LMFDB - Embedded newform 2150.2.a.v.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2150,2,Mod(1,2150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2150, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2150.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2150 = 2 \cdot 5^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2150.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:	$17.1678364346$
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:	$\Z[a_1, a_2, a_3]$
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			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	2150.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} -1.41421 q^{6} -1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} -1.41421 q^{6} -1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{9} +3.41421 q^{11} +1.41421 q^{12} -3.82843 q^{13} +1.00000 q^{14} +1.00000 q^{16} +1.41421 q^{17} +1.00000 q^{18} -1.00000 q^{19} -1.41421 q^{21} -3.41421 q^{22} -9.07107 q^{23} -1.41421 q^{24} +3.82843 q^{26} -5.65685 q^{27} -1.00000 q^{28} -7.24264 q^{29} +2.41421 q^{31} -1.00000 q^{32} +4.82843 q^{33} -1.41421 q^{34} -1.00000 q^{36} +6.24264 q^{37} +1.00000 q^{38} -5.41421 q^{39} +3.82843 q^{41} +1.41421 q^{42} -1.00000 q^{43} +3.41421 q^{44} +9.07107 q^{46} +7.07107 q^{47} +1.41421 q^{48} -6.00000 q^{49} +2.00000 q^{51} -3.82843 q^{52} -5.65685 q^{53} +5.65685 q^{54} +1.00000 q^{56} -1.41421 q^{57} +7.24264 q^{58} -6.82843 q^{59} -14.0711 q^{61} -2.41421 q^{62} +1.00000 q^{63} +1.00000 q^{64} -4.82843 q^{66} -7.24264 q^{67} +1.41421 q^{68} -12.8284 q^{69} -7.89949 q^{71} +1.00000 q^{72} +0.757359 q^{73} -6.24264 q^{74} -1.00000 q^{76} -3.41421 q^{77} +5.41421 q^{78} -5.58579 q^{79} -5.00000 q^{81} -3.82843 q^{82} +7.65685 q^{83} -1.41421 q^{84} +1.00000 q^{86} -10.2426 q^{87} -3.41421 q^{88} +5.07107 q^{89} +3.82843 q^{91} -9.07107 q^{92} +3.41421 q^{93} -7.07107 q^{94} -1.41421 q^{96} +17.5563 q^{97} +6.00000 q^{98} -3.41421 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 2 * q^8 - 2 * q^9	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8} - 2 q^{9} + 4 q^{11} - 2 q^{13} + 2 q^{14} + 2 q^{16} + 2 q^{18} - 2 q^{19} - 4 q^{22} - 4 q^{23} + 2 q^{26} - 2 q^{28} - 6 q^{29} + 2 q^{31} - 2 q^{32} + 4 q^{33} - 2 q^{36} + 4 q^{37} + 2 q^{38} - 8 q^{39} + 2 q^{41} - 2 q^{43} + 4 q^{44} + 4 q^{46} - 12 q^{49} + 4 q^{51} - 2 q^{52} + 2 q^{56} + 6 q^{58} - 8 q^{59} - 14 q^{61} - 2 q^{62} + 2 q^{63} + 2 q^{64} - 4 q^{66} - 6 q^{67} - 20 q^{69} + 4 q^{71} + 2 q^{72} + 10 q^{73} - 4 q^{74} - 2 q^{76} - 4 q^{77} + 8 q^{78} - 14 q^{79} - 10 q^{81} - 2 q^{82} + 4 q^{83} + 2 q^{86} - 12 q^{87} - 4 q^{88} - 4 q^{89} + 2 q^{91} - 4 q^{92} + 4 q^{93} + 4 q^{97} + 12 q^{98} - 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 2 * q^8 - 2 * q^9 + 4 * q^11 - 2 * q^13 + 2 * q^14 + 2 * q^16 + 2 * q^18 - 2 * q^19 - 4 * q^22 - 4 * q^23 + 2 * q^26 - 2 * q^28 - 6 * q^29 + 2 * q^31 - 2 * q^32 + 4 * q^33 - 2 * q^36 + 4 * q^37 + 2 * q^38 - 8 * q^39 + 2 * q^41 - 2 * q^43 + 4 * q^44 + 4 * q^46 - 12 * q^49 + 4 * q^51 - 2 * q^52 + 2 * q^56 + 6 * q^58 - 8 * q^59 - 14 * q^61 - 2 * q^62 + 2 * q^63 + 2 * q^64 - 4 * q^66 - 6 * q^67 - 20 * q^69 + 4 * q^71 + 2 * q^72 + 10 * q^73 - 4 * q^74 - 2 * q^76 - 4 * q^77 + 8 * q^78 - 14 * q^79 - 10 * q^81 - 2 * q^82 + 4 * q^83 + 2 * q^86 - 12 * q^87 - 4 * q^88 - 4 * q^89 + 2 * q^91 - 4 * q^92 + 4 * q^93 + 4 * q^97 + 12 * q^98 - 4 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.41421	0.816497	0.408248	−	0.912871i	$-0.366140\pi$
$3$	1.41421	0.816497	0.408248	+	0.912871i	$0.366140\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.41421	−0.577350
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	−1.00000	−0.353553
$9$	−1.00000	−0.333333
$10$	0	0
$11$	3.41421	1.02942	0.514712	−	0.857363i	$-0.327899\pi$
$11$	3.41421	1.02942	0.514712	+	0.857363i	$0.327899\pi$
$12$	1.41421	0.408248
$13$	−3.82843	−1.06181	−0.530907	−	0.847430i	$-0.678149\pi$
$13$	−3.82843	−1.06181	−0.530907	+	0.847430i	$0.678149\pi$
$14$	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	$0.445139\pi$
$18$	1.00000	0.235702
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	−1.41421	−0.308607
$22$	−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	$0.894646\pi$
$24$	−1.41421	−0.288675
$25$	0	0
$26$	3.82843	0.750816
$27$	−5.65685	−1.08866
$28$	−1.00000	−0.188982
$29$	−7.24264	−1.34492	−0.672462	−	0.740131i	$-0.734763\pi$
$29$	−7.24264	−1.34492	−0.672462	+	0.740131i	$0.734763\pi$
$30$	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
$33$	4.82843	0.840521
$34$	−1.41421	−0.242536
$35$	0	0
$36$	−1.00000	−0.166667
$37$	6.24264	1.02628	0.513142	−	0.858304i	$-0.328481\pi$
$37$	6.24264	1.02628	0.513142	+	0.858304i	$0.328481\pi$
