LMFDB - Embedded newform 8000.2.a.bo.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8000,2,Mod(1,8000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8000, 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("8000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8000 = 2^{6} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8000.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:	$63.8803216170$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.10025.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 11x^{2} + 10x + 20 $ x^4 - x^3 - 11x^2 + 10x + 20
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1000)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.39867$ of defining polynomial
Character	$\chi$	$=$	8000.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.39867 q^{3} -1.13558 q^{7} +2.75361 q^{9} +O(q^{10})$	$q+2.39867 q^{3} -1.13558 q^{7} +2.75361 q^{9} -2.01670 q^{11} -2.26309 q^{13} -1.08379 q^{17} +5.58295 q^{19} -2.72387 q^{21} -8.33656 q^{23} -0.591009 q^{27} -9.11719 q^{29} +8.75193 q^{31} -4.83740 q^{33} +2.54457 q^{37} -5.42841 q^{39} +9.82934 q^{41} -2.91621 q^{43} +2.09017 q^{47} -5.71047 q^{49} -2.59965 q^{51} +10.5326 q^{53} +13.3916 q^{57} +5.53424 q^{59} +1.63474 q^{61} -3.12693 q^{63} -10.5844 q^{67} -19.9966 q^{69} -12.9496 q^{71} -13.2447 q^{73} +2.29012 q^{77} -6.84604 q^{79} -9.67846 q^{81} -2.81798 q^{83} -21.8691 q^{87} -15.1673 q^{89} +2.56991 q^{91} +20.9930 q^{93} -18.2591 q^{97} -5.55321 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{3} - 9 q^{7} + 11 q^{9}+O(q^{10}) $ 4 * q + q^3 - 9 * q^7 + 11 * q^9	$ 4 q + q^{3} - 9 q^{7} + 11 q^{9} + 5 q^{11} + 4 q^{13} - 4 q^{17} - 3 q^{21} - 11 q^{23} - 2 q^{27} - 10 q^{29} - 9 q^{31} - 19 q^{33} + 15 q^{37} - 21 q^{39} + 17 q^{41} - 12 q^{43} - 14 q^{47} + 17 q^{49} + 25 q^{51} - 6 q^{53} - 9 q^{57} + 18 q^{59} - 11 q^{61} - 52 q^{63} + q^{67} + 8 q^{69} - 26 q^{71} - 9 q^{73} - 8 q^{77} + 8 q^{79} - 4 q^{81} - 12 q^{83} - 17 q^{87} + 5 q^{89} - 33 q^{91} + 30 q^{93} - 29 q^{97} + 8 q^{99}+O(q^{100}) $ 4 * q + q^3 - 9 * q^7 + 11 * q^9 + 5 * q^11 + 4 * q^13 - 4 * q^17 - 3 * q^21 - 11 * q^23 - 2 * q^27 - 10 * q^29 - 9 * q^31 - 19 * q^33 + 15 * q^37 - 21 * q^39 + 17 * q^41 - 12 * q^43 - 14 * q^47 + 17 * q^49 + 25 * q^51 - 6 * q^53 - 9 * q^57 + 18 * q^59 - 11 * q^61 - 52 * q^63 + q^67 + 8 * q^69 - 26 * q^71 - 9 * q^73 - 8 * q^77 + 8 * q^79 - 4 * q^81 - 12 * q^83 - 17 * q^87 + 5 * q^89 - 33 * q^91 + 30 * q^93 - 29 * q^97 + 8 * 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
$3$	2.39867	1.38487	0.692436	−	0.721479i	$-0.256538\pi$
$3$	2.39867	1.38487	0.692436	+	0.721479i	$0.256538\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.13558	−0.429207	−0.214604	−	0.976701i	$-0.568846\pi$
$7$	−1.13558	−0.429207	−0.214604	+	0.976701i	$0.568846\pi$
$8$	0	0
$9$	2.75361	0.917870
$10$	0	0
$11$	−2.01670	−0.608059	−0.304029	−	0.952663i	$-0.598332\pi$
$11$	−2.01670	−0.608059	−0.304029	+	0.952663i	$0.598332\pi$
$12$	0	0
$13$	−2.26309	−0.627669	−0.313835	−	0.949478i	$-0.601614\pi$
$13$	−2.26309	−0.627669	−0.313835	+	0.949478i	$0.601614\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.08379	−0.262858	−0.131429	−	0.991326i	$-0.541956\pi$
$17$	−1.08379	−0.262858	−0.131429	+	0.991326i	$0.541956\pi$
$18$	0	0
$19$	5.58295	1.28082	0.640408	−	0.768035i	$-0.278765\pi$
$19$	5.58295	1.28082	0.640408	+	0.768035i	$0.278765\pi$
$20$	0	0
$21$	−2.72387	−0.594397
$22$	0	0
$23$	−8.33656	−1.73829	−0.869147	−	0.494555i	$-0.835331\pi$
$23$	−8.33656	−1.73829	−0.869147	+	0.494555i	$0.835331\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−0.591009	−0.113740
$28$	0	0
$29$	−9.11719	−1.69302	−0.846510	−	0.532372i	$-0.821301\pi$
$29$	−9.11719	−1.69302	−0.846510	+	0.532372i	$0.821301\pi$
$30$	0	0
$31$	8.75193	1.57189	0.785947	−	0.618294i	$-0.212176\pi$
$31$	8.75193	1.57189	0.785947	+	0.618294i	$0.212176\pi$
$32$	0	0
$33$	−4.83740	−0.842083
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.54457	0.418324	0.209162	−	0.977881i	$-0.432926\pi$
$37$	2.54457	0.418324	0.209162	+	0.977881i	$0.432926\pi$
$38$	0	0
$39$	−5.42841	−0.869241
$40$	0	0
$41$	9.82934	1.53509	0.767543	−	0.640998i	$-0.221479\pi$
$41$	9.82934	1.53509	0.767543	+	0.640998i	$0.221479\pi$
$42$	0	0
$43$	−2.91621	−0.444718	−0.222359	−	0.974965i	$-0.571376\pi$
$43$	−2.91621	−0.444718	−0.222359	+	0.974965i	$0.571376\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.09017	0.304883	0.152441	−	0.988313i	$-0.451286\pi$
$47$	2.09017	0.304883	0.152441	+	0.988313i	$0.451286\pi$
$48$	0	0
$49$	−5.71047	−0.815781
$50$	0	0
$51$	−2.59965	−0.364024
$52$	0	0
$53$	10.5326	1.44676	0.723380	−	0.690451i	$-0.242588\pi$
$53$	10.5326	1.44676	0.723380	+	0.690451i	$0.242588\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	13.3916	1.77377
