LMFDB - Embedded newform 4024.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4024,2,Mod(1,4024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4024, 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("4024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4024 = 2^{3} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4024.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:	$32.1318017734$
Analytic rank:	$1$
Dimension:	$29$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.34526 q^{3} -0.499712 q^{5} +1.77499 q^{7} +8.19075 q^{9} +O(q^{10})$	$q-3.34526 q^{3} -0.499712 q^{5} +1.77499 q^{7} +8.19075 q^{9} -4.17709 q^{11} +4.80122 q^{13} +1.67166 q^{15} -7.81552 q^{17} +5.65618 q^{19} -5.93781 q^{21} -1.25017 q^{23} -4.75029 q^{25} -17.3644 q^{27} +2.71351 q^{29} -6.02507 q^{31} +13.9734 q^{33} -0.886984 q^{35} +5.49913 q^{37} -16.0613 q^{39} +6.76048 q^{41} +0.148338 q^{43} -4.09301 q^{45} -2.60866 q^{47} -3.84940 q^{49} +26.1449 q^{51} +9.53625 q^{53} +2.08734 q^{55} -18.9214 q^{57} -6.44639 q^{59} +6.69895 q^{61} +14.5385 q^{63} -2.39923 q^{65} +2.05614 q^{67} +4.18213 q^{69} -13.1917 q^{71} -1.01876 q^{73} +15.8909 q^{75} -7.41430 q^{77} +3.52375 q^{79} +33.5162 q^{81} +14.0162 q^{83} +3.90550 q^{85} -9.07740 q^{87} -9.84101 q^{89} +8.52213 q^{91} +20.1554 q^{93} -2.82646 q^{95} +18.8622 q^{97} -34.2135 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9}+O(q^{10}) $ 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9	$ 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9} - 27 q^{11} + 16 q^{13} - 14 q^{15} - 15 q^{17} - 14 q^{19} + q^{21} - 25 q^{23} + 21 q^{25} - 25 q^{27} - 13 q^{29} - 27 q^{31} - 9 q^{33} - 29 q^{35} + 35 q^{37} - 38 q^{39} - 30 q^{41} - 38 q^{43} + q^{45} - 35 q^{47} + 14 q^{49} - 21 q^{51} + 2 q^{53} - 25 q^{55} - 25 q^{57} - 40 q^{59} + 10 q^{61} - 56 q^{63} - 50 q^{65} - 31 q^{67} + 11 q^{69} - 65 q^{71} - 23 q^{73} - 32 q^{75} + 13 q^{77} - 44 q^{79} - 7 q^{81} - 41 q^{83} + 26 q^{85} - 25 q^{87} - 48 q^{89} - 44 q^{91} + 25 q^{93} - 75 q^{95} - 18 q^{97} - 80 q^{99}+O(q^{100}) $ 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9 - 27 * q^11 + 16 * q^13 - 14 * q^15 - 15 * q^17 - 14 * q^19 + q^21 - 25 * q^23 + 21 * q^25 - 25 * q^27 - 13 * q^29 - 27 * q^31 - 9 * q^33 - 29 * q^35 + 35 * q^37 - 38 * q^39 - 30 * q^41 - 38 * q^43 + q^45 - 35 * q^47 + 14 * q^49 - 21 * q^51 + 2 * q^53 - 25 * q^55 - 25 * q^57 - 40 * q^59 + 10 * q^61 - 56 * q^63 - 50 * q^65 - 31 * q^67 + 11 * q^69 - 65 * q^71 - 23 * q^73 - 32 * q^75 + 13 * q^77 - 44 * q^79 - 7 * q^81 - 41 * q^83 + 26 * q^85 - 25 * q^87 - 48 * q^89 - 44 * q^91 + 25 * q^93 - 75 * q^95 - 18 * q^97 - 80 * 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
$3$	−3.34526	−1.93139	−0.965693	−	0.259687i	$-0.916381\pi$
$3$	−3.34526	−1.93139	−0.965693	+	0.259687i	$0.916381\pi$
$4$	0	0
$5$	−0.499712	−0.223478	−0.111739	−	0.993738i	$-0.535642\pi$
$5$	−0.499712	−0.223478	−0.111739	+	0.993738i	$0.535642\pi$
$6$	0	0
$7$	1.77499	0.670884	0.335442	−	0.942061i	$-0.391114\pi$
$7$	1.77499	0.670884	0.335442	+	0.942061i	$0.391114\pi$
$8$	0	0
$9$	8.19075	2.73025
$10$	0	0
$11$	−4.17709	−1.25944	−0.629720	−	0.776822i	$-0.716830\pi$
$11$	−4.17709	−1.25944	−0.629720	+	0.776822i	$0.716830\pi$
$12$	0	0
$13$	4.80122	1.33162	0.665810	−	0.746122i	$-0.268086\pi$
$13$	4.80122	1.33162	0.665810	+	0.746122i	$0.268086\pi$
$14$	0	0
$15$	1.67166	0.431622
$16$	0	0
$17$	−7.81552	−1.89554	−0.947771	−	0.318952i	$-0.896669\pi$
$17$	−7.81552	−1.89554	−0.947771	+	0.318952i	$0.896669\pi$
$18$	0	0
$19$	5.65618	1.29762	0.648809	−	0.760952i	$-0.275268\pi$
$19$	5.65618	1.29762	0.648809	+	0.760952i	$0.275268\pi$
$20$	0	0
$21$	−5.93781	−1.29574
$22$	0	0
$23$	−1.25017	−0.260678	−0.130339	−	0.991470i	$-0.541607\pi$
$23$	−1.25017	−0.260678	−0.130339	+	0.991470i	$0.541607\pi$
$24$	0	0
$25$	−4.75029	−0.950058
$26$	0	0
$27$	−17.3644	−3.34178
$28$	0	0
$29$	2.71351	0.503887	0.251943	−	0.967742i	$-0.418930\pi$
$29$	2.71351	0.503887	0.251943	+	0.967742i	$0.418930\pi$
$30$	0	0
$31$	−6.02507	−1.08213	−0.541067	−	0.840980i	$-0.681979\pi$
$31$	−6.02507	−1.08213	−0.541067	+	0.840980i	$0.681979\pi$
$32$	0	0
$33$	13.9734	2.43246
$34$	0	0
$35$	−0.886984	−0.149928
$36$	0	0
$37$	5.49913	0.904051	0.452026	−	0.892005i	$-0.350702\pi$
$37$	5.49913	0.904051	0.452026	+	0.892005i	$0.350702\pi$
$38$	0	0
$39$	−16.0613	−2.57187
$40$	0	0
$41$	6.76048	1.05581	0.527905	−	0.849303i	$-0.322978\pi$
$41$	6.76048	1.05581	0.527905	+	0.849303i	$0.322978\pi$
$42$	0	0
$43$	0.148338	0.0226213	0.0113107	−	0.999936i	$-0.496400\pi$
$43$	0.148338	0.0226213	0.0113107	+	0.999936i	$0.496400\pi$
$44$	0	0
$45$	−4.09301	−0.610151
$46$	0	0
$47$	−2.60866	−0.380512	−0.190256	−	0.981734i	$-0.560932\pi$
$47$	−2.60866	−0.380512	−0.190256	+	0.981734i	$0.560932\pi$
