LMFDB - Embedded newform 1840.2.a.v.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1840,2,Mod(1,1840)] 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("1840.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1840.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,0,0,-5,0,2,0,13] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$14.6924739719$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.13955077.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 14x^{3} - x^{2} + 32x + 16 $ x^5 - 14x^3 - x^2 + 32x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 920)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.93283$ of defining polynomial
Character	$\chi$	$=$	1840.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.93283 q^{3} -1.00000 q^{5} -2.38236 q^{7} +0.735829 q^{9} -5.33368 q^{11} -4.53752 q^{13} +1.93283 q^{15} +1.81464 q^{17} -7.00233 q^{19} +4.60469 q^{21} +1.00000 q^{23} +1.00000 q^{25} +4.37626 q^{27} -0.118188 q^{29} +0.884147 q^{31} +10.3091 q^{33} +2.38236 q^{35} +7.51903 q^{37} +8.77026 q^{39} -1.45186 q^{41} -0.735829 q^{45} -10.4389 q^{47} -1.32437 q^{49} -3.50739 q^{51} -9.42167 q^{53} +5.33368 q^{55} +13.5343 q^{57} -7.79239 q^{59} -2.80533 q^{61} -1.75301 q^{63} +4.53752 q^{65} +3.11134 q^{67} -1.93283 q^{69} +13.5909 q^{71} +12.4389 q^{73} -1.93283 q^{75} +12.7067 q^{77} -6.80169 q^{79} -10.6660 q^{81} +13.5190 q^{83} -1.81464 q^{85} +0.228437 q^{87} +2.89906 q^{89} +10.8100 q^{91} -1.70890 q^{93} +7.00233 q^{95} -1.97774 q^{97} -3.92468 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{5} + 2 q^{7} + 13 q^{9} + q^{11} + 4 q^{13} + 4 q^{17} - 7 q^{19} + 6 q^{21} + 5 q^{23} + 5 q^{25} - 3 q^{27} + 4 q^{29} - 19 q^{31} + 17 q^{33} - 2 q^{35} + 15 q^{37} - 19 q^{39} + 25 q^{41}+ \cdots + 65 q^{99}+O(q^{100}) $ 5 * q - 5 * q^5 + 2 * q^7 + 13 * q^9 + q^11 + 4 * q^13 + 4 * q^17 - 7 * q^19 + 6 * q^21 + 5 * q^23 + 5 * q^25 - 3 * q^27 + 4 * q^29 - 19 * q^31 + 17 * q^33 - 2 * q^35 + 15 * q^37 - 19 * q^39 + 25 * q^41 - 13 * q^45 + 11 * q^47 + 25 * q^49 - 19 * q^51 + 3 * q^53 - q^55 + 48 * q^57 + q^59 - 5 * q^61 + 41 * q^63 - 4 * q^65 - 9 * q^67 - q^71 - q^73 + 18 * q^77 + 2 * q^79 + 57 * q^81 + 45 * q^83 - 4 * q^85 + 9 * q^87 + 6 * q^89 - 11 * q^91 - 39 * q^93 + 7 * q^95 + 25 * q^97 + 65 * q^99

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
$3$	−1.93283	−1.11592	−0.557960	−	0.829868i	$-0.688416\pi$
$3$	−1.93283	−1.11592	−0.557960	+	0.829868i	$0.688416\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.38236	−0.900447	−0.450223	−	0.892916i	$-0.648656\pi$
$7$	−2.38236	−0.900447	−0.450223	+	0.892916i	$0.648656\pi$
$8$	0	0
$9$	0.735829	0.245276
$10$	0	0
$11$	−5.33368	−1.60816	−0.804082	−	0.594519i	$-0.797343\pi$
$11$	−5.33368	−1.60816	−0.804082	+	0.594519i	$0.797343\pi$
$12$	0	0
$13$	−4.53752	−1.25848	−0.629241	−	0.777210i	$-0.716634\pi$
$13$	−4.53752	−1.25848	−0.629241	+	0.777210i	$0.716634\pi$
$14$	0	0
$15$	1.93283	0.499054
$16$	0	0
$17$	1.81464	0.440115	0.220058	−	0.975487i	$-0.429375\pi$
$17$	1.81464	0.440115	0.220058	+	0.975487i	$0.429375\pi$
$18$	0	0
$19$	−7.00233	−1.60645	−0.803223	−	0.595679i	$-0.796883\pi$
$19$	−7.00233	−1.60645	−0.803223	+	0.595679i	$0.796883\pi$
$20$	0	0
$21$	4.60469	1.00483
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	4.37626	0.842211
$28$	0	0
$29$	−0.118188	−0.0219469	−0.0109735	−	0.999940i	$-0.503493\pi$
$29$	−0.118188	−0.0219469	−0.0109735	+	0.999940i	$0.503493\pi$
$30$	0	0
$31$	0.884147	0.158797	0.0793987	−	0.996843i	$-0.474700\pi$
$31$	0.884147	0.158797	0.0793987	+	0.996843i	$0.474700\pi$
$32$	0	0
$33$	10.3091	1.79458
$34$	0	0
$35$	2.38236	0.402692
$36$	0	0
$37$	7.51903	1.23612	0.618061	−	0.786130i	$-0.287919\pi$
$37$	7.51903	1.23612	0.618061	+	0.786130i	$0.287919\pi$
$38$	0	0
$39$	8.77026	1.40436
$40$	0	0
$41$	−1.45186	−0.226743	−0.113371	−	0.993553i	$-0.536165\pi$
$41$	−1.45186	−0.226743	−0.113371	+	0.993553i	$0.536165\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	−0.735829	−0.109691
$46$	0	0
$47$	−10.4389	−1.52267	−0.761336	−	0.648357i	$-0.775456\pi$
$47$	−10.4389	−1.52267	−0.761336	+	0.648357i	$0.775456\pi$
$48$	0	0
$49$	−1.32437	−0.189195
$50$	0	0
$51$	−3.50739	−0.491133
$52$	0	0
$53$	−9.42167	−1.29417	−0.647083	−	0.762420i	$-0.724011\pi$
$53$	−9.42167	−1.29417	−0.647083	+	0.762420i	$0.724011\pi$
$54$	0	0
$55$	5.33368	0.719193
$56$	0	0
$57$	13.5343	1.79266
$58$	0	0
$59$	−7.79239	−1.01448	−0.507241	−	0.861804i	$-0.669335\pi$
$59$	−7.79239	−1.01448	−0.507241	+	0.861804i	$0.669335\pi$
$60$	0	0
$61$	−2.80533	−0.359186	−0.179593	−	0.983741i	$-0.557478\pi$
$61$	−2.80533	−0.359186	−0.179593	+	0.983741i	$0.557478\pi$
$62$	0	0
$63$	−1.75301	−0.220858
