LMFDB - Embedded newform 1216.3.f.b.799.36

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,3,Mod(799,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1216.799");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1216.f (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$33.1336001462$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.36
Character	$\chi$	$=$	1216.799
Dual form			1216.3.f.b.799.35

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.18150 q^{3} -2.95407i q^{5} +0.327266i q^{7} +1.12196 q^{9} +O(q^{10})$	$q+3.18150 q^{3} -2.95407i q^{5} +0.327266i q^{7} +1.12196 q^{9} +19.0176 q^{11} +25.3639i q^{13} -9.39837i q^{15} +13.2615 q^{17} -4.35890 q^{19} +1.04120i q^{21} -2.08382i q^{23} +16.2735 q^{25} -25.0640 q^{27} -18.2270i q^{29} +21.5738i q^{31} +60.5047 q^{33} +0.966767 q^{35} +62.8483i q^{37} +80.6954i q^{39} -56.7625 q^{41} +13.8389 q^{43} -3.31433i q^{45} -31.3493i q^{47} +48.8929 q^{49} +42.1914 q^{51} -94.5917i q^{53} -56.1794i q^{55} -13.8678 q^{57} +105.050 q^{59} -6.22105i q^{61} +0.367178i q^{63} +74.9267 q^{65} +98.8837 q^{67} -6.62968i q^{69} +43.1075i q^{71} +71.2332 q^{73} +51.7741 q^{75} +6.22383i q^{77} +130.530i q^{79} -89.8388 q^{81} -78.7228 q^{83} -39.1752i q^{85} -57.9893i q^{87} +78.2993 q^{89} -8.30076 q^{91} +68.6372i q^{93} +12.8765i q^{95} +102.505 q^{97} +21.3370 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 168 q^{9}+O(q^{10}) $ 48 * q + 168 * q^9	$ 48 q + 168 q^{9} + 100 q^{17} - 204 q^{25} - 8 q^{33} - 240 q^{41} - 460 q^{49} - 368 q^{65} - 132 q^{73} + 768 q^{81} + 696 q^{89} - 8 q^{97}+O(q^{100}) $ 48 * q + 168 * q^9 + 100 * q^17 - 204 * q^25 - 8 * q^33 - 240 * q^41 - 460 * q^49 - 368 * q^65 - 132 * q^73 + 768 * q^81 + 696 * q^89 - 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$.

$n$	$191$	$705$	$837$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	3.18150	1.06050	0.530250	−	0.847841i	$-0.322098\pi$
$3$	3.18150	1.06050	0.530250	+	0.847841i	$0.322098\pi$
$4$	0	0
$5$	− 2.95407i	− 0.590813i	−0.955372	−	0.295407i	$-0.904545\pi$
$5$	− 2.95407i	− 0.590813i	0.955372	−	0.295407i	$-0.0954551\pi$
$6$	0	0
$7$	0.327266i	0.0467523i	0.999727	+	0.0233762i	$0.00744154\pi$
$7$	0.327266i	0.0467523i	−0.999727	+	0.0233762i	$0.992558\pi$
$8$	0	0
$9$	1.12196	0.124662
$10$	0	0
$11$	19.0176	1.72888	0.864438	−	0.502739i	$-0.167674\pi$
$11$	19.0176	1.72888	0.864438	+	0.502739i	$0.167674\pi$
$12$	0	0
$13$	25.3639i	1.95107i	0.219841	+	0.975536i	$0.429446\pi$
$13$	25.3639i	1.95107i	−0.219841	+	0.975536i	$0.570554\pi$
$14$	0	0
$15$	− 9.39837i	− 0.626558i
$16$	0	0
$17$	13.2615	0.780086	0.390043	−	0.920797i	$-0.372460\pi$
$17$	13.2615	0.780086	0.390043	+	0.920797i	$0.372460\pi$
$18$	0	0
$19$	−4.35890	−0.229416
$19$	−4.35890	−0.229416
$20$	0	0
$21$	1.04120i	0.0495809i
$22$	0	0
$23$	− 2.08382i	− 0.0906009i	−0.998973	−	0.0453005i	$-0.985575\pi$
$23$	− 2.08382i	− 0.0906009i	0.998973	−	0.0453005i	$-0.0144245\pi$
$24$	0	0
$25$	16.2735	0.650940
$26$	0	0
$27$	−25.0640	−0.928297
$28$	0	0
$29$	− 18.2270i	− 0.628517i	−0.949337	−	0.314259i	$-0.898244\pi$
$29$	− 18.2270i	− 0.628517i	0.949337	−	0.314259i	$-0.101756\pi$
$30$	0	0
$31$	21.5738i	0.695930i	0.937507	+	0.347965i	$0.113127\pi$
$31$	21.5738i	0.695930i	−0.937507	+	0.347965i	$0.886873\pi$
$32$	0	0
$33$	60.5047	1.83347
$34$	0	0
$35$	0.966767	0.0276219
$36$	0	0
$37$	62.8483i	1.69860i	0.527909	+	0.849301i	$0.322976\pi$
$37$	62.8483i	1.69860i	−0.527909	+	0.849301i	$0.677024\pi$
$38$	0	0
$39$	80.6954i	2.06911i
$40$	0	0
$41$	−56.7625	−1.38445	−0.692226	−	0.721681i	$-0.743370\pi$
$41$	−56.7625	−1.38445	−0.692226	+	0.721681i	$0.743370\pi$
$42$	0	0
$43$	13.8389	0.321835	0.160917	−	0.986968i	$-0.448555\pi$
$43$	13.8389	0.321835	0.160917	+	0.986968i	$0.448555\pi$
$44$	0	0
$45$	− 3.31433i	− 0.0736519i
$46$	0	0
$47$	− 31.3493i	− 0.667006i	−0.942749	−	0.333503i	$-0.891769\pi$
$47$	− 31.3493i	− 0.667006i	0.942749	−	0.333503i	$-0.108231\pi$
$48$	0	0
$49$	48.8929	0.997814
$50$	0	0
$51$	42.1914	0.827281
$52$	0	0
$53$	− 94.5917i	− 1.78475i	−0.451296	−	0.892374i	$-0.649038\pi$
$53$	− 94.5917i	− 1.78475i	0.451296	−	0.892374i	$-0.350962\pi$
$54$	0	0
$55$	− 56.1794i	− 1.02144i
$56$	0	0
$57$	−13.8678	−0.243296
$58$	0	0
$59$	105.050	1.78052	0.890258	−	0.455456i	$-0.150524\pi$
$59$	105.050	1.78052	0.890258	+	0.455456i	$0.150524\pi$
$60$	0	0
$61$	− 6.22105i	− 0.101984i	−0.998699	−	0.0509922i	$-0.983762\pi$
$61$	− 6.22105i	− 0.101984i	0.998699	−	0.0509922i	$-0.0162384\pi$
$62$	0	0
$63$	0.367178i	0.00582823i
$64$	0	0
$65$	74.9267	1.15272
$66$	0	0
$67$	98.8837	1.47588	0.737938	−	0.674868i	$-0.235800\pi$
$67$	98.8837	1.47588	0.737938	+	0.674868i	$0.235800\pi$
$68$	0	0
$69$	− 6.62968i	− 0.0960824i
$70$	0	0
$71$	43.1075i	0.607148i	0.952808	+	0.303574i	$0.0981800\pi$
