LMFDB - Embedded newform 6050.2.a.bt.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6050,2,Mod(1,6050)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6050, 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("6050.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6050 = 2 \cdot 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6050.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:	$48.3094932229$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1210)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	6050.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +0.732051 q^{3} +1.00000 q^{4} -0.732051 q^{6} -1.26795 q^{7} -1.00000 q^{8} -2.46410 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +0.732051 q^{3} +1.00000 q^{4} -0.732051 q^{6} -1.26795 q^{7} -1.00000 q^{8} -2.46410 q^{9} +0.732051 q^{12} +2.46410 q^{13} +1.26795 q^{14} +1.00000 q^{16} -1.73205 q^{17} +2.46410 q^{18} +4.19615 q^{19} -0.928203 q^{21} +2.73205 q^{23} -0.732051 q^{24} -2.46410 q^{26} -4.00000 q^{27} -1.26795 q^{28} -3.73205 q^{29} +8.73205 q^{31} -1.00000 q^{32} +1.73205 q^{34} -2.46410 q^{36} -7.92820 q^{37} -4.19615 q^{38} +1.80385 q^{39} -10.4641 q^{41} +0.928203 q^{42} -9.46410 q^{43} -2.73205 q^{46} +6.73205 q^{47} +0.732051 q^{48} -5.39230 q^{49} -1.26795 q^{51} +2.46410 q^{52} +3.00000 q^{53} +4.00000 q^{54} +1.26795 q^{56} +3.07180 q^{57} +3.73205 q^{58} +13.8564 q^{59} -8.92820 q^{61} -8.73205 q^{62} +3.12436 q^{63} +1.00000 q^{64} +7.26795 q^{67} -1.73205 q^{68} +2.00000 q^{69} +6.92820 q^{71} +2.46410 q^{72} -12.9282 q^{73} +7.92820 q^{74} +4.19615 q^{76} -1.80385 q^{78} +7.26795 q^{79} +4.46410 q^{81} +10.4641 q^{82} +3.26795 q^{83} -0.928203 q^{84} +9.46410 q^{86} -2.73205 q^{87} +0.464102 q^{89} -3.12436 q^{91} +2.73205 q^{92} +6.39230 q^{93} -6.73205 q^{94} -0.732051 q^{96} -17.1962 q^{97} +5.39230 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 6 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 6 * q^7 - 2 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 6 q^{7} - 2 q^{8} + 2 q^{9} - 2 q^{12} - 2 q^{13} + 6 q^{14} + 2 q^{16} - 2 q^{18} - 2 q^{19} + 12 q^{21} + 2 q^{23} + 2 q^{24} + 2 q^{26} - 8 q^{27} - 6 q^{28} - 4 q^{29} + 14 q^{31} - 2 q^{32} + 2 q^{36} - 2 q^{37} + 2 q^{38} + 14 q^{39} - 14 q^{41} - 12 q^{42} - 12 q^{43} - 2 q^{46} + 10 q^{47} - 2 q^{48} + 10 q^{49} - 6 q^{51} - 2 q^{52} + 6 q^{53} + 8 q^{54} + 6 q^{56} + 20 q^{57} + 4 q^{58} - 4 q^{61} - 14 q^{62} - 18 q^{63} + 2 q^{64} + 18 q^{67} + 4 q^{69} - 2 q^{72} - 12 q^{73} + 2 q^{74} - 2 q^{76} - 14 q^{78} + 18 q^{79} + 2 q^{81} + 14 q^{82} + 10 q^{83} + 12 q^{84} + 12 q^{86} - 2 q^{87} - 6 q^{89} + 18 q^{91} + 2 q^{92} - 8 q^{93} - 10 q^{94} + 2 q^{96} - 24 q^{97} - 10 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 6 * q^7 - 2 * q^8 + 2 * q^9 - 2 * q^12 - 2 * q^13 + 6 * q^14 + 2 * q^16 - 2 * q^18 - 2 * q^19 + 12 * q^21 + 2 * q^23 + 2 * q^24 + 2 * q^26 - 8 * q^27 - 6 * q^28 - 4 * q^29 + 14 * q^31 - 2 * q^32 + 2 * q^36 - 2 * q^37 + 2 * q^38 + 14 * q^39 - 14 * q^41 - 12 * q^42 - 12 * q^43 - 2 * q^46 + 10 * q^47 - 2 * q^48 + 10 * q^49 - 6 * q^51 - 2 * q^52 + 6 * q^53 + 8 * q^54 + 6 * q^56 + 20 * q^57 + 4 * q^58 - 4 * q^61 - 14 * q^62 - 18 * q^63 + 2 * q^64 + 18 * q^67 + 4 * q^69 - 2 * q^72 - 12 * q^73 + 2 * q^74 - 2 * q^76 - 14 * q^78 + 18 * q^79 + 2 * q^81 + 14 * q^82 + 10 * q^83 + 12 * q^84 + 12 * q^86 - 2 * q^87 - 6 * q^89 + 18 * q^91 + 2 * q^92 - 8 * q^93 - 10 * q^94 + 2 * q^96 - 24 * q^97 - 10 * 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
$3$	0.732051	0.422650	0.211325	−	0.977416i	$-0.432222\pi$
$3$	0.732051	0.422650	0.211325	+	0.977416i	$0.432222\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−0.732051	−0.298858
$7$	−1.26795	−0.479240	−0.239620	−	0.970867i	$-0.577023\pi$
$7$	−1.26795	−0.479240	−0.239620	+	0.970867i	$0.577023\pi$
$8$	−1.00000	−0.353553
$9$	−2.46410	−0.821367
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0.732051	0.211325
$13$	2.46410	0.683419	0.341709	−	0.939806i	$-0.388994\pi$
$13$	2.46410	0.683419	0.341709	+	0.939806i	$0.388994\pi$
$14$	1.26795	0.338874
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.73205	−0.420084	−0.210042	−	0.977692i	$-0.567360\pi$
$17$	−1.73205	−0.420084	−0.210042	+	0.977692i	$0.567360\pi$
$18$	2.46410	0.580794
$19$	4.19615	0.962663	0.481332	−	0.876539i	$-0.340153\pi$
$19$	4.19615	0.962663	0.481332	+	0.876539i	$0.340153\pi$
$20$	0	0
$21$	−0.928203	−0.202551
