LMFDB - Embedded newform 784.4.a.be.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,4,Mod(1,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, 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("784.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	784.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:	$46.2574974445$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1929.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 10x + 13 $ x^3 - x^2 - 10*x + 13
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 56)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.27144$ of defining polynomial
Character	$\chi$	$=$	784.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.54289 q^{3} -15.8094 q^{5} +64.0667 q^{9} +O(q^{10})$	$q+9.54289 q^{3} -15.8094 q^{5} +64.0667 q^{9} -21.7994 q^{11} +21.3423 q^{13} -150.867 q^{15} +75.7910 q^{17} +21.2864 q^{19} +102.267 q^{23} +124.936 q^{25} +353.723 q^{27} +91.0008 q^{29} -66.8486 q^{31} -208.029 q^{33} -168.597 q^{37} +203.667 q^{39} +101.555 q^{41} +314.402 q^{43} -1012.85 q^{45} -269.345 q^{47} +723.265 q^{51} +308.859 q^{53} +344.635 q^{55} +203.133 q^{57} +808.744 q^{59} +653.765 q^{61} -337.408 q^{65} +473.860 q^{67} +975.925 q^{69} -157.998 q^{71} +270.077 q^{73} +1192.25 q^{75} -325.457 q^{79} +1645.74 q^{81} +1033.08 q^{83} -1198.21 q^{85} +868.410 q^{87} +149.769 q^{89} -637.928 q^{93} -336.524 q^{95} -1865.72 q^{97} -1396.62 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 7 q^{3} - 3 q^{5} + 18 q^{9}+O(q^{10}) $ 3 * q + 7 * q^3 - 3 * q^5 + 18 * q^9	$ 3 q + 7 q^{3} - 3 q^{5} + 18 q^{9} + 3 q^{11} - 26 q^{13} - 127 q^{15} - 31 q^{17} + 89 q^{19} + 201 q^{23} + 300 q^{25} + 469 q^{27} + 190 q^{29} + 339 q^{31} - 105 q^{33} - 535 q^{37} + 134 q^{39} + 58 q^{41} - 268 q^{43} - 1410 q^{45} + 205 q^{47} + 965 q^{51} + 757 q^{53} + 1653 q^{55} + 261 q^{57} + 1799 q^{59} + 625 q^{61} - 1750 q^{65} + 495 q^{67} + 973 q^{69} - 640 q^{71} - 443 q^{73} + 1484 q^{75} - 79 q^{79} + 2523 q^{81} + 2372 q^{83} - 977 q^{85} + 910 q^{87} + 821 q^{89} - 1321 q^{93} + 1327 q^{95} - 342 q^{97} - 2310 q^{99}+O(q^{100}) $ 3 * q + 7 * q^3 - 3 * q^5 + 18 * q^9 + 3 * q^11 - 26 * q^13 - 127 * q^15 - 31 * q^17 + 89 * q^19 + 201 * q^23 + 300 * q^25 + 469 * q^27 + 190 * q^29 + 339 * q^31 - 105 * q^33 - 535 * q^37 + 134 * q^39 + 58 * q^41 - 268 * q^43 - 1410 * q^45 + 205 * q^47 + 965 * q^51 + 757 * q^53 + 1653 * q^55 + 261 * q^57 + 1799 * q^59 + 625 * q^61 - 1750 * q^65 + 495 * q^67 + 973 * q^69 - 640 * q^71 - 443 * q^73 + 1484 * q^75 - 79 * q^79 + 2523 * q^81 + 2372 * q^83 - 977 * q^85 + 910 * q^87 + 821 * q^89 - 1321 * q^93 + 1327 * q^95 - 342 * q^97 - 2310 * 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
$3$	9.54289	1.83653	0.918265	−	0.395967i	$-0.129591\pi$
$3$	9.54289	1.83653	0.918265	+	0.395967i	$0.129591\pi$
$4$	0	0
$5$	−15.8094	−1.41403	−0.707016	−	0.707198i	$-0.749959\pi$
$5$	−15.8094	−1.41403	−0.707016	+	0.707198i	$0.749959\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	64.0667	2.37284
$10$	0	0
$11$	−21.7994	−0.597524	−0.298762	−	0.954328i	$-0.596574\pi$
$11$	−21.7994	−0.597524	−0.298762	+	0.954328i	$0.596574\pi$
$12$	0	0
$13$	21.3423	0.455330	0.227665	−	0.973740i	$-0.426891\pi$
$13$	21.3423	0.455330	0.227665	+	0.973740i	$0.426891\pi$
$14$	0	0
$15$	−150.867	−2.59691
$16$	0	0
$17$	75.7910	1.08130	0.540648	−	0.841249i	$-0.318179\pi$
$17$	75.7910	1.08130	0.540648	+	0.841249i	$0.318179\pi$
$18$	0	0
$19$	21.2864	0.257022	0.128511	−	0.991708i	$-0.458980\pi$
$19$	21.2864	0.257022	0.128511	+	0.991708i	$0.458980\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	102.267	0.927139	0.463570	−	0.886061i	$-0.346568\pi$
$23$	102.267	0.927139	0.463570	+	0.886061i	$0.346568\pi$
$24$	0	0
$25$	124.936	0.999486
$26$	0	0
$27$	353.723	2.52126
$28$	0	0
$29$	91.0008	0.582704	0.291352	−	0.956616i	$-0.405895\pi$
$29$	91.0008	0.582704	0.291352	+	0.956616i	$0.405895\pi$
$30$	0	0
$31$	−66.8486	−0.387302	−0.193651	−	0.981071i	$-0.562033\pi$
$31$	−66.8486	−0.387302	−0.193651	+	0.981071i	$0.562033\pi$
$32$	0	0
$33$	−208.029	−1.09737
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−168.597	−0.749114	−0.374557	−	0.927204i	$-0.622205\pi$
$37$	−168.597	−0.749114	−0.374557	+	0.927204i	$0.622205\pi$
$38$	0	0
$39$	203.667	0.836226
$40$	0	0
$41$	101.555	0.386836	0.193418	−	0.981116i	$-0.438043\pi$
$41$	101.555	0.386836	0.193418	+	0.981116i	$0.438043\pi$
$42$	0	0
$43$	314.402	1.11502	0.557509	−	0.830171i	$-0.311757\pi$
$43$	314.402	1.11502	0.557509	+	0.830171i	$0.311757\pi$
$44$	0	0
$45$	−1012.85	−3.35527
$46$	0	0
$47$	−269.345	−0.835915	−0.417957	−	0.908467i	$-0.637254\pi$
$47$	−269.345	−0.835915	−0.417957	+	0.908467i	$0.637254\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	723.265	1.98583
$52$	0	0
$53$	308.859	0.800473	0.400236	−	0.916412i	$-0.368928\pi$
