LMFDB - Embedded newform 3344.2.a.n.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3344,2,Mod(1,3344)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3344.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3344, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3344 = 2^{4} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3344.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,2,0,-2,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$26.7019744359$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 209)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	3344.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.414214 q^{3} -1.00000 q^{5} +0.585786 q^{7} -2.82843 q^{9} +1.00000 q^{11} -6.24264 q^{13} +0.414214 q^{15} +0.585786 q^{17} +1.00000 q^{19} -0.242641 q^{21} +3.00000 q^{23} -4.00000 q^{25} +2.41421 q^{27} +2.24264 q^{29} +3.58579 q^{31} -0.414214 q^{33} -0.585786 q^{35} -4.07107 q^{37} +2.58579 q^{39} +9.65685 q^{41} -11.6569 q^{43} +2.82843 q^{45} -3.17157 q^{47} -6.65685 q^{49} -0.242641 q^{51} +12.4853 q^{53} -1.00000 q^{55} -0.414214 q^{57} +4.41421 q^{59} +3.07107 q^{61} -1.65685 q^{63} +6.24264 q^{65} +7.58579 q^{67} -1.24264 q^{69} +9.58579 q^{71} +12.4853 q^{73} +1.65685 q^{75} +0.585786 q^{77} +17.4142 q^{79} +7.48528 q^{81} -0.585786 q^{83} -0.585786 q^{85} -0.928932 q^{87} -14.8995 q^{89} -3.65685 q^{91} -1.48528 q^{93} -1.00000 q^{95} -0.414214 q^{97} -2.82843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{5} + 4 q^{7} + 2 q^{11} - 4 q^{13} - 2 q^{15} + 4 q^{17} + 2 q^{19} + 8 q^{21} + 6 q^{23} - 8 q^{25} + 2 q^{27} - 4 q^{29} + 10 q^{31} + 2 q^{33} - 4 q^{35} + 6 q^{37} + 8 q^{39} + 8 q^{41}+ \cdots + 2 q^{97}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^5 + 4 * q^7 + 2 * q^11 - 4 * q^13 - 2 * q^15 + 4 * q^17 + 2 * q^19 + 8 * q^21 + 6 * q^23 - 8 * q^25 + 2 * q^27 - 4 * q^29 + 10 * q^31 + 2 * q^33 - 4 * q^35 + 6 * q^37 + 8 * q^39 + 8 * q^41 - 12 * q^43 - 12 * q^47 - 2 * q^49 + 8 * q^51 + 8 * q^53 - 2 * q^55 + 2 * q^57 + 6 * q^59 - 8 * q^61 + 8 * q^63 + 4 * q^65 + 18 * q^67 + 6 * q^69 + 22 * q^71 + 8 * q^73 - 8 * q^75 + 4 * q^77 + 32 * q^79 - 2 * q^81 - 4 * q^83 - 4 * q^85 - 16 * q^87 - 10 * q^89 + 4 * q^91 + 14 * q^93 - 2 * q^95 + 2 * q^97

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−0.414214	−0.239146	−0.119573	−	0.992825i	$-0.538153\pi$
$3$	−0.414214	−0.239146	−0.119573	+	0.992825i	$0.538153\pi$
$4$	0	0
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	0.585786	0.221406	0.110703	−	0.993854i	$-0.464690\pi$
$7$	0.585786	0.221406	0.110703	+	0.993854i	$0.464690\pi$
$8$	0	0
$9$	−2.82843	−0.942809
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−6.24264	−1.73140	−0.865699	−	0.500566i	$-0.833125\pi$
$13$	−6.24264	−1.73140	−0.865699	+	0.500566i	$0.833125\pi$
$14$	0	0
$15$	0.414214	0.106949
$16$	0	0
$17$	0.585786	0.142074	0.0710370	−	0.997474i	$-0.477369\pi$
$17$	0.585786	0.142074	0.0710370	+	0.997474i	$0.477369\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−0.242641	−0.0529485
$22$	0	0
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	2.41421	0.464616
$28$	0	0
$29$	2.24264	0.416448	0.208224	−	0.978081i	$-0.433232\pi$
$29$	2.24264	0.416448	0.208224	+	0.978081i	$0.433232\pi$
$30$	0	0
$31$	3.58579	0.644026	0.322013	−	0.946735i	$-0.395640\pi$
$31$	3.58579	0.644026	0.322013	+	0.946735i	$0.395640\pi$
$32$	0	0
$33$	−0.414214	−0.0721053
$34$	0	0
$35$	−0.585786	−0.0990160
$36$	0	0
$37$	−4.07107	−0.669279	−0.334640	−	0.942346i	$-0.608615\pi$
$37$	−4.07107	−0.669279	−0.334640	+	0.942346i	$0.608615\pi$
$38$	0	0
$39$	2.58579	0.414057
$40$	0	0
$41$	9.65685	1.50815	0.754074	−	0.656790i	$-0.228086\pi$
$41$	9.65685	1.50815	0.754074	+	0.656790i	$0.228086\pi$
$42$	0	0
$43$	−11.6569	−1.77765	−0.888827	−	0.458243i	$-0.848479\pi$
$43$	−11.6569	−1.77765	−0.888827	+	0.458243i	$0.848479\pi$
$44$	0	0
$45$	2.82843	0.421637
$46$	0	0
$47$	−3.17157	−0.462621	−0.231311	−	0.972880i	$-0.574301\pi$
$47$	−3.17157	−0.462621	−0.231311	+	0.972880i	$0.574301\pi$
$48$	0	0
$49$	−6.65685	−0.950979
$50$	0	0
$51$	−0.242641	−0.0339765
$52$	0	0
$53$	12.4853	1.71499	0.857493	−	0.514496i	$-0.172021\pi$
$53$	12.4853	1.71499	0.857493	+	0.514496i	$0.172021\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	−0.414214	−0.0548639
$58$	0	0
$59$	4.41421	0.574682	0.287341	−	0.957828i	$-0.407229\pi$
$59$	4.41421	0.574682	0.287341	+	0.957828i	$0.407229\pi$
$60$	0	0
$61$	3.07107	0.393210	0.196605	−	0.980483i	$-0.437008\pi$
$61$	3.07107	0.393210	0.196605	+	0.980483i	$0.437008\pi$
$62$	0	0
$63$	−1.65685	−0.208744
$64$	0	0
$65$	6.24264	0.774304
$66$	0	0
$67$	7.58579	0.926751	0.463376	−	0.886162i	$-0.346638\pi$
