LMFDB - Embedded newform 8037.2.a.w.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8037,2,Mod(1,8037)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8037, 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("8037.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8037 = 3^{2} \cdot 19 \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8037.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:	$64.1757681045$
Analytic rank:	$0$
Dimension:	$34$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	8037.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.77726 q^{2} +1.15866 q^{4} +0.968051 q^{5} +3.87038 q^{7} +1.49528 q^{8} +O(q^{10})$	\(q-1.77726 q^{2} +1.15866 q^{4} +0.968051 q^{5} +3.87038 q^{7} +1.49528 q^{8} -1.72048 q^{10} +4.37099 q^{11} +4.37159 q^{13} -6.87867 q^{14} -4.97483 q^{16} +2.51730 q^{17} -1.00000 q^{19} +1.12164 q^{20} -7.76839 q^{22} -1.18234 q^{23} -4.06288 q^{25} -7.76946 q^{26} +4.48444 q^{28} -6.43589 q^{29} -7.58350 q^{31} +5.85100 q^{32} -4.47389 q^{34} +3.74672 q^{35} +0.196855 q^{37} +1.77726 q^{38} +1.44751 q^{40} +4.78962 q^{41} -6.40222 q^{43} +5.06449 q^{44} +2.10133 q^{46} +1.00000 q^{47} +7.97981 q^{49} +7.22079 q^{50} +5.06518 q^{52} +6.24829 q^{53} +4.23134 q^{55} +5.78731 q^{56} +11.4383 q^{58} +9.83955 q^{59} -2.75133 q^{61} +13.4779 q^{62} -0.449105 q^{64} +4.23192 q^{65} +1.53906 q^{67} +2.91669 q^{68} -6.65891 q^{70} -0.857468 q^{71} +0.676614 q^{73} -0.349862 q^{74} -1.15866 q^{76} +16.9174 q^{77} +16.8507 q^{79} -4.81589 q^{80} -8.51240 q^{82} +3.62077 q^{83} +2.43687 q^{85} +11.3784 q^{86} +6.53587 q^{88} -2.53519 q^{89} +16.9197 q^{91} -1.36993 q^{92} -1.77726 q^{94} -0.968051 q^{95} -6.12936 q^{97} -14.1822 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 34 q + 5 q^{2} + 31 q^{4} + 6 q^{5} + 15 q^{8}+O(q^{10}) $ 34 * q + 5 * q^2 + 31 * q^4 + 6 * q^5 + 15 * q^8	$ 34 q + 5 q^{2} + 31 q^{4} + 6 q^{5} + 15 q^{8} + 4 q^{10} + 6 q^{11} + 2 q^{13} + 12 q^{14} + 21 q^{16} + 4 q^{17} - 34 q^{19} + 20 q^{20} - 8 q^{22} + 26 q^{23} + 32 q^{25} + 29 q^{26} - 4 q^{28} + 14 q^{29} + 2 q^{31} + 35 q^{32} - 18 q^{34} + 50 q^{35} - 10 q^{37} - 5 q^{38} + 17 q^{40} + 18 q^{41} + 6 q^{43} + 6 q^{44} + 18 q^{46} + 34 q^{47} + 28 q^{49} + 41 q^{50} + 10 q^{52} + 40 q^{53} - 8 q^{55} + 76 q^{56} + 4 q^{58} + 62 q^{59} - 2 q^{61} + 50 q^{62} + 11 q^{64} + 32 q^{65} + 20 q^{67} + 28 q^{68} + 22 q^{70} + 52 q^{71} - 8 q^{73} + 10 q^{74} - 31 q^{76} + 36 q^{77} - 12 q^{79} + 92 q^{80} + 10 q^{82} + 82 q^{83} - 4 q^{85} + 40 q^{86} - 16 q^{88} + 58 q^{89} + 100 q^{92} + 5 q^{94} - 6 q^{95} - 6 q^{97} + 45 q^{98}+O(q^{100}) $ 34 * q + 5 * q^2 + 31 * q^4 + 6 * q^5 + 15 * q^8 + 4 * q^10 + 6 * q^11 + 2 * q^13 + 12 * q^14 + 21 * q^16 + 4 * q^17 - 34 * q^19 + 20 * q^20 - 8 * q^22 + 26 * q^23 + 32 * q^25 + 29 * q^26 - 4 * q^28 + 14 * q^29 + 2 * q^31 + 35 * q^32 - 18 * q^34 + 50 * q^35 - 10 * q^37 - 5 * q^38 + 17 * q^40 + 18 * q^41 + 6 * q^43 + 6 * q^44 + 18 * q^46 + 34 * q^47 + 28 * q^49 + 41 * q^50 + 10 * q^52 + 40 * q^53 - 8 * q^55 + 76 * q^56 + 4 * q^58 + 62 * q^59 - 2 * q^61 + 50 * q^62 + 11 * q^64 + 32 * q^65 + 20 * q^67 + 28 * q^68 + 22 * q^70 + 52 * q^71 - 8 * q^73 + 10 * q^74 - 31 * q^76 + 36 * q^77 - 12 * q^79 + 92 * q^80 + 10 * q^82 + 82 * q^83 - 4 * q^85 + 40 * q^86 - 16 * q^88 + 58 * q^89 + 100 * q^92 + 5 * q^94 - 6 * q^95 - 6 * q^97 + 45 * q^98

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 $

$2$	−1.77726	−1.25671	−0.628357	−	0.777925i	$-0.716272\pi$
$2$	−1.77726	−1.25671	−0.628357	+	0.777925i	$0.716272\pi$
$3$	0	0
$3$	0	0
$4$	1.15866	0.579329
$5$	0.968051	0.432926	0.216463	−	0.976291i	$-0.430548\pi$
$5$	0.968051	0.432926	0.216463	+	0.976291i	$0.430548\pi$
$6$	0	0
$7$	3.87038	1.46286	0.731432	−	0.681914i	$-0.238852\pi$
$7$	3.87038	1.46286	0.731432	+	0.681914i	$0.238852\pi$
$8$	1.49528	0.528663
$9$	0	0
$10$	−1.72048	−0.544064
$11$	4.37099	1.31790	0.658952	−	0.752185i	$-0.271000\pi$
$11$	4.37099	1.31790	0.658952	+	0.752185i	$0.271000\pi$
$12$	0	0
$13$	4.37159	1.21246	0.606230	−	0.795289i	$-0.292681\pi$
$13$	4.37159	1.21246	0.606230	+	0.795289i	$0.292681\pi$
$14$	−6.87867	−1.83840
$15$	0	0
$16$	−4.97483	−1.24371
$17$	2.51730	0.610534	0.305267	−	0.952267i	$-0.401254\pi$
$17$	2.51730	0.610534	0.305267	+	0.952267i	$0.401254\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	1.12164	0.250806
$21$	0	0
$22$	−7.76839	−1.65623
$23$	−1.18234	−0.246535	−0.123267	−	0.992373i	$-0.539337\pi$
$23$	−1.18234	−0.246535	−0.123267	+	0.992373i	$0.539337\pi$
$24$	0	0
$25$	−4.06288	−0.812575
$26$	−7.76946	−1.52372
$27$	0	0
$28$	4.48444	0.847480
$29$	−6.43589	−1.19512	−0.597558	−	0.801826i	$-0.703862\pi$
$29$	−6.43589	−1.19512	−0.597558	+	0.801826i	$0.703862\pi$
$30$	0	0
$31$	−7.58350	−1.36204	−0.681018	−	0.732267i	$-0.738462\pi$
$31$	−7.58350	−1.36204	−0.681018	+	0.732267i	$0.738462\pi$
$32$	5.85100	1.03432
$33$	0	0
$34$	−4.47389	−0.767266
$35$	3.74672	0.633312
$36$	0	0
$37$	0.196855	0.0323627	0.0161813	−	0.999869i	$-0.494849\pi$
$37$	0.196855	0.0323627	0.0161813	+	0.999869i	$0.494849\pi$
$38$	1.77726	0.288310
$39$	0	0
$40$	1.44751	0.228872
$41$	4.78962	0.748013	0.374006	−	0.927426i	$-0.377984\pi$
$41$	4.78962	0.748013	0.374006	+	0.927426i	$0.377984\pi$
$42$	0	0
