LMFDB - Embedded newform 8032.2.a.j.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8032,2,Mod(1,8032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8032, 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("8032.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8032 = 2^{5} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8032.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:	$64.1358429035$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	8032.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.19215 q^{3} -0.449856 q^{5} -2.62487 q^{7} -1.57879 q^{9} +O(q^{10})$	$q-1.19215 q^{3} -0.449856 q^{5} -2.62487 q^{7} -1.57879 q^{9} -0.634913 q^{11} +0.888075 q^{13} +0.536293 q^{15} -4.11019 q^{17} -7.30790 q^{19} +3.12923 q^{21} -3.95421 q^{23} -4.79763 q^{25} +5.45858 q^{27} -9.40706 q^{29} +10.3025 q^{31} +0.756908 q^{33} +1.18081 q^{35} -3.75817 q^{37} -1.05871 q^{39} -10.1974 q^{41} -6.79542 q^{43} +0.710227 q^{45} +4.84233 q^{47} -0.110044 q^{49} +4.89994 q^{51} -7.23156 q^{53} +0.285619 q^{55} +8.71208 q^{57} +8.29926 q^{59} +5.40678 q^{61} +4.14412 q^{63} -0.399505 q^{65} -5.99262 q^{67} +4.71400 q^{69} +3.95575 q^{71} -8.81291 q^{73} +5.71947 q^{75} +1.66657 q^{77} -11.5745 q^{79} -1.77106 q^{81} +13.0122 q^{83} +1.84899 q^{85} +11.2146 q^{87} -8.50047 q^{89} -2.33108 q^{91} -12.2821 q^{93} +3.28750 q^{95} +15.0221 q^{97} +1.00239 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + 9 q^{3} + 5 q^{7} + 35 q^{9}+O(q^{10}) $ 30 * q + 9 * q^3 + 5 * q^7 + 35 * q^9	$ 30 q + 9 q^{3} + 5 q^{7} + 35 q^{9} + 31 q^{11} + 5 q^{13} + 8 q^{15} - q^{17} + 29 q^{19} + 6 q^{21} + 27 q^{23} + 34 q^{25} + 36 q^{27} + q^{29} - 9 q^{31} - 8 q^{33} + 51 q^{35} + 5 q^{37} - 8 q^{39} - 3 q^{41} + 43 q^{43} + 42 q^{47} + 41 q^{49} + 54 q^{51} - 11 q^{53} - 10 q^{55} + 2 q^{57} + 49 q^{59} + 21 q^{61} + 15 q^{63} - 14 q^{65} + 68 q^{67} - 10 q^{69} + 44 q^{71} + 25 q^{73} + 51 q^{75} - 20 q^{77} - 49 q^{79} + 6 q^{81} + 88 q^{83} - 5 q^{85} + 16 q^{87} - 16 q^{89} + 46 q^{91} - 4 q^{93} + 28 q^{95} - 4 q^{97} + 71 q^{99}+O(q^{100}) $ 30 * q + 9 * q^3 + 5 * q^7 + 35 * q^9 + 31 * q^11 + 5 * q^13 + 8 * q^15 - q^17 + 29 * q^19 + 6 * q^21 + 27 * q^23 + 34 * q^25 + 36 * q^27 + q^29 - 9 * q^31 - 8 * q^33 + 51 * q^35 + 5 * q^37 - 8 * q^39 - 3 * q^41 + 43 * q^43 + 42 * q^47 + 41 * q^49 + 54 * q^51 - 11 * q^53 - 10 * q^55 + 2 * q^57 + 49 * q^59 + 21 * q^61 + 15 * q^63 - 14 * q^65 + 68 * q^67 - 10 * q^69 + 44 * q^71 + 25 * q^73 + 51 * q^75 - 20 * q^77 - 49 * q^79 + 6 * q^81 + 88 * q^83 - 5 * q^85 + 16 * q^87 - 16 * q^89 + 46 * q^91 - 4 * q^93 + 28 * q^95 - 4 * q^97 + 71 * 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
$3$	−1.19215	−0.688286	−0.344143	−	0.938917i	$-0.611830\pi$
$3$	−1.19215	−0.688286	−0.344143	+	0.938917i	$0.611830\pi$
$4$	0	0
$5$	−0.449856	−0.201181	−0.100591	−	0.994928i	$-0.532073\pi$
$5$	−0.449856	−0.201181	−0.100591	+	0.994928i	$0.532073\pi$
$6$	0	0
$7$	−2.62487	−0.992109	−0.496054	−	0.868292i	$-0.665218\pi$
$7$	−2.62487	−0.992109	−0.496054	+	0.868292i	$0.665218\pi$
$8$	0	0
$9$	−1.57879	−0.526263
$10$	0	0
$11$	−0.634913	−0.191433	−0.0957167	−	0.995409i	$-0.530514\pi$
$11$	−0.634913	−0.191433	−0.0957167	+	0.995409i	$0.530514\pi$
$12$	0	0
$13$	0.888075	0.246308	0.123154	−	0.992388i	$-0.460699\pi$
$13$	0.888075	0.246308	0.123154	+	0.992388i	$0.460699\pi$
$14$	0	0
$15$	0.536293	0.138470
$16$	0	0
$17$	−4.11019	−0.996867	−0.498434	−	0.866928i	$-0.666091\pi$
$17$	−4.11019	−0.996867	−0.498434	+	0.866928i	$0.666091\pi$
$18$	0	0
$19$	−7.30790	−1.67655	−0.838273	−	0.545250i	$-0.816435\pi$
$19$	−7.30790	−1.67655	−0.838273	+	0.545250i	$0.816435\pi$
$20$	0	0
$21$	3.12923	0.682854
$22$	0	0
$23$	−3.95421	−0.824510	−0.412255	−	0.911068i	$-0.635259\pi$
$23$	−3.95421	−0.824510	−0.412255	+	0.911068i	$0.635259\pi$
$24$	0	0
$25$	−4.79763	−0.959526
$26$	0	0
$27$	5.45858	1.05050
$28$	0	0
$29$	−9.40706	−1.74685	−0.873423	−	0.486962i	$-0.838105\pi$
$29$	−9.40706	−1.74685	−0.873423	+	0.486962i	$0.838105\pi$
$30$	0	0
$31$	10.3025	1.85038	0.925191	−	0.379501i	$-0.123904\pi$
$31$	10.3025	1.85038	0.925191	+	0.379501i	$0.123904\pi$
$32$	0	0
$33$	0.756908	0.131761
$34$	0	0
$35$	1.18081	0.199594
$36$	0	0
$37$	−3.75817	−0.617839	−0.308920	−	0.951088i	$-0.599967\pi$
$37$	−3.75817	−0.617839	−0.308920	+	0.951088i	$0.599967\pi$
$38$	0	0
$39$	−1.05871	−0.169530
$40$	0	0
$41$	−10.1974	−1.59257	−0.796285	−	0.604921i	$-0.793205\pi$
$41$	−10.1974	−1.59257	−0.796285	+	0.604921i	$0.793205\pi$
$42$	0	0
$43$	−6.79542	−1.03629	−0.518146	−	0.855292i	$-0.673378\pi$
$43$	−6.79542	−1.03629	−0.518146	+	0.855292i	$0.673378\pi$
$44$	0	0
$45$	0.710227	0.105874
$46$	0	0
$47$	4.84233	0.706326	0.353163	−	0.935562i	$-0.385106\pi$
$47$	4.84233	0.706326	0.353163	+	0.935562i	$0.385106\pi$
$48$	0	0
