LMFDB - Embedded newform 8024.2.a.bc.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8024,2,Mod(1,8024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8024 = 2^{3} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8024.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:	$64.0719625819$
Analytic rank:	$0$
Dimension:	$33$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	8024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.882440 q^{3} +0.934682 q^{5} +3.89754 q^{7} -2.22130 q^{9} +O(q^{10})$	$q-0.882440 q^{3} +0.934682 q^{5} +3.89754 q^{7} -2.22130 q^{9} +4.45139 q^{11} -1.64706 q^{13} -0.824801 q^{15} -1.00000 q^{17} +4.03591 q^{19} -3.43934 q^{21} -1.54886 q^{23} -4.12637 q^{25} +4.60748 q^{27} -1.52651 q^{29} -8.57540 q^{31} -3.92808 q^{33} +3.64296 q^{35} -5.00519 q^{37} +1.45344 q^{39} +10.8363 q^{41} +9.41474 q^{43} -2.07621 q^{45} +12.0465 q^{47} +8.19080 q^{49} +0.882440 q^{51} +0.848297 q^{53} +4.16063 q^{55} -3.56145 q^{57} -1.00000 q^{59} -9.57103 q^{61} -8.65760 q^{63} -1.53948 q^{65} +14.5600 q^{67} +1.36678 q^{69} -6.70386 q^{71} +13.7898 q^{73} +3.64127 q^{75} +17.3495 q^{77} -1.60930 q^{79} +2.59807 q^{81} +5.56550 q^{83} -0.934682 q^{85} +1.34705 q^{87} -15.1495 q^{89} -6.41949 q^{91} +7.56728 q^{93} +3.77229 q^{95} +8.58778 q^{97} -9.88787 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 33 q - 3 q^{3} + 3 q^{5} + 48 q^{9}+O(q^{10}) $ 33 * q - 3 * q^3 + 3 * q^5 + 48 * q^9	$ 33 q - 3 q^{3} + 3 q^{5} + 48 q^{9} - 7 q^{11} + 9 q^{13} - 7 q^{15} - 33 q^{17} + 5 q^{19} + 14 q^{21} - 19 q^{23} + 52 q^{25} - 24 q^{27} + 14 q^{29} + 19 q^{31} + 34 q^{33} + 15 q^{35} + 13 q^{37} + 26 q^{39} + 32 q^{41} + 11 q^{43} - 27 q^{47} + 93 q^{49} + 3 q^{51} - 7 q^{53} + 31 q^{55} + 6 q^{57} - 33 q^{59} + 33 q^{61} + 10 q^{63} + 21 q^{65} - 10 q^{67} + 78 q^{69} - 29 q^{71} + 30 q^{73} + 31 q^{75} + 6 q^{77} + 66 q^{79} + 97 q^{81} - 9 q^{83} - 3 q^{85} - 27 q^{87} + 68 q^{89} + 28 q^{91} + 34 q^{93} + 8 q^{95} + 62 q^{97} + 15 q^{99}+O(q^{100}) $ 33 * q - 3 * q^3 + 3 * q^5 + 48 * q^9 - 7 * q^11 + 9 * q^13 - 7 * q^15 - 33 * q^17 + 5 * q^19 + 14 * q^21 - 19 * q^23 + 52 * q^25 - 24 * q^27 + 14 * q^29 + 19 * q^31 + 34 * q^33 + 15 * q^35 + 13 * q^37 + 26 * q^39 + 32 * q^41 + 11 * q^43 - 27 * q^47 + 93 * q^49 + 3 * q^51 - 7 * q^53 + 31 * q^55 + 6 * q^57 - 33 * q^59 + 33 * q^61 + 10 * q^63 + 21 * q^65 - 10 * q^67 + 78 * q^69 - 29 * q^71 + 30 * q^73 + 31 * q^75 + 6 * q^77 + 66 * q^79 + 97 * q^81 - 9 * q^83 - 3 * q^85 - 27 * q^87 + 68 * q^89 + 28 * q^91 + 34 * q^93 + 8 * q^95 + 62 * q^97 + 15 * 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
$3$	−0.882440	−0.509477	−0.254739	−	0.967010i	$-0.581989\pi$
$3$	−0.882440	−0.509477	−0.254739	+	0.967010i	$0.581989\pi$
$4$	0	0
$5$	0.934682	0.418003	0.209001	−	0.977915i	$-0.432979\pi$
$5$	0.934682	0.418003	0.209001	+	0.977915i	$0.432979\pi$
$6$	0	0
$7$	3.89754	1.47313	0.736565	−	0.676366i	$-0.236446\pi$
$7$	3.89754	1.47313	0.736565	+	0.676366i	$0.236446\pi$
$8$	0	0
$9$	−2.22130	−0.740433
$10$	0	0
$11$	4.45139	1.34214	0.671072	−	0.741392i	$-0.265834\pi$
$11$	4.45139	1.34214	0.671072	+	0.741392i	$0.265834\pi$
$12$	0	0
$13$	−1.64706	−0.456813	−0.228407	−	0.973566i	$-0.573352\pi$
$13$	−1.64706	−0.456813	−0.228407	+	0.973566i	$0.573352\pi$
$14$	0	0
$15$	−0.824801	−0.212963
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	4.03591	0.925900	0.462950	−	0.886384i	$-0.346791\pi$
$19$	4.03591	0.925900	0.462950	+	0.886384i	$0.346791\pi$
$20$	0	0
$21$	−3.43934	−0.750526
$22$	0	0
$23$	−1.54886	−0.322960	−0.161480	−	0.986876i	$-0.551627\pi$
$23$	−1.54886	−0.322960	−0.161480	+	0.986876i	$0.551627\pi$
$24$	0	0
$25$	−4.12637	−0.825274
$26$	0	0
$27$	4.60748	0.886711
$28$	0	0
$29$	−1.52651	−0.283465	−0.141732	−	0.989905i	$-0.545267\pi$
$29$	−1.52651	−0.283465	−0.141732	+	0.989905i	$0.545267\pi$
$30$	0	0
$31$	−8.57540	−1.54019	−0.770094	−	0.637931i	$-0.779791\pi$
$31$	−8.57540	−1.54019	−0.770094	+	0.637931i	$0.779791\pi$
$32$	0	0
$33$	−3.92808	−0.683792
$34$	0	0
$35$	3.64296	0.615772
$36$	0	0
$37$	−5.00519	−0.822848	−0.411424	−	0.911444i	$-0.634968\pi$
$37$	−5.00519	−0.822848	−0.411424	+	0.911444i	$0.634968\pi$
$38$	0	0
$39$	1.45344	0.232736
$40$	0	0
$41$	10.8363	1.69235	0.846174	−	0.532907i	$-0.178900\pi$
$41$	10.8363	1.69235	0.846174	+	0.532907i	$0.178900\pi$
$42$	0	0
$43$	9.41474	1.43573	0.717867	−	0.696180i	$-0.245118\pi$
$43$	9.41474	1.43573	0.717867	+	0.696180i	$0.245118\pi$
$44$	0	0
$45$	−2.07621	−0.309503
$46$	0	0
$47$	12.0465	1.75716	0.878580	−	0.477595i	$-0.158491\pi$
$47$	12.0465	1.75716	0.878580	+	0.477595i	$0.158491\pi$
$48$	0	0
$49$	8.19080	1.17011
$50$	0	0
$51$	0.882440	0.123566
$52$	0	0
$53$	0.848297	0.116523	0.0582613	−	0.998301i	$-0.481444\pi$