$67$	7.58579	0.926751	0.463376	+	0.886162i	$0.346638\pi$
$68$	0	0
$69$	−1.24264	−0.149596
$70$	0	0
$71$	9.58579	1.13762	0.568812	−	0.822468i	$-0.307403\pi$
$71$	9.58579	1.13762	0.568812	+	0.822468i	$0.307403\pi$
$72$	0	0
$73$	12.4853	1.46129	0.730646	−	0.682757i	$-0.239219\pi$
$73$	12.4853	1.46129	0.730646	+	0.682757i	$0.239219\pi$
$74$	0	0
$75$	1.65685	0.191317
$76$	0	0
$77$	0.585786	0.0667566
$78$	0	0
$79$	17.4142	1.95925	0.979626	−	0.200830i	$-0.0643640\pi$
$79$	17.4142	1.95925	0.979626	+	0.200830i	$0.0643640\pi$
$80$	0	0
$81$	7.48528	0.831698
$82$	0	0
$83$	−0.585786	−0.0642984	−0.0321492	−	0.999483i	$-0.510235\pi$
$83$	−0.585786	−0.0642984	−0.0321492	+	0.999483i	$0.510235\pi$
$84$	0	0
$85$	−0.585786	−0.0635375
$86$	0	0
$87$	−0.928932	−0.0995920
$88$	0	0
$89$	−14.8995	−1.57934	−0.789672	−	0.613530i	$-0.789749\pi$
$89$	−14.8995	−1.57934	−0.789672	+	0.613530i	$0.789749\pi$
$90$	0	0
$91$	−3.65685	−0.383342
$92$	0	0
$93$	−1.48528	−0.154017
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	−0.414214	−0.0420570	−0.0210285	−	0.999779i	$-0.506694\pi$
$97$	−0.414214	−0.0420570	−0.0210285	+	0.999779i	$0.506694\pi$
$98$	0	0
$99$	−2.82843	−0.284268
$100$	0	0
$101$	−2.24264	−0.223151	−0.111576	−	0.993756i	$-0.535590\pi$
$101$	−2.24264	−0.223151	−0.111576	+	0.993756i	$0.535590\pi$
$102$	0	0
$103$	−2.34315	−0.230877	−0.115439	−	0.993315i	$-0.536827\pi$
$103$	−2.34315	−0.230877	−0.115439	+	0.993315i	$0.536827\pi$
$104$	0	0
$105$	0.242641	0.0236793
$106$	0	0
$107$	−4.34315	−0.419868	−0.209934	−	0.977716i	$-0.567325\pi$
$107$	−4.34315	−0.419868	−0.209934	+	0.977716i	$0.567325\pi$
$108$	0	0
$109$	−11.6569	−1.11652	−0.558262	−	0.829665i	$-0.688532\pi$
$109$	−11.6569	−1.11652	−0.558262	+	0.829665i	$0.688532\pi$
$110$	0	0
$111$	1.68629	0.160056
$112$	0	0
$113$	−6.41421	−0.603398	−0.301699	−	0.953403i	$-0.597554\pi$
$113$	−6.41421	−0.603398	−0.301699	+	0.953403i	$0.597554\pi$
$114$	0	0
$115$	−3.00000	−0.279751
$116$	0	0
$117$	17.6569	1.63238
$118$	0	0
$119$	0.343146	0.0314561
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−4.00000	−0.360668
$124$	0	0
$125$	9.00000	0.804984
$126$	0	0
$127$	16.5858	1.47175	0.735875	−	0.677117i	$-0.236771\pi$
$127$	16.5858	1.47175	0.735875	+	0.677117i	$0.236771\pi$
$128$	0	0
$129$	4.82843	0.425119
$130$	0	0
$131$	2.48528	0.217140	0.108570	−	0.994089i	$-0.465373\pi$
$131$	2.48528	0.217140	0.108570	+	0.994089i	$0.465373\pi$
$132$	0	0
$133$	0.585786	0.0507941
$134$	0	0
$135$	−2.41421	−0.207782
$136$	0	0
$137$	8.65685	0.739605	0.369802	−	0.929110i	$-0.379425\pi$
$137$	8.65685	0.739605	0.369802	+	0.929110i	$0.379425\pi$
$138$	0	0
$139$	16.3848	1.38974	0.694869	−	0.719136i	$-0.255462\pi$
$139$	16.3848	1.38974	0.694869	+	0.719136i	$0.255462\pi$
$140$	0	0
$141$	1.31371	0.110634
$142$	0	0
$143$	−6.24264	−0.522036
$144$	0	0
$145$	−2.24264	−0.186241
$146$	0	0
$147$	2.75736	0.227423
$148$	0	0
$149$	−1.31371	−0.107623	−0.0538116	−	0.998551i	$-0.517137\pi$
$149$	−1.31371	−0.107623	−0.0538116	+	0.998551i	$0.517137\pi$
$150$	0	0
$151$	−6.48528	−0.527765	−0.263882	−	0.964555i	$-0.585003\pi$
$151$	−6.48528	−0.527765	−0.263882	+	0.964555i	$0.585003\pi$
$152$	0	0
$153$	−1.65685	−0.133949
$154$	0	0
$155$	−3.58579	−0.288017
$156$	0	0
$157$	5.00000	0.399043	0.199522	−	0.979893i	$-0.436061\pi$
$157$	5.00000	0.399043	0.199522	+	0.979893i	$0.436061\pi$
$158$	0	0
$159$	−5.17157	−0.410132
$160$	0	0
$161$	1.75736	0.138499
$162$	0	0
$163$	−8.14214	−0.637741	−0.318871	−	0.947798i	$-0.603304\pi$
$163$	−8.14214	−0.637741	−0.318871	+	0.947798i	$0.603304\pi$
$164$	0	0
$165$	0.414214	0.0322465
$166$	0	0
$167$	−6.72792	−0.520622	−0.260311	−	0.965525i	$-0.583825\pi$
$167$	−6.72792	−0.520622	−0.260311	+	0.965525i	$0.583825\pi$
$168$	0	0
$169$	25.9706	1.99774
$170$	0	0
$171$	−2.82843	−0.216295
$172$	0	0
$173$	−17.8995	−1.36087	−0.680437	−	0.732807i	$-0.738210\pi$
$173$	−17.8995	−1.36087	−0.680437	+	0.732807i	$0.738210\pi$
$174$	0	0
$175$	−2.34315	−0.177125
$176$	0	0
$177$	−1.82843	−0.137433
$178$	0	0
$179$	−1.92893	−0.144175	−0.0720876	−	0.997398i	$-0.522966\pi$
$179$	−1.92893	−0.144175	−0.0720876	+	0.997398i	$0.522966\pi$
$180$	0	0
$181$	17.3848	1.29220	0.646100	−	0.763253i	$-0.276399\pi$
$181$	17.3848	1.29220	0.646100	+	0.763253i	$0.276399\pi$
$182$	0	0
$183$	−1.27208	−0.0940347
$184$	0	0
$185$	4.07107	0.299311
$186$	0	0
$187$	0.585786	0.0428369
$188$	0	0
$189$	1.41421	0.102869
$190$	0	0
$191$	14.3137	1.03570	0.517852	−	0.855470i	$-0.326732\pi$
$191$	14.3137	1.03570	0.517852	+	0.855470i	$0.326732\pi$
$192$	0	0
$193$	6.82843	0.491521	0.245760	−	0.969331i	$-0.420962\pi$
$193$	6.82843	0.491521	0.245760	+	0.969331i	$0.420962\pi$
$194$	0	0
$195$	−2.58579	−0.185172
$196$	0	0
