LMFDB - Embedded newform 7728.2.a.bp.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7728,2,Mod(1,7728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7728, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("7728.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7728.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:	$61.7083906820$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3864)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.11491$ of defining polynomial
Character	$\chi$	$=$	7728.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.11491 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.11491 q^{5} -1.00000 q^{7} +1.00000 q^{9} +3.94567 q^{11} -5.47283 q^{13} -1.11491 q^{15} -0.926221 q^{17} +2.28415 q^{19} +1.00000 q^{21} +1.00000 q^{23} -3.75698 q^{25} -1.00000 q^{27} +2.58774 q^{29} -8.81756 q^{31} -3.94567 q^{33} -1.11491 q^{35} +1.87189 q^{37} +5.47283 q^{39} -4.28415 q^{41} +9.70265 q^{43} +1.11491 q^{45} +7.81756 q^{47} +1.00000 q^{49} +0.926221 q^{51} +4.70265 q^{53} +4.39905 q^{55} -2.28415 q^{57} +6.93246 q^{59} -13.0606 q^{61} -1.00000 q^{63} -6.10170 q^{65} +8.77643 q^{67} -1.00000 q^{69} +12.6483 q^{71} -7.87189 q^{73} +3.75698 q^{75} -3.94567 q^{77} -8.81756 q^{79} +1.00000 q^{81} -7.04737 q^{83} -1.03265 q^{85} -2.58774 q^{87} +0.0411284 q^{89} +5.47283 q^{91} +8.81756 q^{93} +2.54661 q^{95} +16.5613 q^{97} +3.94567 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 - 3 * q^5 - 3 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9} + q^{11} - 11 q^{13} + 3 q^{15} + 5 q^{19} + 3 q^{21} + 3 q^{23} - 4 q^{25} - 3 q^{27} - 4 q^{29} - 2 q^{31} - q^{33} + 3 q^{35} - 8 q^{37} + 11 q^{39} - 11 q^{41} + 11 q^{43} - 3 q^{45} - q^{47} + 3 q^{49} - 4 q^{53} + 5 q^{55} - 5 q^{57} - 10 q^{59} - 22 q^{61} - 3 q^{63} + 8 q^{65} + 11 q^{67} - 3 q^{69} + 9 q^{71} - 10 q^{73} + 4 q^{75} - q^{77} - 2 q^{79} + 3 q^{81} + 16 q^{83} - 15 q^{85} + 4 q^{87} - 9 q^{89} + 11 q^{91} + 2 q^{93} + 5 q^{95} - 2 q^{97} + q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 3 * q^5 - 3 * q^7 + 3 * q^9 + q^11 - 11 * q^13 + 3 * q^15 + 5 * q^19 + 3 * q^21 + 3 * q^23 - 4 * q^25 - 3 * q^27 - 4 * q^29 - 2 * q^31 - q^33 + 3 * q^35 - 8 * q^37 + 11 * q^39 - 11 * q^41 + 11 * q^43 - 3 * q^45 - q^47 + 3 * q^49 - 4 * q^53 + 5 * q^55 - 5 * q^57 - 10 * q^59 - 22 * q^61 - 3 * q^63 + 8 * q^65 + 11 * q^67 - 3 * q^69 + 9 * q^71 - 10 * q^73 + 4 * q^75 - q^77 - 2 * q^79 + 3 * q^81 + 16 * q^83 - 15 * q^85 + 4 * q^87 - 9 * q^89 + 11 * q^91 + 2 * q^93 + 5 * q^95 - 2 * q^97 + 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.11491	0.498602	0.249301	−	0.968426i	$-0.419799\pi$
$5$	1.11491	0.498602	0.249301	+	0.968426i	$0.419799\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.94567	1.18966	0.594832	−	0.803850i	$-0.297219\pi$
$11$	3.94567	1.18966	0.594832	+	0.803850i	$0.297219\pi$
$12$	0	0
$13$	−5.47283	−1.51789	−0.758946	−	0.651154i	$-0.774285\pi$
$13$	−5.47283	−1.51789	−0.758946	+	0.651154i	$0.774285\pi$
$14$	0	0
$15$	−1.11491	−0.287868
$16$	0	0
$17$	−0.926221	−0.224642	−0.112321	−	0.993672i	$-0.535828\pi$
$17$	−0.926221	−0.224642	−0.112321	+	0.993672i	$0.535828\pi$
$18$	0	0
$19$	2.28415	0.524019	0.262010	−	0.965065i	$-0.415615\pi$
$19$	2.28415	0.524019	0.262010	+	0.965065i	$0.415615\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−3.75698	−0.751396
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	2.58774	0.480532	0.240266	−	0.970707i	$-0.422765\pi$
$29$	2.58774	0.480532	0.240266	+	0.970707i	$0.422765\pi$
$30$	0	0
$31$	−8.81756	−1.58368	−0.791840	−	0.610729i	$-0.790877\pi$
$31$	−8.81756	−1.58368	−0.791840	+	0.610729i	$0.790877\pi$
$32$	0	0
$33$	−3.94567	−0.686853
$34$	0	0
$35$	−1.11491	−0.188454
$36$	0	0
$37$	1.87189	0.307737	0.153868	−	0.988091i	$-0.450827\pi$
$37$	1.87189	0.307737	0.153868	+	0.988091i	$0.450827\pi$
$38$	0	0
$39$	5.47283	0.876355
$40$	0	0
$41$	−4.28415	−0.669071	−0.334536	−	0.942383i	$-0.608579\pi$
$41$	−4.28415	−0.669071	−0.334536	+	0.942383i	$0.608579\pi$
$42$	0	0
$43$	9.70265	1.47964	0.739820	−	0.672805i	$-0.234911\pi$
$43$	9.70265	1.47964	0.739820	+	0.672805i	$0.234911\pi$
$44$	0	0
$45$	1.11491	0.166201
$46$	0	0
$47$	7.81756	1.14031	0.570154	−	0.821538i	$-0.306884\pi$
$47$	7.81756	1.14031	0.570154	+	0.821538i	$0.306884\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.926221	0.129697
$52$	0	0
$53$	4.70265	0.645959	0.322979	−	0.946406i	$-0.395316\pi$
$53$	4.70265	0.645959	0.322979	+	0.946406i	$0.395316\pi$
$54$	0	0
$55$	4.39905	0.593168
$56$	0	0
$57$	−2.28415	−0.302543
$58$	0	0
$59$	6.93246	0.902530	0.451265	−	0.892390i	$-0.350973\pi$
$59$	6.93246	0.902530	0.451265	+	0.892390i	$0.350973\pi$
$60$	0	0
$61$	−13.0606	−1.67224	−0.836118	−	0.548550i	$-0.815180\pi$