$53$	0.848297	0.116523	0.0582613	+	0.998301i	$0.481444\pi$
$54$	0	0
$55$	4.16063	0.561020
$56$	0	0
$57$	−3.56145	−0.471725
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	−9.57103	−1.22545	−0.612723	−	0.790298i	$-0.709926\pi$
$61$	−9.57103	−1.22545	−0.612723	+	0.790298i	$0.709926\pi$
$62$	0	0
$63$	−8.65760	−1.09075
$64$	0	0
$65$	−1.53948	−0.190949
$66$	0	0
$67$	14.5600	1.77878	0.889392	−	0.457145i	$-0.151128\pi$
$67$	14.5600	1.77878	0.889392	+	0.457145i	$0.151128\pi$
$68$	0	0
$69$	1.36678	0.164541
$70$	0	0
$71$	−6.70386	−0.795601	−0.397801	−	0.917472i	$-0.630226\pi$
$71$	−6.70386	−0.795601	−0.397801	+	0.917472i	$0.630226\pi$
$72$	0	0
$73$	13.7898	1.61397	0.806986	−	0.590571i	$-0.201097\pi$
$73$	13.7898	1.61397	0.806986	+	0.590571i	$0.201097\pi$
$74$	0	0
$75$	3.64127	0.420458
$76$	0	0
$77$	17.3495	1.97715
$78$	0	0
$79$	−1.60930	−0.181061	−0.0905304	−	0.995894i	$-0.528856\pi$
$79$	−1.60930	−0.181061	−0.0905304	+	0.995894i	$0.528856\pi$
$80$	0	0
$81$	2.59807	0.288674
$82$	0	0
$83$	5.56550	0.610893	0.305446	−	0.952209i	$-0.401194\pi$
$83$	5.56550	0.610893	0.305446	+	0.952209i	$0.401194\pi$
$84$	0	0
$85$	−0.934682	−0.101381
$86$	0	0
$87$	1.34705	0.144419
$88$	0	0
$89$	−15.1495	−1.60584	−0.802922	−	0.596084i	$-0.796722\pi$
$89$	−15.1495	−1.60584	−0.802922	+	0.596084i	$0.796722\pi$
$90$	0	0
$91$	−6.41949	−0.672946
$92$	0	0
$93$	7.56728	0.784690
$94$	0	0
$95$	3.77229	0.387029
$96$	0	0
$97$	8.58778	0.871957	0.435978	−	0.899957i	$-0.356402\pi$
$97$	8.58778	0.871957	0.435978	+	0.899957i	$0.356402\pi$
$98$	0	0
$99$	−9.88787	−0.993768
$100$	0	0
$101$	−5.05114	−0.502607	−0.251304	−	0.967908i	$-0.580859\pi$
$101$	−5.05114	−0.502607	−0.251304	+	0.967908i	$0.580859\pi$
$102$	0	0
$103$	4.62949	0.456157	0.228079	−	0.973643i	$-0.426756\pi$
$103$	4.62949	0.456157	0.228079	+	0.973643i	$0.426756\pi$
$104$	0	0
$105$	−3.21469	−0.313722
$106$	0	0
$107$	12.7516	1.23274	0.616370	−	0.787457i	$-0.288603\pi$
$107$	12.7516	1.23274	0.616370	+	0.787457i	$0.288603\pi$
$108$	0	0
$109$	0.168474	0.0161369	0.00806845	−	0.999967i	$-0.497432\pi$
$109$	0.168474	0.0161369	0.00806845	+	0.999967i	$0.497432\pi$
$110$	0	0
$111$	4.41678	0.419222
$112$	0	0
$113$	15.2989	1.43920	0.719598	−	0.694391i	$-0.244326\pi$
$113$	15.2989	1.43920	0.719598	+	0.694391i	$0.244326\pi$
$114$	0	0
$115$	−1.44769	−0.134998
$116$	0	0
$117$	3.65862	0.338240
$118$	0	0
$119$	−3.89754	−0.357287
$120$	0	0
$121$	8.81486	0.801351
$122$	0	0
$123$	−9.56240	−0.862213
$124$	0	0
$125$	−8.53026	−0.762969
$126$	0	0
$127$	−18.7464	−1.66347	−0.831735	−	0.555172i	$-0.812652\pi$
$127$	−18.7464	−1.66347	−0.831735	+	0.555172i	$0.812652\pi$
$128$	0	0
$129$	−8.30795	−0.731474
$130$	0	0
$131$	19.8010	1.73003	0.865013	−	0.501750i	$-0.167310\pi$
$131$	19.8010	1.73003	0.865013	+	0.501750i	$0.167310\pi$
$132$	0	0
$133$	15.7301	1.36397
$134$	0	0
$135$	4.30653	0.370647
$136$	0	0
$137$	1.06027	0.0905847	0.0452924	−	0.998974i	$-0.485578\pi$
$137$	1.06027	0.0905847	0.0452924	+	0.998974i	$0.485578\pi$
$138$	0	0
$139$	−10.9247	−0.926618	−0.463309	−	0.886197i	$-0.653338\pi$
$139$	−10.9247	−0.926618	−0.463309	+	0.886197i	$0.653338\pi$
$140$	0	0
$141$	−10.6303	−0.895233
$142$	0	0
$143$	−7.33172	−0.613110
$144$	0	0
$145$	−1.42680	−0.118489
$146$	0	0
$147$	−7.22789	−0.596146
$148$	0	0
$149$	−4.52805	−0.370952	−0.185476	−	0.982649i	$-0.559383\pi$
$149$	−4.52805	−0.370952	−0.185476	+	0.982649i	$0.559383\pi$
$150$	0	0
$151$	15.3396	1.24831	0.624157	−	0.781299i	$-0.285442\pi$
$151$	15.3396	1.24831	0.624157	+	0.781299i	$0.285442\pi$
$152$	0	0
$153$	2.22130	0.179581
$154$	0	0
$155$	−8.01527	−0.643802
$156$	0	0
$157$	−20.1195	−1.60571	−0.802857	−	0.596172i	$-0.796688\pi$
$157$	−20.1195	−1.60571	−0.802857	+	0.596172i	$0.796688\pi$
$158$	0	0
$159$	−0.748571	−0.0593656
$160$	0	0
$161$	−6.03674	−0.475762
$162$	0	0
$163$	0.659626	0.0516659	0.0258330	−	0.999666i	$-0.491776\pi$
$163$	0.659626	0.0516659	0.0258330	+	0.999666i	$0.491776\pi$
$164$	0	0
$165$	−3.67151	−0.285827
$166$	0	0
$167$	−21.4026	−1.65618	−0.828092	−	0.560592i	$-0.810573\pi$
$167$	−21.4026	−1.65618	−0.828092	+	0.560592i	$0.810573\pi$
$168$	0	0
$169$	−10.2872	−0.791322
$170$	0	0
$171$	−8.96496	−0.685567
$172$	0	0
$173$	10.3717	0.788544	0.394272	−	0.918994i	$-0.370997\pi$
$173$	10.3717	0.788544	0.394272	+	0.918994i	$0.370997\pi$
$174$	0	0
$175$	−16.0827	−1.21574
$176$	0	0
$177$	0.882440	0.0663283
$178$	0	0
$179$	8.42084	0.629403	0.314702	−	0.949191i	$-0.398096\pi$
$179$	8.42084	0.629403	0.314702	+	0.949191i	$0.398096\pi$
$180$	0	0
$181$	14.6016	1.08533	0.542664	−	0.839950i	$-0.317416\pi$
$181$	14.6016	1.08533	0.542664	+	0.839950i	$0.317416\pi$
$182$	0	0
$183$	8.44586	0.624336
$184$	0	0
$185$	−4.67826	−0.343953
$186$	0	0
$187$	−4.45139	−0.325518
$188$	0	0
$189$	17.9578	1.30624
$190$	0	0
