LMFDB - Embedded newform 3870.2.a.bc.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3870,2,Mod(1,3870)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3870, 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("3870.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3870 = 2 \cdot 3^{2} \cdot 5 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3870.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:	$30.9021055822$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 430)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	3870.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{10} -3.41421 q^{11} +3.82843 q^{13} -1.00000 q^{14} +1.00000 q^{16} +1.41421 q^{17} -1.00000 q^{19} -1.00000 q^{20} +3.41421 q^{22} -9.07107 q^{23} +1.00000 q^{25} -3.82843 q^{26} +1.00000 q^{28} +7.24264 q^{29} +2.41421 q^{31} -1.00000 q^{32} -1.41421 q^{34} -1.00000 q^{35} -6.24264 q^{37} +1.00000 q^{38} +1.00000 q^{40} -3.82843 q^{41} +1.00000 q^{43} -3.41421 q^{44} +9.07107 q^{46} +7.07107 q^{47} -6.00000 q^{49} -1.00000 q^{50} +3.82843 q^{52} -5.65685 q^{53} +3.41421 q^{55} -1.00000 q^{56} -7.24264 q^{58} +6.82843 q^{59} -14.0711 q^{61} -2.41421 q^{62} +1.00000 q^{64} -3.82843 q^{65} +7.24264 q^{67} +1.41421 q^{68} +1.00000 q^{70} +7.89949 q^{71} -0.757359 q^{73} +6.24264 q^{74} -1.00000 q^{76} -3.41421 q^{77} -5.58579 q^{79} -1.00000 q^{80} +3.82843 q^{82} +7.65685 q^{83} -1.41421 q^{85} -1.00000 q^{86} +3.41421 q^{88} -5.07107 q^{89} +3.82843 q^{91} -9.07107 q^{92} -7.07107 q^{94} +1.00000 q^{95} -17.5563 q^{97} +6.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8} + 2 q^{10} - 4 q^{11} + 2 q^{13} - 2 q^{14} + 2 q^{16} - 2 q^{19} - 2 q^{20} + 4 q^{22} - 4 q^{23} + 2 q^{25} - 2 q^{26} + 2 q^{28} + 6 q^{29} + 2 q^{31} - 2 q^{32} - 2 q^{35} - 4 q^{37} + 2 q^{38} + 2 q^{40} - 2 q^{41} + 2 q^{43} - 4 q^{44} + 4 q^{46} - 12 q^{49} - 2 q^{50} + 2 q^{52} + 4 q^{55} - 2 q^{56} - 6 q^{58} + 8 q^{59} - 14 q^{61} - 2 q^{62} + 2 q^{64} - 2 q^{65} + 6 q^{67} + 2 q^{70} - 4 q^{71} - 10 q^{73} + 4 q^{74} - 2 q^{76} - 4 q^{77} - 14 q^{79} - 2 q^{80} + 2 q^{82} + 4 q^{83} - 2 q^{86} + 4 q^{88} + 4 q^{89} + 2 q^{91} - 4 q^{92} + 2 q^{95} - 4 q^{97} + 12 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 - 2 * q^8 + 2 * q^10 - 4 * q^11 + 2 * q^13 - 2 * q^14 + 2 * q^16 - 2 * q^19 - 2 * q^20 + 4 * q^22 - 4 * q^23 + 2 * q^25 - 2 * q^26 + 2 * q^28 + 6 * q^29 + 2 * q^31 - 2 * q^32 - 2 * q^35 - 4 * q^37 + 2 * q^38 + 2 * q^40 - 2 * q^41 + 2 * q^43 - 4 * q^44 + 4 * q^46 - 12 * q^49 - 2 * q^50 + 2 * q^52 + 4 * q^55 - 2 * q^56 - 6 * q^58 + 8 * q^59 - 14 * q^61 - 2 * q^62 + 2 * q^64 - 2 * q^65 + 6 * q^67 + 2 * q^70 - 4 * q^71 - 10 * q^73 + 4 * q^74 - 2 * q^76 - 4 * q^77 - 14 * q^79 - 2 * q^80 + 2 * q^82 + 4 * q^83 - 2 * q^86 + 4 * q^88 + 4 * q^89 + 2 * q^91 - 4 * q^92 + 2 * q^95 - 4 * q^97 + 12 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	−3.41421	−1.02942	−0.514712	−	0.857363i	$-0.672101\pi$
$11$	−3.41421	−1.02942	−0.514712	+	0.857363i	$0.672101\pi$
$12$	0	0
$13$	3.82843	1.06181	0.530907	−	0.847430i	$-0.321851\pi$
$13$	3.82843	1.06181	0.530907	+	0.847430i	$0.321851\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	1.41421	0.342997	0.171499	−	0.985184i	$-0.445139\pi$
$17$	1.41421	0.342997	0.171499	+	0.985184i	$0.445139\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	3.41421	0.727913
$23$	−9.07107	−1.89145	−0.945724	−	0.324970i	$-0.894646\pi$
$23$	−9.07107	−1.89145	−0.945724	+	0.324970i	$0.894646\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−3.82843	−0.750816
$27$	0	0
$28$	1.00000	0.188982
$29$	7.24264	1.34492	0.672462	−	0.740131i	$-0.265237\pi$
$29$	7.24264	1.34492	0.672462	+	0.740131i	$0.265237\pi$
$30$	0	0
$31$	2.41421	0.433606	0.216803	−	0.976215i	$-0.430437\pi$
$31$	2.41421	0.433606	0.216803	+	0.976215i	$0.430437\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−1.41421	−0.242536
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−6.24264	−1.02628	−0.513142	−	0.858304i	$-0.671519\pi$
$37$	−6.24264	−1.02628	−0.513142	+	0.858304i	$0.671519\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	1.00000	0.158114
$41$	−3.82843	−0.597900	−0.298950	−	0.954269i	$-0.596636\pi$
$41$	−3.82843	−0.597900	−0.298950	+	0.954269i	$0.596636\pi$
$42$	0	0
$43$	1.00000	0.152499
$43$	1.00000	0.152499
$44$	−3.41421	−0.514712
$45$	0	0
$46$	9.07107	1.33746
$47$	7.07107	1.03142	0.515711	−	0.856763i	$-0.327528\pi$
$47$	7.07107	1.03142	0.515711	+	0.856763i	$0.327528\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	−1.00000	−0.141421
$51$	0	0
$52$	3.82843	0.530907
$53$	−5.65685	−0.777029	−0.388514	−	0.921443i	$-0.627012\pi$
$53$	−5.65685	−0.777029	−0.388514	+	0.921443i	$0.627012\pi$
$54$	0	0
$55$	3.41421	0.460372
$56$	−1.00000	−0.133631
$57$	0	0
$58$	−7.24264	−0.951005
$59$	6.82843	0.888985	0.444493	−	0.895782i	$-0.353384\pi$
$59$	6.82843	0.888985	0.444493	+	0.895782i	$0.353384\pi$
$60$	0	0
$61$	−14.0711	−1.80162	−0.900808	−	0.434218i	$-0.857025\pi$
$61$	−14.0711	−1.80162	−0.900808	+	0.434218i	$0.857025\pi$
$62$	−2.41421	−0.306605
$63$	0	0
$64$	1.00000	0.125000
$65$	−3.82843	−0.474858