$197$	−3.89949	−0.277828	−0.138914	−	0.990304i	$-0.544361\pi$
$197$	−3.89949	−0.277828	−0.138914	+	0.990304i	$0.544361\pi$
$198$	0	0
$199$	16.1421	1.14429	0.572143	−	0.820154i	$-0.306112\pi$
$199$	16.1421	1.14429	0.572143	+	0.820154i	$0.306112\pi$
$200$	0	0
$201$	−3.14214	−0.221629
$202$	0	0
$203$	1.31371	0.0922043
$204$	0	0
$205$	−9.65685	−0.674464
$206$	0	0
$207$	−8.48528	−0.589768
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	15.4142	1.06116	0.530579	−	0.847635i	$-0.321974\pi$
$211$	15.4142	1.06116	0.530579	+	0.847635i	$0.321974\pi$
$212$	0	0
$213$	−3.97056	−0.272058
$214$	0	0
$215$	11.6569	0.794991
$216$	0	0
$217$	2.10051	0.142592
$218$	0	0
$219$	−5.17157	−0.349463
$220$	0	0
$221$	−3.65685	−0.245987
$222$	0	0
$223$	13.5858	0.909772	0.454886	−	0.890550i	$-0.349680\pi$
$223$	13.5858	0.909772	0.454886	+	0.890550i	$0.349680\pi$
$224$	0	0
$225$	11.3137	0.754247
$226$	0	0
$227$	−19.0711	−1.26579	−0.632896	−	0.774237i	$-0.718134\pi$
$227$	−19.0711	−1.26579	−0.632896	+	0.774237i	$0.718134\pi$
$228$	0	0
$229$	8.31371	0.549385	0.274693	−	0.961532i	$-0.411424\pi$
$229$	8.31371	0.549385	0.274693	+	0.961532i	$0.411424\pi$
$230$	0	0
$231$	−0.242641	−0.0159646
$232$	0	0
$233$	8.24264	0.539993	0.269997	−	0.962861i	$-0.412977\pi$
$233$	8.24264	0.539993	0.269997	+	0.962861i	$0.412977\pi$
$234$	0	0
$235$	3.17157	0.206891
$236$	0	0
$237$	−7.21320	−0.468548
$238$	0	0
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	16.9706	1.09317	0.546585	−	0.837404i	$-0.315928\pi$
$241$	16.9706	1.09317	0.546585	+	0.837404i	$0.315928\pi$
$242$	0	0
$243$	−10.3431	−0.663513
$244$	0	0
$245$	6.65685	0.425291
$246$	0	0
$247$	−6.24264	−0.397210
$248$	0	0
$249$	0.242641	0.0153767
$250$	0	0
$251$	22.6569	1.43009	0.715044	−	0.699079i	$-0.246407\pi$
$251$	22.6569	1.43009	0.715044	+	0.699079i	$0.246407\pi$
$252$	0	0
$253$	3.00000	0.188608
$254$	0	0
$255$	0.242641	0.0151947
$256$	0	0
$257$	−24.8284	−1.54875	−0.774377	−	0.632724i	$-0.781937\pi$
$257$	−24.8284	−1.54875	−0.774377	+	0.632724i	$0.781937\pi$
$258$	0	0
$259$	−2.38478	−0.148183
$260$	0	0
$261$	−6.34315	−0.392631
$262$	0	0
$263$	26.4853	1.63315	0.816576	−	0.577238i	$-0.195869\pi$
$263$	26.4853	1.63315	0.816576	+	0.577238i	$0.195869\pi$
$264$	0	0
$265$	−12.4853	−0.766965
$266$	0	0
$267$	6.17157	0.377694
$268$	0	0
$269$	−2.00000	−0.121942	−0.0609711	−	0.998140i	$-0.519420\pi$
$269$	−2.00000	−0.121942	−0.0609711	+	0.998140i	$0.519420\pi$
$270$	0	0
$271$	22.1421	1.34504	0.672519	−	0.740079i	$-0.265212\pi$
$271$	22.1421	1.34504	0.672519	+	0.740079i	$0.265212\pi$
$272$	0	0
$273$	1.51472	0.0916749
$274$	0	0
$275$	−4.00000	−0.241209
$276$	0	0
$277$	14.4853	0.870336	0.435168	−	0.900349i	$-0.356689\pi$
$277$	14.4853	0.870336	0.435168	+	0.900349i	$0.356689\pi$
$278$	0	0
$279$	−10.1421	−0.607194
$280$	0	0
$281$	12.3431	0.736330	0.368165	−	0.929760i	$-0.379986\pi$
$281$	12.3431	0.736330	0.368165	+	0.929760i	$0.379986\pi$
$282$	0	0
$283$	−8.72792	−0.518821	−0.259411	−	0.965767i	$-0.583528\pi$
$283$	−8.72792	−0.518821	−0.259411	+	0.965767i	$0.583528\pi$
$284$	0	0
$285$	0.414214	0.0245359
$286$	0	0
$287$	5.65685	0.333914
$288$	0	0
$289$	−16.6569	−0.979815
$290$	0	0
$291$	0.171573	0.0100578
$292$	0	0
$293$	27.6985	1.61816	0.809081	−	0.587697i	$-0.199965\pi$
$293$	27.6985	1.61816	0.809081	+	0.587697i	$0.199965\pi$
$294$	0	0
$295$	−4.41421	−0.257005
$296$	0	0
$297$	2.41421	0.140087
$298$	0	0
$299$	−18.7279	−1.08306
$300$	0	0
$301$	−6.82843	−0.393584
$302$	0	0
$303$	0.928932	0.0533658
$304$	0	0
$305$	−3.07107	−0.175849
$306$	0	0
$307$	−4.58579	−0.261725	−0.130862	−	0.991401i	$-0.541775\pi$
$307$	−4.58579	−0.261725	−0.130862	+	0.991401i	$0.541775\pi$
$308$	0	0
$309$	0.970563	0.0552134
$310$	0	0
$311$	0.343146	0.0194580	0.00972901	−	0.999953i	$-0.496903\pi$
$311$	0.343146	0.0194580	0.00972901	+	0.999953i	$0.496903\pi$
$312$	0	0
$313$	19.9706	1.12880	0.564401	−	0.825500i	$-0.309107\pi$
$313$	19.9706	1.12880	0.564401	+	0.825500i	$0.309107\pi$
$314$	0	0
$315$	1.65685	0.0933532
$316$	0	0
$317$	−25.5858	−1.43704	−0.718520	−	0.695506i	$-0.755180\pi$
$317$	−25.5858	−1.43704	−0.718520	+	0.695506i	$0.755180\pi$
$318$	0	0
$319$	2.24264	0.125564
$320$	0	0
$321$	1.79899	0.100410
$322$	0	0
$323$	0.585786	0.0325940
$324$	0	0
$325$	24.9706	1.38512
$326$	0	0
$327$	4.82843	0.267013
$328$	0	0
$329$	−1.85786	−0.102427
$330$	0	0
$331$	8.21320	0.451438	0.225719	−	0.974192i	$-0.427527\pi$
$331$	8.21320	0.451438	0.225719	+	0.974192i	$0.427527\pi$
$332$	0	0
$333$	11.5147	0.631003
$334$	0	0
$335$	−7.58579	−0.414456
$336$	0	0
$337$	−8.72792	−0.475440	−0.237720	−	0.971334i	$-0.576400\pi$
$337$	−8.72792	−0.475440	−0.237720	+	0.971334i	$0.576400\pi$
$338$	0	0
