LMFDB - Embedded newform 8032.2.a.h.1.16

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:	$1$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
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)

$f(q)$	$=$	$q+0.0692070 q^{3} +0.775693 q^{5} +4.19948 q^{7} -2.99521 q^{9} +O(q^{10})$	$q+0.0692070 q^{3} +0.775693 q^{5} +4.19948 q^{7} -2.99521 q^{9} +0.210429 q^{11} -5.86540 q^{13} +0.0536833 q^{15} -6.81669 q^{17} +7.37312 q^{19} +0.290633 q^{21} -3.64209 q^{23} -4.39830 q^{25} -0.414910 q^{27} -6.37032 q^{29} +8.26252 q^{31} +0.0145631 q^{33} +3.25751 q^{35} +5.38161 q^{37} -0.405926 q^{39} +7.37849 q^{41} +4.77900 q^{43} -2.32336 q^{45} +7.67625 q^{47} +10.6357 q^{49} -0.471763 q^{51} -4.45001 q^{53} +0.163228 q^{55} +0.510271 q^{57} -3.83062 q^{59} +9.08061 q^{61} -12.5783 q^{63} -4.54975 q^{65} -15.5095 q^{67} -0.252058 q^{69} -8.49435 q^{71} -10.1802 q^{73} -0.304393 q^{75} +0.883692 q^{77} +3.41891 q^{79} +8.95692 q^{81} -12.8168 q^{83} -5.28766 q^{85} -0.440871 q^{87} -9.76098 q^{89} -24.6316 q^{91} +0.571824 q^{93} +5.71928 q^{95} -14.2642 q^{97} -0.630278 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q - 3 q^{3} - 13 q^{7} + 35 q^{9}+O(q^{10}) $ 30 * q - 3 * q^3 - 13 * q^7 + 35 * q^9	$ 30 q - 3 q^{3} - 13 q^{7} + 35 q^{9} - 13 q^{11} - 7 q^{13} - 28 q^{15} - 9 q^{17} + 17 q^{19} - 6 q^{21} - 43 q^{23} + 34 q^{25} - 12 q^{27} - q^{29} - 39 q^{31} - 17 q^{35} - q^{37} - 48 q^{39} - 3 q^{41} + 19 q^{43} - 66 q^{47} + 25 q^{49} + 14 q^{51} + 3 q^{53} - 50 q^{55} - 14 q^{57} - 27 q^{59} + 15 q^{61} - 75 q^{63} - 6 q^{65} - 8 q^{67} + 18 q^{69} - 64 q^{71} - 15 q^{73} - 9 q^{75} - 71 q^{79} + 6 q^{81} - 60 q^{83} + 15 q^{85} - 64 q^{87} - 32 q^{89} + 26 q^{91} - 4 q^{93} - 72 q^{95} - 4 q^{97} - 13 q^{99}+O(q^{100}) $ 30 * q - 3 * q^3 - 13 * q^7 + 35 * q^9 - 13 * q^11 - 7 * q^13 - 28 * q^15 - 9 * q^17 + 17 * q^19 - 6 * q^21 - 43 * q^23 + 34 * q^25 - 12 * q^27 - q^29 - 39 * q^31 - 17 * q^35 - q^37 - 48 * q^39 - 3 * q^41 + 19 * q^43 - 66 * q^47 + 25 * q^49 + 14 * q^51 + 3 * q^53 - 50 * q^55 - 14 * q^57 - 27 * q^59 + 15 * q^61 - 75 * q^63 - 6 * q^65 - 8 * q^67 + 18 * q^69 - 64 * q^71 - 15 * q^73 - 9 * q^75 - 71 * q^79 + 6 * q^81 - 60 * q^83 + 15 * q^85 - 64 * q^87 - 32 * q^89 + 26 * q^91 - 4 * q^93 - 72 * q^95 - 4 * q^97 - 13 * 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
$3$	0.0692070	0.0399567	0.0199783	−	0.999800i	$-0.493640\pi$
$3$	0.0692070	0.0399567	0.0199783	+	0.999800i	$0.493640\pi$
$4$	0	0
$5$	0.775693	0.346900	0.173450	−	0.984843i	$-0.444508\pi$
$5$	0.775693	0.346900	0.173450	+	0.984843i	$0.444508\pi$
$6$	0	0
$7$	4.19948	1.58726	0.793628	−	0.608404i	$-0.208190\pi$
$7$	4.19948	1.58726	0.793628	+	0.608404i	$0.208190\pi$
$8$	0	0
$9$	−2.99521	−0.998403
$10$	0	0
$11$	0.210429	0.0634466	0.0317233	−	0.999497i	$-0.489900\pi$
$11$	0.210429	0.0634466	0.0317233	+	0.999497i	$0.489900\pi$
$12$	0	0
$13$	−5.86540	−1.62677	−0.813384	−	0.581727i	$-0.802377\pi$
$13$	−5.86540	−1.62677	−0.813384	+	0.581727i	$0.802377\pi$
$14$	0	0
$15$	0.0536833	0.0138610
$16$	0	0
$17$	−6.81669	−1.65329	−0.826646	−	0.562723i	$-0.809754\pi$
$17$	−6.81669	−1.65329	−0.826646	+	0.562723i	$0.809754\pi$
$18$	0	0
$19$	7.37312	1.69151	0.845755	−	0.533571i	$-0.179150\pi$
$19$	7.37312	1.69151	0.845755	+	0.533571i	$0.179150\pi$
$20$	0	0
$21$	0.290633	0.0634214
$22$	0	0
$23$	−3.64209	−0.759428	−0.379714	−	0.925104i	$-0.623978\pi$
$23$	−3.64209	−0.759428	−0.379714	+	0.925104i	$0.623978\pi$
$24$	0	0
$25$	−4.39830	−0.879660
$26$	0	0
$27$	−0.414910	−0.0798495
$28$	0	0
$29$	−6.37032	−1.18294	−0.591470	−	0.806327i	$-0.701452\pi$
$29$	−6.37032	−1.18294	−0.591470	+	0.806327i	$0.701452\pi$
$30$	0	0
$31$	8.26252	1.48399	0.741996	−	0.670404i	$-0.233879\pi$
$31$	8.26252	1.48399	0.741996	+	0.670404i	$0.233879\pi$
$32$	0	0
$33$	0.0145631	0.00253512
$34$	0	0
$35$	3.25751	0.550620
$36$	0	0
$37$	5.38161	0.884731	0.442366	−	0.896835i	$-0.354139\pi$
$37$	5.38161	0.884731	0.442366	+	0.896835i	$0.354139\pi$
$38$	0	0
$39$	−0.405926	−0.0650002
$40$	0	0
$41$	7.37849	1.15233	0.576163	−	0.817335i	$-0.304549\pi$
$41$	7.37849	1.15233	0.576163	+	0.817335i	$0.304549\pi$
$42$	0	0
$43$	4.77900	0.728791	0.364395	−	0.931244i	$-0.381276\pi$
$43$	4.77900	0.728791	0.364395	+	0.931244i	$0.381276\pi$
$44$	0	0
$45$	−2.32336	−0.346347
$46$	0	0
$47$	7.67625	1.11970	0.559848	−	0.828595i	$-0.310859\pi$
$47$	7.67625	1.11970	0.559848	+	0.828595i	$0.310859\pi$
$48$	0	0
$49$	10.6357	1.51938
$50$	0	0
$51$	−0.471763	−0.0660600
$52$	0	0
$53$	−4.45001	−0.611256	−0.305628	−	0.952151i	$-0.598866\pi$
$53$	−4.45001	−0.611256	−0.305628	+	0.952151i	$0.598866\pi$
$54$	0	0
$55$	0.163228	0.0220097
$56$	0	0
$57$	0.510271	0.0675871
$58$	0	0
$59$	−3.83062	−0.498705	−0.249352	−	0.968413i	$-0.580218\pi$