$71$	43.1075i	0.607148i	−0.952808	+	0.303574i	$0.901820\pi$
$72$	0	0
$73$	71.2332	0.975798	0.487899	−	0.872900i	$-0.337763\pi$
$73$	71.2332	0.975798	0.487899	+	0.872900i	$0.337763\pi$
$74$	0	0
$75$	51.7741	0.690322
$76$	0	0
$77$	6.22383i	0.0808290i
$78$	0	0
$79$	130.530i	1.65227i	0.563469	+	0.826137i	$0.309466\pi$
$79$	130.530i	1.65227i	−0.563469	+	0.826137i	$0.690534\pi$
$80$	0	0
$81$	−89.8388	−1.10912
$82$	0	0
$83$	−78.7228	−0.948467	−0.474234	−	0.880399i	$-0.657275\pi$
$83$	−78.7228	−0.948467	−0.474234	+	0.880399i	$0.657275\pi$
$84$	0	0
$85$	− 39.1752i	− 0.460885i
$86$	0	0
$87$	− 57.9893i	− 0.666543i
$88$	0	0
$89$	78.2993	0.879768	0.439884	−	0.898055i	$-0.355020\pi$
$89$	78.2993	0.879768	0.439884	+	0.898055i	$0.355020\pi$
$90$	0	0
$91$	−8.30076	−0.0912171
$92$	0	0
$93$	68.6372i	0.738035i
$94$	0	0
$95$	12.8765i	0.135542i
$96$	0	0
$97$	102.505	1.05676	0.528379	−	0.849009i	$-0.322800\pi$
$97$	102.505	1.05676	0.528379	+	0.849009i	$0.322800\pi$
$98$	0	0
$99$	21.3370	0.215525
$100$	0	0
$101$	− 150.928i	− 1.49434i	−0.664636	−	0.747168i	$-0.731413\pi$
$101$	− 150.928i	− 1.49434i	0.664636	−	0.747168i	$-0.268587\pi$
$102$	0	0
$103$	− 23.3527i	− 0.226726i	−0.993554	−	0.113363i	$-0.963838\pi$
$103$	− 23.3527i	− 0.226726i	0.993554	−	0.113363i	$-0.0361623\pi$
$104$	0	0
$105$	3.07577	0.0292930
$106$	0	0
$107$	−47.3004	−0.442059	−0.221030	−	0.975267i	$-0.570942\pi$
$107$	−47.3004	−0.442059	−0.221030	+	0.975267i	$0.570942\pi$
$108$	0	0
$109$	− 118.475i	− 1.08692i	−0.839434	−	0.543462i	$-0.817113\pi$
$109$	− 118.475i	− 1.08692i	0.839434	−	0.543462i	$-0.182887\pi$
$110$	0	0
$111$	199.952i	1.80137i
$112$	0	0
$113$	−136.749	−1.21017	−0.605083	−	0.796162i	$-0.706860\pi$
$113$	−136.749	−1.21017	−0.605083	+	0.796162i	$0.706860\pi$
$114$	0	0
$115$	−6.15575	−0.0535282
$116$	0	0
$117$	28.4572i	0.243224i
$118$	0	0
$119$	4.34003i	0.0364708i
$120$	0	0
$121$	240.671	1.98901
$122$	0	0
$123$	−180.590	−1.46821
$124$	0	0
$125$	− 121.925i	− 0.975397i
$126$	0	0
$127$	218.714i	1.72216i	0.508471	+	0.861079i	$0.330211\pi$
$127$	218.714i	1.72216i	−0.508471	+	0.861079i	$0.669789\pi$
$128$	0	0
$129$	44.0285	0.341306
$130$	0	0
$131$	−38.8083	−0.296246	−0.148123	−	0.988969i	$-0.547323\pi$
$131$	−38.8083	−0.296246	−0.148123	+	0.988969i	$0.547323\pi$
$132$	0	0
$133$	− 1.42652i	− 0.0107257i
$134$	0	0
$135$	74.0408i	0.548450i
$136$	0	0
$137$	−32.1959	−0.235006	−0.117503	−	0.993073i	$-0.537489\pi$
$137$	−32.1959	−0.235006	−0.117503	+	0.993073i	$0.537489\pi$
$138$	0	0
$139$	−86.9066	−0.625227	−0.312614	−	0.949880i	$-0.601205\pi$
$139$	−86.9066	−0.625227	−0.312614	+	0.949880i	$0.601205\pi$
$140$	0	0
$141$	− 99.7379i	− 0.707361i
$142$	0	0
$143$	482.362i	3.37316i
$144$	0	0
$145$	−53.8438	−0.371336
$146$	0	0
$147$	155.553	1.05818
$148$	0	0
$149$	− 207.360i	− 1.39168i	−0.718198	−	0.695839i	$-0.755033\pi$
$149$	− 207.360i	− 1.39168i	0.718198	−	0.695839i	$-0.244967\pi$
$150$	0	0
$151$	− 115.148i	− 0.762569i	−0.924458	−	0.381285i	$-0.875482\pi$
$151$	− 115.148i	− 0.762569i	0.924458	−	0.381285i	$-0.124518\pi$
$152$	0	0
$153$	14.8788	0.0972469
$154$	0	0
$155$	63.7306	0.411165
$156$	0	0
$157$	181.120i	1.15363i	0.816875	+	0.576815i	$0.195705\pi$
$157$	181.120i	1.15363i	−0.816875	+	0.576815i	$0.804295\pi$
$158$	0	0
$159$	− 300.944i	− 1.89273i
$160$	0	0
$161$	0.681965	0.00423580
$162$	0	0
$163$	−265.772	−1.63050	−0.815252	−	0.579106i	$-0.803402\pi$
$163$	−265.772	−1.63050	−0.815252	+	0.579106i	$0.803402\pi$
$164$	0	0
$165$	− 178.735i	− 1.08324i
$166$	0	0
$167$	− 160.259i	− 0.959632i	−0.877369	−	0.479816i	$-0.840703\pi$
$167$	− 160.259i	− 0.959632i	0.877369	−	0.479816i	$-0.159297\pi$
$168$	0	0
$169$	−474.329	−2.80668
$170$	0	0
$171$	−4.89049	−0.0285994
$172$	0	0
$173$	52.5124i	0.303540i	0.988416	+	0.151770i	$0.0484973\pi$
$173$	52.5124i	0.303540i	−0.988416	+	0.151770i	$0.951503\pi$
$174$	0	0
$175$	5.32576i	0.0304329i
$176$	0	0
$177$	334.218	1.88824
$178$	0	0
$179$	199.981	1.11721	0.558607	−	0.829432i	$-0.311336\pi$
$179$	199.981	1.11721	0.558607	+	0.829432i	$0.311336\pi$
$180$	0	0
$181$	51.9372i	0.286946i	0.989654	+	0.143473i	$0.0458270\pi$
$181$	51.9372i	0.286946i	−0.989654	+	0.143473i	$0.954173\pi$
$182$	0	0
$183$	− 19.7923i	− 0.108154i
$184$	0	0
$185$	185.658	1.00356
$186$	0	0
$187$	252.202	1.34867
$188$	0	0
$189$	− 8.20261i	− 0.0434000i
$190$	0	0
$191$	134.633i	0.704883i	0.935834	+	0.352441i	$0.114648\pi$
$191$	134.633i	0.704883i	−0.935834	+	0.352441i	$0.885352\pi$
$192$	0	0
$193$	2.63560	0.0136560	0.00682798	−	0.999977i	$-0.497827\pi$
$193$	2.63560	0.0136560	0.00682798	+	0.999977i	$0.497827\pi$
$194$	0	0
$195$	238.380	1.22246
$196$	0	0
$197$	− 32.7230i	− 0.166106i	−0.996545	−	0.0830532i	$-0.973533\pi$
$197$	− 32.7230i	− 0.166106i	0.996545	−	0.0830532i	$-0.0264671\pi$