$49$	−0.110044	−0.0157206
$50$	0	0
$51$	4.89994	0.686129
$52$	0	0
$53$	−7.23156	−0.993332	−0.496666	−	0.867942i	$-0.665443\pi$
$53$	−7.23156	−0.993332	−0.496666	+	0.867942i	$0.665443\pi$
$54$	0	0
$55$	0.285619	0.0385129
$56$	0	0
$57$	8.71208	1.15394
$58$	0	0
$59$	8.29926	1.08047	0.540236	−	0.841514i	$-0.318335\pi$
$59$	8.29926	1.08047	0.540236	+	0.841514i	$0.318335\pi$
$60$	0	0
$61$	5.40678	0.692267	0.346133	−	0.938185i	$-0.387494\pi$
$61$	5.40678	0.692267	0.346133	+	0.938185i	$0.387494\pi$
$62$	0	0
$63$	4.14412	0.522110
$64$	0	0
$65$	−0.399505	−0.0495525
$66$	0	0
$67$	−5.99262	−0.732115	−0.366057	−	0.930592i	$-0.619293\pi$
$67$	−5.99262	−0.732115	−0.366057	+	0.930592i	$0.619293\pi$
$68$	0	0
$69$	4.71400	0.567499
$70$	0	0
$71$	3.95575	0.469462	0.234731	−	0.972060i	$-0.424579\pi$
$71$	3.95575	0.469462	0.234731	+	0.972060i	$0.424579\pi$
$72$	0	0
$73$	−8.81291	−1.03147	−0.515736	−	0.856747i	$-0.672482\pi$
$73$	−8.81291	−1.03147	−0.515736	+	0.856747i	$0.672482\pi$
$74$	0	0
$75$	5.71947	0.660428
$76$	0	0
$77$	1.66657	0.189923
$78$	0	0
$79$	−11.5745	−1.30224	−0.651119	−	0.758976i	$-0.725700\pi$
$79$	−11.5745	−1.30224	−0.651119	+	0.758976i	$0.725700\pi$
$80$	0	0
$81$	−1.77106	−0.196784
$82$	0	0
$83$	13.0122	1.42827	0.714137	−	0.700006i	$-0.246819\pi$
$83$	13.0122	1.42827	0.714137	+	0.700006i	$0.246819\pi$
$84$	0	0
$85$	1.84899	0.200551
$86$	0	0
$87$	11.2146	1.20233
$88$	0	0
$89$	−8.50047	−0.901048	−0.450524	−	0.892764i	$-0.648763\pi$
$89$	−8.50047	−0.901048	−0.450524	+	0.892764i	$0.648763\pi$
$90$	0	0
$91$	−2.33108	−0.244364
$92$	0	0
$93$	−12.2821	−1.27359
$94$	0	0
$95$	3.28750	0.337290
$96$	0	0
$97$	15.0221	1.52527	0.762634	−	0.646831i	$-0.223906\pi$
$97$	15.0221	1.52527	0.762634	+	0.646831i	$0.223906\pi$
$98$	0	0
$99$	1.00239	0.100744
$100$	0	0
$101$	−4.54056	−0.451802	−0.225901	−	0.974150i	$-0.572533\pi$
$101$	−4.54056	−0.451802	−0.225901	+	0.974150i	$0.572533\pi$
$102$	0	0
$103$	−10.9055	−1.07455	−0.537277	−	0.843406i	$-0.680547\pi$
$103$	−10.9055	−1.07455	−0.537277	+	0.843406i	$0.680547\pi$
$104$	0	0
$105$	−1.40770	−0.137378
$106$	0	0
$107$	−15.1910	−1.46857	−0.734287	−	0.678840i	$-0.762483\pi$
$107$	−15.1910	−1.46857	−0.734287	+	0.678840i	$0.762483\pi$
$108$	0	0
$109$	−3.77482	−0.361562	−0.180781	−	0.983523i	$-0.557862\pi$
$109$	−3.77482	−0.361562	−0.180781	+	0.983523i	$0.557862\pi$
$110$	0	0
$111$	4.48029	0.425250
$112$	0	0
$113$	20.8482	1.96123	0.980615	−	0.195942i	$-0.0627764\pi$
$113$	20.8482	1.96123	0.980615	+	0.195942i	$0.0627764\pi$
$114$	0	0
$115$	1.77882	0.165876
$116$	0	0
$117$	−1.40208	−0.129623
$118$	0	0
$119$	10.7887	0.989000
$120$	0	0
$121$	−10.5969	−0.963353
$122$	0	0
$123$	12.1568	1.09614
$124$	0	0
$125$	4.40752	0.394220
$126$	0	0
$127$	−17.9397	−1.59189	−0.795946	−	0.605368i	$-0.793026\pi$
$127$	−17.9397	−1.59189	−0.795946	+	0.605368i	$0.793026\pi$
$128$	0	0
$129$	8.10114	0.713265
$130$	0	0
$131$	−7.29042	−0.636967	−0.318483	−	0.947928i	$-0.603174\pi$
$131$	−7.29042	−0.636967	−0.318483	+	0.947928i	$0.603174\pi$
$132$	0	0
$133$	19.1823	1.66332
$134$	0	0
$135$	−2.45557	−0.211342
$136$	0	0
$137$	−12.5245	−1.07004	−0.535019	−	0.844840i	$-0.679695\pi$
$137$	−12.5245	−1.07004	−0.535019	+	0.844840i	$0.679695\pi$
$138$	0	0
$139$	1.86629	0.158297	0.0791483	−	0.996863i	$-0.474780\pi$
$139$	1.86629	0.158297	0.0791483	+	0.996863i	$0.474780\pi$
$140$	0	0
$141$	−5.77276	−0.486154
$142$	0	0
$143$	−0.563850	−0.0471515
$144$	0	0
$145$	4.23182	0.351433
$146$	0	0
$147$	0.131189	0.0108203
$148$	0	0
$149$	−10.6696	−0.874087	−0.437044	−	0.899440i	$-0.643975\pi$
$149$	−10.6696	−0.874087	−0.437044	+	0.899440i	$0.643975\pi$
$150$	0	0
$151$	6.41381	0.521948	0.260974	−	0.965346i	$-0.415956\pi$
$151$	6.41381	0.521948	0.260974	+	0.965346i	$0.415956\pi$
$152$	0	0
$153$	6.48912	0.524614
$154$	0	0
$155$	−4.63463	−0.372263
$156$	0	0
$157$	−12.5671	−1.00296	−0.501481	−	0.865169i	$-0.667211\pi$
$157$	−12.5671	−1.00296	−0.501481	+	0.865169i	$0.667211\pi$
$158$	0	0
$159$	8.62108	0.683696
$160$	0	0
$161$	10.3793	0.818004
$162$	0	0
$163$	11.8767	0.930258	0.465129	−	0.885243i	$-0.346008\pi$
$163$	11.8767	0.930258	0.465129	+	0.885243i	$0.346008\pi$
$164$	0	0
$165$	−0.340499	−0.0265078
$166$	0	0
$167$	−13.4557	−1.04123	−0.520616	−	0.853791i	$-0.674298\pi$
$167$	−13.4557	−1.04123	−0.520616	+	0.853791i	$0.674298\pi$
$168$	0	0
$169$	−12.2113	−0.939333
$170$	0	0
$171$	11.5376	0.882304
$172$	0	0
$173$	22.2210	1.68943	0.844715	−	0.535217i	$-0.179770\pi$
$173$	22.2210	1.68943	0.844715	+	0.535217i	$0.179770\pi$
$174$	0	0
$175$	12.5932	0.951954
$176$	0	0
$177$	−9.89393	−0.743673
$178$	0	0
$179$	17.1210	1.27968	0.639841	−	0.768507i	$-0.279000\pi$
$179$	17.1210	1.27968	0.639841	+	0.768507i	$0.279000\pi$
$180$	0	0
$181$	23.8565	1.77324	0.886618	−	0.462502i	$-0.153048\pi$