$38$	1.00000	0.162221
$39$	−5.41421	−0.866968
$40$	0	0
$41$	3.82843	0.597900	0.298950	−	0.954269i	$-0.403364\pi$
$41$	3.82843	0.597900	0.298950	+	0.954269i	$0.403364\pi$
$42$	1.41421	0.218218
$43$	−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	$0.327528\pi$
$48$	1.41421	0.204124
$49$	−6.00000	−0.857143
$50$	0	0
$51$	2.00000	0.280056
$52$	−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	$0.627012\pi$
$54$	5.65685	0.769800
$55$	0	0
$56$	1.00000	0.133631
$57$	−1.41421	−0.187317
$58$	7.24264	0.951005
$59$	−6.82843	−0.888985	−0.444493	−	0.895782i	$-0.646616\pi$
$59$	−6.82843	−0.888985	−0.444493	+	0.895782i	$0.646616\pi$
$60$	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
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	−4.82843	−0.594338
$67$	−7.24264	−0.884829	−0.442415	−	0.896811i	$-0.645878\pi$
$67$	−7.24264	−0.884829	−0.442415	+	0.896811i	$0.645878\pi$
$68$	1.41421	0.171499
$69$	−12.8284	−1.54436
$70$	0	0
$71$	−7.89949	−0.937498	−0.468749	−	0.883332i	$-0.655295\pi$
$71$	−7.89949	−0.937498	−0.468749	+	0.883332i	$0.655295\pi$
$72$	1.00000	0.117851
$73$	0.757359	0.0886422	0.0443211	−	0.999017i	$-0.485888\pi$
$73$	0.757359	0.0886422	0.0443211	+	0.999017i	$0.485888\pi$
$74$	−6.24264	−0.725692
$75$	0	0
$76$	−1.00000	−0.114708
$77$	−3.41421	−0.389086
$78$	5.41421	0.613039
$79$	−5.58579	−0.628450	−0.314225	−	0.949349i	$-0.601745\pi$
$79$	−5.58579	−0.628450	−0.314225	+	0.949349i	$0.601745\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	−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	$0.361951\pi$
$84$	−1.41421	−0.154303
$85$	0	0
$86$	1.00000	0.107833
$87$	−10.2426	−1.09813
$88$	−3.41421	−0.363956
$89$	5.07107	0.537532	0.268766	−	0.963205i	$-0.413384\pi$
$89$	5.07107	0.537532	0.268766	+	0.963205i	$0.413384\pi$
$90$	0	0
$91$	3.82843	0.401328
$92$	−9.07107	−0.945724
$93$	3.41421	0.354037
$94$	−7.07107	−0.729325
$95$	0	0
$96$	−1.41421	−0.144338
$97$	17.5563	1.78258	0.891289	−	0.453436i	$-0.149802\pi$
$97$	17.5563	1.78258	0.891289	+	0.453436i	$0.149802\pi$
$98$	6.00000	0.606092
$99$	−3.41421	−0.343141
$100$	0	0
$101$	−1.41421	−0.140720	−0.0703598	−	0.997522i	$-0.522415\pi$
$101$	−1.41421	−0.140720	−0.0703598	+	0.997522i	$0.522415\pi$
$102$	−2.00000	−0.198030
$103$	4.82843	0.475759	0.237880	−	0.971295i	$-0.423548\pi$
$103$	4.82843	0.475759	0.237880	+	0.971295i	$0.423548\pi$
$104$	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	$0.783171\pi$
$108$	−5.65685	−0.544331
$109$	−5.17157	−0.495347	−0.247673	−	0.968844i	$-0.579666\pi$
$109$	−5.17157	−0.495347	−0.247673	+	0.968844i	$0.579666\pi$
$110$	0	0
$111$	8.82843	0.837957
$112$	−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	$0.252821\pi$
$114$	1.41421	0.132453
$115$	0	0
$116$	−7.24264	−0.672462
$117$	3.82843	0.353938
$118$	6.82843	0.628608
$119$	−1.41421	−0.129641
$120$	0	0
$121$	0.656854	0.0597140
$122$	14.0711	1.27393
$123$	5.41421	0.488183
$124$	2.41421	0.216803
$125$	0	0
$126$	−1.00000	−0.0890871
$127$	−20.4853	−1.81777	−0.908887	−	0.417042i	$-0.863067\pi$
$127$	−20.4853	−1.81777	−0.908887	+	0.417042i	$0.863067\pi$
$128$	−1.00000	−0.0883883
$129$	−1.41421	−0.124515
$130$	0	0
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	4.82843	0.420261
$133$	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	$0.780722\pi$
$138$	12.8284	1.09203
$139$	2.24264	0.190218	0.0951092	−	0.995467i	$-0.469680\pi$
$139$	2.24264	0.190218	0.0951092	+	0.995467i	$0.469680\pi$
$140$	0	0
$141$	10.0000	0.842152
$142$	7.89949	0.662911
$143$	−13.0711	−1.09306
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−0.757359	−0.0626795
$147$	−8.48528	−0.699854
$148$	6.24264	0.513142
$149$	20.5563	1.68404	0.842021	−	0.539445i	$-0.181366\pi$
$149$	20.5563	1.68404	0.842021	+	0.539445i	$0.181366\pi$
$150$	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
$153$	−1.41421	−0.114332
$154$	3.41421	0.275125
$155$	0	0
$156$	−5.41421	−0.433484
$157$	−1.75736	−0.140253	−0.0701263	−	0.997538i	$-0.522340\pi$
$157$	−1.75736	−0.140253	−0.0701263	+	0.997538i	$0.522340\pi$
$158$	5.58579	0.444381
$159$	−8.00000	−0.634441
$160$	0	0
$161$	9.07107	0.714900
$162$	5.00000	0.392837
$163$	−2.34315	−0.183529	−0.0917647	−	0.995781i	$-0.529251\pi$
$163$	−2.34315	−0.183529	−0.0917647	+	0.995781i	$0.529251\pi$
$164$	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	$0.262294\pi$
$168$	1.41421	0.109109
$169$	1.65685	0.127450
$170$	0	0
$171$	1.00000	0.0764719
$172$	−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	$0.731437\pi$
$174$	10.2426	0.776493
$175$	0	0
$176$	3.41421	0.257356
$177$	−9.65685	−0.725854
$178$	−5.07107	−0.380093
$179$	0.514719	0.0384719	0.0192359	−	0.999815i	$-0.493877\pi$
$179$	0.514719	0.0384719	0.0192359	+	0.999815i	$0.493877\pi$