$59$	−3.83062	−0.498705	−0.249352	+	0.968413i	$0.580218\pi$
$60$	0	0
$61$	9.08061	1.16265	0.581327	−	0.813670i	$-0.302534\pi$
$61$	9.08061	1.16265	0.581327	+	0.813670i	$0.302534\pi$
$62$	0	0
$63$	−12.5783	−1.58472
$64$	0	0
$65$	−4.54975	−0.564327
$66$	0	0
$67$	−15.5095	−1.89478	−0.947391	−	0.320079i	$-0.896291\pi$
$67$	−15.5095	−1.89478	−0.947391	+	0.320079i	$0.896291\pi$
$68$	0	0
$69$	−0.252058	−0.0303442
$70$	0	0
$71$	−8.49435	−1.00809	−0.504047	−	0.863676i	$-0.668156\pi$
$71$	−8.49435	−1.00809	−0.504047	+	0.863676i	$0.668156\pi$
$72$	0	0
$73$	−10.1802	−1.19151	−0.595753	−	0.803168i	$-0.703146\pi$
$73$	−10.1802	−1.19151	−0.595753	+	0.803168i	$0.703146\pi$
$74$	0	0
$75$	−0.304393	−0.0351483
$76$	0	0
$77$	0.883692	0.100706
$78$	0	0
$79$	3.41891	0.384657	0.192329	−	0.981331i	$-0.438396\pi$
$79$	3.41891	0.384657	0.192329	+	0.981331i	$0.438396\pi$
$80$	0	0
$81$	8.95692	0.995213
$82$	0	0
$83$	−12.8168	−1.40683	−0.703415	−	0.710780i	$-0.748342\pi$
$83$	−12.8168	−1.40683	−0.703415	+	0.710780i	$0.748342\pi$
$84$	0	0
$85$	−5.28766	−0.573527
$86$	0	0
$87$	−0.440871	−0.0472663
$88$	0	0
$89$	−9.76098	−1.03466	−0.517331	−	0.855785i	$-0.673074\pi$
$89$	−9.76098	−1.03466	−0.517331	+	0.855785i	$0.673074\pi$
$90$	0	0
$91$	−24.6316	−2.58210
$92$	0	0
$93$	0.571824	0.0592954
$94$	0	0
$95$	5.71928	0.586786
$96$	0	0
$97$	−14.2642	−1.44831	−0.724153	−	0.689640i	$-0.757769\pi$
$97$	−14.2642	−1.44831	−0.724153	+	0.689640i	$0.757769\pi$
$98$	0	0
$99$	−0.630278	−0.0633454
$100$	0	0
$101$	−9.22211	−0.917634	−0.458817	−	0.888531i	$-0.651727\pi$
$101$	−9.22211	−0.917634	−0.458817	+	0.888531i	$0.651727\pi$
$102$	0	0
$103$	−13.0736	−1.28818	−0.644090	−	0.764949i	$-0.722764\pi$
$103$	−13.0736	−1.28818	−0.644090	+	0.764949i	$0.722764\pi$
$104$	0	0
$105$	0.225442	0.0220009
$106$	0	0
$107$	−5.28095	−0.510529	−0.255264	−	0.966871i	$-0.582162\pi$
$107$	−5.28095	−0.510529	−0.255264	+	0.966871i	$0.582162\pi$
$108$	0	0
$109$	6.92351	0.663152	0.331576	−	0.943428i	$-0.392420\pi$
$109$	6.92351	0.663152	0.331576	+	0.943428i	$0.392420\pi$
$110$	0	0
$111$	0.372445	0.0353509
$112$	0	0
$113$	−11.0723	−1.04159	−0.520795	−	0.853682i	$-0.674365\pi$
$113$	−11.0723	−1.04159	−0.520795	+	0.853682i	$0.674365\pi$
$114$	0	0
$115$	−2.82514	−0.263446
$116$	0	0
$117$	17.5681	1.62417
$118$	0	0
$119$	−28.6266	−2.62420
$120$	0	0
$121$	−10.9557	−0.995975
$122$	0	0
$123$	0.510643	0.0460431
$124$	0	0
$125$	−7.29020	−0.652055
$126$	0	0
$127$	−6.81998	−0.605175	−0.302588	−	0.953122i	$-0.597851\pi$
$127$	−6.81998	−0.605175	−0.302588	+	0.953122i	$0.597851\pi$
$128$	0	0
$129$	0.330740	0.0291200
$130$	0	0
$131$	15.9420	1.39286	0.696430	−	0.717624i	$-0.254771\pi$
$131$	15.9420	1.39286	0.696430	+	0.717624i	$0.254771\pi$
$132$	0	0
$133$	30.9633	2.68486
$134$	0	0
$135$	−0.321843	−0.0276998
$136$	0	0
$137$	4.19887	0.358734	0.179367	−	0.983782i	$-0.442595\pi$
$137$	4.19887	0.358734	0.179367	+	0.983782i	$0.442595\pi$
$138$	0	0
$139$	8.46896	0.718328	0.359164	−	0.933274i	$-0.383062\pi$
$139$	8.46896	0.718328	0.359164	+	0.933274i	$0.383062\pi$
$140$	0	0
$141$	0.531250	0.0447393
$142$	0	0
$143$	−1.23425	−0.103213
$144$	0	0
$145$	−4.94142	−0.410362
$146$	0	0
$147$	0.736062	0.0607094
$148$	0	0
$149$	−15.4331	−1.26433	−0.632163	−	0.774835i	$-0.717833\pi$
$149$	−15.4331	−1.26433	−0.632163	+	0.774835i	$0.717833\pi$
$150$	0	0
$151$	2.07053	0.168497	0.0842485	−	0.996445i	$-0.473151\pi$
$151$	2.07053	0.168497	0.0842485	+	0.996445i	$0.473151\pi$
$152$	0	0
$153$	20.4174	1.65065
$154$	0	0
$155$	6.40918	0.514798
$156$	0	0
$157$	3.26357	0.260462	0.130231	−	0.991484i	$-0.458428\pi$
$157$	3.26357	0.260462	0.130231	+	0.991484i	$0.458428\pi$
$158$	0	0
$159$	−0.307972	−0.0244238
$160$	0	0
$161$	−15.2949	−1.20541
$162$	0	0
$163$	0.306747	0.0240263	0.0120132	−	0.999928i	$-0.496176\pi$
$163$	0.306747	0.0240263	0.0120132	+	0.999928i	$0.496176\pi$
$164$	0	0
$165$	0.0112965	0.000879433 0
$166$	0	0
$167$	−3.14321	−0.243229	−0.121614	−	0.992577i	$-0.538807\pi$
$167$	−3.14321	−0.243229	−0.121614	+	0.992577i	$0.538807\pi$
$168$	0	0
$169$	21.4029	1.64637
$170$	0	0
$171$	−22.0841	−1.68881
$172$	0	0
$173$	−11.6604	−0.886521	−0.443260	−	0.896393i	$-0.646178\pi$
$173$	−11.6604	−0.886521	−0.443260	+	0.896393i	$0.646178\pi$
$174$	0	0
$175$	−18.4706	−1.39625
$176$	0	0
$177$	−0.265106	−0.0199266
$178$	0	0
$179$	11.2220	0.838775	0.419388	−	0.907807i	$-0.362245\pi$
$179$	11.2220	0.838775	0.419388	+	0.907807i	$0.362245\pi$
$180$	0	0
$181$	15.3483	1.14083	0.570413	−	0.821358i	$-0.306783\pi$
$181$	15.3483	1.14083	0.570413	+	0.821358i	$0.306783\pi$
$182$	0	0
$183$	0.628442	0.0464557
$184$	0	0
$185$	4.17448	0.306914
$186$	0	0
$187$	−1.43443	−0.104896
$188$	0	0
$189$	−1.74241	−0.126742
$190$	0	0
$191$	−23.3587	−1.69018	−0.845089	−	0.534625i	$-0.820453\pi$