$58$	0	0
$59$	5.53424	0.720497	0.360249	−	0.932856i	$-0.382692\pi$
$59$	5.53424	0.720497	0.360249	+	0.932856i	$0.382692\pi$
$60$	0	0
$61$	1.63474	0.209307	0.104653	−	0.994509i	$-0.466627\pi$
$61$	1.63474	0.209307	0.104653	+	0.994509i	$0.466627\pi$
$62$	0	0
$63$	−3.12693	−0.393956
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−10.5844	−1.29308	−0.646542	−	0.762878i	$-0.723786\pi$
$67$	−10.5844	−1.29308	−0.646542	+	0.762878i	$0.723786\pi$
$68$	0	0
$69$	−19.9966	−2.40731
$70$	0	0
$71$	−12.9496	−1.53684	−0.768418	−	0.639948i	$-0.778956\pi$
$71$	−12.9496	−1.53684	−0.768418	+	0.639948i	$0.778956\pi$
$72$	0	0
$73$	−13.2447	−1.55018	−0.775088	−	0.631853i	$-0.782295\pi$
$73$	−13.2447	−1.55018	−0.775088	+	0.631853i	$0.782295\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.29012	0.260983
$78$	0	0
$79$	−6.84604	−0.770240	−0.385120	−	0.922866i	$-0.625840\pi$
$79$	−6.84604	−0.770240	−0.385120	+	0.922866i	$0.625840\pi$
$80$	0	0
$81$	−9.67846	−1.07538
$82$	0	0
$83$	−2.81798	−0.309314	−0.154657	−	0.987968i	$-0.549427\pi$
$83$	−2.81798	−0.309314	−0.154657	+	0.987968i	$0.549427\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−21.8691	−2.34462
$88$	0	0
$89$	−15.1673	−1.60773	−0.803865	−	0.594811i	$-0.797227\pi$
$89$	−15.1673	−1.60773	−0.803865	+	0.594811i	$0.797227\pi$
$90$	0	0
$91$	2.56991	0.269400
$92$	0	0
$93$	20.9930	2.17687
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−18.2591	−1.85394	−0.926968	−	0.375141i	$-0.877594\pi$
$97$	−18.2591	−1.85394	−0.926968	+	0.375141i	$0.877594\pi$
$98$	0	0
$99$	−5.55321	−0.558119
$100$	0	0
$101$	6.04871	0.601869	0.300934	−	0.953645i	$-0.402701\pi$
$101$	6.04871	0.601869	0.300934	+	0.953645i	$0.402701\pi$
$102$	0	0
$103$	−6.03735	−0.594878	−0.297439	−	0.954741i	$-0.596132\pi$
$103$	−6.03735	−0.594878	−0.297439	+	0.954741i	$0.596132\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.36359	−0.518517	−0.259259	−	0.965808i	$-0.583478\pi$
$107$	−5.36359	−0.518517	−0.259259	+	0.965808i	$0.583478\pi$
$108$	0	0
$109$	−6.59965	−0.632132	−0.316066	−	0.948737i	$-0.602362\pi$
$109$	−6.59965	−0.632132	−0.316066	+	0.948737i	$0.602362\pi$
$110$	0	0
$111$	6.10357	0.579325
$112$	0	0
$113$	9.77756	0.919795	0.459898	−	0.887972i	$-0.347886\pi$
$113$	9.77756	0.919795	0.459898	+	0.887972i	$0.347886\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−6.23167	−0.576118
$118$	0	0
$119$	1.23073	0.112820
$120$	0	0
$121$	−6.93291	−0.630265
$122$	0	0
$123$	23.5773	2.12590
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−7.20266	−0.639133	−0.319567	−	0.947564i	$-0.603537\pi$
$127$	−7.20266	−0.639133	−0.319567	+	0.947564i	$0.603537\pi$
$128$	0	0
$129$	−6.99502	−0.615877
$130$	0	0
$131$	6.20904	0.542487	0.271243	−	0.962511i	$-0.412565\pi$
$131$	6.20904	0.542487	0.271243	+	0.962511i	$0.412565\pi$
$132$	0	0
$133$	−6.33986	−0.549736
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.94550	−0.593394	−0.296697	−	0.954972i	$-0.595885\pi$
$137$	−6.94550	−0.593394	−0.296697	+	0.954972i	$0.595885\pi$
$138$	0	0
$139$	−7.38695	−0.626553	−0.313276	−	0.949662i	$-0.601427\pi$
$139$	−7.38695	−0.626553	−0.313276	+	0.949662i	$0.601427\pi$
$140$	0	0
$141$	5.01362	0.422223
$142$	0	0
$143$	4.56398	0.381660
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−13.6975	−1.12975
$148$	0	0
$149$	7.09655	0.581372	0.290686	−	0.956819i	$-0.406116\pi$
$149$	7.09655	0.581372	0.290686	+	0.956819i	$0.406116\pi$
$150$	0	0
$151$	−12.3132	−1.00203	−0.501017	−	0.865437i	$-0.667041\pi$
$151$	−12.3132	−1.00203	−0.501017	+	0.865437i	$0.667041\pi$
$152$	0	0
$153$	−2.98434	−0.241269
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−12.4114	−0.990540	−0.495270	−	0.868739i	$-0.664931\pi$
$157$	−12.4114	−0.990540	−0.495270	+	0.868739i	$0.664931\pi$
$158$	0	0
$159$	25.2641	2.00358
$160$	0	0
$161$	9.46679	0.746088
$162$	0	0
$163$	14.2257	1.11425	0.557123	−	0.830430i	$-0.311905\pi$
$163$	14.2257	1.11425	0.557123	+	0.830430i	$0.311905\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.7790	0.834101	0.417050	−	0.908883i	$-0.363064\pi$
$167$	10.7790	0.834101	0.417050	+	0.908883i	$0.363064\pi$
$168$	0	0
$169$	−7.87841	−0.606032
$170$	0	0
$171$	15.3733	1.17562
$172$	0	0
$173$	−9.96632	−0.757725	−0.378863	−	0.925453i	$-0.623685\pi$
$173$	−9.96632	−0.757725	−0.378863	+	0.925453i	$0.623685\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	13.2748	0.997796
$178$	0	0
$179$	1.15622	0.0864200	0.0432100	−	0.999066i	$-0.486242\pi$
$179$	1.15622	0.0864200	0.0432100	+	0.999066i	$0.486242\pi$
$180$	0	0
$181$	1.58191	0.117583	0.0587914	−	0.998270i	$-0.481275\pi$
$181$	1.58191	0.117583	0.0587914	+	0.998270i	$0.481275\pi$
$182$	0	0
$183$	3.92119	0.289863
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2.18568	0.159833
$188$	0	0
$189$	0.671136	0.0488179
$190$	0	0