$66$	0	0
$67$	7.24264	0.884829	0.442415	−	0.896811i	$-0.354122\pi$
$67$	7.24264	0.884829	0.442415	+	0.896811i	$0.354122\pi$
$68$	1.41421	0.171499
$69$	0	0
$70$	1.00000	0.119523
$71$	7.89949	0.937498	0.468749	−	0.883332i	$-0.344705\pi$
$71$	7.89949	0.937498	0.468749	+	0.883332i	$0.344705\pi$
$72$	0	0
$73$	−0.757359	−0.0886422	−0.0443211	−	0.999017i	$-0.514112\pi$
$73$	−0.757359	−0.0886422	−0.0443211	+	0.999017i	$0.514112\pi$
$74$	6.24264	0.725692
$75$	0	0
$76$	−1.00000	−0.114708
$77$	−3.41421	−0.389086
$78$	0	0
$79$	−5.58579	−0.628450	−0.314225	−	0.949349i	$-0.601745\pi$
$79$	−5.58579	−0.628450	−0.314225	+	0.949349i	$0.601745\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	3.82843	0.422779
$83$	7.65685	0.840449	0.420224	−	0.907420i	$-0.361951\pi$
$83$	7.65685	0.840449	0.420224	+	0.907420i	$0.361951\pi$
$84$	0	0
$85$	−1.41421	−0.153393
$86$	−1.00000	−0.107833
$87$	0	0
$88$	3.41421	0.363956
$89$	−5.07107	−0.537532	−0.268766	−	0.963205i	$-0.586616\pi$
$89$	−5.07107	−0.537532	−0.268766	+	0.963205i	$0.586616\pi$
$90$	0	0
$91$	3.82843	0.401328
$92$	−9.07107	−0.945724
$93$	0	0
$94$	−7.07107	−0.729325
$95$	1.00000	0.102598
$96$	0	0
$97$	−17.5563	−1.78258	−0.891289	−	0.453436i	$-0.850198\pi$
$97$	−17.5563	−1.78258	−0.891289	+	0.453436i	$0.850198\pi$
$98$	6.00000	0.606092
$99$	0	0
$100$	1.00000	0.100000
$101$	1.41421	0.140720	0.0703598	−	0.997522i	$-0.477585\pi$
$101$	1.41421	0.140720	0.0703598	+	0.997522i	$0.477585\pi$
$102$	0	0
$103$	−4.82843	−0.475759	−0.237880	−	0.971295i	$-0.576452\pi$
$103$	−4.82843	−0.475759	−0.237880	+	0.971295i	$0.576452\pi$
$104$	−3.82843	−0.375408
$105$	0	0
$106$	5.65685	0.549442
$107$	−16.0711	−1.55365	−0.776824	−	0.629717i	$-0.783171\pi$
$107$	−16.0711	−1.55365	−0.776824	+	0.629717i	$0.783171\pi$
$108$	0	0
$109$	−5.17157	−0.495347	−0.247673	−	0.968844i	$-0.579666\pi$
$109$	−5.17157	−0.495347	−0.247673	+	0.968844i	$0.579666\pi$
$110$	−3.41421	−0.325532
$111$	0	0
$112$	1.00000	0.0944911
$113$	14.8995	1.40163	0.700813	−	0.713345i	$-0.252821\pi$
$113$	14.8995	1.40163	0.700813	+	0.713345i	$0.252821\pi$
$114$	0	0
$115$	9.07107	0.845881
$116$	7.24264	0.672462
$117$	0	0
$118$	−6.82843	−0.628608
$119$	1.41421	0.129641
$120$	0	0
$121$	0.656854	0.0597140
$122$	14.0711	1.27393
$123$	0	0
$124$	2.41421	0.216803
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	20.4853	1.81777	0.908887	−	0.417042i	$-0.136933\pi$
$127$	20.4853	1.81777	0.908887	+	0.417042i	$0.136933\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	3.82843	0.335775
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	−7.24264	−0.625669
$135$	0	0
$136$	−1.41421	−0.121268
$137$	−18.0711	−1.54392	−0.771958	−	0.635674i	$-0.780722\pi$
$137$	−18.0711	−1.54392	−0.771958	+	0.635674i	$0.780722\pi$
$138$	0	0
$139$	2.24264	0.190218	0.0951092	−	0.995467i	$-0.469680\pi$
$139$	2.24264	0.190218	0.0951092	+	0.995467i	$0.469680\pi$
$140$	−1.00000	−0.0845154
$141$	0	0
$142$	−7.89949	−0.662911
$143$	−13.0711	−1.09306
$144$	0	0
$145$	−7.24264	−0.601469
$146$	0.757359	0.0626795
$147$	0	0
$148$	−6.24264	−0.513142
$149$	−20.5563	−1.68404	−0.842021	−	0.539445i	$-0.818634\pi$
$149$	−20.5563	−1.68404	−0.842021	+	0.539445i	$0.818634\pi$
$150$	0	0
$151$	−6.48528	−0.527765	−0.263882	−	0.964555i	$-0.585003\pi$
$151$	−6.48528	−0.527765	−0.263882	+	0.964555i	$0.585003\pi$
$152$	1.00000	0.0811107
$153$	0	0
$154$	3.41421	0.275125
$155$	−2.41421	−0.193914
$156$	0	0
$157$	1.75736	0.140253	0.0701263	−	0.997538i	$-0.477660\pi$
$157$	1.75736	0.140253	0.0701263	+	0.997538i	$0.477660\pi$
$158$	5.58579	0.444381
$159$	0	0
$160$	1.00000	0.0790569
$161$	−9.07107	−0.714900
$162$	0	0
$163$	2.34315	0.183529	0.0917647	−	0.995781i	$-0.470749\pi$
$163$	2.34315	0.183529	0.0917647	+	0.995781i	$0.470749\pi$
$164$	−3.82843	−0.298950
$165$	0	0
$166$	−7.65685	−0.594287
$167$	17.5563	1.35855	0.679276	−	0.733883i	$-0.262294\pi$
$167$	17.5563	1.35855	0.679276	+	0.733883i	$0.262294\pi$
$168$	0	0
$169$	1.65685	0.127450
$170$	1.41421	0.108465
$171$	0	0
$172$	1.00000	0.0762493
$173$	−17.4853	−1.32938	−0.664691	−	0.747119i	$-0.731437\pi$
$173$	−17.4853	−1.32938	−0.664691	+	0.747119i	$0.731437\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	−3.41421	−0.257356
$177$	0	0
$178$	5.07107	0.380093
$179$	−0.514719	−0.0384719	−0.0192359	−	0.999815i	$-0.506123\pi$
$179$	−0.514719	−0.0384719	−0.0192359	+	0.999815i	$0.506123\pi$
$180$	0	0
$181$	−0.928932	−0.0690470	−0.0345235	−	0.999404i	$-0.510991\pi$
$181$	−0.928932	−0.0690470	−0.0345235	+	0.999404i	$0.510991\pi$
$182$	−3.82843	−0.283782
$183$	0	0
$184$	9.07107	0.668728
$185$	6.24264	0.458968
$186$	0	0
$187$	−4.82843	−0.353090
$188$	7.07107	0.515711
$189$	0	0
$190$	−1.00000	−0.0725476
$191$	−0.100505	−0.00727229	−0.00363615	−	0.999993i	$-0.501157\pi$
$191$	−0.100505	−0.00727229	−0.00363615	+	0.999993i	$0.501157\pi$
$192$	0	0
$193$	1.17157	0.0843317	0.0421658	−	0.999111i	$-0.486574\pi$
$193$	1.17157	0.0843317	0.0421658	+	0.999111i	$0.486574\pi$
$194$	17.5563	1.26047
$195$	0	0
$196$	−6.00000	−0.428571
$197$	3.14214	0.223868	0.111934	−	0.993716i	$-0.464295\pi$
$197$	3.14214	0.223868	0.111934	+	0.993716i	$0.464295\pi$
