LMFDB - Embedded newform 2752.2.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2752,2,Mod(1,2752)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2752, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2752.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2752 = 2^{6} \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2752.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$21.9748306363$
Analytic rank:	$1$
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 43)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	2752.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{3} -0.585786 q^{5} +0.585786 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.41421 q^{3} -0.585786 q^{5} +0.585786 q^{7} -1.00000 q^{9} +1.82843 q^{11} -3.82843 q^{13} +0.828427 q^{15} +7.82843 q^{17} -4.82843 q^{19} -0.828427 q^{21} +4.65685 q^{23} -4.65685 q^{25} +5.65685 q^{27} -4.24264 q^{29} +3.00000 q^{31} -2.58579 q^{33} -0.343146 q^{35} +8.48528 q^{37} +5.41421 q^{39} -3.82843 q^{41} +1.00000 q^{43} +0.585786 q^{45} -6.00000 q^{47} -6.65685 q^{49} -11.0711 q^{51} -8.17157 q^{53} -1.07107 q^{55} +6.82843 q^{57} +0.828427 q^{59} -8.24264 q^{61} -0.585786 q^{63} +2.24264 q^{65} +9.48528 q^{67} -6.58579 q^{69} +8.82843 q^{71} -7.75736 q^{73} +6.58579 q^{75} +1.07107 q^{77} +0.828427 q^{79} -5.00000 q^{81} +14.6569 q^{83} -4.58579 q^{85} +6.00000 q^{87} -10.2426 q^{89} -2.24264 q^{91} -4.24264 q^{93} +2.82843 q^{95} -3.82843 q^{97} -1.82843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5} + 4 q^{7} - 2 q^{9}+O(q^{10}) $ 2 * q - 4 * q^5 + 4 * q^7 - 2 * q^9	$ 2 q - 4 q^{5} + 4 q^{7} - 2 q^{9} - 2 q^{11} - 2 q^{13} - 4 q^{15} + 10 q^{17} - 4 q^{19} + 4 q^{21} - 2 q^{23} + 2 q^{25} + 6 q^{31} - 8 q^{33} - 12 q^{35} + 8 q^{39} - 2 q^{41} + 2 q^{43} + 4 q^{45} - 12 q^{47} - 2 q^{49} - 8 q^{51} - 22 q^{53} + 12 q^{55} + 8 q^{57} - 4 q^{59} - 8 q^{61} - 4 q^{63} - 4 q^{65} + 2 q^{67} - 16 q^{69} + 12 q^{71} - 24 q^{73} + 16 q^{75} - 12 q^{77} - 4 q^{79} - 10 q^{81} + 18 q^{83} - 12 q^{85} + 12 q^{87} - 12 q^{89} + 4 q^{91} - 2 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q - 4 * q^5 + 4 * q^7 - 2 * q^9 - 2 * q^11 - 2 * q^13 - 4 * q^15 + 10 * q^17 - 4 * q^19 + 4 * q^21 - 2 * q^23 + 2 * q^25 + 6 * q^31 - 8 * q^33 - 12 * q^35 + 8 * q^39 - 2 * q^41 + 2 * q^43 + 4 * q^45 - 12 * q^47 - 2 * q^49 - 8 * q^51 - 22 * q^53 + 12 * q^55 + 8 * q^57 - 4 * q^59 - 8 * q^61 - 4 * q^63 - 4 * q^65 + 2 * q^67 - 16 * q^69 + 12 * q^71 - 24 * q^73 + 16 * q^75 - 12 * q^77 - 4 * q^79 - 10 * q^81 + 18 * q^83 - 12 * q^85 + 12 * q^87 - 12 * q^89 + 4 * q^91 - 2 * q^97 + 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.41421	−0.816497	−0.408248	−	0.912871i	$-0.633860\pi$
$3$	−1.41421	−0.816497	−0.408248	+	0.912871i	$0.633860\pi$
$4$	0	0
$5$	−0.585786	−0.261972	−0.130986	−	0.991384i	$-0.541814\pi$
$5$	−0.585786	−0.261972	−0.130986	+	0.991384i	$0.541814\pi$
$6$	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
$9$	−1.00000	−0.333333
$10$	0	0
$11$	1.82843	0.551292	0.275646	−	0.961259i	$-0.411108\pi$
$11$	1.82843	0.551292	0.275646	+	0.961259i	$0.411108\pi$
$12$	0	0
$13$	−3.82843	−1.06181	−0.530907	−	0.847430i	$-0.678149\pi$
$13$	−3.82843	−1.06181	−0.530907	+	0.847430i	$0.678149\pi$
$14$	0	0
$15$	0.828427	0.213899
$16$	0	0
$17$	7.82843	1.89867	0.949336	−	0.314262i	$-0.101757\pi$
$17$	7.82843	1.89867	0.949336	+	0.314262i	$0.101757\pi$
$18$	0	0
$19$	−4.82843	−1.10772	−0.553859	−	0.832611i	$-0.686845\pi$
$19$	−4.82843	−1.10772	−0.553859	+	0.832611i	$0.686845\pi$
$20$	0	0
$21$	−0.828427	−0.180778
$22$	0	0
$23$	4.65685	0.971021	0.485511	−	0.874231i	$-0.338634\pi$
$23$	4.65685	0.971021	0.485511	+	0.874231i	$0.338634\pi$
$24$	0	0
$25$	−4.65685	−0.931371
$26$	0	0
$27$	5.65685	1.08866
$28$	0	0
$29$	−4.24264	−0.787839	−0.393919	−	0.919145i	$-0.628881\pi$
$29$	−4.24264	−0.787839	−0.393919	+	0.919145i	$0.628881\pi$
$30$	0	0
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	0	0
$33$	−2.58579	−0.450128
$34$	0	0
$35$	−0.343146	−0.0580022
$36$	0	0
$37$	8.48528	1.39497	0.697486	−	0.716599i	$-0.254302\pi$
$37$	8.48528	1.39497	0.697486	+	0.716599i	$0.254302\pi$
$38$	0	0
$39$	5.41421	0.866968
$40$	0	0
$41$	−3.82843	−0.597900	−0.298950	−	0.954269i	$-0.596636\pi$
$41$	−3.82843	−0.597900	−0.298950	+	0.954269i	$0.596636\pi$
$42$	0	0
$43$	1.00000	0.152499
$43$	1.00000	0.152499
$44$	0	0
$45$	0.585786	0.0873239
$46$	0	0
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	−6.65685	−0.950979
$50$	0	0
$51$	−11.0711	−1.55026
$52$	0	0
$53$	−8.17157	−1.12245	−0.561226	−	0.827663i	$-0.689670\pi$
$53$	−8.17157	−1.12245	−0.561226	+	0.827663i	$0.689670\pi$
$54$	0	0
$55$	−1.07107	−0.144423
$56$	0	0
$57$	6.82843	0.904447
$58$	0	0
$59$	0.828427	0.107852	0.0539260	−	0.998545i	$-0.482826\pi$
$59$	0.828427	0.107852	0.0539260	+	0.998545i	$0.482826\pi$
$60$	0	0