$191$	−8.12629	−0.587998	−0.293999	−	0.955806i	$-0.594986\pi$
$191$	−8.12629	−0.587998	−0.293999	+	0.955806i	$0.594986\pi$
$192$	0	0
$193$	0.279795	0.0201401	0.0100700	−	0.999949i	$-0.496795\pi$
$193$	0.279795	0.0201401	0.0100700	+	0.999949i	$0.496795\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	14.4201	1.02739	0.513694	−	0.857974i	$-0.328277\pi$
$197$	14.4201	1.02739	0.513694	+	0.857974i	$0.328277\pi$
$198$	0	0
$199$	−6.01276	−0.426233	−0.213117	−	0.977027i	$-0.568361\pi$
$199$	−6.01276	−0.426233	−0.213117	+	0.977027i	$0.568361\pi$
$200$	0	0
$201$	−25.3883	−1.79076
$202$	0	0
$203$	10.3533	0.726657
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−22.9556	−1.59553
$208$	0	0
$209$	−11.2591	−0.778812
$210$	0	0
$211$	−21.5528	−1.48376	−0.741880	−	0.670533i	$-0.766065\pi$
$211$	−21.5528	−1.48376	−0.741880	+	0.670533i	$0.766065\pi$
$212$	0	0
$213$	−31.0618	−2.12832
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−9.93848	−0.674668
$218$	0	0
$219$	−31.7697	−2.14680
$220$	0	0
$221$	2.45272	0.164988
$222$	0	0
$223$	−9.70409	−0.649834	−0.324917	−	0.945743i	$-0.605336\pi$
$223$	−9.70409	−0.649834	−0.324917	+	0.945743i	$0.605336\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−10.4969	−0.696703	−0.348352	−	0.937364i	$-0.613259\pi$
$227$	−10.4969	−0.696703	−0.348352	+	0.937364i	$0.613259\pi$
$228$	0	0
$229$	17.1573	1.13378	0.566892	−	0.823792i	$-0.308146\pi$
$229$	17.1573	1.13378	0.566892	+	0.823792i	$0.308146\pi$
$230$	0	0
$231$	5.49323	0.361428
$232$	0	0
$233$	−12.7586	−0.835843	−0.417922	−	0.908483i	$-0.637241\pi$
$233$	−12.7586	−0.835843	−0.417922	+	0.908483i	$0.637241\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−16.4214	−1.06668
$238$	0	0
$239$	−12.6150	−0.815994	−0.407997	−	0.912983i	$-0.633773\pi$
$239$	−12.6150	−0.815994	−0.407997	+	0.912983i	$0.633773\pi$
$240$	0	0
$241$	−3.16532	−0.203896	−0.101948	−	0.994790i	$-0.532508\pi$
$241$	−3.16532	−0.203896	−0.101948	+	0.994790i	$0.532508\pi$
$242$	0	0
$243$	−21.4424	−1.37553
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−12.6347	−0.803929
$248$	0	0
$249$	−6.75940	−0.428360
$250$	0	0
$251$	28.0367	1.76966	0.884831	−	0.465913i	$-0.154274\pi$
$251$	28.0367	1.76966	0.884831	+	0.465913i	$0.154274\pi$
$252$	0	0
$253$	16.8124	1.05698
$254$	0	0
$255$	0	0
$256$	0	0
$257$	20.0119	1.24831	0.624155	−	0.781300i	$-0.285443\pi$
$257$	20.0119	1.24831	0.624155	+	0.781300i	$0.285443\pi$
$258$	0	0
$259$	−2.88955	−0.179548
$260$	0	0
$261$	−25.1052	−1.55397
$262$	0	0
$263$	10.9507	0.675246	0.337623	−	0.941281i	$-0.390377\pi$
$263$	10.9507	0.675246	0.337623	+	0.941281i	$0.390377\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−36.3813	−2.22650
$268$	0	0
$269$	7.61862	0.464515	0.232258	−	0.972654i	$-0.425389\pi$
$269$	7.61862	0.464515	0.232258	+	0.972654i	$0.425389\pi$
$270$	0	0
$271$	28.7205	1.74465	0.872323	−	0.488929i	$-0.162612\pi$
$271$	28.7205	1.74465	0.872323	+	0.488929i	$0.162612\pi$
$272$	0	0
$273$	6.16437	0.373085
$274$	0	0
$275$	0	0
$276$	0	0
$277$	28.3774	1.70503	0.852516	−	0.522701i	$-0.175076\pi$
$277$	28.3774	1.70503	0.852516	+	0.522701i	$0.175076\pi$
$278$	0	0
$279$	24.0994	1.44279
$280$	0	0
$281$	31.6027	1.88526	0.942629	−	0.333843i	$-0.108345\pi$
$281$	31.6027	1.88526	0.942629	+	0.333843i	$0.108345\pi$
$282$	0	0
$283$	−22.4226	−1.33289	−0.666443	−	0.745556i	$-0.732184\pi$
$283$	−22.4226	−1.33289	−0.666443	+	0.745556i	$0.732184\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−11.1620	−0.658870
$288$	0	0
$289$	−15.8254	−0.930906
$290$	0	0
$291$	−43.7976	−2.56746
$292$	0	0
$293$	18.6799	1.09129	0.545645	−	0.838017i	$-0.316285\pi$
$293$	18.6799	1.09129	0.545645	+	0.838017i	$0.316285\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.19189	0.0691604
$298$	0	0
$299$	18.8664	1.09107
$300$	0	0
$301$	3.31158	0.190876
$302$	0	0
$303$	14.5088	0.833511
$304$	0	0
$305$	0	0
$306$	0	0
$307$	5.72762	0.326893	0.163446	−	0.986552i	$-0.447739\pi$
$307$	5.72762	0.326893	0.163446	+	0.986552i	$0.447739\pi$
$308$	0	0
$309$	−14.4816	−0.823829
$310$	0	0
$311$	−8.65452	−0.490753	−0.245376	−	0.969428i	$-0.578912\pi$
$311$	−8.65452	−0.490753	−0.245376	+	0.969428i	$0.578912\pi$
$312$	0	0
$313$	3.34960	0.189331	0.0946653	−	0.995509i	$-0.469822\pi$
$313$	3.34960	0.189331	0.0946653	+	0.995509i	$0.469822\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8.04929	−0.452093	−0.226047	−	0.974116i	$-0.572580\pi$
$317$	−8.04929	−0.452093	−0.226047	+	0.974116i	$0.572580\pi$
$318$	0	0
$319$	18.3867	1.02946
$320$	0	0
$321$	−12.8655	−0.718080
$322$	0	0
$323$	−6.05075	−0.336673
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−15.8304	−0.875422
$328$	0	0
$329$	−2.37355	−0.130858
$330$	0	0
$331$	18.5509	1.01965	0.509826	−	0.860277i	$-0.329710\pi$
$331$	18.5509	1.01965	0.509826	+	0.860277i	$0.329710\pi$