$191$	12.3287	0.892070	0.446035	−	0.895016i	$-0.352836\pi$
$191$	12.3287	0.892070	0.446035	+	0.895016i	$0.352836\pi$
$192$	0	0
$193$	0.844633	0.0607980	0.0303990	−	0.999538i	$-0.490322\pi$
$193$	0.844633	0.0607980	0.0303990	+	0.999538i	$0.490322\pi$
$194$	0	0
$195$	1.35850	0.0972842
$196$	0	0
$197$	−1.43168	−0.102003	−0.0510014	−	0.998699i	$-0.516241\pi$
$197$	−1.43168	−0.102003	−0.0510014	+	0.998699i	$0.516241\pi$
$198$	0	0
$199$	−18.1917	−1.28957	−0.644786	−	0.764363i	$-0.723054\pi$
$199$	−18.1917	−1.28957	−0.644786	+	0.764363i	$0.723054\pi$
$200$	0	0
$201$	−12.8483	−0.906250
$202$	0	0
$203$	−5.94961	−0.417581
$204$	0	0
$205$	10.1285	0.707406
$206$	0	0
$207$	3.44048	0.239130
$208$	0	0
$209$	17.9654	1.24269
$210$	0	0
$211$	3.91698	0.269656	0.134828	−	0.990869i	$-0.456952\pi$
$211$	3.91698	0.269656	0.134828	+	0.990869i	$0.456952\pi$
$212$	0	0
$213$	5.91575	0.405341
$214$	0	0
$215$	8.79979	0.600141
$216$	0	0
$217$	−33.4229	−2.26890
$218$	0	0
$219$	−12.1687	−0.822282
$220$	0	0
$221$	1.64706	0.110794
$222$	0	0
$223$	9.58328	0.641744	0.320872	−	0.947123i	$-0.396024\pi$
$223$	9.58328	0.641744	0.320872	+	0.947123i	$0.396024\pi$
$224$	0	0
$225$	9.16590	0.611060
$226$	0	0
$227$	11.7333	0.778769	0.389384	−	0.921075i	$-0.372688\pi$
$227$	11.7333	0.778769	0.389384	+	0.921075i	$0.372688\pi$
$228$	0	0
$229$	16.6084	1.09751	0.548756	−	0.835982i	$-0.315102\pi$
$229$	16.6084	1.09751	0.548756	+	0.835982i	$0.315102\pi$
$230$	0	0
$231$	−15.3099	−1.00731
$232$	0	0
$233$	8.98456	0.588598	0.294299	−	0.955713i	$-0.404914\pi$
$233$	8.98456	0.588598	0.294299	+	0.955713i	$0.404914\pi$
$234$	0	0
$235$	11.2596	0.734498
$236$	0	0
$237$	1.42011	0.0922463
$238$	0	0
$239$	−17.5963	−1.13821	−0.569106	−	0.822264i	$-0.692711\pi$
$239$	−17.5963	−1.13821	−0.569106	+	0.822264i	$0.692711\pi$
$240$	0	0
$241$	20.9529	1.34969	0.674847	−	0.737958i	$-0.264210\pi$
$241$	20.9529	1.34969	0.674847	+	0.737958i	$0.264210\pi$
$242$	0	0
$243$	−16.1151	−1.03378
$244$	0	0
$245$	7.65579	0.489111
$246$	0	0
$247$	−6.64740	−0.422964
$248$	0	0
$249$	−4.91122	−0.311236
$250$	0	0
$251$	−1.13362	−0.0715532	−0.0357766	−	0.999360i	$-0.511390\pi$
$251$	−1.13362	−0.0715532	−0.0357766	+	0.999360i	$0.511390\pi$
$252$	0	0
$253$	−6.89458	−0.433459
$254$	0	0
$255$	0.824801	0.0516511
$256$	0	0
$257$	5.93660	0.370315	0.185157	−	0.982709i	$-0.440720\pi$
$257$	5.93660	0.370315	0.185157	+	0.982709i	$0.440720\pi$
$258$	0	0
$259$	−19.5079	−1.21216
$260$	0	0
$261$	3.39082	0.209887
$262$	0	0
$263$	−9.88486	−0.609526	−0.304763	−	0.952428i	$-0.598577\pi$
$263$	−9.88486	−0.609526	−0.304763	+	0.952428i	$0.598577\pi$
$264$	0	0
$265$	0.792888	0.0487067
$266$	0	0
$267$	13.3685	0.818141
$268$	0	0
$269$	−15.6994	−0.957209	−0.478605	−	0.878031i	$-0.658857\pi$
$269$	−15.6994	−0.957209	−0.478605	+	0.878031i	$0.658857\pi$
$270$	0	0
$271$	11.8839	0.721894	0.360947	−	0.932586i	$-0.382454\pi$
$271$	11.8839	0.721894	0.360947	+	0.932586i	$0.382454\pi$
$272$	0	0
$273$	5.66482	0.342851
$274$	0	0
$275$	−18.3681	−1.10764
$276$	0	0
$277$	29.2282	1.75615	0.878076	−	0.478522i	$-0.158827\pi$
$277$	29.2282	1.75615	0.878076	+	0.478522i	$0.158827\pi$
$278$	0	0
$279$	19.0485	1.14041
$280$	0	0
$281$	15.2946	0.912399	0.456200	−	0.889877i	$-0.349210\pi$
$281$	15.2946	0.912399	0.456200	+	0.889877i	$0.349210\pi$
$282$	0	0
$283$	11.3022	0.671846	0.335923	−	0.941890i	$-0.390952\pi$
$283$	11.3022	0.671846	0.335923	+	0.941890i	$0.390952\pi$
$284$	0	0
$285$	−3.32882	−0.197182
$286$	0	0
$287$	42.2349	2.49305
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−7.57820	−0.444242
$292$	0	0
$293$	15.5781	0.910085	0.455042	−	0.890470i	$-0.349624\pi$
$293$	15.5781	0.910085	0.455042	+	0.890470i	$0.349624\pi$
$294$	0	0
$295$	−0.934682	−0.0544193
$296$	0	0
$297$	20.5097	1.19009
$298$	0	0
$299$	2.55107	0.147532
$300$	0	0
$301$	36.6943	2.11502
$302$	0	0
$303$	4.45733	0.256067
$304$	0	0
$305$	−8.94587	−0.512239
$306$	0	0
$307$	−26.6807	−1.52275	−0.761375	−	0.648312i	$-0.775475\pi$
$307$	−26.6807	−1.52275	−0.761375	+	0.648312i	$0.775475\pi$
$308$	0	0
$309$	−4.08525	−0.232402
$310$	0	0
$311$	−11.8242	−0.670489	−0.335244	−	0.942131i	$-0.608819\pi$
$311$	−11.8242	−0.670489	−0.335244	+	0.942131i	$0.608819\pi$
$312$	0	0
$313$	28.1549	1.59141	0.795706	−	0.605683i	$-0.207100\pi$
$313$	28.1549	1.59141	0.795706	+	0.605683i	$0.207100\pi$
$314$	0	0
$315$	−8.09210	−0.455938
$316$	0	0
$317$	−28.1087	−1.57874	−0.789372	−	0.613915i	$-0.789594\pi$
$317$	−28.1087	−1.57874	−0.789372	+	0.613915i	$0.789594\pi$
$318$	0	0
$319$	−6.79507	−0.380451
$320$	0	0
$321$	−11.2525	−0.628053
$322$	0	0
$323$	−4.03591	−0.224564
$324$	0	0
$325$	6.79639	0.376996
$326$	0	0
$327$	−0.148668	−0.00822138
$328$	0	0
$329$	46.9516	2.58853
$330$	0	0
$331$	17.8055	0.978677	0.489339	−	0.872094i	$-0.337238\pi$