$181$	23.8565	1.77324	0.886618	+	0.462502i	$0.153048\pi$
$182$	0	0
$183$	−6.44566	−0.476477
$184$	0	0
$185$	1.69063	0.124298
$186$	0	0
$187$	2.60961	0.190834
$188$	0	0
$189$	−14.3281	−1.04221
$190$	0	0
$191$	−13.2473	−0.958538	−0.479269	−	0.877668i	$-0.659098\pi$
$191$	−13.2473	−0.958538	−0.479269	+	0.877668i	$0.659098\pi$
$192$	0	0
$193$	−10.6844	−0.769083	−0.384542	−	0.923108i	$-0.625640\pi$
$193$	−10.6844	−0.769083	−0.384542	+	0.923108i	$0.625640\pi$
$194$	0	0
$195$	0.476268	0.0341063
$196$	0	0
$197$	−11.9482	−0.851273	−0.425636	−	0.904894i	$-0.639950\pi$
$197$	−11.9482	−0.851273	−0.425636	+	0.904894i	$0.639950\pi$
$198$	0	0
$199$	2.25980	0.160193	0.0800964	−	0.996787i	$-0.474477\pi$
$199$	2.25980	0.160193	0.0800964	+	0.996787i	$0.474477\pi$
$200$	0	0
$201$	7.14407	0.503904
$202$	0	0
$203$	24.6923	1.73306
$204$	0	0
$205$	4.58737	0.320396
$206$	0	0
$207$	6.24287	0.433909
$208$	0	0
$209$	4.63988	0.320947
$210$	0	0
$211$	21.7569	1.49781	0.748903	−	0.662679i	$-0.230581\pi$
$211$	21.7569	1.49781	0.748903	+	0.662679i	$0.230581\pi$
$212$	0	0
$213$	−4.71584	−0.323124
$214$	0	0
$215$	3.05696	0.208483
$216$	0	0
$217$	−27.0427	−1.83578
$218$	0	0
$219$	10.5063	0.709948
$220$	0	0
$221$	−3.65015	−0.245536
$222$	0	0
$223$	2.59387	0.173698	0.0868491	−	0.996221i	$-0.472320\pi$
$223$	2.59387	0.173698	0.0868491	+	0.996221i	$0.472320\pi$
$224$	0	0
$225$	7.57445	0.504963
$226$	0	0
$227$	25.6950	1.70544	0.852719	−	0.522369i	$-0.174952\pi$
$227$	25.6950	1.70544	0.852719	+	0.522369i	$0.174952\pi$
$228$	0	0
$229$	−21.7487	−1.43719	−0.718597	−	0.695427i	$-0.755215\pi$
$229$	−21.7487	−1.43719	−0.718597	+	0.695427i	$0.755215\pi$
$230$	0	0
$231$	−1.98679	−0.130721
$232$	0	0
$233$	11.9675	0.784018	0.392009	−	0.919961i	$-0.371780\pi$
$233$	11.9675	0.784018	0.392009	+	0.919961i	$0.371780\pi$
$234$	0	0
$235$	−2.17835	−0.142100
$236$	0	0
$237$	13.7985	0.896311
$238$	0	0
$239$	12.9175	0.835562	0.417781	−	0.908548i	$-0.362808\pi$
$239$	12.9175	0.835562	0.417781	+	0.908548i	$0.362808\pi$
$240$	0	0
$241$	−17.2333	−1.11010	−0.555048	−	0.831819i	$-0.687300\pi$
$241$	−17.2333	−1.11010	−0.555048	+	0.831819i	$0.687300\pi$
$242$	0	0
$243$	−14.2644	−0.915061
$244$	0	0
$245$	0.0495040	0.00316269
$246$	0	0
$247$	−6.48996	−0.412946
$248$	0	0
$249$	−15.5124	−0.983061
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	2.51058	0.157839
$254$	0	0
$255$	−2.20427	−0.138037
$256$	0	0
$257$	7.03132	0.438602	0.219301	−	0.975657i	$-0.429622\pi$
$257$	7.03132	0.438602	0.219301	+	0.975657i	$0.429622\pi$
$258$	0	0
$259$	9.86472	0.612964
$260$	0	0
$261$	14.8518	0.919301
$262$	0	0
$263$	−18.5707	−1.14512	−0.572558	−	0.819864i	$-0.694049\pi$
$263$	−18.5707	−1.14512	−0.572558	+	0.819864i	$0.694049\pi$
$264$	0	0
$265$	3.25316	0.199840
$266$	0	0
$267$	10.1338	0.620178
$268$	0	0
$269$	−3.34289	−0.203820	−0.101910	−	0.994794i	$-0.532495\pi$
$269$	−3.34289	−0.203820	−0.101910	+	0.994794i	$0.532495\pi$
$270$	0	0
$271$	−16.1682	−0.982146	−0.491073	−	0.871119i	$-0.663395\pi$
$271$	−16.1682	−0.982146	−0.491073	+	0.871119i	$0.663395\pi$
$272$	0	0
$273$	2.77899	0.168192
$274$	0	0
$275$	3.04608	0.183685
$276$	0	0
$277$	−6.04331	−0.363107	−0.181554	−	0.983381i	$-0.558113\pi$
$277$	−6.04331	−0.363107	−0.181554	+	0.983381i	$0.558113\pi$
$278$	0	0
$279$	−16.2655	−0.973788
$280$	0	0
$281$	0.523616	0.0312363	0.0156182	−	0.999878i	$-0.495028\pi$
$281$	0.523616	0.0312363	0.0156182	+	0.999878i	$0.495028\pi$
$282$	0	0
$283$	23.1328	1.37510	0.687551	−	0.726136i	$-0.258686\pi$
$283$	23.1328	1.37510	0.687551	+	0.726136i	$0.258686\pi$
$284$	0	0
$285$	−3.91918	−0.232152
$286$	0	0
$287$	26.7669	1.58000
$288$	0	0
$289$	−0.106348	−0.00625576
$290$	0	0
$291$	−17.9086	−1.04982
$292$	0	0
$293$	−8.38079	−0.489611	−0.244805	−	0.969572i	$-0.578724\pi$
$293$	−8.38079	−0.489611	−0.244805	+	0.969572i	$0.578724\pi$
$294$	0	0
$295$	−3.73347	−0.217371
$296$	0	0
$297$	−3.46572	−0.201102
$298$	0	0
$299$	−3.51164	−0.203083
$300$	0	0
$301$	17.8371	1.02811
$302$	0	0
$303$	5.41301	0.310969
$304$	0	0
$305$	−2.43227	−0.139271
$306$	0	0
$307$	−9.57955	−0.546734	−0.273367	−	0.961910i	$-0.588137\pi$
$307$	−9.57955	−0.546734	−0.273367	+	0.961910i	$0.588137\pi$
$308$	0	0
$309$	13.0010	0.739600
$310$	0	0
$311$	11.9554	0.677926	0.338963	−	0.940800i	$-0.389924\pi$
$311$	11.9554	0.677926	0.338963	+	0.940800i	$0.389924\pi$
$312$	0	0
$313$	14.8299	0.838233	0.419117	−	0.907932i	$-0.362340\pi$
$313$	14.8299	0.838233	0.419117	+	0.907932i	$0.362340\pi$
$314$	0	0
$315$	−1.86426	−0.105039
$316$	0	0
$317$	−17.6129	−0.989241	−0.494620	−	0.869109i	$-0.664693\pi$
$317$	−17.6129	−0.989241	−0.494620	+	0.869109i	$0.664693\pi$
$318$	0	0
$319$	5.97266	0.334405
$320$	0	0
$321$	18.1099	1.01080
$322$	0	0
$323$	30.0368	1.67129
$324$	0	0
$325$	−4.26065	−0.236339
$326$	0	0
$327$	4.50013	0.248858
$328$	0	0