$332$	0	0
$333$	7.00674	0.383967
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−25.1300	−1.36892	−0.684458	−	0.729052i	$-0.739961\pi$
$337$	−25.1300	−1.36892	−0.684458	+	0.729052i	$0.739961\pi$
$338$	0	0
$339$	23.4531	1.27380
$340$	0	0
$341$	−17.6500	−0.955803
$342$	0	0
$343$	14.4337	0.779346
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−25.2876	−1.35751	−0.678754	−	0.734366i	$-0.737480\pi$
$347$	−25.2876	−1.35751	−0.678754	+	0.734366i	$0.737480\pi$
$348$	0	0
$349$	−2.69911	−0.144480	−0.0722400	−	0.997387i	$-0.523015\pi$
$349$	−2.69911	−0.144480	−0.0722400	+	0.997387i	$0.523015\pi$
$350$	0	0
$351$	1.33751	0.0713909
$352$	0	0
$353$	13.1225	0.698442	0.349221	−	0.937040i	$-0.386446\pi$
$353$	13.1225	0.698442	0.349221	+	0.937040i	$0.386446\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	2.95210	0.156242
$358$	0	0
$359$	1.72762	0.0911804	0.0455902	−	0.998960i	$-0.485483\pi$
$359$	1.72762	0.0911804	0.0455902	+	0.998960i	$0.485483\pi$
$360$	0	0
$361$	12.1693	0.640492
$362$	0	0
$363$	−16.6298	−0.872836
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−34.4287	−1.79716	−0.898582	−	0.438805i	$-0.855402\pi$
$367$	−34.4287	−1.79716	−0.898582	+	0.438805i	$0.855402\pi$
$368$	0	0
$369$	27.0662	1.40901
$370$	0	0
$371$	−11.9605	−0.620959
$372$	0	0
$373$	−26.5994	−1.37726	−0.688632	−	0.725111i	$-0.741788\pi$
$373$	−26.5994	−1.37726	−0.688632	+	0.725111i	$0.741788\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	20.6331	1.06266
$378$	0	0
$379$	13.0117	0.668367	0.334184	−	0.942508i	$-0.391539\pi$
$379$	13.0117	0.668367	0.334184	+	0.942508i	$0.391539\pi$
$380$	0	0
$381$	−17.2768	−0.885117
$382$	0	0
$383$	17.8421	0.911689	0.455844	−	0.890059i	$-0.349337\pi$
$383$	17.8421	0.911689	0.455844	+	0.890059i	$0.349337\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−8.03010	−0.408193
$388$	0	0
$389$	1.94794	0.0987643	0.0493821	−	0.998780i	$-0.484275\pi$
$389$	1.94794	0.0987643	0.0493821	+	0.998780i	$0.484275\pi$
$390$	0	0
$391$	9.03508	0.456924
$392$	0	0
$393$	14.8934	0.751274
$394$	0	0
$395$	0	0
$396$	0	0
$397$	12.5726	0.631002	0.315501	−	0.948925i	$-0.397827\pi$
$397$	12.5726	0.631002	0.315501	+	0.948925i	$0.397827\pi$
$398$	0	0
$399$	−15.2072	−0.761314
$400$	0	0
$401$	24.2160	1.20929	0.604645	−	0.796495i	$-0.293315\pi$
$401$	24.2160	1.20929	0.604645	+	0.796495i	$0.293315\pi$
$402$	0	0
$403$	−19.8064	−0.986629
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−5.13163	−0.254366
$408$	0	0
$409$	27.0728	1.33867	0.669333	−	0.742963i	$-0.266580\pi$
$409$	27.0728	1.33867	0.669333	+	0.742963i	$0.266580\pi$
$410$	0	0
$411$	−16.6599	−0.821775
$412$	0	0
$413$	−6.28455	−0.309243
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−17.7188	−0.867695
$418$	0	0
$419$	−33.1266	−1.61834	−0.809170	−	0.587574i	$-0.800083\pi$
$419$	−33.1266	−1.61834	−0.809170	+	0.587574i	$0.800083\pi$
$420$	0	0
$421$	37.8125	1.84287	0.921435	−	0.388532i	$-0.127018\pi$
$421$	37.8125	1.84287	0.921435	+	0.388532i	$0.127018\pi$
$422$	0	0
$423$	5.75551	0.279843
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−1.85637	−0.0898359
$428$	0	0
$429$	10.9475	0.528550
$430$	0	0
$431$	16.0656	0.773852	0.386926	−	0.922111i	$-0.373537\pi$
$431$	16.0656	0.773852	0.386926	+	0.922111i	$0.373537\pi$
$432$	0	0
$433$	33.0940	1.59040	0.795198	−	0.606350i	$-0.207367\pi$
$433$	33.0940	1.59040	0.795198	+	0.606350i	$0.207367\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−46.5426	−2.22643
$438$	0	0
$439$	−33.1634	−1.58280	−0.791400	−	0.611298i	$-0.790648\pi$
$439$	−33.1634	−1.58280	−0.791400	+	0.611298i	$0.790648\pi$
$440$	0	0
$441$	−15.7244	−0.748781
$442$	0	0
$443$	5.78036	0.274633	0.137316	−	0.990527i	$-0.456152\pi$
$443$	5.78036	0.274633	0.137316	+	0.990527i	$0.456152\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	17.0223	0.805126
$448$	0	0
$449$	20.6234	0.973277	0.486639	−	0.873603i	$-0.338223\pi$
$449$	20.6234	0.973277	0.486639	+	0.873603i	$0.338223\pi$
$450$	0	0
$451$	−19.8229	−0.933422
$452$	0	0
$453$	−29.5353	−1.38769
$454$	0	0
$455$	0	0
$456$	0	0
$457$	9.42094	0.440693	0.220346	−	0.975422i	$-0.429281\pi$
$457$	9.42094	0.440693	0.220346	+	0.975422i	$0.429281\pi$
$458$	0	0
$459$	0.640530	0.0298974
$460$	0	0
$461$	−24.9744	−1.16317	−0.581586	−	0.813485i	$-0.697568\pi$
$461$	−24.9744	−1.16317	−0.581586	+	0.813485i	$0.697568\pi$
$462$	0	0
$463$	−27.8110	−1.29248	−0.646242	−	0.763132i	$-0.723661\pi$
$463$	−27.8110	−1.29248	−0.646242	+	0.763132i	$0.723661\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−3.80898	−0.176258	−0.0881292	−	0.996109i	$-0.528089\pi$
$467$	−3.80898	−0.176258	−0.0881292	+	0.996109i	$0.528089\pi$
$468$	0	0
$469$	12.0193	0.555001
$470$	0	0
$471$	−29.7709	−1.37177
$472$	0	0
$473$	5.88113	0.270414
$474$	0	0
$475$	0	0
$476$	0	0
$477$	29.0026	1.32794