$43$	−6.40222	−0.976330	−0.488165	−	0.872751i	$-0.662333\pi$
$43$	−6.40222	−0.976330	−0.488165	+	0.872751i	$0.662333\pi$
$44$	5.06449	0.763500
$45$	0	0
$46$	2.10133	0.309824
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	0	0
$49$	7.97981	1.13997
$50$	7.22079	1.02117
$51$	0	0
$52$	5.06518	0.702414
$53$	6.24829	0.858268	0.429134	−	0.903241i	$-0.358819\pi$
$53$	6.24829	0.858268	0.429134	+	0.903241i	$0.358819\pi$
$54$	0	0
$55$	4.23134	0.570554
$56$	5.78731	0.773362
$57$	0	0
$58$	11.4383	1.50192
$59$	9.83955	1.28100	0.640500	−	0.767958i	$-0.278727\pi$
$59$	9.83955	1.28100	0.640500	+	0.767958i	$0.278727\pi$
$60$	0	0
$61$	−2.75133	−0.352271	−0.176136	−	0.984366i	$-0.556360\pi$
$61$	−2.75133	−0.352271	−0.176136	+	0.984366i	$0.556360\pi$
$62$	13.4779	1.71169
$63$	0	0
$64$	−0.449105	−0.0561381
$65$	4.23192	0.524905
$66$	0	0
$67$	1.53906	0.188026	0.0940132	−	0.995571i	$-0.470030\pi$
$67$	1.53906	0.188026	0.0940132	+	0.995571i	$0.470030\pi$
$68$	2.91669	0.353700
$69$	0	0
$70$	−6.65891	−0.795891
$71$	−0.857468	−0.101763	−0.0508813	−	0.998705i	$-0.516203\pi$
$71$	−0.857468	−0.101763	−0.0508813	+	0.998705i	$0.516203\pi$
$72$	0	0
$73$	0.676614	0.0791917	0.0395958	−	0.999216i	$-0.487393\pi$
$73$	0.676614	0.0791917	0.0395958	+	0.999216i	$0.487393\pi$
$74$	−0.349862	−0.0406706
$75$	0	0
$76$	−1.15866	−0.132907
$77$	16.9174	1.92791
$78$	0	0
$79$	16.8507	1.89585	0.947926	−	0.318492i	$-0.103176\pi$
$79$	16.8507	1.89585	0.947926	+	0.318492i	$0.103176\pi$
$80$	−4.81589	−0.538433
$81$	0	0
$82$	−8.51240	−0.940038
$83$	3.62077	0.397431	0.198716	−	0.980057i	$-0.436323\pi$
$83$	3.62077	0.397431	0.198716	+	0.980057i	$0.436323\pi$
$84$	0	0
$85$	2.43687	0.264316
$86$	11.3784	1.22697
$87$	0	0
$88$	6.53587	0.696726
$89$	−2.53519	−0.268730	−0.134365	−	0.990932i	$-0.542899\pi$
$89$	−2.53519	−0.268730	−0.134365	+	0.990932i	$0.542899\pi$
$90$	0	0
$91$	16.9197	1.77367
$92$	−1.36993	−0.142825
$93$	0	0
$94$	−1.77726	−0.183311
$95$	−0.968051	−0.0993200
$96$	0	0
$97$	−6.12936	−0.622342	−0.311171	−	0.950354i	$-0.600721\pi$
$97$	−6.12936	−0.622342	−0.311171	+	0.950354i	$0.600721\pi$
$98$	−14.1822	−1.43262
$99$	0	0
$100$	−4.70749	−0.470749
$101$	2.65436	0.264119	0.132060	−	0.991242i	$-0.457841\pi$
$101$	2.65436	0.264119	0.132060	+	0.991242i	$0.457841\pi$
$102$	0	0
$103$	0.272788	0.0268786	0.0134393	−	0.999910i	$-0.495722\pi$
$103$	0.272788	0.0268786	0.0134393	+	0.999910i	$0.495722\pi$
$104$	6.53677	0.640983
$105$	0	0
$106$	−11.1048	−1.07860
$107$	9.21069	0.890431	0.445216	−	0.895423i	$-0.353127\pi$
$107$	9.21069	0.890431	0.445216	+	0.895423i	$0.353127\pi$
$108$	0	0
$109$	10.9973	1.05335	0.526676	−	0.850066i	$-0.323438\pi$
$109$	10.9973	1.05335	0.526676	+	0.850066i	$0.323438\pi$
$110$	−7.52020	−0.717023
$111$	0	0
$112$	−19.2545	−1.81937
$113$	6.59738	0.620629	0.310315	−	0.950634i	$-0.399566\pi$
$113$	6.59738	0.620629	0.310315	+	0.950634i	$0.399566\pi$
$114$	0	0
$115$	−1.14456	−0.106731
$116$	−7.45700	−0.692365
$117$	0	0
$118$	−17.4875	−1.60985
$119$	9.74288	0.893129
$120$	0	0
$121$	8.10556	0.736869
$122$	4.88982	0.442704
$123$	0	0
$124$	−8.78668	−0.789067
$125$	−8.77333	−0.784710
$126$	0	0
$127$	−14.8840	−1.32074	−0.660369	−	0.750941i	$-0.729600\pi$
$127$	−14.8840	−1.32074	−0.660369	+	0.750941i	$0.729600\pi$
$128$	−10.9038	−0.963771
$129$	0	0
$130$	−7.52123	−0.659656
$131$	0.329386	0.0287786	0.0143893	−	0.999896i	$-0.495420\pi$
$131$	0.329386	0.0287786	0.0143893	+	0.999896i	$0.495420\pi$
$132$	0	0
$133$	−3.87038	−0.335604
$134$	−2.73532	−0.236295
$135$	0	0
$136$	3.76407	0.322766
$137$	9.92883	0.848278	0.424139	−	0.905597i	$-0.360577\pi$
$137$	9.92883	0.848278	0.424139	+	0.905597i	$0.360577\pi$
$138$	0	0
$139$	6.46638	0.548471	0.274236	−	0.961663i	$-0.411575\pi$
$139$	6.46638	0.548471	0.274236	+	0.961663i	$0.411575\pi$
$140$	4.34117	0.366896
$141$	0	0
$142$	1.52394	0.127887
$143$	19.1082	1.59791
$144$	0	0
$145$	−6.23027	−0.517396
$146$	−1.20252	−0.0995213
$147$	0	0
$148$	0.228087	0.0187487
$149$	−1.88233	−0.154206	−0.0771031	−	0.997023i	$-0.524567\pi$
$149$	−1.88233	−0.154206	−0.0771031	+	0.997023i	$0.524567\pi$
$150$	0	0
$151$	3.38163	0.275193	0.137596	−	0.990488i	$-0.456062\pi$
$151$	3.38163	0.275193	0.137596	+	0.990488i	$0.456062\pi$
$152$	−1.49528	−0.121284
$153$	0	0
$154$	−30.0666	−2.42284
$155$	−7.34121	−0.589660
$156$	0	0
$157$	−18.0563	−1.44105	−0.720524	−	0.693430i	$-0.756098\pi$
$157$	−18.0563	−1.44105	−0.720524	+	0.693430i	$0.756098\pi$
$158$	−29.9481	−2.38254
$159$	0	0
$160$	5.66407	0.447784
$161$	−4.57610	−0.360647
$162$	0	0
$163$	−20.7737	−1.62712	−0.813560	−	0.581480i	$-0.802474\pi$
$163$	−20.7737	−1.62712	−0.813560	+	0.581480i	$0.802474\pi$
$164$	5.54953	0.433346
$165$	0	0
$166$	−6.43506	−0.499457
$167$	6.92342	0.535750	0.267875	−	0.963454i	$-0.413679\pi$
$167$	6.92342	0.535750	0.267875	+	0.963454i	$0.413679\pi$
$168$	0	0
$169$	6.11079	0.470061
$170$	−4.33096	−0.332169
$171$	0	0
$172$	−7.41799	−0.565616
$173$	11.7671	0.894639	0.447320	−	0.894374i	$-0.352379\pi$
$173$	11.7671	0.894639	0.447320	+	0.894374i	$0.352379\pi$
$174$	0	0
$175$	−15.7249	−1.18869
$176$	−21.7449	−1.63909
$177$	0	0
$178$	4.50570	0.337716
$179$	15.1795	1.13457	0.567285	−	0.823522i	$-0.307994\pi$
$179$	15.1795	1.13457	0.567285	+	0.823522i	$0.307994\pi$
$180$	0	0
$181$	18.1672	1.35036	0.675178	−	0.737655i	$-0.264067\pi$
