LMFDB - Embedded newform 832.4.a.z.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [832,4,Mod(1,832)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(832, 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("832.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 832 = 2^{6} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	832.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:	$49.0895891248$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	832.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.68466 q^{3} +3.56155 q^{5} +27.1771 q^{7} +48.4233 q^{9} +O(q^{10})$	$q+8.68466 q^{3} +3.56155 q^{5} +27.1771 q^{7} +48.4233 q^{9} +15.2614 q^{11} +13.0000 q^{13} +30.9309 q^{15} +44.5464 q^{17} +23.9697 q^{19} +236.024 q^{21} -122.739 q^{23} -112.315 q^{25} +186.054 q^{27} +219.909 q^{29} -27.0928 q^{31} +132.540 q^{33} +96.7926 q^{35} -94.1922 q^{37} +112.901 q^{39} -160.354 q^{41} -151.302 q^{43} +172.462 q^{45} -466.948 q^{47} +395.594 q^{49} +386.870 q^{51} +120.847 q^{53} +54.3542 q^{55} +208.169 q^{57} -439.633 q^{59} +137.305 q^{61} +1316.00 q^{63} +46.3002 q^{65} +512.280 q^{67} -1065.94 q^{69} -410.719 q^{71} -308.004 q^{73} -975.420 q^{75} +414.759 q^{77} +586.462 q^{79} +308.386 q^{81} +1354.20 q^{83} +158.654 q^{85} +1909.84 q^{87} +439.882 q^{89} +353.302 q^{91} -235.292 q^{93} +85.3693 q^{95} -1511.27 q^{97} +739.006 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 5 q^{3} + 3 q^{5} + 9 q^{7} + 35 q^{9}+O(q^{10}) $ 2 * q + 5 * q^3 + 3 * q^5 + 9 * q^7 + 35 * q^9	$ 2 q + 5 q^{3} + 3 q^{5} + 9 q^{7} + 35 q^{9} + 80 q^{11} + 26 q^{13} + 33 q^{15} + 19 q^{17} - 84 q^{19} + 303 q^{21} - 196 q^{23} - 237 q^{25} + 335 q^{27} + 44 q^{29} + 86 q^{31} - 106 q^{33} + 107 q^{35} - 209 q^{37} + 65 q^{39} - 230 q^{41} + 287 q^{43} + 180 q^{45} - 435 q^{47} + 383 q^{49} + 481 q^{51} + 118 q^{53} + 18 q^{55} + 606 q^{57} - 368 q^{59} + 1058 q^{61} + 1560 q^{63} + 39 q^{65} + 68 q^{67} - 796 q^{69} + 131 q^{71} + 456 q^{73} - 516 q^{75} - 762 q^{77} + 1008 q^{79} + 122 q^{81} + 1958 q^{83} + 173 q^{85} + 2558 q^{87} - 720 q^{89} + 117 q^{91} - 652 q^{93} + 146 q^{95} - 928 q^{97} - 130 q^{99}+O(q^{100}) $ 2 * q + 5 * q^3 + 3 * q^5 + 9 * q^7 + 35 * q^9 + 80 * q^11 + 26 * q^13 + 33 * q^15 + 19 * q^17 - 84 * q^19 + 303 * q^21 - 196 * q^23 - 237 * q^25 + 335 * q^27 + 44 * q^29 + 86 * q^31 - 106 * q^33 + 107 * q^35 - 209 * q^37 + 65 * q^39 - 230 * q^41 + 287 * q^43 + 180 * q^45 - 435 * q^47 + 383 * q^49 + 481 * q^51 + 118 * q^53 + 18 * q^55 + 606 * q^57 - 368 * q^59 + 1058 * q^61 + 1560 * q^63 + 39 * q^65 + 68 * q^67 - 796 * q^69 + 131 * q^71 + 456 * q^73 - 516 * q^75 - 762 * q^77 + 1008 * q^79 + 122 * q^81 + 1958 * q^83 + 173 * q^85 + 2558 * q^87 - 720 * q^89 + 117 * q^91 - 652 * q^93 + 146 * q^95 - 928 * q^97 - 130 * 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
$3$	8.68466	1.67136	0.835682	−	0.549214i	$-0.185073\pi$
$3$	8.68466	1.67136	0.835682	+	0.549214i	$0.185073\pi$
$4$	0	0
$5$	3.56155	0.318555	0.159277	−	0.987234i	$-0.449084\pi$
$5$	3.56155	0.318555	0.159277	+	0.987234i	$0.449084\pi$
$6$	0	0
$7$	27.1771	1.46742	0.733712	−	0.679460i	$-0.237786\pi$
$7$	27.1771	1.46742	0.733712	+	0.679460i	$0.237786\pi$
$8$	0	0
$9$	48.4233	1.79346
$10$	0	0
$11$	15.2614	0.418316	0.209158	−	0.977882i	$-0.432928\pi$
$11$	15.2614	0.418316	0.209158	+	0.977882i	$0.432928\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	30.9309	0.532421
$16$	0	0
$17$	44.5464	0.635535	0.317767	−	0.948169i	$-0.397067\pi$
$17$	44.5464	0.635535	0.317767	+	0.948169i	$0.397067\pi$
$18$	0	0
$19$	23.9697	0.289422	0.144711	−	0.989474i	$-0.453775\pi$
$19$	23.9697	0.289422	0.144711	+	0.989474i	$0.453775\pi$
$20$	0	0
$21$	236.024	2.45260
$22$	0	0
$23$	−122.739	−1.11273	−0.556365	−	0.830938i	$-0.687804\pi$
$23$	−122.739	−1.11273	−0.556365	+	0.830938i	$0.687804\pi$
$24$	0	0
$25$	−112.315	−0.898523
$26$	0	0
$27$	186.054	1.32615
$28$	0	0
$29$	219.909	1.40814	0.704071	−	0.710130i	$-0.251364\pi$
$29$	219.909	1.40814	0.704071	+	0.710130i	$0.251364\pi$
$30$	0	0
$31$	−27.0928	−0.156968	−0.0784840	−	0.996915i	$-0.525008\pi$
$31$	−27.0928	−0.156968	−0.0784840	+	0.996915i	$0.525008\pi$
$32$	0	0
$33$	132.540	0.699158
$34$	0	0
$35$	96.7926	0.467455
$36$	0	0
$37$	−94.1922	−0.418516	−0.209258	−	0.977860i	$-0.567105\pi$
$37$	−94.1922	−0.418516	−0.209258	+	0.977860i	$0.567105\pi$
$38$	0	0
$39$	112.901	0.463553
$40$	0	0
$41$	−160.354	−0.610808	−0.305404	−	0.952223i	$-0.598791\pi$
$41$	−160.354	−0.610808	−0.305404	+	0.952223i	$0.598791\pi$
$42$	0	0
$43$	−151.302	−0.536589	−0.268295	−	0.963337i	$-0.586460\pi$
$43$	−151.302	−0.536589	−0.268295	+	0.963337i	$0.586460\pi$
$44$	0	0
$45$	172.462	0.571314
$46$	0	0
$47$	−466.948	−1.44918	−0.724589	−	0.689181i	$-0.757970\pi$
$47$	−466.948	−1.44918	−0.724589	+	0.689181i	$0.757970\pi$
$48$	0	0
$49$	395.594	1.15333
$50$	0	0
$51$	386.870	1.06221
$52$	0	0
$53$	120.847	0.313199	0.156600	−	0.987662i	$-0.449947\pi$
$53$	120.847	0.313199	0.156600	+	0.987662i	$0.449947\pi$