$64$	0	0
$65$	4.53752	0.562810
$66$	0	0
$67$	3.11134	0.380111	0.190055	−	0.981773i	$-0.439133\pi$
$67$	3.11134	0.380111	0.190055	+	0.981773i	$0.439133\pi$
$68$	0	0
$69$	−1.93283	−0.232685
$70$	0	0
$71$	13.5909	1.61294	0.806470	−	0.591275i	$-0.201375\pi$
$71$	13.5909	1.61294	0.806470	+	0.591275i	$0.201375\pi$
$72$	0	0
$73$	12.4389	1.45586	0.727932	−	0.685649i	$-0.240481\pi$
$73$	12.4389	1.45586	0.727932	+	0.685649i	$0.240481\pi$
$74$	0	0
$75$	−1.93283	−0.223184
$76$	0	0
$77$	12.7067	1.44807
$78$	0	0
$79$	−6.80169	−0.765250	−0.382625	−	0.923904i	$-0.624980\pi$
$79$	−6.80169	−0.765250	−0.382625	+	0.923904i	$0.624980\pi$
$80$	0	0
$81$	−10.6660	−1.18512
$82$	0	0
$83$	13.5190	1.48391	0.741953	−	0.670451i	$-0.233900\pi$
$83$	13.5190	1.48391	0.741953	+	0.670451i	$0.233900\pi$
$84$	0	0
$85$	−1.81464	−0.196826
$86$	0	0
$87$	0.228437	0.0244910
$88$	0	0
$89$	2.89906	0.307300	0.153650	−	0.988125i	$-0.450897\pi$
$89$	2.89906	0.307300	0.153650	+	0.988125i	$0.450897\pi$
$90$	0	0
$91$	10.8100	1.13320
$92$	0	0
$93$	−1.70890	−0.177205
$94$	0	0
$95$	7.00233	0.718424
$96$	0	0
$97$	−1.97774	−0.200809	−0.100405	−	0.994947i	$-0.532014\pi$
$97$	−1.97774	−0.200809	−0.100405	+	0.994947i	$0.532014\pi$
$98$	0	0
$99$	−3.92468	−0.394445
$100$	0	0
$101$	−12.2838	−1.22228	−0.611139	−	0.791523i	$-0.709289\pi$
$101$	−12.2838	−1.22228	−0.611139	+	0.791523i	$0.709289\pi$
$102$	0	0
$103$	12.5061	1.23226	0.616131	−	0.787644i	$-0.288699\pi$
$103$	12.5061	1.23226	0.616131	+	0.787644i	$0.288699\pi$
$104$	0	0
$105$	−4.60469	−0.449372
$106$	0	0
$107$	17.3847	1.68064	0.840321	−	0.542089i	$-0.182367\pi$
$107$	17.3847	1.68064	0.840321	+	0.542089i	$0.182367\pi$
$108$	0	0
$109$	4.30908	0.412735	0.206368	−	0.978475i	$-0.433836\pi$
$109$	4.30908	0.412735	0.206368	+	0.978475i	$0.433836\pi$
$110$	0	0
$111$	−14.5330	−1.37941
$112$	0	0
$113$	−12.1494	−1.14292	−0.571460	−	0.820630i	$-0.693623\pi$
$113$	−12.1494	−1.14292	−0.571460	+	0.820630i	$0.693623\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	−3.33884	−0.308676
$118$	0	0
$119$	−4.32313	−0.396300
$120$	0	0
$121$	17.4481	1.58619
$122$	0	0
$123$	2.80620	0.253027
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	1.22953	0.109103	0.0545515	−	0.998511i	$-0.482627\pi$
$127$	1.22953	0.109103	0.0545515	+	0.998511i	$0.482627\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1.97345	−0.172421	−0.0862104	−	0.996277i	$-0.527476\pi$
$131$	−1.97345	−0.172421	−0.0862104	+	0.996277i	$0.527476\pi$
$132$	0	0
$133$	16.6821	1.44652
$134$	0	0
$135$	−4.37626	−0.376648
$136$	0	0
$137$	13.7671	1.17620	0.588099	−	0.808789i	$-0.299876\pi$
$137$	13.7671	1.17620	0.588099	+	0.808789i	$0.299876\pi$
$138$	0	0
$139$	−5.30280	−0.449778	−0.224889	−	0.974384i	$-0.572202\pi$
$139$	−5.30280	−0.449778	−0.224889	+	0.974384i	$0.572202\pi$
$140$	0	0
$141$	20.1766	1.69918
$142$	0	0
$143$	24.2017	2.02385
$144$	0	0
$145$	0.118188	0.00981497
$146$	0	0
$147$	2.55978	0.211127
$148$	0	0
$149$	7.73256	0.633476	0.316738	−	0.948513i	$-0.397412\pi$
$149$	7.73256	0.633476	0.316738	+	0.948513i	$0.397412\pi$
$150$	0	0
$151$	−21.4156	−1.74277	−0.871387	−	0.490596i	$-0.836779\pi$
$151$	−21.4156	−1.74277	−0.871387	+	0.490596i	$0.836779\pi$
$152$	0	0
$153$	1.33527	0.107950
$154$	0	0
$155$	−0.884147	−0.0710164
$156$	0	0
$157$	−4.04738	−0.323016	−0.161508	−	0.986871i	$-0.551636\pi$
$157$	−4.04738	−0.323016	−0.161508	+	0.986871i	$0.551636\pi$
$158$	0	0
$159$	18.2105	1.44418
$160$	0	0
$161$	−2.38236	−0.187756
$162$	0	0
$163$	−9.04725	−0.708635	−0.354318	−	0.935125i	$-0.615287\pi$
$163$	−9.04725	−0.708635	−0.354318	+	0.935125i	$0.615287\pi$
$164$	0	0
$165$	−10.3091	−0.802561
$166$	0	0
$167$	12.5467	0.970893	0.485446	−	0.874266i	$-0.338657\pi$
$167$	12.5467	0.970893	0.485446	+	0.874266i	$0.338657\pi$
$168$	0	0
$169$	7.58910	0.583777
$170$	0	0
$171$	−5.15252	−0.394023
$172$	0	0
$173$	−14.1354	−1.07469	−0.537346	−	0.843362i	$-0.680573\pi$
$173$	−14.1354	−1.07469	−0.537346	+	0.843362i	$0.680573\pi$
$174$	0	0
$175$	−2.38236	−0.180089
$176$	0	0
$177$	15.0614	1.13208
$178$	0	0
$179$	26.2442	1.96158	0.980791	−	0.195061i	$-0.0624904\pi$
$179$	26.2442	1.96158	0.980791	+	0.195061i	$0.0624904\pi$
$180$	0	0
$181$	−18.3461	−1.36365	−0.681826	−	0.731514i	$-0.738814\pi$
$181$	−18.3461	−1.36365	−0.681826	+	0.731514i	$0.738814\pi$
$182$	0	0
$183$	5.42223	0.400823
$184$	0	0
$185$	−7.51903	−0.552810
$186$	0	0
$187$	−9.67871	−0.707777
$188$	0	0
$189$	−10.4258	−0.758366
$190$	0	0
$191$	3.86566	0.279709	0.139855	−	0.990172i	$-0.455336\pi$
$191$	3.86566	0.279709	0.139855	+	0.990172i	$0.455336\pi$
$192$	0	0
$193$	−23.8599	−1.71747	−0.858737	−	0.512417i	$-0.828750\pi$
$193$	−23.8599	−1.71747	−0.858737	+	0.512417i	$0.828750\pi$