$181$	18.1672	1.35036	0.675178	+	0.737655i	$0.264067\pi$
$182$	−30.0707	−2.22899
$183$	0	0
$184$	−1.76793	−0.130334
$185$	0.190565	0.0140106
$186$	0	0
$187$	11.0031	0.804625
$188$	1.15866	0.0845039
$189$	0	0
$190$	1.72048	0.124817
$191$	20.3462	1.47220	0.736098	−	0.676875i	$-0.236666\pi$
$191$	20.3462	1.47220	0.736098	+	0.676875i	$0.236666\pi$
$192$	0	0
$193$	16.8157	1.21042	0.605210	−	0.796066i	$-0.293089\pi$
$193$	16.8157	1.21042	0.605210	+	0.796066i	$0.293089\pi$
$194$	10.8935	0.782106
$195$	0	0
$196$	9.24588	0.660420
$197$	9.25777	0.659589	0.329794	−	0.944053i	$-0.393021\pi$
$197$	9.25777	0.659589	0.329794	+	0.944053i	$0.393021\pi$
$198$	0	0
$199$	−7.75695	−0.549876	−0.274938	−	0.961462i	$-0.588657\pi$
$199$	−7.75695	−0.549876	−0.274938	+	0.961462i	$0.588657\pi$
$200$	−6.07515	−0.429578
$201$	0	0
$202$	−4.71750	−0.331922
$203$	−24.9093	−1.74829
$204$	0	0
$205$	4.63660	0.323834
$206$	−0.484816	−0.0337787
$207$	0	0
$208$	−21.7479	−1.50795
$209$	−4.37099	−0.302348
$210$	0	0
$211$	−9.33733	−0.642808	−0.321404	−	0.946942i	$-0.604155\pi$
$211$	−9.33733	−0.642808	−0.321404	+	0.946942i	$0.604155\pi$
$212$	7.23963	0.497220
$213$	0	0
$214$	−16.3698	−1.11902
$215$	−6.19768	−0.422678
$216$	0	0
$217$	−29.3510	−1.99247
$218$	−19.5451	−1.32376
$219$	0	0
$220$	4.90268	0.330539
$221$	11.0046	0.740248
$222$	0	0
$223$	4.49035	0.300696	0.150348	−	0.988633i	$-0.451961\pi$
$223$	4.49035	0.300696	0.150348	+	0.988633i	$0.451961\pi$
$224$	22.6456	1.51307
$225$	0	0
$226$	−11.7253	−0.779953
$227$	16.2797	1.08052	0.540262	−	0.841497i	$-0.318325\pi$
$227$	16.2797	1.08052	0.540262	+	0.841497i	$0.318325\pi$
$228$	0	0
$229$	11.1973	0.739940	0.369970	−	0.929044i	$-0.379368\pi$
$229$	11.1973	0.739940	0.369970	+	0.929044i	$0.379368\pi$
$230$	2.03419	0.134131
$231$	0	0
$232$	−9.62349	−0.631813
$233$	3.99762	0.261893	0.130947	−	0.991389i	$-0.458198\pi$
$233$	3.99762	0.261893	0.130947	+	0.991389i	$0.458198\pi$
$234$	0	0
$235$	0.968051	0.0631487
$236$	11.4007	0.742121
$237$	0	0
$238$	−17.3156	−1.12241
$239$	1.76541	0.114195	0.0570974	−	0.998369i	$-0.481815\pi$
$239$	1.76541	0.114195	0.0570974	+	0.998369i	$0.481815\pi$
$240$	0	0
$241$	4.21311	0.271390	0.135695	−	0.990751i	$-0.456673\pi$
$241$	4.21311	0.271390	0.135695	+	0.990751i	$0.456673\pi$
$242$	−14.4057	−0.926034
$243$	0	0
$244$	−3.18785	−0.204081
$245$	7.72487	0.493524
$246$	0	0
$247$	−4.37159	−0.278158
$248$	−11.3395	−0.720058
$249$	0	0
$250$	15.5925	0.986156
$251$	−9.43115	−0.595289	−0.297644	−	0.954677i	$-0.596201\pi$
$251$	−9.43115	−0.595289	−0.297644	+	0.954677i	$0.596201\pi$
$252$	0	0
$253$	−5.16799	−0.324909
$254$	26.4527	1.65979
$255$	0	0
$256$	20.2772	1.26732
$257$	6.18872	0.386041	0.193021	−	0.981195i	$-0.438172\pi$
$257$	6.18872	0.386041	0.193021	+	0.981195i	$0.438172\pi$
$258$	0	0
$259$	0.761901	0.0473423
$260$	4.90335	0.304093
$261$	0	0
$262$	−0.585405	−0.0361664
$263$	8.55941	0.527796	0.263898	−	0.964551i	$-0.414992\pi$
$263$	8.55941	0.527796	0.263898	+	0.964551i	$0.414992\pi$
$264$	0	0
$265$	6.04866	0.371566
$266$	6.87867	0.421758
$267$	0	0
$268$	1.78325	0.108929
$269$	−4.73282	−0.288565	−0.144283	−	0.989537i	$-0.546087\pi$
$269$	−4.73282	−0.288565	−0.144283	+	0.989537i	$0.546087\pi$
$270$	0	0
$271$	26.3893	1.60304	0.801519	−	0.597970i	$-0.204026\pi$
$271$	26.3893	1.60304	0.801519	+	0.597970i	$0.204026\pi$
$272$	−12.5231	−0.759325
$273$	0	0
$274$	−17.6461	−1.06604
$275$	−17.7588	−1.07090
$276$	0	0
$277$	−2.33233	−0.140136	−0.0700679	−	0.997542i	$-0.522322\pi$
$277$	−2.33233	−0.140136	−0.0700679	+	0.997542i	$0.522322\pi$
$278$	−11.4924	−0.689271
$279$	0	0
$280$	5.60241	0.334808
$281$	−10.3004	−0.614469	−0.307235	−	0.951634i	$-0.599404\pi$
$281$	−10.3004	−0.614469	−0.307235	+	0.951634i	$0.599404\pi$
$282$	0	0
$283$	−7.84324	−0.466232	−0.233116	−	0.972449i	$-0.574892\pi$
$283$	−7.84324	−0.466232	−0.233116	+	0.972449i	$0.574892\pi$
$284$	−0.993512	−0.0589541
$285$	0	0
$286$	−33.9602	−2.00811
$287$	18.5376	1.09424
$288$	0	0
$289$	−10.6632	−0.627248
$290$	11.0728	0.650219
$291$	0	0
$292$	0.783965	0.0458781
$293$	5.19708	0.303616	0.151808	−	0.988410i	$-0.451490\pi$
$293$	5.19708	0.303616	0.151808	+	0.988410i	$0.451490\pi$
$294$	0	0
$295$	9.52519	0.554578
$296$	0.294354	0.0171090
$297$	0	0
$298$	3.34539	0.193793
$299$	−5.16870	−0.298914
$300$	0	0
$301$	−24.7790	−1.42824
$302$	−6.01003	−0.345839
$303$	0	0
$304$	4.97483	0.285326
$305$	−2.66342	−0.152507
$306$	0	0
$307$	−0.304242	−0.0173640	−0.00868199	−	0.999962i	$-0.502764\pi$
$307$	−0.304242	−0.0173640	−0.00868199	+	0.999962i	$0.502764\pi$
$308$	19.6015	1.11690
$309$	0	0
$310$	13.0473	0.741034
$311$	4.25162	0.241087	0.120544	−	0.992708i	$-0.461536\pi$
$311$	4.25162	0.241087	0.120544	+	0.992708i	$0.461536\pi$
$312$	0	0
$313$	0.632572	0.0357551	0.0178775	−	0.999840i	$-0.494309\pi$
$313$	0.632572	0.0357551	0.0178775	+	0.999840i	$0.494309\pi$
$314$	32.0907	1.81098
$315$	0	0
$316$	19.5242	1.09832
$317$	−22.2270	−1.24839	−0.624197	−	0.781267i	$-0.714574\pi$
$317$	−22.2270	−1.24839	−0.624197	+	0.781267i	$0.714574\pi$
$318$	0	0
$319$	−28.1312	−1.57505
$320$	−0.434757	−0.0243036
$321$	0	0
$322$	8.13292	0.453230
$323$	−2.51730	−0.140066
$324$	0	0
$325$	−17.7612	−0.985216
$326$	36.9203	2.04483
$327$	0	0
$328$	7.16184	0.395446