$61$	−8.24264	−1.05536	−0.527681	−	0.849443i	$-0.676938\pi$
$61$	−8.24264	−1.05536	−0.527681	+	0.849443i	$0.676938\pi$
$62$	0	0
$63$	−0.585786	−0.0738022
$64$	0	0
$65$	2.24264	0.278165
$66$	0	0
$67$	9.48528	1.15881	0.579406	−	0.815039i	$-0.303285\pi$
$67$	9.48528	1.15881	0.579406	+	0.815039i	$0.303285\pi$
$68$	0	0
$69$	−6.58579	−0.792836
$70$	0	0
$71$	8.82843	1.04774	0.523871	−	0.851798i	$-0.324487\pi$
$71$	8.82843	1.04774	0.523871	+	0.851798i	$0.324487\pi$
$72$	0	0
$73$	−7.75736	−0.907930	−0.453965	−	0.891019i	$-0.649991\pi$
$73$	−7.75736	−0.907930	−0.453965	+	0.891019i	$0.649991\pi$
$74$	0	0
$75$	6.58579	0.760461
$76$	0	0
$77$	1.07107	0.122060
$78$	0	0
$79$	0.828427	0.0932053	0.0466027	−	0.998914i	$-0.485161\pi$
$79$	0.828427	0.0932053	0.0466027	+	0.998914i	$0.485161\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	14.6569	1.60880	0.804399	−	0.594089i	$-0.202487\pi$
$83$	14.6569	1.60880	0.804399	+	0.594089i	$0.202487\pi$
$84$	0	0
$85$	−4.58579	−0.497398
$86$	0	0
$87$	6.00000	0.643268
$88$	0	0
$89$	−10.2426	−1.08572	−0.542859	−	0.839824i	$-0.682658\pi$
$89$	−10.2426	−1.08572	−0.542859	+	0.839824i	$0.682658\pi$
$90$	0	0
$91$	−2.24264	−0.235093
$92$	0	0
$93$	−4.24264	−0.439941
$94$	0	0
$95$	2.82843	0.290191
$96$	0	0
$97$	−3.82843	−0.388718	−0.194359	−	0.980930i	$-0.562263\pi$
$97$	−3.82843	−0.388718	−0.194359	+	0.980930i	$0.562263\pi$
$98$	0	0
$99$	−1.82843	−0.183764
$100$	0	0
$101$	−0.171573	−0.0170721	−0.00853607	−	0.999964i	$-0.502717\pi$
$101$	−0.171573	−0.0170721	−0.00853607	+	0.999964i	$0.502717\pi$
$102$	0	0
$103$	−17.4853	−1.72288	−0.861438	−	0.507863i	$-0.830436\pi$
$103$	−17.4853	−1.72288	−0.861438	+	0.507863i	$0.830436\pi$
$104$	0	0
$105$	0.485281	0.0473586
$106$	0	0
$107$	−11.6569	−1.12691	−0.563455	−	0.826147i	$-0.690528\pi$
$107$	−11.6569	−1.12691	−0.563455	+	0.826147i	$0.690528\pi$
$108$	0	0
$109$	−13.9706	−1.33814	−0.669069	−	0.743201i	$-0.733307\pi$
$109$	−13.9706	−1.33814	−0.669069	+	0.743201i	$0.733307\pi$
$110$	0	0
$111$	−12.0000	−1.13899
$112$	0	0
$113$	−1.17157	−0.110212	−0.0551062	−	0.998481i	$-0.517550\pi$
$113$	−1.17157	−0.110212	−0.0551062	+	0.998481i	$0.517550\pi$
$114$	0	0
$115$	−2.72792	−0.254380
$116$	0	0
$117$	3.82843	0.353938
$118$	0	0
$119$	4.58579	0.420378
$120$	0	0
$121$	−7.65685	−0.696078
$122$	0	0
$123$	5.41421	0.488183
$124$	0	0
$125$	5.65685	0.505964
$126$	0	0
$127$	1.82843	0.162247	0.0811233	−	0.996704i	$-0.474149\pi$
$127$	1.82843	0.162247	0.0811233	+	0.996704i	$0.474149\pi$
$128$	0	0
$129$	−1.41421	−0.124515
$130$	0	0
$131$	−1.65685	−0.144760	−0.0723800	−	0.997377i	$-0.523059\pi$
$131$	−1.65685	−0.144760	−0.0723800	+	0.997377i	$0.523059\pi$
$132$	0	0
$133$	−2.82843	−0.245256
$134$	0	0
$135$	−3.31371	−0.285199
$136$	0	0
$137$	2.48528	0.212332	0.106166	−	0.994348i	$-0.466143\pi$
$137$	2.48528	0.212332	0.106166	+	0.994348i	$0.466143\pi$
$138$	0	0
$139$	−11.4853	−0.974169	−0.487084	−	0.873355i	$-0.661940\pi$
$139$	−11.4853	−0.974169	−0.487084	+	0.873355i	$0.661940\pi$
$140$	0	0
$141$	8.48528	0.714590
$142$	0	0
$143$	−7.00000	−0.585369
$144$	0	0
$145$	2.48528	0.206391
$146$	0	0
$147$	9.41421	0.776471
$148$	0	0
$149$	8.48528	0.695141	0.347571	−	0.937654i	$-0.387007\pi$
$149$	8.48528	0.695141	0.347571	+	0.937654i	$0.387007\pi$
$150$	0	0
$151$	−9.75736	−0.794043	−0.397021	−	0.917809i	$-0.629956\pi$
$151$	−9.75736	−0.794043	−0.397021	+	0.917809i	$0.629956\pi$
$152$	0	0
$153$	−7.82843	−0.632891
$154$	0	0
$155$	−1.75736	−0.141154
$156$	0	0
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	0	0
$159$	11.5563	0.916478
$160$	0	0
$161$	2.72792	0.214990
$162$	0	0
$163$	−20.2426	−1.58553	−0.792763	−	0.609530i	$-0.791358\pi$
$163$	−20.2426	−1.58553	−0.792763	+	0.609530i	$0.791358\pi$
$164$	0	0
$165$	1.51472	0.117921
$166$	0	0
$167$	−8.31371	−0.643334	−0.321667	−	0.946853i	$-0.604243\pi$
$167$	−8.31371	−0.643334	−0.321667	+	0.946853i	$0.604243\pi$
$168$	0	0
$169$	1.65685	0.127450
$170$	0	0
$171$	4.82843	0.369239
$172$	0	0
$173$	−12.3431	−0.938432	−0.469216	−	0.883083i	$-0.655463\pi$
$173$	−12.3431	−0.938432	−0.469216	+	0.883083i	$0.655463\pi$
$174$	0	0
$175$	−2.72792	−0.206212
$176$	0	0
$177$	−1.17157	−0.0880608
$178$	0	0
$179$	−7.41421	−0.554164	−0.277082	−	0.960846i	$-0.589367\pi$
$179$	−7.41421	−0.554164	−0.277082	+	0.960846i	$0.589367\pi$
$180$	0	0
$181$	15.3137	1.13826	0.569129	−	0.822248i	$-0.307280\pi$
$181$	15.3137	1.13826	0.569129	+	0.822248i	$0.307280\pi$
$182$	0	0
$183$	11.6569	0.861699
$184$	0	0
$185$	−4.97056	−0.365443
$186$	0	0
$187$	14.3137	1.04672
$188$	0	0
$189$	3.31371	0.241037
$190$	0	0
$191$	−22.1421	−1.60215	−0.801074	−	0.598565i	$-0.795738\pi$
$191$	−22.1421	−1.60215	−0.801074	+	0.598565i	$0.795738\pi$
$192$	0	0
$193$	−17.9706	−1.29355	−0.646775	−	0.762681i	$-0.723883\pi$
$193$	−17.9706	−1.29355	−0.646775	+	0.762681i	$0.723883\pi$
$194$	0	0
$195$	−3.17157	−0.227121