$194$	0	0
$195$	−8.77026	−0.628051
$196$	0	0
$197$	11.7356	0.836128	0.418064	−	0.908418i	$-0.362709\pi$
$197$	11.7356	0.836128	0.418064	+	0.908418i	$0.362709\pi$
$198$	0	0
$199$	8.89906	0.630838	0.315419	−	0.948953i	$-0.397855\pi$
$199$	8.89906	0.630838	0.315419	+	0.948953i	$0.397855\pi$
$200$	0	0
$201$	−6.01369	−0.424173
$202$	0	0
$203$	0.281566	0.0197620
$204$	0	0
$205$	1.45186	0.101403
$206$	0	0
$207$	0.735829	0.0511437
$208$	0	0
$209$	37.3482	2.58343
$210$	0	0
$211$	−23.1224	−1.59181	−0.795906	−	0.605420i	$-0.793005\pi$
$211$	−23.1224	−1.59181	−0.795906	+	0.605420i	$0.793005\pi$
$212$	0	0
$213$	−26.2688	−1.79991
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−2.10635	−0.142989
$218$	0	0
$219$	−24.0423	−1.62463
$220$	0	0
$221$	−8.23398	−0.553877
$222$	0	0
$223$	−1.50863	−0.101026	−0.0505128	−	0.998723i	$-0.516086\pi$
$223$	−1.50863	−0.101026	−0.0505128	+	0.998723i	$0.516086\pi$
$224$	0	0
$225$	0.735829	0.0490553
$226$	0	0
$227$	22.0047	1.46050	0.730251	−	0.683179i	$-0.239403\pi$
$227$	22.0047	1.46050	0.730251	+	0.683179i	$0.239403\pi$
$228$	0	0
$229$	−11.2931	−0.746266	−0.373133	−	0.927778i	$-0.621717\pi$
$229$	−11.2931	−0.746266	−0.373133	+	0.927778i	$0.621717\pi$
$230$	0	0
$231$	−24.5599	−1.61593
$232$	0	0
$233$	−17.6789	−1.15818	−0.579091	−	0.815263i	$-0.696592\pi$
$233$	−17.6789	−1.15818	−0.579091	+	0.815263i	$0.696592\pi$
$234$	0	0
$235$	10.4389	0.680960
$236$	0	0
$237$	13.1465	0.853958
$238$	0	0
$239$	25.2583	1.63382	0.816911	−	0.576763i	$-0.195684\pi$
$239$	25.2583	1.63382	0.816911	+	0.576763i	$0.195684\pi$
$240$	0	0
$241$	0.790615	0.0509280	0.0254640	−	0.999676i	$-0.491894\pi$
$241$	0.790615	0.0509280	0.0254640	+	0.999676i	$0.491894\pi$
$242$	0	0
$243$	7.48688	0.480283
$244$	0	0
$245$	1.32437	0.0846108
$246$	0	0
$247$	31.7732	2.02168
$248$	0	0
$249$	−26.1300	−1.65592
$250$	0	0
$251$	−12.3461	−0.779276	−0.389638	−	0.920968i	$-0.627400\pi$
$251$	−12.3461	−0.779276	−0.389638	+	0.920968i	$0.627400\pi$
$252$	0	0
$253$	−5.33368	−0.335325
$254$	0	0
$255$	3.50739	0.219641
$256$	0	0
$257$	6.40442	0.399497	0.199748	−	0.979847i	$-0.435988\pi$
$257$	6.40442	0.399497	0.199748	+	0.979847i	$0.435988\pi$
$258$	0	0
$259$	−17.9130	−1.11306
$260$	0	0
$261$	−0.0869661	−0.00538307
$262$	0	0
$263$	−19.6914	−1.21422	−0.607111	−	0.794617i	$-0.707672\pi$
$263$	−19.6914	−1.21422	−0.607111	+	0.794617i	$0.707672\pi$
$264$	0	0
$265$	9.42167	0.578768
$266$	0	0
$267$	−5.60338	−0.342922
$268$	0	0
$269$	−3.42494	−0.208822	−0.104411	−	0.994534i	$-0.533296\pi$
$269$	−3.42494	−0.208822	−0.104411	+	0.994534i	$0.533296\pi$
$270$	0	0
$271$	−13.7197	−0.833411	−0.416705	−	0.909042i	$-0.636815\pi$
$271$	−13.7197	−0.833411	−0.416705	+	0.909042i	$0.636815\pi$
$272$	0	0
$273$	−20.8939	−1.26456
$274$	0	0
$275$	−5.33368	−0.321633
$276$	0	0
$277$	0.636129	0.0382213	0.0191107	−	0.999817i	$-0.493917\pi$
$277$	0.636129	0.0382213	0.0191107	+	0.999817i	$0.493917\pi$
$278$	0	0
$279$	0.650581	0.0389493
$280$	0	0
$281$	18.4124	1.09839	0.549195	−	0.835694i	$-0.314935\pi$
$281$	18.4124	1.09839	0.549195	+	0.835694i	$0.314935\pi$
$282$	0	0
$283$	7.95729	0.473012	0.236506	−	0.971630i	$-0.423998\pi$
$283$	7.95729	0.473012	0.236506	+	0.971630i	$0.423998\pi$
$284$	0	0
$285$	−13.5343	−0.801704
$286$	0	0
$287$	3.45886	0.204170
$288$	0	0
$289$	−13.7071	−0.806299
$290$	0	0
$291$	3.82264	0.224087
$292$	0	0
$293$	−20.9166	−1.22196	−0.610981	−	0.791645i	$-0.709225\pi$
$293$	−20.9166	−1.22196	−0.610981	+	0.791645i	$0.709225\pi$
$294$	0	0
$295$	7.79239	0.453690
$296$	0	0
$297$	−23.3415	−1.35441
$298$	0	0
$299$	−4.53752	−0.262412
$300$	0	0
$301$	0	0
$302$	0	0
$303$	23.7424	1.36396
$304$	0	0
$305$	2.80533	0.160633
$306$	0	0
$307$	5.78807	0.330342	0.165171	−	0.986265i	$-0.447182\pi$
$307$	5.78807	0.330342	0.165171	+	0.986265i	$0.447182\pi$
$308$	0	0
$309$	−24.1721	−1.37510
$310$	0	0
$311$	−18.3674	−1.04152	−0.520761	−	0.853702i	$-0.674352\pi$
$311$	−18.3674	−1.04152	−0.520761	+	0.853702i	$0.674352\pi$
$312$	0	0
$313$	9.85869	0.557246	0.278623	−	0.960401i	$-0.410122\pi$
$313$	9.85869	0.557246	0.278623	+	0.960401i	$0.410122\pi$
$314$	0	0
$315$	1.75301	0.0987709
$316$	0	0
$317$	−32.9488	−1.85059	−0.925294	−	0.379250i	$-0.876182\pi$
$317$	−32.9488	−1.85059	−0.925294	+	0.379250i	$0.876182\pi$
$318$	0	0
$319$	0.630376	0.0352943
$320$	0	0
$321$	−33.6016	−1.87546
$322$	0	0
$323$	−12.7067	−0.707021
$324$	0	0
$325$	−4.53752	−0.251696
$326$	0	0
$327$	−8.32873	−0.460580
$328$	0	0
$329$	24.8692	1.37109
$330$	0	0
$331$	6.85308	0.376679	0.188340	−	0.982104i	$-0.439689\pi$
$331$	6.85308	0.376679	0.188340	+	0.982104i	$0.439689\pi$
$332$	0	0
$333$	5.53273	0.303192
$334$	0	0
$335$	−3.11134	−0.169991
$336$	0	0