$329$	3.87038	0.213381
$330$	0	0
$331$	25.9818	1.42809	0.714043	−	0.700102i	$-0.246862\pi$
$331$	25.9818	1.42809	0.714043	+	0.700102i	$0.246862\pi$
$332$	4.19524	0.230244
$333$	0	0
$334$	−12.3047	−0.673284
$335$	1.48989	0.0814015
$336$	0	0
$337$	20.2715	1.10426	0.552130	−	0.833758i	$-0.313815\pi$
$337$	20.2715	1.10426	0.552130	+	0.833758i	$0.313815\pi$
$338$	−10.8605	−0.590732
$339$	0	0
$340$	2.82350	0.153126
$341$	−33.1474	−1.79503
$342$	0	0
$343$	3.79224	0.204762
$344$	−9.57314	−0.516149
$345$	0	0
$346$	−20.9133	−1.12431
$347$	10.3871	0.557609	0.278804	−	0.960348i	$-0.410062\pi$
$347$	10.3871	0.557609	0.278804	+	0.960348i	$0.410062\pi$
$348$	0	0
$349$	8.07120	0.432041	0.216021	−	0.976389i	$-0.430692\pi$
$349$	8.07120	0.432041	0.216021	+	0.976389i	$0.430692\pi$
$350$	27.9472	1.49384
$351$	0	0
$352$	25.5747	1.36313
$353$	−21.1385	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$353$	−21.1385	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$354$	0	0
$355$	−0.830073	−0.0440557
$356$	−2.93742	−0.155683
$357$	0	0
$358$	−26.9779	−1.42583
$359$	−8.48173	−0.447648	−0.223824	−	0.974630i	$-0.571854\pi$
$359$	−8.48173	−0.447648	−0.223824	+	0.974630i	$0.571854\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−32.2878	−1.69701
$363$	0	0
$364$	19.6041	1.02754
$365$	0.654997	0.0342841
$366$	0	0
$367$	−4.98796	−0.260369	−0.130185	−	0.991490i	$-0.541557\pi$
$367$	−4.98796	−0.260369	−0.130185	+	0.991490i	$0.541557\pi$
$368$	5.88193	0.306617
$369$	0	0
$370$	−0.338684	−0.0176074
$371$	24.1832	1.25553
$372$	0	0
$373$	10.9217	0.565507	0.282753	−	0.959193i	$-0.408752\pi$
$373$	10.9217	0.565507	0.282753	+	0.959193i	$0.408752\pi$
$374$	−19.5553	−1.01118
$375$	0	0
$376$	1.49528	0.0771134
$377$	−28.1351	−1.44903
$378$	0	0
$379$	−19.6994	−1.01189	−0.505946	−	0.862565i	$-0.668856\pi$
$379$	−19.6994	−1.01189	−0.505946	+	0.862565i	$0.668856\pi$
$380$	−1.12164	−0.0575390
$381$	0	0
$382$	−36.1604	−1.85013
$383$	−11.6918	−0.597425	−0.298713	−	0.954343i	$-0.596557\pi$
$383$	−11.6918	−0.597425	−0.298713	+	0.954343i	$0.596557\pi$
$384$	0	0
$385$	16.3769	0.834644
$386$	−29.8859	−1.52115
$387$	0	0
$388$	−7.10184	−0.360541
$389$	−19.7304	−1.00037	−0.500186	−	0.865918i	$-0.666735\pi$
$389$	−19.7304	−1.00037	−0.500186	+	0.865918i	$0.666735\pi$
$390$	0	0
$391$	−2.97630	−0.150518
$392$	11.9321	0.602661
$393$	0	0
$394$	−16.4535	−0.828914
$395$	16.3123	0.820763
$396$	0	0
$397$	−24.3262	−1.22090	−0.610449	−	0.792056i	$-0.709011\pi$
$397$	−24.3262	−1.22090	−0.610449	+	0.792056i	$0.709011\pi$
$398$	13.7861	0.691036
$399$	0	0
$400$	20.2121	1.01061
$401$	27.9016	1.39334	0.696670	−	0.717392i	$-0.254664\pi$
$401$	27.9016	1.39334	0.696670	+	0.717392i	$0.254664\pi$
$402$	0	0
$403$	−33.1519	−1.65141
$404$	3.07550	0.153012
$405$	0	0
$406$	44.2704	2.19710
$407$	0.860450	0.0426509
$408$	0	0
$409$	6.55231	0.323991	0.161995	−	0.986792i	$-0.448207\pi$
$409$	6.55231	0.323991	0.161995	+	0.986792i	$0.448207\pi$
$410$	−8.24044	−0.406966
$411$	0	0
$412$	0.316068	0.0155716
$413$	38.0828	1.87393
$414$	0	0
$415$	3.50509	0.172058
$416$	25.5782	1.25407
$417$	0	0
$418$	7.76839	0.379965
$419$	29.2033	1.42668	0.713338	−	0.700820i	$-0.247182\pi$
$419$	29.2033	1.42668	0.713338	+	0.700820i	$0.247182\pi$
$420$	0	0
$421$	−12.2254	−0.595827	−0.297914	−	0.954593i	$-0.596291\pi$
$421$	−12.2254	−0.595827	−0.297914	+	0.954593i	$0.596291\pi$
$422$	16.5949	0.807826
$423$	0	0
$424$	9.34296	0.453734
$425$	−10.2275	−0.496105
$426$	0	0
$427$	−10.6487	−0.515325
$428$	10.6720	0.515853
$429$	0	0
$430$	11.0149	0.531185
$431$	20.6655	0.995420	0.497710	−	0.867344i	$-0.334174\pi$
$431$	20.6655	0.995420	0.497710	+	0.867344i	$0.334174\pi$
$432$	0	0
$433$	22.6849	1.09017	0.545084	−	0.838382i	$-0.316498\pi$
$433$	22.6849	1.09017	0.545084	+	0.838382i	$0.316498\pi$
$434$	52.1644	2.50397
$435$	0	0
$436$	12.7421	0.610238
$437$	1.18234	0.0565590
$438$	0	0
$439$	4.47874	0.213759	0.106879	−	0.994272i	$-0.465914\pi$
$439$	4.47874	0.213759	0.106879	+	0.994272i	$0.465914\pi$
$440$	6.32706	0.301631
$441$	0	0
$442$	−19.5580	−0.930280
$443$	−35.0018	−1.66298	−0.831492	−	0.555536i	$-0.812513\pi$
$443$	−35.0018	−1.66298	−0.831492	+	0.555536i	$0.812513\pi$
$444$	0	0
$445$	−2.45420	−0.116340
$446$	−7.98052	−0.377889
$447$	0	0
$448$	−1.73821	−0.0821225
$449$	37.5125	1.77032	0.885162	−	0.465283i	$-0.154047\pi$
$449$	37.5125	1.77032	0.885162	+	0.465283i	$0.154047\pi$
$450$	0	0
$451$	20.9354	0.985808
$452$	7.64411	0.359549
$453$	0	0
$454$	−28.9333	−1.35791
$455$	16.3791	0.767865
$456$	0	0
$457$	28.1955	1.31893	0.659464	−	0.751736i	$-0.270783\pi$
$457$	28.1955	1.31893	0.659464	+	0.751736i	$0.270783\pi$
$458$	−19.9006	−0.929893
$459$	0	0
$460$	−1.32616	−0.0618325
$461$	21.2199	0.988310	0.494155	−	0.869374i	$-0.335478\pi$
$461$	21.2199	0.988310	0.494155	+	0.869374i	$0.335478\pi$
$462$	0	0
$463$	−11.5540	−0.536961	−0.268480	−	0.963285i	$-0.586521\pi$
$463$	−11.5540	−0.536961	−0.268480	+	0.963285i	$0.586521\pi$
$464$	32.0175	1.48637
$465$	0	0
$466$	−7.10482	−0.329125
$467$	27.8194	1.28733	0.643665	−	0.765308i	$-0.277413\pi$
$467$	27.8194	1.28733	0.643665	+	0.765308i	$0.277413\pi$
$468$	0	0
$469$	5.95675	0.275057
$470$	−1.72048	−0.0793598
$471$	0	0
$472$	14.7129	0.677217
$473$	−27.9841	−1.28671
$474$	0	0
$475$	4.06288	0.186418