$198$	0	0
$199$	48.0423i	0.241418i	0.992688	+	0.120709i	$0.0385169\pi$
$199$	48.0423i	0.241418i	−0.992688	+	0.120709i	$0.961483\pi$
$200$	0	0
$201$	314.599	1.56517
$202$	0	0
$203$	5.96508	0.0293847
$204$	0	0
$205$	167.680i	0.817953i
$206$	0	0
$207$	− 2.33796i	− 0.0112945i
$208$	0	0
$209$	−82.8960	−0.396631
$210$	0	0
$211$	−320.373	−1.51835	−0.759177	−	0.650884i	$-0.774398\pi$
$211$	−320.373	−1.51835	−0.759177	+	0.650884i	$0.774398\pi$
$212$	0	0
$213$	137.147i	0.643881i
$214$	0	0
$215$	− 40.8810i	− 0.190144i
$216$	0	0
$217$	−7.06039	−0.0325364
$218$	0	0
$219$	226.629	1.03483
$220$	0	0
$221$	336.363i	1.52200i
$222$	0	0
$223$	55.9638i	0.250959i	0.992096	+	0.125479i	$0.0400469\pi$
$223$	55.9638i	0.250959i	−0.992096	+	0.125479i	$0.959953\pi$
$224$	0	0
$225$	18.2581	0.0811473
$226$	0	0
$227$	−242.543	−1.06847	−0.534235	−	0.845336i	$-0.679400\pi$
$227$	−242.543	−1.06847	−0.534235	+	0.845336i	$0.679400\pi$
$228$	0	0
$229$	− 179.312i	− 0.783022i	−0.920173	−	0.391511i	$-0.871952\pi$
$229$	− 179.312i	− 0.783022i	0.920173	−	0.391511i	$-0.128048\pi$
$230$	0	0
$231$	19.8011i	0.0857192i
$232$	0	0
$233$	212.631	0.912581	0.456291	−	0.889831i	$-0.349178\pi$
$233$	212.631	0.912581	0.456291	+	0.889831i	$0.349178\pi$
$234$	0	0
$235$	−92.6079	−0.394076
$236$	0	0
$237$	415.280i	1.75224i
$238$	0	0
$239$	− 287.776i	− 1.20408i	−0.798465	−	0.602042i	$-0.794354\pi$
$239$	− 287.776i	− 1.20408i	0.798465	−	0.602042i	$-0.205646\pi$
$240$	0	0
$241$	−14.3514	−0.0595494	−0.0297747	−	0.999557i	$-0.509479\pi$
$241$	−14.3514	−0.0595494	−0.0297747	+	0.999557i	$0.509479\pi$
$242$	0	0
$243$	−60.2463	−0.247927
$244$	0	0
$245$	− 144.433i	− 0.589522i
$246$	0	0
$247$	− 110.559i	− 0.447607i
$248$	0	0
$249$	−250.457	−1.00585
$250$	0	0
$251$	−134.282	−0.534987	−0.267493	−	0.963560i	$-0.586195\pi$
$251$	−134.282	−0.534987	−0.267493	+	0.963560i	$0.586195\pi$
$252$	0	0
$253$	− 39.6294i	− 0.156638i
$254$	0	0
$255$	− 124.636i	− 0.488769i
$256$	0	0
$257$	53.9448	0.209902	0.104951	−	0.994477i	$-0.466531\pi$
$257$	53.9448	0.209902	0.104951	+	0.994477i	$0.466531\pi$
$258$	0	0
$259$	−20.5681	−0.0794136
$260$	0	0
$261$	− 20.4499i	− 0.0783521i
$262$	0	0
$263$	239.810i	0.911823i	0.890025	+	0.455912i	$0.150687\pi$
$263$	239.810i	0.911823i	−0.890025	+	0.455912i	$0.849313\pi$
$264$	0	0
$265$	−279.430	−1.05445
$266$	0	0
$267$	249.110	0.932994
$268$	0	0
$269$	156.402i	0.581420i	0.956811	+	0.290710i	$0.0938915\pi$
$269$	156.402i	0.581420i	−0.956811	+	0.290710i	$0.906109\pi$
$270$	0	0
$271$	147.578i	0.544569i	0.962217	+	0.272284i	$0.0877791\pi$
$271$	147.578i	0.544569i	−0.962217	+	0.272284i	$0.912221\pi$
$272$	0	0
$273$	−26.4089	−0.0967358
$274$	0	0
$275$	309.483	1.12539
$276$	0	0
$277$	− 462.232i	− 1.66871i	−0.551230	−	0.834353i	$-0.685841\pi$
$277$	− 462.232i	− 1.66871i	0.551230	−	0.834353i	$-0.314159\pi$
$278$	0	0
$279$	24.2049i	0.0867559i
$280$	0	0
$281$	245.672	0.874278	0.437139	−	0.899394i	$-0.355992\pi$
$281$	245.672	0.874278	0.437139	+	0.899394i	$0.355992\pi$
$282$	0	0
$283$	−186.202	−0.657956	−0.328978	−	0.944338i	$-0.606704\pi$
$283$	−186.202	−0.657956	−0.328978	+	0.944338i	$0.606704\pi$
$284$	0	0
$285$	40.9665i	0.143742i
$286$	0	0
$287$	− 18.5765i	− 0.0647264i
$288$	0	0
$289$	−113.134	−0.391466
$290$	0	0
$291$	326.121	1.12069
$292$	0	0
$293$	48.2216i	0.164579i	0.996608	+	0.0822894i	$0.0262232\pi$
$293$	48.2216i	0.164579i	−0.996608	+	0.0822894i	$0.973777\pi$
$294$	0	0
$295$	− 310.326i	− 1.05195i
$296$	0	0
$297$	−476.658	−1.60491
$298$	0	0
$299$	52.8539	0.176769
$300$	0	0
$301$	4.52900i	0.0150465i
$302$	0	0
$303$	− 480.177i	− 1.58474i
$304$	0	0
$305$	−18.3774	−0.0602537
$306$	0	0
$307$	389.254	1.26793	0.633964	−	0.773362i	$-0.281427\pi$
$307$	389.254	1.26793	0.633964	+	0.773362i	$0.281427\pi$
$308$	0	0
$309$	− 74.2968i	− 0.240443i
$310$	0	0
$311$	178.649i	0.574432i	0.957866	+	0.287216i	$0.0927298\pi$
$311$	178.649i	0.574432i	−0.957866	+	0.287216i	$0.907270\pi$
$312$	0	0
$313$	−278.244	−0.888958	−0.444479	−	0.895789i	$-0.646611\pi$
$313$	−278.244	−0.888958	−0.444479	+	0.895789i	$0.646611\pi$
$314$	0	0
$315$	1.08467	0.00344340
$316$	0	0
$317$	133.094i	0.419855i	0.977717	+	0.209928i	$0.0673228\pi$
$317$	133.094i	0.419855i	−0.977717	+	0.209928i	$0.932677\pi$
$318$	0	0
$319$	− 346.635i	− 1.08663i
$320$	0	0
$321$	−150.486	−0.468804
$322$	0	0
$323$	−57.8053	−0.178964
$324$	0	0
$325$	412.760i	1.27003i
$326$	0	0
$327$	− 376.928i	− 1.15268i
$328$	0	0
$329$	10.2596	0.0311841
$330$	0	0
$331$	−347.507	−1.04987	−0.524934	−	0.851143i	$-0.675910\pi$
$331$	−347.507	−1.04987	−0.524934	+	0.851143i	$0.675910\pi$
$332$	0	0
$333$	70.5130i	0.211751i
$334$	0	0
$335$	− 292.109i	− 0.871967i
$336$	0	0
$337$	35.3718	0.104961	0.0524805	−	0.998622i	$-0.483287\pi$