$54$	0	0
$55$	54.3542	0.133257
$56$	0	0
$57$	208.169	0.483730
$58$	0	0
$59$	−439.633	−0.970090	−0.485045	−	0.874489i	$-0.661197\pi$
$59$	−439.633	−0.970090	−0.485045	+	0.874489i	$0.661197\pi$
$60$	0	0
$61$	137.305	0.288198	0.144099	−	0.989563i	$-0.453972\pi$
$61$	137.305	0.288198	0.144099	+	0.989563i	$0.453972\pi$
$62$	0	0
$63$	1316.00	2.63176
$64$	0	0
$65$	46.3002	0.0883513
$66$	0	0
$67$	512.280	0.934104	0.467052	−	0.884230i	$-0.345316\pi$
$67$	512.280	0.934104	0.467052	+	0.884230i	$0.345316\pi$
$68$	0	0
$69$	−1065.94	−1.85977
$70$	0	0
$71$	−410.719	−0.686526	−0.343263	−	0.939239i	$-0.611532\pi$
$71$	−410.719	−0.686526	−0.343263	+	0.939239i	$0.611532\pi$
$72$	0	0
$73$	−308.004	−0.493823	−0.246912	−	0.969038i	$-0.579416\pi$
$73$	−308.004	−0.493823	−0.246912	+	0.969038i	$0.579416\pi$
$74$	0	0
$75$	−975.420	−1.50176
$76$	0	0
$77$	414.759	0.613847
$78$	0	0
$79$	586.462	0.835217	0.417608	−	0.908627i	$-0.362868\pi$
$79$	586.462	0.835217	0.417608	+	0.908627i	$0.362868\pi$
$80$	0	0
$81$	308.386	0.423027
$82$	0	0
$83$	1354.20	1.79088	0.895440	−	0.445182i	$-0.146861\pi$
$83$	1354.20	1.79088	0.895440	+	0.445182i	$0.146861\pi$
$84$	0	0
$85$	158.654	0.202453
$86$	0	0
$87$	1909.84	2.35352
$88$	0	0
$89$	439.882	0.523904	0.261952	−	0.965081i	$-0.415634\pi$
$89$	439.882	0.523904	0.261952	+	0.965081i	$0.415634\pi$
$90$	0	0
$91$	353.302	0.406990
$92$	0	0
$93$	−235.292	−0.262351
$94$	0	0
$95$	85.3693	0.0921969
$96$	0	0
$97$	−1511.27	−1.58192	−0.790959	−	0.611869i	$-0.790418\pi$
$97$	−1511.27	−1.58192	−0.790959	+	0.611869i	$0.790418\pi$
$98$	0	0
$99$	739.006	0.750231
$100$	0	0
$101$	−336.260	−0.331278	−0.165639	−	0.986186i	$-0.552969\pi$
$101$	−336.260	−0.331278	−0.165639	+	0.986186i	$0.552969\pi$
$102$	0	0
$103$	−322.712	−0.308716	−0.154358	−	0.988015i	$-0.549331\pi$
$103$	−322.712	−0.308716	−0.154358	+	0.988015i	$0.549331\pi$
$104$	0	0
$105$	840.611	0.781288
$106$	0	0
$107$	1434.62	1.29617	0.648083	−	0.761570i	$-0.275571\pi$
$107$	1434.62	1.29617	0.648083	+	0.761570i	$0.275571\pi$
$108$	0	0
$109$	−849.147	−0.746179	−0.373089	−	0.927795i	$-0.621702\pi$
$109$	−849.147	−0.746179	−0.373089	+	0.927795i	$0.621702\pi$
$110$	0	0
$111$	−818.027	−0.699493
$112$	0	0
$113$	1614.53	1.34409	0.672044	−	0.740511i	$-0.265417\pi$
$113$	1614.53	1.34409	0.672044	+	0.740511i	$0.265417\pi$
$114$	0	0
$115$	−437.140	−0.354465
$116$	0	0
$117$	629.503	0.497415
$118$	0	0
$119$	1210.64	0.932599
$120$	0	0
$121$	−1098.09	−0.825012
$122$	0	0
$123$	−1392.62	−1.02088
$124$	0	0
$125$	−845.211	−0.604784
$126$	0	0
$127$	−865.174	−0.604502	−0.302251	−	0.953228i	$-0.597738\pi$
$127$	−865.174	−0.604502	−0.302251	+	0.953228i	$0.597738\pi$
$128$	0	0
$129$	−1314.01	−0.896836
$130$	0	0
$131$	−281.400	−0.187680	−0.0938400	−	0.995587i	$-0.529914\pi$
$131$	−281.400	−0.187680	−0.0938400	+	0.995587i	$0.529914\pi$
$132$	0	0
$133$	651.426	0.424705
$134$	0	0
$135$	662.641	0.422452
$136$	0	0
$137$	−2641.43	−1.64725	−0.823624	−	0.567137i	$-0.808051\pi$
$137$	−2641.43	−1.64725	−0.823624	+	0.567137i	$0.808051\pi$
$138$	0	0
$139$	−1998.64	−1.21958	−0.609791	−	0.792562i	$-0.708747\pi$
$139$	−1998.64	−1.21958	−0.609791	+	0.792562i	$0.708747\pi$
$140$	0	0
$141$	−4055.28	−2.42210
$142$	0	0
$143$	198.398	0.116020
$144$	0	0
$145$	783.218	0.448570
$146$	0	0
$147$	3435.60	1.92764
$148$	0	0
$149$	1752.98	0.963824	0.481912	−	0.876220i	$-0.339942\pi$
$149$	1752.98	0.963824	0.481912	+	0.876220i	$0.339942\pi$
$150$	0	0
$151$	2794.64	1.50613	0.753063	−	0.657949i	$-0.228576\pi$
$151$	2794.64	1.50613	0.753063	+	0.657949i	$0.228576\pi$
$152$	0	0
$153$	2157.08	1.13980
$154$	0	0
$155$	−96.4924	−0.0500030
$156$	0	0
$157$	−3244.87	−1.64949	−0.824743	−	0.565508i	$-0.808680\pi$
$157$	−3244.87	−1.64949	−0.824743	+	0.565508i	$0.808680\pi$
$158$	0	0
$159$	1049.51	0.523470
$160$	0	0
$161$	−3335.68	−1.63285
$162$	0	0
$163$	3281.47	1.57684	0.788418	−	0.615139i	$-0.210900\pi$
$163$	3281.47	1.57684	0.788418	+	0.615139i	$0.210900\pi$
$164$	0	0
$165$	472.047	0.222720
$166$	0	0
$167$	3126.52	1.44873	0.724364	−	0.689418i	$-0.242134\pi$
$167$	3126.52	1.44873	0.724364	+	0.689418i	$0.242134\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	1160.69	0.519066
$172$	0	0
$173$	−97.5698	−0.0428792	−0.0214396	−	0.999770i	$-0.506825\pi$
$173$	−97.5698	−0.0428792	−0.0214396	+	0.999770i	$0.506825\pi$
$174$	0	0
$175$	−3052.40	−1.31851
$176$	0	0
$177$	−3818.06	−1.62137
$178$	0	0
$179$	−34.7150	−0.0144956	−0.00724782	−	0.999974i	$-0.502307\pi$
$179$	−34.7150	−0.0144956	−0.00724782	+	0.999974i	$0.502307\pi$
$180$	0	0
$181$	1229.35	0.504843	0.252422	−	0.967617i	$-0.418773\pi$
$181$	1229.35	0.504843	0.252422	+	0.967617i	$0.418773\pi$
$182$	0	0
$183$	1192.45	0.481684
$184$	0	0
$185$	−335.471	−0.133320
$186$	0	0
$187$	679.839	0.265854
$188$	0	0
$189$	5056.40	1.94603
$190$	0	0
$191$	−4280.80	−1.62172	−0.810858	−	0.585243i	$-0.800999\pi$
$191$	−4280.80	−1.62172	−0.810858	+	0.585243i	$0.800999\pi$
$192$	0	0
$193$	472.320	0.176157	0.0880786	−	0.996114i	$-0.471927\pi$