$476$	11.2887	0.517415
$477$	0	0
$478$	−3.13759	−0.143510
$479$	4.95287	0.226302	0.113151	−	0.993578i	$-0.463906\pi$
$479$	4.95287	0.226302	0.113151	+	0.993578i	$0.463906\pi$
$480$	0	0
$481$	0.860567	0.0392385
$482$	−7.48780	−0.341060
$483$	0	0
$484$	9.39158	0.426890
$485$	−5.93354	−0.269428
$486$	0	0
$487$	−37.4098	−1.69520	−0.847599	−	0.530637i	$-0.821953\pi$
$487$	−37.4098	−1.69520	−0.847599	+	0.530637i	$0.821953\pi$
$488$	−4.11401	−0.186233
$489$	0	0
$490$	−13.7291	−0.620218
$491$	−28.3263	−1.27835	−0.639173	−	0.769063i	$-0.720723\pi$
$491$	−28.3263	−1.27835	−0.639173	+	0.769063i	$0.720723\pi$
$492$	0	0
$493$	−16.2010	−0.729658
$494$	7.76946	0.349564
$495$	0	0
$496$	37.7266	1.69397
$497$	−3.31872	−0.148865
$498$	0	0
$499$	19.5135	0.873544	0.436772	−	0.899572i	$-0.356122\pi$
$499$	19.5135	0.873544	0.436772	+	0.899572i	$0.356122\pi$
$500$	−10.1653	−0.454606
$501$	0	0
$502$	16.7616	0.748108
$503$	−31.8354	−1.41947	−0.709734	−	0.704470i	$-0.751185\pi$
$503$	−31.8354	−1.41947	−0.709734	+	0.704470i	$0.751185\pi$
$504$	0	0
$505$	2.56956	0.114344
$506$	9.18488	0.408318
$507$	0	0
$508$	−17.2454	−0.765142
$509$	−30.7975	−1.36507	−0.682537	−	0.730851i	$-0.739123\pi$
$509$	−30.7975	−1.36507	−0.682537	+	0.730851i	$0.739123\pi$
$510$	0	0
$511$	2.61875	0.115847
$512$	−14.2302	−0.628890
$513$	0	0
$514$	−10.9990	−0.485144
$515$	0.264073	0.0116364
$516$	0	0
$517$	4.37099	0.192236
$518$	−1.35410	−0.0594957
$519$	0	0
$520$	6.32792	0.277498
$521$	6.41188	0.280909	0.140455	−	0.990087i	$-0.455144\pi$
$521$	6.41188	0.280909	0.140455	+	0.990087i	$0.455144\pi$
$522$	0	0
$523$	−3.01476	−0.131826	−0.0659131	−	0.997825i	$-0.520996\pi$
$523$	−3.01476	−0.131826	−0.0659131	+	0.997825i	$0.520996\pi$
$524$	0.381646	0.0166723
$525$	0	0
$526$	−15.2123	−0.663289
$527$	−19.0899	−0.831569
$528$	0	0
$529$	−21.6021	−0.939221
$530$	−10.7501	−0.466952
$531$	0	0
$532$	−4.48444	−0.194425
$533$	20.9382	0.906936
$534$	0	0
$535$	8.91642	0.385491
$536$	2.30134	0.0994025
$537$	0	0
$538$	8.41146	0.362644
$539$	34.8797	1.50237
$540$	0	0
$541$	−19.3798	−0.833201	−0.416600	−	0.909090i	$-0.636779\pi$
$541$	−19.3798	−0.833201	−0.416600	+	0.909090i	$0.636779\pi$
$542$	−46.9007	−2.01456
$543$	0	0
$544$	14.7287	0.631488
$545$	10.6460	0.456023
$546$	0	0
$547$	−19.5586	−0.836264	−0.418132	−	0.908386i	$-0.637315\pi$
$547$	−19.5586	−0.836264	−0.418132	+	0.908386i	$0.637315\pi$
$548$	11.5041	0.491432
$549$	0	0
$550$	31.5620	1.34581
$551$	6.43589	0.274178
$552$	0	0
$553$	65.2185	2.77337
$554$	4.14515	0.176111
$555$	0	0
$556$	7.49232	0.317745
$557$	−8.61080	−0.364851	−0.182426	−	0.983220i	$-0.558395\pi$
$557$	−8.61080	−0.364851	−0.182426	+	0.983220i	$0.558395\pi$
$558$	0	0
$559$	−27.9879	−1.18376
$560$	−18.6393	−0.787654
$561$	0	0
$562$	18.3065	0.772212
$563$	−9.78211	−0.412267	−0.206133	−	0.978524i	$-0.566088\pi$
$563$	−9.78211	−0.412267	−0.206133	+	0.978524i	$0.566088\pi$
$564$	0	0
$565$	6.38660	0.268686
$566$	13.9395	0.585920
$567$	0	0
$568$	−1.28216	−0.0537981
$569$	−34.0605	−1.42789	−0.713944	−	0.700202i	$-0.753093\pi$
$569$	−34.0605	−1.42789	−0.713944	+	0.700202i	$0.753093\pi$
$570$	0	0
$571$	24.7091	1.03404	0.517021	−	0.855972i	$-0.327041\pi$
$571$	24.7091	1.03404	0.517021	+	0.855972i	$0.327041\pi$
$572$	22.1398	0.925714
$573$	0	0
$574$	−32.9462	−1.37515
$575$	4.80370	0.200328
$576$	0	0
$577$	34.5591	1.43872	0.719358	−	0.694640i	$-0.244436\pi$
$577$	34.5591	1.43872	0.719358	+	0.694640i	$0.244436\pi$
$578$	18.9513	0.788272
$579$	0	0
$580$	−7.21876	−0.299743
$581$	14.0137	0.581388
$582$	0	0
$583$	27.3112	1.13111
$584$	1.01173	0.0418657
$585$	0	0
$586$	−9.23656	−0.381559
$587$	−12.6220	−0.520966	−0.260483	−	0.965478i	$-0.583882\pi$
$587$	−12.6220	−0.520966	−0.260483	+	0.965478i	$0.583882\pi$
$588$	0	0
$589$	7.58350	0.312472
$590$	−16.9288	−0.696946
$591$	0	0
$592$	−0.979318	−0.0402497
$593$	10.7933	0.443229	0.221615	−	0.975134i	$-0.428867\pi$
$593$	10.7933	0.443229	0.221615	+	0.975134i	$0.428867\pi$
$594$	0	0
$595$	9.43161	0.386658
$596$	−2.18097	−0.0893362
$597$	0	0
$598$	9.18613	0.375649
$599$	−37.2191	−1.52073	−0.760366	−	0.649495i	$-0.774980\pi$
$599$	−37.2191	−1.52073	−0.760366	+	0.649495i	$0.774980\pi$
$600$	0	0
$601$	−5.99709	−0.244626	−0.122313	−	0.992492i	$-0.539031\pi$
$601$	−5.99709	−0.244626	−0.122313	+	0.992492i	$0.539031\pi$
$602$	44.0388	1.79489
$603$	0	0
$604$	3.91815	0.159427
$605$	7.84660	0.319010
$606$	0	0
$607$	44.8753	1.82143	0.910716	−	0.413033i	$-0.135531\pi$
$607$	44.8753	1.82143	0.910716	+	0.413033i	$0.135531\pi$
$608$	−5.85100	−0.237289
$609$	0	0
$610$	4.73360	0.191658
$611$	4.37159	0.176856
$612$	0	0
$613$	−19.9910	−0.807430	−0.403715	−	0.914885i	$-0.632281\pi$
$613$	−19.9910	−0.807430	−0.403715	+	0.914885i	$0.632281\pi$
$614$	0.540717	0.0218216
$615$	0	0
$616$	25.2963	1.01922
$617$	20.0628	0.807696	0.403848	−	0.914826i	$-0.367672\pi$
$617$	20.0628	0.807696	0.403848	+	0.914826i	$0.367672\pi$
$618$	0	0
$619$	16.3271	0.656243	0.328121	−	0.944636i	$-0.393585\pi$
$619$	16.3271	0.656243	0.328121	+	0.944636i	$0.393585\pi$
$620$	−8.50596	−0.341607
$621$	0	0
$622$	−7.55624	−0.302977
$623$	−9.81215	−0.393115
$624$	0	0
$625$	11.8214	0.472854
$626$	−1.12425	−0.0449339
$627$	0	0
$628$	−20.9211	−0.834841
$629$	0.495541	0.0197585
$630$	0	0