$198$	0	0
$199$	−19.8995	−1.41064	−0.705319	−	0.708890i	$-0.749196\pi$
$199$	−19.8995	−1.41064	−0.705319	+	0.708890i	$0.749196\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	−1.41421	−0.0995037
$203$	7.24264	0.508334
$204$	0	0
$205$	3.82843	0.267389
$206$	4.82843	0.336412
$207$	0	0
$208$	3.82843	0.265454
$209$	3.41421	0.236166
$210$	0	0
$211$	9.65685	0.664805	0.332403	−	0.943138i	$-0.392141\pi$
$211$	9.65685	0.664805	0.332403	+	0.943138i	$0.392141\pi$
$212$	−5.65685	−0.388514
$213$	0	0
$214$	16.0711	1.09860
$215$	−1.00000	−0.0681994
$216$	0	0
$217$	2.41421	0.163887
$218$	5.17157	0.350263
$219$	0	0
$220$	3.41421	0.230186
$221$	5.41421	0.364199
$222$	0	0
$223$	−12.1421	−0.813098	−0.406549	−	0.913629i	$-0.633268\pi$
$223$	−12.1421	−0.813098	−0.406549	+	0.913629i	$0.633268\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−14.8995	−0.991100
$227$	−14.8284	−0.984197	−0.492099	−	0.870539i	$-0.663770\pi$
$227$	−14.8284	−0.984197	−0.492099	+	0.870539i	$0.663770\pi$
$228$	0	0
$229$	−4.82843	−0.319071	−0.159536	−	0.987192i	$-0.551000\pi$
$229$	−4.82843	−0.319071	−0.159536	+	0.987192i	$0.551000\pi$
$230$	−9.07107	−0.598128
$231$	0	0
$232$	−7.24264	−0.475503
$233$	8.48528	0.555889	0.277945	−	0.960597i	$-0.410347\pi$
$233$	8.48528	0.555889	0.277945	+	0.960597i	$0.410347\pi$
$234$	0	0
$235$	−7.07107	−0.461266
$236$	6.82843	0.444493
$237$	0	0
$238$	−1.41421	−0.0916698
$239$	−14.4142	−0.932378	−0.466189	−	0.884685i	$-0.654373\pi$
$239$	−14.4142	−0.932378	−0.466189	+	0.884685i	$0.654373\pi$
$240$	0	0
$241$	−18.5858	−1.19722	−0.598608	−	0.801042i	$-0.704279\pi$
$241$	−18.5858	−1.19722	−0.598608	+	0.801042i	$0.704279\pi$
$242$	−0.656854	−0.0422242
$243$	0	0
$244$	−14.0711	−0.900808
$245$	6.00000	0.383326
$246$	0	0
$247$	−3.82843	−0.243597
$248$	−2.41421	−0.153303
$249$	0	0
$250$	1.00000	0.0632456
$251$	−22.9706	−1.44989	−0.724945	−	0.688807i	$-0.758135\pi$
$251$	−22.9706	−1.44989	−0.724945	+	0.688807i	$0.758135\pi$
$252$	0	0
$253$	30.9706	1.94710
$254$	−20.4853	−1.28536
$255$	0	0
$256$	1.00000	0.0625000
$257$	10.8995	0.679892	0.339946	−	0.940445i	$-0.389591\pi$
$257$	10.8995	0.679892	0.339946	+	0.940445i	$0.389591\pi$
$258$	0	0
$259$	−6.24264	−0.387899
$260$	−3.82843	−0.237429
$261$	0	0
$262$	6.00000	0.370681
$263$	−0.514719	−0.0317389	−0.0158695	−	0.999874i	$-0.505052\pi$
$263$	−0.514719	−0.0317389	−0.0158695	+	0.999874i	$0.505052\pi$
$264$	0	0
$265$	5.65685	0.347498
$266$	1.00000	0.0613139
$267$	0	0
$268$	7.24264	0.442415
$269$	−14.8284	−0.904105	−0.452053	−	0.891991i	$-0.649308\pi$
$269$	−14.8284	−0.904105	−0.452053	+	0.891991i	$0.649308\pi$
$270$	0	0
$271$	−10.8995	−0.662097	−0.331049	−	0.943614i	$-0.607402\pi$
$271$	−10.8995	−0.662097	−0.331049	+	0.943614i	$0.607402\pi$
$272$	1.41421	0.0857493
$273$	0	0
$274$	18.0711	1.09171
$275$	−3.41421	−0.205885
$276$	0	0
$277$	16.4853	0.990505	0.495252	−	0.868749i	$-0.335076\pi$
$277$	16.4853	0.990505	0.495252	+	0.868749i	$0.335076\pi$
$278$	−2.24264	−0.134505
$279$	0	0
$280$	1.00000	0.0597614
$281$	12.6569	0.755045	0.377522	−	0.926000i	$-0.376776\pi$
$281$	12.6569	0.755045	0.377522	+	0.926000i	$0.376776\pi$
$282$	0	0
$283$	27.2426	1.61941	0.809703	−	0.586839i	$-0.199628\pi$
$283$	27.2426	1.61941	0.809703	+	0.586839i	$0.199628\pi$
$284$	7.89949	0.468749
$285$	0	0
$286$	13.0711	0.772908
$287$	−3.82843	−0.225985
$288$	0	0
$289$	−15.0000	−0.882353
$290$	7.24264	0.425303
$291$	0	0
$292$	−0.757359	−0.0443211
$293$	22.9706	1.34195	0.670977	−	0.741478i	$-0.265875\pi$
$293$	22.9706	1.34195	0.670977	+	0.741478i	$0.265875\pi$
$294$	0	0
$295$	−6.82843	−0.397566
$296$	6.24264	0.362846
$297$	0	0
$298$	20.5563	1.19080
$299$	−34.7279	−2.00837
$300$	0	0
$301$	1.00000	0.0576390
$302$	6.48528	0.373186
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	14.0711	0.805707
$306$	0	0
$307$	−20.0711	−1.14552	−0.572758	−	0.819724i	$-0.694127\pi$
$307$	−20.0711	−1.14552	−0.572758	+	0.819724i	$0.694127\pi$
$308$	−3.41421	−0.194543
$309$	0	0
$310$	2.41421	0.137118
$311$	−10.0711	−0.571078	−0.285539	−	0.958367i	$-0.592173\pi$
$311$	−10.0711	−0.571078	−0.285539	+	0.958367i	$0.592173\pi$
$312$	0	0
$313$	21.3137	1.20472	0.602361	−	0.798224i	$-0.294227\pi$
$313$	21.3137	1.20472	0.602361	+	0.798224i	$0.294227\pi$
$314$	−1.75736	−0.0991735
$315$	0	0
$316$	−5.58579	−0.314225
$317$	−29.1421	−1.63679	−0.818393	−	0.574659i	$-0.805135\pi$
$317$	−29.1421	−1.63679	−0.818393	+	0.574659i	$0.805135\pi$
$318$	0	0
$319$	−24.7279	−1.38450
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	9.07107	0.505511
$323$	−1.41421	−0.0786889
$324$	0	0
$325$	3.82843	0.212363
$326$	−2.34315	−0.129775
$327$	0	0
$328$	3.82843	0.211390
$329$	7.07107	0.389841
$330$	0	0
$331$	−2.68629	−0.147652	−0.0738260	−	0.997271i	$-0.523521\pi$
$331$	−2.68629	−0.147652	−0.0738260	+	0.997271i	$0.523521\pi$
$332$	7.65685	0.420224
$333$	0	0
$334$	−17.5563	−0.960641
$335$	−7.24264	−0.395708
$336$	0	0
$337$	−19.6569	−1.07078	−0.535389	−	0.844606i	$-0.679835\pi$
$337$	−19.6569	−1.07078	−0.535389	+	0.844606i	$0.679835\pi$
$338$	−1.65685	−0.0901210
$339$	0	0
$340$	−1.41421	−0.0766965
$341$	−8.24264	−0.446364