$331$	17.8055	0.978677	0.489339	+	0.872094i	$0.337238\pi$
$332$	0	0
$333$	11.1180	0.609264
$334$	0	0
$335$	13.6090	0.743536
$336$	0	0
$337$	−7.34626	−0.400176	−0.200088	−	0.979778i	$-0.564123\pi$
$337$	−7.34626	−0.400176	−0.200088	+	0.979778i	$0.564123\pi$
$338$	0	0
$339$	−13.5003	−0.733237
$340$	0	0
$341$	−38.1724	−2.06715
$342$	0	0
$343$	4.64118	0.250600
$344$	0	0
$345$	1.27750	0.0687784
$346$	0	0
$347$	−23.6959	−1.27207	−0.636033	−	0.771662i	$-0.719426\pi$
$347$	−23.6959	−1.27207	−0.636033	+	0.771662i	$0.719426\pi$
$348$	0	0
$349$	7.44879	0.398724	0.199362	−	0.979926i	$-0.436113\pi$
$349$	7.44879	0.398724	0.199362	+	0.979926i	$0.436113\pi$
$350$	0	0
$351$	−7.58882	−0.405061
$352$	0	0
$353$	−5.97618	−0.318080	−0.159040	−	0.987272i	$-0.550840\pi$
$353$	−5.97618	−0.318080	−0.159040	+	0.987272i	$0.550840\pi$
$354$	0	0
$355$	−6.26597	−0.332563
$356$	0	0
$357$	3.43934	0.182029
$358$	0	0
$359$	−10.4121	−0.549528	−0.274764	−	0.961512i	$-0.588600\pi$
$359$	−10.4121	−0.549528	−0.274764	+	0.961512i	$0.588600\pi$
$360$	0	0
$361$	−2.71146	−0.142709
$362$	0	0
$363$	−7.77859	−0.408270
$364$	0	0
$365$	12.8891	0.674644
$366$	0	0
$367$	−28.3457	−1.47963	−0.739817	−	0.672808i	$-0.765088\pi$
$367$	−28.3457	−1.47963	−0.739817	+	0.672808i	$0.765088\pi$
$368$	0	0
$369$	−24.0707	−1.25307
$370$	0	0
$371$	3.30627	0.171653
$372$	0	0
$373$	25.5788	1.32442	0.662209	−	0.749319i	$-0.269619\pi$
$373$	25.5788	1.32442	0.662209	+	0.749319i	$0.269619\pi$
$374$	0	0
$375$	7.52744	0.388715
$376$	0	0
$377$	2.51425	0.129491
$378$	0	0
$379$	9.78459	0.502601	0.251300	−	0.967909i	$-0.419142\pi$
$379$	9.78459	0.502601	0.251300	+	0.967909i	$0.419142\pi$
$380$	0	0
$381$	16.5425	0.847500
$382$	0	0
$383$	12.2148	0.624145	0.312072	−	0.950058i	$-0.398977\pi$
$383$	12.2148	0.624145	0.312072	+	0.950058i	$0.398977\pi$
$384$	0	0
$385$	16.2162	0.826455
$386$	0	0
$387$	−20.9130	−1.06307
$388$	0	0
$389$	10.4886	0.531794	0.265897	−	0.964001i	$-0.414332\pi$
$389$	10.4886	0.531794	0.265897	+	0.964001i	$0.414332\pi$
$390$	0	0
$391$	1.54886	0.0783292
$392$	0	0
$393$	−17.4732	−0.881408
$394$	0	0
$395$	−1.50419	−0.0756839
$396$	0	0
$397$	−4.40024	−0.220842	−0.110421	−	0.993885i	$-0.535220\pi$
$397$	−4.40024	−0.220842	−0.110421	+	0.993885i	$0.535220\pi$
$398$	0	0
$399$	−13.8809	−0.694913
$400$	0	0
$401$	−26.3435	−1.31553	−0.657765	−	0.753223i	$-0.728498\pi$
$401$	−26.3435	−1.31553	−0.657765	+	0.753223i	$0.728498\pi$
$402$	0	0
$403$	14.1242	0.703578
$404$	0	0
$405$	2.42837	0.120667
$406$	0	0
$407$	−22.2801	−1.10438
$408$	0	0
$409$	−15.8872	−0.785570	−0.392785	−	0.919630i	$-0.628488\pi$
$409$	−15.8872	−0.785570	−0.392785	+	0.919630i	$0.628488\pi$
$410$	0	0
$411$	−0.935622	−0.0461508
$412$	0	0
$413$	−3.89754	−0.191785
$414$	0	0
$415$	5.20197	0.255355
$416$	0	0
$417$	9.64036	0.472090
$418$	0	0
$419$	−20.4876	−1.00089	−0.500443	−	0.865770i	$-0.666829\pi$
$419$	−20.4876	−1.00089	−0.500443	+	0.865770i	$0.666829\pi$
$420$	0	0
$421$	12.9295	0.630145	0.315072	−	0.949068i	$-0.397971\pi$
$421$	12.9295	0.630145	0.315072	+	0.949068i	$0.397971\pi$
$422$	0	0
$423$	−26.7588	−1.30106
$424$	0	0
$425$	4.12637	0.200158
$426$	0	0
$427$	−37.3035	−1.80524
$428$	0	0
$429$	6.46981	0.312365
$430$	0	0
$431$	30.2478	1.45699	0.728493	−	0.685053i	$-0.240221\pi$
$431$	30.2478	1.45699	0.728493	+	0.685053i	$0.240221\pi$
$432$	0	0
$433$	−28.0864	−1.34975	−0.674873	−	0.737934i	$-0.735801\pi$
$433$	−28.0864	−1.34975	−0.674873	+	0.737934i	$0.735801\pi$
$434$	0	0
$435$	1.25906	0.0603675
$436$	0	0
$437$	−6.25106	−0.299029
$438$	0	0
$439$	28.8385	1.37639	0.688193	−	0.725528i	$-0.258404\pi$
$439$	28.8385	1.37639	0.688193	+	0.725528i	$0.258404\pi$
$440$	0	0
$441$	−18.1942	−0.866391
$442$	0	0
$443$	−11.9119	−0.565950	−0.282975	−	0.959127i	$-0.591321\pi$
$443$	−11.9119	−0.565950	−0.282975	+	0.959127i	$0.591321\pi$
$444$	0	0
$445$	−14.1600	−0.671247
$446$	0	0
$447$	3.99573	0.188992
$448$	0	0
$449$	18.3362	0.865340	0.432670	−	0.901552i	$-0.357572\pi$
$449$	18.3362	0.865340	0.432670	+	0.901552i	$0.357572\pi$
$450$	0	0
$451$	48.2367	2.27138
$452$	0	0
$453$	−13.5362	−0.635988
$454$	0	0
$455$	−6.00019	−0.281293
$456$	0	0
$457$	20.9716	0.981011	0.490506	−	0.871438i	$-0.336812\pi$
$457$	20.9716	0.981011	0.490506	+	0.871438i	$0.336812\pi$
$458$	0	0
$459$	−4.60748	−0.215059
$460$	0	0
$461$	7.23351	0.336898	0.168449	−	0.985710i	$-0.446124\pi$
$461$	7.23351	0.336898	0.168449	+	0.985710i	$0.446124\pi$
$462$	0	0
$463$	−10.2861	−0.478035	−0.239018	−	0.971015i	$-0.576825\pi$
$463$	−10.2861	−0.478035	−0.239018	+	0.971015i	$0.576825\pi$
$464$	0	0
$465$	7.07300	0.328003
$466$	0	0
$467$	−4.20962	−0.194798	−0.0973989	−	0.995245i	$-0.531052\pi$
$467$	−4.20962	−0.194798	−0.0973989	+	0.995245i	$0.531052\pi$
$468$	0	0
$469$	56.7481	2.62038