$61$	−13.0606	−1.67224	−0.836118	+	0.548550i	$0.815180\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−6.10170	−0.756823
$66$	0	0
$67$	8.77643	1.07221	0.536106	−	0.844151i	$-0.319895\pi$
$67$	8.77643	1.07221	0.536106	+	0.844151i	$0.319895\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	12.6483	1.50108	0.750540	−	0.660826i	$-0.229794\pi$
$71$	12.6483	1.50108	0.750540	+	0.660826i	$0.229794\pi$
$72$	0	0
$73$	−7.87189	−0.921335	−0.460667	−	0.887573i	$-0.652390\pi$
$73$	−7.87189	−0.921335	−0.460667	+	0.887573i	$0.652390\pi$
$74$	0	0
$75$	3.75698	0.433819
$76$	0	0
$77$	−3.94567	−0.449651
$78$	0	0
$79$	−8.81756	−0.992053	−0.496026	−	0.868307i	$-0.665208\pi$
$79$	−8.81756	−0.992053	−0.496026	+	0.868307i	$0.665208\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−7.04737	−0.773550	−0.386775	−	0.922174i	$-0.626411\pi$
$83$	−7.04737	−0.773550	−0.386775	+	0.922174i	$0.626411\pi$
$84$	0	0
$85$	−1.03265	−0.112007
$86$	0	0
$87$	−2.58774	−0.277435
$88$	0	0
$89$	0.0411284	0.00435961	0.00217980	−	0.999998i	$-0.499306\pi$
$89$	0.0411284	0.00435961	0.00217980	+	0.999998i	$0.499306\pi$
$90$	0	0
$91$	5.47283	0.573709
$92$	0	0
$93$	8.81756	0.914338
$94$	0	0
$95$	2.54661	0.261277
$96$	0	0
$97$	16.5613	1.68155	0.840774	−	0.541386i	$-0.182100\pi$
$97$	16.5613	1.68155	0.840774	+	0.541386i	$0.182100\pi$
$98$	0	0
$99$	3.94567	0.396555
$100$	0	0
$101$	12.1079	1.20479	0.602393	−	0.798200i	$-0.294214\pi$
$101$	12.1079	1.20479	0.602393	+	0.798200i	$0.294214\pi$
$102$	0	0
$103$	8.87189	0.874173	0.437087	−	0.899419i	$-0.356010\pi$
$103$	8.87189	0.874173	0.437087	+	0.899419i	$0.356010\pi$
$104$	0	0
$105$	1.11491	0.108804
$106$	0	0
$107$	3.93246	0.380166	0.190083	−	0.981768i	$-0.439124\pi$
$107$	3.93246	0.380166	0.190083	+	0.981768i	$0.439124\pi$
$108$	0	0
$109$	−16.5202	−1.58235	−0.791174	−	0.611591i	$-0.790530\pi$
$109$	−16.5202	−1.58235	−0.791174	+	0.611591i	$0.790530\pi$
$110$	0	0
$111$	−1.87189	−0.177672
$112$	0	0
$113$	15.9130	1.49697	0.748485	−	0.663151i	$-0.230781\pi$
$113$	15.9130	1.49697	0.748485	+	0.663151i	$0.230781\pi$
$114$	0	0
$115$	1.11491	0.103966
$116$	0	0
$117$	−5.47283	−0.505964
$118$	0	0
$119$	0.926221	0.0849065
$120$	0	0
$121$	4.56829	0.415300
$122$	0	0
$123$	4.28415	0.386289
$124$	0	0
$125$	−9.76322	−0.873249
$126$	0	0
$127$	−3.06058	−0.271582	−0.135791	−	0.990737i	$-0.543358\pi$
$127$	−3.06058	−0.271582	−0.135791	+	0.990737i	$0.543358\pi$
$128$	0	0
$129$	−9.70265	−0.854271
$130$	0	0
$131$	10.8913	0.951580	0.475790	−	0.879559i	$-0.342162\pi$
$131$	10.8913	0.951580	0.475790	+	0.879559i	$0.342162\pi$
$132$	0	0
$133$	−2.28415	−0.198061
$134$	0	0
$135$	−1.11491	−0.0959560
$136$	0	0
$137$	11.3774	0.972035	0.486017	−	0.873949i	$-0.338449\pi$
$137$	11.3774	0.972035	0.486017	+	0.873949i	$0.338449\pi$
$138$	0	0
$139$	−11.8044	−1.00123	−0.500616	−	0.865669i	$-0.666893\pi$
$139$	−11.8044	−1.00123	−0.500616	+	0.865669i	$0.666893\pi$
$140$	0	0
$141$	−7.81756	−0.658357
$142$	0	0
$143$	−21.5940	−1.80578
$144$	0	0
$145$	2.88509	0.239594
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	3.35793	0.275092	0.137546	−	0.990495i	$-0.456079\pi$
$149$	3.35793	0.275092	0.137546	+	0.990495i	$0.456079\pi$
$150$	0	0
$151$	−6.99304	−0.569085	−0.284543	−	0.958663i	$-0.591842\pi$
$151$	−6.99304	−0.569085	−0.284543	+	0.958663i	$0.591842\pi$
$152$	0	0
$153$	−0.926221	−0.0748805
$154$	0	0
$155$	−9.83076	−0.789626
$156$	0	0
$157$	−13.2104	−1.05430	−0.527151	−	0.849772i	$-0.676740\pi$
$157$	−13.2104	−1.05430	−0.527151	+	0.849772i	$0.676740\pi$
$158$	0	0
$159$	−4.70265	−0.372944
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	4.47283	0.350339	0.175170	−	0.984538i	$-0.443953\pi$
$163$	4.47283	0.350339	0.175170	+	0.984538i	$0.443953\pi$
$164$	0	0
$165$	−4.39905	−0.342466
$166$	0	0
$167$	−6.06682	−0.469465	−0.234732	−	0.972060i	$-0.575421\pi$
$167$	−6.06682	−0.469465	−0.234732	+	0.972060i	$0.575421\pi$
$168$	0	0
$169$	16.9519	1.30399
$170$	0	0
$171$	2.28415	0.174673
$172$	0	0
$173$	−0.735300	−0.0559038	−0.0279519	−	0.999609i	$-0.508899\pi$
$173$	−0.735300	−0.0559038	−0.0279519	+	0.999609i	$0.508899\pi$
$174$	0	0
$175$	3.75698	0.284001
$176$	0	0
$177$	−6.93246	−0.521076
$178$	0	0
$179$	−12.1428	−0.907598	−0.453799	−	0.891104i	$-0.649932\pi$
$179$	−12.1428	−0.907598	−0.453799	+	0.891104i	$0.649932\pi$
$180$	0	0
$181$	5.99304	0.445459	0.222730	−	0.974880i	$-0.428503\pi$
$181$	5.99304	0.445459	0.222730	+	0.974880i	$0.428503\pi$
$182$	0	0
$183$	13.0606	0.965466
$184$	0	0
$185$	2.08698	0.153438
$186$	0	0
$187$	−3.65456	−0.267248
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	13.5613	0.981264	0.490632	−	0.871367i	$-0.336766\pi$
$191$	13.5613	0.981264	0.490632	+	0.871367i	$0.336766\pi$