$631$	−32.7779	−1.30487	−0.652433	−	0.757846i	$-0.726252\pi$
$631$	−32.7779	−1.30487	−0.652433	+	0.757846i	$0.726252\pi$
$632$	25.1966	1.00227
$633$	0	0
$634$	39.5033	1.56887
$635$	−14.4084	−0.571781
$636$	0	0
$637$	34.8845	1.38217
$638$	49.9966	1.97938
$639$	0	0
$640$	−10.5555	−0.417241
$641$	−15.3350	−0.605696	−0.302848	−	0.953039i	$-0.597937\pi$
$641$	−15.3350	−0.605696	−0.302848	+	0.953039i	$0.597937\pi$
$642$	0	0
$643$	4.84529	0.191080	0.0955399	−	0.995426i	$-0.469542\pi$
$643$	4.84529	0.191080	0.0955399	+	0.995426i	$0.469542\pi$
$644$	−5.30213	−0.208933
$645$	0	0
$646$	4.47389	0.176023
$647$	9.94051	0.390802	0.195401	−	0.980723i	$-0.437399\pi$
$647$	9.94051	0.390802	0.195401	+	0.980723i	$0.437399\pi$
$648$	0	0
$649$	43.0086	1.68824
$650$	31.5663	1.23813
$651$	0	0
$652$	−24.0696	−0.942639
$653$	1.03534	0.0405159	0.0202580	−	0.999795i	$-0.493551\pi$
$653$	1.03534	0.0405159	0.0202580	+	0.999795i	$0.493551\pi$
$654$	0	0
$655$	0.318862	0.0124590
$656$	−23.8275	−0.930309
$657$	0	0
$658$	−6.87867	−0.268159
$659$	−40.5761	−1.58062	−0.790309	−	0.612708i	$-0.790080\pi$
$659$	−40.5761	−1.58062	−0.790309	+	0.612708i	$0.790080\pi$
$660$	0	0
$661$	−26.4679	−1.02948	−0.514741	−	0.857346i	$-0.672112\pi$
$661$	−26.4679	−1.02948	−0.514741	+	0.857346i	$0.672112\pi$
$662$	−46.1764	−1.79470
$663$	0	0
$664$	5.41408	0.210107
$665$	−3.74672	−0.145292
$666$	0	0
$667$	7.60941	0.294637
$668$	8.02188	0.310376
$669$	0	0
$670$	−2.64793	−0.102298
$671$	−12.0260	−0.464259
$672$	0	0
$673$	−9.23155	−0.355850	−0.177925	−	0.984044i	$-0.556938\pi$
$673$	−9.23155	−0.355850	−0.177925	+	0.984044i	$0.556938\pi$
$674$	−36.0278	−1.38774
$675$	0	0
$676$	7.08032	0.272320
$677$	11.3302	0.435457	0.217728	−	0.976009i	$-0.430135\pi$
$677$	11.3302	0.435457	0.217728	+	0.976009i	$0.430135\pi$
$678$	0	0
$679$	−23.7229	−0.910403
$680$	3.64381	0.139734
$681$	0	0
$682$	58.9116	2.25584
$683$	9.35188	0.357840	0.178920	−	0.983864i	$-0.442740\pi$
$683$	9.35188	0.357840	0.178920	+	0.983864i	$0.442740\pi$
$684$	0	0
$685$	9.61162	0.367241
$686$	−6.73980	−0.257327
$687$	0	0
$688$	31.8499	1.21427
$689$	27.3149	1.04062
$690$	0	0
$691$	4.79070	0.182247	0.0911234	−	0.995840i	$-0.470954\pi$
$691$	4.79070	0.182247	0.0911234	+	0.995840i	$0.470954\pi$
$692$	13.6341	0.518291
$693$	0	0
$694$	−18.4606	−0.700755
$695$	6.25979	0.237447
$696$	0	0
$697$	12.0569	0.456687
$698$	−14.3446	−0.542952
$699$	0	0
$700$	−18.2197	−0.688642
$701$	49.5004	1.86960	0.934802	−	0.355169i	$-0.115577\pi$
$701$	49.5004	1.86960	0.934802	+	0.355169i	$0.115577\pi$
$702$	0	0
$703$	−0.196855	−0.00742451
$704$	−1.96303	−0.0739846
$705$	0	0
$706$	37.5686	1.41391
$707$	10.2734	0.386371
$708$	0	0
$709$	−40.0415	−1.50379	−0.751896	−	0.659282i	$-0.770860\pi$
$709$	−40.0415	−1.50379	−0.751896	+	0.659282i	$0.770860\pi$
$710$	1.47526	0.0553654
$711$	0	0
$712$	−3.79083	−0.142067
$713$	8.96626	0.335789
$714$	0	0
$715$	18.4977	0.691774
$716$	17.5879	0.657289
$717$	0	0
$718$	15.0742	0.562566
$719$	−43.8322	−1.63467	−0.817333	−	0.576166i	$-0.804548\pi$
$719$	−43.8322	−1.63467	−0.817333	+	0.576166i	$0.804548\pi$
$720$	0	0
$721$	1.05579	0.0393198
$722$	−1.77726	−0.0661428
$723$	0	0
$724$	21.0495	0.782300
$725$	26.1482	0.971121
$726$	0	0
$727$	−47.7084	−1.76941	−0.884703	−	0.466154i	$-0.845639\pi$
$727$	−47.7084	−1.76941	−0.884703	+	0.466154i	$0.845639\pi$
$728$	25.2997	0.937671
$729$	0	0
$730$	−1.16410	−0.0430853
$731$	−16.1163	−0.596082
$732$	0	0
$733$	−4.48682	−0.165724	−0.0828622	−	0.996561i	$-0.526406\pi$
$733$	−4.48682	−0.165724	−0.0828622	+	0.996561i	$0.526406\pi$
$734$	8.86491	0.327210
$735$	0	0
$736$	−6.91787	−0.254996
$737$	6.72723	0.247801
$738$	0	0
$739$	0.402805	0.0148174	0.00740871	−	0.999973i	$-0.497642\pi$
$739$	0.402805	0.0148174	0.00740871	+	0.999973i	$0.497642\pi$
$740$	0.220800	0.00811678
$741$	0	0
$742$	−42.9799	−1.57784
$743$	−6.86024	−0.251678	−0.125839	−	0.992051i	$-0.540162\pi$
$743$	−6.86024	−0.251678	−0.125839	+	0.992051i	$0.540162\pi$
$744$	0	0
$745$	−1.82219	−0.0667598
$746$	−19.4108	−0.710680
$747$	0	0
$748$	12.7488	0.466143
$749$	35.6488	1.30258
$750$	0	0
$751$	40.9244	1.49335	0.746677	−	0.665187i	$-0.231648\pi$
$751$	40.9244	1.49335	0.746677	+	0.665187i	$0.231648\pi$
$752$	−4.97483	−0.181413
$753$	0	0
$754$	50.0034	1.82102
$755$	3.27359	0.119138
$756$	0	0
$757$	11.4028	0.414441	0.207220	−	0.978294i	$-0.433558\pi$
$757$	11.4028	0.414441	0.207220	+	0.978294i	$0.433558\pi$
$758$	35.0110	1.27166
$759$	0	0
$760$	−1.44751	−0.0525068
$761$	31.5142	1.14239	0.571194	−	0.820815i	$-0.306481\pi$
$761$	31.5142	1.14239	0.571194	+	0.820815i	$0.306481\pi$
$762$	0	0
$763$	42.5638	1.54091
$764$	23.5742	0.852886
$765$	0	0
$766$	20.7795	0.750792
$767$	43.0145	1.55316
$768$	0	0
$769$	−37.2029	−1.34157	−0.670786	−	0.741651i	$-0.734043\pi$
$769$	−37.2029	−1.34157	−0.670786	+	0.741651i	$0.734043\pi$
$770$	−29.1060	−1.04891
$771$	0	0
$772$	19.4836	0.701231
$773$	−16.0295	−0.576541	−0.288270	−	0.957549i	$-0.593080\pi$
$773$	−16.0295	−0.576541	−0.288270	+	0.957549i	$0.593080\pi$
$774$	0	0
$775$	30.8108	1.10676
$776$	−9.16514	−0.329009
$777$	0	0
$778$	35.0661	1.25718
$779$	−4.78962	−0.171606
$780$	0	0
$781$	−3.74798	−0.134113
$782$	5.28966	0.189158
$783$	0	0
$784$	−39.6982	−1.41779
$785$	−17.4794	−0.623866
$786$	0	0
$787$	19.0730	0.679881	0.339940	−	0.940447i	$-0.389593\pi$