$337$	35.3718	0.104961	0.0524805	+	0.998622i	$0.483287\pi$
$338$	0	0
$339$	−435.067	−1.28338
$340$	0	0
$341$	410.284i	1.20318i
$342$	0	0
$343$	32.0370i	0.0934025i
$344$	0	0
$345$	−19.5845	−0.0567667
$346$	0	0
$347$	−439.041	−1.26525	−0.632623	−	0.774460i	$-0.718022\pi$
$347$	−439.041	−1.26525	−0.632623	+	0.774460i	$0.718022\pi$
$348$	0	0
$349$	− 266.762i	− 0.764360i	−0.924088	−	0.382180i	$-0.875173\pi$
$349$	− 266.762i	− 0.764360i	0.924088	−	0.382180i	$-0.124827\pi$
$350$	0	0
$351$	− 635.722i	− 1.81117i
$352$	0	0
$353$	−307.704	−0.871683	−0.435842	−	0.900023i	$-0.643549\pi$
$353$	−307.704	−0.871683	−0.435842	+	0.900023i	$0.643549\pi$
$354$	0	0
$355$	127.342	0.358711
$356$	0	0
$357$	13.8078i	0.0386773i
$358$	0	0
$359$	− 135.734i	− 0.378088i	−0.981969	−	0.189044i	$-0.939461\pi$
$359$	− 135.734i	− 0.378088i	0.981969	−	0.189044i	$-0.0605389\pi$
$360$	0	0
$361$	19.0000	0.0526316
$362$	0	0
$363$	765.694	2.10935
$364$	0	0
$365$	− 210.428i	− 0.576514i
$366$	0	0
$367$	133.270i	0.363134i	0.983379	+	0.181567i	$0.0581169\pi$
$367$	133.270i	0.363134i	−0.983379	+	0.181567i	$0.941883\pi$
$368$	0	0
$369$	−63.6851	−0.172588
$370$	0	0
$371$	30.9567	0.0834411
$372$	0	0
$373$	− 225.178i	− 0.603693i	−0.953356	−	0.301847i	$-0.902397\pi$
$373$	− 225.178i	− 0.603693i	0.953356	−	0.301847i	$-0.0976031\pi$
$374$	0	0
$375$	− 387.904i	− 1.03441i
$376$	0	0
$377$	462.308	1.22628
$378$	0	0
$379$	−404.256	−1.06664	−0.533320	−	0.845914i	$-0.679056\pi$
$379$	−404.256	−1.06664	−0.533320	+	0.845914i	$0.679056\pi$
$380$	0	0
$381$	695.839i	1.82635i
$382$	0	0
$383$	467.071i	1.21951i	0.792591	+	0.609753i	$0.208731\pi$
$383$	467.071i	1.21951i	−0.792591	+	0.609753i	$0.791269\pi$
$384$	0	0
$385$	18.3856	0.0477548
$386$	0	0
$387$	15.5266	0.0401205
$388$	0	0
$389$	− 176.361i	− 0.453371i	−0.973968	−	0.226685i	$-0.927211\pi$
$389$	− 176.361i	− 0.453371i	0.973968	−	0.226685i	$-0.0727889\pi$
$390$	0	0
$391$	− 27.6345i	− 0.0706765i
$392$	0	0
$393$	−123.469	−0.314170
$394$	0	0
$395$	385.593	0.976186
$396$	0	0
$397$	535.118i	1.34790i	0.738775	+	0.673952i	$0.235405\pi$
$397$	535.118i	1.34790i	−0.738775	+	0.673952i	$0.764595\pi$
$398$	0	0
$399$	− 4.53848i	− 0.0113746i
$400$	0	0
$401$	−438.547	−1.09363	−0.546817	−	0.837252i	$-0.684161\pi$
$401$	−438.547	−1.09363	−0.546817	+	0.837252i	$0.684161\pi$
$402$	0	0
$403$	−547.197	−1.35781
$404$	0	0
$405$	265.390i	0.655284i
$406$	0	0
$407$	1195.23i	2.93667i
$408$	0	0
$409$	319.560	0.781321	0.390661	−	0.920535i	$-0.372247\pi$
$409$	319.560	0.781321	0.390661	+	0.920535i	$0.372247\pi$
$410$	0	0
$411$	−102.431	−0.249224
$412$	0	0
$413$	34.3795i	0.0832433i
$414$	0	0
$415$	232.552i	0.560367i
$416$	0	0
$417$	−276.494	−0.663054
$418$	0	0
$419$	−493.329	−1.17740	−0.588698	−	0.808353i	$-0.700359\pi$
$419$	−493.329	−1.17740	−0.588698	+	0.808353i	$0.700359\pi$
$420$	0	0
$421$	− 509.419i	− 1.21002i	−0.796217	−	0.605011i	$-0.793169\pi$
$421$	− 509.419i	− 1.21002i	0.796217	−	0.605011i	$-0.206831\pi$
$422$	0	0
$423$	− 35.1725i	− 0.0831502i
$424$	0	0
$425$	215.810	0.507789
$426$	0	0
$427$	2.03594	0.00476801
$428$	0	0
$429$	1534.64i	3.57724i
$430$	0	0
$431$	− 676.765i	− 1.57022i	−0.619355	−	0.785111i	$-0.712606\pi$
$431$	− 676.765i	− 1.57022i	0.619355	−	0.785111i	$-0.287394\pi$
$432$	0	0
$433$	−379.072	−0.875455	−0.437728	−	0.899108i	$-0.644217\pi$
$433$	−379.072	−0.875455	−0.437728	+	0.899108i	$0.644217\pi$
$434$	0	0
$435$	−171.304	−0.393803
$436$	0	0
$437$	9.08317i	0.0207853i
$438$	0	0
$439$	− 271.578i	− 0.618628i	−0.950960	−	0.309314i	$-0.899901\pi$
$439$	− 271.578i	− 0.618628i	0.950960	−	0.309314i	$-0.100099\pi$
$440$	0	0
$441$	54.8557	0.124389
$442$	0	0
$443$	−758.996	−1.71331	−0.856655	−	0.515890i	$-0.827461\pi$
$443$	−758.996	−1.71331	−0.856655	+	0.515890i	$0.827461\pi$
$444$	0	0
$445$	− 231.301i	− 0.519779i
$446$	0	0
$447$	− 659.716i	− 1.47588i
$448$	0	0
$449$	−68.6412	−0.152876	−0.0764379	−	0.997074i	$-0.524355\pi$
$449$	−68.6412	−0.152876	−0.0764379	+	0.997074i	$0.524355\pi$
$450$	0	0
$451$	−1079.49	−2.39355
$452$	0	0
$453$	− 366.343i	− 0.808705i
$454$	0	0
$455$	24.5210i	0.0538923i
$456$	0	0
$457$	−333.056	−0.728787	−0.364394	−	0.931245i	$-0.618724\pi$
$457$	−333.056	−0.728787	−0.364394	+	0.931245i	$0.618724\pi$
$458$	0	0
$459$	−332.385	−0.724151
$460$	0	0
$461$	− 863.541i	− 1.87319i	−0.350413	−	0.936595i	$-0.613959\pi$
$461$	− 863.541i	− 1.87319i	0.350413	−	0.936595i	$-0.386041\pi$
$462$	0	0
$463$	295.481i	0.638187i	0.947723	+	0.319094i	$0.103378\pi$
$463$	295.481i	0.638187i	−0.947723	+	0.319094i	$0.896622\pi$
$464$	0	0
$465$	202.759	0.436041
$466$	0	0
$467$	633.073	1.35562	0.677808	−	0.735239i	$-0.262930\pi$
$467$	633.073	1.35562	0.677808	+	0.735239i	$0.262930\pi$
$468$	0	0
$469$	32.3613i	0.0690007i
$470$	0	0
$471$	576.233i	1.22343i
$472$	0	0