$48$	0	0
$49$	−3.84940	−0.549915
$50$	0	0
$51$	26.1449	3.66102
$52$	0	0
$53$	9.53625	1.30990	0.654952	−	0.755670i	$-0.272689\pi$
$53$	9.53625	1.30990	0.654952	+	0.755670i	$0.272689\pi$
$54$	0	0
$55$	2.08734	0.281457
$56$	0	0
$57$	−18.9214	−2.50620
$58$	0	0
$59$	−6.44639	−0.839248	−0.419624	−	0.907698i	$-0.637838\pi$
$59$	−6.44639	−0.839248	−0.419624	+	0.907698i	$0.637838\pi$
$60$	0	0
$61$	6.69895	0.857712	0.428856	−	0.903373i	$-0.358917\pi$
$61$	6.69895	0.857712	0.428856	+	0.903373i	$0.358917\pi$
$62$	0	0
$63$	14.5385	1.83168
$64$	0	0
$65$	−2.39923	−0.297587
$66$	0	0
$67$	2.05614	0.251197	0.125599	−	0.992081i	$-0.459915\pi$
$67$	2.05614	0.251197	0.125599	+	0.992081i	$0.459915\pi$
$68$	0	0
$69$	4.18213	0.503469
$70$	0	0
$71$	−13.1917	−1.56557	−0.782786	−	0.622291i	$-0.786202\pi$
$71$	−13.1917	−1.56557	−0.782786	+	0.622291i	$0.786202\pi$
$72$	0	0
$73$	−1.01876	−0.119237	−0.0596185	−	0.998221i	$-0.518988\pi$
$73$	−1.01876	−0.119237	−0.0596185	+	0.998221i	$0.518988\pi$
$74$	0	0
$75$	15.8909	1.83493
$76$	0	0
$77$	−7.41430	−0.844938
$78$	0	0
$79$	3.52375	0.396452	0.198226	−	0.980156i	$-0.436482\pi$
$79$	3.52375	0.396452	0.198226	+	0.980156i	$0.436482\pi$
$80$	0	0
$81$	33.5162	3.72402
$82$	0	0
$83$	14.0162	1.53847	0.769237	−	0.638964i	$-0.220637\pi$
$83$	14.0162	1.53847	0.769237	+	0.638964i	$0.220637\pi$
$84$	0	0
$85$	3.90550	0.423611
$86$	0	0
$87$	−9.07740	−0.973200
$88$	0	0
$89$	−9.84101	−1.04314	−0.521572	−	0.853207i	$-0.674654\pi$
$89$	−9.84101	−1.04314	−0.521572	+	0.853207i	$0.674654\pi$
$90$	0	0
$91$	8.52213	0.893362
$92$	0	0
$93$	20.1554	2.09002
$94$	0	0
$95$	−2.82646	−0.289989
$96$	0	0
$97$	18.8622	1.91516	0.957582	−	0.288160i	$-0.0930435\pi$
$97$	18.8622	1.91516	0.957582	+	0.288160i	$0.0930435\pi$
$98$	0	0
$99$	−34.2135	−3.43859
$100$	0	0
$101$	10.9106	1.08564	0.542822	−	0.839848i	$-0.317356\pi$
$101$	10.9106	1.08564	0.542822	+	0.839848i	$0.317356\pi$
$102$	0	0
$103$	5.68102	0.559768	0.279884	−	0.960034i	$-0.409704\pi$
$103$	5.68102	0.559768	0.279884	+	0.960034i	$0.409704\pi$
$104$	0	0
$105$	2.96719	0.289568
$106$	0	0
$107$	−6.53887	−0.632136	−0.316068	−	0.948737i	$-0.602363\pi$
$107$	−6.53887	−0.632136	−0.316068	+	0.948737i	$0.602363\pi$
$108$	0	0
$109$	−14.9749	−1.43434	−0.717169	−	0.696899i	$-0.754563\pi$
$109$	−14.9749	−1.43434	−0.717169	+	0.696899i	$0.754563\pi$
$110$	0	0
$111$	−18.3960	−1.74607
$112$	0	0
$113$	11.5690	1.08832	0.544160	−	0.838981i	$-0.316848\pi$
$113$	11.5690	1.08832	0.544160	+	0.838981i	$0.316848\pi$
$114$	0	0
$115$	0.624723	0.0582557
$116$	0	0
$117$	39.3256	3.63566
$118$	0	0
$119$	−13.8725	−1.27169
$120$	0	0
$121$	6.44807	0.586188
$122$	0	0
$123$	−22.6156	−2.03918
$124$	0	0
$125$	4.87233	0.435795
$126$	0	0
$127$	−1.72529	−0.153095	−0.0765474	−	0.997066i	$-0.524390\pi$
$127$	−1.72529	−0.153095	−0.0765474	+	0.997066i	$0.524390\pi$
$128$	0	0
$129$	−0.496229	−0.0436905
$130$	0	0
$131$	−4.25063	−0.371379	−0.185690	−	0.982608i	$-0.559452\pi$
$131$	−4.25063	−0.371379	−0.185690	+	0.982608i	$0.559452\pi$
$132$	0	0
$133$	10.0397	0.870550
$134$	0	0
$135$	8.67720	0.746814
$136$	0	0
$137$	−14.4583	−1.23526	−0.617628	−	0.786470i	$-0.711906\pi$
$137$	−14.4583	−1.23526	−0.617628	+	0.786470i	$0.711906\pi$
$138$	0	0
$139$	6.45545	0.547544	0.273772	−	0.961795i	$-0.411729\pi$
$139$	6.45545	0.547544	0.273772	+	0.961795i	$0.411729\pi$
$140$	0	0
$141$	8.72664	0.734916
$142$	0	0
$143$	−20.0551	−1.67709
$144$	0	0
$145$	−1.35597	−0.112608
$146$	0	0
$147$	12.8773	1.06210
$148$	0	0
$149$	−20.8625	−1.70913	−0.854563	−	0.519347i	$-0.826175\pi$
$149$	−20.8625	−1.70913	−0.854563	+	0.519347i	$0.826175\pi$
$150$	0	0
$151$	−16.6181	−1.35236	−0.676180	−	0.736737i	$-0.736366\pi$
$151$	−16.6181	−1.35236	−0.676180	+	0.736737i	$0.736366\pi$
$152$	0	0
$153$	−64.0150	−5.17531
$154$	0	0
$155$	3.01079	0.241833
$156$	0	0
$157$	9.20580	0.734703	0.367351	−	0.930082i	$-0.380265\pi$
$157$	9.20580	0.734703	0.367351	+	0.930082i	$0.380265\pi$
$158$	0	0
$159$	−31.9012	−2.52993
$160$	0	0
$161$	−2.21904	−0.174884
$162$	0	0
$163$	−2.45908	−0.192610	−0.0963048	−	0.995352i	$-0.530702\pi$
$163$	−2.45908	−0.192610	−0.0963048	+	0.995352i	$0.530702\pi$
$164$	0	0
$165$	−6.98269	−0.543602
$166$	0	0
$167$	−22.3440	−1.72903	−0.864514	−	0.502608i	$-0.832374\pi$
$167$	−22.3440	−1.72903	−0.864514	+	0.502608i	$0.832374\pi$
$168$	0	0
$169$	10.0517	0.773210
$170$	0	0
$171$	46.3284	3.54282
$172$	0	0
$173$	−14.3070	−1.08774	−0.543871	−	0.839169i	$-0.683042\pi$
$173$	−14.3070	−1.08774	−0.543871	+	0.839169i	$0.683042\pi$
$174$	0	0
$175$	−8.43172	−0.637378
$176$	0	0
$177$	21.5648	1.62091
$178$	0	0
$179$	12.8032	0.956954	0.478477	−	0.878100i	$-0.341189\pi$
$179$	12.8032	0.956954	0.478477	+	0.878100i	$0.341189\pi$
$180$	0	0
$181$	−16.2805	−1.21012	−0.605062	−	0.796179i	$-0.706851\pi$