$329$	−12.7105	−0.700752
$330$	0	0
$331$	11.4171	0.627542	0.313771	−	0.949499i	$-0.398408\pi$
$331$	11.4171	0.627542	0.313771	+	0.949499i	$0.398408\pi$
$332$	0	0
$333$	5.93336	0.325146
$334$	0	0
$335$	2.69581	0.147288
$336$	0	0
$337$	−29.4201	−1.60262	−0.801308	−	0.598252i	$-0.795862\pi$
$337$	−29.4201	−1.60262	−0.801308	+	0.598252i	$0.795862\pi$
$338$	0	0
$339$	−24.8541	−1.34989
$340$	0	0
$341$	−6.54119	−0.354225
$342$	0	0
$343$	18.6630	1.00771
$344$	0	0
$345$	−2.12062	−0.114170
$346$	0	0
$347$	4.81397	0.258427	0.129214	−	0.991617i	$-0.458755\pi$
$347$	4.81397	0.258427	0.129214	+	0.991617i	$0.458755\pi$
$348$	0	0
$349$	31.5833	1.69062	0.845308	−	0.534279i	$-0.179417\pi$
$349$	31.5833	1.69062	0.845308	+	0.534279i	$0.179417\pi$
$350$	0	0
$351$	4.84763	0.258747
$352$	0	0
$353$	22.0827	1.17535	0.587673	−	0.809099i	$-0.300044\pi$
$353$	22.0827	1.17535	0.587673	+	0.809099i	$0.300044\pi$
$354$	0	0
$355$	−1.77952	−0.0944470
$356$	0	0
$357$	−12.8617	−0.680715
$358$	0	0
$359$	−16.9784	−0.896085	−0.448042	−	0.894012i	$-0.647879\pi$
$359$	−16.9784	−0.896085	−0.448042	+	0.894012i	$0.647879\pi$
$360$	0	0
$361$	34.4054	1.81081
$362$	0	0
$363$	12.6330	0.663062
$364$	0	0
$365$	3.96454	0.207513
$366$	0	0
$367$	−5.43670	−0.283793	−0.141897	−	0.989881i	$-0.545320\pi$
$367$	−5.43670	−0.283793	−0.141897	+	0.989881i	$0.545320\pi$
$368$	0	0
$369$	16.0996	0.838111
$370$	0	0
$371$	18.9819	0.985493
$372$	0	0
$373$	17.5881	0.910677	0.455338	−	0.890318i	$-0.349518\pi$
$373$	17.5881	0.910677	0.455338	+	0.890318i	$0.349518\pi$
$374$	0	0
$375$	−5.25440	−0.271336
$376$	0	0
$377$	−8.35417	−0.430262
$378$	0	0
$379$	2.34670	0.120542	0.0602710	−	0.998182i	$-0.480804\pi$
$379$	2.34670	0.120542	0.0602710	+	0.998182i	$0.480804\pi$
$380$	0	0
$381$	21.3867	1.09568
$382$	0	0
$383$	8.37821	0.428107	0.214053	−	0.976822i	$-0.431333\pi$
$383$	8.37821	0.428107	0.214053	+	0.976822i	$0.431333\pi$
$384$	0	0
$385$	−0.749713	−0.0382089
$386$	0	0
$387$	10.7285	0.545362
$388$	0	0
$389$	21.4128	1.08567	0.542836	−	0.839839i	$-0.317350\pi$
$389$	21.4128	1.08567	0.542836	+	0.839839i	$0.317350\pi$
$390$	0	0
$391$	16.2526	0.821927
$392$	0	0
$393$	8.69124	0.438415
$394$	0	0
$395$	5.20687	0.261986
$396$	0	0
$397$	23.8669	1.19785	0.598923	−	0.800807i	$-0.295596\pi$
$397$	23.8669	1.19785	0.598923	+	0.800807i	$0.295596\pi$
$398$	0	0
$399$	−22.8681	−1.14484
$400$	0	0
$401$	−37.1558	−1.85547	−0.927736	−	0.373237i	$-0.878248\pi$
$401$	−37.1558	−1.85547	−0.927736	+	0.373237i	$0.878248\pi$
$402$	0	0
$403$	9.14939	0.455763
$404$	0	0
$405$	0.796720	0.0395894
$406$	0	0
$407$	2.38611	0.118275
$408$	0	0
$409$	9.89145	0.489101	0.244550	−	0.969637i	$-0.421360\pi$
$409$	9.89145	0.489101	0.244550	+	0.969637i	$0.421360\pi$
$410$	0	0
$411$	14.9310	0.736491
$412$	0	0
$413$	−21.7845	−1.07195
$414$	0	0
$415$	−5.85361	−0.287342
$416$	0	0
$417$	−2.22489	−0.108953
$418$	0	0
$419$	−28.5649	−1.39549	−0.697743	−	0.716348i	$-0.745812\pi$
$419$	−28.5649	−1.39549	−0.697743	+	0.716348i	$0.745812\pi$
$420$	0	0
$421$	−7.18100	−0.349981	−0.174990	−	0.984570i	$-0.555989\pi$
$421$	−7.18100	−0.349981	−0.174990	+	0.984570i	$0.555989\pi$
$422$	0	0
$423$	−7.64501	−0.371713
$424$	0	0
$425$	19.7192	0.956520
$426$	0	0
$427$	−14.1921	−0.686804
$428$	0	0
$429$	0.672191	0.0324537
$430$	0	0
$431$	6.71555	0.323477	0.161738	−	0.986834i	$-0.448290\pi$
$431$	6.71555	0.323477	0.161738	+	0.986834i	$0.448290\pi$
$432$	0	0
$433$	24.1309	1.15966	0.579828	−	0.814739i	$-0.303120\pi$
$433$	24.1309	1.15966	0.579828	+	0.814739i	$0.303120\pi$
$434$	0	0
$435$	−5.04494	−0.241886
$436$	0	0
$437$	28.8970	1.38233
$438$	0	0
$439$	20.8365	0.994471	0.497235	−	0.867616i	$-0.334349\pi$
$439$	20.8365	0.994471	0.497235	+	0.867616i	$0.334349\pi$
$440$	0	0
$441$	0.173737	0.00827317
$442$	0	0
$443$	23.4151	1.11249	0.556243	−	0.831020i	$-0.312243\pi$
$443$	23.4151	1.11249	0.556243	+	0.831020i	$0.312243\pi$
$444$	0	0
$445$	3.82398	0.181274
$446$	0	0
$447$	12.7197	0.601622
$448$	0	0
$449$	−1.47440	−0.0695813	−0.0347907	−	0.999395i	$-0.511076\pi$
$449$	−1.47440	−0.0695813	−0.0347907	+	0.999395i	$0.511076\pi$
$450$	0	0
$451$	6.47448	0.304871
$452$	0	0
$453$	−7.64619	−0.359249
$454$	0	0
$455$	1.04865	0.0491615
$456$	0	0
$457$	−9.26964	−0.433616	−0.216808	−	0.976214i	$-0.569564\pi$
$457$	−9.26964	−0.433616	−0.216808	+	0.976214i	$0.569564\pi$
$458$	0	0
$459$	−22.4358	−1.04721
$460$	0	0
$461$	−18.2062	−0.847945	−0.423973	−	0.905675i	$-0.639365\pi$
$461$	−18.2062	−0.847945	−0.423973	+	0.905675i	$0.639365\pi$
$462$	0	0
$463$	−27.6503	−1.28502	−0.642509	−	0.766278i	$-0.722107\pi$
$463$	−27.6503	−1.28502	−0.642509	+	0.766278i	$0.722107\pi$
$464$	0	0
$465$	5.52516	0.256223
$466$	0	0
$467$	11.2746	0.521728	0.260864	−	0.965376i	$-0.415993\pi$
$467$	11.2746	0.521728	0.260864	+	0.965376i	$0.415993\pi$
$468$	0	0