$53$	308.859	0.800473	0.400236	+	0.916412i	$0.368928\pi$
$54$	0	0
$55$	344.635	0.844918
$56$	0	0
$57$	203.133	0.472029
$58$	0	0
$59$	808.744	1.78457	0.892284	−	0.451475i	$-0.149102\pi$
$59$	808.744	1.78457	0.892284	+	0.451475i	$0.149102\pi$
$60$	0	0
$61$	653.765	1.37223	0.686115	−	0.727493i	$-0.259315\pi$
$61$	653.765	1.37223	0.686115	+	0.727493i	$0.259315\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−337.408	−0.643851
$66$	0	0
$67$	473.860	0.864048	0.432024	−	0.901862i	$-0.357800\pi$
$67$	473.860	0.864048	0.432024	+	0.901862i	$0.357800\pi$
$68$	0	0
$69$	975.925	1.70272
$70$	0	0
$71$	−157.998	−0.264098	−0.132049	−	0.991243i	$-0.542156\pi$
$71$	−157.998	−0.264098	−0.132049	+	0.991243i	$0.542156\pi$
$72$	0	0
$73$	270.077	0.433015	0.216508	−	0.976281i	$-0.430533\pi$
$73$	270.077	0.433015	0.216508	+	0.976281i	$0.430533\pi$
$74$	0	0
$75$	1192.25	1.83558
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−325.457	−0.463504	−0.231752	−	0.972775i	$-0.574446\pi$
$79$	−325.457	−0.463504	−0.231752	+	0.972775i	$0.574446\pi$
$80$	0	0
$81$	1645.74	2.25753
$82$	0	0
$83$	1033.08	1.36621	0.683104	−	0.730321i	$-0.260630\pi$
$83$	1033.08	1.36621	0.683104	+	0.730321i	$0.260630\pi$
$84$	0	0
$85$	−1198.21	−1.52899
$86$	0	0
$87$	868.410	1.07015
$88$	0	0
$89$	149.769	0.178376	0.0891879	−	0.996015i	$-0.471573\pi$
$89$	149.769	0.178376	0.0891879	+	0.996015i	$0.471573\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−637.928	−0.711291
$94$	0	0
$95$	−336.524	−0.363438
$96$	0	0
$97$	−1865.72	−1.95294	−0.976468	−	0.215663i	$-0.930809\pi$
$97$	−1865.72	−1.95294	−0.976468	+	0.215663i	$0.930809\pi$
$98$	0	0
$99$	−1396.62	−1.41783
$100$	0	0
$101$	534.549	0.526630	0.263315	−	0.964710i	$-0.415184\pi$
$101$	534.549	0.526630	0.263315	+	0.964710i	$0.415184\pi$
$102$	0	0
$103$	1319.62	1.26239	0.631193	−	0.775626i	$-0.282566\pi$
$103$	1319.62	1.26239	0.631193	+	0.775626i	$0.282566\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−693.556	−0.626622	−0.313311	−	0.949651i	$-0.601438\pi$
$107$	−693.556	−0.626622	−0.313311	+	0.949651i	$0.601438\pi$
$108$	0	0
$109$	−1965.99	−1.72760	−0.863798	−	0.503838i	$-0.831921\pi$
$109$	−1965.99	−1.72760	−0.863798	+	0.503838i	$0.831921\pi$
$110$	0	0
$111$	−1608.90	−1.37577
$112$	0	0
$113$	2167.14	1.80414	0.902068	−	0.431595i	$-0.142049\pi$
$113$	2167.14	1.80414	0.902068	+	0.431595i	$0.142049\pi$
$114$	0	0
$115$	−1616.78	−1.31100
$116$	0	0
$117$	1367.33	1.08042
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−855.786	−0.642965
$122$	0	0
$123$	969.130	0.710435
$124$	0	0
$125$	1.01640	0.000727276 0
$126$	0	0
$127$	−1443.60	−1.00865	−0.504324	−	0.863514i	$-0.668258\pi$
$127$	−1443.60	−1.00865	−0.504324	+	0.863514i	$0.668258\pi$
$128$	0	0
$129$	3000.30	2.04776
$130$	0	0
$131$	−2125.28	−1.41746	−0.708728	−	0.705481i	$-0.750731\pi$
$131$	−2125.28	−1.41746	−0.708728	+	0.705481i	$0.750731\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−5592.13	−3.56514
$136$	0	0
$137$	−405.475	−0.252862	−0.126431	−	0.991975i	$-0.540352\pi$
$137$	−405.475	−0.252862	−0.126431	+	0.991975i	$0.540352\pi$
$138$	0	0
$139$	1812.24	1.10585	0.552923	−	0.833232i	$-0.313512\pi$
$139$	1812.24	1.10585	0.552923	+	0.833232i	$0.313512\pi$
$140$	0	0
$141$	−2570.33	−1.53518
$142$	0	0
$143$	−465.249	−0.272071
$144$	0	0
$145$	−1438.66	−0.823962
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2892.57	−1.59039	−0.795197	−	0.606351i	$-0.792633\pi$
$149$	−2892.57	−1.59039	−0.795197	+	0.606351i	$0.792633\pi$
$150$	0	0
$151$	−1239.27	−0.667882	−0.333941	−	0.942594i	$-0.608379\pi$
$151$	−1239.27	−0.667882	−0.333941	+	0.942594i	$0.608379\pi$
$152$	0	0
$153$	4855.68	2.56574
$154$	0	0
$155$	1056.83	0.547657
$156$	0	0
$157$	−1783.37	−0.906551	−0.453276	−	0.891370i	$-0.649745\pi$
$157$	−1783.37	−0.906551	−0.453276	+	0.891370i	$0.649745\pi$
$158$	0	0
$159$	2947.41	1.47009
$160$	0	0
$161$	0	0
$162$	0	0
$163$	1279.37	0.614771	0.307385	−	0.951585i	$-0.400546\pi$
$163$	1279.37	0.614771	0.307385	+	0.951585i	$0.400546\pi$
$164$	0	0
$165$	3288.81	1.55172
$166$	0	0
$167$	971.264	0.450052	0.225026	−	0.974353i	$-0.427753\pi$
$167$	971.264	0.450052	0.225026	+	0.974353i	$0.427753\pi$
$168$	0	0
$169$	−1741.51	−0.792675
$170$	0	0
$171$	1363.75	0.609873
$172$	0	0
$173$	528.846	0.232413	0.116207	−	0.993225i	$-0.462927\pi$
$173$	528.846	0.232413	0.116207	+	0.993225i	$0.462927\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	7717.75	3.27741
$178$	0	0
$179$	−149.000	−0.0622168	−0.0311084	−	0.999516i	$-0.509904\pi$
$179$	−149.000	−0.0622168	−0.0311084	+	0.999516i	$0.509904\pi$
$180$	0	0
$181$	3624.17	1.48830	0.744151	−	0.668011i	$-0.232854\pi$
$181$	3624.17	1.48830	0.744151	+	0.668011i	$0.232854\pi$
$182$	0	0
$183$	6238.80	2.52014
$184$	0	0
$185$	2665.41	1.05927
$186$	0	0
$187$	−1652.20	−0.646101