$478$	0	0
$479$	−4.45272	−0.203450	−0.101725	−	0.994813i	$-0.532436\pi$
$479$	−4.45272	−0.203450	−0.101725	+	0.994813i	$0.532436\pi$
$480$	0	0
$481$	−5.75859	−0.262569
$482$	0	0
$483$	22.7077	1.03324
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−9.42116	−0.426914	−0.213457	−	0.976952i	$-0.568472\pi$
$487$	−9.42116	−0.426914	−0.213457	+	0.976952i	$0.568472\pi$
$488$	0	0
$489$	34.1228	1.54309
$490$	0	0
$491$	33.0769	1.49274	0.746369	−	0.665533i	$-0.231796\pi$
$491$	33.0769	1.49274	0.746369	+	0.665533i	$0.231796\pi$
$492$	0	0
$493$	9.88113	0.445024
$494$	0	0
$495$	0	0
$496$	0	0
$497$	14.7053	0.659621
$498$	0	0
$499$	32.2197	1.44235	0.721175	−	0.692753i	$-0.243602\pi$
$499$	32.2197	1.44235	0.721175	+	0.692753i	$0.243602\pi$
$500$	0	0
$501$	25.8551	1.15512
$502$	0	0
$503$	−10.6317	−0.474042	−0.237021	−	0.971505i	$-0.576171\pi$
$503$	−10.6317	−0.474042	−0.237021	+	0.971505i	$0.576171\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−18.8977	−0.839276
$508$	0	0
$509$	7.69824	0.341219	0.170609	−	0.985339i	$-0.445426\pi$
$509$	7.69824	0.341219	0.170609	+	0.985339i	$0.445426\pi$
$510$	0	0
$511$	15.0404	0.665347
$512$	0	0
$513$	−3.29958	−0.145680
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−4.21525	−0.185387
$518$	0	0
$519$	−23.9059	−1.04935
$520$	0	0
$521$	−10.2597	−0.449487	−0.224744	−	0.974418i	$-0.572154\pi$
$521$	−10.2597	−0.449487	−0.224744	+	0.974418i	$0.572154\pi$
$522$	0	0
$523$	−32.7292	−1.43115	−0.715575	−	0.698536i	$-0.753835\pi$
$523$	−32.7292	−1.43115	−0.715575	+	0.698536i	$0.753835\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−9.48526	−0.413184
$528$	0	0
$529$	46.4982	2.02166
$530$	0	0
$531$	15.2391	0.661323
$532$	0	0
$533$	−22.2447	−0.963525
$534$	0	0
$535$	0	0
$536$	0	0
$537$	2.77339	0.119681
$538$	0	0
$539$	11.5163	0.496043
$540$	0	0
$541$	7.94875	0.341743	0.170872	−	0.985293i	$-0.445342\pi$
$541$	7.94875	0.341743	0.170872	+	0.985293i	$0.445342\pi$
$542$	0	0
$543$	3.79449	0.162837
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−2.77593	−0.118690	−0.0593452	−	0.998238i	$-0.518901\pi$
$547$	−2.77593	−0.118690	−0.0593452	+	0.998238i	$0.518901\pi$
$548$	0	0
$549$	4.50143	0.192116
$550$	0	0
$551$	−50.9009	−2.16845
$552$	0	0
$553$	7.77420	0.330593
$554$	0	0
$555$	0	0
$556$	0	0
$557$	28.2457	1.19681	0.598405	−	0.801193i	$-0.295801\pi$
$557$	28.2457	1.19681	0.598405	+	0.801193i	$0.295801\pi$
$558$	0	0
$559$	6.59965	0.279136
$560$	0	0
$561$	5.24273	0.221348
$562$	0	0
$563$	10.4749	0.441466	0.220733	−	0.975334i	$-0.429155\pi$
$563$	10.4749	0.441466	0.220733	+	0.975334i	$0.429155\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	10.9906	0.461563
$568$	0	0
$569$	30.2564	1.26842	0.634208	−	0.773163i	$-0.281326\pi$
$569$	30.2564	1.26842	0.634208	+	0.773163i	$0.281326\pi$
$570$	0	0
$571$	33.0909	1.38481	0.692406	−	0.721508i	$-0.256551\pi$
$571$	33.0909	1.38481	0.692406	+	0.721508i	$0.256551\pi$
$572$	0	0
$573$	−19.4923	−0.814301
$574$	0	0
$575$	0	0
$576$	0	0
$577$	37.3160	1.55349	0.776744	−	0.629817i	$-0.216870\pi$
$577$	37.3160	1.55349	0.776744	+	0.629817i	$0.216870\pi$
$578$	0	0
$579$	0.671136	0.0278914
$580$	0	0
$581$	3.20003	0.132760
$582$	0	0
$583$	−21.2410	−0.879714
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−30.9409	−1.27707	−0.638534	−	0.769594i	$-0.720459\pi$
$587$	−30.9409	−1.27707	−0.638534	+	0.769594i	$0.720459\pi$
$588$	0	0
$589$	48.8616	2.01331
$590$	0	0
$591$	34.5890	1.42280
$592$	0	0
$593$	44.1444	1.81279	0.906396	−	0.422428i	$-0.138822\pi$
$593$	44.1444	1.81279	0.906396	+	0.422428i	$0.138822\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−14.4226	−0.590278
$598$	0	0
$599$	−6.25996	−0.255775	−0.127888	−	0.991789i	$-0.540820\pi$
$599$	−6.25996	−0.255775	−0.127888	+	0.991789i	$0.540820\pi$
$600$	0	0
$601$	−5.31678	−0.216876	−0.108438	−	0.994103i	$-0.534585\pi$
$601$	−5.31678	−0.216876	−0.108438	+	0.994103i	$0.534585\pi$
$602$	0	0
$603$	−29.1452	−1.18688
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−36.0584	−1.46356	−0.731782	−	0.681538i	$-0.761311\pi$
$607$	−36.0584	−1.46356	−0.731782	+	0.681538i	$0.761311\pi$
$608$	0	0
$609$	24.8340	1.00633
$610$	0	0
$611$	−4.73025	−0.191365
$612$	0	0
$613$	−9.04390	−0.365280	−0.182640	−	0.983180i	$-0.558464\pi$
$613$	−9.04390	−0.365280	−0.182640	+	0.983180i	$0.558464\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−43.9318	−1.76863	−0.884314	−	0.466892i	$-0.845374\pi$
$617$	−43.9318	−1.76863	−0.884314	+	0.466892i	$0.845374\pi$
$618$	0	0
$619$	−2.08071	−0.0836309	−0.0418154	−	0.999125i	$-0.513314\pi$
$619$	−2.08071	−0.0836309	−0.0418154	+	0.999125i	$0.513314\pi$
$620$	0	0
$621$	4.92698	0.197713
$622$	0	0
$623$	17.2236	0.690050
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−27.0070	−1.07855
$628$	0	0
$629$	−2.75778	−0.109960
