0.599313\pi$
$997$	−19326.8	−0.613928	−0.306964	+	0.951721i	$0.599313\pi$
$998$	−25.2275	−0.000800162 0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.4.a.r.1.1		2
3.2	odd	2		147.4.a.i.1.2		2
7.2	even	3		441.4.e.p.361.2		4
7.3	odd	6		441.4.e.q.226.2		4
7.4	even	3		441.4.e.p.226.2		4
7.5	odd	6		441.4.e.q.361.2		4
7.6	odd	2		63.4.a.e.1.1		2
12.11	even	2		2352.4.a.bz.1.2		2
21.2	odd	6		147.4.e.m.67.1		4
21.5	even	6		147.4.e.l.67.1		4
21.11	odd	6		147.4.e.m.79.1		4
21.17	even	6		147.4.e.l.79.1		4
21.20	even	2		21.4.a.c.1.2	✓	2
28.27	even	2		1008.4.a.ba.1.2		2
35.34	odd	2		1575.4.a.p.1.2		2
84.83	odd	2		336.4.a.m.1.1		2
105.62	odd	4		525.4.d.g.274.3		4
105.83	odd	4		525.4.d.g.274.2		4
105.104	even	2		525.4.a.n.1.1		2
168.83	odd	2		1344.4.a.bo.1.2		2
168.125	even	2		1344.4.a.bg.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.4.a.c.1.2	✓	2	21.20	even	2
63.4.a.e.1.1		2	7.6	odd	2
147.4.a.i.1.2		2	3.2	odd	2
147.4.e.l.67.1		4	21.5	even	6
147.4.e.l.79.1		4	21.17	even	6
147.4.e.m.67.1		4	21.2	odd	6
147.4.e.m.79.1		4	21.11	odd	6
336.4.a.m.1.1		2	84.83	odd	2
441.4.a.r.1.1		2	1.1	even	1	trivial
441.4.e.p.226.2		4	7.4	even	3
441.4.e.p.361.2		4	7.2	even	3
441.4.e.q.226.2		4	7.3	odd	6
441.4.e.q.361.2		4	7.5	odd	6
525.4.a.n.1.1		2	105.104	even	2
525.4.d.g.274.2		4	105.83	odd	4
525.4.d.g.274.3		4	105.62	odd	4
1008.4.a.ba.1.2		2	28.27	even	2
1344.4.a.bg.1.2		2	168.125	even	2
1344.4.a.bo.1.2		2	168.83	odd	2
1575.4.a.p.1.2		2	35.34	odd	2
2352.4.a.bz.1.2		2	12.11	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:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

\(f(q)\)	\(=\)	\(q-2.27492 q^{2} -2.82475 q^{4} -4.54983 q^{5} +24.6254 q^{8} +O(q^{10})\)	\(q-2.27492 q^{2} -2.82475 q^{4} -4.54983 q^{5} +24.6254 q^{8} +10.3505 q^{10} +40.7492 q^{11} -53.2990 q^{13} -33.4228 q^{16} +4.54983 q^{17} -122.598 q^{19} +12.8522 q^{20} -92.7010 q^{22} -131.347 q^{23} -104.299 q^{25} +121.251 q^{26} +216.598 q^{29} +251.794 q^{31} -120.969 q^{32} -10.3505 q^{34} +11.8970 q^{37} +278.900 q^{38} -112.042 q^{40} -111.752 q^{41} +369.196 q^{43} -115.106 q^{44} +298.804 q^{46} -262.694 q^{47} +237.272 q^{50} +150.556 q^{52} +567.100 q^{53} -185.402 q^{55} -492.743 q^{58} +839.890 q^{59} +485.794 q^{61} -572.811 q^{62} +542.577 q^{64} +242.502 q^{65} -333.691 q^{67} -12.8522 q^{68} -590.248 q^{71} -490.701 q^{73} -27.0647 q^{74} +346.309 q^{76} +121.691 q^{79} +152.068 q^{80} +254.228 q^{82} +609.608 q^{83} -20.7010 q^{85} -839.890 q^{86} +1003.47 q^{88} +719.038 q^{89} +371.023 q^{92} +597.608 q^{94} +557.801 q^{95} +637.877 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{2} + 17 q^{4} + 6 q^{5} + 87 q^{8}+O(q^{10}) \) 2 * q + 3 * q^2 + 17 * q^4 + 6 * q^5 + 87 * q^8	\( 2 q + 3 q^{2} + 17 q^{4} + 6 q^{5} + 87 q^{8} + 66 q^{10} + 6 q^{11} - 16 q^{13} + 137 q^{16} - 6 q^{17} - 64 q^{19} + 222 q^{20} - 276 q^{22} - 6 q^{23} - 118 q^{25} + 318 q^{26} + 252 q^{29} - 40 q^{31} + 279 q^{32} - 66 q^{34} - 248 q^{37} + 588 q^{38} + 546 q^{40} - 450 q^{41} + 376 q^{43} - 804 q^{44} + 960 q^{46} - 12 q^{47} + 165 q^{50} + 890 q^{52} + 1104 q^{53} - 552 q^{55} - 306 q^{58} + 804 q^{59} + 428 q^{61} - 2112 q^{62} + 1289 q^{64} + 636 q^{65} + 148 q^{67} - 222 q^{68} - 954 q^{71} - 1072 q^{73} - 1398 q^{74} + 1508 q^{76} - 572 q^{79} + 1950 q^{80} - 1530 q^{82} + 1944 q^{83} - 132 q^{85} - 804 q^{86} - 1164 q^{88} + 366 q^{89} + 2856 q^{92} + 1920 q^{94} + 1176 q^{95} - 808 q^{97}+O(q^{100}) \) 2 * q + 3 * q^2 + 17 * q^4 + 6 * q^5 + 87 * q^8 + 66 * q^10 + 6 * q^11 - 16 * q^13 + 137 * q^16 - 6 * q^17 - 64 * q^19 + 222 * q^20 - 276 * q^22 - 6 * q^23 - 118 * q^25 + 318 * q^26 + 252 * q^29 - 40 * q^31 + 279 * q^32 - 66 * q^34 - 248 * q^37 + 588 * q^38 + 546 * q^40 - 450 * q^41 + 376 * q^43 - 804 * q^44 + 960 * q^46 - 12 * q^47 + 165 * q^50 + 890 * q^52 + 1104 * q^53 - 552 * q^55 - 306 * q^58 + 804 * q^59 + 428 * q^61 - 2112 * q^62 + 1289 * q^64 + 636 * q^65 + 148 * q^67 - 222 * q^68 - 954 * q^71 - 1072 * q^73 - 1398 * q^74 + 1508 * q^76 - 572 * q^79 + 1950 * q^80 - 1530 * q^82 + 1944 * q^83 - 132 * q^85 - 804 * q^86 - 1164 * q^88 + 366 * q^89 + 2856 * q^92 + 1920 * q^94 + 1176 * q^95 - 808 * q^97

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