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3184.60	1.20644	0.603218	−	0.797576i	$-0.293885\pi$
$191$	3184.60	1.20644	0.603218	+	0.797576i	$0.293885\pi$
$192$	0	0
$193$	−4597.63	−1.71474	−0.857370	−	0.514701i	$-0.827903\pi$
$193$	−4597.63	−1.71474	−0.857370	+	0.514701i	$0.827903\pi$
$194$	0	0
$195$	−3219.85	−1.18245
$196$	0	0
$197$	−3877.84	−1.40246	−0.701230	−	0.712935i	$-0.747365\pi$
$197$	−3877.84	−1.40246	−0.701230	+	0.712935i	$0.747365\pi$
$198$	0	0
$199$	−1976.12	−0.703938	−0.351969	−	0.936012i	$-0.614488\pi$
$199$	−1976.12	−0.703938	−0.351969	+	0.936012i	$0.614488\pi$
$200$	0	0
$201$	4521.99	1.58685
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−1605.52	−0.546998
$206$	0	0
$207$	6551.92	2.19995
$208$	0	0
$209$	−464.030	−0.153577
$210$	0	0
$211$	−4169.50	−1.36038	−0.680190	−	0.733036i	$-0.738103\pi$
$211$	−4169.50	−1.36038	−0.680190	+	0.733036i	$0.738103\pi$
$212$	0	0
$213$	−1507.76	−0.485024
$214$	0	0
$215$	−4970.49	−1.57667
$216$	0	0
$217$	0	0
$218$	0	0
$219$	2577.31	0.795245
$220$	0	0
$221$	1617.55	0.492346
$222$	0	0
$223$	1595.43	0.479093	0.239547	−	0.970885i	$-0.423001\pi$
$223$	1595.43	0.479093	0.239547	+	0.970885i	$0.423001\pi$
$224$	0	0
$225$	8004.21	2.37162
$226$	0	0
$227$	857.809	0.250814	0.125407	−	0.992105i	$-0.459976\pi$
$227$	857.809	0.250814	0.125407	+	0.992105i	$0.459976\pi$
$228$	0	0
$229$	78.4879	0.0226490	0.0113245	−	0.999936i	$-0.496395\pi$
$229$	78.4879	0.0226490	0.0113245	+	0.999936i	$0.496395\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	659.765	0.185505	0.0927524	−	0.995689i	$-0.470433\pi$
$233$	659.765	0.185505	0.0927524	+	0.995689i	$0.470433\pi$
$234$	0	0
$235$	4258.17	1.18201
$236$	0	0
$237$	−3105.80	−0.851239
$238$	0	0
$239$	2879.88	0.779430	0.389715	−	0.920935i	$-0.372573\pi$
$239$	2879.88	0.779430	0.389715	+	0.920935i	$0.372573\pi$
$240$	0	0
$241$	−2334.52	−0.623981	−0.311991	−	0.950085i	$-0.600996\pi$
$241$	−2334.52	−0.623981	−0.311991	+	0.950085i	$0.600996\pi$
$242$	0	0
$243$	6154.57	1.62476
$244$	0	0
$245$	0	0
$246$	0	0
$247$	454.300	0.117030
$248$	0	0
$249$	9858.56	2.50908
$250$	0	0
$251$	1386.81	0.348743	0.174372	−	0.984680i	$-0.444211\pi$
$251$	1386.81	0.348743	0.174372	+	0.984680i	$0.444211\pi$
$252$	0	0
$253$	−2229.37	−0.553988
$254$	0	0
$255$	−11434.4	−2.80803
$256$	0	0
$257$	−6053.37	−1.46926	−0.734628	−	0.678470i	$-0.762643\pi$
$257$	−6053.37	−1.46926	−0.734628	+	0.678470i	$0.762643\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	5830.12	1.38266
$262$	0	0
$263$	1003.42	0.235260	0.117630	−	0.993058i	$-0.462470\pi$
$263$	1003.42	0.235260	0.117630	+	0.993058i	$0.462470\pi$
$264$	0	0
$265$	−4882.86	−1.13189
$266$	0	0
$267$	1429.22	0.327592
$268$	0	0
$269$	3512.07	0.796040	0.398020	−	0.917377i	$-0.369698\pi$
$269$	3512.07	0.796040	0.398020	+	0.917377i	$0.369698\pi$
$270$	0	0
$271$	−38.9172	−0.00872343	−0.00436172	−	0.999990i	$-0.501388\pi$
$271$	−38.9172	−0.00872343	−0.00436172	+	0.999990i	$0.501388\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2723.52	−0.597217
$276$	0	0
$277$	−195.950	−0.0425036	−0.0212518	−	0.999774i	$-0.506765\pi$
$277$	−195.950	−0.0425036	−0.0212518	+	0.999774i	$0.506765\pi$
$278$	0	0
$279$	−4282.76	−0.919005
$280$	0	0
$281$	809.583	0.171871	0.0859354	−	0.996301i	$-0.472612\pi$
$281$	809.583	0.171871	0.0859354	+	0.996301i	$0.472612\pi$
$282$	0	0
$283$	3798.61	0.797894	0.398947	−	0.916974i	$-0.369376\pi$
$283$	3798.61	0.797894	0.398947	+	0.916974i	$0.369376\pi$
$284$	0	0
$285$	−3211.41	−0.667464
$286$	0	0
$287$	0	0
$288$	0	0
$289$	831.283	0.169201
$290$	0	0
$291$	−17804.3	−3.58662
$292$	0	0
$293$	3143.28	0.626731	0.313366	−	0.949633i	$-0.398543\pi$
$293$	3143.28	0.626731	0.313366	+	0.949633i	$0.398543\pi$
$294$	0	0
$295$	−12785.7	−2.52343
$296$	0	0
$297$	−7710.95	−1.50651
$298$	0	0
$299$	2182.62	0.422154
$300$	0	0
$301$	0	0
$302$	0	0
$303$	5101.14	0.967172
$304$	0	0
$305$	−10335.6	−1.94038
$306$	0	0
$307$	7726.54	1.43641	0.718203	−	0.695833i	$-0.244965\pi$
$307$	7726.54	1.43641	0.718203	+	0.695833i	$0.244965\pi$
$308$	0	0
$309$	12593.0	2.31841
$310$	0	0
$311$	1477.91	0.269467	0.134734	−	0.990882i	$-0.456982\pi$
$311$	1477.91	0.269467	0.134734	+	0.990882i	$0.456982\pi$
$312$	0	0
$313$	−6821.72	−1.23191	−0.615953	−	0.787783i	$-0.711229\pi$
$313$	−6821.72	−1.23191	−0.615953	+	0.787783i	$0.711229\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−737.222	−0.130620	−0.0653099	−	0.997865i	$-0.520804\pi$
$317$	−737.222	−0.130620	−0.0653099	+	0.997865i	$0.520804\pi$
$318$	0	0
$319$	−1983.76	−0.348180
$320$	0	0
$321$	−6618.53	−1.15081
$322$	0	0
$323$	1613.32	0.277917
$324$	0	0
$325$	2666.41	0.455096
$326$	0	0
$327$	−18761.3	−3.17278
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−3664.28	−0.608481	−0.304241	−	0.952595i	$-0.598403\pi$
$331$	−3664.28	−0.608481	−0.304241	+	0.952595i	$0.598403\pi$
$332$	0	0
$333$	−10801.5	−1.77753