$787$	19.0730	0.679881	0.339940	+	0.940447i	$0.389593\pi$
$788$	10.7266	0.382119
$789$	0	0
$790$	−28.9913	−1.03146
$791$	25.5343	0.907897
$792$	0	0
$793$	−12.0277	−0.427115
$794$	43.2340	1.53432
$795$	0	0
$796$	−8.98765	−0.318559
$797$	−16.8110	−0.595478	−0.297739	−	0.954647i	$-0.596232\pi$
$797$	−16.8110	−0.595478	−0.297739	+	0.954647i	$0.596232\pi$
$798$	0	0
$799$	2.51730	0.0890555
$800$	−23.7719	−0.840464
$801$	0	0
$802$	−49.5885	−1.75103
$803$	2.95747	0.104367
$804$	0	0
$805$	−4.42990	−0.156133
$806$	58.9196	2.07536
$807$	0	0
$808$	3.96903	0.139630
$809$	−25.3639	−0.891746	−0.445873	−	0.895096i	$-0.647107\pi$
$809$	−25.3639	−0.891746	−0.445873	+	0.895096i	$0.647107\pi$
$810$	0	0
$811$	18.2916	0.642304	0.321152	−	0.947028i	$-0.395930\pi$
$811$	18.2916	0.642304	0.321152	+	0.947028i	$0.395930\pi$
$812$	−28.8614	−1.01284
$813$	0	0
$814$	−1.52924	−0.0536000
$815$	−20.1100	−0.704422
$816$	0	0
$817$	6.40222	0.223985
$818$	−11.6452	−0.407164
$819$	0	0
$820$	5.37223	0.187606
$821$	42.5637	1.48548	0.742741	−	0.669579i	$-0.233525\pi$
$821$	42.5637	1.48548	0.742741	+	0.669579i	$0.233525\pi$
$822$	0	0
$823$	−26.9869	−0.940704	−0.470352	−	0.882479i	$-0.655873\pi$
$823$	−26.9869	−0.940704	−0.470352	+	0.882479i	$0.655873\pi$
$824$	0.407895	0.0142097
$825$	0	0
$826$	−67.6830	−2.35499
$827$	−5.28097	−0.183637	−0.0918187	−	0.995776i	$-0.529268\pi$
$827$	−5.28097	−0.183637	−0.0918187	+	0.995776i	$0.529268\pi$
$828$	0	0
$829$	−14.8102	−0.514381	−0.257190	−	0.966361i	$-0.582797\pi$
$829$	−14.8102	−0.514381	−0.257190	+	0.966361i	$0.582797\pi$
$830$	−6.22946	−0.216228
$831$	0	0
$832$	−1.96330	−0.0680653
$833$	20.0875	0.695992
$834$	0	0
$835$	6.70222	0.231940
$836$	−5.06449	−0.175159
$837$	0	0
$838$	−51.9020	−1.79292
$839$	−29.8378	−1.03012	−0.515058	−	0.857155i	$-0.672230\pi$
$839$	−29.8378	−1.03012	−0.515058	+	0.857155i	$0.672230\pi$
$840$	0	0
$841$	12.4207	0.428300
$842$	21.7276	0.748784
$843$	0	0
$844$	−10.8188	−0.372398
$845$	5.91556	0.203501
$846$	0	0
$847$	31.3716	1.07794
$848$	−31.0841	−1.06743
$849$	0	0
$850$	18.1769	0.623462
$851$	−0.232749	−0.00797853
$852$	0	0
$853$	−31.7021	−1.08546	−0.542729	−	0.839908i	$-0.682609\pi$
$853$	−31.7021	−1.08546	−0.542729	+	0.839908i	$0.682609\pi$
$854$	18.9255	0.647616
$855$	0	0
$856$	13.7726	0.470738
$857$	−18.4856	−0.631457	−0.315729	−	0.948850i	$-0.602249\pi$
$857$	−18.4856	−0.631457	−0.315729	+	0.948850i	$0.602249\pi$
$858$	0	0
$859$	−18.5352	−0.632415	−0.316207	−	0.948690i	$-0.602410\pi$
$859$	−18.5352	−0.632415	−0.316207	+	0.948690i	$0.602410\pi$
$860$	−7.18099	−0.244870
$861$	0	0
$862$	−36.7279	−1.25096
$863$	−19.1278	−0.651118	−0.325559	−	0.945522i	$-0.605553\pi$
$863$	−19.1278	−0.651118	−0.325559	+	0.945522i	$0.605553\pi$
$864$	0	0
$865$	11.3912	0.387312
$866$	−40.3170	−1.37003
$867$	0	0
$868$	−34.0078	−1.15430
$869$	73.6542	2.49855
$870$	0	0
$871$	6.72815	0.227975
$872$	16.4441	0.556868
$873$	0	0
$874$	−2.10133	−0.0710784
$875$	−33.9561	−1.14793
$876$	0	0
$877$	15.6520	0.528530	0.264265	−	0.964450i	$-0.414871\pi$
$877$	15.6520	0.528530	0.264265	+	0.964450i	$0.414871\pi$
$878$	−7.95990	−0.268634
$879$	0	0
$880$	−21.0502	−0.709602
$881$	−41.5337	−1.39931	−0.699653	−	0.714482i	$-0.746662\pi$
$881$	−41.5337	−1.39931	−0.699653	+	0.714482i	$0.746662\pi$
$882$	0	0
$883$	8.96487	0.301692	0.150846	−	0.988557i	$-0.451800\pi$
$883$	8.96487	0.301692	0.150846	+	0.988557i	$0.451800\pi$
$884$	12.7506	0.428847
$885$	0	0
$886$	62.2073	2.08990
$887$	53.2045	1.78643	0.893217	−	0.449627i	$-0.148443\pi$
$887$	53.2045	1.78643	0.893217	+	0.449627i	$0.148443\pi$
$888$	0	0
$889$	−57.6065	−1.93206
$890$	4.36175	0.146206
$891$	0	0
$892$	5.20278	0.174202
$893$	−1.00000	−0.0334637
$894$	0	0
$895$	14.6945	0.491184
$896$	−42.2019	−1.40987
$897$	0	0
$898$	−66.6695	−2.22479
$899$	48.8066	1.62779
$900$	0	0
$901$	15.7288	0.524002
$902$	−37.2076	−1.23888
$903$	0	0
$904$	9.86496	0.328104
$905$	17.5868	0.584603
$906$	0	0
$907$	−33.2652	−1.10455	−0.552277	−	0.833661i	$-0.686241\pi$
$907$	−33.2652	−1.10455	−0.552277	+	0.833661i	$0.686241\pi$
$908$	18.8626	0.625979
$909$	0	0
$910$	−29.1100	−0.964987
$911$	−27.5255	−0.911960	−0.455980	−	0.889990i	$-0.650711\pi$
$911$	−27.5255	−0.911960	−0.455980	+	0.889990i	$0.650711\pi$
$912$	0	0
$913$	15.8264	0.523776
$914$	−50.1107	−1.65752
$915$	0	0
$916$	12.9739	0.428669
$917$	1.27485	0.0420992
$918$	0	0
$919$	44.4155	1.46513	0.732566	−	0.680696i	$-0.238322\pi$
$919$	44.4155	1.46513	0.732566	+	0.680696i	$0.238322\pi$
$920$	−1.71145	−0.0564248
$921$	0	0
$922$	−37.7133	−1.24202
$923$	−3.74850	−0.123383
$924$	0	0
$925$	−0.799796	−0.0262971
$926$	20.5345	0.674806
$927$	0	0
$928$	−37.6564	−1.23613
$929$	−24.4751	−0.803003	−0.401501	−	0.915858i	$-0.631511\pi$
$929$	−24.4751	−0.803003	−0.401501	+	0.915858i	$0.631511\pi$
$930$	0	0
$931$	−7.97981	−0.261528
$932$	4.63188	0.151722
$933$	0	0
$934$	−49.4424	−1.61780
$935$	10.6515	0.348343
$936$	0	0
$937$	−25.5544	−0.834826	−0.417413	−	0.908717i	$-0.637063\pi$
$937$	−25.5544	−0.834826	−0.417413	+	0.908717i	$0.637063\pi$
$938$	−10.5867	−0.345668
$939$	0	0
$940$	1.12164	0.0365839
$941$	−22.9641	−0.748608	−0.374304	−	0.927306i	$-0.622118\pi$
$941$	−22.9641	−0.748608	−0.374304	+	0.927306i	$0.622118\pi$
$942$	0	0
$943$	−5.66295	−0.184411