$339$	2.65685	0.144301
$340$	0	0
$341$	3.58579	0.194181
$342$	0	0
$343$	−8.00000	−0.431959
$344$	0	0
$345$	1.24264	0.0669015
$346$	0	0
$347$	25.6985	1.37957	0.689783	−	0.724016i	$-0.257706\pi$
$347$	25.6985	1.37957	0.689783	+	0.724016i	$0.257706\pi$
$348$	0	0
$349$	−19.2132	−1.02846	−0.514230	−	0.857653i	$-0.671922\pi$
$349$	−19.2132	−1.02846	−0.514230	+	0.857653i	$0.671922\pi$
$350$	0	0
$351$	−15.0711	−0.804434
$352$	0	0
$353$	−1.68629	−0.0897522	−0.0448761	−	0.998993i	$-0.514289\pi$
$353$	−1.68629	−0.0897522	−0.0448761	+	0.998993i	$0.514289\pi$
$354$	0	0
$355$	−9.58579	−0.508761
$356$	0	0
$357$	−0.142136	−0.00752261
$358$	0	0
$359$	0.485281	0.0256122	0.0128061	−	0.999918i	$-0.495924\pi$
$359$	0.485281	0.0256122	0.0128061	+	0.999918i	$0.495924\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−0.414214	−0.0217406
$364$	0	0
$365$	−12.4853	−0.653509
$366$	0	0
$367$	4.17157	0.217754	0.108877	−	0.994055i	$-0.465274\pi$
$367$	4.17157	0.217754	0.108877	+	0.994055i	$0.465274\pi$
$368$	0	0
$369$	−27.3137	−1.42189
$370$	0	0
$371$	7.31371	0.379709
$372$	0	0
$373$	−25.3137	−1.31069	−0.655347	−	0.755328i	$-0.727478\pi$
$373$	−25.3137	−1.31069	−0.655347	+	0.755328i	$0.727478\pi$
$374$	0	0
$375$	−3.72792	−0.192509
$376$	0	0
$377$	−14.0000	−0.721037
$378$	0	0
$379$	−28.6985	−1.47414	−0.737071	−	0.675815i	$-0.763792\pi$
$379$	−28.6985	−1.47414	−0.737071	+	0.675815i	$0.763792\pi$
$380$	0	0
$381$	−6.87006	−0.351964
$382$	0	0
$383$	31.5269	1.61095	0.805475	−	0.592630i	$-0.201910\pi$
$383$	31.5269	1.61095	0.805475	+	0.592630i	$0.201910\pi$
$384$	0	0
$385$	−0.585786	−0.0298544
$386$	0	0
$387$	32.9706	1.67599
$388$	0	0
$389$	−5.68629	−0.288306	−0.144153	−	0.989555i	$-0.546046\pi$
$389$	−5.68629	−0.288306	−0.144153	+	0.989555i	$0.546046\pi$
$390$	0	0
$391$	1.75736	0.0888735
$392$	0	0
$393$	−1.02944	−0.0519282
$394$	0	0
$395$	−17.4142	−0.876204
$396$	0	0
$397$	26.0000	1.30490	0.652451	−	0.757831i	$-0.273741\pi$
$397$	26.0000	1.30490	0.652451	+	0.757831i	$0.273741\pi$
$398$	0	0
$399$	−0.242641	−0.0121472
$400$	0	0
$401$	−1.02944	−0.0514076	−0.0257038	−	0.999670i	$-0.508183\pi$
$401$	−1.02944	−0.0514076	−0.0257038	+	0.999670i	$0.508183\pi$
$402$	0	0
$403$	−22.3848	−1.11507
$404$	0	0
$405$	−7.48528	−0.371947
$406$	0	0
$407$	−4.07107	−0.201795
$408$	0	0
$409$	−24.7279	−1.22272	−0.611359	−	0.791354i	$-0.709377\pi$
$409$	−24.7279	−1.22272	−0.611359	+	0.791354i	$0.709377\pi$
$410$	0	0
$411$	−3.58579	−0.176874
$412$	0	0
$413$	2.58579	0.127238
$414$	0	0
$415$	0.585786	0.0287551
$416$	0	0
$417$	−6.78680	−0.332351
$418$	0	0
$419$	−23.4558	−1.14589	−0.572946	−	0.819593i	$-0.694200\pi$
$419$	−23.4558	−1.14589	−0.572946	+	0.819593i	$0.694200\pi$
$420$	0	0
$421$	0.142136	0.00692727	0.00346363	−	0.999994i	$-0.498897\pi$
$421$	0.142136	0.00692727	0.00346363	+	0.999994i	$0.498897\pi$
$422$	0	0
$423$	8.97056	0.436164
$424$	0	0
$425$	−2.34315	−0.113659
$426$	0	0
$427$	1.79899	0.0870592
$428$	0	0
$429$	2.58579	0.124843
$430$	0	0
$431$	−13.5147	−0.650981	−0.325491	−	0.945545i	$-0.605529\pi$
$431$	−13.5147	−0.650981	−0.325491	+	0.945545i	$0.605529\pi$
$432$	0	0
$433$	−25.3848	−1.21991	−0.609957	−	0.792434i	$-0.708813\pi$
$433$	−25.3848	−1.21991	−0.609957	+	0.792434i	$0.708813\pi$
$434$	0	0
$435$	0.928932	0.0445389
$436$	0	0
$437$	3.00000	0.143509
$438$	0	0
$439$	5.75736	0.274784	0.137392	−	0.990517i	$-0.456128\pi$
$439$	5.75736	0.274784	0.137392	+	0.990517i	$0.456128\pi$
$440$	0	0
$441$	18.8284	0.896592
$442$	0	0
$443$	−21.9706	−1.04385	−0.521926	−	0.852990i	$-0.674786\pi$
$443$	−21.9706	−1.04385	−0.521926	+	0.852990i	$0.674786\pi$
$444$	0	0
$445$	14.8995	0.706304
$446$	0	0
$447$	0.544156	0.0257377
$448$	0	0
$449$	23.3848	1.10360	0.551798	−	0.833978i	$-0.313942\pi$
$449$	23.3848	1.10360	0.551798	+	0.833978i	$0.313942\pi$
$450$	0	0
$451$	9.65685	0.454724
$452$	0	0
$453$	2.68629	0.126213
$454$	0	0
$455$	3.65685	0.171436
$456$	0	0
$457$	31.0711	1.45344	0.726722	−	0.686932i	$-0.241043\pi$
$457$	31.0711	1.45344	0.726722	+	0.686932i	$0.241043\pi$
$458$	0	0
$459$	1.41421	0.0660098
$460$	0	0
$461$	−20.9706	−0.976696	−0.488348	−	0.872649i	$-0.662400\pi$
$461$	−20.9706	−0.976696	−0.488348	+	0.872649i	$0.662400\pi$
$462$	0	0
$463$	38.4558	1.78719	0.893597	−	0.448870i	$-0.148173\pi$
$463$	38.4558	1.78719	0.893597	+	0.448870i	$0.148173\pi$
$464$	0	0
$465$	1.48528	0.0688783
$466$	0	0
$467$	−32.3137	−1.49530	−0.747650	−	0.664093i	$-0.768818\pi$
$467$	−32.3137	−1.49530	−0.747650	+	0.664093i	$0.768818\pi$
$468$	0	0
$469$	4.44365	0.205189
$470$	0	0
$471$	−2.07107	−0.0954298
$472$	0	0
$473$	−11.6569	−0.535983
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	−35.3137	−1.61690
$478$	0	0