$470$	0	0
$471$	17.7543	0.818074
$472$	0	0
$473$	41.9087	1.92696
$474$	0	0
$475$	−16.6536	−0.764121
$476$	0	0
$477$	−1.88432	−0.0862771
$478$	0	0
$479$	−3.80891	−0.174034	−0.0870168	−	0.996207i	$-0.527733\pi$
$479$	−3.80891	−0.174034	−0.0870168	+	0.996207i	$0.527733\pi$
$480$	0	0
$481$	8.24387	0.375888
$482$	0	0
$483$	5.32706	0.242390
$484$	0	0
$485$	8.02684	0.364480
$486$	0	0
$487$	22.6708	1.02731	0.513656	−	0.857996i	$-0.328291\pi$
$487$	22.6708	1.02731	0.513656	+	0.857996i	$0.328291\pi$
$488$	0	0
$489$	−0.582081	−0.0263226
$490$	0	0
$491$	−7.06099	−0.318658	−0.159329	−	0.987226i	$-0.550933\pi$
$491$	−7.06099	−0.318658	−0.159329	+	0.987226i	$0.550933\pi$
$492$	0	0
$493$	1.52651	0.0687503
$494$	0	0
$495$	−9.24201	−0.415398
$496$	0	0
$497$	−26.1285	−1.17202
$498$	0	0
$499$	17.1329	0.766973	0.383487	−	0.923546i	$-0.374723\pi$
$499$	17.1329	0.766973	0.383487	+	0.923546i	$0.374723\pi$
$500$	0	0
$501$	18.8865	0.843788
$502$	0	0
$503$	18.4049	0.820632	0.410316	−	0.911943i	$-0.365418\pi$
$503$	18.4049	0.820632	0.410316	+	0.911943i	$0.365418\pi$
$504$	0	0
$505$	−4.72121	−0.210091
$506$	0	0
$507$	9.07782	0.403160
$508$	0	0
$509$	−21.2155	−0.940361	−0.470181	−	0.882570i	$-0.655811\pi$
$509$	−21.2155	−0.940361	−0.470181	+	0.882570i	$0.655811\pi$
$510$	0	0
$511$	53.7462	2.37759
$512$	0	0
$513$	18.5954	0.821006
$514$	0	0
$515$	4.32710	0.190675
$516$	0	0
$517$	53.6236	2.35836
$518$	0	0
$519$	−9.15238	−0.401745
$520$	0	0
$521$	33.8775	1.48420	0.742100	−	0.670289i	$-0.233830\pi$
$521$	33.8775	1.48420	0.742100	+	0.670289i	$0.233830\pi$
$522$	0	0
$523$	2.26814	0.0991791	0.0495895	−	0.998770i	$-0.484209\pi$
$523$	2.26814	0.0991791	0.0495895	+	0.998770i	$0.484209\pi$
$524$	0	0
$525$	14.1920	0.619390
$526$	0	0
$527$	8.57540	0.373550
$528$	0	0
$529$	−20.6010	−0.895697
$530$	0	0
$531$	2.22130	0.0963962
$532$	0	0
$533$	−17.8481	−0.773087
$534$	0	0
$535$	11.9187	0.515289
$536$	0	0
$537$	−7.43089	−0.320667
$538$	0	0
$539$	36.4604	1.57046
$540$	0	0
$541$	−0.00136087	−5.85084e−5 0	−2.92542e−5	−	1.00000i	$-0.500009\pi$
$541$	−0.00136087	−5.85084e−5 0	−2.92542e−5		1.00000i	$0.500009\pi$
$542$	0	0
$543$	−12.8850	−0.552950
$544$	0	0
$545$	0.157470	0.00674527
$546$	0	0
$547$	−11.2742	−0.482051	−0.241026	−	0.970519i	$-0.577484\pi$
$547$	−11.2742	−0.482051	−0.241026	+	0.970519i	$0.577484\pi$
$548$	0	0
$549$	21.2601	0.907360
$550$	0	0
$551$	−6.16083	−0.262460
$552$	0	0
$553$	−6.27232	−0.266726
$554$	0	0
$555$	4.12829	0.175236
$556$	0	0
$557$	−0.473110	−0.0200463	−0.0100232	−	0.999950i	$-0.503191\pi$
$557$	−0.473110	−0.0200463	−0.0100232	+	0.999950i	$0.503191\pi$
$558$	0	0
$559$	−15.5067	−0.655863
$560$	0	0
$561$	3.92808	0.165844
$562$	0	0
$563$	24.9019	1.04949	0.524744	−	0.851260i	$-0.324161\pi$
$563$	24.9019	1.04949	0.524744	+	0.851260i	$0.324161\pi$
$564$	0	0
$565$	14.2996	0.601588
$566$	0	0
$567$	10.1261	0.425255
$568$	0	0
$569$	19.5874	0.821146	0.410573	−	0.911828i	$-0.365329\pi$
$569$	19.5874	0.821146	0.410573	+	0.911828i	$0.365329\pi$
$570$	0	0
$571$	11.7460	0.491553	0.245776	−	0.969327i	$-0.420957\pi$
$571$	11.7460	0.491553	0.245776	+	0.969327i	$0.420957\pi$
$572$	0	0
$573$	−10.8793	−0.454489
$574$	0	0
$575$	6.39117	0.266530
$576$	0	0
$577$	47.2034	1.96510	0.982551	−	0.185995i	$-0.0595508\pi$
$577$	47.2034	1.96510	0.982551	+	0.185995i	$0.0595508\pi$
$578$	0	0
$579$	−0.745338	−0.0309752
$580$	0	0
$581$	21.6917	0.899925
$582$	0	0
$583$	3.77610	0.156390
$584$	0	0
$585$	3.41965	0.141385
$586$	0	0
$587$	−15.4612	−0.638153	−0.319077	−	0.947729i	$-0.603373\pi$
$587$	−15.4612	−0.638153	−0.319077	+	0.947729i	$0.603373\pi$
$588$	0	0
$589$	−34.6095	−1.42606
$590$	0	0
$591$	1.26337	0.0519681
$592$	0	0
$593$	1.36307	0.0559746	0.0279873	−	0.999608i	$-0.491090\pi$
$593$	1.36307	0.0559746	0.0279873	+	0.999608i	$0.491090\pi$
$594$	0	0
$595$	−3.64296	−0.149347
$596$	0	0
$597$	16.0530	0.657007
$598$	0	0
$599$	45.2646	1.84946	0.924731	−	0.380620i	$-0.124290\pi$
$599$	45.2646	1.84946	0.924731	+	0.380620i	$0.124290\pi$
$600$	0	0
$601$	39.7440	1.62119	0.810596	−	0.585606i	$-0.199143\pi$
$601$	39.7440	1.62119	0.810596	+	0.585606i	$0.199143\pi$
$602$	0	0
$603$	−32.3421	−1.31707
$604$	0	0
$605$	8.23910	0.334967
$606$	0	0
$607$	35.3386	1.43435	0.717176	−	0.696892i	$-0.245435\pi$
$607$	35.3386	1.43435	0.717176	+	0.696892i	$0.245435\pi$
$608$	0	0
$609$	5.25018	0.212748
$610$	0	0
$611$	−19.8413	−0.802695
$612$	0	0
$613$	−29.4435	−1.18921	−0.594607	−	0.804017i	$-0.702692\pi$
$613$	−29.4435	−1.18921	−0.594607	+	0.804017i	$0.702692\pi$
$614$	0	0
$615$	−8.93781	−0.360407
$616$	0	0
$617$	7.78000	0.313211	0.156605	−	0.987661i	$-0.449945\pi$
$617$	7.78000	0.313211	0.156605	+	0.987661i	$0.449945\pi$
$618$	0	0
$619$	−15.8311	−0.636305	−0.318152	−	0.948040i	$-0.603062\pi$