$469$	15.7299	0.726338
$470$	0	0
$471$	14.9818	0.690324
$472$	0	0
$473$	4.31450	0.198381
$474$	0	0
$475$	35.0606	1.60869
$476$	0	0
$477$	11.4171	0.522754
$478$	0	0
$479$	−18.9470	−0.865710	−0.432855	−	0.901464i	$-0.642494\pi$
$479$	−18.9470	−0.865710	−0.432855	+	0.901464i	$0.642494\pi$
$480$	0	0
$481$	−3.33754	−0.152179
$482$	0	0
$483$	−12.3736	−0.563020
$484$	0	0
$485$	−6.75779	−0.306856
$486$	0	0
$487$	0.210212	0.00952563	0.00476282	−	0.999989i	$-0.498484\pi$
$487$	0.210212	0.00952563	0.00476282	+	0.999989i	$0.498484\pi$
$488$	0	0
$489$	−14.1588	−0.640283
$490$	0	0
$491$	15.7828	0.712267	0.356134	−	0.934435i	$-0.384095\pi$
$491$	15.7828	0.712267	0.356134	+	0.934435i	$0.384095\pi$
$492$	0	0
$493$	38.6648	1.74137
$494$	0	0
$495$	−0.450932	−0.0202679
$496$	0	0
$497$	−10.3834	−0.465757
$498$	0	0
$499$	32.4539	1.45284	0.726419	−	0.687252i	$-0.241183\pi$
$499$	32.4539	1.45284	0.726419	+	0.687252i	$0.241183\pi$
$500$	0	0
$501$	16.0411	0.716665
$502$	0	0
$503$	−7.02512	−0.313235	−0.156617	−	0.987659i	$-0.550059\pi$
$503$	−7.02512	−0.313235	−0.156617	+	0.987659i	$0.550059\pi$
$504$	0	0
$505$	2.04260	0.0908943
$506$	0	0
$507$	14.5577	0.646529
$508$	0	0
$509$	22.3264	0.989599	0.494800	−	0.869007i	$-0.335241\pi$
$509$	22.3264	0.989599	0.494800	+	0.869007i	$0.335241\pi$
$510$	0	0
$511$	23.1328	1.02333
$512$	0	0
$513$	−39.8908	−1.76122
$514$	0	0
$515$	4.90592	0.216180
$516$	0	0
$517$	−3.07445	−0.135214
$518$	0	0
$519$	−26.4906	−1.16281
$520$	0	0
$521$	−35.7645	−1.56687	−0.783435	−	0.621474i	$-0.786534\pi$
$521$	−35.7645	−1.56687	−0.783435	+	0.621474i	$0.786534\pi$
$522$	0	0
$523$	−32.0730	−1.40245	−0.701227	−	0.712938i	$-0.747364\pi$
$523$	−32.0730	−1.40245	−0.701227	+	0.712938i	$0.747364\pi$
$524$	0	0
$525$	−15.0129	−0.655216
$526$	0	0
$527$	−42.3452	−1.84459
$528$	0	0
$529$	−7.36420	−0.320183
$530$	0	0
$531$	−13.1028	−0.568613
$532$	0	0
$533$	−9.05608	−0.392262
$534$	0	0
$535$	6.83377	0.295450
$536$	0	0
$537$	−20.4107	−0.880787
$538$	0	0
$539$	0.0698685	0.00300945
$540$	0	0
$541$	1.18399	0.0509035	0.0254518	−	0.999676i	$-0.491898\pi$
$541$	1.18399	0.0509035	0.0254518	+	0.999676i	$0.491898\pi$
$542$	0	0
$543$	−28.4404	−1.22049
$544$	0	0
$545$	1.69812	0.0727395
$546$	0	0
$547$	21.3890	0.914527	0.457264	−	0.889331i	$-0.348830\pi$
$547$	21.3890	0.914527	0.457264	+	0.889331i	$0.348830\pi$
$548$	0	0
$549$	−8.53616	−0.364314
$550$	0	0
$551$	68.7458	2.92867
$552$	0	0
$553$	30.3817	1.29196
$554$	0	0
$555$	−2.01548	−0.0855524
$556$	0	0
$557$	−27.1107	−1.14872	−0.574358	−	0.818605i	$-0.694748\pi$
$557$	−27.1107	−1.14872	−0.574358	+	0.818605i	$0.694748\pi$
$558$	0	0
$559$	−6.03484	−0.255247
$560$	0	0
$561$	−3.11104	−0.131348
$562$	0	0
$563$	−39.7718	−1.67618	−0.838092	−	0.545529i	$-0.816329\pi$
$563$	−39.7718	−1.67618	−0.838092	+	0.545529i	$0.816329\pi$
$564$	0	0
$565$	−9.37866	−0.394563
$566$	0	0
$567$	4.64880	0.195231
$568$	0	0
$569$	−42.5161	−1.78237	−0.891184	−	0.453642i	$-0.850124\pi$
$569$	−42.5161	−1.78237	−0.891184	+	0.453642i	$0.850124\pi$
$570$	0	0
$571$	−5.84621	−0.244656	−0.122328	−	0.992490i	$-0.539036\pi$
$571$	−5.84621	−0.244656	−0.122328	+	0.992490i	$0.539036\pi$
$572$	0	0
$573$	15.7927	0.659748
$574$	0	0
$575$	18.9708	0.791139
$576$	0	0
$577$	−19.3350	−0.804925	−0.402463	−	0.915436i	$-0.631846\pi$
$577$	−19.3350	−0.804925	−0.402463	+	0.915436i	$0.631846\pi$
$578$	0	0
$579$	12.7374	0.529349
$580$	0	0
$581$	−34.1554	−1.41700
$582$	0	0
$583$	4.59141	0.190157
$584$	0	0
$585$	0.630735	0.0260777
$586$	0	0
$587$	21.1491	0.872917	0.436458	−	0.899724i	$-0.356233\pi$
$587$	21.1491	0.872917	0.436458	+	0.899724i	$0.356233\pi$
$588$	0	0
$589$	−75.2896	−3.10225
$590$	0	0
$591$	14.2440	0.585919
$592$	0	0
$593$	−18.9292	−0.777329	−0.388664	−	0.921379i	$-0.627063\pi$
$593$	−18.9292	−0.777329	−0.388664	+	0.921379i	$0.627063\pi$
$594$	0	0
$595$	−4.85337	−0.198969
$596$	0	0
$597$	−2.69401	−0.110258
$598$	0	0
$599$	3.35252	0.136980	0.0684902	−	0.997652i	$-0.478182\pi$
$599$	3.35252	0.136980	0.0684902	+	0.997652i	$0.478182\pi$
$600$	0	0
$601$	−7.58806	−0.309524	−0.154762	−	0.987952i	$-0.549461\pi$
$601$	−7.58806	−0.309524	−0.154762	+	0.987952i	$0.549461\pi$
$602$	0	0
$603$	9.46108	0.385285
$604$	0	0
$605$	4.76707	0.193809
$606$	0	0
$607$	0.461898	0.0187479	0.00937394	−	0.999956i	$-0.497016\pi$
$607$	0.461898	0.0187479	0.00937394	+	0.999956i	$0.497016\pi$
$608$	0	0
$609$	−29.4368	−1.19284
$610$	0	0
$611$	4.30035	0.173973
$612$	0	0
$613$	−7.20512	−0.291012	−0.145506	−	0.989357i	$-0.546481\pi$
$613$	−7.20512	−0.291012	−0.145506	+	0.989357i	$0.546481\pi$
$614$	0	0
$615$	−5.46881	−0.220524
$616$	0	0
$617$	42.4091	1.70733	0.853663	−	0.520825i	$-0.174376\pi$
$617$	42.4091	1.70733	0.853663	+	0.520825i	$0.174376\pi$
$618$	0	0
$619$	−23.6432	−0.950299	−0.475149	−	0.879905i	$-0.657606\pi$