$192$	0	0
$193$	−7.24926	−0.521813	−0.260907	−	0.965364i	$-0.584021\pi$
$193$	−7.24926	−0.521813	−0.260907	+	0.965364i	$0.584021\pi$
$194$	0	0
$195$	6.10170	0.436952
$196$	0	0
$197$	3.62039	0.257942	0.128971	−	0.991648i	$-0.458833\pi$
$197$	3.62039	0.257942	0.128971	+	0.991648i	$0.458833\pi$
$198$	0	0
$199$	5.41850	0.384107	0.192054	−	0.981384i	$-0.438485\pi$
$199$	5.41850	0.384107	0.192054	+	0.981384i	$0.438485\pi$
$200$	0	0
$201$	−8.77643	−0.619042
$202$	0	0
$203$	−2.58774	−0.181624
$204$	0	0
$205$	−4.77643	−0.333600
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	9.01249	0.623407
$210$	0	0
$211$	22.5140	1.54993	0.774963	−	0.632007i	$-0.217769\pi$
$211$	22.5140	1.54993	0.774963	+	0.632007i	$0.217769\pi$
$212$	0	0
$213$	−12.6483	−0.866648
$214$	0	0
$215$	10.8176	0.737751
$216$	0	0
$217$	8.81756	0.598575
$218$	0	0
$219$	7.87189	0.531933
$220$	0	0
$221$	5.06905	0.340981
$222$	0	0
$223$	17.7221	1.18676	0.593380	−	0.804923i	$-0.297793\pi$
$223$	17.7221	1.18676	0.593380	+	0.804923i	$0.297793\pi$
$224$	0	0
$225$	−3.75698	−0.250465
$226$	0	0
$227$	−9.00624	−0.597765	−0.298883	−	0.954290i	$-0.596614\pi$
$227$	−9.00624	−0.597765	−0.298883	+	0.954290i	$0.596614\pi$
$228$	0	0
$229$	6.51173	0.430307	0.215154	−	0.976580i	$-0.430975\pi$
$229$	6.51173	0.430307	0.215154	+	0.976580i	$0.430975\pi$
$230$	0	0
$231$	3.94567	0.259606
$232$	0	0
$233$	7.93942	0.520129	0.260065	−	0.965591i	$-0.416256\pi$
$233$	7.93942	0.520129	0.260065	+	0.965591i	$0.416256\pi$
$234$	0	0
$235$	8.71585	0.568560
$236$	0	0
$237$	8.81756	0.572762
$238$	0	0
$239$	7.42698	0.480411	0.240206	−	0.970722i	$-0.422785\pi$
$239$	7.42698	0.480411	0.240206	+	0.970722i	$0.422785\pi$
$240$	0	0
$241$	2.05433	0.132331	0.0661656	−	0.997809i	$-0.478923\pi$
$241$	2.05433	0.132331	0.0661656	+	0.997809i	$0.478923\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	1.11491	0.0712288
$246$	0	0
$247$	−12.5008	−0.795404
$248$	0	0
$249$	7.04737	0.446609
$250$	0	0
$251$	16.7283	1.05588	0.527942	−	0.849281i	$-0.322964\pi$
$251$	16.7283	1.05588	0.527942	+	0.849281i	$0.322964\pi$
$252$	0	0
$253$	3.94567	0.248062
$254$	0	0
$255$	1.03265	0.0646671
$256$	0	0
$257$	7.67000	0.478441	0.239221	−	0.970965i	$-0.423108\pi$
$257$	7.67000	0.478441	0.239221	+	0.970965i	$0.423108\pi$
$258$	0	0
$259$	−1.87189	−0.116314
$260$	0	0
$261$	2.58774	0.160177
$262$	0	0
$263$	−5.35097	−0.329955	−0.164977	−	0.986297i	$-0.552755\pi$
$263$	−5.35097	−0.329955	−0.164977	+	0.986297i	$0.552755\pi$
$264$	0	0
$265$	5.24302	0.322076
$266$	0	0
$267$	−0.0411284	−0.00251702
$268$	0	0
$269$	−13.5117	−0.823825	−0.411912	−	0.911223i	$-0.635139\pi$
$269$	−13.5117	−0.823825	−0.411912	+	0.911223i	$0.635139\pi$
$270$	0	0
$271$	10.4402	0.634196	0.317098	−	0.948393i	$-0.397292\pi$
$271$	10.4402	0.634196	0.317098	+	0.948393i	$0.397292\pi$
$272$	0	0
$273$	−5.47283	−0.331231
$274$	0	0
$275$	−14.8238	−0.893909
$276$	0	0
$277$	15.4379	0.927576	0.463788	−	0.885946i	$-0.346490\pi$
$277$	15.4379	0.927576	0.463788	+	0.885946i	$0.346490\pi$
$278$	0	0
$279$	−8.81756	−0.527893
$280$	0	0
$281$	7.80908	0.465851	0.232925	−	0.972495i	$-0.425170\pi$
$281$	7.80908	0.465851	0.232925	+	0.972495i	$0.425170\pi$
$282$	0	0
$283$	10.5466	0.626931	0.313466	−	0.949600i	$-0.398510\pi$
$283$	10.5466	0.626931	0.313466	+	0.949600i	$0.398510\pi$
$284$	0	0
$285$	−2.54661	−0.150848
$286$	0	0
$287$	4.28415	0.252885
$288$	0	0
$289$	−16.1421	−0.949536
$290$	0	0
$291$	−16.5613	−0.970843
$292$	0	0
$293$	−13.2144	−0.771992	−0.385996	−	0.922500i	$-0.626142\pi$
$293$	−13.2144	−0.771992	−0.385996	+	0.922500i	$0.626142\pi$
$294$	0	0
$295$	7.72906	0.450003
$296$	0	0
$297$	−3.94567	−0.228951
$298$	0	0
$299$	−5.47283	−0.316502
$300$	0	0
$301$	−9.70265	−0.559251
$302$	0	0
$303$	−12.1079	−0.695583
$304$	0	0
$305$	−14.5613	−0.833780
$306$	0	0
$307$	−8.92622	−0.509446	−0.254723	−	0.967014i	$-0.581984\pi$
$307$	−8.92622	−0.509446	−0.254723	+	0.967014i	$0.581984\pi$
$308$	0	0
$309$	−8.87189	−0.504704
$310$	0	0
$311$	15.1817	0.860877	0.430438	−	0.902620i	$-0.358359\pi$
$311$	15.1817	0.860877	0.430438	+	0.902620i	$0.358359\pi$
$312$	0	0
$313$	22.1560	1.25233	0.626167	−	0.779689i	$-0.284623\pi$
$313$	22.1560	1.25233	0.626167	+	0.779689i	$0.284623\pi$
$314$	0	0
$315$	−1.11491	−0.0628179
$316$	0	0
$317$	12.2081	0.685677	0.342839	−	0.939394i	$-0.388612\pi$
$317$	12.2081	0.685677	0.342839	+	0.939394i	$0.388612\pi$
$318$	0	0
$319$	10.2104	0.571671
$320$	0	0
$321$	−3.93246	−0.219489
$322$	0	0
$323$	−2.11562	−0.117717
$324$	0	0
$325$	20.5613	1.14054
$326$	0	0
$327$	16.5202	0.913569
$328$	0	0
$329$	−7.81756	−0.430996
$330$	0	0
$331$	15.4791	0.850807	0.425404	−	0.905004i	$-0.360132\pi$
$331$	15.4791	0.850807	0.425404	+	0.905004i	$0.360132\pi$
$332$	0	0