$191$	−23.3587	−1.69018	−0.845089	+	0.534625i	$0.820453\pi$
$192$	0	0
$193$	−6.05700	−0.435992	−0.217996	−	0.975950i	$-0.569952\pi$
$193$	−6.05700	−0.435992	−0.217996	+	0.975950i	$0.569952\pi$
$194$	0	0
$195$	−0.314874	−0.0225486
$196$	0	0
$197$	1.86724	0.133036	0.0665178	−	0.997785i	$-0.478811\pi$
$197$	1.86724	0.133036	0.0665178	+	0.997785i	$0.478811\pi$
$198$	0	0
$199$	−0.0349107	−0.00247476	−0.00123738	−	0.999999i	$-0.500394\pi$
$199$	−0.0349107	−0.00247476	−0.00123738	+	0.999999i	$0.500394\pi$
$200$	0	0
$201$	−1.07336	−0.0757091
$202$	0	0
$203$	−26.7521	−1.87763
$204$	0	0
$205$	5.72344	0.399742
$206$	0	0
$207$	10.9088	0.758215
$208$	0	0
$209$	1.55152	0.107321
$210$	0	0
$211$	28.3298	1.95030	0.975150	−	0.221545i	$-0.0711098\pi$
$211$	28.3298	1.95030	0.975150	+	0.221545i	$0.0711098\pi$
$212$	0	0
$213$	−0.587868	−0.0402800
$214$	0	0
$215$	3.70704	0.252818
$216$	0	0
$217$	34.6983	2.35548
$218$	0	0
$219$	−0.704543	−0.0476086
$220$	0	0
$221$	39.9826	2.68952
$222$	0	0
$223$	9.13797	0.611924	0.305962	−	0.952044i	$-0.401022\pi$
$223$	9.13797	0.611924	0.305962	+	0.952044i	$0.401022\pi$
$224$	0	0
$225$	13.1738	0.878256
$226$	0	0
$227$	−6.60876	−0.438639	−0.219319	−	0.975653i	$-0.570384\pi$
$227$	−6.60876	−0.438639	−0.219319	+	0.975653i	$0.570384\pi$
$228$	0	0
$229$	−11.0974	−0.733339	−0.366670	−	0.930351i	$-0.619502\pi$
$229$	−11.0974	−0.733339	−0.366670	+	0.930351i	$0.619502\pi$
$230$	0	0
$231$	0.0611576	0.00402388
$232$	0	0
$233$	−24.2582	−1.58921	−0.794603	−	0.607130i	$-0.792321\pi$
$233$	−24.2582	−1.58921	−0.794603	+	0.607130i	$0.792321\pi$
$234$	0	0
$235$	5.95442	0.388423
$236$	0	0
$237$	0.236612	0.0153696
$238$	0	0
$239$	23.9499	1.54919	0.774594	−	0.632458i	$-0.217954\pi$
$239$	23.9499	1.54919	0.774594	+	0.632458i	$0.217954\pi$
$240$	0	0
$241$	−20.6884	−1.33266	−0.666330	−	0.745657i	$-0.732136\pi$
$241$	−20.6884	−1.33266	−0.666330	+	0.745657i	$0.732136\pi$
$242$	0	0
$243$	1.86461	0.119615
$244$	0	0
$245$	8.25001	0.527074
$246$	0	0
$247$	−43.2463	−2.75169
$248$	0	0
$249$	−0.887013	−0.0562122
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	−0.766400	−0.0481831
$254$	0	0
$255$	−0.365943	−0.0229162
$256$	0	0
$257$	−30.6036	−1.90900	−0.954499	−	0.298215i	$-0.903609\pi$
$257$	−30.6036	−1.90900	−0.954499	+	0.298215i	$0.903609\pi$
$258$	0	0
$259$	22.6000	1.40429
$260$	0	0
$261$	19.0805	1.18105
$262$	0	0
$263$	−12.5884	−0.776235	−0.388117	−	0.921610i	$-0.626874\pi$
$263$	−12.5884	−0.776235	−0.388117	+	0.921610i	$0.626874\pi$
$264$	0	0
$265$	−3.45184	−0.212045
$266$	0	0
$267$	−0.675528	−0.0413416
$268$	0	0
$269$	−12.9303	−0.788374	−0.394187	−	0.919030i	$-0.628974\pi$
$269$	−12.9303	−0.788374	−0.394187	+	0.919030i	$0.628974\pi$
$270$	0	0
$271$	−25.4720	−1.54731	−0.773656	−	0.633606i	$-0.781574\pi$
$271$	−25.4720	−1.54731	−0.773656	+	0.633606i	$0.781574\pi$
$272$	0	0
$273$	−1.70468	−0.103172
$274$	0	0
$275$	−0.925529	−0.0558115
$276$	0	0
$277$	−26.6018	−1.59835	−0.799173	−	0.601101i	$-0.794729\pi$
$277$	−26.6018	−1.59835	−0.799173	+	0.601101i	$0.794729\pi$
$278$	0	0
$279$	−24.7480	−1.48162
$280$	0	0
$281$	−19.4661	−1.16125	−0.580626	−	0.814171i	$-0.697192\pi$
$281$	−19.4661	−1.16125	−0.580626	+	0.814171i	$0.697192\pi$
$282$	0	0
$283$	−29.7787	−1.77016	−0.885079	−	0.465440i	$-0.845896\pi$
$283$	−29.7787	−1.77016	−0.885079	+	0.465440i	$0.845896\pi$
$284$	0	0
$285$	0.395814	0.0234460
$286$	0	0
$287$	30.9858	1.82904
$288$	0	0
$289$	29.4673	1.73337
$290$	0	0
$291$	−0.987179	−0.0578694
$292$	0	0
$293$	−24.9066	−1.45506	−0.727529	−	0.686077i	$-0.759331\pi$
$293$	−24.9066	−1.45506	−0.727529	+	0.686077i	$0.759331\pi$
$294$	0	0
$295$	−2.97139	−0.173001
$296$	0	0
$297$	−0.0873090	−0.00506618
$298$	0	0
$299$	21.3623	1.23541
$300$	0	0
$301$	20.0693	1.15678
$302$	0	0
$303$	−0.638234	−0.0366656
$304$	0	0
$305$	7.04377	0.403325
$306$	0	0
$307$	17.9032	1.02179	0.510894	−	0.859644i	$-0.329314\pi$
$307$	17.9032	1.02179	0.510894	+	0.859644i	$0.329314\pi$
$308$	0	0
$309$	−0.904784	−0.0514714
$310$	0	0
$311$	5.68659	0.322457	0.161228	−	0.986917i	$-0.448454\pi$
$311$	5.68659	0.322457	0.161228	+	0.986917i	$0.448454\pi$
$312$	0	0
$313$	−9.58583	−0.541823	−0.270912	−	0.962604i	$-0.587325\pi$
$313$	−9.58583	−0.541823	−0.270912	+	0.962604i	$0.587325\pi$
$314$	0	0
$315$	−9.75693	−0.549741
$316$	0	0
$317$	13.3110	0.747619	0.373809	−	0.927506i	$-0.378051\pi$
$317$	13.3110	0.747619	0.373809	+	0.927506i	$0.378051\pi$
$318$	0	0
$319$	−1.34050	−0.0750535
$320$	0	0
$321$	−0.365478	−0.0203990
$322$	0	0
$323$	−50.2603	−2.79656
$324$	0	0
$325$	25.7978	1.43100
$326$	0	0
$327$	0.479155	0.0264974
$328$	0	0
$329$	32.2363	1.77724
$330$	0	0
$331$	20.2996	1.11577	0.557884	−	0.829919i	$-0.311613\pi$
$331$	20.2996	1.11577	0.557884	+	0.829919i	$0.311613\pi$
$332$	0	0
$333$	−16.1191	−0.883319