$180$	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
$183$	−19.8995	−1.47101
$184$	9.07107	0.668728
$185$	0	0
$186$	−3.41421	−0.250342
$187$	4.82843	0.353090
$188$	7.07107	0.515711
$189$	5.65685	0.411476
$190$	0	0
$191$	0.100505	0.00727229	0.00363615	−	0.999993i	$-0.498843\pi$
$191$	0.100505	0.00727229	0.00363615	+	0.999993i	$0.498843\pi$
$192$	1.41421	0.102062
$193$	−1.17157	−0.0843317	−0.0421658	−	0.999111i	$-0.513426\pi$
$193$	−1.17157	−0.0843317	−0.0421658	+	0.999111i	$0.513426\pi$
$194$	−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	$0.464295\pi$
$198$	3.41421	0.242638
$199$	−19.8995	−1.41064	−0.705319	−	0.708890i	$-0.749196\pi$
$199$	−19.8995	−1.41064	−0.705319	+	0.708890i	$0.749196\pi$
$200$	0	0
$201$	−10.2426	−0.722460
$202$	1.41421	0.0995037
$203$	7.24264	0.508334
$204$	2.00000	0.140028
$205$	0	0
$206$	−4.82843	−0.336412
$207$	9.07107	0.630483
$208$	−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
$213$	−11.1716	−0.765464
$214$	16.0711	1.09860
$215$	0	0
$216$	5.65685	0.384900
$217$	−2.41421	−0.163887
$218$	5.17157	0.350263
$219$	1.07107	0.0723761
$220$	0	0
$221$	−5.41421	−0.364199
$222$	−8.82843	−0.592525
$223$	12.1421	0.813098	0.406549	−	0.913629i	$-0.366732\pi$
$223$	12.1421	0.813098	0.406549	+	0.913629i	$0.366732\pi$
$224$	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	$0.663770\pi$
$228$	−1.41421	−0.0936586
$229$	−4.82843	−0.319071	−0.159536	−	0.987192i	$-0.551000\pi$
$229$	−4.82843	−0.319071	−0.159536	+	0.987192i	$0.551000\pi$
$230$	0	0
$231$	−4.82843	−0.317687
$232$	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	$0.410347\pi$
$234$	−3.82843	−0.250272
$235$	0	0
$236$	−6.82843	−0.444493
$237$	−7.89949	−0.513127
$238$	1.41421	0.0916698
$239$	14.4142	0.932378	0.466189	−	0.884685i	$-0.345627\pi$
$239$	14.4142	0.932378	0.466189	+	0.884685i	$0.345627\pi$
$240$	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
$243$	9.89949	0.635053
$244$	−14.0711	−0.900808
$245$	0	0
$246$	−5.41421	−0.345198
$247$	3.82843	0.243597
$248$	−2.41421	−0.153303
$249$	10.8284	0.686224
$250$	0	0
$251$	22.9706	1.44989	0.724945	−	0.688807i	$-0.241865\pi$
$251$	22.9706	1.44989	0.724945	+	0.688807i	$0.241865\pi$
$252$	1.00000	0.0629941
$253$	−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	$0.389591\pi$
$258$	1.41421	0.0880451
$259$	−6.24264	−0.387899
$260$	0	0
$261$	7.24264	0.448308
$262$	−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	$0.505052\pi$
$264$	−4.82843	−0.297169
$265$	0	0
$266$	−1.00000	−0.0613139
$267$	7.17157	0.438893
$268$	−7.24264	−0.442415
$269$	14.8284	0.904105	0.452053	−	0.891991i	$-0.350692\pi$
$269$	14.8284	0.904105	0.452053	+	0.891991i	$0.350692\pi$
$270$	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
$273$	5.41421	0.327683
$274$	18.0711	1.09171
$275$	0	0
$276$	−12.8284	−0.772181
$277$	−16.4853	−0.990505	−0.495252	−	0.868749i	$-0.664924\pi$
$277$	−16.4853	−0.990505	−0.495252	+	0.868749i	$0.664924\pi$
$278$	−2.24264	−0.134505
$279$	−2.41421	−0.144535
$280$	0	0
$281$	−12.6569	−0.755045	−0.377522	−	0.926000i	$-0.623224\pi$
$281$	−12.6569	−0.755045	−0.377522	+	0.926000i	$0.623224\pi$
$282$	−10.0000	−0.595491
$283$	−27.2426	−1.61941	−0.809703	−	0.586839i	$-0.800372\pi$
$283$	−27.2426	−1.61941	−0.809703	+	0.586839i	$0.800372\pi$
$284$	−7.89949	−0.468749
$285$	0	0
$286$	13.0711	0.772908
$287$	−3.82843	−0.225985
$288$	1.00000	0.0589256
$289$	−15.0000	−0.882353
$290$	0	0
$291$	24.8284	1.45547
$292$	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	$0.265875\pi$
$294$	8.48528	0.494872
$295$	0	0
$296$	−6.24264	−0.362846
$297$	−19.3137	−1.12070
$298$	−20.5563	−1.19080
$299$	34.7279	2.00837
$300$	0	0
$301$	1.00000	0.0576390
$302$	6.48528	0.373186
$303$	−2.00000	−0.114897
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	1.41421	0.0808452
$307$	20.0711	1.14552	0.572758	−	0.819724i	$-0.305873\pi$
$307$	20.0711	1.14552	0.572758	+	0.819724i	$0.305873\pi$
$308$	−3.41421	−0.194543
$309$	6.82843	0.388456
$310$	0	0
$311$	10.0711	0.571078	0.285539	−	0.958367i	$-0.407827\pi$
$311$	10.0711	0.571078	0.285539	+	0.958367i	$0.407827\pi$
$312$	5.41421	0.306519
$313$	−21.3137	−1.20472	−0.602361	−	0.798224i	$-0.705773\pi$
$313$	−21.3137	−1.20472	−0.602361	+	0.798224i	$0.705773\pi$
$314$	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	$0.805135\pi$
$318$	8.00000	0.448618
$319$	−24.7279	−1.38450
$320$	0	0
$321$	−22.7279	−1.26855
$322$	−9.07107	−0.505511
$323$	−1.41421	−0.0786889
$324$	−5.00000	−0.277778
$325$	0	0
$326$	2.34315	0.129775
$327$	−7.31371	−0.404449
$328$	−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
$333$	−6.24264	−0.342095
$334$	−17.5563	−0.960641
$335$	0	0
$336$	−1.41421	−0.0771517