$334$	0	0
$335$	−7491.42	−1.22179
$336$	0	0
$337$	−3269.01	−0.528411	−0.264206	−	0.964466i	$-0.585110\pi$
$337$	−3269.01	−0.528411	−0.264206	+	0.964466i	$0.585110\pi$
$338$	0	0
$339$	20680.8	3.31335
$340$	0	0
$341$	1457.26	0.231422
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−15428.7	−2.40770
$346$	0	0
$347$	−9042.54	−1.39893	−0.699465	−	0.714666i	$-0.746579\pi$
$347$	−9042.54	−1.39893	−0.699465	+	0.714666i	$0.746579\pi$
$348$	0	0
$349$	−5151.89	−0.790185	−0.395092	−	0.918641i	$-0.629287\pi$
$349$	−5151.89	−0.790185	−0.395092	+	0.918641i	$0.629287\pi$
$350$	0	0
$351$	7549.26	1.14800
$352$	0	0
$353$	2142.35	0.323019	0.161510	−	0.986871i	$-0.448364\pi$
$353$	2142.35	0.323019	0.161510	+	0.986871i	$0.448364\pi$
$354$	0	0
$355$	2497.85	0.373443
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1429.97	0.210226	0.105113	−	0.994460i	$-0.466480\pi$
$359$	1429.97	0.210226	0.105113	+	0.994460i	$0.466480\pi$
$360$	0	0
$361$	−6405.89	−0.933939
$362$	0	0
$363$	−8166.67	−1.18082
$364$	0	0
$365$	−4269.74	−0.612297
$366$	0	0
$367$	3020.31	0.429588	0.214794	−	0.976659i	$-0.431092\pi$
$367$	3020.31	0.429588	0.214794	+	0.976659i	$0.431092\pi$
$368$	0	0
$369$	6506.30	0.917899
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−1647.11	−0.228643	−0.114322	−	0.993444i	$-0.536469\pi$
$373$	−1647.11	−0.228643	−0.114322	+	0.993444i	$0.536469\pi$
$374$	0	0
$375$	9.69938	0.00133566
$376$	0	0
$377$	1942.17	0.265323
$378$	0	0
$379$	4385.44	0.594366	0.297183	−	0.954821i	$-0.403953\pi$
$379$	4385.44	0.594366	0.297183	+	0.954821i	$0.403953\pi$
$380$	0	0
$381$	−13776.1	−1.85241
$382$	0	0
$383$	14389.0	1.91969	0.959845	−	0.280532i	$-0.0905108\pi$
$383$	14389.0	1.91969	0.959845	+	0.280532i	$0.0905108\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	20142.7	2.64576
$388$	0	0
$389$	−2665.56	−0.347428	−0.173714	−	0.984796i	$-0.555577\pi$
$389$	−2665.56	−0.347428	−0.173714	+	0.984796i	$0.555577\pi$
$390$	0	0
$391$	7750.94	1.00251
$392$	0	0
$393$	−20281.3	−2.60320
$394$	0	0
$395$	5145.27	0.655409
$396$	0	0
$397$	−4457.71	−0.563542	−0.281771	−	0.959482i	$-0.590922\pi$
$397$	−4457.71	−0.563542	−0.281771	+	0.959482i	$0.590922\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−13313.8	−1.65800	−0.829001	−	0.559246i	$-0.811091\pi$
$401$	−13313.8	−1.65800	−0.829001	+	0.559246i	$0.811091\pi$
$402$	0	0
$403$	−1426.70	−0.176350
$404$	0	0
$405$	−26018.1	−3.19222
$406$	0	0
$407$	3675.32	0.447614
$408$	0	0
$409$	−4092.23	−0.494738	−0.247369	−	0.968921i	$-0.579566\pi$
$409$	−4092.23	−0.494738	−0.247369	+	0.968921i	$0.579566\pi$
$410$	0	0
$411$	−3869.40	−0.464388
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−16332.3	−1.93186
$416$	0	0
$417$	17294.0	2.03092
$418$	0	0
$419$	−8297.69	−0.967467	−0.483733	−	0.875215i	$-0.660720\pi$
$419$	−8297.69	−0.967467	−0.483733	+	0.875215i	$0.660720\pi$
$420$	0	0
$421$	14729.2	1.70512	0.852561	−	0.522627i	$-0.175048\pi$
$421$	14729.2	1.70512	0.852561	+	0.522627i	$0.175048\pi$
$422$	0	0
$423$	−17256.0	−1.98349
$424$	0	0
$425$	9469.01	1.08074
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−4439.82	−0.499666
$430$	0	0
$431$	−4940.51	−0.552149	−0.276074	−	0.961136i	$-0.589034\pi$
$431$	−4940.51	−0.552149	−0.276074	+	0.961136i	$0.589034\pi$
$432$	0	0
$433$	−8291.22	−0.920209	−0.460105	−	0.887865i	$-0.652188\pi$
$433$	−8291.22	−0.920209	−0.460105	+	0.887865i	$0.652188\pi$
$434$	0	0
$435$	−13729.0	−1.51323
$436$	0	0
$437$	2176.90	0.238296
$438$	0	0
$439$	−8097.71	−0.880370	−0.440185	−	0.897907i	$-0.645087\pi$
$439$	−8097.71	−0.880370	−0.440185	+	0.897907i	$0.645087\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	6459.15	0.692739	0.346370	−	0.938098i	$-0.387414\pi$
$443$	6459.15	0.692739	0.346370	+	0.938098i	$0.387414\pi$
$444$	0	0
$445$	−2367.75	−0.252229
$446$	0	0
$447$	−27603.5	−2.92080
$448$	0	0
$449$	4951.16	0.520401	0.260200	−	0.965555i	$-0.416211\pi$
$449$	4951.16	0.520401	0.260200	+	0.965555i	$0.416211\pi$
$450$	0	0
$451$	−2213.84	−0.231144
$452$	0	0
$453$	−11826.2	−1.22658
$454$	0	0
$455$	0	0
$456$	0	0
$457$	15817.5	1.61906	0.809530	−	0.587078i	$-0.199722\pi$
$457$	15817.5	1.61906	0.809530	+	0.587078i	$0.199722\pi$
$458$	0	0
$459$	26809.0	2.72623
$460$	0	0
$461$	8584.25	0.867263	0.433632	−	0.901090i	$-0.357232\pi$
$461$	8584.25	0.867263	0.433632	+	0.901090i	$0.357232\pi$
$462$	0	0
$463$	15304.5	1.53620	0.768101	−	0.640329i	$-0.221202\pi$
$463$	15304.5	1.53620	0.768101	+	0.640329i	$0.221202\pi$
$464$	0	0
$465$	10085.2	1.00579
$466$	0	0
$467$	−8541.74	−0.846391	−0.423196	−	0.906038i	$-0.639092\pi$
$467$	−8541.74	−0.846391	−0.423196	+	0.906038i	$0.639092\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−17018.5	−1.66491
$472$	0	0
$473$	−6853.77	−0.666251
$474$	0	0
$475$	2659.43	0.256890
$476$	0	0
$477$	19787.6	1.89939
$478$	0	0
$479$	−2298.21	−0.219224	−0.109612	−	0.993974i	$-0.534961\pi$