$619$	−23.6432	−0.950299	−0.475149	+	0.879905i	$0.657606\pi$
$620$	0	0
$621$	−21.5844	−0.866152
$622$	0	0
$623$	22.3127	0.893938
$624$	0	0
$625$	22.0054	0.880216
$626$	0	0
$627$	−5.53141	−0.220903
$628$	0	0
$629$	15.4468	0.615904
$630$	0	0
$631$	−15.6613	−0.623468	−0.311734	−	0.950169i	$-0.600910\pi$
$631$	−15.6613	−0.623468	−0.311734	+	0.950169i	$0.600910\pi$
$632$	0	0
$633$	−25.9374	−1.03092
$634$	0	0
$635$	8.07028	0.320259
$636$	0	0
$637$	−0.0977275	−0.00387210
$638$	0	0
$639$	−6.24530	−0.247060
$640$	0	0
$641$	−8.86799	−0.350265	−0.175132	−	0.984545i	$-0.556035\pi$
$641$	−8.86799	−0.350265	−0.175132	+	0.984545i	$0.556035\pi$
$642$	0	0
$643$	−11.3982	−0.449501	−0.224751	−	0.974416i	$-0.572157\pi$
$643$	−11.3982	−0.449501	−0.224751	+	0.974416i	$0.572157\pi$
$644$	0	0
$645$	−3.64434	−0.143496
$646$	0	0
$647$	45.7467	1.79849	0.899245	−	0.437446i	$-0.144117\pi$
$647$	45.7467	1.79849	0.899245	+	0.437446i	$0.144117\pi$
$648$	0	0
$649$	−5.26931	−0.206838
$650$	0	0
$651$	32.2389	1.26354
$652$	0	0
$653$	44.5127	1.74192	0.870958	−	0.491358i	$-0.163499\pi$
$653$	44.5127	1.74192	0.870958	+	0.491358i	$0.163499\pi$
$654$	0	0
$655$	3.27963	0.128146
$656$	0	0
$657$	13.9137	0.542826
$658$	0	0
$659$	−29.6442	−1.15477	−0.577386	−	0.816471i	$-0.695927\pi$
$659$	−29.6442	−1.15477	−0.577386	+	0.816471i	$0.695927\pi$
$660$	0	0
$661$	15.2046	0.591389	0.295695	−	0.955283i	$-0.404449\pi$
$661$	15.2046	0.591389	0.295695	+	0.955283i	$0.404449\pi$
$662$	0	0
$663$	4.35152	0.168999
$664$	0	0
$665$	−8.62926	−0.334628
$666$	0	0
$667$	37.1975	1.44029
$668$	0	0
$669$	−3.09227	−0.119554
$670$	0	0
$671$	−3.43283	−0.132523
$672$	0	0
$673$	51.2410	1.97520	0.987598	−	0.157006i	$-0.0501842\pi$
$673$	51.2410	1.97520	0.987598	+	0.157006i	$0.0501842\pi$
$674$	0	0
$675$	−26.1883	−1.00799
$676$	0	0
$677$	24.6356	0.946823	0.473412	−	0.880841i	$-0.343022\pi$
$677$	24.6356	0.946823	0.473412	+	0.880841i	$0.343022\pi$
$678$	0	0
$679$	−39.4312	−1.51323
$680$	0	0
$681$	−30.6322	−1.17383
$682$	0	0
$683$	4.35892	0.166789	0.0833947	−	0.996517i	$-0.473424\pi$
$683$	4.35892	0.166789	0.0833947	+	0.996517i	$0.473424\pi$
$684$	0	0
$685$	5.63420	0.215272
$686$	0	0
$687$	25.9276	0.989200
$688$	0	0
$689$	−6.42217	−0.244665
$690$	0	0
$691$	−9.19025	−0.349614	−0.174807	−	0.984603i	$-0.555930\pi$
$691$	−9.19025	−0.349614	−0.174807	+	0.984603i	$0.555930\pi$
$692$	0	0
$693$	−2.63115	−0.0999493
$694$	0	0
$695$	−0.839561	−0.0318464
$696$	0	0
$697$	41.9133	1.58758
$698$	0	0
$699$	−14.2670	−0.539628
$700$	0	0
$701$	24.6693	0.931745	0.465873	−	0.884852i	$-0.345740\pi$
$701$	24.6693	0.931745	0.465873	+	0.884852i	$0.345740\pi$
$702$	0	0
$703$	27.4643	1.03584
$704$	0	0
$705$	2.59691	0.0978051
$706$	0	0
$707$	11.9184	0.448237
$708$	0	0
$709$	3.75300	0.140947	0.0704735	−	0.997514i	$-0.477549\pi$
$709$	3.75300	0.140947	0.0704735	+	0.997514i	$0.477549\pi$
$710$	0	0
$711$	18.2738	0.685319
$712$	0	0
$713$	−40.7383	−1.52566
$714$	0	0
$715$	0.253651	0.00948601
$716$	0	0
$717$	−15.3995	−0.575106
$718$	0	0
$719$	−27.6604	−1.03156	−0.515780	−	0.856721i	$-0.672498\pi$
$719$	−27.6604	−1.03156	−0.515780	+	0.856721i	$0.672498\pi$
$720$	0	0
$721$	28.6256	1.06607
$722$	0	0
$723$	20.5446	0.764062
$724$	0	0
$725$	45.1316	1.67614
$726$	0	0
$727$	−40.8624	−1.51550	−0.757750	−	0.652545i	$-0.773702\pi$
$727$	−40.8624	−1.51550	−0.757750	+	0.652545i	$0.773702\pi$
$728$	0	0
$729$	22.3184	0.826608
$730$	0	0
$731$	27.9305	1.03305
$732$	0	0
$733$	−30.4984	−1.12649	−0.563243	−	0.826292i	$-0.690446\pi$
$733$	−30.4984	−1.12649	−0.563243	+	0.826292i	$0.690446\pi$
$734$	0	0
$735$	−0.0590160	−0.00217684
$736$	0	0
$737$	3.80479	0.140151
$738$	0	0
$739$	49.3073	1.81380	0.906899	−	0.421348i	$-0.138443\pi$
$739$	49.3073	1.81380	0.906899	+	0.421348i	$0.138443\pi$
$740$	0	0
$741$	7.73698	0.284225
$742$	0	0
$743$	−30.2430	−1.10951	−0.554755	−	0.832014i	$-0.687188\pi$
$743$	−30.2430	−1.10951	−0.554755	+	0.832014i	$0.687188\pi$
$744$	0	0
$745$	4.79978	0.175850
$746$	0	0
$747$	−20.5435	−0.751648
$748$	0	0
$749$	39.8745	1.45698
$750$	0	0
$751$	−16.2375	−0.592516	−0.296258	−	0.955108i	$-0.595739\pi$
$751$	−16.2375	−0.592516	−0.296258	+	0.955108i	$0.595739\pi$
$752$	0	0
$753$	1.19215	0.0434442
$754$	0	0
$755$	−2.88529	−0.105006
$756$	0	0
$757$	−4.79010	−0.174099	−0.0870496	−	0.996204i	$-0.527744\pi$
$757$	−4.79010	−0.174099	−0.0870496	+	0.996204i	$0.527744\pi$
$758$	0	0
$759$	−2.99298	−0.108638
$760$	0	0
$761$	26.7280	0.968889	0.484444	−	0.874822i	$-0.339022\pi$
$761$	26.7280	0.968889	0.484444	+	0.874822i	$0.339022\pi$
$762$	0	0
$763$	9.90841	0.358708
$764$	0	0
$765$	−2.91917	−0.105543
$766$	0	0
$767$	7.37037	0.266129
$768$	0	0
$769$	−39.1560	−1.41200	−0.706001	−	0.708210i	$-0.749503\pi$
$769$	−39.1560	−1.41200	−0.706001	+	0.708210i	$0.749503\pi$
$770$	0	0
$771$	−8.38236	−0.301883
$772$	0	0