$333$	1.87189	0.102579
$334$	0	0
$335$	9.78491	0.534607
$336$	0	0
$337$	−10.6483	−0.580051	−0.290025	−	0.957019i	$-0.593664\pi$
$337$	−10.6483	−0.580051	−0.290025	+	0.957019i	$0.593664\pi$
$338$	0	0
$339$	−15.9130	−0.864276
$340$	0	0
$341$	−34.7911	−1.88405
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−1.11491	−0.0600246
$346$	0	0
$347$	29.5334	1.58544	0.792718	−	0.609588i	$-0.208665\pi$
$347$	29.5334	1.58544	0.792718	+	0.609588i	$0.208665\pi$
$348$	0	0
$349$	−13.4270	−0.718730	−0.359365	−	0.933197i	$-0.617007\pi$
$349$	−13.4270	−0.718730	−0.359365	+	0.933197i	$0.617007\pi$
$350$	0	0
$351$	5.47283	0.292118
$352$	0	0
$353$	25.3594	1.34975	0.674873	−	0.737933i	$-0.264198\pi$
$353$	25.3594	1.34975	0.674873	+	0.737933i	$0.264198\pi$
$354$	0	0
$355$	14.1017	0.748441
$356$	0	0
$357$	−0.926221	−0.0490208
$358$	0	0
$359$	−13.9130	−0.734301	−0.367150	−	0.930162i	$-0.619667\pi$
$359$	−13.9130	−0.734301	−0.367150	+	0.930162i	$0.619667\pi$
$360$	0	0
$361$	−13.7827	−0.725404
$362$	0	0
$363$	−4.56829	−0.239773
$364$	0	0
$365$	−8.77643	−0.459379
$366$	0	0
$367$	13.2081	0.689459	0.344729	−	0.938702i	$-0.387971\pi$
$367$	13.2081	0.689459	0.344729	+	0.938702i	$0.387971\pi$
$368$	0	0
$369$	−4.28415	−0.223024
$370$	0	0
$371$	−4.70265	−0.244149
$372$	0	0
$373$	−8.93871	−0.462829	−0.231414	−	0.972855i	$-0.574335\pi$
$373$	−8.93871	−0.462829	−0.231414	+	0.972855i	$0.574335\pi$
$374$	0	0
$375$	9.76322	0.504171
$376$	0	0
$377$	−14.1623	−0.729394
$378$	0	0
$379$	−8.60719	−0.442122	−0.221061	−	0.975260i	$-0.570952\pi$
$379$	−8.60719	−0.442122	−0.221061	+	0.975260i	$0.570952\pi$
$380$	0	0
$381$	3.06058	0.156798
$382$	0	0
$383$	−20.8998	−1.06793	−0.533965	−	0.845506i	$-0.679299\pi$
$383$	−20.8998	−1.06793	−0.533965	+	0.845506i	$0.679299\pi$
$384$	0	0
$385$	−4.39905	−0.224197
$386$	0	0
$387$	9.70265	0.493213
$388$	0	0
$389$	−18.7911	−0.952749	−0.476375	−	0.879242i	$-0.658049\pi$
$389$	−18.7911	−0.952749	−0.476375	+	0.879242i	$0.658049\pi$
$390$	0	0
$391$	−0.926221	−0.0468410
$392$	0	0
$393$	−10.8913	−0.549395
$394$	0	0
$395$	−9.83076	−0.494639
$396$	0	0
$397$	37.9038	1.90234	0.951169	−	0.308670i	$-0.0998839\pi$
$397$	37.9038	1.90234	0.951169	+	0.308670i	$0.0998839\pi$
$398$	0	0
$399$	2.28415	0.114350
$400$	0	0
$401$	31.0558	1.55086	0.775428	−	0.631437i	$-0.217534\pi$
$401$	31.0558	1.55086	0.775428	+	0.631437i	$0.217534\pi$
$402$	0	0
$403$	48.2570	2.40385
$404$	0	0
$405$	1.11491	0.0554002
$406$	0	0
$407$	7.38585	0.366103
$408$	0	0
$409$	1.53341	0.0758222	0.0379111	−	0.999281i	$-0.487930\pi$
$409$	1.53341	0.0758222	0.0379111	+	0.999281i	$0.487930\pi$
$410$	0	0
$411$	−11.3774	−0.561204
$412$	0	0
$413$	−6.93246	−0.341124
$414$	0	0
$415$	−7.85717	−0.385693
$416$	0	0
$417$	11.8044	0.578062
$418$	0	0
$419$	−23.2361	−1.13516	−0.567578	−	0.823320i	$-0.692119\pi$
$419$	−23.2361	−1.13516	−0.567578	+	0.823320i	$0.692119\pi$
$420$	0	0
$421$	25.4031	1.23807	0.619035	−	0.785364i	$-0.287524\pi$
$421$	25.4031	1.23807	0.619035	+	0.785364i	$0.287524\pi$
$422$	0	0
$423$	7.81756	0.380103
$424$	0	0
$425$	3.47979	0.168795
$426$	0	0
$427$	13.0606	0.632046
$428$	0	0
$429$	21.5940	1.04257
$430$	0	0
$431$	−25.8712	−1.24617	−0.623085	−	0.782154i	$-0.714121\pi$
$431$	−25.8712	−1.24617	−0.623085	+	0.782154i	$0.714121\pi$
$432$	0	0
$433$	23.5613	1.13229	0.566143	−	0.824307i	$-0.308435\pi$
$433$	23.5613	1.13229	0.566143	+	0.824307i	$0.308435\pi$
$434$	0	0
$435$	−2.88509	−0.138330
$436$	0	0
$437$	2.28415	0.109266
$438$	0	0
$439$	−6.10170	−0.291218	−0.145609	−	0.989342i	$-0.546514\pi$
$439$	−6.10170	−0.291218	−0.145609	+	0.989342i	$0.546514\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−10.3510	−0.491789	−0.245895	−	0.969297i	$-0.579082\pi$
$443$	−10.3510	−0.491789	−0.245895	+	0.969297i	$0.579082\pi$
$444$	0	0
$445$	0.0458544	0.00217371
$446$	0	0
$447$	−3.35793	−0.158824
$448$	0	0
$449$	6.02168	0.284181	0.142090	−	0.989854i	$-0.454618\pi$
$449$	6.02168	0.284181	0.142090	+	0.989854i	$0.454618\pi$
$450$	0	0
$451$	−16.9038	−0.795970
$452$	0	0
$453$	6.99304	0.328562
$454$	0	0
$455$	6.10170	0.286052
$456$	0	0
$457$	−30.8448	−1.44286	−0.721429	−	0.692489i	$-0.756514\pi$
$457$	−30.8448	−1.44286	−0.721429	+	0.692489i	$0.756514\pi$
$458$	0	0
$459$	0.926221	0.0432323
$460$	0	0
$461$	24.3726	1.13515	0.567574	−	0.823323i	$-0.307882\pi$
$461$	24.3726	1.13515	0.567574	+	0.823323i	$0.307882\pi$
$462$	0	0
$463$	26.8913	1.24975	0.624873	−	0.780726i	$-0.285151\pi$
$463$	26.8913	1.24975	0.624873	+	0.780726i	$0.285151\pi$
$464$	0	0
$465$	9.83076	0.455891
$466$	0	0
$467$	−27.5459	−1.27467	−0.637336	−	0.770586i	$-0.719964\pi$
$467$	−27.5459	−1.27467	−0.637336	+	0.770586i	$0.719964\pi$
$468$	0	0
$469$	−8.77643	−0.405258
$470$	0	0
$471$	13.2104	0.608702
$472$	0	0