$22$	0	0
$23$	2.73205	0.569672	0.284836	−	0.958576i	$-0.408061\pi$
$23$	2.73205	0.569672	0.284836	+	0.958576i	$0.408061\pi$
$24$	−0.732051	−0.149429
$25$	0	0
$26$	−2.46410	−0.483250
$27$	−4.00000	−0.769800
$28$	−1.26795	−0.239620
$29$	−3.73205	−0.693024	−0.346512	−	0.938045i	$-0.612634\pi$
$29$	−3.73205	−0.693024	−0.346512	+	0.938045i	$0.612634\pi$
$30$	0	0
$31$	8.73205	1.56832	0.784161	−	0.620557i	$-0.213093\pi$
$31$	8.73205	1.56832	0.784161	+	0.620557i	$0.213093\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	1.73205	0.297044
$35$	0	0
$36$	−2.46410	−0.410684
$37$	−7.92820	−1.30339	−0.651694	−	0.758482i	$-0.725941\pi$
$37$	−7.92820	−1.30339	−0.651694	+	0.758482i	$0.725941\pi$
$38$	−4.19615	−0.680706
$39$	1.80385	0.288847
$40$	0	0
$41$	−10.4641	−1.63422	−0.817109	−	0.576483i	$-0.804425\pi$
$41$	−10.4641	−1.63422	−0.817109	+	0.576483i	$0.804425\pi$
$42$	0.928203	0.143225
$43$	−9.46410	−1.44326	−0.721631	−	0.692278i	$-0.756607\pi$
$43$	−9.46410	−1.44326	−0.721631	+	0.692278i	$0.756607\pi$
$44$	0	0
$45$	0	0
$46$	−2.73205	−0.402819
$47$	6.73205	0.981971	0.490985	−	0.871168i	$-0.336637\pi$
$47$	6.73205	0.981971	0.490985	+	0.871168i	$0.336637\pi$
$48$	0.732051	0.105662
$49$	−5.39230	−0.770329
$50$	0	0
$51$	−1.26795	−0.177548
$52$	2.46410	0.341709
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	4.00000	0.544331
$55$	0	0
$56$	1.26795	0.169437
$57$	3.07180	0.406869
$58$	3.73205	0.490042
$59$	13.8564	1.80395	0.901975	−	0.431788i	$-0.142117\pi$
$59$	13.8564	1.80395	0.901975	+	0.431788i	$0.142117\pi$
$60$	0	0
$61$	−8.92820	−1.14314	−0.571570	−	0.820554i	$-0.693665\pi$
$61$	−8.92820	−1.14314	−0.571570	+	0.820554i	$0.693665\pi$
$62$	−8.73205	−1.10897
$63$	3.12436	0.393632
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	7.26795	0.887921	0.443961	−	0.896046i	$-0.353573\pi$
$67$	7.26795	0.887921	0.443961	+	0.896046i	$0.353573\pi$
$68$	−1.73205	−0.210042
$69$	2.00000	0.240772
$70$	0	0
$71$	6.92820	0.822226	0.411113	−	0.911584i	$-0.365140\pi$
$71$	6.92820	0.822226	0.411113	+	0.911584i	$0.365140\pi$
$72$	2.46410	0.290397
$73$	−12.9282	−1.51313	−0.756566	−	0.653917i	$-0.773124\pi$
$73$	−12.9282	−1.51313	−0.756566	+	0.653917i	$0.773124\pi$
$74$	7.92820	0.921635
$75$	0	0
$76$	4.19615	0.481332
$77$	0	0
$78$	−1.80385	−0.204246
$79$	7.26795	0.817708	0.408854	−	0.912600i	$-0.365928\pi$
$79$	7.26795	0.817708	0.408854	+	0.912600i	$0.365928\pi$
$80$	0	0
$81$	4.46410	0.496011
$82$	10.4641	1.15557
$83$	3.26795	0.358704	0.179352	−	0.983785i	$-0.442600\pi$
$83$	3.26795	0.358704	0.179352	+	0.983785i	$0.442600\pi$
$84$	−0.928203	−0.101275
$85$	0	0
$86$	9.46410	1.02054
$87$	−2.73205	−0.292907
$88$	0	0
$89$	0.464102	0.0491947	0.0245973	−	0.999697i	$-0.492170\pi$
$89$	0.464102	0.0491947	0.0245973	+	0.999697i	$0.492170\pi$
$90$	0	0
$91$	−3.12436	−0.327521
$92$	2.73205	0.284836
$93$	6.39230	0.662851
$94$	−6.73205	−0.694358
$95$	0	0
$96$	−0.732051	−0.0747146
$97$	−17.1962	−1.74600	−0.873002	−	0.487716i	$-0.837830\pi$
$97$	−17.1962	−1.74600	−0.873002	+	0.487716i	$0.837830\pi$
$98$	5.39230	0.544705
$99$	0	0
$100$	0	0
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	1.26795	0.125546
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	−2.46410	−0.241625
$105$	0	0
$106$	−3.00000	−0.291386
$107$	−11.6603	−1.12724	−0.563620	−	0.826034i	$-0.690592\pi$
$107$	−11.6603	−1.12724	−0.563620	+	0.826034i	$0.690592\pi$
$108$	−4.00000	−0.384900
$109$	8.26795	0.791926	0.395963	−	0.918266i	$-0.370411\pi$
$109$	8.26795	0.791926	0.395963	+	0.918266i	$0.370411\pi$
$110$	0	0
$111$	−5.80385	−0.550877
$112$	−1.26795	−0.119810
$113$	6.80385	0.640052	0.320026	−	0.947409i	$-0.396308\pi$
$113$	6.80385	0.640052	0.320026	+	0.947409i	$0.396308\pi$
$114$	−3.07180	−0.287700
$115$	0	0
$116$	−3.73205	−0.346512
$117$	−6.07180	−0.561338
$118$	−13.8564	−1.27559
$119$	2.19615	0.201321
$120$	0	0
$121$	0	0
$122$	8.92820	0.808322
$123$	−7.66025	−0.690702
$124$	8.73205	0.784161
$125$	0	0
$126$	−3.12436	−0.278340
$127$	−21.8564	−1.93944	−0.969721	−	0.244215i	$-0.921470\pi$
$127$	−21.8564	−1.93944	−0.969721	+	0.244215i	$0.921470\pi$
$128$	−1.00000	−0.0883883
$129$	−6.92820	−0.609994
$130$	0	0
$131$	5.46410	0.477401	0.238700	−	0.971093i	$-0.423279\pi$
$131$	5.46410	0.477401	0.238700	+	0.971093i	$0.423279\pi$
$132$	0	0
$133$	−5.32051	−0.461347
$134$	−7.26795	−0.627855
$135$	0	0
$136$	1.73205	0.148522
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	−2.00000	−0.170251
$139$	−9.26795	−0.786097	−0.393049	−	0.919518i	$-0.628580\pi$