$630$	0	0
$631$	−7.90711	−0.314777	−0.157389	−	0.987537i	$-0.550308\pi$
$631$	−7.90711	−0.314777	−0.157389	+	0.987537i	$0.550308\pi$
$632$	0	0
$633$	−51.6981	−2.05482
$634$	0	0
$635$	0	0
$636$	0	0
$637$	12.9233	0.512041
$638$	0	0
$639$	−35.6582	−1.41062
$640$	0	0
$641$	11.1496	0.440381	0.220191	−	0.975457i	$-0.429332\pi$
$641$	11.1496	0.440381	0.220191	+	0.975457i	$0.429332\pi$
$642$	0	0
$643$	−4.47544	−0.176494	−0.0882470	−	0.996099i	$-0.528126\pi$
$643$	−4.47544	−0.176494	−0.0882470	+	0.996099i	$0.528126\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−44.4289	−1.74668	−0.873341	−	0.487109i	$-0.838051\pi$
$647$	−44.4289	−1.74668	−0.873341	+	0.487109i	$0.838051\pi$
$648$	0	0
$649$	−11.1609	−0.438105
$650$	0	0
$651$	−23.8391	−0.934328
$652$	0	0
$653$	28.7385	1.12463	0.562313	−	0.826925i	$-0.309912\pi$
$653$	28.7385	1.12463	0.562313	+	0.826925i	$0.309912\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−36.4708	−1.42286
$658$	0	0
$659$	18.9254	0.737230	0.368615	−	0.929582i	$-0.379832\pi$
$659$	18.9254	0.737230	0.368615	+	0.929582i	$0.379832\pi$
$660$	0	0
$661$	35.6615	1.38707	0.693535	−	0.720423i	$-0.256052\pi$
$661$	35.6615	1.38707	0.693535	+	0.720423i	$0.256052\pi$
$662$	0	0
$663$	5.88326	0.228487
$664$	0	0
$665$	0	0
$666$	0	0
$667$	76.0060	2.94297
$668$	0	0
$669$	−23.2769	−0.899937
$670$	0	0
$671$	−3.29678	−0.127271
$672$	0	0
$673$	−31.9521	−1.23166	−0.615831	−	0.787879i	$-0.711179\pi$
$673$	−31.9521	−1.23166	−0.615831	+	0.787879i	$0.711179\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−23.9709	−0.921278	−0.460639	−	0.887588i	$-0.652380\pi$
$677$	−23.9709	−0.921278	−0.460639	+	0.887588i	$0.652380\pi$
$678$	0	0
$679$	20.7346	0.795723
$680$	0	0
$681$	−25.1786	−0.964845
$682$	0	0
$683$	−7.07047	−0.270544	−0.135272	−	0.990808i	$-0.543191\pi$
$683$	−7.07047	−0.270544	−0.135272	+	0.990808i	$0.543191\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	41.1546	1.57014
$688$	0	0
$689$	−23.8362	−0.908086
$690$	0	0
$691$	−1.97935	−0.0752982	−0.0376491	−	0.999291i	$-0.511987\pi$
$691$	−1.97935	−0.0752982	−0.0376491	+	0.999291i	$0.511987\pi$
$692$	0	0
$693$	6.30609	0.239549
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−10.6529	−0.403509
$698$	0	0
$699$	−30.6036	−1.15754
$700$	0	0
$701$	19.8745	0.750648	0.375324	−	0.926894i	$-0.377531\pi$
$701$	19.8745	0.750648	0.375324	+	0.926894i	$0.377531\pi$
$702$	0	0
$703$	14.2062	0.535797
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−6.86876	−0.258326
$708$	0	0
$709$	8.79068	0.330141	0.165070	−	0.986282i	$-0.447215\pi$
$709$	8.79068	0.330141	0.165070	+	0.986282i	$0.447215\pi$
$710$	0	0
$711$	−18.8513	−0.706980
$712$	0	0
$713$	−72.9610	−2.73241
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−30.2591	−1.13005
$718$	0	0
$719$	−10.2898	−0.383743	−0.191872	−	0.981420i	$-0.561456\pi$
$719$	−10.2898	−0.383743	−0.191872	+	0.981420i	$0.561456\pi$
$720$	0	0
$721$	6.85586	0.255326
$722$	0	0
$723$	−7.59254	−0.282370
$724$	0	0
$725$	0	0
$726$	0	0
$727$	9.73934	0.361212	0.180606	−	0.983556i	$-0.442194\pi$
$727$	9.73934	0.361212	0.180606	+	0.983556i	$0.442194\pi$
$728$	0	0
$729$	−22.3978	−0.829548
$730$	0	0
$731$	3.16056	0.116898
$732$	0	0
$733$	25.2767	0.933617	0.466808	−	0.884358i	$-0.345404\pi$
$733$	25.2767	0.933617	0.466808	+	0.884358i	$0.345404\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	21.3455	0.786271
$738$	0	0
$739$	−39.6691	−1.45925	−0.729626	−	0.683847i	$-0.760306\pi$
$739$	−39.6691	−1.45925	−0.729626	+	0.683847i	$0.760306\pi$
$740$	0	0
$741$	−30.3065	−1.11334
$742$	0	0
$743$	18.7732	0.688721	0.344360	−	0.938838i	$-0.388096\pi$
$743$	18.7732	0.688721	0.344360	+	0.938838i	$0.388096\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−7.75962	−0.283910
$748$	0	0
$749$	6.09076	0.222551
$750$	0	0
$751$	−35.1761	−1.28359	−0.641797	−	0.766874i	$-0.721811\pi$
$751$	−35.1761	−1.28359	−0.641797	+	0.766874i	$0.721811\pi$
$752$	0	0
$753$	67.2508	2.45075
$754$	0	0
$755$	0	0
$756$	0	0
$757$	25.9332	0.942559	0.471279	−	0.881984i	$-0.343792\pi$
$757$	25.9332	0.942559	0.471279	+	0.881984i	$0.343792\pi$
$758$	0	0
$759$	40.3273	1.46379
$760$	0	0
$761$	−11.6131	−0.420973	−0.210486	−	0.977597i	$-0.567505\pi$
$761$	−11.6131	−0.420973	−0.210486	+	0.977597i	$0.567505\pi$
$762$	0	0
$763$	7.49440	0.271316
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−12.5245	−0.452234
$768$	0	0
$769$	17.6988	0.638236	0.319118	−	0.947715i	$-0.396613\pi$
$769$	17.6988	0.638236	0.319118	+	0.947715i	$0.396613\pi$
$770$	0	0
$771$	48.0020	1.72875
$772$	0	0
$773$	−45.3363	−1.63063	−0.815316	−	0.579016i	$-0.803437\pi$
$773$	−45.3363	−1.63063	−0.815316	+	0.579016i	$0.803437\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−6.93106	−0.248651
$778$	0	0
$779$	54.8767	1.96616
$780$	0	0
$781$	26.1155	0.934487
$782$	0	0