$196$	0	0
$197$	−14.1421	−1.00759	−0.503793	−	0.863825i	$-0.668062\pi$
$197$	−14.1421	−1.00759	−0.503793	+	0.863825i	$0.668062\pi$
$198$	0	0
$199$	3.65685	0.259228	0.129614	−	0.991565i	$-0.458626\pi$
$199$	3.65685	0.259228	0.129614	+	0.991565i	$0.458626\pi$
$200$	0	0
$201$	−13.4142	−0.946166
$202$	0	0
$203$	−2.48528	−0.174433
$204$	0	0
$205$	2.24264	0.156633
$206$	0	0
$207$	−4.65685	−0.323674
$208$	0	0
$209$	−8.82843	−0.610675
$210$	0	0
$211$	28.1421	1.93738	0.968692	−	0.248265i	$-0.0798602\pi$
$211$	28.1421	1.93738	0.968692	+	0.248265i	$0.0798602\pi$
$212$	0	0
$213$	−12.4853	−0.855477
$214$	0	0
$215$	−0.585786	−0.0399503
$216$	0	0
$217$	1.75736	0.119297
$218$	0	0
$219$	10.9706	0.741322
$220$	0	0
$221$	−29.9706	−2.01604
$222$	0	0
$223$	−23.8995	−1.60043	−0.800214	−	0.599714i	$-0.795281\pi$
$223$	−23.8995	−1.60043	−0.800214	+	0.599714i	$0.795281\pi$
$224$	0	0
$225$	4.65685	0.310457
$226$	0	0
$227$	−17.6569	−1.17193	−0.585963	−	0.810338i	$-0.699284\pi$
$227$	−17.6569	−1.17193	−0.585963	+	0.810338i	$0.699284\pi$
$228$	0	0
$229$	−7.97056	−0.526710	−0.263355	−	0.964699i	$-0.584829\pi$
$229$	−7.97056	−0.526710	−0.263355	+	0.964699i	$0.584829\pi$
$230$	0	0
$231$	−1.51472	−0.0996612
$232$	0	0
$233$	−0.828427	−0.0542721	−0.0271360	−	0.999632i	$-0.508639\pi$
$233$	−0.828427	−0.0542721	−0.0271360	+	0.999632i	$0.508639\pi$
$234$	0	0
$235$	3.51472	0.229275
$236$	0	0
$237$	−1.17157	−0.0761018
$238$	0	0
$239$	2.48528	0.160759	0.0803797	−	0.996764i	$-0.474387\pi$
$239$	2.48528	0.160759	0.0803797	+	0.996764i	$0.474387\pi$
$240$	0	0
$241$	4.00000	0.257663	0.128831	−	0.991667i	$-0.458877\pi$
$241$	4.00000	0.257663	0.128831	+	0.991667i	$0.458877\pi$
$242$	0	0
$243$	−9.89949	−0.635053
$244$	0	0
$245$	3.89949	0.249130
$246$	0	0
$247$	18.4853	1.17619
$248$	0	0
$249$	−20.7279	−1.31358
$250$	0	0
$251$	−5.14214	−0.324569	−0.162284	−	0.986744i	$-0.551886\pi$
$251$	−5.14214	−0.324569	−0.162284	+	0.986744i	$0.551886\pi$
$252$	0	0
$253$	8.51472	0.535316
$254$	0	0
$255$	6.48528	0.406124
$256$	0	0
$257$	−10.2426	−0.638918	−0.319459	−	0.947600i	$-0.603501\pi$
$257$	−10.2426	−0.638918	−0.319459	+	0.947600i	$0.603501\pi$
$258$	0	0
$259$	4.97056	0.308856
$260$	0	0
$261$	4.24264	0.262613
$262$	0	0
$263$	28.1421	1.73532	0.867659	−	0.497159i	$-0.165624\pi$
$263$	28.1421	1.73532	0.867659	+	0.497159i	$0.165624\pi$
$264$	0	0
$265$	4.78680	0.294051
$266$	0	0
$267$	14.4853	0.886485
$268$	0	0
$269$	3.00000	0.182913	0.0914566	−	0.995809i	$-0.470848\pi$
$269$	3.00000	0.182913	0.0914566	+	0.995809i	$0.470848\pi$
$270$	0	0
$271$	−7.97056	−0.484177	−0.242089	−	0.970254i	$-0.577832\pi$
$271$	−7.97056	−0.484177	−0.242089	+	0.970254i	$0.577832\pi$
$272$	0	0
$273$	3.17157	0.191952
$274$	0	0
$275$	−8.51472	−0.513457
$276$	0	0
$277$	11.8995	0.714971	0.357486	−	0.933919i	$-0.383634\pi$
$277$	11.8995	0.714971	0.357486	+	0.933919i	$0.383634\pi$
$278$	0	0
$279$	−3.00000	−0.179605
$280$	0	0
$281$	2.65685	0.158495	0.0792473	−	0.996855i	$-0.474748\pi$
$281$	2.65685	0.158495	0.0792473	+	0.996855i	$0.474748\pi$
$282$	0	0
$283$	−4.31371	−0.256423	−0.128212	−	0.991747i	$-0.540924\pi$
$283$	−4.31371	−0.256423	−0.128212	+	0.991747i	$0.540924\pi$
$284$	0	0
$285$	−4.00000	−0.236940
$286$	0	0
$287$	−2.24264	−0.132379
$288$	0	0
$289$	44.2843	2.60496
$290$	0	0
$291$	5.41421	0.317387
$292$	0	0
$293$	−21.6569	−1.26521	−0.632603	−	0.774476i	$-0.718014\pi$
$293$	−21.6569	−1.26521	−0.632603	+	0.774476i	$0.718014\pi$
$294$	0	0
$295$	−0.485281	−0.0282542
$296$	0	0
$297$	10.3431	0.600170
$298$	0	0
$299$	−17.8284	−1.03104
$300$	0	0
$301$	0.585786	0.0337642
$302$	0	0
$303$	0.242641	0.0139393
$304$	0	0
$305$	4.82843	0.276475
$306$	0	0
$307$	−12.7990	−0.730477	−0.365238	−	0.930914i	$-0.619013\pi$
$307$	−12.7990	−0.730477	−0.365238	+	0.930914i	$0.619013\pi$
$308$	0	0
$309$	24.7279	1.40672
$310$	0	0
$311$	−19.9706	−1.13243	−0.566213	−	0.824259i	$-0.691592\pi$
$311$	−19.9706	−1.13243	−0.566213	+	0.824259i	$0.691592\pi$
$312$	0	0
$313$	17.2132	0.972948	0.486474	−	0.873695i	$-0.338283\pi$
$313$	17.2132	0.972948	0.486474	+	0.873695i	$0.338283\pi$
$314$	0	0
$315$	0.343146	0.0193341
$316$	0	0
$317$	−20.1716	−1.13295	−0.566474	−	0.824079i	$-0.691693\pi$
$317$	−20.1716	−1.13295	−0.566474	+	0.824079i	$0.691693\pi$
$318$	0	0
$319$	−7.75736	−0.434329
$320$	0	0
$321$	16.4853	0.920119
$322$	0	0
$323$	−37.7990	−2.10319
$324$	0	0
$325$	17.8284	0.988943
$326$	0	0
$327$	19.7574	1.09258
$328$	0	0
$329$	−3.51472	−0.193773
$330$	0	0
$331$	18.5858	1.02157	0.510784	−	0.859709i	$-0.329355\pi$
$331$	18.5858	1.02157	0.510784	+	0.859709i	$0.329355\pi$
$332$	0	0
$333$	−8.48528	−0.464991
$334$	0	0
$335$	−5.55635	−0.303576
$336$	0	0
$337$	−5.00000	−0.272367	−0.136184	−	0.990684i	$-0.543484\pi$
$337$	−5.00000	−0.272367	−0.136184	+	0.990684i	$0.543484\pi$
$338$	0	0
$339$	1.65685	0.0899880
$340$	0	0
$341$	5.48528	0.297045