$193$	472.320	0.176157	0.0880786	+	0.996114i	$0.471927\pi$
$194$	0	0
$195$	402.101	0.147667
$196$	0	0
$197$	4484.37	1.62182	0.810908	−	0.585173i	$-0.198974\pi$
$197$	4484.37	1.62182	0.810908	+	0.585173i	$0.198974\pi$
$198$	0	0
$199$	366.240	0.130463	0.0652314	−	0.997870i	$-0.479221\pi$
$199$	366.240	0.130463	0.0652314	+	0.997870i	$0.479221\pi$
$200$	0	0
$201$	4448.98	1.56123
$202$	0	0
$203$	5976.49	2.06634
$204$	0	0
$205$	−571.110	−0.194576
$206$	0	0
$207$	−5943.41	−1.99563
$208$	0	0
$209$	365.810	0.121070
$210$	0	0
$211$	2122.55	0.692524	0.346262	−	0.938138i	$-0.387451\pi$
$211$	2122.55	0.692524	0.346262	+	0.938138i	$0.387451\pi$
$212$	0	0
$213$	−3566.95	−1.14743
$214$	0	0
$215$	−538.870	−0.170933
$216$	0	0
$217$	−736.303	−0.230339
$218$	0	0
$219$	−2674.91	−0.825358
$220$	0	0
$221$	579.103	0.176266
$222$	0	0
$223$	5926.42	1.77965	0.889826	−	0.456301i	$-0.150826\pi$
$223$	5926.42	1.77965	0.889826	+	0.456301i	$0.150826\pi$
$224$	0	0
$225$	−5438.68	−1.61146
$226$	0	0
$227$	−895.661	−0.261881	−0.130941	−	0.991390i	$-0.541800\pi$
$227$	−895.661	−0.261881	−0.130941	+	0.991390i	$0.541800\pi$
$228$	0	0
$229$	−627.717	−0.181138	−0.0905692	−	0.995890i	$-0.528869\pi$
$229$	−627.717	−0.181138	−0.0905692	+	0.995890i	$0.528869\pi$
$230$	0	0
$231$	3602.04	1.02596
$232$	0	0
$233$	2303.72	0.647734	0.323867	−	0.946103i	$-0.395017\pi$
$233$	2303.72	0.647734	0.323867	+	0.946103i	$0.395017\pi$
$234$	0	0
$235$	−1663.06	−0.461643
$236$	0	0
$237$	5093.22	1.39595
$238$	0	0
$239$	−544.622	−0.147400	−0.0737001	−	0.997280i	$-0.523481\pi$
$239$	−544.622	−0.147400	−0.0737001	+	0.997280i	$0.523481\pi$
$240$	0	0
$241$	5426.10	1.45031	0.725157	−	0.688584i	$-0.241767\pi$
$241$	5426.10	1.45031	0.725157	+	0.688584i	$0.241767\pi$
$242$	0	0
$243$	−2345.23	−0.619121
$244$	0	0
$245$	1408.93	0.367400
$246$	0	0
$247$	311.606	0.0802713
$248$	0	0
$249$	11760.8	2.99321
$250$	0	0
$251$	−5221.22	−1.31299	−0.656494	−	0.754331i	$-0.727961\pi$
$251$	−5221.22	−1.31299	−0.656494	+	0.754331i	$0.727961\pi$
$252$	0	0
$253$	−1873.16	−0.465472
$254$	0	0
$255$	1377.86	0.338372
$256$	0	0
$257$	658.206	0.159758	0.0798789	−	0.996805i	$-0.474547\pi$
$257$	658.206	0.159758	0.0798789	+	0.996805i	$0.474547\pi$
$258$	0	0
$259$	−2559.87	−0.614141
$260$	0	0
$261$	10648.7	2.52544
$262$	0	0
$263$	−3246.45	−0.761160	−0.380580	−	0.924748i	$-0.624276\pi$
$263$	−3246.45	−0.761160	−0.380580	+	0.924748i	$0.624276\pi$
$264$	0	0
$265$	430.401	0.0997711
$266$	0	0
$267$	3820.23	0.875634
$268$	0	0
$269$	2585.80	0.586093	0.293047	−	0.956098i	$-0.405331\pi$
$269$	2585.80	0.586093	0.293047	+	0.956098i	$0.405331\pi$
$270$	0	0
$271$	−988.933	−0.221673	−0.110836	−	0.993839i	$-0.535353\pi$
$271$	−988.933	−0.221673	−0.110836	+	0.993839i	$0.535353\pi$
$272$	0	0
$273$	3068.31	0.680229
$274$	0	0
$275$	−1714.09	−0.375866
$276$	0	0
$277$	−8142.40	−1.76617	−0.883086	−	0.469211i	$-0.844538\pi$
$277$	−8142.40	−1.76617	−0.883086	+	0.469211i	$0.844538\pi$
$278$	0	0
$279$	−1311.92	−0.281515
$280$	0	0
$281$	1534.21	0.325705	0.162853	−	0.986650i	$-0.447930\pi$
$281$	1534.21	0.325705	0.162853	+	0.986650i	$0.447930\pi$
$282$	0	0
$283$	−6965.00	−1.46299	−0.731495	−	0.681847i	$-0.761177\pi$
$283$	−6965.00	−1.46299	−0.731495	+	0.681847i	$0.761177\pi$
$284$	0	0
$285$	741.403	0.154095
$286$	0	0
$287$	−4357.96	−0.896314
$288$	0	0
$289$	−2928.62	−0.596096
$290$	0	0
$291$	−13124.9	−2.64396
$292$	0	0
$293$	−640.029	−0.127614	−0.0638070	−	0.997962i	$-0.520324\pi$
$293$	−640.029	−0.127614	−0.0638070	+	0.997962i	$0.520324\pi$
$294$	0	0
$295$	−1565.77	−0.309027
$296$	0	0
$297$	2839.44	0.554750
$298$	0	0
$299$	−1595.60	−0.308616
$300$	0	0
$301$	−4111.95	−0.787404
$302$	0	0
$303$	−2920.30	−0.553686
$304$	0	0
$305$	489.019	0.0918070
$306$	0	0
$307$	−100.406	−0.0186660	−0.00933299	−	0.999956i	$-0.502971\pi$
$307$	−100.406	−0.0186660	−0.00933299	+	0.999956i	$0.502971\pi$
$308$	0	0
$309$	−2802.64	−0.515977
$310$	0	0
$311$	3878.92	0.707245	0.353623	−	0.935388i	$-0.384950\pi$
$311$	3878.92	0.707245	0.353623	+	0.935388i	$0.384950\pi$
$312$	0	0
$313$	−3789.39	−0.684311	−0.342155	−	0.939643i	$-0.611157\pi$
$313$	−3789.39	−0.684311	−0.342155	+	0.939643i	$0.611157\pi$
$314$	0	0
$315$	4687.02	0.838360
$316$	0	0
$317$	−4406.81	−0.780791	−0.390396	−	0.920647i	$-0.627662\pi$
$317$	−4406.81	−0.780791	−0.390396	+	0.920647i	$0.627662\pi$
$318$	0	0
$319$	3356.11	0.589048
$320$	0	0
$321$	12459.2	2.16636
$322$	0	0
$323$	1067.76	0.183938
$324$	0	0
$325$	−1460.10	−0.249205
$326$	0	0
$327$	−7374.55	−1.24714
$328$	0	0
$329$	−12690.3	−2.12656
$330$	0	0
$331$	−4131.49	−0.686064	−0.343032	−	0.939324i	$-0.611454\pi$
$331$	−4131.49	−0.686064	−0.343032	+	0.939324i	$0.611454\pi$
$332$	0	0
$333$	−4561.10	−0.750591
$334$	0	0
$335$	1824.51	0.297564
$336$	0	0
$337$	−4560.82	−0.737221	−0.368611	−	0.929584i	$-0.620166\pi$
$337$	−4560.82	−0.737221	−0.368611	+	0.929584i	$0.620166\pi$
$338$	0	0
$339$	14021.6	2.24646
$340$	0	0
$341$	−413.473	−0.0656622
$342$	0	0
$343$	1429.34	0.225007