$334$	0	0
$335$	−12.0306	−0.657301
$336$	0	0
$337$	14.4150	0.785236	0.392618	−	0.919702i	$-0.371569\pi$
$337$	14.4150	0.785236	0.392618	+	0.919702i	$0.371569\pi$
$338$	0	0
$339$	−0.766278	−0.0416185
$340$	0	0
$341$	1.73867	0.0941543
$342$	0	0
$343$	15.2679	0.824390
$344$	0	0
$345$	−0.195519	−0.0105264
$346$	0	0
$347$	17.0670	0.916207	0.458103	−	0.888899i	$-0.348529\pi$
$347$	17.0670	0.916207	0.458103	+	0.888899i	$0.348529\pi$
$348$	0	0
$349$	−11.2910	−0.604392	−0.302196	−	0.953246i	$-0.597720\pi$
$349$	−11.2910	−0.604392	−0.302196	+	0.953246i	$0.597720\pi$
$350$	0	0
$351$	2.43361	0.129897
$352$	0	0
$353$	5.54786	0.295283	0.147641	−	0.989041i	$-0.452832\pi$
$353$	5.54786	0.295283	0.147641	+	0.989041i	$0.452832\pi$
$354$	0	0
$355$	−6.58900	−0.349708
$356$	0	0
$357$	−1.98116	−0.104854
$358$	0	0
$359$	8.63433	0.455703	0.227851	−	0.973696i	$-0.426830\pi$
$359$	8.63433	0.455703	0.227851	+	0.973696i	$0.426830\pi$
$360$	0	0
$361$	35.3629	1.86121
$362$	0	0
$363$	−0.758212	−0.0397958
$364$	0	0
$365$	−7.89673	−0.413334
$366$	0	0
$367$	−21.4158	−1.11790	−0.558948	−	0.829203i	$-0.688795\pi$
$367$	−21.4158	−1.11790	−0.558948	+	0.829203i	$0.688795\pi$
$368$	0	0
$369$	−22.1001	−1.15049
$370$	0	0
$371$	−18.6878	−0.970220
$372$	0	0
$373$	−15.2168	−0.787894	−0.393947	−	0.919133i	$-0.628891\pi$
$373$	−15.2168	−0.787894	−0.393947	+	0.919133i	$0.628891\pi$
$374$	0	0
$375$	−0.504532	−0.0260539
$376$	0	0
$377$	37.3645	1.92437
$378$	0	0
$379$	34.1173	1.75249	0.876244	−	0.481868i	$-0.160041\pi$
$379$	34.1173	1.75249	0.876244	+	0.481868i	$0.160041\pi$
$380$	0	0
$381$	−0.471990	−0.0241808
$382$	0	0
$383$	20.9211	1.06902	0.534510	−	0.845162i	$-0.320496\pi$
$383$	20.9211	1.06902	0.534510	+	0.845162i	$0.320496\pi$
$384$	0	0
$385$	0.685474	0.0349350
$386$	0	0
$387$	−14.3141	−0.727627
$388$	0	0
$389$	−6.01551	−0.304998	−0.152499	−	0.988304i	$-0.548732\pi$
$389$	−6.01551	−0.304998	−0.152499	+	0.988304i	$0.548732\pi$
$390$	0	0
$391$	24.8270	1.25556
$392$	0	0
$393$	1.10330	0.0556540
$394$	0	0
$395$	2.65202	0.133438
$396$	0	0
$397$	−18.7540	−0.941237	−0.470619	−	0.882337i	$-0.655969\pi$
$397$	−18.7540	−0.941237	−0.470619	+	0.882337i	$0.655969\pi$
$398$	0	0
$399$	2.14288	0.107278
$400$	0	0
$401$	14.8115	0.739649	0.369825	−	0.929102i	$-0.379418\pi$
$401$	14.8115	0.739649	0.369825	+	0.929102i	$0.379418\pi$
$402$	0	0
$403$	−48.4629	−2.41411
$404$	0	0
$405$	6.94782	0.345240
$406$	0	0
$407$	1.13245	0.0561332
$408$	0	0
$409$	−24.4023	−1.20662	−0.603308	−	0.797509i	$-0.706151\pi$
$409$	−24.4023	−1.20662	−0.603308	+	0.797509i	$0.706151\pi$
$410$	0	0
$411$	0.290591	0.0143338
$412$	0	0
$413$	−16.0866	−0.791572
$414$	0	0
$415$	−9.94192	−0.488030
$416$	0	0
$417$	0.586111	0.0287020
$418$	0	0
$419$	34.5520	1.68798	0.843988	−	0.536362i	$-0.180202\pi$
$419$	34.5520	1.68798	0.843988	+	0.536362i	$0.180202\pi$
$420$	0	0
$421$	5.18666	0.252782	0.126391	−	0.991980i	$-0.459661\pi$
$421$	5.18666	0.252782	0.126391	+	0.991980i	$0.459661\pi$
$422$	0	0
$423$	−22.9920	−1.11791
$424$	0	0
$425$	29.9819	1.45433
$426$	0	0
$427$	38.1339	1.84543
$428$	0	0
$429$	−0.0854185	−0.00412404
$430$	0	0
$431$	23.0787	1.11166	0.555830	−	0.831296i	$-0.312400\pi$
$431$	23.0787	1.11166	0.555830	+	0.831296i	$0.312400\pi$
$432$	0	0
$433$	33.7554	1.62218	0.811091	−	0.584919i	$-0.198874\pi$
$433$	33.7554	1.62218	0.811091	+	0.584919i	$0.198874\pi$
$434$	0	0
$435$	−0.341980	−0.0163967
$436$	0	0
$437$	−26.8536	−1.28458
$438$	0	0
$439$	−23.4275	−1.11814	−0.559068	−	0.829122i	$-0.688841\pi$
$439$	−23.4275	−1.11814	−0.559068	+	0.829122i	$0.688841\pi$
$440$	0	0
$441$	−31.8560	−1.51695
$442$	0	0
$443$	14.3445	0.681526	0.340763	−	0.940149i	$-0.389315\pi$
$443$	14.3445	0.681526	0.340763	+	0.940149i	$0.389315\pi$
$444$	0	0
$445$	−7.57152	−0.358925
$446$	0	0
$447$	−1.06808	−0.0505182
$448$	0	0
$449$	24.0335	1.13421	0.567104	−	0.823646i	$-0.308064\pi$
$449$	24.0335	1.13421	0.567104	+	0.823646i	$0.308064\pi$
$450$	0	0
$451$	1.55265	0.0731112
$452$	0	0
$453$	0.143295	0.00673257
$454$	0	0
$455$	−19.1066	−0.895730
$456$	0	0
$457$	25.8243	1.20801	0.604005	−	0.796981i	$-0.293571\pi$
$457$	25.8243	1.20801	0.604005	+	0.796981i	$0.293571\pi$
$458$	0	0
$459$	2.82832	0.132015
$460$	0	0
$461$	7.12699	0.331937	0.165968	−	0.986131i	$-0.446925\pi$
$461$	7.12699	0.331937	0.165968	+	0.986131i	$0.446925\pi$
$462$	0	0
$463$	−11.7942	−0.548124	−0.274062	−	0.961712i	$-0.588367\pi$
$463$	−11.7942	−0.548124	−0.274062	+	0.961712i	$0.588367\pi$
$464$	0	0
$465$	0.443560	0.0205696
$466$	0	0
$467$	−17.6347	−0.816038	−0.408019	−	0.912974i	$-0.633780\pi$
$467$	−17.6347	−0.816038	−0.408019	+	0.912974i	$0.633780\pi$
$468$	0	0
$469$	−65.1317	−3.00750
$470$	0	0
$471$	0.225862	0.0104072
$472$	0	0
$473$	1.00564	0.0462393
$474$	0	0
$475$	−32.4292	−1.48795
$476$	0	0