$139$	−9.26795	−0.786097	−0.393049	+	0.919518i	$0.628580\pi$
$140$	0	0
$141$	4.92820	0.415030
$142$	−6.92820	−0.581402
$143$	0	0
$144$	−2.46410	−0.205342
$145$	0	0
$146$	12.9282	1.06995
$147$	−3.94744	−0.325579
$148$	−7.92820	−0.651694
$149$	−13.5885	−1.11321	−0.556605	−	0.830777i	$-0.687896\pi$
$149$	−13.5885	−1.11321	−0.556605	+	0.830777i	$0.687896\pi$
$150$	0	0
$151$	10.9282	0.889325	0.444662	−	0.895698i	$-0.353324\pi$
$151$	10.9282	0.889325	0.444662	+	0.895698i	$0.353324\pi$
$152$	−4.19615	−0.340353
$153$	4.26795	0.345043
$154$	0	0
$155$	0	0
$156$	1.80385	0.144423
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	−7.26795	−0.578207
$159$	2.19615	0.174166
$160$	0	0
$161$	−3.46410	−0.273009
$162$	−4.46410	−0.350733
$163$	−9.12436	−0.714675	−0.357337	−	0.933975i	$-0.616315\pi$
$163$	−9.12436	−0.714675	−0.357337	+	0.933975i	$0.616315\pi$
$164$	−10.4641	−0.817109
$165$	0	0
$166$	−3.26795	−0.253642
$167$	15.3205	1.18554	0.592768	−	0.805373i	$-0.298035\pi$
$167$	15.3205	1.18554	0.592768	+	0.805373i	$0.298035\pi$
$168$	0.928203	0.0716124
$169$	−6.92820	−0.532939
$170$	0	0
$171$	−10.3397	−0.790700
$172$	−9.46410	−0.721631
$173$	6.53590	0.496915	0.248458	−	0.968643i	$-0.420076\pi$
$173$	6.53590	0.496915	0.248458	+	0.968643i	$0.420076\pi$
$174$	2.73205	0.207116
$175$	0	0
$176$	0	0
$177$	10.1436	0.762439
$178$	−0.464102	−0.0347859
$179$	−21.8564	−1.63362	−0.816812	−	0.576904i	$-0.804261\pi$
$179$	−21.8564	−1.63362	−0.816812	+	0.576904i	$0.804261\pi$
$180$	0	0
$181$	−8.66025	−0.643712	−0.321856	−	0.946789i	$-0.604307\pi$
$181$	−8.66025	−0.643712	−0.321856	+	0.946789i	$0.604307\pi$
$182$	3.12436	0.231593
$183$	−6.53590	−0.483148
$184$	−2.73205	−0.201409
$185$	0	0
$186$	−6.39230	−0.468707
$187$	0	0
$188$	6.73205	0.490985
$189$	5.07180	0.368919
$190$	0	0
$191$	1.46410	0.105939	0.0529693	−	0.998596i	$-0.483131\pi$
$191$	1.46410	0.105939	0.0529693	+	0.998596i	$0.483131\pi$
$192$	0.732051	0.0528312
$193$	−13.1962	−0.949880	−0.474940	−	0.880018i	$-0.657530\pi$
$193$	−13.1962	−0.949880	−0.474940	+	0.880018i	$0.657530\pi$
$194$	17.1962	1.23461
$195$	0	0
$196$	−5.39230	−0.385165
$197$	−10.0718	−0.717586	−0.358793	−	0.933417i	$-0.616812\pi$
$197$	−10.0718	−0.717586	−0.358793	+	0.933417i	$0.616812\pi$
$198$	0	0
$199$	−13.0718	−0.926635	−0.463318	−	0.886192i	$-0.653341\pi$
$199$	−13.0718	−0.926635	−0.463318	+	0.886192i	$0.653341\pi$
$200$	0	0
$201$	5.32051	0.375280
$202$	−10.0000	−0.703598
$203$	4.73205	0.332125
$204$	−1.26795	−0.0887742
$205$	0	0
$206$	8.00000	0.557386
$207$	−6.73205	−0.467910
$208$	2.46410	0.170855
$209$	0	0
$210$	0	0
$211$	9.07180	0.624528	0.312264	−	0.949995i	$-0.398913\pi$
$211$	9.07180	0.624528	0.312264	+	0.949995i	$0.398913\pi$
$212$	3.00000	0.206041
$213$	5.07180	0.347514
$214$	11.6603	0.797079
$215$	0	0
$216$	4.00000	0.272166
$217$	−11.0718	−0.751603
$218$	−8.26795	−0.559976
$219$	−9.46410	−0.639525
$220$	0	0
$221$	−4.26795	−0.287093
$222$	5.80385	0.389529
$223$	−20.3923	−1.36557	−0.682785	−	0.730619i	$-0.739231\pi$
$223$	−20.3923	−1.36557	−0.682785	+	0.730619i	$0.739231\pi$
$224$	1.26795	0.0847184
$225$	0	0
$226$	−6.80385	−0.452585
$227$	−28.3923	−1.88446	−0.942232	−	0.334962i	$-0.891277\pi$
$227$	−28.3923	−1.88446	−0.942232	+	0.334962i	$0.891277\pi$
$228$	3.07180	0.203435
$229$	9.33975	0.617188	0.308594	−	0.951194i	$-0.400142\pi$
$229$	9.33975	0.617188	0.308594	+	0.951194i	$0.400142\pi$
$230$	0	0
$231$	0	0
$232$	3.73205	0.245021
$233$	13.1962	0.864509	0.432254	−	0.901752i	$-0.357718\pi$
$233$	13.1962	0.864509	0.432254	+	0.901752i	$0.357718\pi$
$234$	6.07180	0.396926
$235$	0	0
$236$	13.8564	0.901975
$237$	5.32051	0.345604
$238$	−2.19615	−0.142355
$239$	−16.7321	−1.08231	−0.541153	−	0.840924i	$-0.682012\pi$
$239$	−16.7321	−1.08231	−0.541153	+	0.840924i	$0.682012\pi$
$240$	0	0
$241$	19.3205	1.24454	0.622272	−	0.782801i	$-0.286210\pi$
$241$	19.3205	1.24454	0.622272	+	0.782801i	$0.286210\pi$
$242$	0	0
$243$	15.2679	0.979439
$244$	−8.92820	−0.571570
$245$	0	0
$246$	7.66025	0.488400
$247$	10.3397	0.657902
$248$	−8.73205	−0.554486
$249$	2.39230	0.151606
$250$	0	0
$251$	−16.5885	−1.04705	−0.523527	−	0.852009i	$-0.675384\pi$
$251$	−16.5885	−1.04705	−0.523527	+	0.852009i	$0.675384\pi$
$252$	3.12436	0.196816
$253$	0	0
$254$	21.8564	1.37139
$255$	0	0
$256$	1.00000	0.0625000
$257$	−27.9808	−1.74539	−0.872696	−	0.488264i	$-0.837630\pi$
$257$	−27.9808	−1.74539	−0.872696	+	0.488264i	$0.837630\pi$
$258$	6.92820	0.431331