$344$	0	0
$345$	−3796.41	−0.592441
$346$	0	0
$347$	10069.4	1.55779	0.778896	−	0.627153i	$-0.215780\pi$
$347$	10069.4	1.55779	0.778896	+	0.627153i	$0.215780\pi$
$348$	0	0
$349$	−5879.32	−0.901757	−0.450878	−	0.892585i	$-0.648889\pi$
$349$	−5879.32	−0.901757	−0.450878	+	0.892585i	$0.648889\pi$
$350$	0	0
$351$	2418.70	0.367808
$352$	0	0
$353$	−9142.56	−1.37850	−0.689249	−	0.724525i	$-0.742059\pi$
$353$	−9142.56	−1.37850	−0.689249	+	0.724525i	$0.742059\pi$
$354$	0	0
$355$	−1462.80	−0.218696
$356$	0	0
$357$	10514.0	1.55871
$358$	0	0
$359$	2754.32	0.404924	0.202462	−	0.979290i	$-0.435106\pi$
$359$	2754.32	0.404924	0.202462	+	0.979290i	$0.435106\pi$
$360$	0	0
$361$	−6284.45	−0.916235
$362$	0	0
$363$	−9536.54	−1.37889
$364$	0	0
$365$	−1096.97	−0.157310
$366$	0	0
$367$	−3040.19	−0.432416	−0.216208	−	0.976347i	$-0.569369\pi$
$367$	−3040.19	−0.432416	−0.216208	+	0.976347i	$0.569369\pi$
$368$	0	0
$369$	−7764.88	−1.09546
$370$	0	0
$371$	3284.26	0.459596
$372$	0	0
$373$	5384.72	0.747481	0.373740	−	0.927533i	$-0.378075\pi$
$373$	5384.72	0.747481	0.373740	+	0.927533i	$0.378075\pi$
$374$	0	0
$375$	−7340.37	−1.01081
$376$	0	0
$377$	2858.82	0.390548
$378$	0	0
$379$	−3424.27	−0.464097	−0.232049	−	0.972704i	$-0.574543\pi$
$379$	−3424.27	−0.464097	−0.232049	+	0.972704i	$0.574543\pi$
$380$	0	0
$381$	−7513.74	−1.01034
$382$	0	0
$383$	382.985	0.0510956	0.0255478	−	0.999674i	$-0.491867\pi$
$383$	382.985	0.0510956	0.0255478	+	0.999674i	$0.491867\pi$
$384$	0	0
$385$	1477.19	0.195544
$386$	0	0
$387$	−7326.54	−0.962349
$388$	0	0
$389$	−8588.34	−1.11940	−0.559699	−	0.828696i	$-0.689083\pi$
$389$	−8588.34	−1.11940	−0.559699	+	0.828696i	$0.689083\pi$
$390$	0	0
$391$	−5467.56	−0.707178
$392$	0	0
$393$	−2443.87	−0.313681
$394$	0	0
$395$	2088.72	0.266063
$396$	0	0
$397$	7239.16	0.915171	0.457586	−	0.889166i	$-0.348714\pi$
$397$	7239.16	0.915171	0.457586	+	0.889166i	$0.348714\pi$
$398$	0	0
$399$	5657.41	0.709837
$400$	0	0
$401$	4269.62	0.531708	0.265854	−	0.964013i	$-0.414346\pi$
$401$	4269.62	0.531708	0.265854	+	0.964013i	$0.414346\pi$
$402$	0	0
$403$	−352.206	−0.0435351
$404$	0	0
$405$	1098.33	0.134757
$406$	0	0
$407$	−1437.50	−0.175072
$408$	0	0
$409$	13562.5	1.63967	0.819834	−	0.572602i	$-0.194066\pi$
$409$	13562.5	1.63967	0.819834	+	0.572602i	$0.194066\pi$
$410$	0	0
$411$	−22939.9	−2.75315
$412$	0	0
$413$	−11947.9	−1.42353
$414$	0	0
$415$	4823.06	0.570494
$416$	0	0
$417$	−17357.5	−2.03837
$418$	0	0
$419$	−14576.9	−1.69959	−0.849794	−	0.527114i	$-0.823274\pi$
$419$	−14576.9	−1.69959	−0.849794	+	0.527114i	$0.823274\pi$
$420$	0	0
$421$	−15848.4	−1.83469	−0.917343	−	0.398099i	$-0.869670\pi$
$421$	−15848.4	−1.83469	−0.917343	+	0.398099i	$0.869670\pi$
$422$	0	0
$423$	−22611.2	−2.59904
$424$	0	0
$425$	−5003.24	−0.571042
$426$	0	0
$427$	3731.55	0.422909
$428$	0	0
$429$	1723.02	0.193912
$430$	0	0
$431$	−10694.7	−1.19524	−0.597618	−	0.801781i	$-0.703886\pi$
$431$	−10694.7	−1.19524	−0.597618	+	0.801781i	$0.703886\pi$
$432$	0	0
$433$	−16079.0	−1.78454	−0.892272	−	0.451498i	$-0.850890\pi$
$433$	−16079.0	−1.78454	−0.892272	+	0.451498i	$0.850890\pi$
$434$	0	0
$435$	6801.98	0.749724
$436$	0	0
$437$	−2942.01	−0.322049
$438$	0	0
$439$	−6035.80	−0.656203	−0.328101	−	0.944643i	$-0.606409\pi$
$439$	−6035.80	−0.656203	−0.328101	+	0.944643i	$0.606409\pi$
$440$	0	0
$441$	19156.0	2.06845
$442$	0	0
$443$	10201.3	1.09409	0.547043	−	0.837105i	$-0.315753\pi$
$443$	10201.3	1.09409	0.547043	+	0.837105i	$0.315753\pi$
$444$	0	0
$445$	1566.66	0.166892
$446$	0	0
$447$	15224.0	1.61090
$448$	0	0
$449$	−5822.54	−0.611988	−0.305994	−	0.952033i	$-0.598989\pi$
$449$	−5822.54	−0.611988	−0.305994	+	0.952033i	$0.598989\pi$
$450$	0	0
$451$	−2447.22	−0.255511
$452$	0	0
$453$	24270.5	2.51728
$454$	0	0
$455$	1258.30	0.129649
$456$	0	0
$457$	4621.60	0.473062	0.236531	−	0.971624i	$-0.423990\pi$
$457$	4621.60	0.473062	0.236531	+	0.971624i	$0.423990\pi$
$458$	0	0
$459$	8288.03	0.842816
$460$	0	0
$461$	−5127.77	−0.518056	−0.259028	−	0.965870i	$-0.583402\pi$
$461$	−5127.77	−0.518056	−0.259028	+	0.965870i	$0.583402\pi$
$462$	0	0
$463$	−6486.27	−0.651064	−0.325532	−	0.945531i	$-0.605543\pi$
$463$	−6486.27	−0.651064	−0.325532	+	0.945531i	$0.605543\pi$
$464$	0	0
$465$	−838.004	−0.0835731
$466$	0	0
$467$	12978.0	1.28598	0.642990	−	0.765875i	$-0.277694\pi$
$467$	12978.0	1.28598	0.642990	+	0.765875i	$0.277694\pi$
$468$	0	0
$469$	13922.3	1.37073
$470$	0	0
$471$	−28180.6	−2.75689
$472$	0	0
$473$	−2309.08	−0.224464
$474$	0	0
$475$	−2692.16	−0.260053
$476$	0	0
$477$	5851.79	0.561709
$478$	0	0
$479$	5808.96	0.554109	0.277055	−	0.960854i	$-0.410642\pi$
$479$	5808.96	0.554109	0.277055	+	0.960854i	$0.410642\pi$
$480$	0	0
$481$	−1224.50	−0.116076
$482$	0	0
$483$	−28969.2	−2.72908
$484$	0	0
$485$	−5382.46	−0.503928
$486$	0	0
$487$	5387.14	0.501262	0.250631	−	0.968083i	$-0.419362\pi$
$487$	5387.14	0.501262	0.250631	+	0.968083i	$0.419362\pi$
$488$	0	0
$489$	28498.4	2.63547
$490$	0	0
$491$	15259.1	1.40251	0.701255	−	0.712911i	$-0.252624\pi$