$181$	−16.2805	−1.21012	−0.605062	+	0.796179i	$0.706851\pi$
$182$	0	0
$183$	−22.4097	−1.65657
$184$	0	0
$185$	−2.74798	−0.202035
$186$	0	0
$187$	32.6461	2.38732
$188$	0	0
$189$	−30.8217	−2.24195
$190$	0	0
$191$	17.9310	1.29744	0.648721	−	0.761026i	$-0.275304\pi$
$191$	17.9310	1.29744	0.648721	+	0.761026i	$0.275304\pi$
$192$	0	0
$193$	5.19248	0.373763	0.186881	−	0.982382i	$-0.440162\pi$
$193$	5.19248	0.373763	0.186881	+	0.982382i	$0.440162\pi$
$194$	0	0
$195$	8.02603	0.574756
$196$	0	0
$197$	−20.7284	−1.47684	−0.738420	−	0.674341i	$-0.764428\pi$
$197$	−20.7284	−1.47684	−0.738420	+	0.674341i	$0.764428\pi$
$198$	0	0
$199$	−10.5053	−0.744704	−0.372352	−	0.928092i	$-0.621449\pi$
$199$	−10.5053	−0.744704	−0.372352	+	0.928092i	$0.621449\pi$
$200$	0	0
$201$	−6.87831	−0.485158
$202$	0	0
$203$	4.81646	0.338049
$204$	0	0
$205$	−3.37829	−0.235950
$206$	0	0
$207$	−10.2398	−0.711716
$208$	0	0
$209$	−23.6264	−1.63427
$210$	0	0
$211$	−16.1332	−1.11065	−0.555327	−	0.831632i	$-0.687407\pi$
$211$	−16.1332	−1.11065	−0.555327	+	0.831632i	$0.687407\pi$
$212$	0	0
$213$	44.1298	3.02372
$214$	0	0
$215$	−0.0741262	−0.00505537
$216$	0	0
$217$	−10.6944	−0.725986
$218$	0	0
$219$	3.40802	0.230293
$220$	0	0
$221$	−37.5240	−2.52414
$222$	0	0
$223$	6.50284	0.435462	0.217731	−	0.976009i	$-0.430134\pi$
$223$	6.50284	0.435462	0.217731	+	0.976009i	$0.430134\pi$
$224$	0	0
$225$	−38.9084	−2.59390
$226$	0	0
$227$	−26.4789	−1.75747	−0.878733	−	0.477314i	$-0.841611\pi$
$227$	−26.4789	−1.75747	−0.878733	+	0.477314i	$0.841611\pi$
$228$	0	0
$229$	6.64558	0.439152	0.219576	−	0.975595i	$-0.429533\pi$
$229$	6.64558	0.439152	0.219576	+	0.975595i	$0.429533\pi$
$230$	0	0
$231$	24.8027	1.63190
$232$	0	0
$233$	8.10362	0.530886	0.265443	−	0.964127i	$-0.414482\pi$
$233$	8.10362	0.530886	0.265443	+	0.964127i	$0.414482\pi$
$234$	0	0
$235$	1.30358	0.0850360
$236$	0	0
$237$	−11.7878	−0.765703
$238$	0	0
$239$	−5.10538	−0.330239	−0.165120	−	0.986274i	$-0.552801\pi$
$239$	−5.10538	−0.330239	−0.165120	+	0.986274i	$0.552801\pi$
$240$	0	0
$241$	−21.1581	−1.36291	−0.681456	−	0.731859i	$-0.738653\pi$
$241$	−21.1581	−1.36291	−0.681456	+	0.731859i	$0.738653\pi$
$242$	0	0
$243$	−60.0271	−3.85074
$244$	0	0
$245$	1.92359	0.122894
$246$	0	0
$247$	27.1566	1.72793
$248$	0	0
$249$	−46.8877	−2.97139
$250$	0	0
$251$	−0.0773743	−0.00488382	−0.00244191	−	0.999997i	$-0.500777\pi$
$251$	−0.0773743	−0.00488382	−0.00244191	+	0.999997i	$0.500777\pi$
$252$	0	0
$253$	5.22206	0.328308
$254$	0	0
$255$	−13.0649	−0.818157
$256$	0	0
$257$	−10.2816	−0.641351	−0.320676	−	0.947189i	$-0.603910\pi$
$257$	−10.2816	−0.641351	−0.320676	+	0.947189i	$0.603910\pi$
$258$	0	0
$259$	9.76091	0.606513
$260$	0	0
$261$	22.2257	1.37574
$262$	0	0
$263$	22.5369	1.38969	0.694843	−	0.719162i	$-0.255474\pi$
$263$	22.5369	1.38969	0.694843	+	0.719162i	$0.255474\pi$
$264$	0	0
$265$	−4.76537	−0.292735
$266$	0	0
$267$	32.9207	2.01472
$268$	0	0
$269$	10.6089	0.646836	0.323418	−	0.946256i	$-0.395168\pi$
$269$	10.6089	0.646836	0.323418	+	0.946256i	$0.395168\pi$
$270$	0	0
$271$	11.0497	0.671222	0.335611	−	0.942001i	$-0.391057\pi$
$271$	11.0497	0.671222	0.335611	+	0.942001i	$0.391057\pi$
$272$	0	0
$273$	−28.5087	−1.72543
$274$	0	0
$275$	19.8424	1.19654
$276$	0	0
$277$	12.4887	0.750375	0.375188	−	0.926949i	$-0.377578\pi$
$277$	12.4887	0.750375	0.375188	+	0.926949i	$0.377578\pi$
$278$	0	0
$279$	−49.3498	−2.95450
$280$	0	0
$281$	−4.22945	−0.252308	−0.126154	−	0.992011i	$-0.540263\pi$
$281$	−4.22945	−0.252308	−0.126154	+	0.992011i	$0.540263\pi$
$282$	0	0
$283$	−6.44902	−0.383354	−0.191677	−	0.981458i	$-0.561393\pi$
$283$	−6.44902	−0.383354	−0.191677	+	0.981458i	$0.561393\pi$
$284$	0	0
$285$	9.45524	0.560080
$286$	0	0
$287$	11.9998	0.708326
$288$	0	0
$289$	44.0823	2.59308
$290$	0	0
$291$	−63.0989	−3.69892
$292$	0	0
$293$	−24.8416	−1.45126	−0.725630	−	0.688085i	$-0.758452\pi$
$293$	−24.8416	−1.45126	−0.725630	+	0.688085i	$0.758452\pi$
$294$	0	0
$295$	3.22133	0.187553
$296$	0	0
$297$	72.5327	4.20877
$298$	0	0
$299$	−6.00233	−0.347124
$300$	0	0
$301$	0.263299	0.0151763
$302$	0	0
$303$	−36.4987	−2.09680
$304$	0	0
$305$	−3.34754	−0.191680
$306$	0	0
$307$	16.9860	0.969440	0.484720	−	0.874669i	$-0.338922\pi$
$307$	16.9860	0.969440	0.484720	+	0.874669i	$0.338922\pi$
$308$	0	0
$309$	−19.0045	−1.08113
$310$	0	0
$311$	−5.49443	−0.311560	−0.155780	−	0.987792i	$-0.549789\pi$
$311$	−5.49443	−0.311560	−0.155780	+	0.987792i	$0.549789\pi$
$312$	0	0
$313$	−32.7854	−1.85314	−0.926571	−	0.376120i	$-0.877258\pi$
$313$	−32.7854	−1.85314	−0.926571	+	0.376120i	$0.877258\pi$
$314$	0	0
$315$	−7.26507	−0.409340
$316$	0	0
$317$	−2.49305	−0.140023	−0.0700117	−	0.997546i	$-0.522304\pi$
$317$	−2.49305	−0.140023	−0.0700117	+	0.997546i	$0.522304\pi$
$318$	0	0
$319$	−11.3346	−0.634615
$320$	0	0
$321$	21.8742	1.22090
$322$	0	0