$619$	−15.8311	−0.636305	−0.318152	+	0.948040i	$0.603062\pi$
$620$	0	0
$621$	−7.13635	−0.286372
$622$	0	0
$623$	−59.0457	−2.36562
$624$	0	0
$625$	12.6588	0.506351
$626$	0	0
$627$	−15.8534	−0.633123
$628$	0	0
$629$	5.00519	0.199570
$630$	0	0
$631$	6.13046	0.244050	0.122025	−	0.992527i	$-0.461061\pi$
$631$	6.13046	0.244050	0.122025	+	0.992527i	$0.461061\pi$
$632$	0	0
$633$	−3.45650	−0.137384
$634$	0	0
$635$	−17.5219	−0.695335
$636$	0	0
$637$	−13.4908	−0.534524
$638$	0	0
$639$	14.8913	0.589090
$640$	0	0
$641$	25.4701	1.00601	0.503005	−	0.864284i	$-0.332228\pi$
$641$	25.4701	1.00601	0.503005	+	0.864284i	$0.332228\pi$
$642$	0	0
$643$	−46.3783	−1.82898	−0.914490	−	0.404609i	$-0.867408\pi$
$643$	−46.3783	−1.82898	−0.914490	+	0.404609i	$0.867408\pi$
$644$	0	0
$645$	−7.76529	−0.305758
$646$	0	0
$647$	8.92699	0.350956	0.175478	−	0.984483i	$-0.443853\pi$
$647$	8.92699	0.350956	0.175478	+	0.984483i	$0.443853\pi$
$648$	0	0
$649$	−4.45139	−0.174732
$650$	0	0
$651$	29.4938	1.15595
$652$	0	0
$653$	24.3798	0.954054	0.477027	−	0.878889i	$-0.341714\pi$
$653$	24.3798	0.954054	0.477027	+	0.878889i	$0.341714\pi$
$654$	0	0
$655$	18.5077	0.723155
$656$	0	0
$657$	−30.6312	−1.19504
$658$	0	0
$659$	−9.29119	−0.361933	−0.180967	−	0.983489i	$-0.557923\pi$
$659$	−9.29119	−0.361933	−0.180967	+	0.983489i	$0.557923\pi$
$660$	0	0
$661$	−26.3538	−1.02504	−0.512522	−	0.858674i	$-0.671289\pi$
$661$	−26.3538	−1.02504	−0.512522	+	0.858674i	$0.671289\pi$
$662$	0	0
$663$	−1.45344	−0.0564468
$664$	0	0
$665$	14.7026	0.570144
$666$	0	0
$667$	2.36434	0.0915477
$668$	0	0
$669$	−8.45667	−0.326954
$670$	0	0
$671$	−42.6044	−1.64472
$672$	0	0
$673$	42.3264	1.63156	0.815781	−	0.578361i	$-0.196308\pi$
$673$	42.3264	1.63156	0.815781	+	0.578361i	$0.196308\pi$
$674$	0	0
$675$	−19.0122	−0.731779
$676$	0	0
$677$	−44.6680	−1.71673	−0.858366	−	0.513037i	$-0.828520\pi$
$677$	−44.6680	−1.71673	−0.858366	+	0.513037i	$0.828520\pi$
$678$	0	0
$679$	33.4712	1.28451
$680$	0	0
$681$	−10.3540	−0.396765
$682$	0	0
$683$	−38.9367	−1.48987	−0.744935	−	0.667137i	$-0.767520\pi$
$683$	−38.9367	−1.48987	−0.744935	+	0.667137i	$0.767520\pi$
$684$	0	0
$685$	0.991013	0.0378647
$686$	0	0
$687$	−14.6559	−0.559157
$688$	0	0
$689$	−1.39720	−0.0532291
$690$	0	0
$691$	5.19299	0.197551	0.0987754	−	0.995110i	$-0.468507\pi$
$691$	5.19299	0.197551	0.0987754	+	0.995110i	$0.468507\pi$
$692$	0	0
$693$	−38.5383	−1.46395
$694$	0	0
$695$	−10.2111	−0.387329
$696$	0	0
$697$	−10.8363	−0.410455
$698$	0	0
$699$	−7.92834	−0.299877
$700$	0	0
$701$	−33.5141	−1.26581	−0.632906	−	0.774229i	$-0.718138\pi$
$701$	−33.5141	−1.26581	−0.632906	+	0.774229i	$0.718138\pi$
$702$	0	0
$703$	−20.2005	−0.761875
$704$	0	0
$705$	−9.93596	−0.374210
$706$	0	0
$707$	−19.6870	−0.740406
$708$	0	0
$709$	21.9364	0.823838	0.411919	−	0.911220i	$-0.364859\pi$
$709$	21.9364	0.823838	0.411919	+	0.911220i	$0.364859\pi$
$710$	0	0
$711$	3.57474	0.134063
$712$	0	0
$713$	13.2821	0.497419
$714$	0	0
$715$	−6.85283	−0.256281
$716$	0	0
$717$	15.5277	0.579893
$718$	0	0
$719$	−5.53557	−0.206442	−0.103221	−	0.994658i	$-0.532915\pi$
$719$	−5.53557	−0.206442	−0.103221	+	0.994658i	$0.532915\pi$
$720$	0	0
$721$	18.0436	0.671979
$722$	0	0
$723$	−18.4897	−0.687638
$724$	0	0
$725$	6.29892	0.233936
$726$	0	0
$727$	19.9307	0.739190	0.369595	−	0.929193i	$-0.379496\pi$
$727$	19.9307	0.739190	0.369595	+	0.929193i	$0.379496\pi$
$728$	0	0
$729$	6.42640	0.238015
$730$	0	0
$731$	−9.41474	−0.348217
$732$	0	0
$733$	−30.0627	−1.11039	−0.555195	−	0.831720i	$-0.687357\pi$
$733$	−30.0627	−1.11039	−0.555195	+	0.831720i	$0.687357\pi$
$734$	0	0
$735$	−6.75578	−0.249191
$736$	0	0
$737$	64.8121	2.38739
$738$	0	0
$739$	24.9025	0.916054	0.458027	−	0.888938i	$-0.348556\pi$
$739$	24.9025	0.916054	0.458027	+	0.888938i	$0.348556\pi$
$740$	0	0
$741$	5.86593	0.215490
$742$	0	0
$743$	7.04808	0.258569	0.129285	−	0.991608i	$-0.458732\pi$
$743$	7.04808	0.258569	0.129285	+	0.991608i	$0.458732\pi$
$744$	0	0
$745$	−4.23229	−0.155059
$746$	0	0
$747$	−12.3626	−0.452325
$748$	0	0
$749$	49.6997	1.81599
$750$	0	0
$751$	44.7484	1.63289	0.816446	−	0.577422i	$-0.195941\pi$
$751$	44.7484	1.63289	0.816446	+	0.577422i	$0.195941\pi$
$752$	0	0
$753$	1.00035	0.0364547
$754$	0	0
$755$	14.3376	0.521799
$756$	0	0
$757$	7.38463	0.268399	0.134199	−	0.990954i	$-0.457154\pi$
$757$	7.38463	0.268399	0.134199	+	0.990954i	$0.457154\pi$
$758$	0	0
$759$	6.08406	0.220837
$760$	0	0
$761$	−32.7177	−1.18602	−0.593008	−	0.805196i	$-0.702060\pi$
$761$	−32.7177	−1.18602	−0.593008	+	0.805196i	$0.702060\pi$
$762$	0	0
$763$	0.656635	0.0237718
$764$	0	0
$765$	2.07621	0.0750655
$766$	0	0
$767$	1.64706	0.0594720
$768$	0	0
$769$	−24.4742	−0.882563	−0.441281	−	0.897369i	$-0.645476\pi$
$769$	−24.4742	−0.882563	−0.441281	+	0.897369i	$0.645476\pi$
$770$	0	0
$771$	−5.23870	−0.188667