$342$	0	0
$343$	−8.00000	−0.431959
$344$	0	0
$345$	3.85786	0.207700
$346$	0	0
$347$	−7.41421	−0.398016	−0.199008	−	0.979998i	$-0.563772\pi$
$347$	−7.41421	−0.398016	−0.199008	+	0.979998i	$0.563772\pi$
$348$	0	0
$349$	11.7574	0.629357	0.314679	−	0.949198i	$-0.398103\pi$
$349$	11.7574	0.629357	0.314679	+	0.949198i	$0.398103\pi$
$350$	0	0
$351$	−21.6569	−1.15596
$352$	0	0
$353$	0.171573	0.00913190	0.00456595	−	0.999990i	$-0.498547\pi$
$353$	0.171573	0.00913190	0.00456595	+	0.999990i	$0.498547\pi$
$354$	0	0
$355$	−5.17157	−0.274479
$356$	0	0
$357$	−6.48528	−0.343237
$358$	0	0
$359$	32.6569	1.72356	0.861781	−	0.507280i	$-0.169349\pi$
$359$	32.6569	1.72356	0.861781	+	0.507280i	$0.169349\pi$
$360$	0	0
$361$	4.31371	0.227037
$362$	0	0
$363$	10.8284	0.568345
$364$	0	0
$365$	4.54416	0.237852
$366$	0	0
$367$	29.7990	1.55549	0.777747	−	0.628577i	$-0.216362\pi$
$367$	29.7990	1.55549	0.777747	+	0.628577i	$0.216362\pi$
$368$	0	0
$369$	3.82843	0.199300
$370$	0	0
$371$	−4.78680	−0.248518
$372$	0	0
$373$	8.48528	0.439351	0.219676	−	0.975573i	$-0.429500\pi$
$373$	8.48528	0.439351	0.219676	+	0.975573i	$0.429500\pi$
$374$	0	0
$375$	−8.00000	−0.413118
$376$	0	0
$377$	16.2426	0.836539
$378$	0	0
$379$	7.68629	0.394818	0.197409	−	0.980321i	$-0.436747\pi$
$379$	7.68629	0.394818	0.197409	+	0.980321i	$0.436747\pi$
$380$	0	0
$381$	−2.58579	−0.132474
$382$	0	0
$383$	3.51472	0.179594	0.0897969	−	0.995960i	$-0.471378\pi$
$383$	3.51472	0.179594	0.0897969	+	0.995960i	$0.471378\pi$
$384$	0	0
$385$	−0.627417	−0.0319761
$386$	0	0
$387$	−1.00000	−0.0508329
$388$	0	0
$389$	−28.6274	−1.45147	−0.725734	−	0.687976i	$-0.758500\pi$
$389$	−28.6274	−1.45147	−0.725734	+	0.687976i	$0.758500\pi$
$390$	0	0
$391$	36.4558	1.84365
$392$	0	0
$393$	2.34315	0.118196
$394$	0	0
$395$	−0.485281	−0.0244172
$396$	0	0
$397$	21.4558	1.07684	0.538419	−	0.842677i	$-0.319022\pi$
$397$	21.4558	1.07684	0.538419	+	0.842677i	$0.319022\pi$
$398$	0	0
$399$	4.00000	0.200250
$400$	0	0
$401$	−29.4853	−1.47242	−0.736212	−	0.676751i	$-0.763388\pi$
$401$	−29.4853	−1.47242	−0.736212	+	0.676751i	$0.763388\pi$
$402$	0	0
$403$	−11.4853	−0.572123
$404$	0	0
$405$	2.92893	0.145540
$406$	0	0
$407$	15.5147	0.769036
$408$	0	0
$409$	36.8701	1.82311	0.911554	−	0.411181i	$-0.134884\pi$
$409$	36.8701	1.82311	0.911554	+	0.411181i	$0.134884\pi$
$410$	0	0
$411$	−3.51472	−0.173368
$412$	0	0
$413$	0.485281	0.0238791
$414$	0	0
$415$	−8.58579	−0.421460
$416$	0	0
$417$	16.2426	0.795406
$418$	0	0
$419$	−4.10051	−0.200323	−0.100161	−	0.994971i	$-0.531936\pi$
$419$	−4.10051	−0.200323	−0.100161	+	0.994971i	$0.531936\pi$
$420$	0	0
$421$	4.34315	0.211672	0.105836	−	0.994384i	$-0.466248\pi$
$421$	4.34315	0.211672	0.105836	+	0.994384i	$0.466248\pi$
$422$	0	0
$423$	6.00000	0.291730
$424$	0	0
$425$	−36.4558	−1.76837
$426$	0	0
$427$	−4.82843	−0.233664
$428$	0	0
$429$	9.89949	0.477952
$430$	0	0
$431$	33.2843	1.60325	0.801623	−	0.597829i	$-0.203970\pi$
$431$	33.2843	1.60325	0.801623	+	0.597829i	$0.203970\pi$
$432$	0	0
$433$	30.2426	1.45337	0.726684	−	0.686972i	$-0.241060\pi$
$433$	30.2426	1.45337	0.726684	+	0.686972i	$0.241060\pi$
$434$	0	0
$435$	−3.51472	−0.168518
$436$	0	0
$437$	−22.4853	−1.07562
$438$	0	0
$439$	5.48528	0.261798	0.130899	−	0.991396i	$-0.458214\pi$
$439$	5.48528	0.261798	0.130899	+	0.991396i	$0.458214\pi$
$440$	0	0
$441$	6.65685	0.316993
$442$	0	0
$443$	36.1421	1.71716	0.858582	−	0.512676i	$-0.171346\pi$
$443$	36.1421	1.71716	0.858582	+	0.512676i	$0.171346\pi$
$444$	0	0
$445$	6.00000	0.284427
$446$	0	0
$447$	−12.0000	−0.567581
$448$	0	0
$449$	−9.21320	−0.434798	−0.217399	−	0.976083i	$-0.569757\pi$
$449$	−9.21320	−0.434798	−0.217399	+	0.976083i	$0.569757\pi$
$450$	0	0
$451$	−7.00000	−0.329617
$452$	0	0
$453$	13.7990	0.648333
$454$	0	0
$455$	1.31371	0.0615876
$456$	0	0
$457$	−24.7279	−1.15672	−0.578362	−	0.815780i	$-0.696308\pi$
$457$	−24.7279	−1.15672	−0.578362	+	0.815780i	$0.696308\pi$
$458$	0	0
$459$	44.2843	2.06701
$460$	0	0
$461$	2.62742	0.122371	0.0611855	−	0.998126i	$-0.480512\pi$
$461$	2.62742	0.122371	0.0611855	+	0.998126i	$0.480512\pi$
$462$	0	0
$463$	5.27208	0.245014	0.122507	−	0.992468i	$-0.460907\pi$
$463$	5.27208	0.245014	0.122507	+	0.992468i	$0.460907\pi$
$464$	0	0
$465$	2.48528	0.115252
$466$	0	0
$467$	12.3431	0.571173	0.285586	−	0.958353i	$-0.407812\pi$
$467$	12.3431	0.571173	0.285586	+	0.958353i	$0.407812\pi$
$468$	0	0
$469$	5.55635	0.256568
$470$	0	0
$471$	−14.1421	−0.651635
$472$	0	0
$473$	1.82843	0.0840712
$474$	0	0
$475$	22.4853	1.03170
$476$	0	0
$477$	8.17157	0.374151
$478$	0	0
$479$	−4.65685	−0.212777	−0.106389	−	0.994325i	$-0.533929\pi$
$479$	−4.65685	−0.212777	−0.106389	+	0.994325i	$0.533929\pi$
$480$	0	0
$481$	−32.4853	−1.48120
$482$	0	0
$483$	−3.85786	−0.175539
$484$	0	0
$485$	2.24264	0.101833
$486$	0	0
$487$	−33.6569	−1.52514	−0.762569	−	0.646907i	$-0.776062\pi$