$323$	−44.2060	−2.45969
$324$	0	0
$325$	−22.8072	−1.26512
$326$	0	0
$327$	50.0950	2.77026
$328$	0	0
$329$	−4.63035	−0.255279
$330$	0	0
$331$	−31.7646	−1.74594	−0.872971	−	0.487772i	$-0.837810\pi$
$331$	−31.7646	−1.74594	−0.872971	+	0.487772i	$0.837810\pi$
$332$	0	0
$333$	45.0420	2.46829
$334$	0	0
$335$	−1.02747	−0.0561370
$336$	0	0
$337$	−24.0384	−1.30945	−0.654727	−	0.755865i	$-0.727217\pi$
$337$	−24.0384	−1.30945	−0.654727	+	0.755865i	$0.727217\pi$
$338$	0	0
$339$	−38.7013	−2.10197
$340$	0	0
$341$	25.1672	1.36288
$342$	0	0
$343$	−19.2576	−1.03981
$344$	0	0
$345$	−2.08986	−0.112514
$346$	0	0
$347$	11.0627	0.593878	0.296939	−	0.954897i	$-0.404034\pi$
$347$	11.0627	0.593878	0.296939	+	0.954897i	$0.404034\pi$
$348$	0	0
$349$	−31.6718	−1.69535	−0.847676	−	0.530514i	$-0.821999\pi$
$349$	−31.6718	−1.69535	−0.847676	+	0.530514i	$0.821999\pi$
$350$	0	0
$351$	−83.3704	−4.44998
$352$	0	0
$353$	2.88442	0.153522	0.0767612	−	0.997050i	$-0.475542\pi$
$353$	2.88442	0.153522	0.0767612	+	0.997050i	$0.475542\pi$
$354$	0	0
$355$	6.59207	0.349871
$356$	0	0
$357$	46.4070	2.45612
$358$	0	0
$359$	−25.3075	−1.33568	−0.667840	−	0.744305i	$-0.732781\pi$
$359$	−25.3075	−1.33568	−0.667840	+	0.744305i	$0.732781\pi$
$360$	0	0
$361$	12.9924	0.683810
$362$	0	0
$363$	−21.5705	−1.13216
$364$	0	0
$365$	0.509087	0.0266468
$366$	0	0
$367$	0.786853	0.0410734	0.0205367	−	0.999789i	$-0.493463\pi$
$367$	0.786853	0.0410734	0.0205367	+	0.999789i	$0.493463\pi$
$368$	0	0
$369$	55.3735	2.88263
$370$	0	0
$371$	16.9268	0.878794
$372$	0	0
$373$	12.8550	0.665607	0.332804	−	0.942996i	$-0.392005\pi$
$373$	12.8550	0.665607	0.332804	+	0.942996i	$0.392005\pi$
$374$	0	0
$375$	−16.2992	−0.841687
$376$	0	0
$377$	13.0282	0.670985
$378$	0	0
$379$	−3.45784	−0.177617	−0.0888086	−	0.996049i	$-0.528306\pi$
$379$	−3.45784	−0.177617	−0.0888086	+	0.996049i	$0.528306\pi$
$380$	0	0
$381$	5.77155	0.295685
$382$	0	0
$383$	−19.6423	−1.00368	−0.501838	−	0.864962i	$-0.667343\pi$
$383$	−19.6423	−1.00368	−0.501838	+	0.864962i	$0.667343\pi$
$384$	0	0
$385$	3.70501	0.188825
$386$	0	0
$387$	1.21500	0.0617619
$388$	0	0
$389$	−1.13441	−0.0575169	−0.0287584	−	0.999586i	$-0.509155\pi$
$389$	−1.13441	−0.0575169	−0.0287584	+	0.999586i	$0.509155\pi$
$390$	0	0
$391$	9.77070	0.494126
$392$	0	0
$393$	14.2195	0.717277
$394$	0	0
$395$	−1.76086	−0.0885983
$396$	0	0
$397$	5.58270	0.280188	0.140094	−	0.990138i	$-0.455260\pi$
$397$	5.58270	0.280188	0.140094	+	0.990138i	$0.455260\pi$
$398$	0	0
$399$	−33.5853	−1.68137
$400$	0	0
$401$	−19.1628	−0.956943	−0.478471	−	0.878103i	$-0.658809\pi$
$401$	−19.1628	−0.956943	−0.478471	+	0.878103i	$0.658809\pi$
$402$	0	0
$403$	−28.9277	−1.44099
$404$	0	0
$405$	−16.7484	−0.832236
$406$	0	0
$407$	−22.9704	−1.13860
$408$	0	0
$409$	13.0252	0.644054	0.322027	−	0.946731i	$-0.395636\pi$
$409$	13.0252	0.644054	0.322027	+	0.946731i	$0.395636\pi$
$410$	0	0
$411$	48.3668	2.38576
$412$	0	0
$413$	−11.4423	−0.563038
$414$	0	0
$415$	−7.00403	−0.343815
$416$	0	0
$417$	−21.5952	−1.05752
$418$	0	0
$419$	25.0770	1.22509	0.612546	−	0.790435i	$-0.290146\pi$
$419$	25.0770	1.22509	0.612546	+	0.790435i	$0.290146\pi$
$420$	0	0
$421$	−30.6428	−1.49344	−0.746719	−	0.665140i	$-0.768372\pi$
$421$	−30.6428	−1.49344	−0.746719	+	0.665140i	$0.768372\pi$
$422$	0	0
$423$	−21.3669	−1.03889
$424$	0	0
$425$	37.1260	1.80087
$426$	0	0
$427$	11.8906	0.575425
$428$	0	0
$429$	67.0896	3.23912
$430$	0	0
$431$	7.29638	0.351454	0.175727	−	0.984439i	$-0.443772\pi$
$431$	7.29638	0.351454	0.175727	+	0.984439i	$0.443772\pi$
$432$	0	0
$433$	25.4822	1.22460	0.612299	−	0.790627i	$-0.290245\pi$
$433$	25.4822	1.22460	0.612299	+	0.790627i	$0.290245\pi$
$434$	0	0
$435$	4.53608	0.217489
$436$	0	0
$437$	−7.07117	−0.338260
$438$	0	0
$439$	21.1894	1.01132	0.505658	−	0.862734i	$-0.331250\pi$
$439$	21.1894	1.01132	0.505658	+	0.862734i	$0.331250\pi$
$440$	0	0
$441$	−31.5295	−1.50141
$442$	0	0
$443$	6.38591	0.303404	0.151702	−	0.988426i	$-0.451525\pi$
$443$	6.38591	0.303404	0.151702	+	0.988426i	$0.451525\pi$
$444$	0	0
$445$	4.91767	0.233120
$446$	0	0
$447$	69.7906	3.30098
$448$	0	0
$449$	−7.44667	−0.351430	−0.175715	−	0.984441i	$-0.556224\pi$
$449$	−7.44667	−0.351430	−0.175715	+	0.984441i	$0.556224\pi$
$450$	0	0
$451$	−28.2391	−1.32973
$452$	0	0
$453$	55.5917	2.61193
$454$	0	0
$455$	−4.25861	−0.199647
$456$	0	0
$457$	−28.4774	−1.33211	−0.666057	−	0.745901i	$-0.732019\pi$
$457$	−28.4774	−1.33211	−0.666057	+	0.745901i	$0.732019\pi$
$458$	0	0
$459$	135.712	6.33449
$460$	0	0
$461$	9.77534	0.455283	0.227641	−	0.973745i	$-0.426899\pi$
$461$	9.77534	0.455283	0.227641	+	0.973745i	$0.426899\pi$
$462$	0	0
$463$	−15.6969	−0.729497	−0.364748	−	0.931106i	$-0.618845\pi$
$463$	−15.6969	−0.729497	−0.364748	+	0.931106i	$0.618845\pi$
$464$	0	0
$465$	−10.0719	−0.467073
$466$	0	0