$259$	10.0526	0.624636
$260$	0	0
$261$	9.19615	0.569228
$262$	−5.46410	−0.337573
$263$	−10.0526	−0.619867	−0.309934	−	0.950758i	$-0.600307\pi$
$263$	−10.0526	−0.619867	−0.309934	+	0.950758i	$0.600307\pi$
$264$	0	0
$265$	0	0
$266$	5.32051	0.326221
$267$	0.339746	0.0207921
$268$	7.26795	0.443961
$269$	−5.19615	−0.316815	−0.158408	−	0.987374i	$-0.550636\pi$
$269$	−5.19615	−0.316815	−0.158408	+	0.987374i	$0.550636\pi$
$270$	0	0
$271$	−17.8564	−1.08470	−0.542350	−	0.840153i	$-0.682465\pi$
$271$	−17.8564	−1.08470	−0.542350	+	0.840153i	$0.682465\pi$
$272$	−1.73205	−0.105021
$273$	−2.28719	−0.138427
$274$	6.00000	0.362473
$275$	0	0
$276$	2.00000	0.120386
$277$	−13.9282	−0.836865	−0.418432	−	0.908248i	$-0.637420\pi$
$277$	−13.9282	−0.836865	−0.418432	+	0.908248i	$0.637420\pi$
$278$	9.26795	0.555855
$279$	−21.5167	−1.28817
$280$	0	0
$281$	28.3923	1.69374	0.846871	−	0.531798i	$-0.178483\pi$
$281$	28.3923	1.69374	0.846871	+	0.531798i	$0.178483\pi$
$282$	−4.92820	−0.293470
$283$	−10.5359	−0.626294	−0.313147	−	0.949705i	$-0.601383\pi$
$283$	−10.5359	−0.626294	−0.313147	+	0.949705i	$0.601383\pi$
$284$	6.92820	0.411113
$285$	0	0
$286$	0	0
$287$	13.2679	0.783182
$288$	2.46410	0.145199
$289$	−14.0000	−0.823529
$290$	0	0
$291$	−12.5885	−0.737948
$292$	−12.9282	−0.756566
$293$	−1.53590	−0.0897281	−0.0448641	−	0.998993i	$-0.514285\pi$
$293$	−1.53590	−0.0897281	−0.0448641	+	0.998993i	$0.514285\pi$
$294$	3.94744	0.230219
$295$	0	0
$296$	7.92820	0.460817
$297$	0	0
$298$	13.5885	0.787158
$299$	6.73205	0.389325
$300$	0	0
$301$	12.0000	0.691669
$302$	−10.9282	−0.628847
$303$	7.32051	0.420552
$304$	4.19615	0.240666
$305$	0	0
$306$	−4.26795	−0.243982
$307$	8.73205	0.498364	0.249182	−	0.968457i	$-0.419838\pi$
$307$	8.73205	0.498364	0.249182	+	0.968457i	$0.419838\pi$
$308$	0	0
$309$	−5.85641	−0.333159
$310$	0	0
$311$	12.3923	0.702703	0.351352	−	0.936244i	$-0.385722\pi$
$311$	12.3923	0.702703	0.351352	+	0.936244i	$0.385722\pi$
$312$	−1.80385	−0.102123
$313$	6.26795	0.354285	0.177143	−	0.984185i	$-0.443315\pi$
$313$	6.26795	0.354285	0.177143	+	0.984185i	$0.443315\pi$
$314$	−10.0000	−0.564333
$315$	0	0
$316$	7.26795	0.408854
$317$	−25.4641	−1.43021	−0.715103	−	0.699019i	$-0.753620\pi$
$317$	−25.4641	−1.43021	−0.715103	+	0.699019i	$0.753620\pi$
$318$	−2.19615	−0.123154
$319$	0	0
$320$	0	0
$321$	−8.53590	−0.476427
$322$	3.46410	0.193047
$323$	−7.26795	−0.404400
$324$	4.46410	0.248006
$325$	0	0
$326$	9.12436	0.505351
$327$	6.05256	0.334707
$328$	10.4641	0.577783
$329$	−8.53590	−0.470599
$330$	0	0
$331$	−9.46410	−0.520194	−0.260097	−	0.965582i	$-0.583755\pi$
$331$	−9.46410	−0.520194	−0.260097	+	0.965582i	$0.583755\pi$
$332$	3.26795	0.179352
$333$	19.5359	1.07056
$334$	−15.3205	−0.838301
$335$	0	0
$336$	−0.928203	−0.0506376
$337$	−22.6603	−1.23438	−0.617191	−	0.786813i	$-0.711730\pi$
$337$	−22.6603	−1.23438	−0.617191	+	0.786813i	$0.711730\pi$
$338$	6.92820	0.376845
$339$	4.98076	0.270518
$340$	0	0
$341$	0	0
$342$	10.3397	0.559109
$343$	15.7128	0.848412
$344$	9.46410	0.510270
$345$	0	0
$346$	−6.53590	−0.351372
$347$	−21.4641	−1.15225	−0.576127	−	0.817360i	$-0.695437\pi$
$347$	−21.4641	−1.15225	−0.576127	+	0.817360i	$0.695437\pi$
$348$	−2.73205	−0.146453
$349$	−26.1244	−1.39840	−0.699202	−	0.714924i	$-0.746461\pi$
$349$	−26.1244	−1.39840	−0.699202	+	0.714924i	$0.746461\pi$
$350$	0	0
$351$	−9.85641	−0.526096
$352$	0	0
$353$	8.12436	0.432416	0.216208	−	0.976347i	$-0.430631\pi$
$353$	8.12436	0.432416	0.216208	+	0.976347i	$0.430631\pi$
$354$	−10.1436	−0.539126
$355$	0	0
$356$	0.464102	0.0245973
$357$	1.60770	0.0850883
$358$	21.8564	1.15515
$359$	−13.5167	−0.713382	−0.356691	−	0.934222i	$-0.616095\pi$
$359$	−13.5167	−0.713382	−0.356691	+	0.934222i	$0.616095\pi$
$360$	0	0
$361$	−1.39230	−0.0732792
$362$	8.66025	0.455173
$363$	0	0
$364$	−3.12436	−0.163761
$365$	0	0
$366$	6.53590	0.341637
$367$	−3.41154	−0.178081	−0.0890405	−	0.996028i	$-0.528380\pi$
$367$	−3.41154	−0.178081	−0.0890405	+	0.996028i	$0.528380\pi$
$368$	2.73205	0.142418
$369$	25.7846	1.34229
$370$	0	0
$371$	−3.80385	−0.197486
$372$	6.39230	0.331426
$373$	12.3923	0.641649	0.320825	−	0.947139i	$-0.396040\pi$
$373$	12.3923	0.641649	0.320825	+	0.947139i	$0.396040\pi$
$374$	0	0
$375$	0	0
$376$	−6.73205	−0.347179
$377$	−9.19615	−0.473626
$378$	−5.07180	−0.260865
$379$	14.2487	0.731907	0.365954	−	0.930633i	$-0.380743\pi$
$379$	14.2487	0.731907	0.365954	+	0.930633i	$0.380743\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	−1.46410	−0.0749100