$772$	0	0
$773$	−28.9977	−1.04297	−0.521487	−	0.853259i	$-0.674622\pi$
$773$	−28.9977	−1.04297	−0.521487	+	0.853259i	$0.674622\pi$
$774$	0	0
$775$	35.3853	1.27108
$776$	0	0
$777$	17.2146	0.617569
$778$	0	0
$779$	43.7343	1.56695
$780$	0	0
$781$	−29.8415	−1.06781
$782$	0	0
$783$	−7.03335	−0.251351
$784$	0	0
$785$	−18.8054	−0.671193
$786$	0	0
$787$	3.98696	0.142120	0.0710599	−	0.997472i	$-0.477362\pi$
$787$	3.98696	0.142120	0.0710599	+	0.997472i	$0.477362\pi$
$788$	0	0
$789$	8.72279	0.310540
$790$	0	0
$791$	59.6279	2.12012
$792$	0	0
$793$	15.7641	0.559800
$794$	0	0
$795$	−0.699676	−0.0248150
$796$	0	0
$797$	5.26656	0.186551	0.0932755	−	0.995640i	$-0.470266\pi$
$797$	5.26656	0.186551	0.0932755	+	0.995640i	$0.470266\pi$
$798$	0	0
$799$	−12.0465	−0.426174
$800$	0	0
$801$	33.6516	1.18902
$802$	0	0
$803$	61.3837	2.16618
$804$	0	0
$805$	−5.64244	−0.198870
$806$	0	0
$807$	13.8538	0.487676
$808$	0	0
$809$	−5.38813	−0.189437	−0.0947183	−	0.995504i	$-0.530195\pi$
$809$	−5.38813	−0.189437	−0.0947183	+	0.995504i	$0.530195\pi$
$810$	0	0
$811$	36.8238	1.29306	0.646529	−	0.762890i	$-0.276220\pi$
$811$	36.8238	1.29306	0.646529	+	0.762890i	$0.276220\pi$
$812$	0	0
$813$	−10.4868	−0.367788
$814$	0	0
$815$	0.616541	0.0215965
$816$	0	0
$817$	37.9970	1.32935
$818$	0	0
$819$	14.2596	0.498271
$820$	0	0
$821$	−21.3238	−0.744206	−0.372103	−	0.928191i	$-0.621363\pi$
$821$	−21.3238	−0.744206	−0.372103	+	0.928191i	$0.621363\pi$
$822$	0	0
$823$	43.6395	1.52118	0.760588	−	0.649235i	$-0.224911\pi$
$823$	43.6395	1.52118	0.760588	+	0.649235i	$0.224911\pi$
$824$	0	0
$825$	16.2087	0.564315
$826$	0	0
$827$	−11.2111	−0.389849	−0.194924	−	0.980818i	$-0.562446\pi$
$827$	−11.2111	−0.389849	−0.194924	+	0.980818i	$0.562446\pi$
$828$	0	0
$829$	13.7654	0.478092	0.239046	−	0.971008i	$-0.423165\pi$
$829$	13.7654	0.478092	0.239046	+	0.971008i	$0.423165\pi$
$830$	0	0
$831$	−25.7921	−0.894719
$832$	0	0
$833$	−8.19080	−0.283794
$834$	0	0
$835$	−20.0046	−0.692289
$836$	0	0
$837$	−39.5110	−1.36570
$838$	0	0
$839$	−2.21277	−0.0763931	−0.0381966	−	0.999270i	$-0.512161\pi$
$839$	−2.21277	−0.0763931	−0.0381966	+	0.999270i	$0.512161\pi$
$840$	0	0
$841$	−26.6698	−0.919648
$842$	0	0
$843$	−13.4966	−0.464846
$844$	0	0
$845$	−9.61524	−0.330774
$846$	0	0
$847$	34.3563	1.18050
$848$	0	0
$849$	−9.97351	−0.342290
$850$	0	0
$851$	7.75234	0.265747
$852$	0	0
$853$	40.0832	1.37242	0.686211	−	0.727403i	$-0.259273\pi$
$853$	40.0832	1.37242	0.686211	+	0.727403i	$0.259273\pi$
$854$	0	0
$855$	−8.37938	−0.286569
$856$	0	0
$857$	−15.3365	−0.523886	−0.261943	−	0.965083i	$-0.584363\pi$
$857$	−15.3365	−0.523886	−0.261943	+	0.965083i	$0.584363\pi$
$858$	0	0
$859$	−37.7871	−1.28928	−0.644639	−	0.764487i	$-0.722992\pi$
$859$	−37.7871	−1.28928	−0.644639	+	0.764487i	$0.722992\pi$
$860$	0	0
$861$	−37.2698	−1.27015
$862$	0	0
$863$	−18.7997	−0.639950	−0.319975	−	0.947426i	$-0.603674\pi$
$863$	−18.7997	−0.639950	−0.319975	+	0.947426i	$0.603674\pi$
$864$	0	0
$865$	9.69422	0.329613
$866$	0	0
$867$	−0.882440	−0.0299692
$868$	0	0
$869$	−7.16364	−0.243010
$870$	0	0
$871$	−23.9812	−0.812573
$872$	0	0
$873$	−19.0760	−0.645626
$874$	0	0
$875$	−33.2470	−1.12395
$876$	0	0
$877$	−29.5658	−0.998367	−0.499183	−	0.866496i	$-0.666367\pi$
$877$	−29.5658	−0.998367	−0.499183	+	0.866496i	$0.666367\pi$
$878$	0	0
$879$	−13.7468	−0.463667
$880$	0	0
$881$	−28.2122	−0.950494	−0.475247	−	0.879853i	$-0.657641\pi$
$881$	−28.2122	−0.950494	−0.475247	+	0.879853i	$0.657641\pi$
$882$	0	0
$883$	−35.0919	−1.18094	−0.590468	−	0.807061i	$-0.701057\pi$
$883$	−35.0919	−1.18094	−0.590468	+	0.807061i	$0.701057\pi$
$884$	0	0
$885$	0.824801	0.0277254
$886$	0	0
$887$	17.3974	0.584147	0.292073	−	0.956396i	$-0.405655\pi$
$887$	17.3974	0.584147	0.292073	+	0.956396i	$0.405655\pi$
$888$	0	0
$889$	−73.0647	−2.45051
$890$	0	0
$891$	11.5650	0.387442
$892$	0	0
$893$	48.6185	1.62696
$894$	0	0
$895$	7.87081	0.263092
$896$	0	0
$897$	−2.25117	−0.0751643
$898$	0	0
$899$	13.0904	0.436589
$900$	0	0
$901$	−0.848297	−0.0282609
$902$	0	0
$903$	−32.3805	−1.07756
$904$	0	0
$905$	13.6478	0.453670
$906$	0	0
$907$	22.6755	0.752929	0.376465	−	0.926431i	$-0.377140\pi$
$907$	22.6755	0.752929	0.376465	+	0.926431i	$0.377140\pi$
$908$	0	0
$909$	11.2201	0.372147
$910$	0	0
$911$	36.0750	1.19522	0.597609	−	0.801788i	$-0.296118\pi$
$911$	36.0750	1.19522	0.597609	+	0.801788i	$0.296118\pi$
$912$	0	0
$913$	24.7742	0.819906
$914$	0	0
$915$	7.89420	0.260974
$916$	0	0
$917$	77.1753	2.54855
$918$	0	0
$919$	−45.5015	−1.50096	−0.750479	−	0.660895i	$-0.770177\pi$
$919$	−45.5015	−1.50096	−0.750479	+	0.660895i	$0.770177\pi$
$920$	0	0
$921$	23.5442	0.775806
$922$	0	0
$923$	11.0417	0.363441
$924$	0	0
$925$	20.6533	0.679075
$926$	0	0
$927$	−10.2835	−0.337754
$928$	0	0