$479$	−2298.21	−0.219224	−0.109612	+	0.993974i	$0.534961\pi$
$480$	0	0
$481$	−3598.25	−0.341094
$482$	0	0
$483$	0	0
$484$	0	0
$485$	29495.8	2.76151
$486$	0	0
$487$	−11019.2	−1.02531	−0.512657	−	0.858593i	$-0.671339\pi$
$487$	−11019.2	−1.02531	−0.512657	+	0.858593i	$0.671339\pi$
$488$	0	0
$489$	12208.8	1.12904
$490$	0	0
$491$	−17105.3	−1.57221	−0.786103	−	0.618096i	$-0.787904\pi$
$491$	−17105.3	−1.57221	−0.786103	+	0.618096i	$0.787904\pi$
$492$	0	0
$493$	6897.05	0.630076
$494$	0	0
$495$	22079.6	2.00486
$496$	0	0
$497$	0	0
$498$	0	0
$499$	1293.00	0.115997	0.0579987	−	0.998317i	$-0.481528\pi$
$499$	1293.00	0.115997	0.0579987	+	0.998317i	$0.481528\pi$
$500$	0	0
$501$	9268.66	0.826534
$502$	0	0
$503$	258.772	0.0229385	0.0114692	−	0.999934i	$-0.496349\pi$
$503$	258.772	0.0229385	0.0114692	+	0.999934i	$0.496349\pi$
$504$	0	0
$505$	−8450.88	−0.744672
$506$	0	0
$507$	−16619.0	−1.45577
$508$	0	0
$509$	16265.0	1.41637	0.708186	−	0.706026i	$-0.249514\pi$
$509$	16265.0	1.41637	0.708186	+	0.706026i	$0.249514\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	7529.48	0.648020
$514$	0	0
$515$	−20862.3	−1.78505
$516$	0	0
$517$	5871.56	0.499479
$518$	0	0
$519$	5046.72	0.426833
$520$	0	0
$521$	−16644.7	−1.39965	−0.699826	−	0.714313i	$-0.746739\pi$
$521$	−16644.7	−1.39965	−0.699826	+	0.714313i	$0.746739\pi$
$522$	0	0
$523$	10274.4	0.859018	0.429509	−	0.903063i	$-0.358687\pi$
$523$	10274.4	0.859018	0.429509	+	0.903063i	$0.358687\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5066.52	−0.418788
$528$	0	0
$529$	−1708.41	−0.140413
$530$	0	0
$531$	51813.5	4.23449
$532$	0	0
$533$	2167.42	0.176138
$534$	0	0
$535$	10964.7	0.886064
$536$	0	0
$537$	−1421.89	−0.114263
$538$	0	0
$539$	0	0
$540$	0	0
$541$	17436.3	1.38567	0.692833	−	0.721098i	$-0.256362\pi$
$541$	17436.3	1.38567	0.692833	+	0.721098i	$0.256362\pi$
$542$	0	0
$543$	34585.1	2.73331
$544$	0	0
$545$	31081.1	2.44288
$546$	0	0
$547$	−3708.40	−0.289871	−0.144936	−	0.989441i	$-0.546297\pi$
$547$	−3708.40	−0.289871	−0.144936	+	0.989441i	$0.546297\pi$
$548$	0	0
$549$	41884.5	3.25608
$550$	0	0
$551$	1937.08	0.149768
$552$	0	0
$553$	0	0
$554$	0	0
$555$	25435.7	1.94538
$556$	0	0
$557$	19921.6	1.51545	0.757725	−	0.652574i	$-0.226311\pi$
$557$	19921.6	1.51545	0.757725	+	0.652574i	$0.226311\pi$
$558$	0	0
$559$	6710.05	0.507701
$560$	0	0
$561$	−15766.8	−1.18658
$562$	0	0
$563$	−6852.74	−0.512981	−0.256491	−	0.966547i	$-0.582566\pi$
$563$	−6852.74	−0.512981	−0.256491	+	0.966547i	$0.582566\pi$
$564$	0	0
$565$	−34261.1	−2.55110
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−5473.04	−0.403237	−0.201618	−	0.979464i	$-0.564620\pi$
$569$	−5473.04	−0.403237	−0.201618	+	0.979464i	$0.564620\pi$
$570$	0	0
$571$	21607.0	1.58358	0.791792	−	0.610791i	$-0.209149\pi$
$571$	21607.0	1.58358	0.791792	+	0.610791i	$0.209149\pi$
$572$	0	0
$573$	30390.2	2.21565
$574$	0	0
$575$	12776.8	0.926662
$576$	0	0
$577$	1160.90	0.0837587	0.0418794	−	0.999123i	$-0.486665\pi$
$577$	1160.90	0.0837587	0.0418794	+	0.999123i	$0.486665\pi$
$578$	0	0
$579$	−43874.7	−3.14917
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−6732.95	−0.478302
$584$	0	0
$585$	−21616.6	−1.52775
$586$	0	0
$587$	6275.52	0.441258	0.220629	−	0.975358i	$-0.429189\pi$
$587$	6275.52	0.441258	0.220629	+	0.975358i	$0.429189\pi$
$588$	0	0
$589$	−1422.96	−0.0995453
$590$	0	0
$591$	−37005.8	−2.57566
$592$	0	0
$593$	14027.2	0.971378	0.485689	−	0.874132i	$-0.338569\pi$
$593$	14027.2	0.971378	0.485689	+	0.874132i	$0.338569\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−18857.9	−1.29280
$598$	0	0
$599$	−22827.0	−1.55707	−0.778535	−	0.627601i	$-0.784037\pi$
$599$	−22827.0	−1.55707	−0.778535	+	0.627601i	$0.784037\pi$
$600$	0	0
$601$	−14056.5	−0.954036	−0.477018	−	0.878894i	$-0.658282\pi$
$601$	−14056.5	−0.954036	−0.477018	+	0.878894i	$0.658282\pi$
$602$	0	0
$603$	30358.6	2.05025
$604$	0	0
$605$	13529.4	0.909172
$606$	0	0
$607$	−20502.3	−1.37094	−0.685472	−	0.728099i	$-0.740404\pi$
$607$	−20502.3	−1.37094	−0.685472	+	0.728099i	$0.740404\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−5748.44	−0.380617
$612$	0	0
$613$	−12496.9	−0.823403	−0.411702	−	0.911319i	$-0.635065\pi$
$613$	−12496.9	−0.823403	−0.411702	+	0.911319i	$0.635065\pi$
$614$	0	0
$615$	−15321.3	−1.00458
$616$	0	0
$617$	3715.94	0.242460	0.121230	−	0.992624i	$-0.461316\pi$
$617$	3715.94	0.242460	0.121230	+	0.992624i	$0.461316\pi$
$618$	0	0
$619$	1497.30	0.0972236	0.0486118	−	0.998818i	$-0.484520\pi$
$619$	1497.30	0.0972236	0.0486118	+	0.998818i	$0.484520\pi$
$620$	0	0
$621$	36174.3	2.33756
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−15633.0	−1.00051
$626$	0	0
$627$	−4428.19	−0.282049
$628$	0	0
$629$	−12778.2	−0.810014
$630$	0	0
$631$	−27717.6	−1.74869	−0.874343	−	0.485309i	$-0.838707\pi$
$631$	−27717.6	−1.74869	−0.874343	+	0.485309i	$0.838707\pi$