$473$	263.183	0.556413
$474$	0	0
$475$	−70.9345	−0.149336
$476$	0	0
$477$	− 106.128i	− 0.222490i
$478$	0	0
$479$	− 109.004i	− 0.227566i	−0.993506	−	0.113783i	$-0.963703\pi$
$479$	− 109.004i	− 0.227566i	0.993506	−	0.113783i	$-0.0362969\pi$
$480$	0	0
$481$	−1594.08	−3.31409
$482$	0	0
$483$	2.16967	0.00449207
$484$	0	0
$485$	− 302.808i	− 0.624347i
$486$	0	0
$487$	− 250.022i	− 0.513392i	−0.966492	−	0.256696i	$-0.917366\pi$
$487$	− 250.022i	− 0.513392i	0.966492	−	0.256696i	$-0.0826339\pi$
$488$	0	0
$489$	−845.555	−1.72915
$490$	0	0
$491$	113.285	0.230724	0.115362	−	0.993324i	$-0.463197\pi$
$491$	113.285	0.230724	0.115362	+	0.993324i	$0.463197\pi$
$492$	0	0
$493$	− 241.717i	− 0.490297i
$494$	0	0
$495$	− 63.0308i	− 0.127335i
$496$	0	0
$497$	−14.1076	−0.0283856
$498$	0	0
$499$	588.287	1.17893	0.589466	−	0.807793i	$-0.299338\pi$
$499$	588.287	1.17893	0.589466	+	0.807793i	$0.299338\pi$
$500$	0	0
$501$	− 509.863i	− 1.01769i
$502$	0	0
$503$	− 419.065i	− 0.833132i	−0.909105	−	0.416566i	$-0.863233\pi$
$503$	− 419.065i	− 0.833132i	0.909105	−	0.416566i	$-0.136767\pi$
$504$	0	0
$505$	−445.851	−0.882873
$506$	0	0
$507$	−1509.08	−2.97649
$508$	0	0
$509$	438.284i	0.861069i	0.902574	+	0.430535i	$0.141675\pi$
$509$	438.284i	0.861069i	−0.902574	+	0.430535i	$0.858325\pi$
$510$	0	0
$511$	23.3122i	0.0456208i
$512$	0	0
$513$	109.252	0.212966
$514$	0	0
$515$	−68.9856	−0.133953
$516$	0	0
$517$	− 596.190i	− 1.15317i
$518$	0	0
$519$	167.068i	0.321904i
$520$	0	0
$521$	−516.120	−0.990633	−0.495316	−	0.868713i	$-0.664948\pi$
$521$	−516.120	−0.990633	−0.495316	+	0.868713i	$0.664948\pi$
$522$	0	0
$523$	−33.1871	−0.0634552	−0.0317276	−	0.999497i	$-0.510101\pi$
$523$	−33.1871	−0.0634552	−0.0317276	+	0.999497i	$0.510101\pi$
$524$	0	0
$525$	16.9439i	0.0322742i
$526$	0	0
$527$	286.101i	0.542885i
$528$	0	0
$529$	524.658	0.991791
$530$	0	0
$531$	117.862	0.221962
$532$	0	0
$533$	− 1439.72i	− 2.70117i
$534$	0	0
$535$	139.728i	0.261175i
$536$	0	0
$537$	636.241	1.18481
$538$	0	0
$539$	929.827	1.72510
$540$	0	0
$541$	706.200i	1.30536i	0.757633	+	0.652680i	$0.226356\pi$
$541$	706.200i	1.30536i	−0.757633	+	0.652680i	$0.773644\pi$
$542$	0	0
$543$	165.238i	0.304306i
$544$	0	0
$545$	−349.982	−0.642169
$546$	0	0
$547$	33.1572	0.0606165	0.0303082	−	0.999541i	$-0.490351\pi$
$547$	33.1572	0.0606165	0.0303082	+	0.999541i	$0.490351\pi$
$548$	0	0
$549$	− 6.97974i	− 0.0127136i
$550$	0	0
$551$	79.4497i	0.144192i
$552$	0	0
$553$	−42.7180	−0.0772477
$554$	0	0
$555$	590.671	1.06427
$556$	0	0
$557$	192.697i	0.345955i	0.984926	+	0.172978i	$0.0553388\pi$
$557$	192.697i	0.345955i	−0.984926	+	0.172978i	$0.944661\pi$
$558$	0	0
$559$	351.009i	0.627923i
$560$	0	0
$561$	802.380	1.43027
$562$	0	0
$563$	−740.912	−1.31601	−0.658004	−	0.753014i	$-0.728599\pi$
$563$	−740.912	−1.31601	−0.658004	+	0.753014i	$0.728599\pi$
$564$	0	0
$565$	403.965i	0.714983i
$566$	0	0
$567$	− 29.4012i	− 0.0518540i
$568$	0	0
$569$	−152.452	−0.267929	−0.133965	−	0.990986i	$-0.542771\pi$
$569$	−152.452	−0.267929	−0.133965	+	0.990986i	$0.542771\pi$
$570$	0	0
$571$	244.246	0.427751	0.213876	−	0.976861i	$-0.431391\pi$
$571$	244.246	0.427751	0.213876	+	0.976861i	$0.431391\pi$
$572$	0	0
$573$	428.334i	0.747529i
$574$	0	0
$575$	− 33.9110i	− 0.0589757i
$576$	0	0
$577$	731.465	1.26770	0.633852	−	0.773455i	$-0.281473\pi$
$577$	731.465	1.26770	0.633852	+	0.773455i	$0.281473\pi$
$578$	0	0
$579$	8.38516	0.0144821
$580$	0	0
$581$	− 25.7633i	− 0.0443430i
$582$	0	0
$583$	− 1798.91i	− 3.08561i
$584$	0	0
$585$	84.0645	0.143700
$586$	0	0
$587$	239.530	0.408058	0.204029	−	0.978965i	$-0.434596\pi$
$587$	239.530	0.408058	0.204029	+	0.978965i	$0.434596\pi$
$588$	0	0
$589$	− 94.0382i	− 0.159657i
$590$	0	0
$591$	− 104.108i	− 0.176156i
$592$	0	0
$593$	720.684	1.21532	0.607659	−	0.794198i	$-0.292109\pi$
$593$	720.684	1.21532	0.607659	+	0.794198i	$0.292109\pi$
$594$	0	0
$595$	12.8207	0.0215474
$596$	0	0
$597$	152.847i	0.256024i
$598$	0	0
$599$	− 22.6100i	− 0.0377462i	−0.999822	−	0.0188731i	$-0.993992\pi$
$599$	− 22.6100i	− 0.0377462i	0.999822	−	0.0188731i	$-0.00600785\pi$
$600$	0	0
$601$	346.941	0.577272	0.288636	−	0.957439i	$-0.406798\pi$
$601$	346.941	0.577272	0.288636	+	0.957439i	$0.406798\pi$
$602$	0	0
$603$	110.943	0.183985
$604$	0	0
$605$	− 710.957i	− 1.17514i
$606$	0	0
$607$	− 496.637i	− 0.818183i	−0.912493	−	0.409092i	$-0.865846\pi$
$607$	− 496.637i	− 0.818183i	0.912493	−	0.409092i	$-0.134154\pi$
$608$	0	0
$609$	18.9779	0.0311624
$610$	0	0
$611$	795.141	1.30138
$612$	0	0
$613$	534.048i	0.871204i	0.900139	+	0.435602i	$0.143465\pi$
$613$	534.048i	0.871204i	−0.900139	+	0.435602i	$0.856535\pi$
$614$	0	0
$615$	533.475i	0.867440i
$616$	0	0
$617$	−765.679	−1.24097	−0.620485	−	0.784218i	$-0.713064\pi$
$617$	−765.679	−1.24097	−0.620485	+	0.784218i	$0.713064\pi$