$383$	−23.3205	−1.19162	−0.595811	−	0.803125i	$-0.703169\pi$
$383$	−23.3205	−1.19162	−0.595811	+	0.803125i	$0.703169\pi$
$384$	−0.732051	−0.0373573
$385$	0	0
$386$	13.1962	0.671666
$387$	23.3205	1.18545
$388$	−17.1962	−0.873002
$389$	28.2679	1.43324	0.716621	−	0.697463i	$-0.245688\pi$
$389$	28.2679	1.43324	0.716621	+	0.697463i	$0.245688\pi$
$390$	0	0
$391$	−4.73205	−0.239310
$392$	5.39230	0.272353
$393$	4.00000	0.201773
$394$	10.0718	0.507410
$395$	0	0
$396$	0	0
$397$	26.7128	1.34068	0.670339	−	0.742055i	$-0.266149\pi$
$397$	26.7128	1.34068	0.670339	+	0.742055i	$0.266149\pi$
$398$	13.0718	0.655230
$399$	−3.89488	−0.194988
$400$	0	0
$401$	2.32051	0.115881	0.0579403	−	0.998320i	$-0.481547\pi$
$401$	2.32051	0.115881	0.0579403	+	0.998320i	$0.481547\pi$
$402$	−5.32051	−0.265363
$403$	21.5167	1.07182
$404$	10.0000	0.497519
$405$	0	0
$406$	−4.73205	−0.234848
$407$	0	0
$408$	1.26795	0.0627728
$409$	17.3923	0.859994	0.429997	−	0.902830i	$-0.358515\pi$
$409$	17.3923	0.859994	0.429997	+	0.902830i	$0.358515\pi$
$410$	0	0
$411$	−4.39230	−0.216656
$412$	−8.00000	−0.394132
$413$	−17.5692	−0.864525
$414$	6.73205	0.330862
$415$	0	0
$416$	−2.46410	−0.120813
$417$	−6.78461	−0.332244
$418$	0	0
$419$	3.41154	0.166665	0.0833324	−	0.996522i	$-0.473444\pi$
$419$	3.41154	0.166665	0.0833324	+	0.996522i	$0.473444\pi$
$420$	0	0
$421$	−17.0526	−0.831091	−0.415545	−	0.909572i	$-0.636409\pi$
$421$	−17.0526	−0.831091	−0.415545	+	0.909572i	$0.636409\pi$
$422$	−9.07180	−0.441608
$423$	−16.5885	−0.806558
$424$	−3.00000	−0.145693
$425$	0	0
$426$	−5.07180	−0.245729
$427$	11.3205	0.547838
$428$	−11.6603	−0.563620
$429$	0	0
$430$	0	0
$431$	17.8564	0.860113	0.430056	−	0.902802i	$-0.358494\pi$
$431$	17.8564	0.860113	0.430056	+	0.902802i	$0.358494\pi$
$432$	−4.00000	−0.192450
$433$	−19.5885	−0.941361	−0.470681	−	0.882304i	$-0.655992\pi$
$433$	−19.5885	−0.941361	−0.470681	+	0.882304i	$0.655992\pi$
$434$	11.0718	0.531463
$435$	0	0
$436$	8.26795	0.395963
$437$	11.4641	0.548402
$438$	9.46410	0.452212
$439$	17.1244	0.817301	0.408650	−	0.912691i	$-0.366000\pi$
$439$	17.1244	0.817301	0.408650	+	0.912691i	$0.366000\pi$
$440$	0	0
$441$	13.2872	0.632723
$442$	4.26795	0.203006
$443$	12.7846	0.607415	0.303708	−	0.952765i	$-0.401775\pi$
$443$	12.7846	0.607415	0.303708	+	0.952765i	$0.401775\pi$
$444$	−5.80385	−0.275438
$445$	0	0
$446$	20.3923	0.965604
$447$	−9.94744	−0.470498
$448$	−1.26795	−0.0599050
$449$	−21.9282	−1.03486	−0.517428	−	0.855727i	$-0.673110\pi$
$449$	−21.9282	−1.03486	−0.517428	+	0.855727i	$0.673110\pi$
$450$	0	0
$451$	0	0
$452$	6.80385	0.320026
$453$	8.00000	0.375873
$454$	28.3923	1.33252
$455$	0	0
$456$	−3.07180	−0.143850
$457$	15.5885	0.729197	0.364599	−	0.931165i	$-0.381206\pi$
$457$	15.5885	0.729197	0.364599	+	0.931165i	$0.381206\pi$
$458$	−9.33975	−0.436418
$459$	6.92820	0.323381
$460$	0	0
$461$	8.51666	0.396660	0.198330	−	0.980135i	$-0.436448\pi$
$461$	8.51666	0.396660	0.198330	+	0.980135i	$0.436448\pi$
$462$	0	0
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	−3.73205	−0.173256
$465$	0	0
$466$	−13.1962	−0.611300
$467$	25.1769	1.16505	0.582524	−	0.812813i	$-0.302065\pi$
$467$	25.1769	1.16505	0.582524	+	0.812813i	$0.302065\pi$
$468$	−6.07180	−0.280669
$469$	−9.21539	−0.425527
$470$	0	0
$471$	7.32051	0.337311
$472$	−13.8564	−0.637793
$473$	0	0
$474$	−5.32051	−0.244379
$475$	0	0
$476$	2.19615	0.100660
$477$	−7.39230	−0.338470
$478$	16.7321	0.765306
$479$	2.14359	0.0979433	0.0489716	−	0.998800i	$-0.484406\pi$
$479$	2.14359	0.0979433	0.0489716	+	0.998800i	$0.484406\pi$
$480$	0	0
$481$	−19.5359	−0.890760
$482$	−19.3205	−0.880025
$483$	−2.53590	−0.115387
$484$	0	0
$485$	0	0
$486$	−15.2679	−0.692568
$487$	38.8372	1.75988	0.879940	−	0.475085i	$-0.157583\pi$
$487$	38.8372	1.75988	0.879940	+	0.475085i	$0.157583\pi$
$488$	8.92820	0.404161
$489$	−6.67949	−0.302057
$490$	0	0
$491$	35.9090	1.62055	0.810274	−	0.586051i	$-0.199318\pi$
$491$	35.9090	1.62055	0.810274	+	0.586051i	$0.199318\pi$
$492$	−7.66025	−0.345351
$493$	6.46410	0.291128
$494$	−10.3397	−0.465207
$495$	0	0
$496$	8.73205	0.392081
$497$	−8.78461	−0.394044
$498$	−2.39230	−0.107202
$499$	−10.9282	−0.489214	−0.244607	−	0.969622i	$-0.578659\pi$
$499$	−10.9282	−0.489214	−0.244607	+	0.969622i	$0.578659\pi$
$500$	0	0
$501$	11.2154	0.501067
$502$	16.5885	0.740379
$503$	−15.1244	−0.674362	−0.337181	−	0.941440i	$-0.609473\pi$
$503$	−15.1244	−0.674362	−0.337181	+	0.941440i	$0.609473\pi$
$504$	−3.12436	−0.139170