$487$	−33.6569	−1.52514	−0.762569	+	0.646907i	$0.776062\pi$
$488$	0	0
$489$	28.6274	1.29458
$490$	0	0
$491$	10.5858	0.477730	0.238865	−	0.971053i	$-0.423225\pi$
$491$	10.5858	0.477730	0.238865	+	0.971053i	$0.423225\pi$
$492$	0	0
$493$	−33.2132	−1.49585
$494$	0	0
$495$	1.07107	0.0481409
$496$	0	0
$497$	5.17157	0.231977
$498$	0	0
$499$	−34.2426	−1.53291	−0.766456	−	0.642297i	$-0.777981\pi$
$499$	−34.2426	−1.53291	−0.766456	+	0.642297i	$0.777981\pi$
$500$	0	0
$501$	11.7574	0.525280
$502$	0	0
$503$	−42.7696	−1.90700	−0.953500	−	0.301393i	$-0.902548\pi$
$503$	−42.7696	−1.90700	−0.953500	+	0.301393i	$0.902548\pi$
$504$	0	0
$505$	0.100505	0.00447242
$506$	0	0
$507$	−2.34315	−0.104063
$508$	0	0
$509$	−17.4853	−0.775021	−0.387511	−	0.921865i	$-0.626665\pi$
$509$	−17.4853	−0.775021	−0.387511	+	0.921865i	$0.626665\pi$
$510$	0	0
$511$	−4.54416	−0.201022
$512$	0	0
$513$	−27.3137	−1.20593
$514$	0	0
$515$	10.2426	0.451345
$516$	0	0
$517$	−10.9706	−0.482485
$518$	0	0
$519$	17.4558	0.766227
$520$	0	0
$521$	31.0711	1.36125	0.680624	−	0.732633i	$-0.261709\pi$
$521$	31.0711	1.36125	0.680624	+	0.732633i	$0.261709\pi$
$522$	0	0
$523$	3.21320	0.140504	0.0702518	−	0.997529i	$-0.477620\pi$
$523$	3.21320	0.140504	0.0702518	+	0.997529i	$0.477620\pi$
$524$	0	0
$525$	3.85786	0.168371
$526$	0	0
$527$	23.4853	1.02303
$528$	0	0
$529$	−1.31371	−0.0571178
$530$	0	0
$531$	−0.828427	−0.0359507
$532$	0	0
$533$	14.6569	0.634859
$534$	0	0
$535$	6.82843	0.295219
$536$	0	0
$537$	10.4853	0.452473
$538$	0	0
$539$	−12.1716	−0.524267
$540$	0	0
$541$	30.7990	1.32415	0.662076	−	0.749437i	$-0.269676\pi$
$541$	30.7990	1.32415	0.662076	+	0.749437i	$0.269676\pi$
$542$	0	0
$543$	−21.6569	−0.929385
$544$	0	0
$545$	8.18377	0.350554
$546$	0	0
$547$	9.00000	0.384812	0.192406	−	0.981315i	$-0.438371\pi$
$547$	9.00000	0.384812	0.192406	+	0.981315i	$0.438371\pi$
$548$	0	0
$549$	8.24264	0.351787
$550$	0	0
$551$	20.4853	0.872702
$552$	0	0
$553$	0.485281	0.0206363
$554$	0	0
$555$	7.02944	0.298383
$556$	0	0
$557$	9.00000	0.381342	0.190671	−	0.981654i	$-0.438934\pi$
$557$	9.00000	0.381342	0.190671	+	0.981654i	$0.438934\pi$
$558$	0	0
$559$	−3.82843	−0.161925
$560$	0	0
$561$	−20.2426	−0.854645
$562$	0	0
$563$	−5.62742	−0.237167	−0.118584	−	0.992944i	$-0.537835\pi$
$563$	−5.62742	−0.237167	−0.118584	+	0.992944i	$0.537835\pi$
$564$	0	0
$565$	0.686292	0.0288725
$566$	0	0
$567$	−2.92893	−0.123004
$568$	0	0
$569$	24.6569	1.03367	0.516835	−	0.856085i	$-0.327110\pi$
$569$	24.6569	1.03367	0.516835	+	0.856085i	$0.327110\pi$
$570$	0	0
$571$	11.0711	0.463310	0.231655	−	0.972798i	$-0.425586\pi$
$571$	11.0711	0.463310	0.231655	+	0.972798i	$0.425586\pi$
$572$	0	0
$573$	31.3137	1.30815
$574$	0	0
$575$	−21.6863	−0.904381
$576$	0	0
$577$	18.9289	0.788022	0.394011	−	0.919106i	$-0.371087\pi$
$577$	18.9289	0.788022	0.394011	+	0.919106i	$0.371087\pi$
$578$	0	0
$579$	25.4142	1.05618
$580$	0	0
$581$	8.58579	0.356198
$582$	0	0
$583$	−14.9411	−0.618798
$584$	0	0
$585$	−2.24264	−0.0927218
$586$	0	0
$587$	−5.79899	−0.239350	−0.119675	−	0.992813i	$-0.538185\pi$
$587$	−5.79899	−0.239350	−0.119675	+	0.992813i	$0.538185\pi$
$588$	0	0
$589$	−14.4853	−0.596856
$590$	0	0
$591$	20.0000	0.822690
$592$	0	0
$593$	5.07107	0.208244	0.104122	−	0.994565i	$-0.466797\pi$
$593$	5.07107	0.208244	0.104122	+	0.994565i	$0.466797\pi$
$594$	0	0
$595$	−2.68629	−0.110127
$596$	0	0
$597$	−5.17157	−0.211658
$598$	0	0
$599$	−25.3431	−1.03549	−0.517746	−	0.855534i	$-0.673229\pi$
$599$	−25.3431	−1.03549	−0.517746	+	0.855534i	$0.673229\pi$
$600$	0	0
$601$	8.97056	0.365917	0.182958	−	0.983121i	$-0.441433\pi$
$601$	8.97056	0.365917	0.182958	+	0.983121i	$0.441433\pi$
$602$	0	0
$603$	−9.48528	−0.386271
$604$	0	0
$605$	4.48528	0.182353
$606$	0	0
$607$	−0.970563	−0.0393939	−0.0196970	−	0.999806i	$-0.506270\pi$
$607$	−0.970563	−0.0393939	−0.0196970	+	0.999806i	$0.506270\pi$
$608$	0	0
$609$	3.51472	0.142424
$610$	0	0
$611$	22.9706	0.929289
$612$	0	0
$613$	−1.17157	−0.0473194	−0.0236597	−	0.999720i	$-0.507532\pi$
$613$	−1.17157	−0.0473194	−0.0236597	+	0.999720i	$0.507532\pi$
$614$	0	0
$615$	−3.17157	−0.127890
$616$	0	0
$617$	−13.9706	−0.562434	−0.281217	−	0.959644i	$-0.590738\pi$
$617$	−13.9706	−0.562434	−0.281217	+	0.959644i	$0.590738\pi$
$618$	0	0
$619$	−4.97056	−0.199784	−0.0998919	−	0.994998i	$-0.531850\pi$
$619$	−4.97056	−0.199784	−0.0998919	+	0.994998i	$0.531850\pi$
$620$	0	0
$621$	26.3431	1.05711
$622$	0	0
$623$	−6.00000	−0.240385
$624$	0	0
$625$	19.9706	0.798823
$626$	0	0
$627$	12.4853	0.498614
$628$	0	0
$629$	66.4264	2.64859
$630$	0	0
$631$	26.7696	1.06568	0.532840	−	0.846216i	$-0.321125\pi$
$631$	26.7696	1.06568	0.532840	+	0.846216i	$0.321125\pi$
$632$	0	0
$633$	−39.7990	−1.58187
$634$	0	0
$635$	−1.07107	−0.0425040
$636$	0	0
$637$	25.4853	1.00976
$638$	0	0
$639$	−8.82843	−0.349247
$640$	0	0
$641$	1.55635	0.0614721	0.0307360	−	0.999528i	$-0.490215\pi$