$783$	5.38835	0.192564
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−1.40446	−0.0500637	−0.0250318	−	0.999687i	$-0.507969\pi$
$787$	−1.40446	−0.0500637	−0.0250318	+	0.999687i	$0.507969\pi$
$788$	0	0
$789$	26.2670	0.935129
$790$	0	0
$791$	−11.1032	−0.394783
$792$	0	0
$793$	−3.69956	−0.131375
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−41.1825	−1.45876	−0.729379	−	0.684109i	$-0.760191\pi$
$797$	−41.1825	−1.45876	−0.729379	+	0.684109i	$0.760191\pi$
$798$	0	0
$799$	−2.26531	−0.0801408
$800$	0	0
$801$	−41.7648	−1.47569
$802$	0	0
$803$	26.7106	0.942598
$804$	0	0
$805$	0	0
$806$	0	0
$807$	18.2745	0.643294
$808$	0	0
$809$	6.98976	0.245747	0.122873	−	0.992422i	$-0.460789\pi$
$809$	6.98976	0.245747	0.122873	+	0.992422i	$0.460789\pi$
$810$	0	0
$811$	−10.9870	−0.385804	−0.192902	−	0.981218i	$-0.561790\pi$
$811$	−10.9870	−0.385804	−0.192902	+	0.981218i	$0.561790\pi$
$812$	0	0
$813$	68.8910	2.41611
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−16.2811	−0.569602
$818$	0	0
$819$	7.07654	0.247274
$820$	0	0
$821$	−35.2863	−1.23150	−0.615751	−	0.787941i	$-0.711147\pi$
$821$	−35.2863	−1.23150	−0.615751	+	0.787941i	$0.711147\pi$
$822$	0	0
$823$	38.8164	1.35306	0.676528	−	0.736417i	$-0.263484\pi$
$823$	38.8164	1.35306	0.676528	+	0.736417i	$0.263484\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−15.6781	−0.545181	−0.272590	−	0.962130i	$-0.587880\pi$
$827$	−15.6781	−0.545181	−0.272590	+	0.962130i	$0.587880\pi$
$828$	0	0
$829$	−27.8171	−0.966126	−0.483063	−	0.875585i	$-0.660476\pi$
$829$	−27.8171	−0.966126	−0.483063	+	0.875585i	$0.660476\pi$
$830$	0	0
$831$	68.0679	2.36125
$832$	0	0
$833$	6.18895	0.214434
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−5.17247	−0.178787
$838$	0	0
$839$	38.1703	1.31779	0.658893	−	0.752237i	$-0.271025\pi$
$839$	38.1703	1.31779	0.658893	+	0.752237i	$0.271025\pi$
$840$	0	0
$841$	54.1232	1.86632
$842$	0	0
$843$	75.8043	2.61084
$844$	0	0
$845$	0	0
$846$	0	0
$847$	7.87284	0.270514
$848$	0	0
$849$	−53.7844	−1.84588
$850$	0	0
$851$	−21.2129	−0.727170
$852$	0	0
$853$	23.4987	0.804580	0.402290	−	0.915512i	$-0.368214\pi$
$853$	23.4987	0.804580	0.402290	+	0.915512i	$0.368214\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	20.7940	0.710309	0.355154	−	0.934808i	$-0.384428\pi$
$857$	20.7940	0.710309	0.355154	+	0.934808i	$0.384428\pi$
$858$	0	0
$859$	−17.4860	−0.596616	−0.298308	−	0.954470i	$-0.596422\pi$
$859$	−17.4860	−0.596616	−0.298308	+	0.954470i	$0.596422\pi$
$860$	0	0
$861$	−26.7738	−0.912450
$862$	0	0
$863$	−3.93490	−0.133945	−0.0669727	−	0.997755i	$-0.521334\pi$
$863$	−3.93490	−0.133945	−0.0669727	+	0.997755i	$0.521334\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−37.9599	−1.28919
$868$	0	0
$869$	13.8064	0.468351
$870$	0	0
$871$	23.9534	0.811629
$872$	0	0
$873$	−50.2786	−1.70167
$874$	0	0
$875$	0	0
$876$	0	0
$877$	30.3758	1.02572	0.512859	−	0.858473i	$-0.328586\pi$
$877$	30.3758	1.02572	0.512859	+	0.858473i	$0.328586\pi$
$878$	0	0
$879$	44.8068	1.51130
$880$	0	0
$881$	13.4095	0.451778	0.225889	−	0.974153i	$-0.427471\pi$
$881$	13.4095	0.451778	0.225889	+	0.974153i	$0.427471\pi$
$882$	0	0
$883$	47.5905	1.60155	0.800774	−	0.598967i	$-0.204422\pi$
$883$	47.5905	1.60155	0.800774	+	0.598967i	$0.204422\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	13.6312	0.457692	0.228846	−	0.973463i	$-0.426505\pi$
$887$	13.6312	0.457692	0.228846	+	0.973463i	$0.426505\pi$
$888$	0	0
$889$	8.17917	0.274320
$890$	0	0
$891$	19.5186	0.653897
$892$	0	0
$893$	11.6693	0.390499
$894$	0	0
$895$	0	0
$896$	0	0
$897$	45.2543	1.51100
$898$	0	0
$899$	−79.7931	−2.66125
$900$	0	0
$901$	−11.4151	−0.380292
$902$	0	0
$903$	7.94337	0.264339
$904$	0	0
$905$	0	0
$906$	0	0
$907$	12.5356	0.416239	0.208120	−	0.978103i	$-0.433266\pi$
$907$	12.5356	0.416239	0.208120	+	0.978103i	$0.433266\pi$
$908$	0	0
$909$	16.6558	0.552437
$910$	0	0
$911$	34.1877	1.13269	0.566344	−	0.824169i	$-0.308357\pi$
$911$	34.1877	1.13269	0.566344	+	0.824169i	$0.308357\pi$
$912$	0	0
$913$	5.68303	0.188081
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−7.05084	−0.232839
$918$	0	0
$919$	−23.6793	−0.781109	−0.390554	−	0.920580i	$-0.627717\pi$
$919$	−23.6793	−0.781109	−0.390554	+	0.920580i	$0.627717\pi$
$920$	0	0
$921$	13.7387	0.452704
$922$	0	0
$923$	29.3062	0.964625
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−16.6245	−0.546020
$928$	0	0
$929$	−9.48509	−0.311196	−0.155598	−	0.987820i	$-0.549730\pi$
$929$	−9.48509	−0.311196	−0.155598	+	0.987820i	$0.549730\pi$
$930$	0	0
$931$	−31.8813	−1.04487
$932$	0	0
$933$	−20.7593	−0.679629
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−40.5790	−1.32566	−0.662829	−	0.748771i	$-0.730644\pi$
$937$	−40.5790	−1.32566	−0.662829	+	0.748771i	$0.730644\pi$
$938$	0	0
$939$	8.03458	0.262198
$940$	0	0