$337$	35.8432	1.95250	0.976251	−	0.216641i	$-0.0695100\pi$
$337$	35.8432	1.95250	0.976251	+	0.216641i	$0.0695100\pi$
$338$	0	0
$339$	23.4827	1.27541
$340$	0	0
$341$	−4.71575	−0.255372
$342$	0	0
$343$	19.8316	1.07081
$344$	0	0
$345$	1.93283	0.104060
$346$	0	0
$347$	4.09198	0.219669	0.109835	−	0.993950i	$-0.464968\pi$
$347$	4.09198	0.219669	0.109835	+	0.993950i	$0.464968\pi$
$348$	0	0
$349$	−11.3229	−0.606101	−0.303051	−	0.952974i	$-0.598005\pi$
$349$	−11.3229	−0.606101	−0.303051	+	0.952974i	$0.598005\pi$
$350$	0	0
$351$	−19.8574	−1.05991
$352$	0	0
$353$	26.1443	1.39152	0.695761	−	0.718273i	$-0.255067\pi$
$353$	26.1443	1.39152	0.695761	+	0.718273i	$0.255067\pi$
$354$	0	0
$355$	−13.5909	−0.721329
$356$	0	0
$357$	8.35587	0.442239
$358$	0	0
$359$	12.2058	0.644198	0.322099	−	0.946706i	$-0.395612\pi$
$359$	12.2058	0.644198	0.322099	+	0.946706i	$0.395612\pi$
$360$	0	0
$361$	30.0327	1.58067
$362$	0	0
$363$	−33.7242	−1.77006
$364$	0	0
$365$	−12.4389	−0.651083
$366$	0	0
$367$	3.15299	0.164585	0.0822923	−	0.996608i	$-0.473776\pi$
$367$	3.15299	0.164585	0.0822923	+	0.996608i	$0.473776\pi$
$368$	0	0
$369$	−1.06832	−0.0556147
$370$	0	0
$371$	22.4458	1.16533
$372$	0	0
$373$	9.59872	0.497003	0.248501	−	0.968632i	$-0.420062\pi$
$373$	9.59872	0.497003	0.248501	+	0.968632i	$0.420062\pi$
$374$	0	0
$375$	1.93283	0.0998109
$376$	0	0
$377$	0.536280	0.0276198
$378$	0	0
$379$	−32.7841	−1.68401	−0.842003	−	0.539473i	$-0.818624\pi$
$379$	−32.7841	−1.68401	−0.842003	+	0.539473i	$0.818624\pi$
$380$	0	0
$381$	−2.37647	−0.121750
$382$	0	0
$383$	−34.7378	−1.77502	−0.887508	−	0.460792i	$-0.847565\pi$
$383$	−34.7378	−1.77502	−0.887508	+	0.460792i	$0.847565\pi$
$384$	0	0
$385$	−12.7067	−0.647595
$386$	0	0
$387$	0	0
$388$	0	0
$389$	13.8965	0.704581	0.352290	−	0.935891i	$-0.385403\pi$
$389$	13.8965	0.704581	0.352290	+	0.935891i	$0.385403\pi$
$390$	0	0
$391$	1.81464	0.0917704
$392$	0	0
$393$	3.81434	0.192408
$394$	0	0
$395$	6.80169	0.342230
$396$	0	0
$397$	−26.1135	−1.31060	−0.655299	−	0.755370i	$-0.727457\pi$
$397$	−26.1135	−1.31060	−0.655299	+	0.755370i	$0.727457\pi$
$398$	0	0
$399$	−32.2436	−1.61420
$400$	0	0
$401$	18.0715	0.902446	0.451223	−	0.892411i	$-0.350988\pi$
$401$	18.0715	0.902446	0.451223	+	0.892411i	$0.350988\pi$
$402$	0	0
$403$	−4.01183	−0.199844
$404$	0	0
$405$	10.6660	0.530000
$406$	0	0
$407$	−40.1041	−1.98789
$408$	0	0
$409$	−0.865532	−0.0427978	−0.0213989	−	0.999771i	$-0.506812\pi$
$409$	−0.865532	−0.0427978	−0.0213989	+	0.999771i	$0.506812\pi$
$410$	0	0
$411$	−26.6094	−1.31254
$412$	0	0
$413$	18.5643	0.913487
$414$	0	0
$415$	−13.5190	−0.663623
$416$	0	0
$417$	10.2494	0.501916
$418$	0	0
$419$	20.5675	1.00479	0.502394	−	0.864639i	$-0.332453\pi$
$419$	20.5675	1.00479	0.502394	+	0.864639i	$0.332453\pi$
$420$	0	0
$421$	26.2423	1.27897	0.639485	−	0.768803i	$-0.279147\pi$
$421$	26.2423	1.27897	0.639485	+	0.768803i	$0.279147\pi$
$422$	0	0
$423$	−7.68126	−0.373476
$424$	0	0
$425$	1.81464	0.0880230
$426$	0	0
$427$	6.68331	0.323428
$428$	0	0
$429$	−46.7777	−2.25845
$430$	0	0
$431$	−23.2654	−1.12066	−0.560328	−	0.828271i	$-0.689325\pi$
$431$	−23.2654	−1.12066	−0.560328	+	0.828271i	$0.689325\pi$
$432$	0	0
$433$	−34.7365	−1.66933	−0.834665	−	0.550758i	$-0.814339\pi$
$433$	−34.7365	−1.66933	−0.834665	+	0.550758i	$0.814339\pi$
$434$	0	0
$435$	−0.228437	−0.0109527
$436$	0	0
$437$	−7.00233	−0.334967
$438$	0	0
$439$	−28.5238	−1.36137	−0.680684	−	0.732577i	$-0.738317\pi$
$439$	−28.5238	−1.36137	−0.680684	+	0.732577i	$0.738317\pi$
$440$	0	0
$441$	−0.974509	−0.0464052
$442$	0	0
$443$	−26.3428	−1.25158	−0.625792	−	0.779990i	$-0.715224\pi$
$443$	−26.3428	−1.25158	−0.625792	+	0.779990i	$0.715224\pi$
$444$	0	0
$445$	−2.89906	−0.137429
$446$	0	0
$447$	−14.9457	−0.706908
$448$	0	0
$449$	29.5411	1.39413	0.697065	−	0.717008i	$-0.254489\pi$
$449$	29.5411	1.39413	0.697065	+	0.717008i	$0.254489\pi$
$450$	0	0
$451$	7.74377	0.364640
$452$	0	0
$453$	41.3926	1.94480
$454$	0	0
$455$	−10.8100	−0.506781
$456$	0	0
$457$	3.16668	0.148131	0.0740655	−	0.997253i	$-0.476403\pi$
$457$	3.16668	0.148131	0.0740655	+	0.997253i	$0.476403\pi$
$458$	0	0
$459$	7.94133	0.370670
$460$	0	0
$461$	8.43891	0.393039	0.196520	−	0.980500i	$-0.437036\pi$
$461$	8.43891	0.393039	0.196520	+	0.980500i	$0.437036\pi$
$462$	0	0
$463$	−9.62461	−0.447294	−0.223647	−	0.974670i	$-0.571796\pi$
$463$	−9.62461	−0.447294	−0.223647	+	0.974670i	$0.571796\pi$
$464$	0	0
$465$	1.70890	0.0792485
$466$	0	0
$467$	23.1041	1.06913	0.534564	−	0.845128i	$-0.320476\pi$
$467$	23.1041	1.06913	0.534564	+	0.845128i	$0.320476\pi$
$468$	0	0
$469$	−7.41233	−0.342270
$470$	0	0
$471$	7.82289	0.360460
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−7.00233	−0.321289