$641$	1.55635	0.0614721	0.0307360	+	0.999528i	$0.490215\pi$
$642$	0	0
$643$	−9.51472	−0.375224	−0.187612	−	0.982243i	$-0.560075\pi$
$643$	−9.51472	−0.375224	−0.187612	+	0.982243i	$0.560075\pi$
$644$	0	0
$645$	0.828427	0.0326193
$646$	0	0
$647$	6.82843	0.268453	0.134227	−	0.990951i	$-0.457145\pi$
$647$	6.82843	0.268453	0.134227	+	0.990951i	$0.457145\pi$
$648$	0	0
$649$	1.51472	0.0594579
$650$	0	0
$651$	−2.48528	−0.0974059
$652$	0	0
$653$	−20.8284	−0.815079	−0.407540	−	0.913188i	$-0.633613\pi$
$653$	−20.8284	−0.815079	−0.407540	+	0.913188i	$0.633613\pi$
$654$	0	0
$655$	0.970563	0.0379230
$656$	0	0
$657$	7.75736	0.302643
$658$	0	0
$659$	−6.31371	−0.245947	−0.122974	−	0.992410i	$-0.539243\pi$
$659$	−6.31371	−0.245947	−0.122974	+	0.992410i	$0.539243\pi$
$660$	0	0
$661$	17.0000	0.661223	0.330612	−	0.943767i	$-0.392745\pi$
$661$	17.0000	0.661223	0.330612	+	0.943767i	$0.392745\pi$
$662$	0	0
$663$	42.3848	1.64609
$664$	0	0
$665$	1.65685	0.0642501
$666$	0	0
$667$	−19.7574	−0.765008
$668$	0	0
$669$	33.7990	1.30674
$670$	0	0
$671$	−15.0711	−0.581812
$672$	0	0
$673$	8.72792	0.336437	0.168218	−	0.985750i	$-0.446199\pi$
$673$	8.72792	0.336437	0.168218	+	0.985750i	$0.446199\pi$
$674$	0	0
$675$	−26.3431	−1.01395
$676$	0	0
$677$	−16.8284	−0.646769	−0.323384	−	0.946268i	$-0.604821\pi$
$677$	−16.8284	−0.646769	−0.323384	+	0.946268i	$0.604821\pi$
$678$	0	0
$679$	−2.24264	−0.0860647
$680$	0	0
$681$	24.9706	0.956874
$682$	0	0
$683$	4.45584	0.170498	0.0852491	−	0.996360i	$-0.472831\pi$
$683$	4.45584	0.170498	0.0852491	+	0.996360i	$0.472831\pi$
$684$	0	0
$685$	−1.45584	−0.0556249
$686$	0	0
$687$	11.2721	0.430057
$688$	0	0
$689$	31.2843	1.19184
$690$	0	0
$691$	−1.27208	−0.0483921	−0.0241961	−	0.999707i	$-0.507703\pi$
$691$	−1.27208	−0.0483921	−0.0241961	+	0.999707i	$0.507703\pi$
$692$	0	0
$693$	−1.07107	−0.0406865
$694$	0	0
$695$	6.72792	0.255205
$696$	0	0
$697$	−29.9706	−1.13522
$698$	0	0
$699$	1.17157	0.0443130
$700$	0	0
$701$	18.3431	0.692811	0.346406	−	0.938085i	$-0.387402\pi$
$701$	18.3431	0.692811	0.346406	+	0.938085i	$0.387402\pi$
$702$	0	0
$703$	−40.9706	−1.54523
$704$	0	0
$705$	−4.97056	−0.187202
$706$	0	0
$707$	−0.100505	−0.00377988
$708$	0	0
$709$	−24.1127	−0.905571	−0.452786	−	0.891619i	$-0.649570\pi$
$709$	−24.1127	−0.905571	−0.452786	+	0.891619i	$0.649570\pi$
$710$	0	0
$711$	−0.828427	−0.0310684
$712$	0	0
$713$	13.9706	0.523202
$714$	0	0
$715$	4.10051	0.153350
$716$	0	0
$717$	−3.51472	−0.131260
$718$	0	0
$719$	−22.6274	−0.843860	−0.421930	−	0.906628i	$-0.638647\pi$
$719$	−22.6274	−0.843860	−0.421930	+	0.906628i	$0.638647\pi$
$720$	0	0
$721$	−10.2426	−0.381456
$722$	0	0
$723$	−5.65685	−0.210381
$724$	0	0
$725$	19.7574	0.733770
$726$	0	0
$727$	−8.97056	−0.332700	−0.166350	−	0.986067i	$-0.553198\pi$
$727$	−8.97056	−0.332700	−0.166350	+	0.986067i	$0.553198\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	0	0
$731$	7.82843	0.289545
$732$	0	0
$733$	16.9706	0.626822	0.313411	−	0.949618i	$-0.398528\pi$
$733$	16.9706	0.626822	0.313411	+	0.949618i	$0.398528\pi$
$734$	0	0
$735$	−5.51472	−0.203413
$736$	0	0
$737$	17.3431	0.638843
$738$	0	0
$739$	11.4558	0.421410	0.210705	−	0.977550i	$-0.432424\pi$
$739$	11.4558	0.421410	0.210705	+	0.977550i	$0.432424\pi$
$740$	0	0
$741$	−26.1421	−0.960355
$742$	0	0
$743$	47.1127	1.72840	0.864199	−	0.503151i	$-0.167826\pi$
$743$	47.1127	1.72840	0.864199	+	0.503151i	$0.167826\pi$
$744$	0	0
$745$	−4.97056	−0.182107
$746$	0	0
$747$	−14.6569	−0.536266
$748$	0	0
$749$	−6.82843	−0.249505
$750$	0	0
$751$	20.2426	0.738664	0.369332	−	0.929297i	$-0.379586\pi$
$751$	20.2426	0.738664	0.369332	+	0.929297i	$0.379586\pi$
$752$	0	0
$753$	7.27208	0.265009
$754$	0	0
$755$	5.71573	0.208017
$756$	0	0
$757$	−20.4853	−0.744550	−0.372275	−	0.928122i	$-0.621422\pi$
$757$	−20.4853	−0.744550	−0.372275	+	0.928122i	$0.621422\pi$
$758$	0	0
$759$	−12.0416	−0.437083
$760$	0	0
$761$	−43.1127	−1.56283	−0.781417	−	0.624009i	$-0.785503\pi$
$761$	−43.1127	−1.56283	−0.781417	+	0.624009i	$0.785503\pi$
$762$	0	0
$763$	−8.18377	−0.296272
$764$	0	0
$765$	4.58579	0.165799
$766$	0	0
$767$	−3.17157	−0.114519
$768$	0	0
$769$	−28.7696	−1.03746	−0.518728	−	0.854939i	$-0.673594\pi$
$769$	−28.7696	−1.03746	−0.518728	+	0.854939i	$0.673594\pi$
$770$	0	0
$771$	14.4853	0.521675
$772$	0	0
$773$	−27.8995	−1.00348	−0.501738	−	0.865020i	$-0.667306\pi$
$773$	−27.8995	−1.00348	−0.501738	+	0.865020i	$0.667306\pi$
$774$	0	0
$775$	−13.9706	−0.501837
$776$	0	0
$777$	−7.02944	−0.252180
$778$	0	0
$779$	18.4853	0.662304
$780$	0	0
$781$	16.1421	0.577611
$782$	0	0
$783$	−24.0000	−0.857690
$784$	0	0
$785$	−5.85786	−0.209076
$786$	0	0
$787$	−29.7990	−1.06222	−0.531110	−	0.847303i	$-0.678225\pi$
$787$	−29.7990	−1.06222	−0.531110	+	0.847303i	$0.678225\pi$
$788$	0	0
$789$	−39.7990	−1.41688
$790$	0	0
$791$	−0.686292	−0.0244017
$792$	0	0
$793$	31.5563	1.12060