$479$	−24.5269	−1.12066	−0.560332	−	0.828268i	$-0.689326\pi$
$479$	−24.5269	−1.12066	−0.560332	+	0.828268i	$0.689326\pi$
$480$	0	0
$481$	25.4142	1.15879
$482$	0	0
$483$	−0.727922	−0.0331216
$484$	0	0
$485$	0.414214	0.0188085
$486$	0	0
$487$	18.5563	0.840868	0.420434	−	0.907323i	$-0.361878\pi$
$487$	18.5563	0.840868	0.420434	+	0.907323i	$0.361878\pi$
$488$	0	0
$489$	3.37258	0.152513
$490$	0	0
$491$	−36.2426	−1.63561	−0.817804	−	0.575497i	$-0.804809\pi$
$491$	−36.2426	−1.63561	−0.817804	+	0.575497i	$0.804809\pi$
$492$	0	0
$493$	1.31371	0.0591665
$494$	0	0
$495$	2.82843	0.127128
$496$	0	0
$497$	5.61522	0.251877
$498$	0	0
$499$	−12.6274	−0.565281	−0.282640	−	0.959226i	$-0.591210\pi$
$499$	−12.6274	−0.565281	−0.282640	+	0.959226i	$0.591210\pi$
$500$	0	0
$501$	2.78680	0.124505
$502$	0	0
$503$	−0.142136	−0.00633751	−0.00316876	−	0.999995i	$-0.501009\pi$
$503$	−0.142136	−0.00633751	−0.00316876	+	0.999995i	$0.501009\pi$
$504$	0	0
$505$	2.24264	0.0997962
$506$	0	0
$507$	−10.7574	−0.477751
$508$	0	0
$509$	34.2132	1.51647	0.758237	−	0.651979i	$-0.226061\pi$
$509$	34.2132	1.51647	0.758237	+	0.651979i	$0.226061\pi$
$510$	0	0
$511$	7.31371	0.323539
$512$	0	0
$513$	2.41421	0.106590
$514$	0	0
$515$	2.34315	0.103251
$516$	0	0
$517$	−3.17157	−0.139486
$518$	0	0
$519$	7.41421	0.325448
$520$	0	0
$521$	−30.5563	−1.33870	−0.669349	−	0.742948i	$-0.733427\pi$
$521$	−30.5563	−1.33870	−0.669349	+	0.742948i	$0.733427\pi$
$522$	0	0
$523$	−13.6569	−0.597173	−0.298586	−	0.954383i	$-0.596515\pi$
$523$	−13.6569	−0.597173	−0.298586	+	0.954383i	$0.596515\pi$
$524$	0	0
$525$	0.970563	0.0423588
$526$	0	0
$527$	2.10051	0.0914994
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	−12.4853	−0.541815
$532$	0	0
$533$	−60.2843	−2.61120
$534$	0	0
$535$	4.34315	0.187771
$536$	0	0
$537$	0.798990	0.0344790
$538$	0	0
$539$	−6.65685	−0.286731
$540$	0	0
$541$	−33.2132	−1.42795	−0.713974	−	0.700173i	$-0.753106\pi$
$541$	−33.2132	−1.42795	−0.713974	+	0.700173i	$0.753106\pi$
$542$	0	0
$543$	−7.20101	−0.309025
$544$	0	0
$545$	11.6569	0.499325
$546$	0	0
$547$	−10.7279	−0.458693	−0.229346	−	0.973345i	$-0.573659\pi$
$547$	−10.7279	−0.458693	−0.229346	+	0.973345i	$0.573659\pi$
$548$	0	0
$549$	−8.68629	−0.370722
$550$	0	0
$551$	2.24264	0.0955397
$552$	0	0
$553$	10.2010	0.433791
$554$	0	0
$555$	−1.68629	−0.0715791
$556$	0	0
$557$	39.9411	1.69236	0.846180	−	0.532897i	$-0.178897\pi$
$557$	39.9411	1.69236	0.846180	+	0.532897i	$0.178897\pi$
$558$	0	0
$559$	72.7696	3.07782
$560$	0	0
$561$	−0.242641	−0.0102443
$562$	0	0
$563$	−19.2132	−0.809740	−0.404870	−	0.914374i	$-0.632683\pi$
$563$	−19.2132	−0.809740	−0.404870	+	0.914374i	$0.632683\pi$
$564$	0	0
$565$	6.41421	0.269848
$566$	0	0
$567$	4.38478	0.184143
$568$	0	0
$569$	−17.7574	−0.744427	−0.372214	−	0.928147i	$-0.621401\pi$
$569$	−17.7574	−0.744427	−0.372214	+	0.928147i	$0.621401\pi$
$570$	0	0
$571$	25.6985	1.07545	0.537724	−	0.843121i	$-0.319284\pi$
$571$	25.6985	1.07545	0.537724	+	0.843121i	$0.319284\pi$
$572$	0	0
$573$	−5.92893	−0.247685
$574$	0	0
$575$	−12.0000	−0.500435
$576$	0	0
$577$	−3.97056	−0.165297	−0.0826483	−	0.996579i	$-0.526338\pi$
$577$	−3.97056	−0.165297	−0.0826483	+	0.996579i	$0.526338\pi$
$578$	0	0
$579$	−2.82843	−0.117545
$580$	0	0
$581$	−0.343146	−0.0142361
$582$	0	0
$583$	12.4853	0.517088
$584$	0	0
$585$	−17.6569	−0.730021
$586$	0	0
$587$	3.65685	0.150935	0.0754673	−	0.997148i	$-0.475955\pi$
$587$	3.65685	0.150935	0.0754673	+	0.997148i	$0.475955\pi$
$588$	0	0
$589$	3.58579	0.147750
$590$	0	0
$591$	1.61522	0.0664414
$592$	0	0
$593$	3.02944	0.124404	0.0622020	−	0.998064i	$-0.480188\pi$
$593$	3.02944	0.124404	0.0622020	+	0.998064i	$0.480188\pi$
$594$	0	0
$595$	−0.343146	−0.0140676
$596$	0	0
$597$	−6.68629	−0.273652
$598$	0	0
$599$	9.31371	0.380548	0.190274	−	0.981731i	$-0.439062\pi$
$599$	9.31371	0.380548	0.190274	+	0.981731i	$0.439062\pi$
$600$	0	0
$601$	−19.5563	−0.797720	−0.398860	−	0.917012i	$-0.630594\pi$
$601$	−19.5563	−0.797720	−0.398860	+	0.917012i	$0.630594\pi$
$602$	0	0
$603$	−21.4558	−0.873750
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	21.8995	0.888873	0.444437	−	0.895810i	$-0.353404\pi$
$607$	21.8995	0.888873	0.444437	+	0.895810i	$0.353404\pi$
$608$	0	0
$609$	−0.544156	−0.0220503
$610$	0	0
$611$	19.7990	0.800981
$612$	0	0
$613$	−10.5858	−0.427556	−0.213778	−	0.976882i	$-0.568577\pi$
$613$	−10.5858	−0.427556	−0.213778	+	0.976882i	$0.568577\pi$
$614$	0	0
$615$	4.00000	0.161296
$616$	0	0
$617$	17.1716	0.691301	0.345651	−	0.938363i	$-0.387658\pi$
$617$	17.1716	0.691301	0.345651	+	0.938363i	$0.387658\pi$
$618$	0	0
$619$	−6.51472	−0.261849	−0.130924	−	0.991392i	$-0.541794\pi$
$619$	−6.51472	−0.261849	−0.130924	+	0.991392i	$0.541794\pi$