$618$	0	0
$619$	−44.8187	−0.0724050	−0.0362025	−	0.999344i	$-0.511526\pi$
$619$	−44.8187	−0.0724050	−0.0362025	+	0.999344i	$0.511526\pi$
$620$	0	0
$621$	52.2289i	0.0841046i
$622$	0	0
$623$	25.6247i	0.0411312i
$624$	0	0
$625$	46.6637	0.0746619
$626$	0	0
$627$	−263.734	−0.420628
$628$	0	0
$629$	833.460i	1.32505i
$630$	0	0
$631$	396.162i	0.627832i	0.949451	+	0.313916i	$0.101641\pi$
$631$	396.162i	0.627832i	−0.949451	+	0.313916i	$0.898359\pi$
$632$	0	0
$633$	−1019.27	−1.61022
$634$	0	0
$635$	646.096	1.01747
$636$	0	0
$637$	1240.12i	1.94681i
$638$	0	0
$639$	48.3647i	0.0756882i
$640$	0	0
$641$	−524.768	−0.818671	−0.409335	−	0.912384i	$-0.634239\pi$
$641$	−524.768	−0.818671	−0.409335	+	0.912384i	$0.634239\pi$
$642$	0	0
$643$	194.356	0.302265	0.151132	−	0.988514i	$-0.451708\pi$
$643$	194.356	0.302265	0.151132	+	0.988514i	$0.451708\pi$
$644$	0	0
$645$	− 130.063i	− 0.201648i
$646$	0	0
$647$	660.834i	1.02138i	0.859764	+	0.510691i	$0.170610\pi$
$647$	660.834i	1.02138i	−0.859764	+	0.510691i	$0.829390\pi$
$648$	0	0
$649$	1997.81	3.07829
$650$	0	0
$651$	−22.4626	−0.0345048
$652$	0	0
$653$	149.332i	0.228686i	0.993441	+	0.114343i	$0.0364763\pi$
$653$	149.332i	0.228686i	−0.993441	+	0.114343i	$0.963524\pi$
$654$	0	0
$655$	114.642i	0.175026i
$656$	0	0
$657$	79.9206	0.121645
$658$	0	0
$659$	429.902	0.652355	0.326178	−	0.945308i	$-0.394239\pi$
$659$	429.902	0.652355	0.326178	+	0.945308i	$0.394239\pi$
$660$	0	0
$661$	112.440i	0.170106i	0.996376	+	0.0850528i	$0.0271059\pi$
$661$	112.440i	0.170106i	−0.996376	+	0.0850528i	$0.972894\pi$
$662$	0	0
$663$	1070.14i	1.61409i
$664$	0	0
$665$	−4.21404	−0.00633690
$666$	0	0
$667$	−37.9818	−0.0569443
$668$	0	0
$669$	178.049i	0.266142i
$670$	0	0
$671$	− 118.310i	− 0.176318i
$672$	0	0
$673$	−320.778	−0.476639	−0.238319	−	0.971187i	$-0.576596\pi$
$673$	−320.778	−0.476639	−0.238319	+	0.971187i	$0.576596\pi$
$674$	0	0
$675$	−407.879	−0.604265
$676$	0	0
$677$	− 546.257i	− 0.806878i	−0.915006	−	0.403439i	$-0.867815\pi$
$677$	− 546.257i	− 0.806878i	0.915006	−	0.403439i	$-0.132185\pi$
$678$	0	0
$679$	33.5466i	0.0494059i
$680$	0	0
$681$	−771.650	−1.13311
$682$	0	0
$683$	−924.923	−1.35421	−0.677103	−	0.735888i	$-0.736765\pi$
$683$	−924.923	−1.35421	−0.677103	+	0.735888i	$0.736765\pi$
$684$	0	0
$685$	95.1088i	0.138845i
$686$	0	0
$687$	− 570.482i	− 0.830396i
$688$	0	0
$689$	2399.22	3.48217
$690$	0	0
$691$	362.238	0.524224	0.262112	−	0.965038i	$-0.415581\pi$
$691$	362.238	0.524224	0.262112	+	0.965038i	$0.415581\pi$
$692$	0	0
$693$	6.98287i	0.0100763i
$694$	0	0
$695$	256.728i	0.369393i
$696$	0	0
$697$	−752.754	−1.07999
$698$	0	0
$699$	676.487	0.967793
$700$	0	0
$701$	− 1023.52i	− 1.46008i	−0.683403	−	0.730042i	$-0.739501\pi$
$701$	− 1023.52i	− 1.46008i	0.683403	−	0.730042i	$-0.260499\pi$
$702$	0	0
$703$	− 273.949i	− 0.389686i
$704$	0	0
$705$	−294.632	−0.417918
$706$	0	0
$707$	49.3936	0.0698637
$708$	0	0
$709$	526.649i	0.742805i	0.928472	+	0.371403i	$0.121123\pi$
$709$	526.649i	0.742805i	−0.928472	+	0.371403i	$0.878877\pi$
$710$	0	0
$711$	146.449i	0.205976i
$712$	0	0
$713$	44.9560	0.0630519
$714$	0	0
$715$	1424.93	1.99291
$716$	0	0
$717$	− 915.560i	− 1.27693i
$718$	0	0
$719$	− 67.2606i	− 0.0935474i	−0.998906	−	0.0467737i	$-0.985106\pi$
$719$	− 67.2606i	− 0.0935474i	0.998906	−	0.0467737i	$-0.0148940\pi$
$720$	0	0
$721$	7.64257	0.0106000
$722$	0	0
$723$	−45.6590	−0.0631521
$724$	0	0
$725$	− 296.617i	− 0.409127i
$726$	0	0
$727$	614.959i	0.845886i	0.906156	+	0.422943i	$0.139003\pi$
$727$	614.959i	0.845886i	−0.906156	+	0.422943i	$0.860997\pi$
$728$	0	0
$729$	616.876	0.846194
$730$	0	0
$731$	183.524	0.251059
$732$	0	0
$733$	− 82.7464i	− 0.112887i	−0.998406	−	0.0564437i	$-0.982024\pi$
$733$	− 82.7464i	− 0.112887i	0.998406	−	0.0564437i	$-0.0179761\pi$
$734$	0	0
$735$	− 459.514i	− 0.625188i
$736$	0	0
$737$	1880.53	2.55161
$738$	0	0
$739$	504.689	0.682935	0.341468	−	0.939894i	$-0.389076\pi$
$739$	504.689	0.682935	0.341468	+	0.939894i	$0.389076\pi$
$740$	0	0
$741$	− 351.743i	− 0.474687i
$742$	0	0
$743$	− 166.827i	− 0.224531i	−0.993678	−	0.112266i	$-0.964189\pi$
$743$	− 166.827i	− 0.224531i	0.993678	−	0.112266i	$-0.0358107\pi$
$744$	0	0
$745$	−612.555	−0.822222
$746$	0	0
$747$	−88.3235	−0.118238
$748$	0	0
$749$	− 15.4798i	− 0.0206673i
$750$	0	0
$751$	− 1275.98i	− 1.69905i	−0.527551	−	0.849524i	$-0.676889\pi$
$751$	− 1275.98i	− 1.69905i	0.527551	−	0.849524i	$-0.323111\pi$
$752$	0	0
$753$	−427.217	−0.567354
$754$	0	0
$755$	−340.155	−0.450536
$756$	0	0
$757$	− 1275.37i	− 1.68477i	−0.538878	−	0.842384i	$-0.681151\pi$
$757$	− 1275.37i	− 1.68477i	0.538878	−	0.842384i	$-0.318849\pi$
$758$	0	0
$759$	− 126.081i	− 0.166115i
$760$	0	0
$761$	734.573	0.965274	0.482637	−	0.875821i	$-0.339679\pi$