$467$	34.1309	1.57939	0.789694	−	0.613500i	$-0.210239\pi$
$467$	34.1309	1.57939	0.789694	+	0.613500i	$0.210239\pi$
$468$	0	0
$469$	3.64962	0.168524
$470$	0	0
$471$	−30.7958	−1.41899
$472$	0	0
$473$	−0.619621	−0.0284902
$474$	0	0
$475$	−26.8685	−1.23281
$476$	0	0
$477$	78.1091	3.57637
$478$	0	0
$479$	−1.48455	−0.0678308	−0.0339154	−	0.999425i	$-0.510798\pi$
$479$	−1.48455	−0.0678308	−0.0339154	+	0.999425i	$0.510798\pi$
$480$	0	0
$481$	26.4025	1.20385
$482$	0	0
$483$	7.42325	0.337769
$484$	0	0
$485$	−9.42565	−0.427997
$486$	0	0
$487$	−38.4851	−1.74393	−0.871964	−	0.489570i	$-0.837154\pi$
$487$	−38.4851	−1.74393	−0.871964	+	0.489570i	$0.837154\pi$
$488$	0	0
$489$	8.22624	0.372004
$490$	0	0
$491$	10.3963	0.469177	0.234588	−	0.972095i	$-0.424626\pi$
$491$	10.3963	0.469177	0.234588	+	0.972095i	$0.424626\pi$
$492$	0	0
$493$	−21.2075	−0.955138
$494$	0	0
$495$	17.0969	0.768448
$496$	0	0
$497$	−23.4152	−1.05032
$498$	0	0
$499$	12.3621	0.553403	0.276702	−	0.960956i	$-0.410759\pi$
$499$	12.3621	0.553403	0.276702	+	0.960956i	$0.410759\pi$
$500$	0	0
$501$	74.7464	3.33942
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−5.45215	−0.242617
$506$	0	0
$507$	−33.6256	−1.49337
$508$	0	0
$509$	−11.9134	−0.528054	−0.264027	−	0.964515i	$-0.585051\pi$
$509$	−11.9134	−0.528054	−0.264027	+	0.964515i	$0.585051\pi$
$510$	0	0
$511$	−1.80829	−0.0799942
$512$	0	0
$513$	−98.2163	−4.33636
$514$	0	0
$515$	−2.83887	−0.125096
$516$	0	0
$517$	10.8966	0.479232
$518$	0	0
$519$	47.8607	2.10085
$520$	0	0
$521$	39.8103	1.74412	0.872061	−	0.489398i	$-0.162783\pi$
$521$	39.8103	1.74412	0.872061	+	0.489398i	$0.162783\pi$
$522$	0	0
$523$	−6.88210	−0.300933	−0.150467	−	0.988615i	$-0.548078\pi$
$523$	−6.88210	−0.300933	−0.150467	+	0.988615i	$0.548078\pi$
$524$	0	0
$525$	28.2063	1.23102
$526$	0	0
$527$	47.0890	2.05123
$528$	0	0
$529$	−21.4371	−0.932047
$530$	0	0
$531$	−52.8008	−2.29136
$532$	0	0
$533$	32.4586	1.40594
$534$	0	0
$535$	3.26755	0.141268
$536$	0	0
$537$	−42.8299	−1.84825
$538$	0	0
$539$	16.0793	0.692585
$540$	0	0
$541$	−39.0853	−1.68041	−0.840204	−	0.542271i	$-0.817565\pi$
$541$	−39.0853	−1.68041	−0.840204	+	0.542271i	$0.817565\pi$
$542$	0	0
$543$	54.4626	2.33721
$544$	0	0
$545$	7.48315	0.320543
$546$	0	0
$547$	2.08505	0.0891501	0.0445751	−	0.999006i	$-0.485807\pi$
$547$	2.08505	0.0891501	0.0445751	+	0.999006i	$0.485807\pi$
$548$	0	0
$549$	54.8694	2.34177
$550$	0	0
$551$	15.3481	0.653852
$552$	0	0
$553$	6.25462	0.265973
$554$	0	0
$555$	9.19270	0.390208
$556$	0	0
$557$	−3.66023	−0.155089	−0.0775444	−	0.996989i	$-0.524708\pi$
$557$	−3.66023	−0.155089	−0.0775444	+	0.996989i	$0.524708\pi$
$558$	0	0
$559$	0.712204	0.0301230
$560$	0	0
$561$	−109.210	−4.61084
$562$	0	0
$563$	−29.5571	−1.24568	−0.622842	−	0.782347i	$-0.714022\pi$
$563$	−29.5571	−1.24568	−0.622842	+	0.782347i	$0.714022\pi$
$564$	0	0
$565$	−5.78117	−0.243215
$566$	0	0
$567$	59.4910	2.49839
$568$	0	0
$569$	−2.79804	−0.117300	−0.0586500	−	0.998279i	$-0.518680\pi$
$569$	−2.79804	−0.117300	−0.0586500	+	0.998279i	$0.518680\pi$
$570$	0	0
$571$	24.1077	1.00888	0.504439	−	0.863448i	$-0.331699\pi$
$571$	24.1077	1.00888	0.504439	+	0.863448i	$0.331699\pi$
$572$	0	0
$573$	−59.9839	−2.50586
$574$	0	0
$575$	5.93865	0.247659
$576$	0	0
$577$	14.6828	0.611255	0.305627	−	0.952151i	$-0.401134\pi$
$577$	14.6828	0.611255	0.305627	+	0.952151i	$0.401134\pi$
$578$	0	0
$579$	−17.3702	−0.721880
$580$	0	0
$581$	24.8786	1.03214
$582$	0	0
$583$	−39.8338	−1.64975
$584$	0	0
$585$	−19.6515	−0.812488
$586$	0	0
$587$	23.8165	0.983011	0.491505	−	0.870874i	$-0.336447\pi$
$587$	23.8165	0.983011	0.491505	+	0.870874i	$0.336447\pi$
$588$	0	0
$589$	−34.0789	−1.40420
$590$	0	0
$591$	69.3420	2.85235
$592$	0	0
$593$	17.9788	0.738300	0.369150	−	0.929370i	$-0.379649\pi$
$593$	17.9788	0.738300	0.369150	+	0.929370i	$0.379649\pi$
$594$	0	0
$595$	6.93224	0.284194
$596$	0	0
$597$	35.1431	1.43831
$598$	0	0
$599$	−37.8753	−1.54754	−0.773772	−	0.633465i	$-0.781632\pi$
$599$	−37.8753	−1.54754	−0.773772	+	0.633465i	$0.781632\pi$
$600$	0	0
$601$	18.3188	0.747240	0.373620	−	0.927582i	$-0.378116\pi$
$601$	18.3188	0.747240	0.373620	+	0.927582i	$0.378116\pi$
$602$	0	0
$603$	16.8413	0.685831
$604$	0	0
$605$	−3.22218	−0.131000
$606$	0	0
$607$	19.4328	0.788753	0.394376	−	0.918949i	$-0.370961\pi$
$607$	19.4328	0.788753	0.394376	+	0.918949i	$0.370961\pi$
$608$	0	0
$609$	−16.1123	−0.652904
$610$	0	0
$611$	−12.5248	−0.506697
$612$	0	0
$613$	−29.3526	−1.18554	−0.592770	−	0.805372i	$-0.701966\pi$
$613$	−29.3526	−1.18554	−0.592770	+	0.805372i	$0.701966\pi$
$614$	0	0
$615$	11.3013	0.455711
$616$	0	0
$617$	−34.5881	−1.39246	−0.696232	−	0.717817i	$-0.745142\pi$
$617$	−34.5881	−1.39246	−0.696232	+	0.717817i	$0.745142\pi$
$618$	0	0
$619$	−30.8400	−1.23956	−0.619782	−	0.784774i	$-0.712779\pi$
$619$	−30.8400	−1.23956	−0.619782	+	0.784774i	$0.712779\pi$