$505$	0	0
$506$	0	0
$507$	−5.07180	−0.225246
$508$	−21.8564	−0.969721
$509$	−8.92820	−0.395736	−0.197868	−	0.980229i	$-0.563402\pi$
$509$	−8.92820	−0.395736	−0.197868	+	0.980229i	$0.563402\pi$
$510$	0	0
$511$	16.3923	0.725153
$512$	−1.00000	−0.0441942
$513$	−16.7846	−0.741059
$514$	27.9808	1.23418
$515$	0	0
$516$	−6.92820	−0.304997
$517$	0	0
$518$	−10.0526	−0.441684
$519$	4.78461	0.210021
$520$	0	0
$521$	−21.1769	−0.927777	−0.463889	−	0.885893i	$-0.653546\pi$
$521$	−21.1769	−0.927777	−0.463889	+	0.885893i	$0.653546\pi$
$522$	−9.19615	−0.402505
$523$	−26.9282	−1.17749	−0.588744	−	0.808320i	$-0.700377\pi$
$523$	−26.9282	−1.17749	−0.588744	+	0.808320i	$0.700377\pi$
$524$	5.46410	0.238700
$525$	0	0
$526$	10.0526	0.438312
$527$	−15.1244	−0.658827
$528$	0	0
$529$	−15.5359	−0.675474
$530$	0	0
$531$	−34.1436	−1.48171
$532$	−5.32051	−0.230673
$533$	−25.7846	−1.11686
$534$	−0.339746	−0.0147022
$535$	0	0
$536$	−7.26795	−0.313928
$537$	−16.0000	−0.690451
$538$	5.19615	0.224022
$539$	0	0
$540$	0	0
$541$	−25.0718	−1.07792	−0.538960	−	0.842331i	$-0.681183\pi$
$541$	−25.0718	−1.07792	−0.538960	+	0.842331i	$0.681183\pi$
$542$	17.8564	0.766998
$543$	−6.33975	−0.272065
$544$	1.73205	0.0742611
$545$	0	0
$546$	2.28719	0.0978826
$547$	−31.3205	−1.33917	−0.669584	−	0.742736i	$-0.733528\pi$
$547$	−31.3205	−1.33917	−0.669584	+	0.742736i	$0.733528\pi$
$548$	−6.00000	−0.256307
$549$	22.0000	0.938937
$550$	0	0
$551$	−15.6603	−0.667149
$552$	−2.00000	−0.0851257
$553$	−9.21539	−0.391878
$554$	13.9282	0.591753
$555$	0	0
$556$	−9.26795	−0.393049
$557$	−1.46410	−0.0620360	−0.0310180	−	0.999519i	$-0.509875\pi$
$557$	−1.46410	−0.0620360	−0.0310180	+	0.999519i	$0.509875\pi$
$558$	21.5167	0.910873
$559$	−23.3205	−0.986352
$560$	0	0
$561$	0	0
$562$	−28.3923	−1.19766
$563$	−43.3731	−1.82796	−0.913978	−	0.405763i	$-0.867006\pi$
$563$	−43.3731	−1.82796	−0.913978	+	0.405763i	$0.867006\pi$
$564$	4.92820	0.207515
$565$	0	0
$566$	10.5359	0.442857
$567$	−5.66025	−0.237708
$568$	−6.92820	−0.290701
$569$	3.32051	0.139203	0.0696015	−	0.997575i	$-0.477827\pi$
$569$	3.32051	0.139203	0.0696015	+	0.997575i	$0.477827\pi$
$570$	0	0
$571$	−38.4449	−1.60887	−0.804434	−	0.594042i	$-0.797531\pi$
$571$	−38.4449	−1.60887	−0.804434	+	0.594042i	$0.797531\pi$
$572$	0	0
$573$	1.07180	0.0447750
$574$	−13.2679	−0.553793
$575$	0	0
$576$	−2.46410	−0.102671
$577$	12.2679	0.510721	0.255361	−	0.966846i	$-0.417806\pi$
$577$	12.2679	0.510721	0.255361	+	0.966846i	$0.417806\pi$
$578$	14.0000	0.582323
$579$	−9.66025	−0.401466
$580$	0	0
$581$	−4.14359	−0.171905
$582$	12.5885	0.521808
$583$	0	0
$584$	12.9282	0.534973
$585$	0	0
$586$	1.53590	0.0634474
$587$	24.7321	1.02080	0.510400	−	0.859937i	$-0.329497\pi$
$587$	24.7321	1.02080	0.510400	+	0.859937i	$0.329497\pi$
$588$	−3.94744	−0.162790
$589$	36.6410	1.50977
$590$	0	0
$591$	−7.37307	−0.303287
$592$	−7.92820	−0.325847
$593$	38.9090	1.59780	0.798900	−	0.601464i	$-0.205416\pi$
$593$	38.9090	1.59780	0.798900	+	0.601464i	$0.205416\pi$
$594$	0	0
$595$	0	0
$596$	−13.5885	−0.556605
$597$	−9.56922	−0.391642
$598$	−6.73205	−0.275294
$599$	−36.7321	−1.50083	−0.750415	−	0.660966i	$-0.770147\pi$
$599$	−36.7321	−1.50083	−0.750415	+	0.660966i	$0.770147\pi$
$600$	0	0
$601$	15.7846	0.643868	0.321934	−	0.946762i	$-0.395667\pi$
$601$	15.7846	0.643868	0.321934	+	0.946762i	$0.395667\pi$
$602$	−12.0000	−0.489083
$603$	−17.9090	−0.729309
$604$	10.9282	0.444662
$605$	0	0
$606$	−7.32051	−0.297375
$607$	−13.6603	−0.554453	−0.277226	−	0.960805i	$-0.589415\pi$
$607$	−13.6603	−0.554453	−0.277226	+	0.960805i	$0.589415\pi$
$608$	−4.19615	−0.170176
$609$	3.46410	0.140372
$610$	0	0
$611$	16.5885	0.671097
$612$	4.26795	0.172522
$613$	−27.0000	−1.09052	−0.545260	−	0.838267i	$-0.683569\pi$
$613$	−27.0000	−1.09052	−0.545260	+	0.838267i	$0.683569\pi$
$614$	−8.73205	−0.352397
$615$	0	0
$616$	0	0
$617$	9.33975	0.376004	0.188002	−	0.982169i	$-0.439799\pi$
$617$	9.33975	0.376004	0.188002	+	0.982169i	$0.439799\pi$
$618$	5.85641	0.235579
$619$	2.33975	0.0940423	0.0470212	−	0.998894i	$-0.485027\pi$
$619$	2.33975	0.0940423	0.0470212	+	0.998894i	$0.485027\pi$
$620$	0	0
$621$	−10.9282	−0.438534
$622$	−12.3923	−0.496886
$623$	−0.588457	−0.0235760
$624$	1.80385	0.0722117
$625$	0	0
$626$	−6.26795	−0.250518
$627$	0	0
$628$	10.0000	0.399043
$629$	13.7321	0.547533
$630$	0	0
$631$	−11.2679	−0.448570	−0.224285	−	0.974524i	$-0.572005\pi$
$631$	−11.2679	−0.448570	−0.224285	+	0.974524i	$0.572005\pi$
$632$	−7.26795	−0.289103