$337$	19.6569	1.07078	0.535389	−	0.844606i	$-0.320165\pi$
$337$	19.6569	1.07078	0.535389	+	0.844606i	$0.320165\pi$
$338$	−1.65685	−0.0901210
$339$	21.0711	1.14442
$340$	0	0
$341$	8.24264	0.446364
$342$	−1.00000	−0.0540738
$343$	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	$0.379609\pi$
$348$	−10.2426	−0.549063
$349$	−27.4558	−1.46968	−0.734839	−	0.678242i	$-0.762742\pi$
$349$	−27.4558	−1.46968	−0.734839	+	0.678242i	$0.762742\pi$
$350$	0	0
$351$	21.6569	1.15596
$352$	−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	$0.537228\pi$
$354$	9.65685	0.513256
$355$	0	0
$356$	5.07107	0.268766
$357$	−2.00000	−0.105851
$358$	−0.514719	−0.0272037
$359$	−37.5269	−1.98059	−0.990297	−	0.138965i	$-0.955623\pi$
$359$	−37.5269	−1.98059	−0.990297	+	0.138965i	$0.955623\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0.928932	0.0488236
$363$	0.928932	0.0487563
$364$	3.82843	0.200664
$365$	0	0
$366$	19.8995	1.04016
$367$	8.68629	0.453421	0.226710	−	0.973962i	$-0.427203\pi$
$367$	8.68629	0.453421	0.226710	+	0.973962i	$0.427203\pi$
$368$	−9.07107	−0.472862
$369$	−3.82843	−0.199300
$370$	0	0
$371$	5.65685	0.293689
$372$	3.41421	0.177019
$373$	10.7279	0.555471	0.277735	−	0.960658i	$-0.410416\pi$
$373$	10.7279	0.555471	0.277735	+	0.960658i	$0.410416\pi$
$374$	−4.82843	−0.249672
$375$	0	0
$376$	−7.07107	−0.364662
$377$	27.7279	1.42806
$378$	−5.65685	−0.290957
$379$	−24.1421	−1.24010	−0.620049	−	0.784563i	$-0.712887\pi$
$379$	−24.1421	−1.24010	−0.620049	+	0.784563i	$0.712887\pi$
$380$	0	0
$381$	−28.9706	−1.48421
$382$	−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	$0.619170\pi$
$384$	−1.41421	−0.0721688
$385$	0	0
$386$	1.17157	0.0596315
$387$	1.00000	0.0508329
$388$	17.5563	0.891289
$389$	15.1716	0.769229	0.384615	−	0.923077i	$-0.374334\pi$
$389$	15.1716	0.769229	0.384615	+	0.923077i	$0.374334\pi$
$390$	0	0
$391$	−12.8284	−0.648761
$392$	6.00000	0.303046
$393$	8.48528	0.428026
$394$	−3.14214	−0.158299
$395$	0	0
$396$	−3.41421	−0.171571
$397$	3.51472	0.176399	0.0881993	−	0.996103i	$-0.471889\pi$
$397$	3.51472	0.176399	0.0881993	+	0.996103i	$0.471889\pi$
$398$	19.8995	0.997472
$399$	1.41421	0.0707992
$400$	0	0
$401$	38.1127	1.90326	0.951629	−	0.307251i	$-0.0994090\pi$
$401$	38.1127	1.90326	0.951629	+	0.307251i	$0.0994090\pi$
$402$	10.2426	0.510856
$403$	−9.24264	−0.460409
$404$	−1.41421	−0.0703598
$405$	0	0
$406$	−7.24264	−0.359446
$407$	21.3137	1.05648
$408$	−2.00000	−0.0990148
$409$	15.2132	0.752244	0.376122	−	0.926570i	$-0.377257\pi$
$409$	15.2132	0.752244	0.376122	+	0.926570i	$0.377257\pi$
$410$	0	0
$411$	−25.5563	−1.26060
$412$	4.82843	0.237880
$413$	6.82843	0.336005
$414$	−9.07107	−0.445819
$415$	0	0
$416$	3.82843	0.187704
$417$	3.17157	0.155313
$418$	3.41421	0.166995
$419$	26.7990	1.30922	0.654608	−	0.755968i	$-0.272834\pi$
$419$	26.7990	1.30922	0.654608	+	0.755968i	$0.272834\pi$
$420$	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
$423$	−7.07107	−0.343807
$424$	5.65685	0.274721
$425$	0	0
$426$	11.1716	0.541264
$427$	14.0711	0.680947
$428$	−16.0711	−0.776824
$429$	−18.4853	−0.892478
$430$	0	0
$431$	34.9706	1.68447	0.842236	−	0.539108i	$-0.181239\pi$
$431$	34.9706	1.68447	0.842236	+	0.539108i	$0.181239\pi$
$432$	−5.65685	−0.272166
$433$	21.9289	1.05384	0.526919	−	0.849916i	$-0.323347\pi$
$433$	21.9289	1.05384	0.526919	+	0.849916i	$0.323347\pi$
$434$	2.41421	0.115886
$435$	0	0
$436$	−5.17157	−0.247673
$437$	9.07107	0.433928
$438$	−1.07107	−0.0511776
$439$	7.31371	0.349064	0.174532	−	0.984651i	$-0.444159\pi$
$439$	7.31371	0.349064	0.174532	+	0.984651i	$0.444159\pi$
$440$	0	0
$441$	6.00000	0.285714
$442$	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	$0.306235\pi$
$444$	8.82843	0.418979
$445$	0	0
$446$	−12.1421	−0.574947
$447$	29.0711	1.37501
$448$	−1.00000	−0.0472456
$449$	13.7574	0.649250	0.324625	−	0.945843i	$-0.394762\pi$
$449$	13.7574	0.649250	0.324625	+	0.945843i	$0.394762\pi$
$450$	0	0
$451$	13.0711	0.615493
$452$	14.8995	0.700813
$453$	−9.17157	−0.430918
$454$	14.8284	0.695933
$455$	0	0
$456$	1.41421	0.0662266
$457$	13.5147	0.632192	0.316096	−	0.948727i	$-0.397628\pi$
$457$	13.5147	0.632192	0.316096	+	0.948727i	$0.397628\pi$
$458$	4.82843	0.225618
$459$	−8.00000	−0.373408
$460$	0	0
$461$	−0.928932	−0.0432647	−0.0216323	−	0.999766i	$-0.506886\pi$
$461$	−0.928932	−0.0432647	−0.0216323	+	0.999766i	$0.506886\pi$
$462$	4.82843	0.224639
$463$	−7.68629	−0.357212	−0.178606	−	0.983921i	$-0.557159\pi$
$463$	−7.68629	−0.357212	−0.178606	+	0.983921i	$0.557159\pi$
$464$	−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	$0.198680\pi$
$468$	3.82843	0.176969
$469$	7.24264	0.334434
$470$	0	0
$471$	−2.48528	−0.114516
$472$	6.82843	0.314304
$473$	−3.41421	−0.156986
$474$	7.89949	0.362836
$475$	0	0