$342$	0	0
$343$	−13.0000	−0.701934
$344$	−1.00000	−0.0539164
$345$	0	0
$346$	17.4853	0.940015
$347$	13.7574	0.738534	0.369267	−	0.929323i	$-0.379609\pi$
$347$	13.7574	0.738534	0.369267	+	0.929323i	$0.379609\pi$
$348$	0	0
$349$	−27.4558	−1.46968	−0.734839	−	0.678242i	$-0.762742\pi$
$349$	−27.4558	−1.46968	−0.734839	+	0.678242i	$0.762742\pi$
$350$	−1.00000	−0.0534522
$351$	0	0
$352$	3.41421	0.181978
$353$	−4.38478	−0.233378	−0.116689	−	0.993168i	$-0.537228\pi$
$353$	−4.38478	−0.233378	−0.116689	+	0.993168i	$0.537228\pi$
$354$	0	0
$355$	−7.89949	−0.419262
$356$	−5.07107	−0.268766
$357$	0	0
$358$	0.514719	0.0272037
$359$	37.5269	1.98059	0.990297	−	0.138965i	$-0.0443775\pi$
$359$	37.5269	1.98059	0.990297	+	0.138965i	$0.0443775\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0.928932	0.0488236
$363$	0	0
$364$	3.82843	0.200664
$365$	0.757359	0.0396420
$366$	0	0
$367$	−8.68629	−0.453421	−0.226710	−	0.973962i	$-0.572797\pi$
$367$	−8.68629	−0.453421	−0.226710	+	0.973962i	$0.572797\pi$
$368$	−9.07107	−0.472862
$369$	0	0
$370$	−6.24264	−0.324539
$371$	−5.65685	−0.293689
$372$	0	0
$373$	−10.7279	−0.555471	−0.277735	−	0.960658i	$-0.589584\pi$
$373$	−10.7279	−0.555471	−0.277735	+	0.960658i	$0.589584\pi$
$374$	4.82843	0.249672
$375$	0	0
$376$	−7.07107	−0.364662
$377$	27.7279	1.42806
$378$	0	0
$379$	−24.1421	−1.24010	−0.620049	−	0.784563i	$-0.712887\pi$
$379$	−24.1421	−1.24010	−0.620049	+	0.784563i	$0.712887\pi$
$380$	1.00000	0.0512989
$381$	0	0
$382$	0.100505	0.00514229
$383$	−14.3137	−0.731396	−0.365698	−	0.930734i	$-0.619170\pi$
$383$	−14.3137	−0.731396	−0.365698	+	0.930734i	$0.619170\pi$
$384$	0	0
$385$	3.41421	0.174004
$386$	−1.17157	−0.0596315
$387$	0	0
$388$	−17.5563	−0.891289
$389$	−15.1716	−0.769229	−0.384615	−	0.923077i	$-0.625666\pi$
$389$	−15.1716	−0.769229	−0.384615	+	0.923077i	$0.625666\pi$
$390$	0	0
$391$	−12.8284	−0.648761
$392$	6.00000	0.303046
$393$	0	0
$394$	−3.14214	−0.158299
$395$	5.58579	0.281051
$396$	0	0
$397$	−3.51472	−0.176399	−0.0881993	−	0.996103i	$-0.528111\pi$
$397$	−3.51472	−0.176399	−0.0881993	+	0.996103i	$0.528111\pi$
$398$	19.8995	0.997472
$399$	0	0
$400$	1.00000	0.0500000
$401$	−38.1127	−1.90326	−0.951629	−	0.307251i	$-0.900591\pi$
$401$	−38.1127	−1.90326	−0.951629	+	0.307251i	$0.900591\pi$
$402$	0	0
$403$	9.24264	0.460409
$404$	1.41421	0.0703598
$405$	0	0
$406$	−7.24264	−0.359446
$407$	21.3137	1.05648
$408$	0	0
$409$	15.2132	0.752244	0.376122	−	0.926570i	$-0.377257\pi$
$409$	15.2132	0.752244	0.376122	+	0.926570i	$0.377257\pi$
$410$	−3.82843	−0.189073
$411$	0	0
$412$	−4.82843	−0.237880
$413$	6.82843	0.336005
$414$	0	0
$415$	−7.65685	−0.375860
$416$	−3.82843	−0.187704
$417$	0	0
$418$	−3.41421	−0.166995
$419$	−26.7990	−1.30922	−0.654608	−	0.755968i	$-0.727166\pi$
$419$	−26.7990	−1.30922	−0.654608	+	0.755968i	$0.727166\pi$
$420$	0	0
$421$	−7.24264	−0.352985	−0.176492	−	0.984302i	$-0.556475\pi$
$421$	−7.24264	−0.352985	−0.176492	+	0.984302i	$0.556475\pi$
$422$	−9.65685	−0.470088
$423$	0	0
$424$	5.65685	0.274721
$425$	1.41421	0.0685994
$426$	0	0
$427$	−14.0711	−0.680947
$428$	−16.0711	−0.776824
$429$	0	0
$430$	1.00000	0.0482243
$431$	−34.9706	−1.68447	−0.842236	−	0.539108i	$-0.818761\pi$
$431$	−34.9706	−1.68447	−0.842236	+	0.539108i	$0.818761\pi$
$432$	0	0
$433$	−21.9289	−1.05384	−0.526919	−	0.849916i	$-0.676653\pi$
$433$	−21.9289	−1.05384	−0.526919	+	0.849916i	$0.676653\pi$
$434$	−2.41421	−0.115886
$435$	0	0
$436$	−5.17157	−0.247673
$437$	9.07107	0.433928
$438$	0	0
$439$	7.31371	0.349064	0.174532	−	0.984651i	$-0.444159\pi$
$439$	7.31371	0.349064	0.174532	+	0.984651i	$0.444159\pi$
$440$	−3.41421	−0.162766
$441$	0	0
$442$	−5.41421	−0.257528
$443$	24.0711	1.14365	0.571825	−	0.820375i	$-0.306235\pi$
$443$	24.0711	1.14365	0.571825	+	0.820375i	$0.306235\pi$
$444$	0	0
$445$	5.07107	0.240392
$446$	12.1421	0.574947
$447$	0	0
$448$	1.00000	0.0472456
$449$	−13.7574	−0.649250	−0.324625	−	0.945843i	$-0.605238\pi$
$449$	−13.7574	−0.649250	−0.324625	+	0.945843i	$0.605238\pi$
$450$	0	0
$451$	13.0711	0.615493
$452$	14.8995	0.700813
$453$	0	0
$454$	14.8284	0.695933
$455$	−3.82843	−0.179479
$456$	0	0
$457$	−13.5147	−0.632192	−0.316096	−	0.948727i	$-0.602372\pi$
$457$	−13.5147	−0.632192	−0.316096	+	0.948727i	$0.602372\pi$
$458$	4.82843	0.225618
$459$	0	0
$460$	9.07107	0.422941
$461$	0.928932	0.0432647	0.0216323	−	0.999766i	$-0.493114\pi$
$461$	0.928932	0.0432647	0.0216323	+	0.999766i	$0.493114\pi$
$462$	0	0
$463$	7.68629	0.357212	0.178606	−	0.983921i	$-0.442841\pi$
$463$	7.68629	0.357212	0.178606	+	0.983921i	$0.442841\pi$
$464$	7.24264	0.336231
$465$	0	0
$466$	−8.48528	−0.393073
$467$	35.0711	1.62290	0.811448	−	0.584425i	$-0.198680\pi$
$467$	35.0711	1.62290	0.811448	+	0.584425i	$0.198680\pi$
$468$	0	0
$469$	7.24264	0.334434
$470$	7.07107	0.326164
$471$	0	0
$472$	−6.82843	−0.314304
$473$	−3.41421	−0.156986
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	1.41421	0.0648204
$477$	0	0
$478$	14.4142	0.659291
$479$	−14.8284	−0.677528	−0.338764	−	0.940871i	$-0.610009\pi$
$479$	−14.8284	−0.677528	−0.338764	+	0.940871i	$0.610009\pi$
$480$	0	0
$481$	−23.8995	−1.08972
$482$	18.5858	0.846559