$473$	38.2834	1.76027
$474$	0	0
$475$	−8.58150	−0.393746
$476$	0	0
$477$	4.70265	0.215320
$478$	0	0
$479$	31.7368	1.45009	0.725046	−	0.688700i	$-0.241818\pi$
$479$	31.7368	1.45009	0.725046	+	0.688700i	$0.241818\pi$
$480$	0	0
$481$	−10.2445	−0.467111
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	18.4644	0.838423
$486$	0	0
$487$	24.7911	1.12339	0.561697	−	0.827343i	$-0.310149\pi$
$487$	24.7911	1.12339	0.561697	+	0.827343i	$0.310149\pi$
$488$	0	0
$489$	−4.47283	−0.202269
$490$	0	0
$491$	−35.9450	−1.62217	−0.811086	−	0.584926i	$-0.801123\pi$
$491$	−35.9450	−1.62217	−0.811086	+	0.584926i	$0.801123\pi$
$492$	0	0
$493$	−2.39682	−0.107947
$494$	0	0
$495$	4.39905	0.197723
$496$	0	0
$497$	−12.6483	−0.567355
$498$	0	0
$499$	−30.4938	−1.36509	−0.682545	−	0.730844i	$-0.739127\pi$
$499$	−30.4938	−1.36509	−0.682545	+	0.730844i	$0.739127\pi$
$500$	0	0
$501$	6.06682	0.271045
$502$	0	0
$503$	9.65679	0.430575	0.215288	−	0.976551i	$-0.430931\pi$
$503$	9.65679	0.430575	0.215288	+	0.976551i	$0.430931\pi$
$504$	0	0
$505$	13.4992	0.600708
$506$	0	0
$507$	−16.9519	−0.752861
$508$	0	0
$509$	31.9123	1.41449	0.707244	−	0.706970i	$-0.249938\pi$
$509$	31.9123	1.41449	0.707244	+	0.706970i	$0.249938\pi$
$510$	0	0
$511$	7.87189	0.348232
$512$	0	0
$513$	−2.28415	−0.100848
$514$	0	0
$515$	9.89134	0.435864
$516$	0	0
$517$	30.8455	1.35658
$518$	0	0
$519$	0.735300	0.0322761
$520$	0	0
$521$	−0.108664	−0.00476067	−0.00238034	−	0.999997i	$-0.500758\pi$
$521$	−0.108664	−0.00476067	−0.00238034	+	0.999997i	$0.500758\pi$
$522$	0	0
$523$	24.2508	1.06041	0.530206	−	0.847869i	$-0.322114\pi$
$523$	24.2508	1.06041	0.530206	+	0.847869i	$0.322114\pi$
$524$	0	0
$525$	−3.75698	−0.163968
$526$	0	0
$527$	8.16701	0.355760
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	6.93246	0.300843
$532$	0	0
$533$	23.4464	1.01558
$534$	0	0
$535$	4.38433	0.189551
$536$	0	0
$537$	12.1428	0.524002
$538$	0	0
$539$	3.94567	0.169952
$540$	0	0
$541$	14.5334	0.624840	0.312420	−	0.949944i	$-0.398860\pi$
$541$	14.5334	0.624840	0.312420	+	0.949944i	$0.398860\pi$
$542$	0	0
$543$	−5.99304	−0.257186
$544$	0	0
$545$	−18.4185	−0.788962
$546$	0	0
$547$	41.0187	1.75383	0.876917	−	0.480642i	$-0.159596\pi$
$547$	41.0187	1.75383	0.876917	+	0.480642i	$0.159596\pi$
$548$	0	0
$549$	−13.0606	−0.557412
$550$	0	0
$551$	5.91078	0.251808
$552$	0	0
$553$	8.81756	0.374961
$554$	0	0
$555$	−2.08698	−0.0885875
$556$	0	0
$557$	−8.65055	−0.366536	−0.183268	−	0.983063i	$-0.558668\pi$
$557$	−8.65055	−0.366536	−0.183268	+	0.983063i	$0.558668\pi$
$558$	0	0
$559$	−53.1010	−2.24593
$560$	0	0
$561$	3.65456	0.154296
$562$	0	0
$563$	17.0257	0.717547	0.358774	−	0.933425i	$-0.383195\pi$
$563$	17.0257	0.717547	0.358774	+	0.933425i	$0.383195\pi$
$564$	0	0
$565$	17.7415	0.746392
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	29.8440	1.25112	0.625562	−	0.780174i	$-0.284870\pi$
$569$	29.8440	1.25112	0.625562	+	0.780174i	$0.284870\pi$
$570$	0	0
$571$	12.7228	0.532433	0.266217	−	0.963913i	$-0.414226\pi$
$571$	12.7228	0.532433	0.266217	+	0.963913i	$0.414226\pi$
$572$	0	0
$573$	−13.5613	−0.566533
$574$	0	0
$575$	−3.75698	−0.156677
$576$	0	0
$577$	16.5683	0.689747	0.344874	−	0.938649i	$-0.387922\pi$
$577$	16.5683	0.689747	0.344874	+	0.938649i	$0.387922\pi$
$578$	0	0
$579$	7.24926	0.301269
$580$	0	0
$581$	7.04737	0.292374
$582$	0	0
$583$	18.5551	0.768473
$584$	0	0
$585$	−6.10170	−0.252274
$586$	0	0
$587$	30.7779	1.27034	0.635171	−	0.772372i	$-0.280930\pi$
$587$	30.7779	1.27034	0.635171	+	0.772372i	$0.280930\pi$
$588$	0	0
$589$	−20.1406	−0.829879
$590$	0	0
$591$	−3.62039	−0.148923
$592$	0	0
$593$	−25.5698	−1.05003	−0.525013	−	0.851094i	$-0.675940\pi$
$593$	−25.5698	−1.05003	−0.525013	+	0.851094i	$0.675940\pi$
$594$	0	0
$595$	1.03265	0.0423345
$596$	0	0
$597$	−5.41850	−0.221765
$598$	0	0
$599$	0.974310	0.0398092	0.0199046	−	0.999802i	$-0.493664\pi$
$599$	0.974310	0.0398092	0.0199046	+	0.999802i	$0.493664\pi$
$600$	0	0
$601$	−23.0885	−0.941800	−0.470900	−	0.882187i	$-0.656071\pi$
$601$	−23.0885	−0.941800	−0.470900	+	0.882187i	$0.656071\pi$
$602$	0	0
$603$	8.77643	0.357404
$604$	0	0
$605$	5.09323	0.207069
$606$	0	0
$607$	6.93095	0.281318	0.140659	−	0.990058i	$-0.455078\pi$
$607$	6.93095	0.281318	0.140659	+	0.990058i	$0.455078\pi$
$608$	0	0
$609$	2.58774	0.104861
$610$	0	0
$611$	−42.7842	−1.73086
$612$	0	0
$613$	31.5893	1.27588	0.637939	−	0.770087i	$-0.279787\pi$
$613$	31.5893	1.27588	0.637939	+	0.770087i	$0.279787\pi$
$614$	0	0
$615$	4.77643	0.192604
$616$	0	0
$617$	35.5676	1.43190	0.715948	−	0.698153i	$-0.245995\pi$
$617$	35.5676	1.43190	0.715948	+	0.698153i	$0.245995\pi$
$618$	0	0
$619$	32.8128	1.31886	0.659430	−	0.751766i	$-0.270798\pi$