$477$	13.3287	0.610280
$478$	0	0
$479$	12.1081	0.553233	0.276616	−	0.960980i	$-0.410787\pi$
$479$	12.1081	0.553233	0.276616	+	0.960980i	$0.410787\pi$
$480$	0	0
$481$	−31.5653	−1.43925
$482$	0	0
$483$	−1.05851	−0.0481640
$484$	0	0
$485$	−11.0646	−0.502418
$486$	0	0
$487$	−32.5399	−1.47452	−0.737262	−	0.675607i	$-0.763882\pi$
$487$	−32.5399	−1.47452	−0.737262	+	0.675607i	$0.763882\pi$
$488$	0	0
$489$	0.0212291	0.000960011 0
$490$	0	0
$491$	26.1411	1.17973	0.589865	−	0.807502i	$-0.299181\pi$
$491$	26.1411	1.17973	0.589865	+	0.807502i	$0.299181\pi$
$492$	0	0
$493$	43.4246	1.95574
$494$	0	0
$495$	−0.488902	−0.0219745
$496$	0	0
$497$	−35.6719	−1.60010
$498$	0	0
$499$	10.8538	0.485881	0.242941	−	0.970041i	$-0.421888\pi$
$499$	10.8538	0.485881	0.242941	+	0.970041i	$0.421888\pi$
$500$	0	0
$501$	−0.217532	−0.00971861
$502$	0	0
$503$	1.38641	0.0618171	0.0309086	−	0.999522i	$-0.490160\pi$
$503$	1.38641	0.0618171	0.0309086	+	0.999522i	$0.490160\pi$
$504$	0	0
$505$	−7.15352	−0.318328
$506$	0	0
$507$	1.48123	0.0657836
$508$	0	0
$509$	16.1097	0.714049	0.357024	−	0.934095i	$-0.383791\pi$
$509$	16.1097	0.714049	0.357024	+	0.934095i	$0.383791\pi$
$510$	0	0
$511$	−42.7517	−1.89122
$512$	0	0
$513$	−3.05918	−0.135066
$514$	0	0
$515$	−10.1411	−0.446870
$516$	0	0
$517$	1.61530	0.0710410
$518$	0	0
$519$	−0.806978	−0.0354224
$520$	0	0
$521$	5.88288	0.257734	0.128867	−	0.991662i	$-0.458866\pi$
$521$	5.88288	0.257734	0.128867	+	0.991662i	$0.458866\pi$
$522$	0	0
$523$	17.6311	0.770953	0.385477	−	0.922718i	$-0.374037\pi$
$523$	17.6311	0.770953	0.385477	+	0.922718i	$0.374037\pi$
$524$	0	0
$525$	−1.27829	−0.0557893
$526$	0	0
$527$	−56.3231	−2.45347
$528$	0	0
$529$	−9.73520	−0.423269
$530$	0	0
$531$	11.4735	0.497908
$532$	0	0
$533$	−43.2777	−1.87457
$534$	0	0
$535$	−4.09640	−0.177103
$536$	0	0
$537$	0.776644	0.0335146
$538$	0	0
$539$	2.23805	0.0963996
$540$	0	0
$541$	5.94447	0.255573	0.127786	−	0.991802i	$-0.459213\pi$
$541$	5.94447	0.255573	0.127786	+	0.991802i	$0.459213\pi$
$542$	0	0
$543$	1.06221	0.0455836
$544$	0	0
$545$	5.37052	0.230048
$546$	0	0
$547$	26.6895	1.14116	0.570581	−	0.821241i	$-0.306718\pi$
$547$	26.6895	1.14116	0.570581	+	0.821241i	$0.306718\pi$
$548$	0	0
$549$	−27.1984	−1.16080
$550$	0	0
$551$	−46.9692	−2.00095
$552$	0	0
$553$	14.3577	0.610550
$554$	0	0
$555$	0.288903	0.0122632
$556$	0	0
$557$	45.9253	1.94592	0.972959	−	0.230976i	$-0.0741920\pi$
$557$	45.9253	1.94592	0.972959	+	0.230976i	$0.0741920\pi$
$558$	0	0
$559$	−28.0307	−1.18557
$560$	0	0
$561$	−0.0992724	−0.00419128
$562$	0	0
$563$	21.0439	0.886896	0.443448	−	0.896300i	$-0.353755\pi$
$563$	21.0439	0.886896	0.443448	+	0.896300i	$0.353755\pi$
$564$	0	0
$565$	−8.58868	−0.361328
$566$	0	0
$567$	37.6144	1.57966
$568$	0	0
$569$	15.7311	0.659481	0.329741	−	0.944072i	$-0.393039\pi$
$569$	15.7311	0.659481	0.329741	+	0.944072i	$0.393039\pi$
$570$	0	0
$571$	−35.3361	−1.47877	−0.739385	−	0.673283i	$-0.764883\pi$
$571$	−35.3361	−1.47877	−0.739385	+	0.673283i	$0.764883\pi$
$572$	0	0
$573$	−1.61659	−0.0675339
$574$	0	0
$575$	16.0190	0.668038
$576$	0	0
$577$	−46.9962	−1.95648	−0.978240	−	0.207478i	$-0.933474\pi$
$577$	−46.9962	−1.95648	−0.978240	+	0.207478i	$0.933474\pi$
$578$	0	0
$579$	−0.419187	−0.0174208
$580$	0	0
$581$	−53.8240	−2.23300
$582$	0	0
$583$	−0.936411	−0.0387822
$584$	0	0
$585$	13.6274	0.563426
$586$	0	0
$587$	25.6596	1.05909	0.529544	−	0.848283i	$-0.322363\pi$
$587$	25.6596	1.05909	0.529544	+	0.848283i	$0.322363\pi$
$588$	0	0
$589$	60.9206	2.51019
$590$	0	0
$591$	0.129226	0.00531566
$592$	0	0
$593$	−41.8400	−1.71816	−0.859082	−	0.511838i	$-0.828965\pi$
$593$	−41.8400	−1.71816	−0.859082	+	0.511838i	$0.828965\pi$
$594$	0	0
$595$	−22.2055	−0.910335
$596$	0	0
$597$	−0.00241606	−9.88830e−5 0
$598$	0	0
$599$	−25.1392	−1.02716	−0.513581	−	0.858041i	$-0.671681\pi$
$599$	−25.1392	−1.02716	−0.513581	+	0.858041i	$0.671681\pi$
$600$	0	0
$601$	1.43174	0.0584019	0.0292009	−	0.999574i	$-0.490704\pi$
$601$	1.43174	0.0584019	0.0292009	+	0.999574i	$0.490704\pi$
$602$	0	0
$603$	46.4541	1.89176
$604$	0	0
$605$	−8.49827	−0.345504
$606$	0	0
$607$	−19.1312	−0.776513	−0.388257	−	0.921551i	$-0.626923\pi$
$607$	−19.1312	−0.776513	−0.388257	+	0.921551i	$0.626923\pi$
$608$	0	0
$609$	−1.85143	−0.0750237
$610$	0	0
$611$	−45.0243	−1.82149
$612$	0	0
$613$	19.9549	0.805972	0.402986	−	0.915206i	$-0.367972\pi$
$613$	19.9549	0.805972	0.402986	+	0.915206i	$0.367972\pi$
$614$	0	0
$615$	0.396102	0.0159724
$616$	0	0
$617$	−12.6157	−0.507890	−0.253945	−	0.967219i	$-0.581728\pi$
$617$	−12.6157	−0.507890	−0.253945	+	0.967219i	$0.581728\pi$
$618$	0	0
$619$	−28.6717	−1.15241	−0.576207	−	0.817304i	$-0.695468\pi$
$619$	−28.6717	−1.15241	−0.576207	+	0.817304i	$0.695468\pi$
$620$	0	0
$621$	1.51114	0.0606399
$622$	0	0
$623$	−40.9911	−1.64227