$483$	0	0
$484$	0.656854	0.0298570
$485$	17.5563	0.797193
$486$	0	0
$487$	9.21320	0.417490	0.208745	−	0.977970i	$-0.433062\pi$
$487$	9.21320	0.417490	0.208745	+	0.977970i	$0.433062\pi$
$488$	14.0711	0.636967
$489$	0	0
$490$	−6.00000	−0.271052
$491$	28.6274	1.29194	0.645969	−	0.763364i	$-0.276454\pi$
$491$	28.6274	1.29194	0.645969	+	0.763364i	$0.276454\pi$
$492$	0	0
$493$	10.2426	0.461305
$494$	3.82843	0.172249
$495$	0	0
$496$	2.41421	0.108401
$497$	7.89949	0.354341
$498$	0	0
$499$	16.4558	0.736665	0.368332	−	0.929694i	$-0.379929\pi$
$499$	16.4558	0.736665	0.368332	+	0.929694i	$0.379929\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	22.9706	1.02523
$503$	20.3431	0.907056	0.453528	−	0.891242i	$-0.350165\pi$
$503$	20.3431	0.907056	0.453528	+	0.891242i	$0.350165\pi$
$504$	0	0
$505$	−1.41421	−0.0629317
$506$	−30.9706	−1.37681
$507$	0	0
$508$	20.4853	0.908887
$509$	3.51472	0.155787	0.0778936	−	0.996962i	$-0.475181\pi$
$509$	3.51472	0.155787	0.0778936	+	0.996962i	$0.475181\pi$
$510$	0	0
$511$	−0.757359	−0.0335036
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−10.8995	−0.480756
$515$	4.82843	0.212766
$516$	0	0
$517$	−24.1421	−1.06177
$518$	6.24264	0.274286
$519$	0	0
$520$	3.82843	0.167888
$521$	−3.07107	−0.134546	−0.0672730	−	0.997735i	$-0.521430\pi$
$521$	−3.07107	−0.134546	−0.0672730	+	0.997735i	$0.521430\pi$
$522$	0	0
$523$	−14.7279	−0.644007	−0.322004	−	0.946738i	$-0.604356\pi$
$523$	−14.7279	−0.644007	−0.322004	+	0.946738i	$0.604356\pi$
$524$	−6.00000	−0.262111
$525$	0	0
$526$	0.514719	0.0224428
$527$	3.41421	0.148725
$528$	0	0
$529$	59.2843	2.57758
$530$	−5.65685	−0.245718
$531$	0	0
$532$	−1.00000	−0.0433555
$533$	−14.6569	−0.634859
$534$	0	0
$535$	16.0711	0.694813
$536$	−7.24264	−0.312834
$537$	0	0
$538$	14.8284	0.639299
$539$	20.4853	0.882364
$540$	0	0
$541$	36.1421	1.55387	0.776936	−	0.629580i	$-0.216773\pi$
$541$	36.1421	1.55387	0.776936	+	0.629580i	$0.216773\pi$
$542$	10.8995	0.468173
$543$	0	0
$544$	−1.41421	−0.0606339
$545$	5.17157	0.221526
$546$	0	0
$547$	−12.0000	−0.513083	−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000	−0.513083	−0.256541	+	0.966533i	$0.582583\pi$
$548$	−18.0711	−0.771958
$549$	0	0
$550$	3.41421	0.145583
$551$	−7.24264	−0.308547
$552$	0	0
$553$	−5.58579	−0.237532
$554$	−16.4853	−0.700392
$555$	0	0
$556$	2.24264	0.0951092
$557$	−36.1716	−1.53264	−0.766319	−	0.642460i	$-0.777914\pi$
$557$	−36.1716	−1.53264	−0.766319	+	0.642460i	$0.777914\pi$
$558$	0	0
$559$	3.82843	0.161925
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	−12.6569	−0.533897
$563$	16.0711	0.677315	0.338657	−	0.940910i	$-0.390027\pi$
$563$	16.0711	0.677315	0.338657	+	0.940910i	$0.390027\pi$
$564$	0	0
$565$	−14.8995	−0.626826
$566$	−27.2426	−1.14509
$567$	0	0
$568$	−7.89949	−0.331455
$569$	16.7990	0.704250	0.352125	−	0.935953i	$-0.385459\pi$
$569$	16.7990	0.704250	0.352125	+	0.935953i	$0.385459\pi$
$570$	0	0
$571$	33.0000	1.38101	0.690504	−	0.723329i	$-0.257389\pi$
$571$	33.0000	1.38101	0.690504	+	0.723329i	$0.257389\pi$
$572$	−13.0711	−0.546529
$573$	0	0
$574$	3.82843	0.159795
$575$	−9.07107	−0.378290
$576$	0	0
$577$	−8.41421	−0.350288	−0.175144	−	0.984543i	$-0.556039\pi$
$577$	−8.41421	−0.350288	−0.175144	+	0.984543i	$0.556039\pi$
$578$	15.0000	0.623918
$579$	0	0
$580$	−7.24264	−0.300734
$581$	7.65685	0.317660
$582$	0	0
$583$	19.3137	0.799892
$584$	0.757359	0.0313398
$585$	0	0
$586$	−22.9706	−0.948905
$587$	−3.55635	−0.146786	−0.0733931	−	0.997303i	$-0.523383\pi$
$587$	−3.55635	−0.146786	−0.0733931	+	0.997303i	$0.523383\pi$
$588$	0	0
$589$	−2.41421	−0.0994759
$590$	6.82843	0.281122
$591$	0	0
$592$	−6.24264	−0.256571
$593$	−1.24264	−0.0510291	−0.0255146	−	0.999674i	$-0.508122\pi$
$593$	−1.24264	−0.0510291	−0.0255146	+	0.999674i	$0.508122\pi$
$594$	0	0
$595$	−1.41421	−0.0579771
$596$	−20.5563	−0.842021
$597$	0	0
$598$	34.7279	1.42013
$599$	−38.1421	−1.55845	−0.779223	−	0.626747i	$-0.784386\pi$
$599$	−38.1421	−1.55845	−0.779223	+	0.626747i	$0.784386\pi$
$600$	0	0
$601$	−28.0000	−1.14214	−0.571072	−	0.820900i	$-0.693472\pi$
$601$	−28.0000	−1.14214	−0.571072	+	0.820900i	$0.693472\pi$
$602$	−1.00000	−0.0407570
$603$	0	0
$604$	−6.48528	−0.263882
$605$	−0.656854	−0.0267049
$606$	0	0
$607$	6.00000	0.243532	0.121766	−	0.992559i	$-0.461144\pi$
$607$	6.00000	0.243532	0.121766	+	0.992559i	$0.461144\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	−14.0711	−0.569721
$611$	27.0711	1.09518
$612$	0	0
$613$	43.8284	1.77021	0.885107	−	0.465388i	$-0.154085\pi$
$613$	43.8284	1.77021	0.885107	+	0.465388i	$0.154085\pi$
$614$	20.0711	0.810002
$615$	0	0
$616$	3.41421	0.137563
$617$	−40.2843	−1.62178	−0.810892	−	0.585196i	$-0.801018\pi$
$617$	−40.2843	−1.62178	−0.810892	+	0.585196i	$0.801018\pi$
$618$	0	0
$619$	−19.8995	−0.799828	−0.399914	−	0.916553i	$-0.630960\pi$
$619$	−19.8995	−0.799828	−0.399914	+	0.916553i	$0.630960\pi$
$620$	−2.41421	−0.0969571
$621$	0	0
$622$	10.0711	0.403813
$623$	−5.07107	−0.203168
$624$	0	0
$625$	1.00000	0.0400000
$626$	−21.3137	−0.851867
$627$	0	0
$628$	1.75736	0.0701263
$629$	−8.82843	−0.352012
$630$	0	0
$631$	45.6985	1.81923	0.909614	−	0.415454i	$-0.136377\pi$