$476$	0	0
$477$	−6.93274	−0.317428
$478$	0	0
$479$	−20.5981	−0.941150	−0.470575	−	0.882360i	$-0.655953\pi$
$479$	−20.5981	−0.941150	−0.470575	+	0.882360i	$0.655953\pi$
$480$	0	0
$481$	−34.1178	−1.55564
$482$	0	0
$483$	4.60469	0.209521
$484$	0	0
$485$	1.97774	0.0898047
$486$	0	0
$487$	−40.1145	−1.81776	−0.908881	−	0.417056i	$-0.863062\pi$
$487$	−40.1145	−1.81776	−0.908881	+	0.417056i	$0.863062\pi$
$488$	0	0
$489$	17.4868	0.790780
$490$	0	0
$491$	18.4799	0.833985	0.416993	−	0.908910i	$-0.363084\pi$
$491$	18.4799	0.833985	0.416993	+	0.908910i	$0.363084\pi$
$492$	0	0
$493$	−0.214469	−0.00965918
$494$	0	0
$495$	3.92468	0.176401
$496$	0	0
$497$	−32.3783	−1.45237
$498$	0	0
$499$	−0.0506452	−0.00226719	−0.00113360	−	0.999999i	$-0.500361\pi$
$499$	−0.0506452	−0.00226719	−0.00113360	+	0.999999i	$0.500361\pi$
$500$	0	0
$501$	−24.2506	−1.08344
$502$	0	0
$503$	31.1887	1.39064	0.695318	−	0.718702i	$-0.255263\pi$
$503$	31.1887	1.39064	0.695318	+	0.718702i	$0.255263\pi$
$504$	0	0
$505$	12.2838	0.546620
$506$	0	0
$507$	−14.6684	−0.651448
$508$	0	0
$509$	3.42426	0.151778	0.0758888	−	0.997116i	$-0.475821\pi$
$509$	3.42426	0.151778	0.0758888	+	0.997116i	$0.475821\pi$
$510$	0	0
$511$	−29.6340	−1.31093
$512$	0	0
$513$	−30.6440	−1.35297
$514$	0	0
$515$	−12.5061	−0.551084
$516$	0	0
$517$	55.6778	2.44871
$518$	0	0
$519$	27.3213	1.19927
$520$	0	0
$521$	−9.78442	−0.428663	−0.214332	−	0.976761i	$-0.568757\pi$
$521$	−9.78442	−0.428663	−0.214332	+	0.976761i	$0.568757\pi$
$522$	0	0
$523$	−6.22884	−0.272368	−0.136184	−	0.990684i	$-0.543484\pi$
$523$	−6.22884	−0.272368	−0.136184	+	0.990684i	$0.543484\pi$
$524$	0	0
$525$	4.60469	0.200965
$526$	0	0
$527$	1.60441	0.0698892
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−5.73387	−0.248829
$532$	0	0
$533$	6.58786	0.285352
$534$	0	0
$535$	−17.3847	−0.751606
$536$	0	0
$537$	−50.7255	−2.18897
$538$	0	0
$539$	7.06375	0.304257
$540$	0	0
$541$	19.8477	0.853319	0.426660	−	0.904412i	$-0.359690\pi$
$541$	19.8477	0.853319	0.426660	+	0.904412i	$0.359690\pi$
$542$	0	0
$543$	35.4598	1.52173
$544$	0	0
$545$	−4.30908	−0.184581
$546$	0	0
$547$	7.68024	0.328383	0.164192	−	0.986428i	$-0.447498\pi$
$547$	7.68024	0.328383	0.164192	+	0.986428i	$0.447498\pi$
$548$	0	0
$549$	−2.06425	−0.0880999
$550$	0	0
$551$	0.827591	0.0352566
$552$	0	0
$553$	16.2041	0.689067
$554$	0	0
$555$	14.5330	0.616892
$556$	0	0
$557$	18.1260	0.768023	0.384012	−	0.923328i	$-0.374542\pi$
$557$	18.1260	0.768023	0.384012	+	0.923328i	$0.374542\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	18.7073	0.789823
$562$	0	0
$563$	16.8772	0.711287	0.355644	−	0.934622i	$-0.384262\pi$
$563$	16.8772	0.711287	0.355644	+	0.934622i	$0.384262\pi$
$564$	0	0
$565$	12.1494	0.511130
$566$	0	0
$567$	25.4103	1.06713
$568$	0	0
$569$	36.0022	1.50929	0.754645	−	0.656133i	$-0.227809\pi$
$569$	36.0022	1.50929	0.754645	+	0.656133i	$0.227809\pi$
$570$	0	0
$571$	−26.6267	−1.11429	−0.557147	−	0.830414i	$-0.688104\pi$
$571$	−26.6267	−1.11429	−0.557147	+	0.830414i	$0.688104\pi$
$572$	0	0
$573$	−7.47166	−0.312133
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	30.0190	1.24971	0.624854	−	0.780742i	$-0.285158\pi$
$577$	30.0190	1.24971	0.624854	+	0.780742i	$0.285158\pi$
$578$	0	0
$579$	46.1171	1.91656
$580$	0	0
$581$	−32.2072	−1.33618
$582$	0	0
$583$	50.2521	2.08123
$584$	0	0
$585$	3.33884	0.138044
$586$	0	0
$587$	31.1814	1.28699	0.643497	−	0.765449i	$-0.277483\pi$
$587$	31.1814	1.28699	0.643497	+	0.765449i	$0.277483\pi$
$588$	0	0
$589$	−6.19109	−0.255099
$590$	0	0
$591$	−22.6829	−0.933051
$592$	0	0
$593$	15.0334	0.617348	0.308674	−	0.951168i	$-0.400115\pi$
$593$	15.0334	0.617348	0.308674	+	0.951168i	$0.400115\pi$
$594$	0	0
$595$	4.32313	0.177231
$596$	0	0
$597$	−17.2004	−0.703964
$598$	0	0
$599$	−20.3534	−0.831617	−0.415808	−	0.909452i	$-0.636501\pi$
$599$	−20.3534	−0.831617	−0.415808	+	0.909452i	$0.636501\pi$
$600$	0	0
$601$	46.5338	1.89816	0.949078	−	0.315043i	$-0.102019\pi$
$601$	46.5338	1.89816	0.949078	+	0.315043i	$0.102019\pi$
$602$	0	0
$603$	2.28942	0.0932323
$604$	0	0
$605$	−17.4481	−0.709366
$606$	0	0
$607$	7.52725	0.305522	0.152761	−	0.988263i	$-0.451184\pi$
$607$	7.52725	0.305522	0.152761	+	0.988263i	$0.451184\pi$
$608$	0	0
$609$	−0.544219	−0.0220529
$610$	0	0
$611$	47.3668	1.91626
$612$	0	0
$613$	32.2643	1.30314	0.651572	−	0.758587i	$-0.274110\pi$
$613$	32.2643	1.30314	0.651572	+	0.758587i	$0.274110\pi$
$614$	0	0
$615$	−2.80620	−0.113157
$616$	0	0
$617$	4.52158	0.182032	0.0910161	−	0.995849i	$-0.470989\pi$
$617$	4.52158	0.182032	0.0910161	+	0.995849i	$0.470989\pi$
$618$	0	0
$619$	−17.7807	−0.714668	−0.357334	−	0.933977i	$-0.616314\pi$
$619$	−17.7807	−0.714668	−0.357334	+	0.933977i	$0.616314\pi$