$624$	0	0
$625$	16.3365	0.653462
$626$	0	0
$627$	0.107376	0.00428817
$628$	0	0
$629$	−36.6848	−1.46272
$630$	0	0
$631$	11.6711	0.464619	0.232309	−	0.972642i	$-0.425372\pi$
$631$	11.6711	0.464619	0.232309	+	0.972642i	$0.425372\pi$
$632$	0	0
$633$	1.96062	0.0779275
$634$	0	0
$635$	−5.29021	−0.209936
$636$	0	0
$637$	−62.3824	−2.47168
$638$	0	0
$639$	25.4424	1.00648
$640$	0	0
$641$	38.4576	1.51898	0.759491	−	0.650517i	$-0.225448\pi$
$641$	38.4576	1.51898	0.759491	+	0.650517i	$0.225448\pi$
$642$	0	0
$643$	28.3532	1.11814	0.559070	−	0.829121i	$-0.311158\pi$
$643$	28.3532	1.11814	0.559070	+	0.829121i	$0.311158\pi$
$644$	0	0
$645$	0.256553	0.0101018
$646$	0	0
$647$	−40.9339	−1.60928	−0.804638	−	0.593766i	$-0.797641\pi$
$647$	−40.9339	−1.60928	−0.804638	+	0.593766i	$0.797641\pi$
$648$	0	0
$649$	−0.806073	−0.0316411
$650$	0	0
$651$	2.40136	0.0941169
$652$	0	0
$653$	−30.2157	−1.18243	−0.591216	−	0.806513i	$-0.701352\pi$
$653$	−30.2157	−1.18243	−0.591216	+	0.806513i	$0.701352\pi$
$654$	0	0
$655$	12.3661	0.483184
$656$	0	0
$657$	30.4919	1.18960
$658$	0	0
$659$	−39.6483	−1.54448	−0.772238	−	0.635333i	$-0.780863\pi$
$659$	−39.6483	−1.54448	−0.772238	+	0.635333i	$0.780863\pi$
$660$	0	0
$661$	−11.2438	−0.437332	−0.218666	−	0.975800i	$-0.570170\pi$
$661$	−11.2438	−0.437332	−0.218666	+	0.975800i	$0.570170\pi$
$662$	0	0
$663$	2.76707	0.107464
$664$	0	0
$665$	24.0180	0.931379
$666$	0	0
$667$	23.2013	0.898357
$668$	0	0
$669$	0.632411	0.0244504
$670$	0	0
$671$	1.91082	0.0737665
$672$	0	0
$673$	19.1182	0.736953	0.368476	−	0.929637i	$-0.379880\pi$
$673$	19.1182	0.736953	0.368476	+	0.929637i	$0.379880\pi$
$674$	0	0
$675$	1.82490	0.0702404
$676$	0	0
$677$	27.5297	1.05805	0.529025	−	0.848606i	$-0.322558\pi$
$677$	27.5297	1.05805	0.529025	+	0.848606i	$0.322558\pi$
$678$	0	0
$679$	−59.9021	−2.29883
$680$	0	0
$681$	−0.457372	−0.0175265
$682$	0	0
$683$	−17.1368	−0.655721	−0.327860	−	0.944726i	$-0.606328\pi$
$683$	−17.1368	−0.655721	−0.327860	+	0.944726i	$0.606328\pi$
$684$	0	0
$685$	3.25703	0.124445
$686$	0	0
$687$	−0.768020	−0.0293018
$688$	0	0
$689$	26.1011	0.994372
$690$	0	0
$691$	3.04409	0.115803	0.0579013	−	0.998322i	$-0.481559\pi$
$691$	3.04409	0.115803	0.0579013	+	0.998322i	$0.481559\pi$
$692$	0	0
$693$	−2.64684	−0.100545
$694$	0	0
$695$	6.56931	0.249188
$696$	0	0
$697$	−50.2969	−1.90513
$698$	0	0
$699$	−1.67883	−0.0634993
$700$	0	0
$701$	6.31631	0.238564	0.119282	−	0.992860i	$-0.461941\pi$
$701$	6.31631	0.238564	0.119282	+	0.992860i	$0.461941\pi$
$702$	0	0
$703$	39.6793	1.49653
$704$	0	0
$705$	0.412087	0.0155201
$706$	0	0
$707$	−38.7281	−1.45652
$708$	0	0
$709$	−24.5850	−0.923307	−0.461654	−	0.887060i	$-0.652744\pi$
$709$	−24.5850	−0.923307	−0.461654	+	0.887060i	$0.652744\pi$
$710$	0	0
$711$	−10.2404	−0.384043
$712$	0	0
$713$	−30.0928	−1.12699
$714$	0	0
$715$	−0.957397	−0.0358046
$716$	0	0
$717$	1.65750	0.0619004
$718$	0	0
$719$	19.1473	0.714074	0.357037	−	0.934090i	$-0.383787\pi$
$719$	19.1473	0.714074	0.357037	+	0.934090i	$0.383787\pi$
$720$	0	0
$721$	−54.9024	−2.04467
$722$	0	0
$723$	−1.43178	−0.0532486
$724$	0	0
$725$	28.0186	1.04058
$726$	0	0
$727$	44.0078	1.63216	0.816079	−	0.577941i	$-0.196144\pi$
$727$	44.0078	1.63216	0.816079	+	0.577941i	$0.196144\pi$
$728$	0	0
$729$	−26.7417	−0.990434
$730$	0	0
$731$	−32.5770	−1.20490
$732$	0	0
$733$	−8.23897	−0.304313	−0.152157	−	0.988356i	$-0.548622\pi$
$733$	−8.23897	−0.304313	−0.152157	+	0.988356i	$0.548622\pi$
$734$	0	0
$735$	0.570958	0.0210601
$736$	0	0
$737$	−3.26364	−0.120218
$738$	0	0
$739$	35.0954	1.29101	0.645503	−	0.763758i	$-0.276648\pi$
$739$	35.0954	1.29101	0.645503	+	0.763758i	$0.276648\pi$
$740$	0	0
$741$	−2.99294	−0.109949
$742$	0	0
$743$	−28.8727	−1.05924	−0.529618	−	0.848236i	$-0.677665\pi$
$743$	−28.8727	−1.05924	−0.529618	+	0.848236i	$0.677665\pi$
$744$	0	0
$745$	−11.9713	−0.438595
$746$	0	0
$747$	38.3891	1.40458
$748$	0	0
$749$	−22.1773	−0.810340
$750$	0	0
$751$	2.39807	0.0875069	0.0437535	−	0.999042i	$-0.486068\pi$
$751$	2.39807	0.0875069	0.0437535	+	0.999042i	$0.486068\pi$
$752$	0	0
$753$	−0.0692070	−0.00252204
$754$	0	0
$755$	1.60609	0.0584517
$756$	0	0
$757$	36.0017	1.30851	0.654253	−	0.756276i	$-0.272983\pi$
$757$	36.0017	1.30851	0.654253	+	0.756276i	$0.272983\pi$
$758$	0	0
$759$	−0.0530402	−0.00192524
$760$	0	0
$761$	−41.2342	−1.49474	−0.747369	−	0.664409i	$-0.768683\pi$
$761$	−41.2342	−1.49474	−0.747369	+	0.664409i	$0.768683\pi$
$762$	0	0
$763$	29.0752	1.05259
$764$	0	0
$765$	15.8377	0.572612
$766$	0	0
$767$	22.4681	0.811277
$768$	0	0
$769$	−14.6825	−0.529465	−0.264733	−	0.964322i	$-0.585284\pi$
$769$	−14.6825	−0.529465	−0.264733	+	0.964322i	$0.585284\pi$
$770$	0	0
$771$	−2.11798	−0.0762771
$772$	0	0
$773$	14.1545	0.509103	0.254551	−	0.967059i	$-0.418072\pi$