$631$	45.6985	1.81923	0.909614	+	0.415454i	$0.136377\pi$
$632$	5.58579	0.222191
$633$	0	0
$634$	29.1421	1.15738
$635$	−20.4853	−0.812934
$636$	0	0
$637$	−22.9706	−0.910127
$638$	24.7279	0.978988
$639$	0	0
$640$	1.00000	0.0395285
$641$	30.7696	1.21532	0.607662	−	0.794196i	$-0.292107\pi$
$641$	30.7696	1.21532	0.607662	+	0.794196i	$0.292107\pi$
$642$	0	0
$643$	14.2721	0.562836	0.281418	−	0.959585i	$-0.409195\pi$
$643$	14.2721	0.562836	0.281418	+	0.959585i	$0.409195\pi$
$644$	−9.07107	−0.357450
$645$	0	0
$646$	1.41421	0.0556415
$647$	−44.7990	−1.76123	−0.880615	−	0.473832i	$-0.842870\pi$
$647$	−44.7990	−1.76123	−0.880615	+	0.473832i	$0.842870\pi$
$648$	0	0
$649$	−23.3137	−0.915143
$650$	−3.82843	−0.150163
$651$	0	0
$652$	2.34315	0.0917647
$653$	9.41421	0.368407	0.184203	−	0.982888i	$-0.441030\pi$
$653$	9.41421	0.368407	0.184203	+	0.982888i	$0.441030\pi$
$654$	0	0
$655$	6.00000	0.234439
$656$	−3.82843	−0.149475
$657$	0	0
$658$	−7.07107	−0.275659
$659$	12.3848	0.482442	0.241221	−	0.970470i	$-0.422452\pi$
$659$	12.3848	0.482442	0.241221	+	0.970470i	$0.422452\pi$
$660$	0	0
$661$	−28.8701	−1.12292	−0.561458	−	0.827506i	$-0.689759\pi$
$661$	−28.8701	−1.12292	−0.561458	+	0.827506i	$0.689759\pi$
$662$	2.68629	0.104406
$663$	0	0
$664$	−7.65685	−0.297144
$665$	1.00000	0.0387783
$666$	0	0
$667$	−65.6985	−2.54386
$668$	17.5563	0.679276
$669$	0	0
$670$	7.24264	0.279808
$671$	48.0416	1.85463
$672$	0	0
$673$	17.5858	0.677882	0.338941	−	0.940808i	$-0.389931\pi$
$673$	17.5858	0.677882	0.338941	+	0.940808i	$0.389931\pi$
$674$	19.6569	0.757154
$675$	0	0
$676$	1.65685	0.0637252
$677$	7.45584	0.286551	0.143276	−	0.989683i	$-0.454236\pi$
$677$	7.45584	0.286551	0.143276	+	0.989683i	$0.454236\pi$
$678$	0	0
$679$	−17.5563	−0.673751
$680$	1.41421	0.0542326
$681$	0	0
$682$	8.24264	0.315627
$683$	20.4853	0.783848	0.391924	−	0.919998i	$-0.371810\pi$
$683$	20.4853	0.783848	0.391924	+	0.919998i	$0.371810\pi$
$684$	0	0
$685$	18.0711	0.690460
$686$	13.0000	0.496342
$687$	0	0
$688$	1.00000	0.0381246
$689$	−21.6569	−0.825060
$690$	0	0
$691$	1.51472	0.0576226	0.0288113	−	0.999585i	$-0.490828\pi$
$691$	1.51472	0.0576226	0.0288113	+	0.999585i	$0.490828\pi$
$692$	−17.4853	−0.664691
$693$	0	0
$694$	−13.7574	−0.522222
$695$	−2.24264	−0.0850682
$696$	0	0
$697$	−5.41421	−0.205078
$698$	27.4558	1.03922
$699$	0	0
$700$	1.00000	0.0377964
$701$	−36.7279	−1.38719	−0.693597	−	0.720363i	$-0.743975\pi$
$701$	−36.7279	−1.38719	−0.693597	+	0.720363i	$0.743975\pi$
$702$	0	0
$703$	6.24264	0.235446
$704$	−3.41421	−0.128678
$705$	0	0
$706$	4.38478	0.165023
$707$	1.41421	0.0531870
$708$	0	0
$709$	35.2132	1.32246	0.661230	−	0.750183i	$-0.270035\pi$
$709$	35.2132	1.32246	0.661230	+	0.750183i	$0.270035\pi$
$710$	7.89949	0.296463
$711$	0	0
$712$	5.07107	0.190046
$713$	−21.8995	−0.820143
$714$	0	0
$715$	13.0711	0.488830
$716$	−0.514719	−0.0192359
$717$	0	0
$718$	−37.5269	−1.40049
$719$	−8.62742	−0.321748	−0.160874	−	0.986975i	$-0.551431\pi$
$719$	−8.62742	−0.321748	−0.160874	+	0.986975i	$0.551431\pi$
$720$	0	0
$721$	−4.82843	−0.179820
$722$	18.0000	0.669891
$723$	0	0
$724$	−0.928932	−0.0345235
$725$	7.24264	0.268985
$726$	0	0
$727$	−14.0000	−0.519231	−0.259616	−	0.965712i	$-0.583596\pi$
$727$	−14.0000	−0.519231	−0.259616	+	0.965712i	$0.583596\pi$
$728$	−3.82843	−0.141891
$729$	0	0
$730$	−0.757359	−0.0280311
$731$	1.41421	0.0523066
$732$	0	0
$733$	36.2426	1.33865	0.669326	−	0.742969i	$-0.266583\pi$
$733$	36.2426	1.33865	0.669326	+	0.742969i	$0.266583\pi$
$734$	8.68629	0.320617
$735$	0	0
$736$	9.07107	0.334364
$737$	−24.7279	−0.910865
$738$	0	0
$739$	41.9706	1.54391	0.771956	−	0.635676i	$-0.219279\pi$
$739$	41.9706	1.54391	0.771956	+	0.635676i	$0.219279\pi$
$740$	6.24264	0.229484
$741$	0	0
$742$	5.65685	0.207670
$743$	20.3137	0.745238	0.372619	−	0.927984i	$-0.378460\pi$
$743$	20.3137	0.745238	0.372619	+	0.927984i	$0.378460\pi$
$744$	0	0
$745$	20.5563	0.753126
$746$	10.7279	0.392777
$747$	0	0
$748$	−4.82843	−0.176545
$749$	−16.0711	−0.587224
$750$	0	0
$751$	16.7279	0.610411	0.305205	−	0.952287i	$-0.401275\pi$
$751$	16.7279	0.610411	0.305205	+	0.952287i	$0.401275\pi$
$752$	7.07107	0.257855
$753$	0	0
$754$	−27.7279	−1.00979
$755$	6.48528	0.236024
$756$	0	0
$757$	−48.4264	−1.76009	−0.880044	−	0.474892i	$-0.842487\pi$
$757$	−48.4264	−1.76009	−0.880044	+	0.474892i	$0.842487\pi$
$758$	24.1421	0.876882
$759$	0	0
$760$	−1.00000	−0.0362738
$761$	39.1716	1.41997	0.709984	−	0.704218i	$-0.248702\pi$
$761$	39.1716	1.41997	0.709984	+	0.704218i	$0.248702\pi$
$762$	0	0
$763$	−5.17157	−0.187224
$764$	−0.100505	−0.00363615
$765$	0	0
$766$	14.3137	0.517175
$767$	26.1421	0.943938
$768$	0	0
$769$	21.0000	0.757279	0.378640	−	0.925544i	$-0.376392\pi$
$769$	21.0000	0.757279	0.378640	+	0.925544i	$0.376392\pi$
$770$	−3.41421	−0.123040
$771$	0	0
$772$	1.17157	0.0421658
$773$	10.9289	0.393086	0.196543	−	0.980495i	$-0.437028\pi$
$773$	10.9289	0.393086	0.196543	+	0.980495i	$0.437028\pi$
$774$	0	0
$775$	2.41421	0.0867211
$776$	17.5563	0.630236
$777$	0	0
$778$	15.1716	0.543927
$779$	3.82843	0.137168
$780$	0	0
$781$	−26.9706	−0.965083
$782$	12.8284	0.458744