$620$	0	0
$621$	4.37626	0.175613
$622$	0	0
$623$	−6.90660	−0.276707
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−72.1877	−2.88290
$628$	0	0
$629$	13.6444	0.544036
$630$	0	0
$631$	−27.4554	−1.09298	−0.546490	−	0.837466i	$-0.684036\pi$
$631$	−27.4554	−1.09298	−0.546490	+	0.837466i	$0.684036\pi$
$632$	0	0
$633$	44.6917	1.77634
$634$	0	0
$635$	−1.22953	−0.0487924
$636$	0	0
$637$	6.00935	0.238099
$638$	0	0
$639$	10.0006	0.395616
$640$	0	0
$641$	−48.9270	−1.93250	−0.966250	−	0.257605i	$-0.917067\pi$
$641$	−48.9270	−1.93250	−0.966250	+	0.257605i	$0.917067\pi$
$642$	0	0
$643$	−2.89333	−0.114102	−0.0570508	−	0.998371i	$-0.518170\pi$
$643$	−2.89333	−0.114102	−0.0570508	+	0.998371i	$0.518170\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−15.0877	−0.593157	−0.296578	−	0.955009i	$-0.595846\pi$
$647$	−15.0877	−0.593157	−0.296578	+	0.955009i	$0.595846\pi$
$648$	0	0
$649$	41.5621	1.63145
$650$	0	0
$651$	4.07122	0.159564
$652$	0	0
$653$	−13.3255	−0.521468	−0.260734	−	0.965411i	$-0.583965\pi$
$653$	−13.3255	−0.521468	−0.260734	+	0.965411i	$0.583965\pi$
$654$	0	0
$655$	1.97345	0.0771090
$656$	0	0
$657$	9.15292	0.357089
$658$	0	0
$659$	39.9000	1.55428	0.777142	−	0.629325i	$-0.216669\pi$
$659$	39.9000	1.55428	0.777142	+	0.629325i	$0.216669\pi$
$660$	0	0
$661$	−37.3060	−1.45104	−0.725518	−	0.688203i	$-0.758400\pi$
$661$	−37.3060	−1.45104	−0.725518	+	0.688203i	$0.758400\pi$
$662$	0	0
$663$	15.9149	0.618082
$664$	0	0
$665$	−16.6821	−0.646903
$666$	0	0
$667$	−0.118188	−0.00457625
$668$	0	0
$669$	2.91593	0.112736
$670$	0	0
$671$	14.9627	0.577630
$672$	0	0
$673$	−18.6641	−0.719447	−0.359724	−	0.933059i	$-0.617129\pi$
$673$	−18.6641	−0.719447	−0.359724	+	0.933059i	$0.617129\pi$
$674$	0	0
$675$	4.37626	0.168442
$676$	0	0
$677$	19.4978	0.749361	0.374681	−	0.927154i	$-0.377752\pi$
$677$	19.4978	0.749361	0.374681	+	0.927154i	$0.377752\pi$
$678$	0	0
$679$	4.71169	0.180818
$680$	0	0
$681$	−42.5313	−1.62980
$682$	0	0
$683$	41.6331	1.59305	0.796523	−	0.604608i	$-0.206670\pi$
$683$	41.6331	1.59305	0.796523	+	0.604608i	$0.206670\pi$
$684$	0	0
$685$	−13.7671	−0.526012
$686$	0	0
$687$	21.8276	0.832773
$688$	0	0
$689$	42.7510	1.62868
$690$	0	0
$691$	9.53192	0.362611	0.181306	−	0.983427i	$-0.441968\pi$
$691$	9.53192	0.362611	0.181306	+	0.983427i	$0.441968\pi$
$692$	0	0
$693$	9.34998	0.355177
$694$	0	0
$695$	5.30280	0.201147
$696$	0	0
$697$	−2.63461	−0.0997930
$698$	0	0
$699$	34.1702	1.29244
$700$	0	0
$701$	27.9897	1.05716	0.528579	−	0.848884i	$-0.322725\pi$
$701$	27.9897	1.05716	0.528579	+	0.848884i	$0.322725\pi$
$702$	0	0
$703$	−52.6508	−1.98576
$704$	0	0
$705$	−20.1766	−0.759896
$706$	0	0
$707$	29.2643	1.10060
$708$	0	0
$709$	27.6832	1.03966	0.519831	−	0.854269i	$-0.325995\pi$
$709$	27.6832	1.03966	0.519831	+	0.854269i	$0.325995\pi$
$710$	0	0
$711$	−5.00489	−0.187698
$712$	0	0
$713$	0.884147	0.0331115
$714$	0	0
$715$	−24.2017	−0.905091
$716$	0	0
$717$	−48.8200	−1.82322
$718$	0	0
$719$	−50.8364	−1.89588	−0.947938	−	0.318454i	$-0.896836\pi$
$719$	−50.8364	−1.89588	−0.947938	+	0.318454i	$0.896836\pi$
$720$	0	0
$721$	−29.7940	−1.10959
$722$	0	0
$723$	−1.52812	−0.0568316
$724$	0	0
$725$	−0.118188	−0.00438939
$726$	0	0
$727$	−37.5061	−1.39102	−0.695512	−	0.718514i	$-0.744823\pi$
$727$	−37.5061	−1.39102	−0.695512	+	0.718514i	$0.744823\pi$
$728$	0	0
$729$	17.5273	0.649158
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−36.9374	−1.36431	−0.682157	−	0.731206i	$-0.738958\pi$
$733$	−36.9374	−1.36431	−0.682157	+	0.731206i	$0.738958\pi$
$734$	0	0
$735$	−2.55978	−0.0944188
$736$	0	0
$737$	−16.5949	−0.611281
$738$	0	0
$739$	12.3287	0.453517	0.226759	−	0.973951i	$-0.427187\pi$
$739$	12.3287	0.453517	0.226759	+	0.973951i	$0.427187\pi$
$740$	0	0
$741$	−61.4123	−2.25604
$742$	0	0
$743$	29.7534	1.09154	0.545772	−	0.837933i	$-0.316236\pi$
$743$	29.7534	1.09154	0.545772	+	0.837933i	$0.316236\pi$
$744$	0	0
$745$	−7.73256	−0.283299
$746$	0	0
$747$	9.94770	0.363967
$748$	0	0
$749$	−41.4166	−1.51333
$750$	0	0
$751$	41.9834	1.53200	0.765999	−	0.642842i	$-0.222245\pi$
$751$	41.9834	1.53200	0.765999	+	0.642842i	$0.222245\pi$
$752$	0	0
$753$	23.8628	0.869610
$754$	0	0
$755$	21.4156	0.779392
$756$	0	0
$757$	11.4708	0.416915	0.208458	−	0.978031i	$-0.433156\pi$
$757$	11.4708	0.416915	0.208458	+	0.978031i	$0.433156\pi$
$758$	0	0
$759$	10.3091	0.374196
$760$	0	0
$761$	−13.4122	−0.486193	−0.243097	−	0.970002i	$-0.578163\pi$
$761$	−13.4122	−0.486193	−0.243097	+	0.970002i	$0.578163\pi$
$762$	0	0
$763$	−10.2658	−0.371646
$764$	0	0
$765$	−1.33527	−0.0482767
$766$	0	0
$767$	35.3581	1.27671
$768$	0	0
$769$	2.18327	0.0787307	0.0393654	−	0.999225i	$-0.487466\pi$
$769$	2.18327	0.0787307	0.0393654	+	0.999225i	$0.487466\pi$