$620$	0	0
$621$	7.24264	0.290637
$622$	0	0
$623$	−8.72792	−0.349677
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	−0.414214	−0.0165421
$628$	0	0
$629$	−2.38478	−0.0950873
$630$	0	0
$631$	41.9706	1.67082	0.835411	−	0.549626i	$-0.185230\pi$
$631$	41.9706	1.67082	0.835411	+	0.549626i	$0.185230\pi$
$632$	0	0
$633$	−6.38478	−0.253772
$634$	0	0
$635$	−16.5858	−0.658187
$636$	0	0
$637$	41.5563	1.64652
$638$	0	0
$639$	−27.1127	−1.07256
$640$	0	0
$641$	−18.8995	−0.746485	−0.373243	−	0.927734i	$-0.621754\pi$
$641$	−18.8995	−0.746485	−0.373243	+	0.927734i	$0.621754\pi$
$642$	0	0
$643$	−13.3431	−0.526202	−0.263101	−	0.964768i	$-0.584745\pi$
$643$	−13.3431	−0.526202	−0.263101	+	0.964768i	$0.584745\pi$
$644$	0	0
$645$	−4.82843	−0.190119
$646$	0	0
$647$	−36.9411	−1.45231	−0.726153	−	0.687533i	$-0.758693\pi$
$647$	−36.9411	−1.45231	−0.726153	+	0.687533i	$0.758693\pi$
$648$	0	0
$649$	4.41421	0.173273
$650$	0	0
$651$	−0.870058	−0.0341002
$652$	0	0
$653$	−25.4853	−0.997316	−0.498658	−	0.866799i	$-0.666174\pi$
$653$	−25.4853	−0.997316	−0.498658	+	0.866799i	$0.666174\pi$
$654$	0	0
$655$	−2.48528	−0.0971080
$656$	0	0
$657$	−35.3137	−1.37772
$658$	0	0
$659$	17.7990	0.693350	0.346675	−	0.937985i	$-0.387311\pi$
$659$	17.7990	0.693350	0.346675	+	0.937985i	$0.387311\pi$
$660$	0	0
$661$	3.87006	0.150528	0.0752639	−	0.997164i	$-0.476020\pi$
$661$	3.87006	0.150528	0.0752639	+	0.997164i	$0.476020\pi$
$662$	0	0
$663$	1.51472	0.0588268
$664$	0	0
$665$	−0.585786	−0.0227158
$666$	0	0
$667$	6.72792	0.260506
$668$	0	0
$669$	−5.62742	−0.217569
$670$	0	0
$671$	3.07107	0.118557
$672$	0	0
$673$	−32.1421	−1.23899	−0.619494	−	0.785001i	$-0.712662\pi$
$673$	−32.1421	−1.23899	−0.619494	+	0.785001i	$0.712662\pi$
$674$	0	0
$675$	−9.65685	−0.371692
$676$	0	0
$677$	−1.31371	−0.0504899	−0.0252450	−	0.999681i	$-0.508037\pi$
$677$	−1.31371	−0.0504899	−0.0252450	+	0.999681i	$0.508037\pi$
$678$	0	0
$679$	−0.242641	−0.00931169
$680$	0	0
$681$	7.89949	0.302709
$682$	0	0
$683$	39.1127	1.49661	0.748303	−	0.663357i	$-0.230869\pi$
$683$	39.1127	1.49661	0.748303	+	0.663357i	$0.230869\pi$
$684$	0	0
$685$	−8.65685	−0.330761
$686$	0	0
$687$	−3.44365	−0.131383
$688$	0	0
$689$	−77.9411	−2.96932
$690$	0	0
$691$	7.97056	0.303214	0.151607	−	0.988441i	$-0.451555\pi$
$691$	7.97056	0.303214	0.151607	+	0.988441i	$0.451555\pi$
$692$	0	0
$693$	−1.65685	−0.0629387
$694$	0	0
$695$	−16.3848	−0.621510
$696$	0	0
$697$	5.65685	0.214269
$698$	0	0
$699$	−3.41421	−0.129137
$700$	0	0
$701$	21.3137	0.805008	0.402504	−	0.915418i	$-0.368140\pi$
$701$	21.3137	0.805008	0.402504	+	0.915418i	$0.368140\pi$
$702$	0	0
$703$	−4.07107	−0.153543
$704$	0	0
$705$	−1.31371	−0.0494771
$706$	0	0
$707$	−1.31371	−0.0494071
$708$	0	0
$709$	31.2843	1.17491	0.587453	−	0.809258i	$-0.300131\pi$
$709$	31.2843	1.17491	0.587453	+	0.809258i	$0.300131\pi$
$710$	0	0
$711$	−49.2548	−1.84720
$712$	0	0
$713$	10.7574	0.402866
$714$	0	0
$715$	6.24264	0.233462
$716$	0	0
$717$	2.48528	0.0928145
$718$	0	0
$719$	−43.9706	−1.63983	−0.819913	−	0.572489i	$-0.805978\pi$
$719$	−43.9706	−1.63983	−0.819913	+	0.572489i	$0.805978\pi$
$720$	0	0
$721$	−1.37258	−0.0511177
$722$	0	0
$723$	−7.02944	−0.261428
$724$	0	0
$725$	−8.97056	−0.333158
$726$	0	0
$727$	−44.1127	−1.63605	−0.818025	−	0.575183i	$-0.804931\pi$
$727$	−44.1127	−1.63605	−0.818025	+	0.575183i	$0.804931\pi$
$728$	0	0
$729$	−18.1716	−0.673021
$730$	0	0
$731$	−6.82843	−0.252559
$732$	0	0
$733$	10.5858	0.390995	0.195497	−	0.980704i	$-0.437368\pi$
$733$	10.5858	0.390995	0.195497	+	0.980704i	$0.437368\pi$
$734$	0	0
$735$	−2.75736	−0.101707
$736$	0	0
$737$	7.58579	0.279426
$738$	0	0
$739$	24.5858	0.904403	0.452201	−	0.891916i	$-0.350639\pi$
$739$	24.5858	0.904403	0.452201	+	0.891916i	$0.350639\pi$
$740$	0	0
$741$	2.58579	0.0949912
$742$	0	0
$743$	21.0711	0.773023	0.386511	−	0.922285i	$-0.373680\pi$
$743$	21.0711	0.773023	0.386511	+	0.922285i	$0.373680\pi$
$744$	0	0
$745$	1.31371	0.0481306
$746$	0	0
$747$	1.65685	0.0606211
$748$	0	0
$749$	−2.54416	−0.0929614
$750$	0	0
$751$	−34.5563	−1.26098	−0.630490	−	0.776198i	$-0.717146\pi$
$751$	−34.5563	−1.26098	−0.630490	+	0.776198i	$0.717146\pi$
$752$	0	0
$753$	−9.38478	−0.342000
$754$	0	0
$755$	6.48528	0.236024
$756$	0	0
$757$	−25.9411	−0.942846	−0.471423	−	0.881907i	$-0.656260\pi$
$757$	−25.9411	−0.942846	−0.471423	+	0.881907i	$0.656260\pi$
$758$	0	0
$759$	−1.24264	−0.0451050
$760$	0	0
$761$	34.1421	1.23765	0.618826	−	0.785528i	$-0.287609\pi$
$761$	34.1421	1.23765	0.618826	+	0.785528i	$0.287609\pi$
$762$	0	0
$763$	−6.82843	−0.247206
$764$	0	0
$765$	1.65685	0.0599037
$766$	0	0
$767$	−27.5563	−0.995002
$768$	0	0
$769$	30.1421	1.08695	0.543477	−	0.839424i	$-0.317108\pi$