$476$	−1.41421	−0.0648204
$477$	5.65685	0.259010
$478$	−14.4142	−0.659291
$479$	14.8284	0.677528	0.338764	−	0.940871i	$-0.389991\pi$
$479$	14.8284	0.677528	0.338764	+	0.940871i	$0.389991\pi$
$480$	0	0
$481$	−23.8995	−1.08972
$482$	18.5858	0.846559
$483$	12.8284	0.583714
$484$	0.656854	0.0298570
$485$	0	0
$486$	−9.89949	−0.449050
$487$	−9.21320	−0.417490	−0.208745	−	0.977970i	$-0.566938\pi$
$487$	−9.21320	−0.417490	−0.208745	+	0.977970i	$0.566938\pi$
$488$	14.0711	0.636967
$489$	−3.31371	−0.149851
$490$	0	0
$491$	−28.6274	−1.29194	−0.645969	−	0.763364i	$-0.723546\pi$
$491$	−28.6274	−1.29194	−0.645969	+	0.763364i	$0.723546\pi$
$492$	5.41421	0.244092
$493$	−10.2426	−0.461305
$494$	−3.82843	−0.172249
$495$	0	0
$496$	2.41421	0.108401
$497$	7.89949	0.354341
$498$	−10.8284	−0.485233
$499$	16.4558	0.736665	0.368332	−	0.929694i	$-0.379929\pi$
$499$	16.4558	0.736665	0.368332	+	0.929694i	$0.379929\pi$
$500$	0	0
$501$	24.8284	1.10925
$502$	−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	$0.350165\pi$
$504$	−1.00000	−0.0445435
$505$	0	0
$506$	30.9706	1.37681
$507$	2.34315	0.104063
$508$	−20.4853	−0.908887
$509$	−3.51472	−0.155787	−0.0778936	−	0.996962i	$-0.524819\pi$
$509$	−3.51472	−0.155787	−0.0778936	+	0.996962i	$0.524819\pi$
$510$	0	0
$511$	−0.757359	−0.0335036
$512$	−1.00000	−0.0441942
$513$	5.65685	0.249756
$514$	−10.8995	−0.480756
$515$	0	0
$516$	−1.41421	−0.0622573
$517$	24.1421	1.06177
$518$	6.24264	0.274286
$519$	−24.7279	−1.08544
$520$	0	0
$521$	3.07107	0.134546	0.0672730	−	0.997735i	$-0.478570\pi$
$521$	3.07107	0.134546	0.0672730	+	0.997735i	$0.478570\pi$
$522$	−7.24264	−0.317002
$523$	14.7279	0.644007	0.322004	−	0.946738i	$-0.395644\pi$
$523$	14.7279	0.644007	0.322004	+	0.946738i	$0.395644\pi$
$524$	6.00000	0.262111
$525$	0	0
$526$	0.514719	0.0224428
$527$	3.41421	0.148725
$528$	4.82843	0.210130
$529$	59.2843	2.57758
$530$	0	0
$531$	6.82843	0.296328
$532$	1.00000	0.0433555
$533$	−14.6569	−0.634859
$534$	−7.17157	−0.310344
$535$	0	0
$536$	7.24264	0.312834
$537$	0.727922	0.0314122
$538$	−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
$543$	−1.31371	−0.0563766
$544$	−1.41421	−0.0606339
$545$	0	0
$546$	−5.41421	−0.231707
$547$	12.0000	0.513083	0.256541	−	0.966533i	$-0.417417\pi$
$547$	12.0000	0.513083	0.256541	+	0.966533i	$0.417417\pi$
$548$	−18.0711	−0.771958
$549$	14.0711	0.600539
$550$	0	0
$551$	7.24264	0.308547
$552$	12.8284	0.546014
$553$	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	$0.777914\pi$
$558$	2.41421	0.102202
$559$	3.82843	0.161925
$560$	0	0
$561$	6.82843	0.288296
$562$	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	$0.390027\pi$
$564$	10.0000	0.421076
$565$	0	0
$566$	27.2426	1.14509
$567$	5.00000	0.209980
$568$	7.89949	0.331455
$569$	−16.7990	−0.704250	−0.352125	−	0.935953i	$-0.614541\pi$
$569$	−16.7990	−0.704250	−0.352125	+	0.935953i	$0.614541\pi$
$570$	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
$573$	0.142136	0.00593780
$574$	3.82843	0.159795
$575$	0	0
$576$	−1.00000	−0.0416667
$577$	8.41421	0.350288	0.175144	−	0.984543i	$-0.443961\pi$
$577$	8.41421	0.350288	0.175144	+	0.984543i	$0.443961\pi$
$578$	15.0000	0.623918
$579$	−1.65685	−0.0688565
$580$	0	0
$581$	−7.65685	−0.317660
$582$	−24.8284	−1.02917
$583$	−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	$0.523383\pi$
$588$	−8.48528	−0.349927
$589$	−2.41421	−0.0994759
$590$	0	0
$591$	4.44365	0.182787
$592$	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	$0.508122\pi$
$594$	19.3137	0.792451
$595$	0	0
$596$	20.5563	0.842021
$597$	−28.1421	−1.15178
$598$	−34.7279	−1.42013
$599$	38.1421	1.55845	0.779223	−	0.626747i	$-0.215614\pi$
$599$	38.1421	1.55845	0.779223	+	0.626747i	$0.215614\pi$
$600$	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
$603$	7.24264	0.294943
$604$	−6.48528	−0.263882
$605$	0	0
$606$	2.00000	0.0812444
$607$	−6.00000	−0.243532	−0.121766	−	0.992559i	$-0.538856\pi$
$607$	−6.00000	−0.243532	−0.121766	+	0.992559i	$0.538856\pi$
$608$	1.00000	0.0405554
$609$	10.2426	0.415053
$610$	0	0
$611$	−27.0711	−1.09518
$612$	−1.41421	−0.0571662
$613$	−43.8284	−1.77021	−0.885107	−	0.465388i	$-0.845915\pi$
$613$	−43.8284	−1.77021	−0.885107	+	0.465388i	$0.845915\pi$
$614$	−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	$0.801018\pi$
$618$	−6.82843	−0.274680
$619$	−19.8995	−0.799828	−0.399914	−	0.916553i	$-0.630960\pi$
$619$	−19.8995	−0.799828	−0.399914	+	0.916553i	$0.630960\pi$
$620$	0	0
$621$	51.3137	2.05915
$622$	−10.0711	−0.403813
$623$	−5.07107	−0.203168
$624$	−5.41421	−0.216742
$625$	0	0
$626$	21.3137	0.851867
$627$	−4.82843	−0.192829
$628$	−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
$633$	13.6569	0.542811
$634$	29.1421	1.15738
$635$	0	0
$636$	−8.00000	−0.317221