$620$	0	0
$621$	21.7084	0.871129
$622$	0	0
$623$	−17.4677	−0.699829
$624$	0	0
$625$	21.3167	0.852667
$626$	0	0
$627$	79.0363	3.15641
$628$	0	0
$629$	−42.9786	−1.71367
$630$	0	0
$631$	−44.4491	−1.76949	−0.884746	−	0.466074i	$-0.845668\pi$
$631$	−44.4491	−1.76949	−0.884746	+	0.466074i	$0.845668\pi$
$632$	0	0
$633$	53.9697	2.14510
$634$	0	0
$635$	0.862148	0.0342133
$636$	0	0
$637$	−18.4818	−0.732277
$638$	0	0
$639$	−108.050	−4.27441
$640$	0	0
$641$	−33.3573	−1.31754	−0.658768	−	0.752346i	$-0.728922\pi$
$641$	−33.3573	−1.31754	−0.658768	+	0.752346i	$0.728922\pi$
$642$	0	0
$643$	6.17735	0.243611	0.121805	−	0.992554i	$-0.461132\pi$
$643$	6.17735	0.243611	0.121805	+	0.992554i	$0.461132\pi$
$644$	0	0
$645$	0.247971	0.00976386
$646$	0	0
$647$	−10.1277	−0.398163	−0.199081	−	0.979983i	$-0.563796\pi$
$647$	−10.1277	−0.398163	−0.199081	+	0.979983i	$0.563796\pi$
$648$	0	0
$649$	26.9271	1.05698
$650$	0	0
$651$	35.7757	1.40216
$652$	0	0
$653$	−2.78229	−0.108880	−0.0544398	−	0.998517i	$-0.517337\pi$
$653$	−2.78229	−0.108880	−0.0544398	+	0.998517i	$0.517337\pi$
$654$	0	0
$655$	2.12409	0.0829950
$656$	0	0
$657$	−8.34442	−0.325547
$658$	0	0
$659$	−11.8788	−0.462732	−0.231366	−	0.972867i	$-0.574320\pi$
$659$	−11.8788	−0.462732	−0.231366	+	0.972867i	$0.574320\pi$
$660$	0	0
$661$	−15.8865	−0.617914	−0.308957	−	0.951076i	$-0.599980\pi$
$661$	−15.8865	−0.617914	−0.308957	+	0.951076i	$0.599980\pi$
$662$	0	0
$663$	125.528	4.87509
$664$	0	0
$665$	−5.01694	−0.194549
$666$	0	0
$667$	−3.39234	−0.131352
$668$	0	0
$669$	−21.7537	−0.841046
$670$	0	0
$671$	−27.9821	−1.08024
$672$	0	0
$673$	19.5000	0.751669	0.375835	−	0.926687i	$-0.377356\pi$
$673$	19.5000	0.751669	0.375835	+	0.926687i	$0.377356\pi$
$674$	0	0
$675$	82.4860	3.17489
$676$	0	0
$677$	−49.4240	−1.89952	−0.949759	−	0.312982i	$-0.898672\pi$
$677$	−49.4240	−1.89952	−0.949759	+	0.312982i	$0.898672\pi$
$678$	0	0
$679$	33.4802	1.28485
$680$	0	0
$681$	88.5788	3.39435
$682$	0	0
$683$	42.6572	1.63223	0.816116	−	0.577889i	$-0.196123\pi$
$683$	42.6572	1.63223	0.816116	+	0.577889i	$0.196123\pi$
$684$	0	0
$685$	7.22498	0.276052
$686$	0	0
$687$	−22.2312	−0.848173
$688$	0	0
$689$	45.7857	1.74429
$690$	0	0
$691$	−24.7106	−0.940034	−0.470017	−	0.882657i	$-0.655752\pi$
$691$	−24.7106	−0.940034	−0.470017	+	0.882657i	$0.655752\pi$
$692$	0	0
$693$	−60.7287	−2.30689
$694$	0	0
$695$	−3.22586	−0.122364
$696$	0	0
$697$	−52.8367	−2.00133
$698$	0	0
$699$	−27.1087	−1.02535
$700$	0	0
$701$	49.8963	1.88456	0.942278	−	0.334832i	$-0.108679\pi$
$701$	49.8963	1.88456	0.942278	+	0.334832i	$0.108679\pi$
$702$	0	0
$703$	31.1041	1.17311
$704$	0	0
$705$	−4.36080	−0.164237
$706$	0	0
$707$	19.3662	0.728341
$708$	0	0
$709$	−23.5068	−0.882818	−0.441409	−	0.897306i	$-0.645521\pi$
$709$	−23.5068	−0.882818	−0.441409	+	0.897306i	$0.645521\pi$
$710$	0	0
$711$	28.8621	1.08241
$712$	0	0
$713$	7.53234	0.282088
$714$	0	0
$715$	10.0218	0.374793
$716$	0	0
$717$	17.0788	0.637819
$718$	0	0
$719$	−5.20803	−0.194227	−0.0971135	−	0.995273i	$-0.530961\pi$
$719$	−5.20803	−0.194227	−0.0971135	+	0.995273i	$0.530961\pi$
$720$	0	0
$721$	10.0838	0.375539
$722$	0	0
$723$	70.7793	2.63231
$724$	0	0
$725$	−12.8900	−0.478722
$726$	0	0
$727$	32.3715	1.20059	0.600297	−	0.799778i	$-0.295049\pi$
$727$	32.3715	1.20059	0.600297	+	0.799778i	$0.295049\pi$
$728$	0	0
$729$	100.258	3.71324
$730$	0	0
$731$	−1.15934	−0.0428797
$732$	0	0
$733$	27.9619	1.03279	0.516397	−	0.856349i	$-0.327273\pi$
$733$	27.9619	1.03279	0.516397	+	0.856349i	$0.327273\pi$
$734$	0	0
$735$	−6.43491	−0.237355
$736$	0	0
$737$	−8.58866	−0.316368
$738$	0	0
$739$	−13.7500	−0.505801	−0.252901	−	0.967492i	$-0.581385\pi$
$739$	−13.7500	−0.505801	−0.252901	+	0.967492i	$0.581385\pi$
$740$	0	0
$741$	−90.8458	−3.33730
$742$	0	0
$743$	1.13983	0.0418163	0.0209082	−	0.999781i	$-0.493344\pi$
$743$	1.13983	0.0418163	0.0209082	+	0.999781i	$0.493344\pi$
$744$	0	0
$745$	10.4253	0.381952
$746$	0	0
$747$	114.803	4.20042
$748$	0	0
$749$	−11.6064	−0.424090
$750$	0	0
$751$	17.1626	0.626272	0.313136	−	0.949708i	$-0.398620\pi$
$751$	17.1626	0.626272	0.313136	+	0.949708i	$0.398620\pi$
$752$	0	0
$753$	0.258837	0.00943254
$754$	0	0
$755$	8.30424	0.302222
$756$	0	0
$757$	−9.54955	−0.347084	−0.173542	−	0.984826i	$-0.555521\pi$
$757$	−9.54955	−0.347084	−0.173542	+	0.984826i	$0.555521\pi$
$758$	0	0
$759$	−17.4691	−0.634089
$760$	0	0
$761$	−9.40251	−0.340841	−0.170420	−	0.985371i	$-0.554513\pi$
$761$	−9.40251	−0.340841	−0.170420	+	0.985371i	$0.554513\pi$
$762$	0	0
$763$	−26.5804	−0.962275
$764$	0	0
$765$	31.9890	1.15657
$766$	0	0
$767$	−30.9505	−1.11756
$768$	0	0
$769$	−4.04629	−0.145913	−0.0729565	−	0.997335i	$-0.523243\pi$
$769$	−4.04629	−0.145913	−0.0729565	+	0.997335i	$0.523243\pi$
$770$	0	0
$771$	34.3947	1.23870
$772$	0	0