$773$	−32.0519	−1.15283	−0.576414	−	0.817158i	$-0.695548\pi$
$773$	−32.0519	−1.15283	−0.576414	+	0.817158i	$0.695548\pi$
$774$	0	0
$775$	−49.4276	−1.77549
$776$	0	0
$777$	−11.7602	−0.421894
$778$	0	0
$779$	74.5217	2.67002
$780$	0	0
$781$	−2.51156	−0.0898707
$782$	0	0
$783$	−51.3492	−1.83507
$784$	0	0
$785$	5.65337	0.201777
$786$	0	0
$787$	−12.9859	−0.462899	−0.231449	−	0.972847i	$-0.574347\pi$
$787$	−12.9859	−0.462899	−0.231449	+	0.972847i	$0.574347\pi$
$788$	0	0
$789$	22.1389	0.788167
$790$	0	0
$791$	−54.7238	−1.94575
$792$	0	0
$793$	4.80162	0.170511
$794$	0	0
$795$	−3.87824	−0.137547
$796$	0	0
$797$	2.83909	0.100566	0.0502828	−	0.998735i	$-0.483988\pi$
$797$	2.83909	0.100566	0.0502828	+	0.998735i	$0.483988\pi$
$798$	0	0
$799$	−19.9029	−0.704113
$800$	0	0
$801$	13.4204	0.474188
$802$	0	0
$803$	5.59543	0.197458
$804$	0	0
$805$	−4.66919	−0.164567
$806$	0	0
$807$	3.98521	0.140286
$808$	0	0
$809$	7.93281	0.278903	0.139451	−	0.990229i	$-0.455466\pi$
$809$	7.93281	0.278903	0.139451	+	0.990229i	$0.455466\pi$
$810$	0	0
$811$	15.8287	0.555821	0.277911	−	0.960607i	$-0.410358\pi$
$811$	15.8287	0.555821	0.277911	+	0.960607i	$0.410358\pi$
$812$	0	0
$813$	19.2748	0.675997
$814$	0	0
$815$	−5.34282	−0.187151
$816$	0	0
$817$	49.6603	1.73739
$818$	0	0
$819$	3.68029	0.128600
$820$	0	0
$821$	−13.5142	−0.471649	−0.235824	−	0.971796i	$-0.575779\pi$
$821$	−13.5142	−0.471649	−0.235824	+	0.971796i	$0.575779\pi$
$822$	0	0
$823$	−20.8645	−0.727291	−0.363645	−	0.931537i	$-0.618468\pi$
$823$	−20.8645	−0.727291	−0.363645	+	0.931537i	$0.618468\pi$
$824$	0	0
$825$	−3.63137	−0.126428
$826$	0	0
$827$	53.4782	1.85962	0.929810	−	0.368041i	$-0.119971\pi$
$827$	53.4782	1.85962	0.929810	+	0.368041i	$0.119971\pi$
$828$	0	0
$829$	−13.9859	−0.485751	−0.242876	−	0.970057i	$-0.578091\pi$
$829$	−13.9859	−0.485751	−0.242876	+	0.970057i	$0.578091\pi$
$830$	0	0
$831$	7.20451	0.249922
$832$	0	0
$833$	0.452302	0.0156714
$834$	0	0
$835$	6.05312	0.209477
$836$	0	0
$837$	56.2370	1.94384
$838$	0	0
$839$	0.978960	0.0337974	0.0168987	−	0.999857i	$-0.494621\pi$
$839$	0.978960	0.0337974	0.0168987	+	0.999857i	$0.494621\pi$
$840$	0	0
$841$	59.4927	2.05147
$842$	0	0
$843$	−0.624227	−0.0214995
$844$	0	0
$845$	5.49333	0.188976
$846$	0	0
$847$	27.8155	0.955751
$848$	0	0
$849$	−27.5777	−0.946463
$850$	0	0
$851$	14.8606	0.509415
$852$	0	0
$853$	−55.3949	−1.89668	−0.948342	−	0.317249i	$-0.897241\pi$
$853$	−55.3949	−1.89668	−0.948342	+	0.317249i	$0.897241\pi$
$854$	0	0
$855$	−5.19027	−0.177503
$856$	0	0
$857$	26.7605	0.914122	0.457061	−	0.889435i	$-0.348902\pi$
$857$	26.7605	0.914122	0.457061	+	0.889435i	$0.348902\pi$
$858$	0	0
$859$	16.0751	0.548475	0.274237	−	0.961662i	$-0.411575\pi$
$859$	16.0751	0.548475	0.274237	+	0.961662i	$0.411575\pi$
$860$	0	0
$861$	−31.9101	−1.08749
$862$	0	0
$863$	−26.4778	−0.901315	−0.450658	−	0.892697i	$-0.648810\pi$
$863$	−26.4778	−0.901315	−0.450658	+	0.892697i	$0.648810\pi$
$864$	0	0
$865$	−9.99623	−0.339882
$866$	0	0
$867$	0.126782	0.00430575
$868$	0	0
$869$	7.34882	0.249292
$870$	0	0
$871$	−5.32189	−0.180325
$872$	0	0
$873$	−23.7168	−0.802692
$874$	0	0
$875$	−11.5692	−0.391109
$876$	0	0
$877$	−43.7184	−1.47626	−0.738132	−	0.674656i	$-0.764292\pi$
$877$	−43.7184	−1.47626	−0.738132	+	0.674656i	$0.764292\pi$
$878$	0	0
$879$	9.99112	0.336992
$880$	0	0
$881$	31.2197	1.05182	0.525909	−	0.850541i	$-0.323725\pi$
$881$	31.2197	1.05182	0.525909	+	0.850541i	$0.323725\pi$
$882$	0	0
$883$	−10.5012	−0.353393	−0.176696	−	0.984265i	$-0.556541\pi$
$883$	−10.5012	−0.353393	−0.176696	+	0.984265i	$0.556541\pi$
$884$	0	0
$885$	4.45084	0.149613
$886$	0	0
$887$	35.6202	1.19601	0.598004	−	0.801493i	$-0.295961\pi$
$887$	35.6202	1.19601	0.598004	+	0.801493i	$0.295961\pi$
$888$	0	0
$889$	47.0894	1.57933
$890$	0	0
$891$	1.12447	0.0376711
$892$	0	0
$893$	−35.3872	−1.18419
$894$	0	0
$895$	−7.70197	−0.257448
$896$	0	0
$897$	4.18638	0.139779
$898$	0	0
$899$	−96.9161	−3.23233
$900$	0	0
$901$	29.7231	0.990220
$902$	0	0
$903$	−21.2644	−0.707637
$904$	0	0
$905$	−10.7320	−0.356742
$906$	0	0
$907$	−5.61693	−0.186507	−0.0932535	−	0.995642i	$-0.529727\pi$
$907$	−5.61693	−0.186507	−0.0932535	+	0.995642i	$0.529727\pi$
$908$	0	0
$909$	7.16858	0.237767
$910$	0	0
$911$	−34.0928	−1.12955	−0.564773	−	0.825246i	$-0.691036\pi$
$911$	−34.0928	−1.12955	−0.564773	+	0.825246i	$0.691036\pi$
$912$	0	0
$913$	−8.26161	−0.273420
$914$	0	0
$915$	2.89962	0.0958584
$916$	0	0
$917$	19.1364	0.631940
$918$	0	0
$919$	−17.1532	−0.565832	−0.282916	−	0.959145i	$-0.591302\pi$
$919$	−17.1532	−0.565832	−0.282916	+	0.959145i	$0.591302\pi$
$920$	0	0
$921$	11.4202	0.376309
$922$	0	0
$923$	3.51301	0.115632
$924$	0	0
$925$	18.0303	0.592833
$926$	0	0
$927$	17.2175	0.565498
$928$	0	0
$929$	2.86495	0.0939959	0.0469980	−	0.998895i	$-0.485035\pi$