$783$	0	0
$784$	−6.00000	−0.214286
$785$	−1.75736	−0.0627228
$786$	0	0
$787$	22.7574	0.811212	0.405606	−	0.914048i	$-0.367060\pi$
$787$	22.7574	0.811212	0.405606	+	0.914048i	$0.367060\pi$
$788$	3.14214	0.111934
$789$	0	0
$790$	−5.58579	−0.198733
$791$	14.8995	0.529765
$792$	0	0
$793$	−53.8701	−1.91298
$794$	3.51472	0.124733
$795$	0	0
$796$	−19.8995	−0.705319
$797$	−24.9411	−0.883460	−0.441730	−	0.897148i	$-0.645635\pi$
$797$	−24.9411	−0.883460	−0.441730	+	0.897148i	$0.645635\pi$
$798$	0	0
$799$	10.0000	0.353775
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	38.1127	1.34581
$803$	2.58579	0.0912504
$804$	0	0
$805$	9.07107	0.319713
$806$	−9.24264	−0.325558
$807$	0	0
$808$	−1.41421	−0.0497519
$809$	−10.7990	−0.379672	−0.189836	−	0.981816i	$-0.560796\pi$
$809$	−10.7990	−0.379672	−0.189836	+	0.981816i	$0.560796\pi$
$810$	0	0
$811$	14.1127	0.495564	0.247782	−	0.968816i	$-0.420298\pi$
$811$	14.1127	0.495564	0.247782	+	0.968816i	$0.420298\pi$
$812$	7.24264	0.254167
$813$	0	0
$814$	−21.3137	−0.747045
$815$	−2.34315	−0.0820768
$816$	0	0
$817$	−1.00000	−0.0349856
$818$	−15.2132	−0.531917
$819$	0	0
$820$	3.82843	0.133694
$821$	−8.14214	−0.284162	−0.142081	−	0.989855i	$-0.545379\pi$
$821$	−8.14214	−0.284162	−0.142081	+	0.989855i	$0.545379\pi$
$822$	0	0
$823$	30.4853	1.06265	0.531325	−	0.847168i	$-0.321694\pi$
$823$	30.4853	1.06265	0.531325	+	0.847168i	$0.321694\pi$
$824$	4.82843	0.168206
$825$	0	0
$826$	−6.82843	−0.237591
$827$	−21.5269	−0.748564	−0.374282	−	0.927315i	$-0.622111\pi$
$827$	−21.5269	−0.748564	−0.374282	+	0.927315i	$0.622111\pi$
$828$	0	0
$829$	−5.24264	−0.182084	−0.0910422	−	0.995847i	$-0.529020\pi$
$829$	−5.24264	−0.182084	−0.0910422	+	0.995847i	$0.529020\pi$
$830$	7.65685	0.265773
$831$	0	0
$832$	3.82843	0.132727
$833$	−8.48528	−0.293998
$834$	0	0
$835$	−17.5563	−0.607563
$836$	3.41421	0.118083
$837$	0	0
$838$	26.7990	0.925756
$839$	−31.0122	−1.07066	−0.535330	−	0.844643i	$-0.679813\pi$
$839$	−31.0122	−1.07066	−0.535330	+	0.844643i	$0.679813\pi$
$840$	0	0
$841$	23.4558	0.808822
$842$	7.24264	0.249598
$843$	0	0
$844$	9.65685	0.332403
$845$	−1.65685	−0.0569975
$846$	0	0
$847$	0.656854	0.0225698
$848$	−5.65685	−0.194257
$849$	0	0
$850$	−1.41421	−0.0485071
$851$	56.6274	1.94116
$852$	0	0
$853$	31.9411	1.09364	0.546822	−	0.837249i	$-0.315838\pi$
$853$	31.9411	1.09364	0.546822	+	0.837249i	$0.315838\pi$
$854$	14.0711	0.481502
$855$	0	0
$856$	16.0711	0.549298
$857$	−32.9289	−1.12483	−0.562415	−	0.826855i	$-0.690128\pi$
$857$	−32.9289	−1.12483	−0.562415	+	0.826855i	$0.690128\pi$
$858$	0	0
$859$	−33.4853	−1.14250	−0.571252	−	0.820775i	$-0.693542\pi$
$859$	−33.4853	−1.14250	−0.571252	+	0.820775i	$0.693542\pi$
$860$	−1.00000	−0.0340997
$861$	0	0
$862$	34.9706	1.19110
$863$	35.1716	1.19725	0.598627	−	0.801028i	$-0.295713\pi$
$863$	35.1716	1.19725	0.598627	+	0.801028i	$0.295713\pi$
$864$	0	0
$865$	17.4853	0.594517
$866$	21.9289	0.745175
$867$	0	0
$868$	2.41421	0.0819437
$869$	19.0711	0.646942
$870$	0	0
$871$	27.7279	0.939525
$872$	5.17157	0.175132
$873$	0	0
$874$	−9.07107	−0.306833
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	26.6274	0.899144	0.449572	−	0.893244i	$-0.351577\pi$
$877$	26.6274	0.899144	0.449572	+	0.893244i	$0.351577\pi$
$878$	−7.31371	−0.246826
$879$	0	0
$880$	3.41421	0.115093
$881$	40.7990	1.37455	0.687276	−	0.726396i	$-0.258806\pi$
$881$	40.7990	1.37455	0.687276	+	0.726396i	$0.258806\pi$
$882$	0	0
$883$	−16.2132	−0.545618	−0.272809	−	0.962068i	$-0.587953\pi$
$883$	−16.2132	−0.545618	−0.272809	+	0.962068i	$0.587953\pi$
$884$	5.41421	0.182100
$885$	0	0
$886$	−24.0711	−0.808683
$887$	−6.11270	−0.205244	−0.102622	−	0.994720i	$-0.532723\pi$
$887$	−6.11270	−0.205244	−0.102622	+	0.994720i	$0.532723\pi$
$888$	0	0
$889$	20.4853	0.687054
$890$	−5.07107	−0.169983
$891$	0	0
$892$	−12.1421	−0.406549
$893$	−7.07107	−0.236624
$894$	0	0
$895$	0.514719	0.0172051
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	13.7574	0.459089
$899$	17.4853	0.583167
$900$	0	0
$901$	−8.00000	−0.266519
$902$	−13.0711	−0.435219
$903$	0	0
$904$	−14.8995	−0.495550
$905$	0.928932	0.0308788
$906$	0	0
$907$	−4.89949	−0.162685	−0.0813425	−	0.996686i	$-0.525921\pi$
$907$	−4.89949	−0.162685	−0.0813425	+	0.996686i	$0.525921\pi$
$908$	−14.8284	−0.492099
$909$	0	0
$910$	3.82843	0.126911
$911$	−16.3848	−0.542852	−0.271426	−	0.962459i	$-0.587495\pi$
$911$	−16.3848	−0.542852	−0.271426	+	0.962459i	$0.587495\pi$
$912$	0	0
$913$	−26.1421	−0.865178
$914$	13.5147	0.447027
$915$	0	0
$916$	−4.82843	−0.159536
$917$	−6.00000	−0.198137
$918$	0	0
$919$	−15.1005	−0.498120	−0.249060	−	0.968488i	$-0.580122\pi$
$919$	−15.1005	−0.498120	−0.249060	+	0.968488i	$0.580122\pi$
$920$	−9.07107	−0.299064
$921$	0	0
$922$	−0.928932	−0.0305928
$923$	30.2426	0.995449
$924$	0	0
$925$	−6.24264	−0.205257
$926$	−7.68629	−0.252587
$927$	0	0
$928$	−7.24264	−0.237751
$929$	−25.7990	−0.846437	−0.423219	−	0.906028i	$-0.639100\pi$
$929$	−25.7990	−0.846437	−0.423219	+	0.906028i	$0.639100\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	8.48528	0.277945
$933$	0	0
$934$	−35.0711	−1.14756
$935$	4.82843	0.157906
$936$	0	0