$632$	0	0
$633$	−39789.0	−2.49838
$634$	0	0
$635$	22822.3	1.42626
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−10122.4	−0.626662
$640$	0	0
$641$	11878.5	0.731941	0.365971	−	0.930626i	$-0.380737\pi$
$641$	11878.5	0.731941	0.365971	+	0.930626i	$0.380737\pi$
$642$	0	0
$643$	−21166.1	−1.29815	−0.649073	−	0.760726i	$-0.724843\pi$
$643$	−21166.1	−1.29815	−0.649073	+	0.760726i	$0.724843\pi$
$644$	0	0
$645$	−47432.8	−2.89560
$646$	0	0
$647$	17912.6	1.08844	0.544218	−	0.838944i	$-0.316827\pi$
$647$	17912.6	1.08844	0.544218	+	0.838944i	$0.316827\pi$
$648$	0	0
$649$	−17630.1	−1.06632
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−4606.94	−0.276085	−0.138043	−	0.990426i	$-0.544081\pi$
$653$	−4606.94	−0.276085	−0.138043	+	0.990426i	$0.544081\pi$
$654$	0	0
$655$	33599.4	2.00433
$656$	0	0
$657$	17302.9	1.02748
$658$	0	0
$659$	−6753.57	−0.399213	−0.199607	−	0.979876i	$-0.563966\pi$
$659$	−6753.57	−0.399213	−0.199607	+	0.979876i	$0.563966\pi$
$660$	0	0
$661$	−31888.8	−1.87645	−0.938223	−	0.346031i	$-0.887529\pi$
$661$	−31888.8	−1.87645	−0.938223	+	0.346031i	$0.887529\pi$
$662$	0	0
$663$	15436.1	0.904208
$664$	0	0
$665$	0	0
$666$	0	0
$667$	9306.40	0.540248
$668$	0	0
$669$	15225.0	0.879868
$670$	0	0
$671$	−14251.7	−0.819940
$672$	0	0
$673$	4332.88	0.248173	0.124086	−	0.992271i	$-0.460400\pi$
$673$	4332.88	0.248173	0.124086	+	0.992271i	$0.460400\pi$
$674$	0	0
$675$	44192.6	2.51996
$676$	0	0
$677$	−1484.13	−0.0842536	−0.0421268	−	0.999112i	$-0.513413\pi$
$677$	−1484.13	−0.0842536	−0.0421268	+	0.999112i	$0.513413\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	8185.98	0.460627
$682$	0	0
$683$	3706.31	0.207640	0.103820	−	0.994596i	$-0.466893\pi$
$683$	3706.31	0.207640	0.103820	+	0.994596i	$0.466893\pi$
$684$	0	0
$685$	6410.29	0.357554
$686$	0	0
$687$	749.001	0.0415956
$688$	0	0
$689$	6591.76	0.364479
$690$	0	0
$691$	25803.2	1.42055	0.710276	−	0.703923i	$-0.248570\pi$
$691$	25803.2	1.42055	0.710276	+	0.703923i	$0.248570\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−28650.4	−1.56370
$696$	0	0
$697$	7696.98	0.418284
$698$	0	0
$699$	6296.06	0.340685
$700$	0	0
$701$	−16440.9	−0.885827	−0.442914	−	0.896564i	$-0.646055\pi$
$701$	−16440.9	−0.885827	−0.442914	+	0.896564i	$0.646055\pi$
$702$	0	0
$703$	−3588.82	−0.192539
$704$	0	0
$705$	40635.2	2.17080
$706$	0	0
$707$	0	0
$708$	0	0
$709$	16621.6	0.880447	0.440223	−	0.897888i	$-0.354899\pi$
$709$	16621.6	0.880447	0.440223	+	0.897888i	$0.354899\pi$
$710$	0	0
$711$	−20851.0	−1.09982
$712$	0	0
$713$	−6836.42	−0.359083
$714$	0	0
$715$	7355.29	0.384717
$716$	0	0
$717$	27482.3	1.43145
$718$	0	0
$719$	−17426.7	−0.903904	−0.451952	−	0.892042i	$-0.649272\pi$
$719$	−17426.7	−0.903904	−0.451952	+	0.892042i	$0.649272\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−22278.0	−1.14596
$724$	0	0
$725$	11369.2	0.582405
$726$	0	0
$727$	19950.2	1.01776	0.508879	−	0.860838i	$-0.330060\pi$
$727$	19950.2	1.01776	0.508879	+	0.860838i	$0.330060\pi$
$728$	0	0
$729$	14297.4	0.726385
$730$	0	0
$731$	23828.8	1.20566
$732$	0	0
$733$	−12706.3	−0.640270	−0.320135	−	0.947372i	$-0.603728\pi$
$733$	−12706.3	−0.640270	−0.320135	+	0.947372i	$0.603728\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−10329.9	−0.516290
$738$	0	0
$739$	15512.8	0.772188	0.386094	−	0.922460i	$-0.373824\pi$
$739$	15512.8	0.772188	0.386094	+	0.922460i	$0.373824\pi$
$740$	0	0
$741$	4335.33	0.214929
$742$	0	0
$743$	−1559.42	−0.0769981	−0.0384991	−	0.999259i	$-0.512258\pi$
$743$	−1559.42	−0.0769981	−0.0384991	+	0.999259i	$0.512258\pi$
$744$	0	0
$745$	45729.7	2.24887
$746$	0	0
$747$	66186.0	3.24179
$748$	0	0
$749$	0	0
$750$	0	0
$751$	10347.3	0.502766	0.251383	−	0.967888i	$-0.419115\pi$
$751$	10347.3	0.502766	0.251383	+	0.967888i	$0.419115\pi$
$752$	0	0
$753$	13234.2	0.640477
$754$	0	0
$755$	19592.0	0.944406
$756$	0	0
$757$	40281.7	1.93403	0.967017	−	0.254711i	$-0.0819805\pi$
$757$	40281.7	1.93403	0.967017	+	0.254711i	$0.0819805\pi$
$758$	0	0
$759$	−21274.6	−1.01742
$760$	0	0
$761$	8577.78	0.408600	0.204300	−	0.978908i	$-0.434508\pi$
$761$	8577.78	0.408600	0.204300	+	0.978908i	$0.434508\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−76765.2	−3.62804
$766$	0	0
$767$	17260.4	0.812567
$768$	0	0
$769$	−10747.8	−0.504001	−0.252000	−	0.967727i	$-0.581088\pi$
$769$	−10747.8	−0.504001	−0.252000	+	0.967727i	$0.581088\pi$
$770$	0	0
$771$	−57766.6	−2.69833
$772$	0	0
$773$	−11788.9	−0.548536	−0.274268	−	0.961653i	$-0.588436\pi$
$773$	−11788.9	−0.548536	−0.274268	+	0.961653i	$0.588436\pi$
$774$	0	0
$775$	−8351.77	−0.387103
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2161.74	0.0994254
$780$	0	0
$781$	3444.27	0.157805
$782$	0	0
$783$	32189.1	1.46915
$784$	0	0
$785$	28194.0	1.28189
$786$	0	0
$787$	11312.5	0.512386	0.256193	−	0.966626i	$-0.417532\pi$
$787$	11312.5	0.512386	0.256193	+	0.966626i	$0.417532\pi$