$770$	0	0
$771$	−12.3787	−0.445806
$772$	0	0
$773$	5.73306	0.206204	0.103102	−	0.994671i	$-0.467123\pi$
$773$	5.73306	0.206204	0.103102	+	0.994671i	$0.467123\pi$
$774$	0	0
$775$	0.884147	0.0317595
$776$	0	0
$777$	34.6228	1.24209
$778$	0	0
$779$	10.1664	0.364250
$780$	0	0
$781$	−72.4893	−2.59387
$782$	0	0
$783$	−0.517220	−0.0184839
$784$	0	0
$785$	4.04738	0.144457
$786$	0	0
$787$	8.13502	0.289982	0.144991	−	0.989433i	$-0.453685\pi$
$787$	8.13502	0.289982	0.144991	+	0.989433i	$0.453685\pi$
$788$	0	0
$789$	38.0601	1.35497
$790$	0	0
$791$	28.9442	1.02914
$792$	0	0
$793$	12.7293	0.452029
$794$	0	0
$795$	−18.2105	−0.645859
$796$	0	0
$797$	−41.5139	−1.47050	−0.735250	−	0.677797i	$-0.762935\pi$
$797$	−41.5139	−1.47050	−0.735250	+	0.677797i	$0.762935\pi$
$798$	0	0
$799$	−18.9429	−0.670151
$800$	0	0
$801$	2.13321	0.0753733
$802$	0	0
$803$	−66.3451	−2.34127
$804$	0	0
$805$	2.38236	0.0839671
$806$	0	0
$807$	6.61982	0.233029
$808$	0	0
$809$	−22.8566	−0.803596	−0.401798	−	0.915728i	$-0.631615\pi$
$809$	−22.8566	−0.803596	−0.401798	+	0.915728i	$0.631615\pi$
$810$	0	0
$811$	42.1917	1.48155	0.740776	−	0.671753i	$-0.234458\pi$
$811$	42.1917	1.48155	0.740776	+	0.671753i	$0.234458\pi$
$812$	0	0
$813$	26.5178	0.930020
$814$	0	0
$815$	9.04725	0.316911
$816$	0	0
$817$	0	0
$818$	0	0
$819$	7.95432	0.277946
$820$	0	0
$821$	27.0195	0.942985	0.471493	−	0.881870i	$-0.343715\pi$
$821$	27.0195	0.942985	0.471493	+	0.881870i	$0.343715\pi$
$822$	0	0
$823$	−27.2435	−0.949650	−0.474825	−	0.880080i	$-0.657489\pi$
$823$	−27.2435	−0.949650	−0.474825	+	0.880080i	$0.657489\pi$
$824$	0	0
$825$	10.3091	0.358916
$826$	0	0
$827$	−10.5732	−0.367667	−0.183834	−	0.982957i	$-0.558851\pi$
$827$	−10.5732	−0.367667	−0.183834	+	0.982957i	$0.558851\pi$
$828$	0	0
$829$	−1.16445	−0.0404429	−0.0202215	−	0.999796i	$-0.506437\pi$
$829$	−1.16445	−0.0404429	−0.0202215	+	0.999796i	$0.506437\pi$
$830$	0	0
$831$	−1.22953	−0.0426519
$832$	0	0
$833$	−2.40325	−0.0832678
$834$	0	0
$835$	−12.5467	−0.434197
$836$	0	0
$837$	3.86925	0.133741
$838$	0	0
$839$	20.2920	0.700556	0.350278	−	0.936646i	$-0.386087\pi$
$839$	20.2920	0.700556	0.350278	+	0.936646i	$0.386087\pi$
$840$	0	0
$841$	−28.9860	−0.999518
$842$	0	0
$843$	−35.5880	−1.22571
$844$	0	0
$845$	−7.58910	−0.261073
$846$	0	0
$847$	−41.5676	−1.42828
$848$	0	0
$849$	−15.3801	−0.527843
$850$	0	0
$851$	7.51903	0.257749
$852$	0	0
$853$	26.6154	0.911295	0.455648	−	0.890160i	$-0.349408\pi$
$853$	26.6154	0.911295	0.455648	+	0.890160i	$0.349408\pi$
$854$	0	0
$855$	5.15252	0.176213
$856$	0	0
$857$	−10.4342	−0.356427	−0.178214	−	0.983992i	$-0.557032\pi$
$857$	−10.4342	−0.356427	−0.178214	+	0.983992i	$0.557032\pi$
$858$	0	0
$859$	24.6623	0.841466	0.420733	−	0.907185i	$-0.361773\pi$
$859$	24.6623	0.841466	0.420733	+	0.907185i	$0.361773\pi$
$860$	0	0
$861$	−6.68538	−0.227837
$862$	0	0
$863$	−8.48317	−0.288771	−0.144385	−	0.989522i	$-0.546120\pi$
$863$	−8.48317	−0.288771	−0.144385	+	0.989522i	$0.546120\pi$
$864$	0	0
$865$	14.1354	0.480617
$866$	0	0
$867$	26.4934	0.899764
$868$	0	0
$869$	36.2780	1.23065
$870$	0	0
$871$	−14.1178	−0.478363
$872$	0	0
$873$	−1.45528	−0.0492538
$874$	0	0
$875$	2.38236	0.0805384
$876$	0	0
$877$	3.08796	0.104273	0.0521365	−	0.998640i	$-0.483397\pi$
$877$	3.08796	0.104273	0.0521365	+	0.998640i	$0.483397\pi$
$878$	0	0
$879$	40.4282	1.36361
$880$	0	0
$881$	42.5879	1.43482	0.717412	−	0.696649i	$-0.245327\pi$
$881$	42.5879	1.43482	0.717412	+	0.696649i	$0.245327\pi$
$882$	0	0
$883$	48.3342	1.62657	0.813287	−	0.581863i	$-0.197676\pi$
$883$	48.3342	1.62657	0.813287	+	0.581863i	$0.197676\pi$
$884$	0	0
$885$	−15.0614	−0.506282
$886$	0	0
$887$	2.12173	0.0712408	0.0356204	−	0.999365i	$-0.488659\pi$
$887$	2.12173	0.0712408	0.0356204	+	0.999365i	$0.488659\pi$
$888$	0	0
$889$	−2.92918	−0.0982415
$890$	0	0
$891$	56.8892	1.90586
$892$	0	0
$893$	73.0968	2.44609
$894$	0	0
$895$	−26.2442	−0.877246
$896$	0	0
$897$	8.77026	0.292830
$898$	0	0
$899$	−0.104495	−0.00348512
$900$	0	0
$901$	−17.0970	−0.569582
$902$	0	0
$903$	0	0
$904$	0	0
$905$	18.3461	0.609844
$906$	0	0
$907$	−28.6914	−0.952684	−0.476342	−	0.879260i	$-0.658038\pi$
$907$	−28.6914	−0.952684	−0.476342	+	0.879260i	$0.658038\pi$
$908$	0	0
$909$	−9.03875	−0.299796
$910$	0	0
$911$	1.14899	0.0380679	0.0190339	−	0.999819i	$-0.493941\pi$
$911$	1.14899	0.0380679	0.0190339	+	0.999819i	$0.493941\pi$
$912$	0	0
$913$	−72.1061	−2.38636
$914$	0	0
$915$	−5.42223	−0.179253
$916$	0	0
$917$	4.70146	0.155256
$918$	0	0
$919$	55.4798	1.83011	0.915055	−	0.403329i	$-0.132147\pi$
$919$	55.4798	1.83011	0.915055	+	0.403329i	$0.132147\pi$
$920$	0	0
$921$	−11.1873	−0.368636
$922$	0	0
$923$	−61.6689	−2.02986
$924$	0	0