$769$	30.1421	1.08695	0.543477	+	0.839424i	$0.317108\pi$
$770$	0	0
$771$	10.2843	0.370379
$772$	0	0
$773$	8.34315	0.300082	0.150041	−	0.988680i	$-0.452059\pi$
$773$	8.34315	0.300082	0.150041	+	0.988680i	$0.452059\pi$
$774$	0	0
$775$	−14.3431	−0.515221
$776$	0	0
$777$	0.987807	0.0354374
$778$	0	0
$779$	9.65685	0.345993
$780$	0	0
$781$	9.58579	0.343006
$782$	0	0
$783$	5.41421	0.193488
$784$	0	0
$785$	−5.00000	−0.178458
$786$	0	0
$787$	−36.0000	−1.28326	−0.641631	−	0.767014i	$-0.721742\pi$
$787$	−36.0000	−1.28326	−0.641631	+	0.767014i	$0.721742\pi$
$788$	0	0
$789$	−10.9706	−0.390562
$790$	0	0
$791$	−3.75736	−0.133596
$792$	0	0
$793$	−19.1716	−0.680803
$794$	0	0
$795$	5.17157	0.183417
$796$	0	0
$797$	−4.75736	−0.168514	−0.0842572	−	0.996444i	$-0.526852\pi$
$797$	−4.75736	−0.168514	−0.0842572	+	0.996444i	$0.526852\pi$
$798$	0	0
$799$	−1.85786	−0.0657265
$800$	0	0
$801$	42.1421	1.48902
$802$	0	0
$803$	12.4853	0.440596
$804$	0	0
$805$	−1.75736	−0.0619388
$806$	0	0
$807$	0.828427	0.0291620
$808$	0	0
$809$	52.4853	1.84528	0.922642	−	0.385657i	$-0.126025\pi$
$809$	52.4853	1.84528	0.922642	+	0.385657i	$0.126025\pi$
$810$	0	0
$811$	−52.2426	−1.83449	−0.917244	−	0.398327i	$-0.869591\pi$
$811$	−52.2426	−1.83449	−0.917244	+	0.398327i	$0.869591\pi$
$812$	0	0
$813$	−9.17157	−0.321661
$814$	0	0
$815$	8.14214	0.285207
$816$	0	0
$817$	−11.6569	−0.407822
$818$	0	0
$819$	10.3431	0.361419
$820$	0	0
$821$	26.5269	0.925796	0.462898	−	0.886412i	$-0.346810\pi$
$821$	26.5269	0.925796	0.462898	+	0.886412i	$0.346810\pi$
$822$	0	0
$823$	23.0000	0.801730	0.400865	−	0.916137i	$-0.368710\pi$
$823$	23.0000	0.801730	0.400865	+	0.916137i	$0.368710\pi$
$824$	0	0
$825$	1.65685	0.0576843
$826$	0	0
$827$	−4.10051	−0.142589	−0.0712943	−	0.997455i	$-0.522713\pi$
$827$	−4.10051	−0.142589	−0.0712943	+	0.997455i	$0.522713\pi$
$828$	0	0
$829$	11.0416	0.383492	0.191746	−	0.981445i	$-0.438585\pi$
$829$	11.0416	0.383492	0.191746	+	0.981445i	$0.438585\pi$
$830$	0	0
$831$	−6.00000	−0.208138
$832$	0	0
$833$	−3.89949	−0.135109
$834$	0	0
$835$	6.72792	0.232829
$836$	0	0
$837$	8.65685	0.299225
$838$	0	0
$839$	−40.5563	−1.40016	−0.700080	−	0.714064i	$-0.746852\pi$
$839$	−40.5563	−1.40016	−0.700080	+	0.714064i	$0.746852\pi$
$840$	0	0
$841$	−23.9706	−0.826571
$842$	0	0
$843$	−5.11270	−0.176091
$844$	0	0
$845$	−25.9706	−0.893415
$846$	0	0
$847$	0.585786	0.0201279
$848$	0	0
$849$	3.61522	0.124074
$850$	0	0
$851$	−12.2132	−0.418663
$852$	0	0
$853$	−14.0000	−0.479351	−0.239675	−	0.970853i	$-0.577041\pi$
$853$	−14.0000	−0.479351	−0.239675	+	0.970853i	$0.577041\pi$
$854$	0	0
$855$	2.82843	0.0967302
$856$	0	0
$857$	47.6985	1.62935	0.814675	−	0.579918i	$-0.196916\pi$
$857$	47.6985	1.62935	0.814675	+	0.579918i	$0.196916\pi$
$858$	0	0
$859$	29.2843	0.999166	0.499583	−	0.866266i	$-0.333487\pi$
$859$	29.2843	0.999166	0.499583	+	0.866266i	$0.333487\pi$
$860$	0	0
$861$	−2.34315	−0.0798542
$862$	0	0
$863$	−47.5980	−1.62025	−0.810127	−	0.586254i	$-0.800602\pi$
$863$	−47.5980	−1.62025	−0.810127	+	0.586254i	$0.800602\pi$
$864$	0	0
$865$	17.8995	0.608601
$866$	0	0
$867$	6.89949	0.234319
$868$	0	0
$869$	17.4142	0.590737
$870$	0	0
$871$	−47.3553	−1.60457
$872$	0	0
$873$	1.17157	0.0396517
$874$	0	0
$875$	5.27208	0.178229
$876$	0	0
$877$	−9.85786	−0.332876	−0.166438	−	0.986052i	$-0.553227\pi$
$877$	−9.85786	−0.332876	−0.166438	+	0.986052i	$0.553227\pi$
$878$	0	0
$879$	−11.4731	−0.386978
$880$	0	0
$881$	21.7696	0.733435	0.366717	−	0.930332i	$-0.380482\pi$
$881$	21.7696	0.733435	0.366717	+	0.930332i	$0.380482\pi$
$882$	0	0
$883$	−33.5147	−1.12786	−0.563930	−	0.825823i	$-0.690711\pi$
$883$	−33.5147	−1.12786	−0.563930	+	0.825823i	$0.690711\pi$
$884$	0	0
$885$	1.82843	0.0614619
$886$	0	0
$887$	51.2548	1.72097	0.860484	−	0.509477i	$-0.170161\pi$
$887$	51.2548	1.72097	0.860484	+	0.509477i	$0.170161\pi$
$888$	0	0
$889$	9.71573	0.325855
$890$	0	0
$891$	7.48528	0.250766
$892$	0	0
$893$	−3.17157	−0.106133
$894$	0	0
$895$	1.92893	0.0644771
$896$	0	0
$897$	7.75736	0.259011
$898$	0	0
$899$	8.04163	0.268203
$900$	0	0
$901$	7.31371	0.243655
$902$	0	0
$903$	2.82843	0.0941242
$904$	0	0
$905$	−17.3848	−0.577890
$906$	0	0
$907$	50.1421	1.66494	0.832471	−	0.554068i	$-0.186925\pi$
$907$	50.1421	1.66494	0.832471	+	0.554068i	$0.186925\pi$
$908$	0	0
$909$	6.34315	0.210389
$910$	0	0
$911$	46.4264	1.53818	0.769088	−	0.639143i	$-0.220711\pi$
$911$	46.4264	1.53818	0.769088	+	0.639143i	$0.220711\pi$
$912$	0	0
$913$	−0.585786	−0.0193867
$914$	0	0
$915$	1.27208	0.0420536
$916$	0	0
$917$	1.45584	0.0480762
$918$	0	0
$919$	19.6152	0.647047	0.323523	−	0.946220i	$-0.395133\pi$
$919$	19.6152	0.647047	0.323523	+	0.946220i	$0.395133\pi$
$920$	0	0
$921$	1.89949	0.0625905