$794$	0	0
$795$	−6.76955	−0.240091
$796$	0	0
$797$	−12.6863	−0.449372	−0.224686	−	0.974431i	$-0.572136\pi$
$797$	−12.6863	−0.449372	−0.224686	+	0.974431i	$0.572136\pi$
$798$	0	0
$799$	−46.9706	−1.66170
$800$	0	0
$801$	10.2426	0.361906
$802$	0	0
$803$	−14.1838	−0.500534
$804$	0	0
$805$	−1.59798	−0.0563214
$806$	0	0
$807$	−4.24264	−0.149348
$808$	0	0
$809$	5.65685	0.198884	0.0994422	−	0.995043i	$-0.468294\pi$
$809$	5.65685	0.198884	0.0994422	+	0.995043i	$0.468294\pi$
$810$	0	0
$811$	48.7279	1.71107	0.855534	−	0.517746i	$-0.173229\pi$
$811$	48.7279	1.71107	0.855534	+	0.517746i	$0.173229\pi$
$812$	0	0
$813$	11.2721	0.395329
$814$	0	0
$815$	11.8579	0.415363
$816$	0	0
$817$	−4.82843	−0.168925
$818$	0	0
$819$	2.24264	0.0783642
$820$	0	0
$821$	−10.1127	−0.352936	−0.176468	−	0.984306i	$-0.556467\pi$
$821$	−10.1127	−0.352936	−0.176468	+	0.984306i	$0.556467\pi$
$822$	0	0
$823$	43.3431	1.51085	0.755424	−	0.655237i	$-0.227431\pi$
$823$	43.3431	1.51085	0.755424	+	0.655237i	$0.227431\pi$
$824$	0	0
$825$	12.0416	0.419236
$826$	0	0
$827$	9.65685	0.335802	0.167901	−	0.985804i	$-0.446301\pi$
$827$	9.65685	0.335802	0.167901	+	0.985804i	$0.446301\pi$
$828$	0	0
$829$	15.7990	0.548722	0.274361	−	0.961627i	$-0.411534\pi$
$829$	15.7990	0.548722	0.274361	+	0.961627i	$0.411534\pi$
$830$	0	0
$831$	−16.8284	−0.583772
$832$	0	0
$833$	−52.1127	−1.80560
$834$	0	0
$835$	4.87006	0.168535
$836$	0	0
$837$	16.9706	0.586588
$838$	0	0
$839$	−4.87006	−0.168133	−0.0840665	−	0.996460i	$-0.526791\pi$
$839$	−4.87006	−0.168133	−0.0840665	+	0.996460i	$0.526791\pi$
$840$	0	0
$841$	−11.0000	−0.379310
$842$	0	0
$843$	−3.75736	−0.129410
$844$	0	0
$845$	−0.970563	−0.0333884
$846$	0	0
$847$	−4.48528	−0.154116
$848$	0	0
$849$	6.10051	0.209369
$850$	0	0
$851$	39.5147	1.35455
$852$	0	0
$853$	32.5980	1.11613	0.558067	−	0.829796i	$-0.311543\pi$
$853$	32.5980	1.11613	0.558067	+	0.829796i	$0.311543\pi$
$854$	0	0
$855$	−2.82843	−0.0967302
$856$	0	0
$857$	−3.65685	−0.124916	−0.0624579	−	0.998048i	$-0.519894\pi$
$857$	−3.65685	−0.124916	−0.0624579	+	0.998048i	$0.519894\pi$
$858$	0	0
$859$	−16.9706	−0.579028	−0.289514	−	0.957174i	$-0.593494\pi$
$859$	−16.9706	−0.579028	−0.289514	+	0.957174i	$0.593494\pi$
$860$	0	0
$861$	3.17157	0.108087
$862$	0	0
$863$	−43.2548	−1.47241	−0.736206	−	0.676758i	$-0.763384\pi$
$863$	−43.2548	−1.47241	−0.736206	+	0.676758i	$0.763384\pi$
$864$	0	0
$865$	7.23045	0.245843
$866$	0	0
$867$	−62.6274	−2.12694
$868$	0	0
$869$	1.51472	0.0513833
$870$	0	0
$871$	−36.3137	−1.23044
$872$	0	0
$873$	3.82843	0.129573
$874$	0	0
$875$	3.31371	0.112024
$876$	0	0
$877$	2.79899	0.0945152	0.0472576	−	0.998883i	$-0.484952\pi$
$877$	2.79899	0.0945152	0.0472576	+	0.998883i	$0.484952\pi$
$878$	0	0
$879$	30.6274	1.03304
$880$	0	0
$881$	−54.2548	−1.82789	−0.913946	−	0.405836i	$-0.866980\pi$
$881$	−54.2548	−1.82789	−0.913946	+	0.405836i	$0.866980\pi$
$882$	0	0
$883$	−41.9706	−1.41242	−0.706211	−	0.708001i	$-0.749597\pi$
$883$	−41.9706	−1.41242	−0.706211	+	0.708001i	$0.749597\pi$
$884$	0	0
$885$	0.686292	0.0230694
$886$	0	0
$887$	−25.0294	−0.840406	−0.420203	−	0.907430i	$-0.638041\pi$
$887$	−25.0294	−0.840406	−0.420203	+	0.907430i	$0.638041\pi$
$888$	0	0
$889$	1.07107	0.0359225
$890$	0	0
$891$	−9.14214	−0.306273
$892$	0	0
$893$	28.9706	0.969463
$894$	0	0
$895$	4.34315	0.145175
$896$	0	0
$897$	25.2132	0.841844
$898$	0	0
$899$	−12.7279	−0.424500
$900$	0	0
$901$	−63.9706	−2.13117
$902$	0	0
$903$	−0.828427	−0.0275683
$904$	0	0
$905$	−8.97056	−0.298192
$906$	0	0
$907$	13.9706	0.463885	0.231942	−	0.972730i	$-0.425492\pi$
$907$	13.9706	0.463885	0.231942	+	0.972730i	$0.425492\pi$
$908$	0	0
$909$	0.171573	0.00569071
$910$	0	0
$911$	−4.24264	−0.140565	−0.0702825	−	0.997527i	$-0.522390\pi$
$911$	−4.24264	−0.140565	−0.0702825	+	0.997527i	$0.522390\pi$
$912$	0	0
$913$	26.7990	0.886917
$914$	0	0
$915$	−6.82843	−0.225741
$916$	0	0
$917$	−0.970563	−0.0320508
$918$	0	0
$919$	29.4853	0.972630	0.486315	−	0.873784i	$-0.338341\pi$
$919$	29.4853	0.972630	0.486315	+	0.873784i	$0.338341\pi$
$920$	0	0
$921$	18.1005	0.596432
$922$	0	0
$923$	−33.7990	−1.11251
$924$	0	0
$925$	−39.5147	−1.29924
$926$	0	0
$927$	17.4853	0.574292
$928$	0	0
$929$	44.8284	1.47077	0.735386	−	0.677648i	$-0.237001\pi$
$929$	44.8284	1.47077	0.735386	+	0.677648i	$0.237001\pi$
$930$	0	0
$931$	32.1421	1.05342
$932$	0	0
$933$	28.2426	0.924623
$934$	0	0
$935$	−8.38478	−0.274212
$936$	0	0
$937$	31.0122	1.01312	0.506562	−	0.862203i	$-0.330916\pi$
$937$	31.0122	1.01312	0.506562	+	0.862203i	$0.330916\pi$
$938$	0	0
$939$	−24.3431	−0.794409
$940$	0	0
$941$	−13.6274	−0.444241	−0.222121	−	0.975019i	$-0.571298\pi$
$941$	−13.6274	−0.444241	−0.222121	+	0.975019i	$0.571298\pi$
$942$	0	0
$943$	−17.8284	−0.580573
$944$	0	0
$945$	−1.94113	−0.0631448
$946$	0	0
$947$	−0.171573	−0.00557537	−0.00278768	−	0.999996i	$-0.500887\pi$