$937$	50.4264	1.64736	0.823679	−	0.567056i	$-0.191918\pi$
$937$	50.4264	1.64736	0.823679	+	0.567056i	$0.191918\pi$
$938$	−7.24264	−0.236481
$939$	0	0
$940$	−7.07107	−0.230633
$941$	38.9289	1.26905	0.634523	−	0.772904i	$-0.281196\pi$
$941$	38.9289	1.26905	0.634523	+	0.772904i	$0.281196\pi$
$942$	0	0
$943$	34.7279	1.13090
$944$	6.82843	0.222246
$945$	0	0
$946$	3.41421	0.111006
$947$	17.1838	0.558397	0.279199	−	0.960233i	$-0.409931\pi$
$947$	17.1838	0.558397	0.279199	+	0.960233i	$0.409931\pi$
$948$	0	0
$949$	−2.89949	−0.0941216
$950$	1.00000	0.0324443
$951$	0	0
$952$	−1.41421	−0.0458349
$953$	28.8995	0.936146	0.468073	−	0.883690i	$-0.344948\pi$
$953$	28.8995	0.936146	0.468073	+	0.883690i	$0.344948\pi$
$954$	0	0
$955$	0.100505	0.00325227
$956$	−14.4142	−0.466189
$957$	0	0
$958$	14.8284	0.479085
$959$	−18.0711	−0.583545
$960$	0	0
$961$	−25.1716	−0.811986
$962$	23.8995	0.770551
$963$	0	0
$964$	−18.5858	−0.598608
$965$	−1.17157	−0.0377143
$966$	0	0
$967$	27.7574	0.892616	0.446308	−	0.894879i	$-0.352738\pi$
$967$	27.7574	0.892616	0.446308	+	0.894879i	$0.352738\pi$
$968$	−0.656854	−0.0211121
$969$	0	0
$970$	−17.5563	−0.563700
$971$	−6.92893	−0.222360	−0.111180	−	0.993800i	$-0.535463\pi$
$971$	−6.92893	−0.222360	−0.111180	+	0.993800i	$0.535463\pi$
$972$	0	0
$973$	2.24264	0.0718958
$974$	−9.21320	−0.295210
$975$	0	0
$976$	−14.0711	−0.450404
$977$	2.92893	0.0937048	0.0468524	−	0.998902i	$-0.485081\pi$
$977$	2.92893	0.0937048	0.0468524	+	0.998902i	$0.485081\pi$
$978$	0	0
$979$	17.3137	0.553349
$980$	6.00000	0.191663
$981$	0	0
$982$	−28.6274	−0.913538
$983$	38.6569	1.23296	0.616481	−	0.787370i	$-0.288558\pi$
$983$	38.6569	1.23296	0.616481	+	0.787370i	$0.288558\pi$
$984$	0	0
$985$	−3.14214	−0.100117
$986$	−10.2426	−0.326192
$987$	0	0
$988$	−3.82843	−0.121798
$989$	−9.07107	−0.288443
$990$	0	0
$991$	−49.2548	−1.56463	−0.782316	−	0.622882i	$-0.785962\pi$
$991$	−49.2548	−1.56463	−0.782316	+	0.622882i	$0.785962\pi$
$992$	−2.41421	−0.0766514
$993$	0	0
$994$	−7.89949	−0.250557
$995$	19.8995	0.630856
$996$	0	0
$997$	42.8284	1.35639	0.678195	−	0.734882i	$-0.262762\pi$
$997$	42.8284	1.35639	0.678195	+	0.734882i	$0.262762\pi$
$998$	−16.4558	−0.520901
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3870.2.a.bc.1.1		2
3.2	odd	2		430.2.a.g.1.1	✓	2
12.11	even	2		3440.2.a.j.1.2		2
15.2	even	4		2150.2.b.o.1549.4		4
15.8	even	4		2150.2.b.o.1549.1		4
15.14	odd	2		2150.2.a.v.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
430.2.a.g.1.1	✓	2	3.2	odd	2
2150.2.a.v.1.2		2	15.14	odd	2
2150.2.b.o.1549.1		4	15.8	even	4
2150.2.b.o.1549.4		4	15.2	even	4
3440.2.a.j.1.2		2	12.11	even	2
3870.2.a.bc.1.1		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 3870 = 2 \cdot 3^{2} \cdot 5 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3870.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{10} -3.41421 q^{11} +3.82843 q^{13} -1.00000 q^{14} +1.00000 q^{16} +1.41421 q^{17} -1.00000 q^{19} -1.00000 q^{20} +3.41421 q^{22} -9.07107 q^{23} +1.00000 q^{25} -3.82843 q^{26} +1.00000 q^{28} +7.24264 q^{29} +2.41421 q^{31} -1.00000 q^{32} -1.41421 q^{34} -1.00000 q^{35} -6.24264 q^{37} +1.00000 q^{38} +1.00000 q^{40} -3.82843 q^{41} +1.00000 q^{43} -3.41421 q^{44} +9.07107 q^{46} +7.07107 q^{47} -6.00000 q^{49} -1.00000 q^{50} +3.82843 q^{52} -5.65685 q^{53} +3.41421 q^{55} -1.00000 q^{56} -7.24264 q^{58} +6.82843 q^{59} -14.0711 q^{61} -2.41421 q^{62} +1.00000 q^{64} -3.82843 q^{65} +7.24264 q^{67} +1.41421 q^{68} +1.00000 q^{70} +7.89949 q^{71} -0.757359 q^{73} +6.24264 q^{74} -1.00000 q^{76} -3.41421 q^{77} -5.58579 q^{79} -1.00000 q^{80} +3.82843 q^{82} +7.65685 q^{83} -1.41421 q^{85} -1.00000 q^{86} +3.41421 q^{88} -5.07107 q^{89} +3.82843 q^{91} -9.07107 q^{92} -7.07107 q^{94} +1.00000 q^{95} -17.5563 q^{97} +6.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8} + 2 q^{10} - 4 q^{11} + 2 q^{13} - 2 q^{14} + 2 q^{16} - 2 q^{19} - 2 q^{20} + 4 q^{22} - 4 q^{23} + 2 q^{25} - 2 q^{26} + 2 q^{28} + 6 q^{29} + 2 q^{31} - 2 q^{32} - 2 q^{35} - 4 q^{37} + 2 q^{38} + 2 q^{40} - 2 q^{41} + 2 q^{43} - 4 q^{44} + 4 q^{46} - 12 q^{49} - 2 q^{50} + 2 q^{52} + 4 q^{55} - 2 q^{56} - 6 q^{58} + 8 q^{59} - 14 q^{61} - 2 q^{62} + 2 q^{64} - 2 q^{65} + 6 q^{67} + 2 q^{70} - 4 q^{71} - 10 q^{73} + 4 q^{74} - 2 q^{76} - 4 q^{77} - 14 q^{79} - 2 q^{80} + 2 q^{82} + 4 q^{83} - 2 q^{86} + 4 q^{88} + 4 q^{89} + 2 q^{91} - 4 q^{92} + 2 q^{95} - 4 q^{97} + 12 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 - 2 * q^8 + 2 * q^10 - 4 * q^11 + 2 * q^13 - 2 * q^14 + 2 * q^16 - 2 * q^19 - 2 * q^20 + 4 * q^22 - 4 * q^23 + 2 * q^25 - 2 * q^26 + 2 * q^28 + 6 * q^29 + 2 * q^31 - 2 * q^32 - 2 * q^35 - 4 * q^37 + 2 * q^38 + 2 * q^40 - 2 * q^41 + 2 * q^43 - 4 * q^44 + 4 * q^46 - 12 * q^49 - 2 * q^50 + 2 * q^52 + 4 * q^55 - 2 * q^56 - 6 * q^58 + 8 * q^59 - 14 * q^61 - 2 * q^62 + 2 * q^64 - 2 * q^65 + 6 * q^67 + 2 * q^70 - 4 * q^71 - 10 * q^73 + 4 * q^74 - 2 * q^76 - 4 * q^77 - 14 * q^79 - 2 * q^80 + 2 * q^82 + 4 * q^83 - 2 * q^86 + 4 * q^88 + 4 * q^89 + 2 * q^91 - 4 * q^92 + 2 * q^95 - 4 * q^97 + 12 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more