$773$	14.1545	0.509103	0.254551	+	0.967059i	$0.418072\pi$
$774$	0	0
$775$	−36.3410	−1.30541
$776$	0	0
$777$	1.56408	0.0561109
$778$	0	0
$779$	54.4025	1.94917
$780$	0	0
$781$	−1.78745	−0.0639601
$782$	0	0
$783$	2.64311	0.0944571
$784$	0	0
$785$	2.53153	0.0903542
$786$	0	0
$787$	28.1095	1.00200	0.500998	−	0.865449i	$-0.332966\pi$
$787$	28.1095	1.00200	0.500998	+	0.865449i	$0.332966\pi$
$788$	0	0
$789$	−0.871206	−0.0310157
$790$	0	0
$791$	−46.4978	−1.65327
$792$	0	0
$793$	−53.2614	−1.89137
$794$	0	0
$795$	−0.238892	−0.00847261
$796$	0	0
$797$	−0.152801	−0.00541250	−0.00270625	−	0.999996i	$-0.500861\pi$
$797$	−0.152801	−0.00541250	−0.00270625	+	0.999996i	$0.500861\pi$
$798$	0	0
$799$	−52.3267	−1.85118
$800$	0	0
$801$	29.2362	1.03301
$802$	0	0
$803$	−2.14221	−0.0755971
$804$	0	0
$805$	−11.8641	−0.418156
$806$	0	0
$807$	−0.894867	−0.0315008
$808$	0	0
$809$	12.7952	0.449855	0.224928	−	0.974375i	$-0.427785\pi$
$809$	12.7952	0.449855	0.224928	+	0.974375i	$0.427785\pi$
$810$	0	0
$811$	−23.2650	−0.816943	−0.408472	−	0.912771i	$-0.633938\pi$
$811$	−23.2650	−0.816943	−0.408472	+	0.912771i	$0.633938\pi$
$812$	0	0
$813$	−1.76284	−0.0618254
$814$	0	0
$815$	0.237942	0.00833474
$816$	0	0
$817$	35.2362	1.23276
$818$	0	0
$819$	73.7769	2.57797
$820$	0	0
$821$	20.4830	0.714862	0.357431	−	0.933940i	$-0.383653\pi$
$821$	20.4830	0.714862	0.357431	+	0.933940i	$0.383653\pi$
$822$	0	0
$823$	−8.92637	−0.311154	−0.155577	−	0.987824i	$-0.549724\pi$
$823$	−8.92637	−0.311154	−0.155577	+	0.987824i	$0.549724\pi$
$824$	0	0
$825$	−0.0640530	−0.00223004
$826$	0	0
$827$	−20.5117	−0.713261	−0.356631	−	0.934245i	$-0.616075\pi$
$827$	−20.5117	−0.713261	−0.356631	+	0.934245i	$0.616075\pi$
$828$	0	0
$829$	−39.2764	−1.36413	−0.682063	−	0.731294i	$-0.738917\pi$
$829$	−39.2764	−1.36413	−0.682063	+	0.731294i	$0.738917\pi$
$830$	0	0
$831$	−1.84103	−0.0638646
$832$	0	0
$833$	−72.5001	−2.51198
$834$	0	0
$835$	−2.43817	−0.0843762
$836$	0	0
$837$	−3.42820	−0.118496
$838$	0	0
$839$	22.7251	0.784558	0.392279	−	0.919846i	$-0.371687\pi$
$839$	22.7251	0.784558	0.392279	+	0.919846i	$0.371687\pi$
$840$	0	0
$841$	11.5810	0.399346
$842$	0	0
$843$	−1.34719	−0.0463997
$844$	0	0
$845$	16.6020	0.571128
$846$	0	0
$847$	−46.0084	−1.58087
$848$	0	0
$849$	−2.06089	−0.0707296
$850$	0	0
$851$	−19.6003	−0.671889
$852$	0	0
$853$	3.45379	0.118256	0.0591278	−	0.998250i	$-0.481168\pi$
$853$	3.45379	0.118256	0.0591278	+	0.998250i	$0.481168\pi$
$854$	0	0
$855$	−17.1304	−0.585849
$856$	0	0
$857$	−0.792349	−0.0270661	−0.0135331	−	0.999908i	$-0.504308\pi$
$857$	−0.792349	−0.0270661	−0.0135331	+	0.999908i	$0.504308\pi$
$858$	0	0
$859$	20.5855	0.702367	0.351184	−	0.936307i	$-0.385779\pi$
$859$	20.5855	0.702367	0.351184	+	0.936307i	$0.385779\pi$
$860$	0	0
$861$	2.14443	0.0730821
$862$	0	0
$863$	46.8872	1.59606	0.798030	−	0.602618i	$-0.205876\pi$
$863$	46.8872	1.59606	0.798030	+	0.602618i	$0.205876\pi$
$864$	0	0
$865$	−9.04486	−0.307534
$866$	0	0
$867$	2.03934	0.0692597
$868$	0	0
$869$	0.719437	0.0244052
$870$	0	0
$871$	90.9691	3.08237
$872$	0	0
$873$	42.7241	1.44599
$874$	0	0
$875$	−30.6151	−1.03498
$876$	0	0
$877$	37.4198	1.26358	0.631788	−	0.775141i	$-0.282321\pi$
$877$	37.4198	1.26358	0.631788	+	0.775141i	$0.282321\pi$
$878$	0	0
$879$	−1.72371	−0.0581392
$880$	0	0
$881$	22.4578	0.756622	0.378311	−	0.925679i	$-0.376505\pi$
$881$	22.4578	0.756622	0.378311	+	0.925679i	$0.376505\pi$
$882$	0	0
$883$	22.6243	0.761370	0.380685	−	0.924705i	$-0.375688\pi$
$883$	22.6243	0.761370	0.380685	+	0.924705i	$0.375688\pi$
$884$	0	0
$885$	−0.205641	−0.00691254
$886$	0	0
$887$	36.3680	1.22112	0.610559	−	0.791971i	$-0.290945\pi$
$887$	36.3680	1.22112	0.610559	+	0.791971i	$0.290945\pi$
$888$	0	0
$889$	−28.6404	−0.960568
$890$	0	0
$891$	1.88479	0.0631429
$892$	0	0
$893$	56.5980	1.89398
$894$	0	0
$895$	8.70486	0.290971
$896$	0	0
$897$	1.47842	0.0493630
$898$	0	0
$899$	−52.6349	−1.75547
$900$	0	0
$901$	30.3344	1.01058
$902$	0	0
$903$	1.38894	0.0462209
$904$	0	0
$905$	11.9055	0.395753
$906$	0	0
$907$	33.8371	1.12354	0.561771	−	0.827293i	$-0.310120\pi$
$907$	33.8371	1.12354	0.561771	+	0.827293i	$0.310120\pi$
$908$	0	0
$909$	27.6222	0.916169
$910$	0	0
$911$	5.76626	0.191045	0.0955223	−	0.995427i	$-0.469548\pi$
$911$	5.76626	0.191045	0.0955223	+	0.995427i	$0.469548\pi$
$912$	0	0
$913$	−2.69703	−0.0892586
$914$	0	0
$915$	0.487478	0.0161155
$916$	0	0
$917$	66.9482	2.21083
$918$	0	0
$919$	−46.7214	−1.54120	−0.770599	−	0.637321i	$-0.780043\pi$
$919$	−46.7214	−1.54120	−0.770599	+	0.637321i	$0.780043\pi$
$920$	0	0
$921$	1.23902	0.0408273
$922$	0	0
$923$	49.8227	1.63993
$924$	0	0
$925$	−23.6699	−0.778263
$926$	0	0
$927$	39.1582	1.28612
$928$	0	0
$929$	23.3107	0.764800	0.382400	−	0.923997i	$-0.375098\pi$
$929$	23.3107	0.764800	0.382400	+	0.923997i	$0.375098\pi$
$930$	0	0