$491$	15259.1	1.40251	0.701255	+	0.712911i	$0.252624\pi$
$492$	0	0
$493$	9796.16	0.894922
$494$	0	0
$495$	2632.01	0.238990
$496$	0	0
$497$	−11162.1	−1.00742
$498$	0	0
$499$	1856.04	0.166509	0.0832544	−	0.996528i	$-0.473469\pi$
$499$	1856.04	0.166509	0.0832544	+	0.996528i	$0.473469\pi$
$500$	0	0
$501$	27152.8	2.42135
$502$	0	0
$503$	−1049.46	−0.0930283	−0.0465142	−	0.998918i	$-0.514811\pi$
$503$	−1049.46	−0.0930283	−0.0465142	+	0.998918i	$0.514811\pi$
$504$	0	0
$505$	−1197.61	−0.105530
$506$	0	0
$507$	1467.71	0.128566
$508$	0	0
$509$	551.106	0.0479909	0.0239954	−	0.999712i	$-0.492361\pi$
$509$	551.106	0.0479909	0.0239954	+	0.999712i	$0.492361\pi$
$510$	0	0
$511$	−8370.64	−0.724649
$512$	0	0
$513$	4459.66	0.383818
$514$	0	0
$515$	−1149.36	−0.0983431
$516$	0	0
$517$	−7126.26	−0.606214
$518$	0	0
$519$	−847.361	−0.0716667
$520$	0	0
$521$	−8995.30	−0.756413	−0.378206	−	0.925721i	$-0.623459\pi$
$521$	−8995.30	−0.756413	−0.378206	+	0.925721i	$0.623459\pi$
$522$	0	0
$523$	2663.91	0.222724	0.111362	−	0.993780i	$-0.464479\pi$
$523$	2663.91	0.222724	0.111362	+	0.993780i	$0.464479\pi$
$524$	0	0
$525$	−26509.1	−2.20372
$526$	0	0
$527$	−1206.89	−0.0997586
$528$	0	0
$529$	2897.77	0.238167
$530$	0	0
$531$	−21288.5	−1.73981
$532$	0	0
$533$	−2084.60	−0.169408
$534$	0	0
$535$	5109.47	0.412900
$536$	0	0
$537$	−301.488	−0.0242275
$538$	0	0
$539$	6037.30	0.482458
$540$	0	0
$541$	6169.23	0.490270	0.245135	−	0.969489i	$-0.421168\pi$
$541$	6169.23	0.490270	0.245135	+	0.969489i	$0.421168\pi$
$542$	0	0
$543$	10676.5	0.843776
$544$	0	0
$545$	−3024.28	−0.237699
$546$	0	0
$547$	5140.42	0.401807	0.200904	−	0.979611i	$-0.435612\pi$
$547$	5140.42	0.401807	0.200904	+	0.979611i	$0.435612\pi$
$548$	0	0
$549$	6648.76	0.516871
$550$	0	0
$551$	5271.15	0.407547
$552$	0	0
$553$	15938.3	1.22562
$554$	0	0
$555$	−2913.45	−0.222827
$556$	0	0
$557$	−2778.56	−0.211367	−0.105683	−	0.994400i	$-0.533703\pi$
$557$	−2778.56	−0.211367	−0.105683	+	0.994400i	$0.533703\pi$
$558$	0	0
$559$	−1966.93	−0.148823
$560$	0	0
$561$	5904.17	0.444339
$562$	0	0
$563$	4906.14	0.367263	0.183632	−	0.982995i	$-0.441215\pi$
$563$	4906.14	0.367263	0.183632	+	0.982995i	$0.441215\pi$
$564$	0	0
$565$	5750.22	0.428166
$566$	0	0
$567$	8381.04	0.620759
$568$	0	0
$569$	−9363.15	−0.689849	−0.344924	−	0.938631i	$-0.612095\pi$
$569$	−9363.15	−0.689849	−0.344924	+	0.938631i	$0.612095\pi$
$570$	0	0
$571$	7199.32	0.527640	0.263820	−	0.964572i	$-0.415018\pi$
$571$	7199.32	0.527640	0.263820	+	0.964572i	$0.415018\pi$
$572$	0	0
$573$	−37177.3	−2.71048
$574$	0	0
$575$	13785.4	0.999813
$576$	0	0
$577$	−11449.6	−0.826086	−0.413043	−	0.910711i	$-0.635534\pi$
$577$	−11449.6	−0.826086	−0.413043	+	0.910711i	$0.635534\pi$
$578$	0	0
$579$	4101.94	0.294423
$580$	0	0
$581$	36803.3	2.62798
$582$	0	0
$583$	1844.28	0.131016
$584$	0	0
$585$	2242.01	0.158454
$586$	0	0
$587$	−5439.39	−0.382466	−0.191233	−	0.981545i	$-0.561249\pi$
$587$	−5439.39	−0.382466	−0.191233	+	0.981545i	$0.561249\pi$
$588$	0	0
$589$	−649.406	−0.0454301
$590$	0	0
$591$	38945.2	2.71064
$592$	0	0
$593$	−28405.8	−1.96709	−0.983547	−	0.180651i	$-0.942180\pi$
$593$	−28405.8	−1.96709	−0.983547	+	0.180651i	$0.942180\pi$
$594$	0	0
$595$	4311.76	0.297084
$596$	0	0
$597$	3180.67	0.218051
$598$	0	0
$599$	10482.3	0.715020	0.357510	−	0.933909i	$-0.383626\pi$
$599$	10482.3	0.715020	0.357510	+	0.933909i	$0.383626\pi$
$600$	0	0
$601$	3199.54	0.217158	0.108579	−	0.994088i	$-0.465370\pi$
$601$	3199.54	0.217158	0.108579	+	0.994088i	$0.465370\pi$
$602$	0	0
$603$	24806.3	1.67527
$604$	0	0
$605$	−3910.91	−0.262812
$606$	0	0
$607$	−11342.8	−0.758468	−0.379234	−	0.925301i	$-0.623812\pi$
$607$	−11342.8	−0.758468	−0.379234	+	0.925301i	$0.623812\pi$
$608$	0	0
$609$	51903.7	3.45361
$610$	0	0
$611$	−6070.32	−0.401930
$612$	0	0
$613$	−14385.4	−0.947831	−0.473916	−	0.880570i	$-0.657160\pi$
$613$	−14385.4	−0.947831	−0.473916	+	0.880570i	$0.657160\pi$
$614$	0	0
$615$	−4959.89	−0.325207
$616$	0	0
$617$	22056.8	1.43918	0.719588	−	0.694401i	$-0.244331\pi$
$617$	22056.8	1.43918	0.719588	+	0.694401i	$0.244331\pi$
$618$	0	0
$619$	13621.4	0.884477	0.442238	−	0.896898i	$-0.354185\pi$
$619$	13621.4	0.884477	0.442238	+	0.896898i	$0.354185\pi$
$620$	0	0
$621$	−22836.0	−1.47565
$622$	0	0
$623$	11954.7	0.768789
$624$	0	0
$625$	11029.2	0.705866
$626$	0	0
$627$	3176.94	0.202352
$628$	0	0
$629$	−4195.92	−0.265982
$630$	0	0
$631$	18737.5	1.18214	0.591068	−	0.806622i	$-0.298707\pi$
$631$	18737.5	1.18214	0.591068	+	0.806622i	$0.298707\pi$
$632$	0	0
$633$	18433.7	1.15746
$634$	0	0
$635$	−3081.36	−0.192567
$636$	0	0
$637$	5142.72	0.319877
$638$	0	0
$639$	−19888.4	−1.23125
$640$	0	0
$641$	29798.7	1.83616	0.918081	−	0.396394i	$-0.129739\pi$
$641$	29798.7	1.83616	0.918081	+	0.396394i	$0.129739\pi$
$642$	0	0
$643$	−22983.5	−1.40961	−0.704807	−	0.709399i	$-0.748966\pi$
$643$	−22983.5	−1.40961	−0.704807	+	0.709399i	$0.748966\pi$
$644$	0	0
$645$	−4679.90	−0.285692
$646$	0	0
$647$	24905.4	1.51334	0.756672	−	0.653794i	$-0.226824\pi$