$929$	12.5484	0.411701	0.205850	−	0.978584i	$-0.434004\pi$
$929$	12.5484	0.411701	0.205850	+	0.978584i	$0.434004\pi$
$930$	0	0
$931$	33.0573	1.08341
$932$	0	0
$933$	10.4342	0.341599
$934$	0	0
$935$	−4.16063	−0.136067
$936$	0	0
$937$	−29.6860	−0.969799	−0.484899	−	0.874570i	$-0.661144\pi$
$937$	−29.6860	−0.969799	−0.484899	+	0.874570i	$0.661144\pi$
$938$	0	0
$939$	−24.8451	−0.810788
$940$	0	0
$941$	−57.7919	−1.88396	−0.941981	−	0.335667i	$-0.891038\pi$
$941$	−57.7919	−1.88396	−0.941981	+	0.335667i	$0.891038\pi$
$942$	0	0
$943$	−16.7839	−0.546560
$944$	0	0
$945$	16.7849	0.546012
$946$	0	0
$947$	2.69524	0.0875836	0.0437918	−	0.999041i	$-0.486056\pi$
$947$	2.69524	0.0875836	0.0437918	+	0.999041i	$0.486056\pi$
$948$	0	0
$949$	−22.7127	−0.737284
$950$	0	0
$951$	24.8043	0.804334
$952$	0	0
$953$	15.3687	0.497841	0.248920	−	0.968524i	$-0.419924\pi$
$953$	15.3687	0.497841	0.248920	+	0.968524i	$0.419924\pi$
$954$	0	0
$955$	11.5234	0.372888
$956$	0	0
$957$	5.99624	0.193831
$958$	0	0
$959$	4.13243	0.133443
$960$	0	0
$961$	42.5375	1.37218
$962$	0	0
$963$	−28.3250	−0.912762
$964$	0	0
$965$	0.789463	0.0254137
$966$	0	0
$967$	24.2641	0.780280	0.390140	−	0.920756i	$-0.372427\pi$
$967$	24.2641	0.780280	0.390140	+	0.920756i	$0.372427\pi$
$968$	0	0
$969$	3.56145	0.114410
$970$	0	0
$971$	48.1282	1.54451	0.772253	−	0.635315i	$-0.219130\pi$
$971$	48.1282	1.54451	0.772253	+	0.635315i	$0.219130\pi$
$972$	0	0
$973$	−42.5793	−1.36503
$974$	0	0
$975$	−5.99741	−0.192071
$976$	0	0
$977$	22.2434	0.711628	0.355814	−	0.934557i	$-0.384204\pi$
$977$	22.2434	0.711628	0.355814	+	0.934557i	$0.384204\pi$
$978$	0	0
$979$	−67.4363	−2.15527
$980$	0	0
$981$	−0.374232	−0.0119483
$982$	0	0
$983$	−7.39958	−0.236010	−0.118005	−	0.993013i	$-0.537650\pi$
$983$	−7.39958	−0.236010	−0.118005	+	0.993013i	$0.537650\pi$
$984$	0	0
$985$	−1.33816	−0.0426374
$986$	0	0
$987$	−41.4320	−1.31880
$988$	0	0
$989$	−14.5821	−0.463685
$990$	0	0
$991$	−24.1516	−0.767201	−0.383601	−	0.923499i	$-0.625316\pi$
$991$	−24.1516	−0.767201	−0.383601	+	0.923499i	$0.625316\pi$
$992$	0	0
$993$	−15.7123	−0.498614
$994$	0	0
$995$	−17.0034	−0.539044
$996$	0	0
$997$	12.2449	0.387801	0.193901	−	0.981021i	$-0.437886\pi$
$997$	12.2449	0.387801	0.193901	+	0.981021i	$0.437886\pi$
$998$	0	0
$999$	−23.0613	−0.729628

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.bc.1.14	✓	33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.bc.1.14	✓	33	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q-0.882440 q^{3} +0.934682 q^{5} +3.89754 q^{7} -2.22130 q^{9} +O(q^{10})\)	\(q-0.882440 q^{3} +0.934682 q^{5} +3.89754 q^{7} -2.22130 q^{9} +4.45139 q^{11} -1.64706 q^{13} -0.824801 q^{15} -1.00000 q^{17} +4.03591 q^{19} -3.43934 q^{21} -1.54886 q^{23} -4.12637 q^{25} +4.60748 q^{27} -1.52651 q^{29} -8.57540 q^{31} -3.92808 q^{33} +3.64296 q^{35} -5.00519 q^{37} +1.45344 q^{39} +10.8363 q^{41} +9.41474 q^{43} -2.07621 q^{45} +12.0465 q^{47} +8.19080 q^{49} +0.882440 q^{51} +0.848297 q^{53} +4.16063 q^{55} -3.56145 q^{57} -1.00000 q^{59} -9.57103 q^{61} -8.65760 q^{63} -1.53948 q^{65} +14.5600 q^{67} +1.36678 q^{69} -6.70386 q^{71} +13.7898 q^{73} +3.64127 q^{75} +17.3495 q^{77} -1.60930 q^{79} +2.59807 q^{81} +5.56550 q^{83} -0.934682 q^{85} +1.34705 q^{87} -15.1495 q^{89} -6.41949 q^{91} +7.56728 q^{93} +3.77229 q^{95} +8.58778 q^{97} -9.88787 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q - 3 q^{3} + 3 q^{5} + 48 q^{9}+O(q^{10}) \) 33 * q - 3 * q^3 + 3 * q^5 + 48 * q^9	\( 33 q - 3 q^{3} + 3 q^{5} + 48 q^{9} - 7 q^{11} + 9 q^{13} - 7 q^{15} - 33 q^{17} + 5 q^{19} + 14 q^{21} - 19 q^{23} + 52 q^{25} - 24 q^{27} + 14 q^{29} + 19 q^{31} + 34 q^{33} + 15 q^{35} + 13 q^{37} + 26 q^{39} + 32 q^{41} + 11 q^{43} - 27 q^{47} + 93 q^{49} + 3 q^{51} - 7 q^{53} + 31 q^{55} + 6 q^{57} - 33 q^{59} + 33 q^{61} + 10 q^{63} + 21 q^{65} - 10 q^{67} + 78 q^{69} - 29 q^{71} + 30 q^{73} + 31 q^{75} + 6 q^{77} + 66 q^{79} + 97 q^{81} - 9 q^{83} - 3 q^{85} - 27 q^{87} + 68 q^{89} + 28 q^{91} + 34 q^{93} + 8 q^{95} + 62 q^{97} + 15 q^{99}+O(q^{100}) \) 33 * q - 3 * q^3 + 3 * q^5 + 48 * q^9 - 7 * q^11 + 9 * q^13 - 7 * q^15 - 33 * q^17 + 5 * q^19 + 14 * q^21 - 19 * q^23 + 52 * q^25 - 24 * q^27 + 14 * q^29 + 19 * q^31 + 34 * q^33 + 15 * q^35 + 13 * q^37 + 26 * q^39 + 32 * q^41 + 11 * q^43 - 27 * q^47 + 93 * q^49 + 3 * q^51 - 7 * q^53 + 31 * q^55 + 6 * q^57 - 33 * q^59 + 33 * q^61 + 10 * q^63 + 21 * q^65 - 10 * q^67 + 78 * q^69 - 29 * q^71 + 30 * q^73 + 31 * q^75 + 6 * q^77 + 66 * q^79 + 97 * q^81 - 9 * q^83 - 3 * q^85 - 27 * q^87 + 68 * q^89 + 28 * q^91 + 34 * q^93 + 8 * q^95 + 62 * q^97 + 15 * q^99

Introduction

L-functions

Modular forms

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Database

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