$773$	2.05888	0.0740528	0.0370264	−	0.999314i	$-0.488211\pi$
$773$	2.05888	0.0740528	0.0370264	+	0.999314i	$0.488211\pi$
$774$	0	0
$775$	28.6208	1.02809
$776$	0	0
$777$	−32.6528	−1.17141
$778$	0	0
$779$	38.2385	1.37004
$780$	0	0
$781$	55.1031	1.97174
$782$	0	0
$783$	−47.1186	−1.68388
$784$	0	0
$785$	−4.60024	−0.164190
$786$	0	0
$787$	−34.4198	−1.22693	−0.613467	−	0.789720i	$-0.710226\pi$
$787$	−34.4198	−1.22693	−0.613467	+	0.789720i	$0.710226\pi$
$788$	0	0
$789$	−75.3918	−2.68402
$790$	0	0
$791$	20.5349	0.730137
$792$	0	0
$793$	32.1631	1.14215
$794$	0	0
$795$	15.9414	0.565383
$796$	0	0
$797$	39.7205	1.40697	0.703486	−	0.710709i	$-0.251626\pi$
$797$	39.7205	1.40697	0.703486	+	0.710709i	$0.251626\pi$
$798$	0	0
$799$	20.3880	0.721277
$800$	0	0
$801$	−80.6053	−2.84805
$802$	0	0
$803$	4.25546	0.150172
$804$	0	0
$805$	1.10888	0.0390828
$806$	0	0
$807$	−35.4895	−1.24929
$808$	0	0
$809$	−24.1652	−0.849603	−0.424802	−	0.905286i	$-0.639656\pi$
$809$	−24.1652	−0.849603	−0.424802	+	0.905286i	$0.639656\pi$
$810$	0	0
$811$	35.3665	1.24189	0.620944	−	0.783855i	$-0.286750\pi$
$811$	35.3665	1.24189	0.620944	+	0.783855i	$0.286750\pi$
$812$	0	0
$813$	−36.9641	−1.29639
$814$	0	0
$815$	1.22883	0.0430440
$816$	0	0
$817$	0.839027	0.0293538
$818$	0	0
$819$	69.8027	2.43910
$820$	0	0
$821$	−40.0467	−1.39764	−0.698820	−	0.715297i	$-0.746291\pi$
$821$	−40.0467	−1.39764	−0.698820	+	0.715297i	$0.746291\pi$
$822$	0	0
$823$	16.3237	0.569007	0.284503	−	0.958675i	$-0.408171\pi$
$823$	16.3237	0.569007	0.284503	+	0.958675i	$0.408171\pi$
$824$	0	0
$825$	−66.3779	−2.31098
$826$	0	0
$827$	−15.2282	−0.529535	−0.264768	−	0.964312i	$-0.585295\pi$
$827$	−15.2282	−0.529535	−0.264768	+	0.964312i	$0.585295\pi$
$828$	0	0
$829$	−30.0445	−1.04349	−0.521745	−	0.853102i	$-0.674719\pi$
$829$	−30.0445	−1.04349	−0.521745	+	0.853102i	$0.674719\pi$
$830$	0	0
$831$	−41.7780	−1.44926
$832$	0	0
$833$	30.0851	1.04239
$834$	0	0
$835$	11.1655	0.386399
$836$	0	0
$837$	104.622	3.61626
$838$	0	0
$839$	−6.80429	−0.234910	−0.117455	−	0.993078i	$-0.537474\pi$
$839$	−6.80429	−0.234910	−0.117455	+	0.993078i	$0.537474\pi$
$840$	0	0
$841$	−21.6368	−0.746098
$842$	0	0
$843$	14.1486	0.487303
$844$	0	0
$845$	−5.02297	−0.172795
$846$	0	0
$847$	11.4453	0.393264
$848$	0	0
$849$	21.5736	0.740405
$850$	0	0
$851$	−6.87483	−0.235666
$852$	0	0
$853$	42.4755	1.45433	0.727166	−	0.686461i	$-0.240837\pi$
$853$	42.4755	1.45433	0.727166	+	0.686461i	$0.240837\pi$
$854$	0	0
$855$	−23.1508	−0.791742
$856$	0	0
$857$	46.8992	1.60204	0.801022	−	0.598635i	$-0.204290\pi$
$857$	46.8992	1.60204	0.801022	+	0.598635i	$0.204290\pi$
$858$	0	0
$859$	18.8425	0.642896	0.321448	−	0.946927i	$-0.395830\pi$
$859$	18.8425	0.642896	0.321448	+	0.946927i	$0.395830\pi$
$860$	0	0
$861$	−40.1424	−1.36805
$862$	0	0
$863$	−7.66671	−0.260978	−0.130489	−	0.991450i	$-0.541655\pi$
$863$	−7.66671	−0.260978	−0.130489	+	0.991450i	$0.541655\pi$
$864$	0	0
$865$	7.14938	0.243086
$866$	0	0
$867$	−147.467	−5.00824
$868$	0	0
$869$	−14.7190	−0.499308
$870$	0	0
$871$	9.87197	0.334499
$872$	0	0
$873$	154.496	5.22888
$874$	0	0
$875$	8.64835	0.292368
$876$	0	0
$877$	23.0017	0.776711	0.388356	−	0.921510i	$-0.373043\pi$
$877$	23.0017	0.776711	0.388356	+	0.921510i	$0.373043\pi$
$878$	0	0
$879$	83.1015	2.80294
$880$	0	0
$881$	−11.6273	−0.391735	−0.195867	−	0.980630i	$-0.562752\pi$
$881$	−11.6273	−0.391735	−0.195867	+	0.980630i	$0.562752\pi$
$882$	0	0
$883$	−35.2757	−1.18712	−0.593561	−	0.804789i	$-0.702279\pi$
$883$	−35.2757	−1.18712	−0.593561	+	0.804789i	$0.702279\pi$
$884$	0	0
$885$	−10.7762	−0.362238
$886$	0	0
$887$	−15.1847	−0.509852	−0.254926	−	0.966961i	$-0.582051\pi$
$887$	−15.1847	−0.509852	−0.254926	+	0.966961i	$0.582051\pi$
$888$	0	0
$889$	−3.06238	−0.102709
$890$	0	0
$891$	−140.000	−4.69018
$892$	0	0
$893$	−14.7551	−0.493759
$894$	0	0
$895$	−6.39790	−0.213858
$896$	0	0
$897$	20.0793	0.670430
$898$	0	0
$899$	−16.3491	−0.545273
$900$	0	0
$901$	−74.5307	−2.48298
$902$	0	0
$903$	−0.880803	−0.0293113
$904$	0	0
$905$	8.13557	0.270436
$906$	0	0
$907$	−47.6730	−1.58296	−0.791478	−	0.611198i	$-0.790688\pi$
$907$	−47.6730	−1.58296	−0.791478	+	0.611198i	$0.790688\pi$
$908$	0	0
$909$	89.3659	2.96408
$910$	0	0
$911$	−11.9027	−0.394354	−0.197177	−	0.980368i	$-0.563177\pi$
$911$	−11.9027	−0.394354	−0.197177	+	0.980368i	$0.563177\pi$
$912$	0	0
$913$	−58.5467	−1.93761
$914$	0	0
$915$	11.1984	0.370207
$916$	0	0
$917$	−7.54483	−0.249152
$918$	0	0
$919$	−5.30835	−0.175106	−0.0875531	−	0.996160i	$-0.527905\pi$
$919$	−5.30835	−0.175106	−0.0875531	+	0.996160i	$0.527905\pi$
$920$	0	0
$921$	−56.8224	−1.87236
$922$	0	0
$923$	−63.3365	−2.08475
$924$	0	0
$925$	−26.1225	−0.858901
$926$	0	0
$927$	46.5319	1.52831
$928$	0	0
$929$	−50.1657	−1.64588	−0.822942	−	0.568125i	$-0.807669\pi$
$929$	−50.1657	−1.64588	−0.822942	+	0.568125i	$0.807669\pi$