$633$	6.64102	0.263957
$634$	25.4641	1.01131
$635$	0	0
$636$	2.19615	0.0870831
$637$	−13.2872	−0.526458
$638$	0	0
$639$	−17.0718	−0.675350
$640$	0	0
$641$	−2.32051	−0.0916546	−0.0458273	−	0.998949i	$-0.514592\pi$
$641$	−2.32051	−0.0916546	−0.0458273	+	0.998949i	$0.514592\pi$
$642$	8.53590	0.336885
$643$	26.5885	1.04855	0.524273	−	0.851550i	$-0.324337\pi$
$643$	26.5885	1.04855	0.524273	+	0.851550i	$0.324337\pi$
$644$	−3.46410	−0.136505
$645$	0	0
$646$	7.26795	0.285954
$647$	25.8564	1.01652	0.508260	−	0.861204i	$-0.330289\pi$
$647$	25.8564	1.01652	0.508260	+	0.861204i	$0.330289\pi$
$648$	−4.46410	−0.175366
$649$	0	0
$650$	0	0
$651$	−8.10512	−0.317665
$652$	−9.12436	−0.357337
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	−6.05256	−0.236674
$655$	0	0
$656$	−10.4641	−0.408555
$657$	31.8564	1.24284
$658$	8.53590	0.332764
$659$	−26.7321	−1.04133	−0.520666	−	0.853760i	$-0.674316\pi$
$659$	−26.7321	−1.04133	−0.520666	+	0.853760i	$0.674316\pi$
$660$	0	0
$661$	42.3731	1.64812	0.824061	−	0.566502i	$-0.191703\pi$
$661$	42.3731	1.64812	0.824061	+	0.566502i	$0.191703\pi$
$662$	9.46410	0.367833
$663$	−3.12436	−0.121340
$664$	−3.26795	−0.126821
$665$	0	0
$666$	−19.5359	−0.757001
$667$	−10.1962	−0.394797
$668$	15.3205	0.592768
$669$	−14.9282	−0.577158
$670$	0	0
$671$	0	0
$672$	0.928203	0.0358062
$673$	−10.7846	−0.415716	−0.207858	−	0.978159i	$-0.566649\pi$
$673$	−10.7846	−0.415716	−0.207858	+	0.978159i	$0.566649\pi$
$674$	22.6603	0.872840
$675$	0	0
$676$	−6.92820	−0.266469
$677$	5.14359	0.197684	0.0988422	−	0.995103i	$-0.468486\pi$
$677$	5.14359	0.197684	0.0988422	+	0.995103i	$0.468486\pi$
$678$	−4.98076	−0.191285
$679$	21.8038	0.836755
$680$	0	0
$681$	−20.7846	−0.796468
$682$	0	0
$683$	−8.33975	−0.319112	−0.159556	−	0.987189i	$-0.551006\pi$
$683$	−8.33975	−0.319112	−0.159556	+	0.987189i	$0.551006\pi$
$684$	−10.3397	−0.395350
$685$	0	0
$686$	−15.7128	−0.599918
$687$	6.83717	0.260854
$688$	−9.46410	−0.360815
$689$	7.39230	0.281624
$690$	0	0
$691$	−31.3205	−1.19149	−0.595744	−	0.803174i	$-0.703143\pi$
$691$	−31.3205	−1.19149	−0.595744	+	0.803174i	$0.703143\pi$
$692$	6.53590	0.248458
$693$	0	0
$694$	21.4641	0.814766
$695$	0	0
$696$	2.73205	0.103558
$697$	18.1244	0.686509
$698$	26.1244	0.988821
$699$	9.66025	0.365384
$700$	0	0
$701$	−47.4449	−1.79197	−0.895984	−	0.444087i	$-0.853528\pi$
$701$	−47.4449	−1.79197	−0.895984	+	0.444087i	$0.853528\pi$
$702$	9.85641	0.372006
$703$	−33.2679	−1.25472
$704$	0	0
$705$	0	0
$706$	−8.12436	−0.305764
$707$	−12.6795	−0.476861
$708$	10.1436	0.381220
$709$	−14.0000	−0.525781	−0.262891	−	0.964826i	$-0.584676\pi$
$709$	−14.0000	−0.525781	−0.262891	+	0.964826i	$0.584676\pi$
$710$	0	0
$711$	−17.9090	−0.671639
$712$	−0.464102	−0.0173929
$713$	23.8564	0.893429
$714$	−1.60770	−0.0601665
$715$	0	0
$716$	−21.8564	−0.816812
$717$	−12.2487	−0.457437
$718$	13.5167	0.504437
$719$	32.4449	1.20999	0.604995	−	0.796230i	$-0.293175\pi$
$719$	32.4449	1.20999	0.604995	+	0.796230i	$0.293175\pi$
$720$	0	0
$721$	10.1436	0.377767
$722$	1.39230	0.0518162
$723$	14.1436	0.526006
$724$	−8.66025	−0.321856
$725$	0	0
$726$	0	0
$727$	3.12436	0.115876	0.0579380	−	0.998320i	$-0.481547\pi$
$727$	3.12436	0.115876	0.0579380	+	0.998320i	$0.481547\pi$
$728$	3.12436	0.115796
$729$	−2.21539	−0.0820515
$730$	0	0
$731$	16.3923	0.606291
$732$	−6.53590	−0.241574
$733$	−2.21539	−0.0818273	−0.0409137	−	0.999163i	$-0.513027\pi$
$733$	−2.21539	−0.0818273	−0.0409137	+	0.999163i	$0.513027\pi$
$734$	3.41154	0.125922
$735$	0	0
$736$	−2.73205	−0.100705
$737$	0	0
$738$	−25.7846	−0.949145
$739$	23.8038	0.875639	0.437819	−	0.899063i	$-0.355751\pi$
$739$	23.8038	0.875639	0.437819	+	0.899063i	$0.355751\pi$
$740$	0	0
$741$	7.56922	0.278062
$742$	3.80385	0.139644
$743$	44.9808	1.65018	0.825092	−	0.564998i	$-0.191123\pi$
$743$	44.9808	1.65018	0.825092	+	0.564998i	$0.191123\pi$
$744$	−6.39230	−0.234353
$745$	0	0
$746$	−12.3923	−0.453715
$747$	−8.05256	−0.294628
$748$	0	0
$749$	14.7846	0.540218
$750$	0	0
$751$	−40.3923	−1.47394	−0.736968	−	0.675928i	$-0.763743\pi$
$751$	−40.3923	−1.47394	−0.736968	+	0.675928i	$0.763743\pi$
$752$	6.73205	0.245493
$753$	−12.1436	−0.442537
$754$	9.19615	0.334904
$755$	0	0
$756$	5.07180	0.184459
$757$	18.4641	0.671089	0.335545	−	0.942024i	$-0.391080\pi$
$757$	18.4641	0.671089	0.335545	+	0.942024i	$0.391080\pi$
$758$	−14.2487	−0.517537
$759$	0	0
$760$	0	0
$761$	3.24871	0.117766	0.0588828	−	0.998265i	$-0.481246\pi$