$619$	32.8128	1.31886	0.659430	+	0.751766i	$0.270798\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−0.0411284	−0.00164778
$624$	0	0
$625$	7.89981	0.315993
$626$	0	0
$627$	−9.01249	−0.359924
$628$	0	0
$629$	−1.73378	−0.0691304
$630$	0	0
$631$	32.6825	1.30107	0.650535	−	0.759477i	$-0.274545\pi$
$631$	32.6825	1.30107	0.650535	+	0.759477i	$0.274545\pi$
$632$	0	0
$633$	−22.5140	−0.894850
$634$	0	0
$635$	−3.41226	−0.135411
$636$	0	0
$637$	−5.47283	−0.216842
$638$	0	0
$639$	12.6483	0.500360
$640$	0	0
$641$	−15.6942	−0.619882	−0.309941	−	0.950756i	$-0.600309\pi$
$641$	−15.6942	−0.619882	−0.309941	+	0.950756i	$0.600309\pi$
$642$	0	0
$643$	7.62887	0.300853	0.150427	−	0.988621i	$-0.451935\pi$
$643$	7.62887	0.300853	0.150427	+	0.988621i	$0.451935\pi$
$644$	0	0
$645$	−10.8176	−0.425941
$646$	0	0
$647$	−3.64431	−0.143273	−0.0716363	−	0.997431i	$-0.522822\pi$
$647$	−3.64431	−0.143273	−0.0716363	+	0.997431i	$0.522822\pi$
$648$	0	0
$649$	27.3532	1.07371
$650$	0	0
$651$	−8.81756	−0.345587
$652$	0	0
$653$	34.9673	1.36838	0.684189	−	0.729305i	$-0.260156\pi$
$653$	34.9673	1.36838	0.684189	+	0.729305i	$0.260156\pi$
$654$	0	0
$655$	12.1428	0.474460
$656$	0	0
$657$	−7.87189	−0.307112
$658$	0	0
$659$	−43.7368	−1.70374	−0.851872	−	0.523750i	$-0.824533\pi$
$659$	−43.7368	−1.70374	−0.851872	+	0.523750i	$0.824533\pi$
$660$	0	0
$661$	−46.9986	−1.82803	−0.914016	−	0.405678i	$-0.867036\pi$
$661$	−46.9986	−1.82803	−0.914016	+	0.405678i	$0.867036\pi$
$662$	0	0
$663$	−5.06905	−0.196866
$664$	0	0
$665$	−2.54661	−0.0987534
$666$	0	0
$667$	2.58774	0.100198
$668$	0	0
$669$	−17.7221	−0.685176
$670$	0	0
$671$	−51.5327	−1.98940
$672$	0	0
$673$	0.245253	0.00945382	0.00472691	−	0.999989i	$-0.498495\pi$
$673$	0.245253	0.00945382	0.00472691	+	0.999989i	$0.498495\pi$
$674$	0	0
$675$	3.75698	0.144606
$676$	0	0
$677$	17.2819	0.664198	0.332099	−	0.943245i	$-0.392243\pi$
$677$	17.2819	0.664198	0.332099	+	0.943245i	$0.392243\pi$
$678$	0	0
$679$	−16.5613	−0.635566
$680$	0	0
$681$	9.00624	0.345120
$682$	0	0
$683$	−15.1127	−0.578270	−0.289135	−	0.957288i	$-0.593368\pi$
$683$	−15.1127	−0.578270	−0.289135	+	0.957288i	$0.593368\pi$
$684$	0	0
$685$	12.6847	0.484658
$686$	0	0
$687$	−6.51173	−0.248438
$688$	0	0
$689$	−25.7368	−0.980495
$690$	0	0
$691$	−14.1234	−0.537279	−0.268639	−	0.963241i	$-0.586574\pi$
$691$	−14.1234	−0.537279	−0.268639	+	0.963241i	$0.586574\pi$
$692$	0	0
$693$	−3.94567	−0.149884
$694$	0	0
$695$	−13.1608	−0.499216
$696$	0	0
$697$	3.96807	0.150301
$698$	0	0
$699$	−7.93942	−0.300297
$700$	0	0
$701$	−48.1204	−1.81748	−0.908742	−	0.417359i	$-0.862956\pi$
$701$	−48.1204	−1.81748	−0.908742	+	0.417359i	$0.862956\pi$
$702$	0	0
$703$	4.27567	0.161260
$704$	0	0
$705$	−8.71585	−0.328258
$706$	0	0
$707$	−12.1079	−0.455366
$708$	0	0
$709$	−3.83772	−0.144129	−0.0720643	−	0.997400i	$-0.522959\pi$
$709$	−3.83772	−0.144129	−0.0720643	+	0.997400i	$0.522959\pi$
$710$	0	0
$711$	−8.81756	−0.330684
$712$	0	0
$713$	−8.81756	−0.330220
$714$	0	0
$715$	−24.0753	−0.900365
$716$	0	0
$717$	−7.42698	−0.277366
$718$	0	0
$719$	53.4108	1.99189	0.995944	−	0.0899774i	$-0.0286795\pi$
$719$	53.4108	1.99189	0.995944	+	0.0899774i	$0.0286795\pi$
$720$	0	0
$721$	−8.87189	−0.330406
$722$	0	0
$723$	−2.05433	−0.0764014
$724$	0	0
$725$	−9.72210	−0.361070
$726$	0	0
$727$	−3.20341	−0.118808	−0.0594039	−	0.998234i	$-0.518920\pi$
$727$	−3.20341	−0.118808	−0.0594039	+	0.998234i	$0.518920\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−8.98680	−0.332389
$732$	0	0
$733$	−46.6197	−1.72194	−0.860968	−	0.508658i	$-0.830142\pi$
$733$	−46.6197	−1.72194	−0.860968	+	0.508658i	$0.830142\pi$
$734$	0	0
$735$	−1.11491	−0.0411240
$736$	0	0
$737$	34.6289	1.27557
$738$	0	0
$739$	34.7787	1.27935	0.639677	−	0.768644i	$-0.279068\pi$
$739$	34.7787	1.27935	0.639677	+	0.768644i	$0.279068\pi$
$740$	0	0
$741$	12.5008	0.459227
$742$	0	0
$743$	−1.09394	−0.0401329	−0.0200664	−	0.999799i	$-0.506388\pi$
$743$	−1.09394	−0.0401329	−0.0200664	+	0.999799i	$0.506388\pi$
$744$	0	0
$745$	3.74378	0.137161
$746$	0	0
$747$	−7.04737	−0.257850
$748$	0	0
$749$	−3.93246	−0.143689
$750$	0	0
$751$	−11.6009	−0.423325	−0.211662	−	0.977343i	$-0.567888\pi$
$751$	−11.6009	−0.423325	−0.211662	+	0.977343i	$0.567888\pi$
$752$	0	0
$753$	−16.7283	−0.609615
$754$	0	0
$755$	−7.79659	−0.283747
$756$	0	0
$757$	26.6825	0.969791	0.484896	−	0.874572i	$-0.338858\pi$
$757$	26.6825	0.969791	0.484896	+	0.874572i	$0.338858\pi$
$758$	0	0
$759$	−3.94567	−0.143219
$760$	0	0
$761$	20.9497	0.759425	0.379713	−	0.925105i	$-0.376023\pi$
$761$	20.9497	0.759425	0.379713	+	0.925105i	$0.376023\pi$
$762$	0	0
$763$	16.5202	0.598072
$764$	0	0
$765$	−1.03265	−0.0373356
$766$	0	0
$767$	−37.9402	−1.36994
$768$	0	0
$769$	11.5918	0.418009	0.209005	−	0.977915i	$-0.432978\pi$