$930$	0	0
$931$	−21.7729	−0.713579
$932$	0	0
$933$	18.3803	0.601743
$934$	0	0
$935$	−16.3136	−0.533513
$936$	0	0
$937$	19.7093	0.643874	0.321937	−	0.946761i	$-0.395666\pi$
$937$	19.7093	0.643874	0.321937	+	0.946761i	$0.395666\pi$
$938$	0	0
$939$	109.676	3.57913
$940$	0	0
$941$	−2.81144	−0.0916504	−0.0458252	−	0.998949i	$-0.514592\pi$
$941$	−2.81144	−0.0916504	−0.0458252	+	0.998949i	$0.514592\pi$
$942$	0	0
$943$	−8.45173	−0.275226
$944$	0	0
$945$	15.4020	0.501026
$946$	0	0
$947$	14.3549	0.466470	0.233235	−	0.972420i	$-0.425069\pi$
$947$	14.3549	0.466470	0.233235	+	0.972420i	$0.425069\pi$
$948$	0	0
$949$	−4.89130	−0.158778
$950$	0	0
$951$	8.33989	0.270439
$952$	0	0
$953$	−1.11525	−0.0361266	−0.0180633	−	0.999837i	$-0.505750\pi$
$953$	−1.11525	−0.0361266	−0.0180633	+	0.999837i	$0.505750\pi$
$954$	0	0
$955$	−8.96033	−0.289949
$956$	0	0
$957$	37.9171	1.22569
$958$	0	0
$959$	−25.6634	−0.828714
$960$	0	0
$961$	5.30143	0.171014
$962$	0	0
$963$	−53.5582	−1.72589
$964$	0	0
$965$	−2.59474	−0.0835277
$966$	0	0
$967$	46.7287	1.50269	0.751346	−	0.659908i	$-0.229405\pi$
$967$	46.7287	1.50269	0.751346	+	0.659908i	$0.229405\pi$
$968$	0	0
$969$	147.880	4.75061
$970$	0	0
$971$	−36.3244	−1.16571	−0.582853	−	0.812577i	$-0.698064\pi$
$971$	−36.3244	−1.16571	−0.582853	+	0.812577i	$0.698064\pi$
$972$	0	0
$973$	11.4584	0.367339
$974$	0	0
$975$	76.2959	2.44343
$976$	0	0
$977$	57.6443	1.84420	0.922102	−	0.386947i	$-0.126471\pi$
$977$	57.6443	1.84420	0.922102	+	0.386947i	$0.126471\pi$
$978$	0	0
$979$	41.1068	1.31378
$980$	0	0
$981$	−122.656	−3.91611
$982$	0	0
$983$	27.2235	0.868296	0.434148	−	0.900842i	$-0.357049\pi$
$983$	27.2235	0.868296	0.434148	+	0.900842i	$0.357049\pi$
$984$	0	0
$985$	10.3582	0.330041
$986$	0	0
$987$	15.4897	0.493043
$988$	0	0
$989$	−0.185447	−0.00589688
$990$	0	0
$991$	33.3376	1.05900	0.529502	−	0.848309i	$-0.322379\pi$
$991$	33.3376	1.05900	0.529502	+	0.848309i	$0.322379\pi$
$992$	0	0
$993$	106.261	3.37209
$994$	0	0
$995$	5.24964	0.166425
$996$	0	0
$997$	−41.6670	−1.31961	−0.659803	−	0.751439i	$-0.729360\pi$
$997$	−41.6670	−1.31961	−0.659803	+	0.751439i	$0.729360\pi$
$998$	0	0
$999$	−95.4892	−3.02114

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4024.2.a.e.1.1	✓	29
4.3	odd	2		8048.2.a.w.1.29		29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.e.1.1	✓	29	1.1	even	1	trivial
8048.2.a.w.1.29		29	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4024 = 2^{3} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4024.a (trivial)

\(f(q)\)	\(=\)	\(q-3.34526 q^{3} -0.499712 q^{5} +1.77499 q^{7} +8.19075 q^{9} +O(q^{10})\)	\(q-3.34526 q^{3} -0.499712 q^{5} +1.77499 q^{7} +8.19075 q^{9} -4.17709 q^{11} +4.80122 q^{13} +1.67166 q^{15} -7.81552 q^{17} +5.65618 q^{19} -5.93781 q^{21} -1.25017 q^{23} -4.75029 q^{25} -17.3644 q^{27} +2.71351 q^{29} -6.02507 q^{31} +13.9734 q^{33} -0.886984 q^{35} +5.49913 q^{37} -16.0613 q^{39} +6.76048 q^{41} +0.148338 q^{43} -4.09301 q^{45} -2.60866 q^{47} -3.84940 q^{49} +26.1449 q^{51} +9.53625 q^{53} +2.08734 q^{55} -18.9214 q^{57} -6.44639 q^{59} +6.69895 q^{61} +14.5385 q^{63} -2.39923 q^{65} +2.05614 q^{67} +4.18213 q^{69} -13.1917 q^{71} -1.01876 q^{73} +15.8909 q^{75} -7.41430 q^{77} +3.52375 q^{79} +33.5162 q^{81} +14.0162 q^{83} +3.90550 q^{85} -9.07740 q^{87} -9.84101 q^{89} +8.52213 q^{91} +20.1554 q^{93} -2.82646 q^{95} +18.8622 q^{97} -34.2135 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9}+O(q^{10}) \) 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9	\( 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9} - 27 q^{11} + 16 q^{13} - 14 q^{15} - 15 q^{17} - 14 q^{19} + q^{21} - 25 q^{23} + 21 q^{25} - 25 q^{27} - 13 q^{29} - 27 q^{31} - 9 q^{33} - 29 q^{35} + 35 q^{37} - 38 q^{39} - 30 q^{41} - 38 q^{43} + q^{45} - 35 q^{47} + 14 q^{49} - 21 q^{51} + 2 q^{53} - 25 q^{55} - 25 q^{57} - 40 q^{59} + 10 q^{61} - 56 q^{63} - 50 q^{65} - 31 q^{67} + 11 q^{69} - 65 q^{71} - 23 q^{73} - 32 q^{75} + 13 q^{77} - 44 q^{79} - 7 q^{81} - 41 q^{83} + 26 q^{85} - 25 q^{87} - 48 q^{89} - 44 q^{91} + 25 q^{93} - 75 q^{95} - 18 q^{97} - 80 q^{99}+O(q^{100}) \) 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9 - 27 * q^11 + 16 * q^13 - 14 * q^15 - 15 * q^17 - 14 * q^19 + q^21 - 25 * q^23 + 21 * q^25 - 25 * q^27 - 13 * q^29 - 27 * q^31 - 9 * q^33 - 29 * q^35 + 35 * q^37 - 38 * q^39 - 30 * q^41 - 38 * q^43 + q^45 - 35 * q^47 + 14 * q^49 - 21 * q^51 + 2 * q^53 - 25 * q^55 - 25 * q^57 - 40 * q^59 + 10 * q^61 - 56 * q^63 - 50 * q^65 - 31 * q^67 + 11 * q^69 - 65 * q^71 - 23 * q^73 - 32 * q^75 + 13 * q^77 - 44 * q^79 - 7 * q^81 - 41 * q^83 + 26 * q^85 - 25 * q^87 - 48 * q^89 - 44 * q^91 + 25 * q^93 - 75 * q^95 - 18 * q^97 - 80 * q^99

Introduction

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