$647$	24905.4	1.51334	0.756672	+	0.653794i	$0.226824\pi$
$648$	0	0
$649$	−6709.39	−0.405804
$650$	0	0
$651$	−6394.54	−0.384980
$652$	0	0
$653$	−10077.8	−0.603946	−0.301973	−	0.953316i	$-0.597645\pi$
$653$	−10077.8	−0.603946	−0.301973	+	0.953316i	$0.597645\pi$
$654$	0	0
$655$	−1002.22	−0.0597864
$656$	0	0
$657$	−14914.6	−0.885650
$658$	0	0
$659$	12334.6	0.729116	0.364558	−	0.931181i	$-0.381220\pi$
$659$	12334.6	0.729116	0.364558	+	0.931181i	$0.381220\pi$
$660$	0	0
$661$	12749.1	0.750202	0.375101	−	0.926984i	$-0.377608\pi$
$661$	12749.1	0.750202	0.375101	+	0.926984i	$0.377608\pi$
$662$	0	0
$663$	5029.31	0.294604
$664$	0	0
$665$	2320.09	0.135292
$666$	0	0
$667$	−26991.3	−1.56688
$668$	0	0
$669$	51468.9	2.97444
$670$	0	0
$671$	2095.46	0.120558
$672$	0	0
$673$	−13618.2	−0.780007	−0.390004	−	0.920813i	$-0.627526\pi$
$673$	−13618.2	−0.780007	−0.390004	+	0.920813i	$0.627526\pi$
$674$	0	0
$675$	−20896.7	−1.19158
$676$	0	0
$677$	−9655.67	−0.548150	−0.274075	−	0.961708i	$-0.588372\pi$
$677$	−9655.67	−0.548150	−0.274075	+	0.961708i	$0.588372\pi$
$678$	0	0
$679$	−41071.9	−2.32135
$680$	0	0
$681$	−7778.51	−0.437699
$682$	0	0
$683$	16316.8	0.914119	0.457060	−	0.889436i	$-0.348903\pi$
$683$	16316.8	0.914119	0.457060	+	0.889436i	$0.348903\pi$
$684$	0	0
$685$	−9407.61	−0.524739
$686$	0	0
$687$	−5451.51	−0.302748
$688$	0	0
$689$	1571.01	0.0868658
$690$	0	0
$691$	2350.84	0.129421	0.0647106	−	0.997904i	$-0.479388\pi$
$691$	2350.84	0.129421	0.0647106	+	0.997904i	$0.479388\pi$
$692$	0	0
$693$	20084.0	1.10091
$694$	0	0
$695$	−7118.24	−0.388504
$696$	0	0
$697$	−7143.20	−0.388189
$698$	0	0
$699$	20007.1	1.08260
$700$	0	0
$701$	8076.90	0.435179	0.217589	−	0.976040i	$-0.430181\pi$
$701$	8076.90	0.435179	0.217589	+	0.976040i	$0.430181\pi$
$702$	0	0
$703$	−2257.76	−0.121128
$704$	0	0
$705$	−14443.1	−0.771573
$706$	0	0
$707$	−9138.55	−0.486125
$708$	0	0
$709$	13624.9	0.721712	0.360856	−	0.932622i	$-0.382485\pi$
$709$	13624.9	0.721712	0.360856	+	0.932622i	$0.382485\pi$
$710$	0	0
$711$	28398.4	1.49792
$712$	0	0
$713$	3325.33	0.174663
$714$	0	0
$715$	706.604	0.0369587
$716$	0	0
$717$	−4729.86	−0.246359
$718$	0	0
$719$	−16235.8	−0.842131	−0.421066	−	0.907030i	$-0.638344\pi$
$719$	−16235.8	−0.842131	−0.421066	+	0.907030i	$0.638344\pi$
$720$	0	0
$721$	−8770.37	−0.453018
$722$	0	0
$723$	47123.8	2.42400
$724$	0	0
$725$	−24699.2	−1.26525
$726$	0	0
$727$	−24181.2	−1.23361	−0.616803	−	0.787118i	$-0.711572\pi$
$727$	−24181.2	−1.23361	−0.616803	+	0.787118i	$0.711572\pi$
$728$	0	0
$729$	−28693.9	−1.45780
$730$	0	0
$731$	−6739.96	−0.341021
$732$	0	0
$733$	−3053.70	−0.153876	−0.0769379	−	0.997036i	$-0.524514\pi$
$733$	−3053.70	−0.153876	−0.0769379	+	0.997036i	$0.524514\pi$
$734$	0	0
$735$	12236.1	0.614060
$736$	0	0
$737$	7818.10	0.390751
$738$	0	0
$739$	−8033.62	−0.399894	−0.199947	−	0.979807i	$-0.564077\pi$
$739$	−8033.62	−0.399894	−0.199947	+	0.979807i	$0.564077\pi$
$740$	0	0
$741$	2706.19	0.134163
$742$	0	0
$743$	16139.6	0.796912	0.398456	−	0.917187i	$-0.369546\pi$
$743$	16139.6	0.796912	0.398456	+	0.917187i	$0.369546\pi$
$744$	0	0
$745$	6243.33	0.307031
$746$	0	0
$747$	65574.9	3.21186
$748$	0	0
$749$	38988.7	1.90202
$750$	0	0
$751$	18491.1	0.898469	0.449235	−	0.893414i	$-0.351697\pi$
$751$	18491.1	0.898469	0.449235	+	0.893414i	$0.351697\pi$
$752$	0	0
$753$	−45344.5	−2.19448
$754$	0	0
$755$	9953.28	0.479784
$756$	0	0
$757$	−160.630	−0.00771227	−0.00385613	−	0.999993i	$-0.501227\pi$
$757$	−160.630	−0.00771227	−0.00385613	+	0.999993i	$0.501227\pi$
$758$	0	0
$759$	−16267.7	−0.777973
$760$	0	0
$761$	26799.1	1.27656	0.638282	−	0.769803i	$-0.279645\pi$
$761$	26799.1	1.27656	0.638282	+	0.769803i	$0.279645\pi$
$762$	0	0
$763$	−23077.3	−1.09496
$764$	0	0
$765$	7682.57	0.363090
$766$	0	0
$767$	−5715.22	−0.269054
$768$	0	0
$769$	−5145.82	−0.241304	−0.120652	−	0.992695i	$-0.538499\pi$
$769$	−5145.82	−0.241304	−0.120652	+	0.992695i	$0.538499\pi$
$770$	0	0
$771$	5716.29	0.267013
$772$	0	0
$773$	−12810.6	−0.596072	−0.298036	−	0.954555i	$-0.596332\pi$
$773$	−12810.6	−0.596072	−0.298036	+	0.954555i	$0.596332\pi$
$774$	0	0
$775$	3042.94	0.141039
$776$	0	0
$777$	−22231.6	−1.02645
$778$	0	0
$779$	−3843.64	−0.176781
$780$	0	0
$781$	−6268.13	−0.287185
$782$	0	0
$783$	40915.0	1.86741
$784$	0	0
$785$	−11556.8	−0.525452
$786$	0	0
$787$	28073.0	1.27153	0.635764	−	0.771883i	$-0.280685\pi$
$787$	28073.0	1.27153	0.635764	+	0.771883i	$0.280685\pi$
$788$	0	0
$789$	−28194.3	−1.27217
$790$	0	0
$791$	43878.1	1.97235
$792$	0	0
$793$	1784.96	0.0799318
$794$	0	0
$795$	3737.89	0.166754
$796$	0	0
$797$	−30093.1	−1.33746	−0.668729	−	0.743507i	$-0.733161\pi$
$797$	−30093.1	−1.33746	−0.668729	+	0.743507i	$0.733161\pi$
$798$	0	0
$799$	−20800.8	−0.921003
$800$	0	0
$801$	21300.6	0.939598
$802$	0	0
$803$	−4700.56	−0.206574
$804$	0	0
$805$	−11880.2	−0.520151
$806$	0	0
$807$	22456.8	0.979575
$808$	0	0
$809$	−24337.1	−1.05766	−0.528831	−	0.848727i	$-0.677369\pi$
$809$	−24337.1	−1.05766	−0.528831	+	0.848727i	$0.677369\pi$
$810$	0	0