$637$	22.9706	0.910127
$638$	24.7279	0.978988
$639$	7.89949	0.312499
$640$	0	0
$641$	−30.7696	−1.21532	−0.607662	−	0.794196i	$-0.707893\pi$
$641$	−30.7696	−1.21532	−0.607662	+	0.794196i	$0.707893\pi$
$642$	22.7279	0.897000
$643$	−14.2721	−0.562836	−0.281418	−	0.959585i	$-0.590805\pi$
$643$	−14.2721	−0.562836	−0.281418	+	0.959585i	$0.590805\pi$
$644$	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	$0.842870\pi$
$648$	5.00000	0.196419
$649$	−23.3137	−0.915143
$650$	0	0
$651$	−3.41421	−0.133814
$652$	−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	$0.441030\pi$
$654$	7.31371	0.285989
$655$	0	0
$656$	3.82843	0.149475
$657$	−0.757359	−0.0295474
$658$	7.07107	0.275659
$659$	−12.3848	−0.482442	−0.241221	−	0.970470i	$-0.577548\pi$
$659$	−12.3848	−0.482442	−0.241221	+	0.970470i	$0.577548\pi$
$660$	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
$663$	−7.65685	−0.297368
$664$	−7.65685	−0.297144
$665$	0	0
$666$	6.24264	0.241897
$667$	65.6985	2.54386
$668$	17.5563	0.679276
$669$	17.1716	0.663891
$670$	0	0
$671$	−48.0416	−1.85463
$672$	1.41421	0.0545545
$673$	−17.5858	−0.677882	−0.338941	−	0.940808i	$-0.610069\pi$
$673$	−17.5858	−0.677882	−0.338941	+	0.940808i	$0.610069\pi$
$674$	−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	$0.454236\pi$
$678$	−21.0711	−0.809229
$679$	−17.5563	−0.673751
$680$	0	0
$681$	−20.9706	−0.803594
$682$	−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	$0.371810\pi$
$684$	1.00000	0.0382360
$685$	0	0
$686$	−13.0000	−0.496342
$687$	−6.82843	−0.260521
$688$	−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
$693$	3.41421	0.129695
$694$	−13.7574	−0.522222
$695$	0	0
$696$	10.2426	0.388246
$697$	5.41421	0.205078
$698$	27.4558	1.03922
$699$	12.0000	0.453882
$700$	0	0
$701$	36.7279	1.38719	0.693597	−	0.720363i	$-0.256025\pi$
$701$	36.7279	1.38719	0.693597	+	0.720363i	$0.256025\pi$
$702$	−21.6569	−0.817385
$703$	−6.24264	−0.235446
$704$	3.41421	0.128678
$705$	0	0
$706$	4.38478	0.165023
$707$	1.41421	0.0531870
$708$	−9.65685	−0.362927
$709$	35.2132	1.32246	0.661230	−	0.750183i	$-0.270035\pi$
$709$	35.2132	1.32246	0.661230	+	0.750183i	$0.270035\pi$
$710$	0	0
$711$	5.58579	0.209483
$712$	−5.07107	−0.190046
$713$	−21.8995	−0.820143
$714$	2.00000	0.0748481
$715$	0	0
$716$	0.514719	0.0192359
$717$	20.3848	0.761283
$718$	37.5269	1.40049
$719$	8.62742	0.321748	0.160874	−	0.986975i	$-0.448569\pi$
$719$	8.62742	0.321748	0.160874	+	0.986975i	$0.448569\pi$
$720$	0	0
$721$	−4.82843	−0.179820
$722$	18.0000	0.669891
$723$	−26.2843	−0.977523
$724$	−0.928932	−0.0345235
$725$	0	0
$726$	−0.928932	−0.0344759
$727$	14.0000	0.519231	0.259616	−	0.965712i	$-0.416404\pi$
$727$	14.0000	0.519231	0.259616	+	0.965712i	$0.416404\pi$
$728$	−3.82843	−0.141891
$729$	29.0000	1.07407
$730$	0	0
$731$	−1.41421	−0.0523066
$732$	−19.8995	−0.735506
$733$	−36.2426	−1.33865	−0.669326	−	0.742969i	$-0.733417\pi$
$733$	−36.2426	−1.33865	−0.669326	+	0.742969i	$0.733417\pi$
$734$	−8.68629	−0.320617
$735$	0	0
$736$	9.07107	0.334364
$737$	−24.7279	−0.910865
$738$	3.82843	0.140926
$739$	41.9706	1.54391	0.771956	−	0.635676i	$-0.219279\pi$
$739$	41.9706	1.54391	0.771956	+	0.635676i	$0.219279\pi$
$740$	0	0
$741$	5.41421	0.198896
$742$	−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	$0.378460\pi$
$744$	−3.41421	−0.125171
$745$	0	0
$746$	−10.7279	−0.392777
$747$	−7.65685	−0.280150
$748$	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
$753$	32.4853	1.18383
$754$	−27.7279	−1.00979
$755$	0	0
$756$	5.65685	0.205738
$757$	48.4264	1.76009	0.880044	−	0.474892i	$-0.157513\pi$
$757$	48.4264	1.76009	0.880044	+	0.474892i	$0.157513\pi$
$758$	24.1421	0.876882
$759$	−43.7990	−1.58980
$760$	0	0
$761$	−39.1716	−1.41997	−0.709984	−	0.704218i	$-0.751298\pi$
$761$	−39.1716	−1.41997	−0.709984	+	0.704218i	$0.751298\pi$
$762$	28.9706	1.04949
$763$	5.17157	0.187224
$764$	0.100505	0.00363615
$765$	0	0
$766$	14.3137	0.517175
$767$	26.1421	0.943938
$768$	1.41421	0.0510310
$769$	21.0000	0.757279	0.378640	−	0.925544i	$-0.376392\pi$
$769$	21.0000	0.757279	0.378640	+	0.925544i	$0.376392\pi$
$770$	0	0
$771$	15.4142	0.555129
$772$	−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	$0.437028\pi$
$774$	−1.00000	−0.0359443
$775$	0	0
$776$	−17.5563	−0.630236
$777$	−8.82843	−0.316718
$778$	−15.1716	−0.543927
$779$	−3.82843	−0.137168
$780$	0	0
$781$	−26.9706	−0.965083
$782$	12.8284	0.458744
$783$	40.9706	1.46417
$784$	−6.00000	−0.214286
$785$	0	0
$786$	−8.48528	−0.302660
$787$	−22.7574	−0.811212	−0.405606	−	0.914048i	$-0.632940\pi$
$787$	−22.7574	−0.811212	−0.405606	+	0.914048i	$0.632940\pi$
$788$	3.14214	0.111934
$789$	−0.727922	−0.0259147
$790$	0	0
$791$	−14.8995	−0.529765
$792$	3.41421	0.121319