$925$	7.51903	0.247224
$926$	0	0
$927$	9.20235	0.302245
$928$	0	0
$929$	−4.05780	−0.133132	−0.0665660	−	0.997782i	$-0.521204\pi$
$929$	−4.05780	−0.133132	−0.0665660	+	0.997782i	$0.521204\pi$
$930$	0	0
$931$	9.27367	0.303932
$932$	0	0
$933$	35.5011	1.16226
$934$	0	0
$935$	9.67871	0.316528
$936$	0	0
$937$	16.0588	0.524617	0.262308	−	0.964984i	$-0.415516\pi$
$937$	16.0588	0.524617	0.262308	+	0.964984i	$0.415516\pi$
$938$	0	0
$939$	−19.0552	−0.621842
$940$	0	0
$941$	4.73779	0.154448	0.0772238	−	0.997014i	$-0.475394\pi$
$941$	4.73779	0.154448	0.0772238	+	0.997014i	$0.475394\pi$
$942$	0	0
$943$	−1.45186	−0.0472792
$944$	0	0
$945$	10.4258	0.339152
$946$	0	0
$947$	3.38634	0.110041	0.0550207	−	0.998485i	$-0.482478\pi$
$947$	3.38634	0.110041	0.0550207	+	0.998485i	$0.482478\pi$
$948$	0	0
$949$	−56.4418	−1.83218
$950$	0	0
$951$	63.6844	2.06511
$952$	0	0
$953$	29.2967	0.949013	0.474507	−	0.880252i	$-0.342627\pi$
$953$	29.2967	0.949013	0.474507	+	0.880252i	$0.342627\pi$
$954$	0	0
$955$	−3.86566	−0.125090
$956$	0	0
$957$	−1.21841	−0.0393856
$958$	0	0
$959$	−32.7981	−1.05910
$960$	0	0
$961$	−30.2183	−0.974783
$962$	0	0
$963$	12.7922	0.412222
$964$	0	0
$965$	23.8599	0.768078
$966$	0	0
$967$	15.5743	0.500837	0.250419	−	0.968138i	$-0.419432\pi$
$967$	15.5743	0.500837	0.250419	+	0.968138i	$0.419432\pi$
$968$	0	0
$969$	24.5599	0.788979
$970$	0	0
$971$	−51.5951	−1.65577	−0.827883	−	0.560900i	$-0.810455\pi$
$971$	−51.5951	−1.65577	−0.827883	+	0.560900i	$0.810455\pi$
$972$	0	0
$973$	12.6332	0.405001
$974$	0	0
$975$	8.77026	0.280873
$976$	0	0
$977$	9.04984	0.289530	0.144765	−	0.989466i	$-0.453757\pi$
$977$	9.04984	0.289530	0.144765	+	0.989466i	$0.453757\pi$
$978$	0	0
$979$	−15.4626	−0.494188
$980$	0	0
$981$	3.17075	0.101234
$982$	0	0
$983$	39.3855	1.25620	0.628101	−	0.778132i	$-0.283833\pi$
$983$	39.3855	1.25620	0.628101	+	0.778132i	$0.283833\pi$
$984$	0	0
$985$	−11.7356	−0.373928
$986$	0	0
$987$	−48.0680	−1.53002
$988$	0	0
$989$	0	0
$990$	0	0
$991$	18.9152	0.600860	0.300430	−	0.953804i	$-0.402870\pi$
$991$	18.9152	0.600860	0.300430	+	0.953804i	$0.402870\pi$
$992$	0	0
$993$	−13.2458	−0.420344
$994$	0	0
$995$	−8.89906	−0.282119
$996$	0	0
$997$	−12.2966	−0.389438	−0.194719	−	0.980859i	$-0.562380\pi$
$997$	−12.2966	−0.389438	−0.194719	+	0.980859i	$0.562380\pi$
$998$	0	0
$999$	32.9052	1.04107

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.2.a.v.1.2		5
4.3	odd	2		920.2.a.j.1.4	✓	5
5.4	even	2		9200.2.a.cu.1.4		5
8.3	odd	2		7360.2.a.co.1.2		5
8.5	even	2		7360.2.a.cp.1.4		5
12.11	even	2		8280.2.a.bs.1.4		5
20.3	even	4		4600.2.e.u.4049.8		10
20.7	even	4		4600.2.e.u.4049.3		10
20.19	odd	2		4600.2.a.be.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.j.1.4	✓	5	4.3	odd	2
1840.2.a.v.1.2		5	1.1	even	1	trivial
4600.2.a.be.1.2		5	20.19	odd	2
4600.2.e.u.4049.3		10	20.7	even	4
4600.2.e.u.4049.8		10	20.3	even	4
7360.2.a.co.1.2		5	8.3	odd	2
7360.2.a.cp.1.4		5	8.5	even	2
8280.2.a.bs.1.4		5	12.11	even	2
9200.2.a.cu.1.4		5	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1840.a (trivial)

\(f(q)\)	\(=\)	\(q-1.93283 q^{3} -1.00000 q^{5} -2.38236 q^{7} +0.735829 q^{9} -5.33368 q^{11} -4.53752 q^{13} +1.93283 q^{15} +1.81464 q^{17} -7.00233 q^{19} +4.60469 q^{21} +1.00000 q^{23} +1.00000 q^{25} +4.37626 q^{27} -0.118188 q^{29} +0.884147 q^{31} +10.3091 q^{33} +2.38236 q^{35} +7.51903 q^{37} +8.77026 q^{39} -1.45186 q^{41} -0.735829 q^{45} -10.4389 q^{47} -1.32437 q^{49} -3.50739 q^{51} -9.42167 q^{53} +5.33368 q^{55} +13.5343 q^{57} -7.79239 q^{59} -2.80533 q^{61} -1.75301 q^{63} +4.53752 q^{65} +3.11134 q^{67} -1.93283 q^{69} +13.5909 q^{71} +12.4389 q^{73} -1.93283 q^{75} +12.7067 q^{77} -6.80169 q^{79} -10.6660 q^{81} +13.5190 q^{83} -1.81464 q^{85} +0.228437 q^{87} +2.89906 q^{89} +10.8100 q^{91} -1.70890 q^{93} +7.00233 q^{95} -1.97774 q^{97} -3.92468 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{5} + 2 q^{7} + 13 q^{9} + q^{11} + 4 q^{13} + 4 q^{17} - 7 q^{19} + 6 q^{21} + 5 q^{23} + 5 q^{25} - 3 q^{27} + 4 q^{29} - 19 q^{31} + 17 q^{33} - 2 q^{35} + 15 q^{37} - 19 q^{39} + 25 q^{41}+ \cdots + 65 q^{99}+O(q^{100}) \) 5 * q - 5 * q^5 + 2 * q^7 + 13 * q^9 + q^11 + 4 * q^13 + 4 * q^17 - 7 * q^19 + 6 * q^21 + 5 * q^23 + 5 * q^25 - 3 * q^27 + 4 * q^29 - 19 * q^31 + 17 * q^33 - 2 * q^35 + 15 * q^37 - 19 * q^39 + 25 * q^41 - 13 * q^45 + 11 * q^47 + 25 * q^49 - 19 * q^51 + 3 * q^53 - q^55 + 48 * q^57 + q^59 - 5 * q^61 + 41 * q^63 - 4 * q^65 - 9 * q^67 - q^71 - q^73 + 18 * q^77 + 2 * q^79 + 57 * q^81 + 45 * q^83 - 4 * q^85 + 9 * q^87 + 6 * q^89 - 11 * q^91 - 39 * q^93 + 7 * q^95 + 25 * q^97 + 65 * q^99

Introduction

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