$769$	11.5918	0.418009	0.209005	+	0.977915i	$0.432978\pi$
$770$	0	0
$771$	−7.67000	−0.276228
$772$	0	0
$773$	−45.9277	−1.65191	−0.825953	−	0.563739i	$-0.809362\pi$
$773$	−45.9277	−1.65191	−0.825953	+	0.563739i	$0.809362\pi$
$774$	0	0
$775$	33.1274	1.18997
$776$	0	0
$777$	1.87189	0.0671536
$778$	0	0
$779$	−9.78562	−0.350606
$780$	0	0
$781$	49.9061	1.78578
$782$	0	0
$783$	−2.58774	−0.0924783
$784$	0	0
$785$	−14.7283	−0.525677
$786$	0	0
$787$	32.7585	1.16771	0.583857	−	0.811856i	$-0.301543\pi$
$787$	32.7585	1.16771	0.583857	+	0.811856i	$0.301543\pi$
$788$	0	0
$789$	5.35097	0.190499
$790$	0	0
$791$	−15.9130	−0.565802
$792$	0	0
$793$	71.4784	2.53827
$794$	0	0
$795$	−5.24302	−0.185951
$796$	0	0
$797$	−24.1770	−0.856393	−0.428197	−	0.903686i	$-0.640851\pi$
$797$	−24.1770	−0.856393	−0.428197	+	0.903686i	$0.640851\pi$
$798$	0	0
$799$	−7.24078	−0.256161
$800$	0	0
$801$	0.0411284	0.00145320
$802$	0	0
$803$	−31.0599	−1.09608
$804$	0	0
$805$	−1.11491	−0.0392953
$806$	0	0
$807$	13.5117	0.475635
$808$	0	0
$809$	−29.1777	−1.02583	−0.512917	−	0.858438i	$-0.671435\pi$
$809$	−29.1777	−1.02583	−0.512917	+	0.858438i	$0.671435\pi$
$810$	0	0
$811$	−16.9915	−0.596653	−0.298327	−	0.954464i	$-0.596428\pi$
$811$	−16.9915	−0.596653	−0.298327	+	0.954464i	$0.596428\pi$
$812$	0	0
$813$	−10.4402	−0.366153
$814$	0	0
$815$	4.98680	0.174680
$816$	0	0
$817$	22.1623	0.775360
$818$	0	0
$819$	5.47283	0.191236
$820$	0	0
$821$	37.0793	1.29408	0.647038	−	0.762457i	$-0.276007\pi$
$821$	37.0793	1.29408	0.647038	+	0.762457i	$0.276007\pi$
$822$	0	0
$823$	32.9450	1.14839	0.574194	−	0.818719i	$-0.305315\pi$
$823$	32.9450	1.14839	0.574194	+	0.818719i	$0.305315\pi$
$824$	0	0
$825$	14.8238	0.516098
$826$	0	0
$827$	−30.8936	−1.07427	−0.537137	−	0.843495i	$-0.680494\pi$
$827$	−30.8936	−1.07427	−0.537137	+	0.843495i	$0.680494\pi$
$828$	0	0
$829$	20.1406	0.699512	0.349756	−	0.936841i	$-0.386265\pi$
$829$	20.1406	0.699512	0.349756	+	0.936841i	$0.386265\pi$
$830$	0	0
$831$	−15.4379	−0.535537
$832$	0	0
$833$	−0.926221	−0.0320917
$834$	0	0
$835$	−6.76394	−0.234076
$836$	0	0
$837$	8.81756	0.304779
$838$	0	0
$839$	32.2401	1.11305	0.556525	−	0.830831i	$-0.312134\pi$
$839$	32.2401	1.11305	0.556525	+	0.830831i	$0.312134\pi$
$840$	0	0
$841$	−22.3036	−0.769089
$842$	0	0
$843$	−7.80908	−0.268959
$844$	0	0
$845$	18.8998	0.650173
$846$	0	0
$847$	−4.56829	−0.156968
$848$	0	0
$849$	−10.5466	−0.361959
$850$	0	0
$851$	1.87189	0.0641675
$852$	0	0
$853$	−51.8510	−1.77534	−0.887672	−	0.460476i	$-0.847679\pi$
$853$	−51.8510	−1.77534	−0.887672	+	0.460476i	$0.847679\pi$
$854$	0	0
$855$	2.54661	0.0870923
$856$	0	0
$857$	49.8425	1.70259	0.851294	−	0.524689i	$-0.175818\pi$
$857$	49.8425	1.70259	0.851294	+	0.524689i	$0.175818\pi$
$858$	0	0
$859$	42.3121	1.44367	0.721835	−	0.692066i	$-0.243299\pi$
$859$	42.3121	1.44367	0.721835	+	0.692066i	$0.243299\pi$
$860$	0	0
$861$	−4.28415	−0.146003
$862$	0	0
$863$	−8.48604	−0.288868	−0.144434	−	0.989514i	$-0.546136\pi$
$863$	−8.48604	−0.288868	−0.144434	+	0.989514i	$0.546136\pi$
$864$	0	0
$865$	−0.819791	−0.0278737
$866$	0	0
$867$	16.1421	0.548215
$868$	0	0
$869$	−34.7911	−1.18021
$870$	0	0
$871$	−48.0319	−1.62750
$872$	0	0
$873$	16.5613	0.560516
$874$	0	0
$875$	9.76322	0.330057
$876$	0	0
$877$	−17.5488	−0.592582	−0.296291	−	0.955098i	$-0.595750\pi$
$877$	−17.5488	−0.592582	−0.296291	+	0.955098i	$0.595750\pi$
$878$	0	0
$879$	13.2144	0.445710
$880$	0	0
$881$	19.7313	0.664764	0.332382	−	0.943145i	$-0.392148\pi$
$881$	19.7313	0.664764	0.332382	+	0.943145i	$0.392148\pi$
$882$	0	0
$883$	−46.1873	−1.55432	−0.777162	−	0.629300i	$-0.783342\pi$
$883$	−46.1873	−1.55432	−0.777162	+	0.629300i	$0.783342\pi$
$884$	0	0
$885$	−7.72906	−0.259809
$886$	0	0
$887$	39.8113	1.33673	0.668367	−	0.743832i	$-0.266994\pi$
$887$	39.8113	1.33673	0.668367	+	0.743832i	$0.266994\pi$
$888$	0	0
$889$	3.06058	0.102648
$890$	0	0
$891$	3.94567	0.132185
$892$	0	0
$893$	17.8565	0.597543
$894$	0	0
$895$	−13.5381	−0.452530
$896$	0	0
$897$	5.47283	0.182733
$898$	0	0
$899$	−22.8176	−0.761008
$900$	0	0
$901$	−4.35569	−0.145109
$902$	0	0
$903$	9.70265	0.322884
$904$	0	0
$905$	6.68168	0.222107
$906$	0	0
$907$	−51.5481	−1.71163	−0.855814	−	0.517284i	$-0.826943\pi$
$907$	−51.5481	−1.71163	−0.855814	+	0.517284i	$0.826943\pi$
$908$	0	0
$909$	12.1079	0.401595
$910$	0	0
$911$	−16.7842	−0.556085	−0.278042	−	0.960569i	$-0.589686\pi$
$911$	−16.7842	−0.556085	−0.278042	+	0.960569i	$0.589686\pi$
$912$	0	0
$913$	−27.8066	−0.920264
$914$	0	0
$915$	14.5613	0.481383
$916$	0	0
$917$	−10.8913	−0.359664
$918$	0	0
$919$	27.2508	0.898920	0.449460	−	0.893300i	$-0.351616\pi$
$919$	27.2508	0.898920	0.449460	+	0.893300i	$0.351616\pi$
$920$	0	0
$921$	8.92622	0.294129
$922$	0	0
$923$	−69.2221	−2.27847
$924$	0	0