$931$	78.4181	2.57005
$932$	0	0
$933$	0.393551	0.0128843
$934$	0	0
$935$	−1.11268	−0.0363884
$936$	0	0
$937$	13.2306	0.432225	0.216113	−	0.976368i	$-0.430662\pi$
$937$	13.2306	0.432225	0.216113	+	0.976368i	$0.430662\pi$
$938$	0	0
$939$	−0.663406	−0.0216494
$940$	0	0
$941$	−2.79636	−0.0911588	−0.0455794	−	0.998961i	$-0.514513\pi$
$941$	−2.79636	−0.0911588	−0.0455794	+	0.998961i	$0.514513\pi$
$942$	0	0
$943$	−26.8731	−0.875108
$944$	0	0
$945$	−1.35157	−0.0439667
$946$	0	0
$947$	−54.4701	−1.77004	−0.885020	−	0.465553i	$-0.845856\pi$
$947$	−54.4701	−1.77004	−0.885020	+	0.465553i	$0.845856\pi$
$948$	0	0
$949$	59.7111	1.93830
$950$	0	0
$951$	0.921212	0.0298723
$952$	0	0
$953$	18.4756	0.598485	0.299243	−	0.954177i	$-0.403266\pi$
$953$	18.4756	0.598485	0.299243	+	0.954177i	$0.403266\pi$
$954$	0	0
$955$	−18.1192	−0.586324
$956$	0	0
$957$	−0.0927719	−0.00299889
$958$	0	0
$959$	17.6331	0.569402
$960$	0	0
$961$	37.2692	1.20223
$962$	0	0
$963$	15.8176	0.509714
$964$	0	0
$965$	−4.69837	−0.151246
$966$	0	0
$967$	−57.8946	−1.86176	−0.930882	−	0.365321i	$-0.880959\pi$
$967$	−57.8946	−1.86176	−0.930882	+	0.365321i	$0.880959\pi$
$968$	0	0
$969$	−3.47836	−0.111741
$970$	0	0
$971$	8.52061	0.273439	0.136720	−	0.990610i	$-0.456344\pi$
$971$	8.52061	0.273439	0.136720	+	0.990610i	$0.456344\pi$
$972$	0	0
$973$	35.5653	1.14017
$974$	0	0
$975$	1.78538	0.0571781
$976$	0	0
$977$	23.7051	0.758394	0.379197	−	0.925316i	$-0.376200\pi$
$977$	23.7051	0.758394	0.379197	+	0.925316i	$0.376200\pi$
$978$	0	0
$979$	−2.05399	−0.0656458
$980$	0	0
$981$	−20.7374	−0.662094
$982$	0	0
$983$	−45.9747	−1.46637	−0.733183	−	0.680031i	$-0.761966\pi$
$983$	−45.9747	−1.46637	−0.733183	+	0.680031i	$0.761966\pi$
$984$	0	0
$985$	1.44841	0.0461501
$986$	0	0
$987$	2.23098	0.0710128
$988$	0	0
$989$	−17.4055	−0.553464
$990$	0	0
$991$	−45.1192	−1.43326	−0.716629	−	0.697454i	$-0.754316\pi$
$991$	−45.1192	−1.43326	−0.716629	+	0.697454i	$0.754316\pi$
$992$	0	0
$993$	1.40487	0.0445823
$994$	0	0
$995$	−0.0270800	−0.000858494 0
$996$	0	0
$997$	43.6920	1.38374	0.691869	−	0.722023i	$-0.256787\pi$
$997$	43.6920	1.38374	0.691869	+	0.722023i	$0.256787\pi$
$998$	0	0
$999$	−2.23288	−0.0706454

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8032.2.a.h.1.16	✓	30
4.3	odd	2		8032.2.a.i.1.15	yes	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8032.2.a.h.1.16	✓	30	1.1	even	1	trivial
8032.2.a.i.1.15	yes	30	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:	\( N \)	\(=\)	\( 8032 = 2^{5} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8032.a (trivial)

\(f(q)\)	\(=\)	\(q+0.0692070 q^{3} +0.775693 q^{5} +4.19948 q^{7} -2.99521 q^{9} +O(q^{10})\)	\(q+0.0692070 q^{3} +0.775693 q^{5} +4.19948 q^{7} -2.99521 q^{9} +0.210429 q^{11} -5.86540 q^{13} +0.0536833 q^{15} -6.81669 q^{17} +7.37312 q^{19} +0.290633 q^{21} -3.64209 q^{23} -4.39830 q^{25} -0.414910 q^{27} -6.37032 q^{29} +8.26252 q^{31} +0.0145631 q^{33} +3.25751 q^{35} +5.38161 q^{37} -0.405926 q^{39} +7.37849 q^{41} +4.77900 q^{43} -2.32336 q^{45} +7.67625 q^{47} +10.6357 q^{49} -0.471763 q^{51} -4.45001 q^{53} +0.163228 q^{55} +0.510271 q^{57} -3.83062 q^{59} +9.08061 q^{61} -12.5783 q^{63} -4.54975 q^{65} -15.5095 q^{67} -0.252058 q^{69} -8.49435 q^{71} -10.1802 q^{73} -0.304393 q^{75} +0.883692 q^{77} +3.41891 q^{79} +8.95692 q^{81} -12.8168 q^{83} -5.28766 q^{85} -0.440871 q^{87} -9.76098 q^{89} -24.6316 q^{91} +0.571824 q^{93} +5.71928 q^{95} -14.2642 q^{97} -0.630278 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 3 q^{3} - 13 q^{7} + 35 q^{9}+O(q^{10}) \) 30 * q - 3 * q^3 - 13 * q^7 + 35 * q^9	\( 30 q - 3 q^{3} - 13 q^{7} + 35 q^{9} - 13 q^{11} - 7 q^{13} - 28 q^{15} - 9 q^{17} + 17 q^{19} - 6 q^{21} - 43 q^{23} + 34 q^{25} - 12 q^{27} - q^{29} - 39 q^{31} - 17 q^{35} - q^{37} - 48 q^{39} - 3 q^{41} + 19 q^{43} - 66 q^{47} + 25 q^{49} + 14 q^{51} + 3 q^{53} - 50 q^{55} - 14 q^{57} - 27 q^{59} + 15 q^{61} - 75 q^{63} - 6 q^{65} - 8 q^{67} + 18 q^{69} - 64 q^{71} - 15 q^{73} - 9 q^{75} - 71 q^{79} + 6 q^{81} - 60 q^{83} + 15 q^{85} - 64 q^{87} - 32 q^{89} + 26 q^{91} - 4 q^{93} - 72 q^{95} - 4 q^{97} - 13 q^{99}+O(q^{100}) \) 30 * q - 3 * q^3 - 13 * q^7 + 35 * q^9 - 13 * q^11 - 7 * q^13 - 28 * q^15 - 9 * q^17 + 17 * q^19 - 6 * q^21 - 43 * q^23 + 34 * q^25 - 12 * q^27 - q^29 - 39 * q^31 - 17 * q^35 - q^37 - 48 * q^39 - 3 * q^41 + 19 * q^43 - 66 * q^47 + 25 * q^49 + 14 * q^51 + 3 * q^53 - 50 * q^55 - 14 * q^57 - 27 * q^59 + 15 * q^61 - 75 * q^63 - 6 * q^65 - 8 * q^67 + 18 * q^69 - 64 * q^71 - 15 * q^73 - 9 * q^75 - 71 * q^79 + 6 * q^81 - 60 * q^83 + 15 * q^85 - 64 * q^87 - 32 * q^89 + 26 * q^91 - 4 * q^93 - 72 * q^95 - 4 * q^97 - 13 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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