$929$	2.86495	0.0939959	0.0469980	+	0.998895i	$0.485035\pi$
$930$	0	0
$931$	0.804192	0.0263563
$932$	0	0
$933$	−14.2525	−0.466607
$934$	0	0
$935$	−1.17395	−0.0383922
$936$	0	0
$937$	13.8816	0.453491	0.226745	−	0.973954i	$-0.427191\pi$
$937$	13.8816	0.453491	0.226745	+	0.973954i	$0.427191\pi$
$938$	0	0
$939$	−17.6793	−0.576944
$940$	0	0
$941$	−44.4658	−1.44954	−0.724772	−	0.688989i	$-0.758055\pi$
$941$	−44.4658	−1.44954	−0.724772	+	0.688989i	$0.758055\pi$
$942$	0	0
$943$	40.3228	1.31309
$944$	0	0
$945$	6.44557	0.209674
$946$	0	0
$947$	−59.9018	−1.94655	−0.973274	−	0.229647i	$-0.926243\pi$
$947$	−59.9018	−1.94655	−0.973274	+	0.229647i	$0.926243\pi$
$948$	0	0
$949$	−7.82652	−0.254060
$950$	0	0
$951$	20.9972	0.680880
$952$	0	0
$953$	54.4641	1.76427	0.882133	−	0.471001i	$-0.156107\pi$
$953$	54.4641	1.76427	0.882133	+	0.471001i	$0.156107\pi$
$954$	0	0
$955$	5.95935	0.192840
$956$	0	0
$957$	−7.12028	−0.230166
$958$	0	0
$959$	32.8751	1.06159
$960$	0	0
$961$	75.1414	2.42392
$962$	0	0
$963$	23.9834	0.772856
$964$	0	0
$965$	4.80646	0.154725
$966$	0	0
$967$	−58.9325	−1.89514	−0.947571	−	0.319546i	$-0.896470\pi$
$967$	−58.9325	−1.89514	−0.947571	+	0.319546i	$0.896470\pi$
$968$	0	0
$969$	−35.8083	−1.15033
$970$	0	0
$971$	23.6222	0.758071	0.379036	−	0.925382i	$-0.376256\pi$
$971$	23.6222	0.758071	0.379036	+	0.925382i	$0.376256\pi$
$972$	0	0
$973$	−4.89877	−0.157047
$974$	0	0
$975$	5.07932	0.162668
$976$	0	0
$977$	8.04677	0.257439	0.128719	−	0.991681i	$-0.458913\pi$
$977$	8.04677	0.257439	0.128719	+	0.991681i	$0.458913\pi$
$978$	0	0
$979$	5.39706	0.172491
$980$	0	0
$981$	5.95964	0.190277
$982$	0	0
$983$	−7.84901	−0.250345	−0.125172	−	0.992135i	$-0.539948\pi$
$983$	−7.84901	−0.250345	−0.125172	+	0.992135i	$0.539948\pi$
$984$	0	0
$985$	5.37496	0.171260
$986$	0	0
$987$	15.1527	0.482317
$988$	0	0
$989$	26.8706	0.854434
$990$	0	0
$991$	29.5460	0.938561	0.469280	−	0.883049i	$-0.344513\pi$
$991$	29.5460	0.938561	0.469280	+	0.883049i	$0.344513\pi$
$992$	0	0
$993$	−13.6109	−0.431928
$994$	0	0
$995$	−1.01658	−0.0322278
$996$	0	0
$997$	−12.6667	−0.401159	−0.200580	−	0.979677i	$-0.564283\pi$
$997$	−12.6667	−0.401159	−0.200580	+	0.979677i	$0.564283\pi$
$998$	0	0
$999$	−20.5143	−0.649043

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8032.2.a.j.1.10	yes	30
4.3	odd	2		8032.2.a.g.1.21	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8032.2.a.g.1.21	✓	30	4.3	odd	2
8032.2.a.j.1.10	yes	30	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:	\( N \)	\(=\)	\( 8032 = 2^{5} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8032.a (trivial)

\(f(q)\)	\(=\)	\(q-1.19215 q^{3} -0.449856 q^{5} -2.62487 q^{7} -1.57879 q^{9} +O(q^{10})\)	\(q-1.19215 q^{3} -0.449856 q^{5} -2.62487 q^{7} -1.57879 q^{9} -0.634913 q^{11} +0.888075 q^{13} +0.536293 q^{15} -4.11019 q^{17} -7.30790 q^{19} +3.12923 q^{21} -3.95421 q^{23} -4.79763 q^{25} +5.45858 q^{27} -9.40706 q^{29} +10.3025 q^{31} +0.756908 q^{33} +1.18081 q^{35} -3.75817 q^{37} -1.05871 q^{39} -10.1974 q^{41} -6.79542 q^{43} +0.710227 q^{45} +4.84233 q^{47} -0.110044 q^{49} +4.89994 q^{51} -7.23156 q^{53} +0.285619 q^{55} +8.71208 q^{57} +8.29926 q^{59} +5.40678 q^{61} +4.14412 q^{63} -0.399505 q^{65} -5.99262 q^{67} +4.71400 q^{69} +3.95575 q^{71} -8.81291 q^{73} +5.71947 q^{75} +1.66657 q^{77} -11.5745 q^{79} -1.77106 q^{81} +13.0122 q^{83} +1.84899 q^{85} +11.2146 q^{87} -8.50047 q^{89} -2.33108 q^{91} -12.2821 q^{93} +3.28750 q^{95} +15.0221 q^{97} +1.00239 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 9 q^{3} + 5 q^{7} + 35 q^{9}+O(q^{10}) \) 30 * q + 9 * q^3 + 5 * q^7 + 35 * q^9	\( 30 q + 9 q^{3} + 5 q^{7} + 35 q^{9} + 31 q^{11} + 5 q^{13} + 8 q^{15} - q^{17} + 29 q^{19} + 6 q^{21} + 27 q^{23} + 34 q^{25} + 36 q^{27} + q^{29} - 9 q^{31} - 8 q^{33} + 51 q^{35} + 5 q^{37} - 8 q^{39} - 3 q^{41} + 43 q^{43} + 42 q^{47} + 41 q^{49} + 54 q^{51} - 11 q^{53} - 10 q^{55} + 2 q^{57} + 49 q^{59} + 21 q^{61} + 15 q^{63} - 14 q^{65} + 68 q^{67} - 10 q^{69} + 44 q^{71} + 25 q^{73} + 51 q^{75} - 20 q^{77} - 49 q^{79} + 6 q^{81} + 88 q^{83} - 5 q^{85} + 16 q^{87} - 16 q^{89} + 46 q^{91} - 4 q^{93} + 28 q^{95} - 4 q^{97} + 71 q^{99}+O(q^{100}) \) 30 * q + 9 * q^3 + 5 * q^7 + 35 * q^9 + 31 * q^11 + 5 * q^13 + 8 * q^15 - q^17 + 29 * q^19 + 6 * q^21 + 27 * q^23 + 34 * q^25 + 36 * q^27 + q^29 - 9 * q^31 - 8 * q^33 + 51 * q^35 + 5 * q^37 - 8 * q^39 - 3 * q^41 + 43 * q^43 + 42 * q^47 + 41 * q^49 + 54 * q^51 - 11 * q^53 - 10 * q^55 + 2 * q^57 + 49 * q^59 + 21 * q^61 + 15 * q^63 - 14 * q^65 + 68 * q^67 - 10 * q^69 + 44 * q^71 + 25 * q^73 + 51 * q^75 - 20 * q^77 - 49 * q^79 + 6 * q^81 + 88 * q^83 - 5 * q^85 + 16 * q^87 - 16 * q^89 + 46 * q^91 - 4 * q^93 + 28 * q^95 - 4 * q^97 + 71 * q^99

Introduction

L-functions

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