$925$	−7.03265	−0.231232
$926$	0	0
$927$	8.87189	0.291391
$928$	0	0
$929$	5.04514	0.165526	0.0827628	−	0.996569i	$-0.473626\pi$
$929$	5.04514	0.165526	0.0827628	+	0.996569i	$0.473626\pi$
$930$	0	0
$931$	2.28415	0.0748599
$932$	0	0
$933$	−15.1817	−0.497027
$934$	0	0
$935$	−4.07450	−0.133250
$936$	0	0
$937$	2.64903	0.0865402	0.0432701	−	0.999063i	$-0.486222\pi$
$937$	2.64903	0.0865402	0.0432701	+	0.999063i	$0.486222\pi$
$938$	0	0
$939$	−22.1560	−0.723035
$940$	0	0
$941$	−32.7019	−1.06605	−0.533026	−	0.846099i	$-0.678945\pi$
$941$	−32.7019	−1.06605	−0.533026	+	0.846099i	$0.678945\pi$
$942$	0	0
$943$	−4.28415	−0.139511
$944$	0	0
$945$	1.11491	0.0362679
$946$	0	0
$947$	15.8216	0.514132	0.257066	−	0.966394i	$-0.417244\pi$
$947$	15.8216	0.514132	0.257066	+	0.966394i	$0.417244\pi$
$948$	0	0
$949$	43.0815	1.39849
$950$	0	0
$951$	−12.2081	−0.395876
$952$	0	0
$953$	26.7655	0.867018	0.433509	−	0.901149i	$-0.357275\pi$
$953$	26.7655	0.867018	0.433509	+	0.901149i	$0.357275\pi$
$954$	0	0
$955$	15.1196	0.489260
$956$	0	0
$957$	−10.2104	−0.330054
$958$	0	0
$959$	−11.3774	−0.367395
$960$	0	0
$961$	46.7493	1.50804
$962$	0	0
$963$	3.93246	0.126722
$964$	0	0
$965$	−8.08226	−0.260177
$966$	0	0
$967$	−29.3400	−0.943511	−0.471755	−	0.881729i	$-0.656379\pi$
$967$	−29.3400	−0.943511	−0.471755	+	0.881729i	$0.656379\pi$
$968$	0	0
$969$	2.11562	0.0679637
$970$	0	0
$971$	−20.9309	−0.671706	−0.335853	−	0.941914i	$-0.609025\pi$
$971$	−20.9309	−0.671706	−0.335853	+	0.941914i	$0.609025\pi$
$972$	0	0
$973$	11.8044	0.378430
$974$	0	0
$975$	−20.5613	−0.658490
$976$	0	0
$977$	26.0496	0.833401	0.416700	−	0.909044i	$-0.363186\pi$
$977$	26.0496	0.833401	0.416700	+	0.909044i	$0.363186\pi$
$978$	0	0
$979$	0.162279	0.00518646
$980$	0	0
$981$	−16.5202	−0.527450
$982$	0	0
$983$	−59.7882	−1.90695	−0.953474	−	0.301476i	$-0.902521\pi$
$983$	−59.7882	−1.90695	−0.953474	+	0.301476i	$0.902521\pi$
$984$	0	0
$985$	4.03640	0.128610
$986$	0	0
$987$	7.81756	0.248836
$988$	0	0
$989$	9.70265	0.308526
$990$	0	0
$991$	−30.0062	−0.953180	−0.476590	−	0.879126i	$-0.658127\pi$
$991$	−30.0062	−0.953180	−0.476590	+	0.879126i	$0.658127\pi$
$992$	0	0
$993$	−15.4791	−0.491214
$994$	0	0
$995$	6.04113	0.191517
$996$	0	0
$997$	7.82853	0.247932	0.123966	−	0.992286i	$-0.460439\pi$
$997$	7.82853	0.247932	0.123966	+	0.992286i	$0.460439\pi$
$998$	0	0
$999$	−1.87189	−0.0592239

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.bp.1.3		3
4.3	odd	2		3864.2.a.p.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3864.2.a.p.1.3	✓	3	4.3	odd	2
7728.2.a.bp.1.3		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7728.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.11491 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.11491 q^{5} -1.00000 q^{7} +1.00000 q^{9} +3.94567 q^{11} -5.47283 q^{13} -1.11491 q^{15} -0.926221 q^{17} +2.28415 q^{19} +1.00000 q^{21} +1.00000 q^{23} -3.75698 q^{25} -1.00000 q^{27} +2.58774 q^{29} -8.81756 q^{31} -3.94567 q^{33} -1.11491 q^{35} +1.87189 q^{37} +5.47283 q^{39} -4.28415 q^{41} +9.70265 q^{43} +1.11491 q^{45} +7.81756 q^{47} +1.00000 q^{49} +0.926221 q^{51} +4.70265 q^{53} +4.39905 q^{55} -2.28415 q^{57} +6.93246 q^{59} -13.0606 q^{61} -1.00000 q^{63} -6.10170 q^{65} +8.77643 q^{67} -1.00000 q^{69} +12.6483 q^{71} -7.87189 q^{73} +3.75698 q^{75} -3.94567 q^{77} -8.81756 q^{79} +1.00000 q^{81} -7.04737 q^{83} -1.03265 q^{85} -2.58774 q^{87} +0.0411284 q^{89} +5.47283 q^{91} +8.81756 q^{93} +2.54661 q^{95} +16.5613 q^{97} +3.94567 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 - 3 * q^5 - 3 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9} + q^{11} - 11 q^{13} + 3 q^{15} + 5 q^{19} + 3 q^{21} + 3 q^{23} - 4 q^{25} - 3 q^{27} - 4 q^{29} - 2 q^{31} - q^{33} + 3 q^{35} - 8 q^{37} + 11 q^{39} - 11 q^{41} + 11 q^{43} - 3 q^{45} - q^{47} + 3 q^{49} - 4 q^{53} + 5 q^{55} - 5 q^{57} - 10 q^{59} - 22 q^{61} - 3 q^{63} + 8 q^{65} + 11 q^{67} - 3 q^{69} + 9 q^{71} - 10 q^{73} + 4 q^{75} - q^{77} - 2 q^{79} + 3 q^{81} + 16 q^{83} - 15 q^{85} + 4 q^{87} - 9 q^{89} + 11 q^{91} + 2 q^{93} + 5 q^{95} - 2 q^{97} + q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 3 * q^5 - 3 * q^7 + 3 * q^9 + q^11 - 11 * q^13 + 3 * q^15 + 5 * q^19 + 3 * q^21 + 3 * q^23 - 4 * q^25 - 3 * q^27 - 4 * q^29 - 2 * q^31 - q^33 + 3 * q^35 - 8 * q^37 + 11 * q^39 - 11 * q^41 + 11 * q^43 - 3 * q^45 - q^47 + 3 * q^49 - 4 * q^53 + 5 * q^55 - 5 * q^57 - 10 * q^59 - 22 * q^61 - 3 * q^63 + 8 * q^65 + 11 * q^67 - 3 * q^69 + 9 * q^71 - 10 * q^73 + 4 * q^75 - q^77 - 2 * q^79 + 3 * q^81 + 16 * q^83 - 15 * q^85 + 4 * q^87 - 9 * q^89 + 11 * q^91 + 2 * q^93 + 5 * q^95 - 2 * q^97 + q^99

Introduction

L-functions

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