$811$	19078.7	0.826071	0.413035	−	0.910715i	$-0.364469\pi$
$811$	19078.7	0.826071	0.413035	+	0.910715i	$0.364469\pi$
$812$	0	0
$813$	−8588.54	−0.370496
$814$	0	0
$815$	11687.1	0.502309
$816$	0	0
$817$	−3626.66	−0.155301
$818$	0	0
$819$	17108.0	0.729919
$820$	0	0
$821$	−2013.92	−0.0856104	−0.0428052	−	0.999083i	$-0.513629\pi$
$821$	−2013.92	−0.0856104	−0.0428052	+	0.999083i	$0.513629\pi$
$822$	0	0
$823$	7692.10	0.325795	0.162898	−	0.986643i	$-0.447916\pi$
$823$	7692.10	0.325795	0.162898	+	0.986643i	$0.447916\pi$
$824$	0	0
$825$	−14886.2	−0.628209
$826$	0	0
$827$	−4762.76	−0.200263	−0.100131	−	0.994974i	$-0.531926\pi$
$827$	−4762.76	−0.200263	−0.100131	+	0.994974i	$0.531926\pi$
$828$	0	0
$829$	−19977.7	−0.836976	−0.418488	−	0.908222i	$-0.637440\pi$
$829$	−19977.7	−0.836976	−0.418488	+	0.908222i	$0.637440\pi$
$830$	0	0
$831$	−70714.0	−2.95191
$832$	0	0
$833$	17622.3	0.732984
$834$	0	0
$835$	11135.3	0.461499
$836$	0	0
$837$	−5040.72	−0.208164
$838$	0	0
$839$	30615.8	1.25980	0.629901	−	0.776676i	$-0.283095\pi$
$839$	30615.8	1.25980	0.629901	+	0.776676i	$0.283095\pi$
$840$	0	0
$841$	23971.0	0.982861
$842$	0	0
$843$	13324.1	0.544372
$844$	0	0
$845$	601.902	0.0245042
$846$	0	0
$847$	−29842.9	−1.21064
$848$	0	0
$849$	−60488.6	−2.44519
$850$	0	0
$851$	11561.0	0.465696
$852$	0	0
$853$	−5660.88	−0.227227	−0.113614	−	0.993525i	$-0.536243\pi$
$853$	−5660.88	−0.227227	−0.113614	+	0.993525i	$0.536243\pi$
$854$	0	0
$855$	4133.86	0.165351
$856$	0	0
$857$	41346.1	1.64802	0.824012	−	0.566572i	$-0.191731\pi$
$857$	41346.1	1.64802	0.824012	+	0.566572i	$0.191731\pi$
$858$	0	0
$859$	−34810.5	−1.38268	−0.691339	−	0.722530i	$-0.742979\pi$
$859$	−34810.5	−1.38268	−0.691339	+	0.722530i	$0.742979\pi$
$860$	0	0
$861$	−37847.4	−1.49807
$862$	0	0
$863$	8360.51	0.329774	0.164887	−	0.986312i	$-0.447274\pi$
$863$	8360.51	0.329774	0.164887	+	0.986312i	$0.447274\pi$
$864$	0	0
$865$	−347.500	−0.0136594
$866$	0	0
$867$	−25434.1	−0.996293
$868$	0	0
$869$	8950.21	0.349385
$870$	0	0
$871$	6659.64	0.259074
$872$	0	0
$873$	−73180.6	−2.83710
$874$	0	0
$875$	−22970.4	−0.887475
$876$	0	0
$877$	40579.3	1.56245	0.781223	−	0.624251i	$-0.214596\pi$
$877$	40579.3	1.56245	0.781223	+	0.624251i	$0.214596\pi$
$878$	0	0
$879$	−5558.43	−0.213289
$880$	0	0
$881$	10445.2	0.399442	0.199721	−	0.979853i	$-0.435996\pi$
$881$	10445.2	0.399442	0.199721	+	0.979853i	$0.435996\pi$
$882$	0	0
$883$	18227.6	0.694685	0.347343	−	0.937738i	$-0.387084\pi$
$883$	18227.6	0.694685	0.347343	+	0.937738i	$0.387084\pi$
$884$	0	0
$885$	−13598.2	−0.516496
$886$	0	0
$887$	23517.7	0.890245	0.445122	−	0.895470i	$-0.353160\pi$
$887$	23517.7	0.890245	0.445122	+	0.895470i	$0.353160\pi$
$888$	0	0
$889$	−23512.9	−0.887061
$890$	0	0
$891$	4706.40	0.176959
$892$	0	0
$893$	−11192.6	−0.419424
$894$	0	0
$895$	−123.639	−0.00461766
$896$	0	0
$897$	−13857.3	−0.515809
$898$	0	0
$899$	−5957.95	−0.221033
$900$	0	0
$901$	5383.28	0.199049
$902$	0	0
$903$	−35710.9	−1.31604
$904$	0	0
$905$	4378.38	0.160820
$906$	0	0
$907$	−30564.6	−1.11894	−0.559471	−	0.828850i	$-0.688996\pi$
$907$	−30564.6	−1.11894	−0.559471	+	0.828850i	$0.688996\pi$
$908$	0	0
$909$	−16282.8	−0.594132
$910$	0	0
$911$	32766.5	1.19166	0.595831	−	0.803110i	$-0.296823\pi$
$911$	32766.5	1.19166	0.595831	+	0.803110i	$0.296823\pi$
$912$	0	0
$913$	20667.0	0.749154
$914$	0	0
$915$	4246.96	0.153443
$916$	0	0
$917$	−7647.64	−0.275406
$918$	0	0
$919$	−20686.7	−0.742538	−0.371269	−	0.928525i	$-0.621077\pi$
$919$	−20686.7	−0.742538	−0.371269	+	0.928525i	$0.621077\pi$
$920$	0	0
$921$	−871.989	−0.0311976
$922$	0	0
$923$	−5339.34	−0.190408
$924$	0	0
$925$	10579.2	0.376047
$926$	0	0
$927$	−15626.8	−0.553669
$928$	0	0
$929$	45632.2	1.61156	0.805782	−	0.592212i	$-0.201745\pi$
$929$	45632.2	1.61156	0.805782	+	0.592212i	$0.201745\pi$
$930$	0	0
$931$	9482.26	0.333801
$932$	0	0
$933$	33687.1	1.18206
$934$	0	0
$935$	2421.28	0.0846892
$936$	0	0
$937$	−17761.4	−0.619253	−0.309626	−	0.950858i	$-0.600204\pi$
$937$	−17761.4	−0.619253	−0.309626	+	0.950858i	$0.600204\pi$
$938$	0	0
$939$	−32909.6	−1.14373
$940$	0	0
$941$	44888.3	1.55507	0.777534	−	0.628841i	$-0.216470\pi$
$941$	44888.3	1.55507	0.777534	+	0.628841i	$0.216470\pi$
$942$	0	0
$943$	19681.7	0.679664
$944$	0	0
$945$	18008.6	0.619917
$946$	0	0
$947$	16069.6	0.551415	0.275708	−	0.961242i	$-0.411088\pi$
$947$	16069.6	0.551415	0.275708	+	0.961242i	$0.411088\pi$
$948$	0	0
$949$	−4004.05	−0.136962
$950$	0	0
$951$	−38271.6	−1.30499
$952$	0	0
$953$	3512.03	0.119377	0.0596883	−	0.998217i	$-0.480989\pi$
$953$	3512.03	0.119377	0.0596883	+	0.998217i	$0.480989\pi$
$954$	0	0
$955$	−15246.3	−0.516605
$956$	0	0
$957$	29146.7	0.984513
$958$	0	0
$959$	−71786.5	−2.41721
$960$	0	0
$961$	−29057.0	−0.975361
$962$	0	0
$963$	69468.9	2.32461
$964$	0	0
$965$	1682.19	0.0561158
$966$	0	0
$967$	37011.9	1.23084	0.615421	−	0.788199i	$-0.288986\pi$
$967$	37011.9	1.23084	0.615421	+	0.788199i	$0.288986\pi$
$968$	0	0
$969$	9273.16	0.307427
$970$	0	0
$971$	19532.3	0.645542	0.322771	−	0.946477i	$-0.395386\pi$