$941$	8.37604	0.273051	0.136526	−	0.990637i	$-0.456406\pi$
$941$	8.37604	0.273051	0.136526	+	0.990637i	$0.456406\pi$
$942$	0	0
$943$	−81.9429	−2.66843
$944$	0	0
$945$	0	0
$946$	0	0
$947$	37.3750	1.21453	0.607263	−	0.794501i	$-0.292268\pi$
$947$	37.3750	1.21453	0.607263	+	0.794501i	$0.292268\pi$
$948$	0	0
$949$	29.9740	0.972998
$950$	0	0
$951$	−19.3076	−0.626091
$952$	0	0
$953$	15.5258	0.502931	0.251465	−	0.967866i	$-0.419088\pi$
$953$	15.5258	0.502931	0.251465	+	0.967866i	$0.419088\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	44.1035	1.42566
$958$	0	0
$959$	7.88714	0.254689
$960$	0	0
$961$	45.5963	1.47085
$962$	0	0
$963$	−14.7692	−0.475931
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−48.6865	−1.56565	−0.782826	−	0.622240i	$-0.786223\pi$
$967$	−48.6865	−1.56565	−0.782826	+	0.622240i	$0.786223\pi$
$968$	0	0
$969$	−14.5137	−0.466248
$970$	0	0
$971$	−14.4960	−0.465199	−0.232599	−	0.972573i	$-0.574723\pi$
$971$	−14.4960	−0.465199	−0.232599	+	0.972573i	$0.574723\pi$
$972$	0	0
$973$	8.38843	0.268921
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−7.35164	−0.235200	−0.117600	−	0.993061i	$-0.537520\pi$
$977$	−7.35164	−0.235200	−0.117600	+	0.993061i	$0.537520\pi$
$978$	0	0
$979$	30.5879	0.977595
$980$	0	0
$981$	−18.1729	−0.580215
$982$	0	0
$983$	−32.6724	−1.04209	−0.521044	−	0.853530i	$-0.674457\pi$
$983$	−32.6724	−1.04209	−0.521044	+	0.853530i	$0.674457\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−5.69335	−0.181221
$988$	0	0
$989$	24.3112	0.773050
$990$	0	0
$991$	−18.7552	−0.595780	−0.297890	−	0.954600i	$-0.596283\pi$
$991$	−18.7552	−0.595780	−0.297890	+	0.954600i	$0.596283\pi$
$992$	0	0
$993$	44.4976	1.41209
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−3.42813	−0.108570	−0.0542850	−	0.998525i	$-0.517288\pi$
$997$	−3.42813	−0.108570	−0.0542850	+	0.998525i	$0.517288\pi$
$998$	0	0
$999$	−1.50386	−0.0475801

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8000.2.a.bo.1.3		4
4.3	odd	2		8000.2.a.bd.1.2		4
5.4	even	2		8000.2.a.be.1.2		4
8.3	odd	2		1000.2.a.g.1.3	yes	4
8.5	even	2		2000.2.a.n.1.2		4
20.19	odd	2		8000.2.a.bn.1.3		4
24.11	even	2		9000.2.a.bb.1.2		4
40.3	even	4		1000.2.c.c.249.6		8
40.13	odd	4		2000.2.c.i.1249.3		8
40.19	odd	2		1000.2.a.f.1.2	✓	4
40.27	even	4		1000.2.c.c.249.3		8
40.29	even	2		2000.2.a.q.1.3		4
40.37	odd	4		2000.2.c.i.1249.6		8
120.59	even	2		9000.2.a.q.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1000.2.a.f.1.2	✓	4	40.19	odd	2
1000.2.a.g.1.3	yes	4	8.3	odd	2
1000.2.c.c.249.3		8	40.27	even	4
1000.2.c.c.249.6		8	40.3	even	4
2000.2.a.n.1.2		4	8.5	even	2
2000.2.a.q.1.3		4	40.29	even	2
2000.2.c.i.1249.3		8	40.13	odd	4
2000.2.c.i.1249.6		8	40.37	odd	4
8000.2.a.bd.1.2		4	4.3	odd	2
8000.2.a.be.1.2		4	5.4	even	2
8000.2.a.bn.1.3		4	20.19	odd	2
8000.2.a.bo.1.3		4	1.1	even	1	trivial
9000.2.a.q.1.3		4	120.59	even	2
9000.2.a.bb.1.2		4	24.11	even	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:	\( N \)	\(=\)	\( 8000 = 2^{6} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8000.a (trivial)

\(f(q)\)	\(=\)	\(q+2.39867 q^{3} -1.13558 q^{7} +2.75361 q^{9} +O(q^{10})\)	\(q+2.39867 q^{3} -1.13558 q^{7} +2.75361 q^{9} -2.01670 q^{11} -2.26309 q^{13} -1.08379 q^{17} +5.58295 q^{19} -2.72387 q^{21} -8.33656 q^{23} -0.591009 q^{27} -9.11719 q^{29} +8.75193 q^{31} -4.83740 q^{33} +2.54457 q^{37} -5.42841 q^{39} +9.82934 q^{41} -2.91621 q^{43} +2.09017 q^{47} -5.71047 q^{49} -2.59965 q^{51} +10.5326 q^{53} +13.3916 q^{57} +5.53424 q^{59} +1.63474 q^{61} -3.12693 q^{63} -10.5844 q^{67} -19.9966 q^{69} -12.9496 q^{71} -13.2447 q^{73} +2.29012 q^{77} -6.84604 q^{79} -9.67846 q^{81} -2.81798 q^{83} -21.8691 q^{87} -15.1673 q^{89} +2.56991 q^{91} +20.9930 q^{93} -18.2591 q^{97} -5.55321 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{3} - 9 q^{7} + 11 q^{9}+O(q^{10}) \) 4 * q + q^3 - 9 * q^7 + 11 * q^9	\( 4 q + q^{3} - 9 q^{7} + 11 q^{9} + 5 q^{11} + 4 q^{13} - 4 q^{17} - 3 q^{21} - 11 q^{23} - 2 q^{27} - 10 q^{29} - 9 q^{31} - 19 q^{33} + 15 q^{37} - 21 q^{39} + 17 q^{41} - 12 q^{43} - 14 q^{47} + 17 q^{49} + 25 q^{51} - 6 q^{53} - 9 q^{57} + 18 q^{59} - 11 q^{61} - 52 q^{63} + q^{67} + 8 q^{69} - 26 q^{71} - 9 q^{73} - 8 q^{77} + 8 q^{79} - 4 q^{81} - 12 q^{83} - 17 q^{87} + 5 q^{89} - 33 q^{91} + 30 q^{93} - 29 q^{97} + 8 q^{99}+O(q^{100}) \) 4 * q + q^3 - 9 * q^7 + 11 * q^9 + 5 * q^11 + 4 * q^13 - 4 * q^17 - 3 * q^21 - 11 * q^23 - 2 * q^27 - 10 * q^29 - 9 * q^31 - 19 * q^33 + 15 * q^37 - 21 * q^39 + 17 * q^41 - 12 * q^43 - 14 * q^47 + 17 * q^49 + 25 * q^51 - 6 * q^53 - 9 * q^57 + 18 * q^59 - 11 * q^61 - 52 * q^63 + q^67 + 8 * q^69 - 26 * q^71 - 9 * q^73 - 8 * q^77 + 8 * q^79 - 4 * q^81 - 12 * q^83 - 17 * q^87 + 5 * q^89 - 33 * q^91 + 30 * q^93 - 29 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more