$788$	0	0
$789$	9575.48	0.432061
$790$	0	0
$791$	0	0
$792$	0	0
$793$	13952.8	0.624817
$794$	0	0
$795$	−46596.6	−2.07876
$796$	0	0
$797$	−32275.4	−1.43444	−0.717222	−	0.696845i	$-0.754587\pi$
$797$	−32275.4	−1.43444	−0.717222	+	0.696845i	$0.754587\pi$
$798$	0	0
$799$	−20413.9	−0.903871
$800$	0	0
$801$	9595.18	0.423257
$802$	0	0
$803$	−5887.52	−0.258737
$804$	0	0
$805$	0	0
$806$	0	0
$807$	33515.3	1.46195
$808$	0	0
$809$	−10310.8	−0.448095	−0.224047	−	0.974578i	$-0.571927\pi$
$809$	−10310.8	−0.448095	−0.224047	+	0.974578i	$0.571927\pi$
$810$	0	0
$811$	−13890.7	−0.601439	−0.300720	−	0.953713i	$-0.597227\pi$
$811$	−13890.7	−0.601439	−0.300720	+	0.953713i	$0.597227\pi$
$812$	0	0
$813$	−371.382	−0.0160208
$814$	0	0
$815$	−20225.9	−0.869305
$816$	0	0
$817$	6692.47	0.286585
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−31610.8	−1.34376	−0.671878	−	0.740662i	$-0.734512\pi$
$821$	−31610.8	−1.34376	−0.671878	+	0.740662i	$0.734512\pi$
$822$	0	0
$823$	−11508.8	−0.487448	−0.243724	−	0.969845i	$-0.578369\pi$
$823$	−11508.8	−0.487448	−0.243724	+	0.969845i	$0.578369\pi$
$824$	0	0
$825$	−25990.3	−1.09681
$826$	0	0
$827$	42336.9	1.78017	0.890083	−	0.455798i	$-0.150646\pi$
$827$	42336.9	1.78017	0.890083	+	0.455798i	$0.150646\pi$
$828$	0	0
$829$	−35928.6	−1.50525	−0.752625	−	0.658450i	$-0.771213\pi$
$829$	−35928.6	−1.50525	−0.752625	+	0.658450i	$0.771213\pi$
$830$	0	0
$831$	−1869.93	−0.0780592
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−15355.1	−0.636388
$836$	0	0
$837$	−23645.9	−0.976489
$838$	0	0
$839$	−1043.87	−0.0429541	−0.0214771	−	0.999769i	$-0.506837\pi$
$839$	−1043.87	−0.0429541	−0.0214771	+	0.999769i	$0.506837\pi$
$840$	0	0
$841$	−16107.9	−0.660456
$842$	0	0
$843$	7725.76	0.315646
$844$	0	0
$845$	27532.1	1.12087
$846$	0	0
$847$	0	0
$848$	0	0
$849$	36249.7	1.46536
$850$	0	0
$851$	−17242.0	−0.694533
$852$	0	0
$853$	−23453.0	−0.941403	−0.470702	−	0.882292i	$-0.655999\pi$
$853$	−23453.0	−0.941403	−0.470702	+	0.882292i	$0.655999\pi$
$854$	0	0
$855$	−21560.0	−0.862380
$856$	0	0
$857$	39863.5	1.58893	0.794464	−	0.607312i	$-0.207752\pi$
$857$	39863.5	1.58893	0.794464	+	0.607312i	$0.207752\pi$
$858$	0	0
$859$	7907.93	0.314104	0.157052	−	0.987590i	$-0.449801\pi$
$859$	7907.93	0.314104	0.157052	+	0.987590i	$0.449801\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−6769.67	−0.267025	−0.133512	−	0.991047i	$-0.542626\pi$
$863$	−6769.67	−0.267025	−0.133512	+	0.991047i	$0.542626\pi$
$864$	0	0
$865$	−8360.72	−0.328639
$866$	0	0
$867$	7932.84	0.310742
$868$	0	0
$869$	7094.78	0.276955
$870$	0	0
$871$	10113.3	0.393427
$872$	0	0
$873$	−119530.	−4.63400
$874$	0	0
$875$	0	0
$876$	0	0
$877$	17317.7	0.666791	0.333396	−	0.942787i	$-0.391806\pi$
$877$	17317.7	0.666791	0.333396	+	0.942787i	$0.391806\pi$
$878$	0	0
$879$	29996.0	1.15101
$880$	0	0
$881$	33759.7	1.29103	0.645513	−	0.763750i	$-0.276644\pi$
$881$	33759.7	1.29103	0.645513	+	0.763750i	$0.276644\pi$
$882$	0	0
$883$	20233.7	0.771142	0.385571	−	0.922678i	$-0.374004\pi$
$883$	20233.7	0.771142	0.385571	+	0.922678i	$0.374004\pi$
$884$	0	0
$885$	−122013.	−4.63436
$886$	0	0
$887$	27806.7	1.05260	0.526300	−	0.850299i	$-0.323579\pi$
$887$	27806.7	1.05260	0.526300	+	0.850299i	$0.323579\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−35876.1	−1.34893
$892$	0	0
$893$	−5733.37	−0.214849
$894$	0	0
$895$	2355.60	0.0879765
$896$	0	0
$897$	20828.5	0.775298
$898$	0	0
$899$	−6083.27	−0.225682
$900$	0	0
$901$	23408.8	0.865548
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−57295.9	−2.10451
$906$	0	0
$907$	19192.9	0.702635	0.351318	−	0.936256i	$-0.385734\pi$
$907$	19192.9	0.702635	0.351318	+	0.936256i	$0.385734\pi$
$908$	0	0
$909$	34246.8	1.24961
$910$	0	0
$911$	39131.9	1.42316	0.711579	−	0.702607i	$-0.247981\pi$
$911$	39131.9	1.42316	0.711579	+	0.702607i	$0.247981\pi$
$912$	0	0
$913$	−22520.5	−0.816343
$914$	0	0
$915$	−98631.4	−3.56356
$916$	0	0
$917$	0	0
$918$	0	0
$919$	50478.2	1.81188	0.905942	−	0.423402i	$-0.139164\pi$
$919$	50478.2	1.81188	0.905942	+	0.423402i	$0.139164\pi$
$920$	0	0
$921$	73733.5	2.63800
$922$	0	0
$923$	−3372.05	−0.120252
$924$	0	0
$925$	−21063.8	−0.748729
$926$	0	0
$927$	84543.5	2.99544
$928$	0	0
$929$	19092.5	0.674277	0.337138	−	0.941455i	$-0.390541\pi$
$929$	19092.5	0.674277	0.337138	+	0.941455i	$0.390541\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	14103.5	0.494885
$934$	0	0
$935$	26120.2	0.913607
$936$	0	0
$937$	−19922.3	−0.694592	−0.347296	−	0.937756i	$-0.612900\pi$
$937$	−19922.3	−0.694592	−0.347296	+	0.937756i	$0.612900\pi$
$938$	0	0
$939$	−65098.9	−2.26243
$940$	0	0
$941$	15325.0	0.530906	0.265453	−	0.964124i	$-0.414479\pi$
$941$	15325.0	0.530906	0.265453	+	0.964124i	$0.414479\pi$
$942$	0	0
$943$	10385.8	0.358650
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2260.56	−0.0775696	−0.0387848	−	0.999248i	$-0.512349\pi$