$793$	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	$0.645635\pi$
$798$	−1.41421	−0.0500626
$799$	10.0000	0.353775
$800$	0	0
$801$	−5.07107	−0.179177
$802$	−38.1127	−1.34581
$803$	2.58579	0.0912504
$804$	−10.2426	−0.361230
$805$	0	0
$806$	9.24264	0.325558
$807$	20.9706	0.738199
$808$	1.41421	0.0497519
$809$	10.7990	0.379672	0.189836	−	0.981816i	$-0.439204\pi$
$809$	10.7990	0.379672	0.189836	+	0.981816i	$0.439204\pi$
$810$	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
$813$	−15.4142	−0.540600
$814$	−21.3137	−0.747045
$815$	0	0
$816$	2.00000	0.0700140
$817$	1.00000	0.0349856
$818$	−15.2132	−0.531917
$819$	−3.82843	−0.133776
$820$	0	0
$821$	8.14214	0.284162	0.142081	−	0.989855i	$-0.454621\pi$
$821$	8.14214	0.284162	0.142081	+	0.989855i	$0.454621\pi$
$822$	25.5563	0.891380
$823$	−30.4853	−1.06265	−0.531325	−	0.847168i	$-0.678306\pi$
$823$	−30.4853	−1.06265	−0.531325	+	0.847168i	$0.678306\pi$
$824$	−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	$0.622111\pi$
$828$	9.07107	0.315241
$829$	−5.24264	−0.182084	−0.0910422	−	0.995847i	$-0.529020\pi$
$829$	−5.24264	−0.182084	−0.0910422	+	0.995847i	$0.529020\pi$
$830$	0	0
$831$	−23.3137	−0.808744
$832$	−3.82843	−0.132727
$833$	−8.48528	−0.293998
$834$	−3.17157	−0.109823
$835$	0	0
$836$	−3.41421	−0.118083
$837$	−13.6569	−0.472050
$838$	−26.7990	−0.925756
$839$	31.0122	1.07066	0.535330	−	0.844643i	$-0.320187\pi$
$839$	31.0122	1.07066	0.535330	+	0.844643i	$0.320187\pi$
$840$	0	0
$841$	23.4558	0.808822
$842$	7.24264	0.249598
$843$	−17.8995	−0.616491
$844$	9.65685	0.332403
$845$	0	0
$846$	7.07107	0.243108
$847$	−0.656854	−0.0225698
$848$	−5.65685	−0.194257
$849$	−38.5269	−1.32224
$850$	0	0
$851$	−56.6274	−1.94116
$852$	−11.1716	−0.382732
$853$	−31.9411	−1.09364	−0.546822	−	0.837249i	$-0.684162\pi$
$853$	−31.9411	−1.09364	−0.546822	+	0.837249i	$0.684162\pi$
$854$	−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	$0.690128\pi$
$858$	18.4853	0.631077
$859$	−33.4853	−1.14250	−0.571252	−	0.820775i	$-0.693542\pi$
$859$	−33.4853	−1.14250	−0.571252	+	0.820775i	$0.693542\pi$
$860$	0	0
$861$	−5.41421	−0.184516
$862$	−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	$0.295713\pi$
$864$	5.65685	0.192450
$865$	0	0
$866$	−21.9289	−0.745175
$867$	−21.2132	−0.720438
$868$	−2.41421	−0.0819437
$869$	−19.0711	−0.646942
$870$	0	0
$871$	27.7279	0.939525
$872$	5.17157	0.175132
$873$	−17.5563	−0.594192
$874$	−9.07107	−0.306833
$875$	0	0
$876$	1.07107	0.0361880
$877$	−26.6274	−0.899144	−0.449572	−	0.893244i	$-0.648423\pi$
$877$	−26.6274	−0.899144	−0.449572	+	0.893244i	$0.648423\pi$
$878$	−7.31371	−0.246826
$879$	32.4853	1.09570
$880$	0	0
$881$	−40.7990	−1.37455	−0.687276	−	0.726396i	$-0.741194\pi$
$881$	−40.7990	−1.37455	−0.687276	+	0.726396i	$0.741194\pi$
$882$	−6.00000	−0.202031
$883$	16.2132	0.545618	0.272809	−	0.962068i	$-0.412047\pi$
$883$	16.2132	0.545618	0.272809	+	0.962068i	$0.412047\pi$
$884$	−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	$0.532723\pi$
$888$	−8.82843	−0.296263
$889$	20.4853	0.687054
$890$	0	0
$891$	−17.0711	−0.571902
$892$	12.1421	0.406549
$893$	−7.07107	−0.236624
$894$	−29.0711	−0.972282
$895$	0	0
$896$	1.00000	0.0334077
$897$	49.1127	1.63983
$898$	−13.7574	−0.459089
$899$	−17.4853	−0.583167
$900$	0	0
$901$	−8.00000	−0.266519
$902$	−13.0711	−0.435219
$903$	1.41421	0.0470621
$904$	−14.8995	−0.495550
$905$	0	0
$906$	9.17157	0.304705
$907$	4.89949	0.162685	0.0813425	−	0.996686i	$-0.474079\pi$
$907$	4.89949	0.162685	0.0813425	+	0.996686i	$0.474079\pi$
$908$	−14.8284	−0.492099
$909$	1.41421	0.0469065
$910$	0	0
$911$	16.3848	0.542852	0.271426	−	0.962459i	$-0.412505\pi$
$911$	16.3848	0.542852	0.271426	+	0.962459i	$0.412505\pi$
$912$	−1.41421	−0.0468293
$913$	26.1421	0.865178
$914$	−13.5147	−0.447027
$915$	0	0
$916$	−4.82843	−0.159536
$917$	−6.00000	−0.198137
$918$	8.00000	0.264039
$919$	−15.1005	−0.498120	−0.249060	−	0.968488i	$-0.580122\pi$
$919$	−15.1005	−0.498120	−0.249060	+	0.968488i	$0.580122\pi$
$920$	0	0
$921$	28.3848	0.935310
$922$	0.928932	0.0305928
$923$	30.2426	0.995449
$924$	−4.82843	−0.158844
$925$	0	0
$926$	7.68629	0.252587
$927$	−4.82843	−0.158586
$928$	7.24264	0.237751
$929$	25.7990	0.846437	0.423219	−	0.906028i	$-0.360900\pi$
$929$	25.7990	0.846437	0.423219	+	0.906028i	$0.360900\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	8.48528	0.277945
$933$	14.2426	0.466283
$934$	−35.0711	−1.14756
$935$	0	0
$936$	−3.82843	−0.125136
$937$	−50.4264	−1.64736	−0.823679	−	0.567056i	$-0.808082\pi$
$937$	−50.4264	−1.64736	−0.823679	+	0.567056i	$0.808082\pi$
$938$	−7.24264	−0.236481
$939$	−30.1421	−0.983651
$940$	0	0
$941$	−38.9289	−1.26905	−0.634523	−	0.772904i	$-0.718804\pi$
$941$	−38.9289	−1.26905	−0.634523	+	0.772904i	$0.718804\pi$