$947$	−0.171573	−0.00557537	−0.00278768	+	0.999996i	$0.500887\pi$
$948$	0	0
$949$	29.6985	0.964054
$950$	0	0
$951$	28.5269	0.925048
$952$	0	0
$953$	24.0416	0.778785	0.389392	−	0.921072i	$-0.372685\pi$
$953$	24.0416	0.778785	0.389392	+	0.921072i	$0.372685\pi$
$954$	0	0
$955$	12.9706	0.419718
$956$	0	0
$957$	10.9706	0.354628
$958$	0	0
$959$	1.45584	0.0470117
$960$	0	0
$961$	−22.0000	−0.709677
$962$	0	0
$963$	11.6569	0.375637
$964$	0	0
$965$	10.5269	0.338873
$966$	0	0
$967$	−26.9411	−0.866368	−0.433184	−	0.901305i	$-0.642610\pi$
$967$	−26.9411	−0.866368	−0.433184	+	0.901305i	$0.642610\pi$
$968$	0	0
$969$	53.4558	1.71725
$970$	0	0
$971$	25.1421	0.806850	0.403425	−	0.915013i	$-0.367820\pi$
$971$	25.1421	0.806850	0.403425	+	0.915013i	$0.367820\pi$
$972$	0	0
$973$	−6.72792	−0.215687
$974$	0	0
$975$	−25.2132	−0.807469
$976$	0	0
$977$	−0.686292	−0.0219564	−0.0109782	−	0.999940i	$-0.503495\pi$
$977$	−0.686292	−0.0219564	−0.0109782	+	0.999940i	$0.503495\pi$
$978$	0	0
$979$	−18.7279	−0.598547
$980$	0	0
$981$	13.9706	0.446046
$982$	0	0
$983$	−2.52691	−0.0805960	−0.0402980	−	0.999188i	$-0.512831\pi$
$983$	−2.52691	−0.0805960	−0.0402980	+	0.999188i	$0.512831\pi$
$984$	0	0
$985$	8.28427	0.263959
$986$	0	0
$987$	4.97056	0.158215
$988$	0	0
$989$	4.65685	0.148079
$990$	0	0
$991$	19.5563	0.621228	0.310614	−	0.950536i	$-0.399465\pi$
$991$	19.5563	0.621228	0.310614	+	0.950536i	$0.399465\pi$
$992$	0	0
$993$	−26.2843	−0.834106
$994$	0	0
$995$	−2.14214	−0.0679103
$996$	0	0
$997$	−17.0711	−0.540646	−0.270323	−	0.962770i	$-0.587131\pi$
$997$	−17.0711	−0.540646	−0.270323	+	0.962770i	$0.587131\pi$
$998$	0	0
$999$	48.0000	1.51865

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2752.2.a.m.1.1		2
4.3	odd	2		2752.2.a.l.1.2		2
8.3	odd	2		43.2.a.b.1.2	✓	2
8.5	even	2		688.2.a.f.1.2		2
24.5	odd	2		6192.2.a.bd.1.2		2
24.11	even	2		387.2.a.h.1.1		2
40.3	even	4		1075.2.b.f.474.2		4
40.19	odd	2		1075.2.a.i.1.1		2
40.27	even	4		1075.2.b.f.474.3		4
56.27	even	2		2107.2.a.b.1.2		2
88.43	even	2		5203.2.a.f.1.1		2
104.51	odd	2		7267.2.a.b.1.1		2
120.59	even	2		9675.2.a.bf.1.2		2
344.171	even	2		1849.2.a.f.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.2.a.b.1.2	✓	2	8.3	odd	2
387.2.a.h.1.1		2	24.11	even	2
688.2.a.f.1.2		2	8.5	even	2
1075.2.a.i.1.1		2	40.19	odd	2
1075.2.b.f.474.2		4	40.3	even	4
1075.2.b.f.474.3		4	40.27	even	4
1849.2.a.f.1.1		2	344.171	even	2
2107.2.a.b.1.2		2	56.27	even	2
2752.2.a.l.1.2		2	4.3	odd	2
2752.2.a.m.1.1		2	1.1	even	1	trivial
5203.2.a.f.1.1		2	88.43	even	2
6192.2.a.bd.1.2		2	24.5	odd	2
7267.2.a.b.1.1		2	104.51	odd	2
9675.2.a.bf.1.2		2	120.59	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2752 = 2^{6} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2752.a (trivial)

\(f(q)\)	\(=\)	\(q-1.41421 q^{3} -0.585786 q^{5} +0.585786 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.41421 q^{3} -0.585786 q^{5} +0.585786 q^{7} -1.00000 q^{9} +1.82843 q^{11} -3.82843 q^{13} +0.828427 q^{15} +7.82843 q^{17} -4.82843 q^{19} -0.828427 q^{21} +4.65685 q^{23} -4.65685 q^{25} +5.65685 q^{27} -4.24264 q^{29} +3.00000 q^{31} -2.58579 q^{33} -0.343146 q^{35} +8.48528 q^{37} +5.41421 q^{39} -3.82843 q^{41} +1.00000 q^{43} +0.585786 q^{45} -6.00000 q^{47} -6.65685 q^{49} -11.0711 q^{51} -8.17157 q^{53} -1.07107 q^{55} +6.82843 q^{57} +0.828427 q^{59} -8.24264 q^{61} -0.585786 q^{63} +2.24264 q^{65} +9.48528 q^{67} -6.58579 q^{69} +8.82843 q^{71} -7.75736 q^{73} +6.58579 q^{75} +1.07107 q^{77} +0.828427 q^{79} -5.00000 q^{81} +14.6569 q^{83} -4.58579 q^{85} +6.00000 q^{87} -10.2426 q^{89} -2.24264 q^{91} -4.24264 q^{93} +2.82843 q^{95} -3.82843 q^{97} -1.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5} + 4 q^{7} - 2 q^{9}+O(q^{10}) \) 2 * q - 4 * q^5 + 4 * q^7 - 2 * q^9	\( 2 q - 4 q^{5} + 4 q^{7} - 2 q^{9} - 2 q^{11} - 2 q^{13} - 4 q^{15} + 10 q^{17} - 4 q^{19} + 4 q^{21} - 2 q^{23} + 2 q^{25} + 6 q^{31} - 8 q^{33} - 12 q^{35} + 8 q^{39} - 2 q^{41} + 2 q^{43} + 4 q^{45} - 12 q^{47} - 2 q^{49} - 8 q^{51} - 22 q^{53} + 12 q^{55} + 8 q^{57} - 4 q^{59} - 8 q^{61} - 4 q^{63} - 4 q^{65} + 2 q^{67} - 16 q^{69} + 12 q^{71} - 24 q^{73} + 16 q^{75} - 12 q^{77} - 4 q^{79} - 10 q^{81} + 18 q^{83} - 12 q^{85} + 12 q^{87} - 12 q^{89} + 4 q^{91} - 2 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q - 4 * q^5 + 4 * q^7 - 2 * q^9 - 2 * q^11 - 2 * q^13 - 4 * q^15 + 10 * q^17 - 4 * q^19 + 4 * q^21 - 2 * q^23 + 2 * q^25 + 6 * q^31 - 8 * q^33 - 12 * q^35 + 8 * q^39 - 2 * q^41 + 2 * q^43 + 4 * q^45 - 12 * q^47 - 2 * q^49 - 8 * q^51 - 22 * q^53 + 12 * q^55 + 8 * q^57 - 4 * q^59 - 8 * q^61 - 4 * q^63 - 4 * q^65 + 2 * q^67 - 16 * q^69 + 12 * q^71 - 24 * q^73 + 16 * q^75 - 12 * q^77 - 4 * q^79 - 10 * q^81 + 18 * q^83 - 12 * q^85 + 12 * q^87 - 12 * q^89 + 4 * q^91 - 2 * q^97 + 2 * q^99

Introduction

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