$922$	0	0
$923$	−59.8406	−1.96968
$924$	0	0
$925$	16.2843	0.535424
$926$	0	0
$927$	6.62742	0.217673
$928$	0	0
$929$	−56.2843	−1.84663	−0.923314	−	0.384047i	$-0.874530\pi$
$929$	−56.2843	−1.84663	−0.923314	+	0.384047i	$0.874530\pi$
$930$	0	0
$931$	−6.65685	−0.218170
$932$	0	0
$933$	−0.142136	−0.00465331
$934$	0	0
$935$	−0.585786	−0.0191573
$936$	0	0
$937$	23.5563	0.769552	0.384776	−	0.923010i	$-0.374279\pi$
$937$	23.5563	0.769552	0.384776	+	0.923010i	$0.374279\pi$
$938$	0	0
$939$	−8.27208	−0.269949
$940$	0	0
$941$	19.7574	0.644072	0.322036	−	0.946728i	$-0.395633\pi$
$941$	19.7574	0.644072	0.322036	+	0.946728i	$0.395633\pi$
$942$	0	0
$943$	28.9706	0.943411
$944$	0	0
$945$	−1.41421	−0.0460044
$946$	0	0
$947$	−5.97056	−0.194017	−0.0970086	−	0.995284i	$-0.530927\pi$
$947$	−5.97056	−0.194017	−0.0970086	+	0.995284i	$0.530927\pi$
$948$	0	0
$949$	−77.9411	−2.53008
$950$	0	0
$951$	10.5980	0.343663
$952$	0	0
$953$	25.8995	0.838967	0.419483	−	0.907763i	$-0.362211\pi$
$953$	25.8995	0.838967	0.419483	+	0.907763i	$0.362211\pi$
$954$	0	0
$955$	−14.3137	−0.463181
$956$	0	0
$957$	−0.928932	−0.0300281
$958$	0	0
$959$	5.07107	0.163753
$960$	0	0
$961$	−18.1421	−0.585230
$962$	0	0
$963$	12.2843	0.395855
$964$	0	0
$965$	−6.82843	−0.219815
$966$	0	0
$967$	27.5563	0.886152	0.443076	−	0.896484i	$-0.353887\pi$
$967$	27.5563	0.886152	0.443076	+	0.896484i	$0.353887\pi$
$968$	0	0
$969$	−0.242641	−0.00779474
$970$	0	0
$971$	−25.7279	−0.825648	−0.412824	−	0.910811i	$-0.635458\pi$
$971$	−25.7279	−0.825648	−0.412824	+	0.910811i	$0.635458\pi$
$972$	0	0
$973$	9.59798	0.307697
$974$	0	0
$975$	−10.3431	−0.331246
$976$	0	0
$977$	−44.8406	−1.43458	−0.717289	−	0.696776i	$-0.754617\pi$
$977$	−44.8406	−1.43458	−0.717289	+	0.696776i	$0.754617\pi$
$978$	0	0
$979$	−14.8995	−0.476190
$980$	0	0
$981$	32.9706	1.05267
$982$	0	0
$983$	−13.1005	−0.417841	−0.208921	−	0.977933i	$-0.566995\pi$
$983$	−13.1005	−0.417841	−0.208921	+	0.977933i	$0.566995\pi$
$984$	0	0
$985$	3.89949	0.124248
$986$	0	0
$987$	0.769553	0.0244951
$988$	0	0
$989$	−34.9706	−1.11200
$990$	0	0
$991$	8.68629	0.275929	0.137965	−	0.990437i	$-0.455944\pi$
$991$	8.68629	0.275929	0.137965	+	0.990437i	$0.455944\pi$
$992$	0	0
$993$	−3.40202	−0.107960
$994$	0	0
$995$	−16.1421	−0.511740
$996$	0	0
$997$	−37.5563	−1.18942	−0.594711	−	0.803940i	$-0.702733\pi$
$997$	−37.5563	−1.18942	−0.594711	+	0.803940i	$0.702733\pi$
$998$	0	0
$999$	−9.82843	−0.310958

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.n.1.1		2
4.3	odd	2		209.2.a.b.1.1	✓	2
12.11	even	2		1881.2.a.d.1.2		2
20.19	odd	2		5225.2.a.f.1.2		2
44.43	even	2		2299.2.a.f.1.2		2
76.75	even	2		3971.2.a.d.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
209.2.a.b.1.1	✓	2	4.3	odd	2
1881.2.a.d.1.2		2	12.11	even	2
2299.2.a.f.1.2		2	44.43	even	2
3344.2.a.n.1.1		2	1.1	even	1	trivial
3971.2.a.d.1.2		2	76.75	even	2
5225.2.a.f.1.2		2	20.19	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3344.a (trivial)

\(f(q)\)	\(=\)	\(q-0.414214 q^{3} -1.00000 q^{5} +0.585786 q^{7} -2.82843 q^{9} +1.00000 q^{11} -6.24264 q^{13} +0.414214 q^{15} +0.585786 q^{17} +1.00000 q^{19} -0.242641 q^{21} +3.00000 q^{23} -4.00000 q^{25} +2.41421 q^{27} +2.24264 q^{29} +3.58579 q^{31} -0.414214 q^{33} -0.585786 q^{35} -4.07107 q^{37} +2.58579 q^{39} +9.65685 q^{41} -11.6569 q^{43} +2.82843 q^{45} -3.17157 q^{47} -6.65685 q^{49} -0.242641 q^{51} +12.4853 q^{53} -1.00000 q^{55} -0.414214 q^{57} +4.41421 q^{59} +3.07107 q^{61} -1.65685 q^{63} +6.24264 q^{65} +7.58579 q^{67} -1.24264 q^{69} +9.58579 q^{71} +12.4853 q^{73} +1.65685 q^{75} +0.585786 q^{77} +17.4142 q^{79} +7.48528 q^{81} -0.585786 q^{83} -0.585786 q^{85} -0.928932 q^{87} -14.8995 q^{89} -3.65685 q^{91} -1.48528 q^{93} -1.00000 q^{95} -0.414214 q^{97} -2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5} + 4 q^{7} + 2 q^{11} - 4 q^{13} - 2 q^{15} + 4 q^{17} + 2 q^{19} + 8 q^{21} + 6 q^{23} - 8 q^{25} + 2 q^{27} - 4 q^{29} + 10 q^{31} + 2 q^{33} - 4 q^{35} + 6 q^{37} + 8 q^{39} + 8 q^{41}+ \cdots + 2 q^{97}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^5 + 4 * q^7 + 2 * q^11 - 4 * q^13 - 2 * q^15 + 4 * q^17 + 2 * q^19 + 8 * q^21 + 6 * q^23 - 8 * q^25 + 2 * q^27 - 4 * q^29 + 10 * q^31 + 2 * q^33 - 4 * q^35 + 6 * q^37 + 8 * q^39 + 8 * q^41 - 12 * q^43 - 12 * q^47 - 2 * q^49 + 8 * q^51 + 8 * q^53 - 2 * q^55 + 2 * q^57 + 6 * q^59 - 8 * q^61 + 8 * q^63 + 4 * q^65 + 18 * q^67 + 6 * q^69 + 22 * q^71 + 8 * q^73 - 8 * q^75 + 4 * q^77 + 32 * q^79 - 2 * q^81 - 4 * q^83 - 4 * q^85 - 16 * q^87 - 10 * q^89 + 4 * q^91 + 14 * q^93 - 2 * q^95 + 2 * q^97

Introduction

L-functions

Modular forms

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