$942$	2.48528	0.0809748
$943$	−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	$0.409931\pi$
$948$	−7.89949	−0.256564
$949$	−2.89949	−0.0941216
$950$	0	0
$951$	−41.2132	−1.33643
$952$	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	$0.344948\pi$
$954$	−5.65685	−0.183147
$955$	0	0
$956$	14.4142	0.466189
$957$	−34.9706	−1.13044
$958$	−14.8284	−0.479085
$959$	18.0711	0.583545
$960$	0	0
$961$	−25.1716	−0.811986
$962$	23.8995	0.770551
$963$	16.0711	0.517883
$964$	−18.5858	−0.598608
$965$	0	0
$966$	−12.8284	−0.412748
$967$	−27.7574	−0.892616	−0.446308	−	0.894879i	$-0.647262\pi$
$967$	−27.7574	−0.892616	−0.446308	+	0.894879i	$0.647262\pi$
$968$	−0.656854	−0.0211121
$969$	−2.00000	−0.0642493
$970$	0	0
$971$	6.92893	0.222360	0.111180	−	0.993800i	$-0.464537\pi$
$971$	6.92893	0.222360	0.111180	+	0.993800i	$0.464537\pi$
$972$	9.89949	0.317526
$973$	−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	$0.485081\pi$
$978$	3.31371	0.105961
$979$	17.3137	0.553349
$980$	0	0
$981$	5.17157	0.165116
$982$	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	$0.288558\pi$
$984$	−5.41421	−0.172599
$985$	0	0
$986$	10.2426	0.326192
$987$	−10.0000	−0.318304
$988$	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
$993$	−3.79899	−0.120557
$994$	−7.89949	−0.250557
$995$	0	0
$996$	10.8284	0.343112
$997$	−42.8284	−1.35639	−0.678195	−	0.734882i	$-0.737238\pi$
$997$	−42.8284	−1.35639	−0.678195	+	0.734882i	$0.737238\pi$
$998$	−16.4558	−0.520901
$999$	−35.3137	−1.11728

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
430.2.a.g.1.1	✓	2	5.4	even	2
2150.2.a.v.1.2		2	1.1	even	1	trivial
2150.2.b.o.1549.1		4	5.2	odd	4
2150.2.b.o.1549.4		4	5.3	odd	4
3440.2.a.j.1.2		2	20.19	odd	2
3870.2.a.bc.1.1		2	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2150 = 2 \cdot 5^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2150.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} -1.41421 q^{6} -1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} -1.41421 q^{6} -1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{9} +3.41421 q^{11} +1.41421 q^{12} -3.82843 q^{13} +1.00000 q^{14} +1.00000 q^{16} +1.41421 q^{17} +1.00000 q^{18} -1.00000 q^{19} -1.41421 q^{21} -3.41421 q^{22} -9.07107 q^{23} -1.41421 q^{24} +3.82843 q^{26} -5.65685 q^{27} -1.00000 q^{28} -7.24264 q^{29} +2.41421 q^{31} -1.00000 q^{32} +4.82843 q^{33} -1.41421 q^{34} -1.00000 q^{36} +6.24264 q^{37} +1.00000 q^{38} -5.41421 q^{39} +3.82843 q^{41} +1.41421 q^{42} -1.00000 q^{43} +3.41421 q^{44} +9.07107 q^{46} +7.07107 q^{47} +1.41421 q^{48} -6.00000 q^{49} +2.00000 q^{51} -3.82843 q^{52} -5.65685 q^{53} +5.65685 q^{54} +1.00000 q^{56} -1.41421 q^{57} +7.24264 q^{58} -6.82843 q^{59} -14.0711 q^{61} -2.41421 q^{62} +1.00000 q^{63} +1.00000 q^{64} -4.82843 q^{66} -7.24264 q^{67} +1.41421 q^{68} -12.8284 q^{69} -7.89949 q^{71} +1.00000 q^{72} +0.757359 q^{73} -6.24264 q^{74} -1.00000 q^{76} -3.41421 q^{77} +5.41421 q^{78} -5.58579 q^{79} -5.00000 q^{81} -3.82843 q^{82} +7.65685 q^{83} -1.41421 q^{84} +1.00000 q^{86} -10.2426 q^{87} -3.41421 q^{88} +5.07107 q^{89} +3.82843 q^{91} -9.07107 q^{92} +3.41421 q^{93} -7.07107 q^{94} -1.41421 q^{96} +17.5563 q^{97} +6.00000 q^{98} -3.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 2 * q^8 - 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8} - 2 q^{9} + 4 q^{11} - 2 q^{13} + 2 q^{14} + 2 q^{16} + 2 q^{18} - 2 q^{19} - 4 q^{22} - 4 q^{23} + 2 q^{26} - 2 q^{28} - 6 q^{29} + 2 q^{31} - 2 q^{32} + 4 q^{33} - 2 q^{36} + 4 q^{37} + 2 q^{38} - 8 q^{39} + 2 q^{41} - 2 q^{43} + 4 q^{44} + 4 q^{46} - 12 q^{49} + 4 q^{51} - 2 q^{52} + 2 q^{56} + 6 q^{58} - 8 q^{59} - 14 q^{61} - 2 q^{62} + 2 q^{63} + 2 q^{64} - 4 q^{66} - 6 q^{67} - 20 q^{69} + 4 q^{71} + 2 q^{72} + 10 q^{73} - 4 q^{74} - 2 q^{76} - 4 q^{77} + 8 q^{78} - 14 q^{79} - 10 q^{81} - 2 q^{82} + 4 q^{83} + 2 q^{86} - 12 q^{87} - 4 q^{88} - 4 q^{89} + 2 q^{91} - 4 q^{92} + 4 q^{93} + 4 q^{97} + 12 q^{98} - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 2 * q^8 - 2 * q^9 + 4 * q^11 - 2 * q^13 + 2 * q^14 + 2 * q^16 + 2 * q^18 - 2 * q^19 - 4 * q^22 - 4 * q^23 + 2 * q^26 - 2 * q^28 - 6 * q^29 + 2 * q^31 - 2 * q^32 + 4 * q^33 - 2 * q^36 + 4 * q^37 + 2 * q^38 - 8 * q^39 + 2 * q^41 - 2 * q^43 + 4 * q^44 + 4 * q^46 - 12 * q^49 + 4 * q^51 - 2 * q^52 + 2 * q^56 + 6 * q^58 - 8 * q^59 - 14 * q^61 - 2 * q^62 + 2 * q^63 + 2 * q^64 - 4 * q^66 - 6 * q^67 - 20 * q^69 + 4 * q^71 + 2 * q^72 + 10 * q^73 - 4 * q^74 - 2 * q^76 - 4 * q^77 + 8 * q^78 - 14 * q^79 - 10 * q^81 - 2 * q^82 + 4 * q^83 + 2 * q^86 - 12 * q^87 - 4 * q^88 - 4 * q^89 + 2 * q^91 - 4 * q^92 + 4 * q^93 + 4 * q^97 + 12 * q^98 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more