$944$	−48.9501	−1.59319
$945$	0	0
$946$	49.7350	1.61702
$947$	5.57182	0.181060	0.0905298	−	0.995894i	$-0.471144\pi$
$947$	5.57182	0.181060	0.0905298	+	0.995894i	$0.471144\pi$
$948$	0	0
$949$	2.95788	0.0960168
$950$	−7.22079	−0.234274
$951$	0	0
$952$	14.5684	0.472164
$953$	−15.0218	−0.486603	−0.243301	−	0.969951i	$-0.578230\pi$
$953$	−15.0218	−0.486603	−0.243301	+	0.969951i	$0.578230\pi$
$954$	0	0
$955$	19.6961	0.637351
$956$	2.04551	0.0661564
$957$	0	0
$958$	−8.80255	−0.284397
$959$	38.4283	1.24092
$960$	0	0
$961$	26.5094	0.855142
$962$	−1.52945	−0.0493116
$963$	0	0
$964$	4.88156	0.157224
$965$	16.2784	0.524022
$966$	0	0
$967$	29.0545	0.934329	0.467164	−	0.884171i	$-0.345276\pi$
$967$	29.0545	0.934329	0.467164	+	0.884171i	$0.345276\pi$
$968$	12.1201	0.389555
$969$	0	0
$970$	10.5454	0.338594
$971$	−7.12730	−0.228726	−0.114363	−	0.993439i	$-0.536483\pi$
$971$	−7.12730	−0.228726	−0.114363	+	0.993439i	$0.536483\pi$
$972$	0	0
$973$	25.0273	0.802339
$974$	66.4870	2.13038
$975$	0	0
$976$	13.6874	0.438122
$977$	33.0294	1.05670	0.528351	−	0.849026i	$-0.322810\pi$
$977$	33.0294	1.05670	0.528351	+	0.849026i	$0.322810\pi$
$978$	0	0
$979$	−11.0813	−0.354160
$980$	8.95048	0.285913
$981$	0	0
$982$	50.3432	1.60652
$983$	49.4845	1.57831	0.789156	−	0.614193i	$-0.210518\pi$
$983$	49.4845	1.57831	0.789156	+	0.614193i	$0.210518\pi$
$984$	0	0
$985$	8.96199	0.285553
$986$	28.7935	0.916972
$987$	0	0
$988$	−5.06518	−0.161145
$989$	7.56960	0.240699
$990$	0	0
$991$	−5.17223	−0.164301	−0.0821506	−	0.996620i	$-0.526179\pi$
$991$	−5.17223	−0.164301	−0.0821506	+	0.996620i	$0.526179\pi$
$992$	−44.3710	−1.40878
$993$	0	0
$994$	5.89824	0.187081
$995$	−7.50912	−0.238055
$996$	0	0
$997$	−19.7958	−0.626940	−0.313470	−	0.949598i	$-0.601491\pi$
$997$	−19.7958	−0.626940	−0.313470	+	0.949598i	$0.601491\pi$
$998$	−34.6806	−1.09779
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.w.1.6	yes	34
3.2	odd	2		8037.2.a.v.1.29	✓	34

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8037.2.a.v.1.29	✓	34	3.2	odd	2
8037.2.a.w.1.6	yes	34	1.1	even	1	trivial

Overview	Random
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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:	\( N \)	\(=\)	\( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8037.a (trivial)

\(f(q)\)	\(=\)	\(q-1.77726 q^{2} +1.15866 q^{4} +0.968051 q^{5} +3.87038 q^{7} +1.49528 q^{8} +O(q^{10})\)	\(q-1.77726 q^{2} +1.15866 q^{4} +0.968051 q^{5} +3.87038 q^{7} +1.49528 q^{8} -1.72048 q^{10} +4.37099 q^{11} +4.37159 q^{13} -6.87867 q^{14} -4.97483 q^{16} +2.51730 q^{17} -1.00000 q^{19} +1.12164 q^{20} -7.76839 q^{22} -1.18234 q^{23} -4.06288 q^{25} -7.76946 q^{26} +4.48444 q^{28} -6.43589 q^{29} -7.58350 q^{31} +5.85100 q^{32} -4.47389 q^{34} +3.74672 q^{35} +0.196855 q^{37} +1.77726 q^{38} +1.44751 q^{40} +4.78962 q^{41} -6.40222 q^{43} +5.06449 q^{44} +2.10133 q^{46} +1.00000 q^{47} +7.97981 q^{49} +7.22079 q^{50} +5.06518 q^{52} +6.24829 q^{53} +4.23134 q^{55} +5.78731 q^{56} +11.4383 q^{58} +9.83955 q^{59} -2.75133 q^{61} +13.4779 q^{62} -0.449105 q^{64} +4.23192 q^{65} +1.53906 q^{67} +2.91669 q^{68} -6.65891 q^{70} -0.857468 q^{71} +0.676614 q^{73} -0.349862 q^{74} -1.15866 q^{76} +16.9174 q^{77} +16.8507 q^{79} -4.81589 q^{80} -8.51240 q^{82} +3.62077 q^{83} +2.43687 q^{85} +11.3784 q^{86} +6.53587 q^{88} -2.53519 q^{89} +16.9197 q^{91} -1.36993 q^{92} -1.77726 q^{94} -0.968051 q^{95} -6.12936 q^{97} -14.1822 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q + 5 q^{2} + 31 q^{4} + 6 q^{5} + 15 q^{8}+O(q^{10}) \) 34 * q + 5 * q^2 + 31 * q^4 + 6 * q^5 + 15 * q^8	\( 34 q + 5 q^{2} + 31 q^{4} + 6 q^{5} + 15 q^{8} + 4 q^{10} + 6 q^{11} + 2 q^{13} + 12 q^{14} + 21 q^{16} + 4 q^{17} - 34 q^{19} + 20 q^{20} - 8 q^{22} + 26 q^{23} + 32 q^{25} + 29 q^{26} - 4 q^{28} + 14 q^{29} + 2 q^{31} + 35 q^{32} - 18 q^{34} + 50 q^{35} - 10 q^{37} - 5 q^{38} + 17 q^{40} + 18 q^{41} + 6 q^{43} + 6 q^{44} + 18 q^{46} + 34 q^{47} + 28 q^{49} + 41 q^{50} + 10 q^{52} + 40 q^{53} - 8 q^{55} + 76 q^{56} + 4 q^{58} + 62 q^{59} - 2 q^{61} + 50 q^{62} + 11 q^{64} + 32 q^{65} + 20 q^{67} + 28 q^{68} + 22 q^{70} + 52 q^{71} - 8 q^{73} + 10 q^{74} - 31 q^{76} + 36 q^{77} - 12 q^{79} + 92 q^{80} + 10 q^{82} + 82 q^{83} - 4 q^{85} + 40 q^{86} - 16 q^{88} + 58 q^{89} + 100 q^{92} + 5 q^{94} - 6 q^{95} - 6 q^{97} + 45 q^{98}+O(q^{100}) \) 34 * q + 5 * q^2 + 31 * q^4 + 6 * q^5 + 15 * q^8 + 4 * q^10 + 6 * q^11 + 2 * q^13 + 12 * q^14 + 21 * q^16 + 4 * q^17 - 34 * q^19 + 20 * q^20 - 8 * q^22 + 26 * q^23 + 32 * q^25 + 29 * q^26 - 4 * q^28 + 14 * q^29 + 2 * q^31 + 35 * q^32 - 18 * q^34 + 50 * q^35 - 10 * q^37 - 5 * q^38 + 17 * q^40 + 18 * q^41 + 6 * q^43 + 6 * q^44 + 18 * q^46 + 34 * q^47 + 28 * q^49 + 41 * q^50 + 10 * q^52 + 40 * q^53 - 8 * q^55 + 76 * q^56 + 4 * q^58 + 62 * q^59 - 2 * q^61 + 50 * q^62 + 11 * q^64 + 32 * q^65 + 20 * q^67 + 28 * q^68 + 22 * q^70 + 52 * q^71 - 8 * q^73 + 10 * q^74 - 31 * q^76 + 36 * q^77 - 12 * q^79 + 92 * q^80 + 10 * q^82 + 82 * q^83 - 4 * q^85 + 40 * q^86 - 16 * q^88 + 58 * q^89 + 100 * q^92 + 5 * q^94 - 6 * q^95 - 6 * q^97 + 45 * q^98

Introduction

L-functions

Modular forms

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Database

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