$947$	−2260.56	−0.0775696	−0.0387848	+	0.999248i	$0.512349\pi$
$948$	0	0
$949$	5764.06	0.197165
$950$	0	0
$951$	−7035.22	−0.239887
$952$	0	0
$953$	24905.1	0.846543	0.423272	−	0.906003i	$-0.360882\pi$
$953$	24905.1	0.846543	0.423272	+	0.906003i	$0.360882\pi$
$954$	0	0
$955$	−50346.4	−1.70594
$956$	0	0
$957$	−18930.8	−0.639443
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−25322.3	−0.849997
$962$	0	0
$963$	−44433.8	−1.48687
$964$	0	0
$965$	72685.6	2.42470
$966$	0	0
$967$	28438.6	0.945734	0.472867	−	0.881134i	$-0.343219\pi$
$967$	28438.6	0.945734	0.472867	+	0.881134i	$0.343219\pi$
$968$	0	0
$969$	15395.7	0.510403
$970$	0	0
$971$	15280.4	0.505016	0.252508	−	0.967595i	$-0.418745\pi$
$971$	15280.4	0.505016	0.252508	+	0.967595i	$0.418745\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	25445.3	0.835796
$976$	0	0
$977$	−31083.5	−1.01786	−0.508930	−	0.860808i	$-0.669959\pi$
$977$	−31083.5	−1.01786	−0.508930	+	0.860808i	$0.669959\pi$
$978$	0	0
$979$	−3264.87	−0.106584
$980$	0	0
$981$	−125955.	−4.09931
$982$	0	0
$983$	9506.14	0.308442	0.154221	−	0.988036i	$-0.450713\pi$
$983$	9506.14	0.308442	0.154221	+	0.988036i	$0.450713\pi$
$984$	0	0
$985$	61306.1	1.98312
$986$	0	0
$987$	0	0
$988$	0	0
$989$	32153.0	1.03378
$990$	0	0
$991$	−41313.2	−1.32427	−0.662137	−	0.749383i	$-0.730350\pi$
$991$	−41313.2	−1.32427	−0.662137	+	0.749383i	$0.730350\pi$
$992$	0	0
$993$	−34967.8	−1.11749
$994$	0	0
$995$	31241.2	0.995391
$996$	0	0
$997$	29778.9	0.945946	0.472973	−	0.881077i	$-0.343181\pi$
$997$	29778.9	0.945946	0.472973	+	0.881077i	$0.343181\pi$
$998$	0	0
$999$	−59636.7	−1.88871

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.4.a.be.1.3		3
4.3	odd	2		392.4.a.i.1.1		3
7.2	even	3		112.4.i.e.81.1		6
7.4	even	3		112.4.i.e.65.1		6
7.6	odd	2		784.4.a.bb.1.1		3
28.3	even	6		392.4.i.m.177.1		6
28.11	odd	6		56.4.i.b.9.3	✓	6
28.19	even	6		392.4.i.m.361.1		6
28.23	odd	6		56.4.i.b.25.3	yes	6
28.27	even	2		392.4.a.l.1.3		3
56.11	odd	6		448.4.i.j.65.1		6
56.37	even	6		448.4.i.m.193.3		6
56.51	odd	6		448.4.i.j.193.1		6
56.53	even	6		448.4.i.m.65.3		6
84.11	even	6		504.4.s.h.289.1		6
84.23	even	6		504.4.s.h.361.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.4.i.b.9.3	✓	6	28.11	odd	6
56.4.i.b.25.3	yes	6	28.23	odd	6
112.4.i.e.65.1		6	7.4	even	3
112.4.i.e.81.1		6	7.2	even	3
392.4.a.i.1.1		3	4.3	odd	2
392.4.a.l.1.3		3	28.27	even	2
392.4.i.m.177.1		6	28.3	even	6
392.4.i.m.361.1		6	28.19	even	6
448.4.i.j.65.1		6	56.11	odd	6
448.4.i.j.193.1		6	56.51	odd	6
448.4.i.m.65.3		6	56.53	even	6
448.4.i.m.193.3		6	56.37	even	6
504.4.s.h.289.1		6	84.11	even	6
504.4.s.h.361.1		6	84.23	even	6
784.4.a.bb.1.1		3	7.6	odd	2
784.4.a.be.1.3		3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	784.a (trivial)

\(f(q)\)	\(=\)	\(q+9.54289 q^{3} -15.8094 q^{5} +64.0667 q^{9} +O(q^{10})\)	\(q+9.54289 q^{3} -15.8094 q^{5} +64.0667 q^{9} -21.7994 q^{11} +21.3423 q^{13} -150.867 q^{15} +75.7910 q^{17} +21.2864 q^{19} +102.267 q^{23} +124.936 q^{25} +353.723 q^{27} +91.0008 q^{29} -66.8486 q^{31} -208.029 q^{33} -168.597 q^{37} +203.667 q^{39} +101.555 q^{41} +314.402 q^{43} -1012.85 q^{45} -269.345 q^{47} +723.265 q^{51} +308.859 q^{53} +344.635 q^{55} +203.133 q^{57} +808.744 q^{59} +653.765 q^{61} -337.408 q^{65} +473.860 q^{67} +975.925 q^{69} -157.998 q^{71} +270.077 q^{73} +1192.25 q^{75} -325.457 q^{79} +1645.74 q^{81} +1033.08 q^{83} -1198.21 q^{85} +868.410 q^{87} +149.769 q^{89} -637.928 q^{93} -336.524 q^{95} -1865.72 q^{97} -1396.62 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 7 q^{3} - 3 q^{5} + 18 q^{9}+O(q^{10}) \) 3 * q + 7 * q^3 - 3 * q^5 + 18 * q^9	\( 3 q + 7 q^{3} - 3 q^{5} + 18 q^{9} + 3 q^{11} - 26 q^{13} - 127 q^{15} - 31 q^{17} + 89 q^{19} + 201 q^{23} + 300 q^{25} + 469 q^{27} + 190 q^{29} + 339 q^{31} - 105 q^{33} - 535 q^{37} + 134 q^{39} + 58 q^{41} - 268 q^{43} - 1410 q^{45} + 205 q^{47} + 965 q^{51} + 757 q^{53} + 1653 q^{55} + 261 q^{57} + 1799 q^{59} + 625 q^{61} - 1750 q^{65} + 495 q^{67} + 973 q^{69} - 640 q^{71} - 443 q^{73} + 1484 q^{75} - 79 q^{79} + 2523 q^{81} + 2372 q^{83} - 977 q^{85} + 910 q^{87} + 821 q^{89} - 1321 q^{93} + 1327 q^{95} - 342 q^{97} - 2310 q^{99}+O(q^{100}) \) 3 * q + 7 * q^3 - 3 * q^5 + 18 * q^9 + 3 * q^11 - 26 * q^13 - 127 * q^15 - 31 * q^17 + 89 * q^19 + 201 * q^23 + 300 * q^25 + 469 * q^27 + 190 * q^29 + 339 * q^31 - 105 * q^33 - 535 * q^37 + 134 * q^39 + 58 * q^41 - 268 * q^43 - 1410 * q^45 + 205 * q^47 + 965 * q^51 + 757 * q^53 + 1653 * q^55 + 261 * q^57 + 1799 * q^59 + 625 * q^61 - 1750 * q^65 + 495 * q^67 + 973 * q^69 - 640 * q^71 - 443 * q^73 + 1484 * q^75 - 79 * q^79 + 2523 * q^81 + 2372 * q^83 - 977 * q^85 + 910 * q^87 + 821 * q^89 - 1321 * q^93 + 1327 * q^95 - 342 * q^97 - 2310 * q^99

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