$971$	19532.3	0.645542	0.322771	+	0.946477i	$0.395386\pi$
$972$	0	0
$973$	−54317.1	−1.78965
$974$	0	0
$975$	−12680.5	−0.416513
$976$	0	0
$977$	30201.2	0.988970	0.494485	−	0.869186i	$-0.335357\pi$
$977$	30201.2	0.988970	0.494485	+	0.869186i	$0.335357\pi$
$978$	0	0
$979$	6713.21	0.219157
$980$	0	0
$981$	−41118.5	−1.33824
$982$	0	0
$983$	38774.9	1.25812	0.629058	−	0.777359i	$-0.283441\pi$
$983$	38774.9	1.25812	0.629058	+	0.777359i	$0.283441\pi$
$984$	0	0
$985$	15971.3	0.516638
$986$	0	0
$987$	−110211.	−3.55425
$988$	0	0
$989$	18570.6	0.597079
$990$	0	0
$991$	27728.9	0.888838	0.444419	−	0.895819i	$-0.353410\pi$
$991$	27728.9	0.888838	0.444419	+	0.895819i	$0.353410\pi$
$992$	0	0
$993$	−35880.6	−1.14666
$994$	0	0
$995$	1304.38	0.0415596
$996$	0	0
$997$	48918.2	1.55392	0.776958	−	0.629552i	$-0.216761\pi$
$997$	48918.2	1.55392	0.776958	+	0.629552i	$0.216761\pi$
$998$	0	0
$999$	−17524.8	−0.555016

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	832.4.a.z.1.2		2
4.3	odd	2		832.4.a.s.1.1		2
8.3	odd	2		13.4.a.b.1.1	✓	2
8.5	even	2		208.4.a.h.1.1		2
24.5	odd	2		1872.4.a.bb.1.2		2
24.11	even	2		117.4.a.d.1.2		2
40.3	even	4		325.4.b.e.274.3		4
40.19	odd	2		325.4.a.f.1.2		2
40.27	even	4		325.4.b.e.274.2		4
56.27	even	2		637.4.a.b.1.1		2
88.43	even	2		1573.4.a.b.1.2		2
104.3	odd	6		169.4.c.g.22.2		4
104.11	even	12		169.4.e.f.147.3		8
104.19	even	12		169.4.e.f.23.3		8
104.35	odd	6		169.4.c.g.146.2		4
104.43	odd	6		169.4.c.j.146.1		4
104.51	odd	2		169.4.a.g.1.2		2
104.59	even	12		169.4.e.f.23.2		8
104.67	even	12		169.4.e.f.147.2		8
104.75	odd	6		169.4.c.j.22.1		4
104.83	even	4		169.4.b.f.168.3		4
104.99	even	4		169.4.b.f.168.2		4
312.155	even	2		1521.4.a.r.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.4.a.b.1.1	✓	2	8.3	odd	2
117.4.a.d.1.2		2	24.11	even	2
169.4.a.g.1.2		2	104.51	odd	2
169.4.b.f.168.2		4	104.99	even	4
169.4.b.f.168.3		4	104.83	even	4
169.4.c.g.22.2		4	104.3	odd	6
169.4.c.g.146.2		4	104.35	odd	6
169.4.c.j.22.1		4	104.75	odd	6
169.4.c.j.146.1		4	104.43	odd	6
169.4.e.f.23.2		8	104.59	even	12
169.4.e.f.23.3		8	104.19	even	12
169.4.e.f.147.2		8	104.67	even	12
169.4.e.f.147.3		8	104.11	even	12
208.4.a.h.1.1		2	8.5	even	2
325.4.a.f.1.2		2	40.19	odd	2
325.4.b.e.274.2		4	40.27	even	4
325.4.b.e.274.3		4	40.3	even	4
637.4.a.b.1.1		2	56.27	even	2
832.4.a.s.1.1		2	4.3	odd	2
832.4.a.z.1.2		2	1.1	even	1	trivial
1521.4.a.r.1.1		2	312.155	even	2
1573.4.a.b.1.2		2	88.43	even	2
1872.4.a.bb.1.2		2	24.5	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:	\( N \)	\(=\)	\( 832 = 2^{6} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	832.a (trivial)

\(f(q)\)	\(=\)	\(q+8.68466 q^{3} +3.56155 q^{5} +27.1771 q^{7} +48.4233 q^{9} +O(q^{10})\)	\(q+8.68466 q^{3} +3.56155 q^{5} +27.1771 q^{7} +48.4233 q^{9} +15.2614 q^{11} +13.0000 q^{13} +30.9309 q^{15} +44.5464 q^{17} +23.9697 q^{19} +236.024 q^{21} -122.739 q^{23} -112.315 q^{25} +186.054 q^{27} +219.909 q^{29} -27.0928 q^{31} +132.540 q^{33} +96.7926 q^{35} -94.1922 q^{37} +112.901 q^{39} -160.354 q^{41} -151.302 q^{43} +172.462 q^{45} -466.948 q^{47} +395.594 q^{49} +386.870 q^{51} +120.847 q^{53} +54.3542 q^{55} +208.169 q^{57} -439.633 q^{59} +137.305 q^{61} +1316.00 q^{63} +46.3002 q^{65} +512.280 q^{67} -1065.94 q^{69} -410.719 q^{71} -308.004 q^{73} -975.420 q^{75} +414.759 q^{77} +586.462 q^{79} +308.386 q^{81} +1354.20 q^{83} +158.654 q^{85} +1909.84 q^{87} +439.882 q^{89} +353.302 q^{91} -235.292 q^{93} +85.3693 q^{95} -1511.27 q^{97} +739.006 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 5 q^{3} + 3 q^{5} + 9 q^{7} + 35 q^{9}+O(q^{10}) \) 2 * q + 5 * q^3 + 3 * q^5 + 9 * q^7 + 35 * q^9	\( 2 q + 5 q^{3} + 3 q^{5} + 9 q^{7} + 35 q^{9} + 80 q^{11} + 26 q^{13} + 33 q^{15} + 19 q^{17} - 84 q^{19} + 303 q^{21} - 196 q^{23} - 237 q^{25} + 335 q^{27} + 44 q^{29} + 86 q^{31} - 106 q^{33} + 107 q^{35} - 209 q^{37} + 65 q^{39} - 230 q^{41} + 287 q^{43} + 180 q^{45} - 435 q^{47} + 383 q^{49} + 481 q^{51} + 118 q^{53} + 18 q^{55} + 606 q^{57} - 368 q^{59} + 1058 q^{61} + 1560 q^{63} + 39 q^{65} + 68 q^{67} - 796 q^{69} + 131 q^{71} + 456 q^{73} - 516 q^{75} - 762 q^{77} + 1008 q^{79} + 122 q^{81} + 1958 q^{83} + 173 q^{85} + 2558 q^{87} - 720 q^{89} + 117 q^{91} - 652 q^{93} + 146 q^{95} - 928 q^{97} - 130 q^{99}+O(q^{100}) \) 2 * q + 5 * q^3 + 3 * q^5 + 9 * q^7 + 35 * q^9 + 80 * q^11 + 26 * q^13 + 33 * q^15 + 19 * q^17 - 84 * q^19 + 303 * q^21 - 196 * q^23 - 237 * q^25 + 335 * q^27 + 44 * q^29 + 86 * q^31 - 106 * q^33 + 107 * q^35 - 209 * q^37 + 65 * q^39 - 230 * q^41 + 287 * q^43 + 180 * q^45 - 435 * q^47 + 383 * q^49 + 481 * q^51 + 118 * q^53 + 18 * q^55 + 606 * q^57 - 368 * q^59 + 1058 * q^61 + 1560 * q^63 + 39 * q^65 + 68 * q^67 - 796 * q^69 + 131 * q^71 + 456 * q^73 - 516 * q^75 - 762 * q^77 + 1008 * q^79 + 122 * q^81 + 1958 * q^83 + 173 * q^85 + 2558 * q^87 - 720 * q^89 + 117 * q^91 - 652 * q^93 + 146 * q^95 - 928 * q^97 - 130 * q^99

Introduction

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