$761$	734.573	0.965274	0.482637	+	0.875821i	$0.339679\pi$
$762$	0	0
$763$	38.7728	0.0508162
$764$	0	0
$765$	− 43.9529i	− 0.0574548i
$766$	0	0
$767$	2664.49i	3.47392i
$768$	0	0
$769$	−589.509	−0.766591	−0.383296	−	0.923626i	$-0.625211\pi$
$769$	−589.509	−0.766591	−0.383296	+	0.923626i	$0.625211\pi$
$770$	0	0
$771$	171.625	0.222601
$772$	0	0
$773$	− 253.428i	− 0.327850i	−0.986473	−	0.163925i	$-0.947585\pi$
$773$	− 253.428i	− 0.327850i	0.986473	−	0.163925i	$-0.0524154\pi$
$774$	0	0
$775$	351.082i	0.453009i
$776$	0	0
$777$	−65.4375	−0.0842182
$778$	0	0
$779$	247.422	0.317615
$780$	0	0
$781$	819.803i	1.04968i
$782$	0	0
$783$	456.842i	0.583451i
$784$	0	0
$785$	535.040	0.681580
$786$	0	0
$787$	617.729	0.784917	0.392458	−	0.919770i	$-0.371625\pi$
$787$	617.729	0.784917	0.392458	+	0.919770i	$0.371625\pi$
$788$	0	0
$789$	762.955i	0.966989i
$790$	0	0
$791$	− 44.7533i	− 0.0565781i
$792$	0	0
$793$	157.790	0.198979
$794$	0	0
$795$	−889.007	−1.11825
$796$	0	0
$797$	− 1329.45i	− 1.66807i	−0.551709	−	0.834036i	$-0.686024\pi$
$797$	− 1329.45i	− 1.66807i	0.551709	−	0.834036i	$-0.313976\pi$
$798$	0	0
$799$	− 415.737i	− 0.520322i
$800$	0	0
$801$	87.8484	0.109673
$802$	0	0
$803$	1354.69	1.68703
$804$	0	0
$805$	− 2.01457i	− 0.00250257i
$806$	0	0
$807$	497.593i	0.616596i
$808$	0	0
$809$	−325.542	−0.402401	−0.201200	−	0.979550i	$-0.564484\pi$
$809$	−325.542	−0.402401	−0.201200	+	0.979550i	$0.564484\pi$
$810$	0	0
$811$	−1092.05	−1.34655	−0.673273	−	0.739394i	$-0.735112\pi$
$811$	−1092.05	−1.34655	−0.673273	+	0.739394i	$0.735112\pi$
$812$	0	0
$813$	469.520i	0.577515i
$814$	0	0
$815$	785.109i	0.963324i
$816$	0	0
$817$	−60.3224	−0.0738340
$818$	0	0
$819$	−9.31309	−0.0113713
$820$	0	0
$821$	125.979i	0.153446i	0.997052	+	0.0767231i	$0.0244458\pi$
$821$	125.979i	0.153446i	−0.997052	+	0.0767231i	$0.975554\pi$
$822$	0	0
$823$	− 1254.58i	− 1.52440i	−0.647342	−	0.762199i	$-0.724120\pi$
$823$	− 1254.58i	− 1.52440i	0.647342	−	0.762199i	$-0.275880\pi$
$824$	0	0
$825$	984.622	1.19348
$826$	0	0
$827$	939.804	1.13640	0.568201	−	0.822890i	$-0.307640\pi$
$827$	939.804	1.13640	0.568201	+	0.822890i	$0.307640\pi$
$828$	0	0
$829$	1389.36i	1.67595i	0.545711	+	0.837974i	$0.316260\pi$
$829$	1389.36i	1.67595i	−0.545711	+	0.837974i	$0.683740\pi$
$830$	0	0
$831$	− 1470.59i	− 1.76966i
$832$	0	0
$833$	648.391	0.778381
$834$	0	0
$835$	−473.414	−0.566963
$836$	0	0
$837$	− 540.727i	− 0.646030i
$838$	0	0
$839$	1006.46i	1.19959i	0.800152	+	0.599797i	$0.204752\pi$
$839$	1006.46i	1.19959i	−0.800152	+	0.599797i	$0.795248\pi$
$840$	0	0
$841$	508.776	0.604966
$842$	0	0
$843$	781.607	0.927173
$844$	0	0
$845$	1401.20i	1.65822i
$846$	0	0
$847$	78.7634i	0.0929910i
$848$	0	0
$849$	−592.401	−0.697763
$850$	0	0
$851$	130.965	0.153895
$852$	0	0
$853$	− 1042.56i	− 1.22223i	−0.791544	−	0.611113i	$-0.790722\pi$
$853$	− 1042.56i	− 1.22223i	0.791544	−	0.611113i	$-0.209278\pi$
$854$	0	0
$855$	14.4468i	0.0168969i
$856$	0	0
$857$	1604.77	1.87255	0.936274	−	0.351271i	$-0.114250\pi$
$857$	1604.77	1.87255	0.936274	+	0.351271i	$0.114250\pi$
$858$	0	0
$859$	−288.343	−0.335673	−0.167836	−	0.985815i	$-0.553678\pi$
$859$	−288.343	−0.335673	−0.167836	+	0.985815i	$0.553678\pi$
$860$	0	0
$861$	− 59.1011i	− 0.0686424i
$862$	0	0
$863$	1330.02i	1.54116i	0.637344	+	0.770579i	$0.280033\pi$
$863$	1330.02i	1.54116i	−0.637344	+	0.770579i	$0.719967\pi$
$864$	0	0
$865$	155.125	0.179335
$866$	0	0
$867$	−359.935	−0.415150
$868$	0	0
$869$	2482.37i	2.85658i
$870$	0	0
$871$	2508.08i	2.87954i
$872$	0	0
$873$	115.007	0.131737
$874$	0	0
$875$	39.9018	0.0456021
$876$	0	0
$877$	− 1091.50i	− 1.24459i	−0.782783	−	0.622294i	$-0.786201\pi$
$877$	− 1091.50i	− 1.24459i	0.782783	−	0.622294i	$-0.213799\pi$
$878$	0	0
$879$	153.417i	0.174536i
$880$	0	0
$881$	1084.33	1.23080	0.615398	−	0.788216i	$-0.288995\pi$
$881$	1084.33	1.23080	0.615398	+	0.788216i	$0.288995\pi$
$882$	0	0
$883$	698.592	0.791158	0.395579	−	0.918432i	$-0.370544\pi$
$883$	698.592	0.791158	0.395579	+	0.918432i	$0.370544\pi$
$884$	0	0
$885$	− 987.303i	− 1.11560i
$886$	0	0
$887$	1094.01i	1.23338i	0.787205	+	0.616691i	$0.211527\pi$
$887$	1094.01i	1.23338i	−0.787205	+	0.616691i	$0.788473\pi$
$888$	0	0
$889$	−71.5778	−0.0805149
$890$	0	0
$891$	−1708.52	−1.91753
$892$	0	0
$893$	136.648i	0.153022i
$894$	0	0
$895$	− 590.758i	− 0.660065i
$896$	0	0
$897$	168.155	0.187464
$898$	0	0
$899$	393.226	0.437404
$900$	0	0
$901$	− 1254.42i	− 1.39226i
$902$	0	0
$903$	14.4090i	0.0159569i
$904$	0	0
$905$	153.426	0.169531
$906$	0	0
$907$	1342.65	1.48032	0.740161	−	0.672430i	$-0.234749\pi$
$907$	1342.65	1.48032	0.740161	+	0.672430i	$0.234749\pi$
$908$	0	0
$909$	− 169.334i	− 0.186287i
$910$	0	0
$911$	147.927i	0.162379i	0.996699	+	0.0811894i	$0.0258719\pi$
$911$	147.927i	0.162379i	−0.996699	+	0.0811894i	$0.974128\pi$
$912$	0	0
$913$	−1497.12	−1.63978
$914$	0	0
$915$	−58.4677	−0.0638991
$916$	0	0
$917$	− 12.7006i	− 0.0138502i
$918$	0	0
$919$	− 1550.70i	− 1.68738i	−0.536829	−	0.843691i	$-0.680378\pi$
$919$	− 1550.70i	− 1.68738i	0.536829	−	0.843691i	$-0.319622\pi$
$920$	0	0
$921$	1238.41	1.34464
$922$	0	0
$923$	−1093.38	−1.18459
$924$	0	0
$925$	1022.76i	1.10569i
$926$	0	0
$927$	− 26.2008i	− 0.0282640i
$928$	0	0
$929$	−511.095	−0.550156	−0.275078	−	0.961422i	$-0.588704\pi$
$929$	−511.095	−0.550156	−0.275078	+	0.961422i	$0.588704\pi$
$930$	0	0
$931$	−213.119	−0.228914
$932$	0	0
$933$	568.371i	0.609186i
$934$	0	0
$935$	− 745.020i	− 0.796813i
$936$	0	0
$937$	517.012	0.551774	0.275887	−	0.961190i	$-0.411028\pi$
$937$	517.012	0.551774	0.275887	+	0.961190i	$0.411028\pi$
$938$	0	0
$939$	−885.233	−0.942741
$940$	0	0
$941$	− 677.774i	− 0.720270i	−0.932900	−	0.360135i	$-0.882731\pi$
$941$	− 677.774i	− 0.720270i	0.932900	−	0.360135i	$-0.117269\pi$
$942$	0	0
$943$	118.283i	0.125433i
$944$	0	0
$945$	−24.2310	−0.0256413
$946$	0	0
$947$	−1131.47	−1.19479	−0.597397	−	0.801946i	$-0.703798\pi$
$947$	−1131.47	−1.19479	−0.597397	+	0.801946i	$0.703798\pi$
$948$	0	0
$949$	1806.75i	1.90385i
$950$	0	0
$951$	423.439i	0.445257i
$952$	0	0
$953$	−401.743	−0.421556	−0.210778	−	0.977534i	$-0.567600\pi$
$953$	−401.743	−0.421556	−0.210778	+	0.977534i	$0.567600\pi$
$954$	0	0
$955$	397.714	0.416454
$956$	0	0
$957$	− 1102.82i	− 1.15237i
$958$	0	0
$959$	− 10.5366i	− 0.0109871i
$960$	0	0
$961$	495.569	0.515681
$962$	0	0
$963$	−53.0689	−0.0551079
$964$	0	0
$965$	− 7.78573i	− 0.00806812i
$966$	0	0
$967$	− 161.040i	− 0.166536i	−0.996527	−	0.0832679i	$-0.973464\pi$
$967$	− 161.040i	− 0.166536i	0.996527	−	0.0832679i	$-0.0265357\pi$
$968$	0	0
$969$	−183.908	−0.189791
$970$	0	0
$971$	1592.36	1.63992	0.819960	−	0.572420i	$-0.193995\pi$
$971$	1592.36	1.63992	0.819960	+	0.572420i	$0.193995\pi$
$972$	0	0
$973$	− 28.4416i	− 0.0292308i
$974$	0	0
$975$	1313.20i	1.34687i
$976$	0	0
$977$	371.849	0.380603	0.190301	−	0.981726i	$-0.439054\pi$
$977$	371.849	0.380603	0.190301	+	0.981726i	$0.439054\pi$
$978$	0	0
$979$	1489.07	1.52101
$980$	0	0
$981$	− 132.923i	− 0.135498i
$982$	0	0
$983$	− 280.721i	− 0.285575i	−0.989753	−	0.142788i	$-0.954393\pi$
$983$	− 280.721i	− 0.285575i	0.989753	−	0.142788i	$-0.0456066\pi$
$984$	0	0
$985$	−96.6658	−0.0981379
$986$	0	0
$987$	32.6408	0.0330708
$988$	0	0
$989$	− 28.8378i	− 0.0291585i
$990$	0	0
$991$	992.938i	1.00196i	0.865460	+	0.500978i	$0.167026\pi$
$991$	992.938i	1.00196i	−0.865460	+	0.500978i	$0.832974\pi$
$992$	0	0
$993$	−1105.59	−1.11339
$994$	0	0
$995$	141.920	0.142633
$996$	0	0
$997$	− 1144.50i	− 1.14795i	−0.818874	−	0.573974i	$-0.805401\pi$
$997$	− 1144.50i	− 1.14795i	0.818874	−	0.573974i	$-0.194599\pi$
$998$	0	0
$999$	− 1575.23i	− 1.57681i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.3.f.b.799.36	yes	48
4.3	odd	2	inner	1216.3.f.b.799.14	yes	48
8.3	odd	2	inner	1216.3.f.b.799.35	yes	48
8.5	even	2	inner	1216.3.f.b.799.13	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1216.3.f.b.799.13	✓	48	8.5	even	2	inner
1216.3.f.b.799.14	yes	48	4.3	odd	2	inner
1216.3.f.b.799.35	yes	48	8.3	odd	2	inner
1216.3.f.b.799.36	yes	48	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:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1216.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.18150 q^{3} -2.95407i q^{5} +0.327266i q^{7} +1.12196 q^{9} +O(q^{10})\)	\(q+3.18150 q^{3} -2.95407i q^{5} +0.327266i q^{7} +1.12196 q^{9} +19.0176 q^{11} +25.3639i q^{13} -9.39837i q^{15} +13.2615 q^{17} -4.35890 q^{19} +1.04120i q^{21} -2.08382i q^{23} +16.2735 q^{25} -25.0640 q^{27} -18.2270i q^{29} +21.5738i q^{31} +60.5047 q^{33} +0.966767 q^{35} +62.8483i q^{37} +80.6954i q^{39} -56.7625 q^{41} +13.8389 q^{43} -3.31433i q^{45} -31.3493i q^{47} +48.8929 q^{49} +42.1914 q^{51} -94.5917i q^{53} -56.1794i q^{55} -13.8678 q^{57} +105.050 q^{59} -6.22105i q^{61} +0.367178i q^{63} +74.9267 q^{65} +98.8837 q^{67} -6.62968i q^{69} +43.1075i q^{71} +71.2332 q^{73} +51.7741 q^{75} +6.22383i q^{77} +130.530i q^{79} -89.8388 q^{81} -78.7228 q^{83} -39.1752i q^{85} -57.9893i q^{87} +78.2993 q^{89} -8.30076 q^{91} +68.6372i q^{93} +12.8765i q^{95} +102.505 q^{97} +21.3370 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 168 q^{9}+O(q^{10}) \) 48 * q + 168 * q^9	\( 48 q + 168 q^{9} + 100 q^{17} - 204 q^{25} - 8 q^{33} - 240 q^{41} - 460 q^{49} - 368 q^{65} - 132 q^{73} + 768 q^{81} + 696 q^{89} - 8 q^{97}+O(q^{100}) \) 48 * q + 168 * q^9 + 100 * q^17 - 204 * q^25 - 8 * q^33 - 240 * q^41 - 460 * q^49 - 368 * q^65 - 132 * q^73 + 768 * q^81 + 696 * q^89 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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