$761$	3.24871	0.117766	0.0588828	+	0.998265i	$0.481246\pi$
$762$	16.0000	0.579619
$763$	−10.4833	−0.379522
$764$	1.46410	0.0529693
$765$	0	0
$766$	23.3205	0.842604
$767$	34.1436	1.23285
$768$	0.732051	0.0264156
$769$	9.00000	0.324548	0.162274	−	0.986746i	$-0.448117\pi$
$769$	9.00000	0.324548	0.162274	+	0.986746i	$0.448117\pi$
$770$	0	0
$771$	−20.4833	−0.737689
$772$	−13.1962	−0.474940
$773$	13.4641	0.484270	0.242135	−	0.970243i	$-0.422152\pi$
$773$	13.4641	0.484270	0.242135	+	0.970243i	$0.422152\pi$
$774$	−23.3205	−0.838238
$775$	0	0
$776$	17.1962	0.617306
$777$	7.35898	0.264002
$778$	−28.2679	−1.01346
$779$	−43.9090	−1.57320
$780$	0	0
$781$	0	0
$782$	4.73205	0.169218
$783$	14.9282	0.533490
$784$	−5.39230	−0.192582
$785$	0	0
$786$	−4.00000	−0.142675
$787$	32.3923	1.15466	0.577330	−	0.816511i	$-0.304094\pi$
$787$	32.3923	1.15466	0.577330	+	0.816511i	$0.304094\pi$
$788$	−10.0718	−0.358793
$789$	−7.35898	−0.261987
$790$	0	0
$791$	−8.62693	−0.306738
$792$	0	0
$793$	−22.0000	−0.781243
$794$	−26.7128	−0.948002
$795$	0	0
$796$	−13.0718	−0.463318
$797$	−42.2487	−1.49653	−0.748263	−	0.663402i	$-0.769112\pi$
$797$	−42.2487	−1.49653	−0.748263	+	0.663402i	$0.769112\pi$
$798$	3.89488	0.137877
$799$	−11.6603	−0.412510
$800$	0	0
$801$	−1.14359	−0.0404069
$802$	−2.32051	−0.0819400
$803$	0	0
$804$	5.32051	0.187640
$805$	0	0
$806$	−21.5167	−0.757892
$807$	−3.80385	−0.133902
$808$	−10.0000	−0.351799
$809$	−20.3923	−0.716955	−0.358478	−	0.933538i	$-0.616704\pi$
$809$	−20.3923	−0.716955	−0.358478	+	0.933538i	$0.616704\pi$
$810$	0	0
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	4.73205	0.166062
$813$	−13.0718	−0.458448
$814$	0	0
$815$	0	0
$816$	−1.26795	−0.0443871
$817$	−39.7128	−1.38938
$818$	−17.3923	−0.608108
$819$	7.69873	0.269015
$820$	0	0
$821$	31.0718	1.08441	0.542207	−	0.840245i	$-0.317589\pi$
$821$	31.0718	1.08441	0.542207	+	0.840245i	$0.317589\pi$
$822$	4.39230	0.153199
$823$	13.4641	0.469329	0.234665	−	0.972076i	$-0.424601\pi$
$823$	13.4641	0.469329	0.234665	+	0.972076i	$0.424601\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	17.5692	0.611311
$827$	31.3731	1.09095	0.545474	−	0.838128i	$-0.316350\pi$
$827$	31.3731	1.09095	0.545474	+	0.838128i	$0.316350\pi$
$828$	−6.73205	−0.233955
$829$	−1.87564	−0.0651438	−0.0325719	−	0.999469i	$-0.510370\pi$
$829$	−1.87564	−0.0651438	−0.0325719	+	0.999469i	$0.510370\pi$
$830$	0	0
$831$	−10.1962	−0.353701
$832$	2.46410	0.0854274
$833$	9.33975	0.323603
$834$	6.78461	0.234932
$835$	0	0
$836$	0	0
$837$	−34.9282	−1.20730
$838$	−3.41154	−0.117850
$839$	−23.6603	−0.816843	−0.408421	−	0.912794i	$-0.633921\pi$
$839$	−23.6603	−0.816843	−0.408421	+	0.912794i	$0.633921\pi$
$840$	0	0
$841$	−15.0718	−0.519717
$842$	17.0526	0.587670
$843$	20.7846	0.715860
$844$	9.07180	0.312264
$845$	0	0
$846$	16.5885	0.570323
$847$	0	0
$848$	3.00000	0.103020
$849$	−7.71281	−0.264703
$850$	0	0
$851$	−21.6603	−0.742504
$852$	5.07180	0.173757
$853$	−21.3923	−0.732459	−0.366229	−	0.930525i	$-0.619351\pi$
$853$	−21.3923	−0.732459	−0.366229	+	0.930525i	$0.619351\pi$
$854$	−11.3205	−0.387380
$855$	0	0
$856$	11.6603	0.398539
$857$	32.0000	1.09310	0.546550	−	0.837427i	$-0.315941\pi$
$857$	32.0000	1.09310	0.546550	+	0.837427i	$0.315941\pi$
$858$	0	0
$859$	−13.4641	−0.459389	−0.229695	−	0.973263i	$-0.573773\pi$
$859$	−13.4641	−0.459389	−0.229695	+	0.973263i	$0.573773\pi$
$860$	0	0
$861$	9.71281	0.331012
$862$	−17.8564	−0.608192
$863$	−38.4449	−1.30868	−0.654339	−	0.756201i	$-0.727053\pi$
$863$	−38.4449	−1.30868	−0.654339	+	0.756201i	$0.727053\pi$
$864$	4.00000	0.136083
$865$	0	0
$866$	19.5885	0.665643
$867$	−10.2487	−0.348064
$868$	−11.0718	−0.375801
$869$	0	0
$870$	0	0
$871$	17.9090	0.606822
$872$	−8.26795	−0.279988
$873$	42.3731	1.43411
$874$	−11.4641	−0.387779
$875$	0	0
$876$	−9.46410	−0.319762
$877$	−33.7846	−1.14083	−0.570413	−	0.821358i	$-0.693217\pi$
$877$	−33.7846	−1.14083	−0.570413	+	0.821358i	$0.693217\pi$
$878$	−17.1244	−0.577919
$879$	−1.12436	−0.0379236
$880$	0	0
$881$	7.39230	0.249053	0.124527	−	0.992216i	$-0.460259\pi$
$881$	7.39230	0.249053	0.124527	+	0.992216i	$0.460259\pi$
$882$	−13.2872	−0.447403
$883$	17.0718	0.574512	0.287256	−	0.957854i	$-0.407257\pi$
$883$	17.0718	0.574512	0.287256	+	0.957854i	$0.407257\pi$
$884$	−4.26795	−0.143547
$885$	0	0
$886$	−12.7846	−0.429507
$887$	48.9808	1.64461	0.822307	−	0.569045i	$-0.192687\pi$
$887$	48.9808	1.64461	0.822307	+	0.569045i	$0.192687\pi$
$888$	5.80385	0.194764
$889$	27.7128	0.929458
$890$	0	0
$891$	0	0
$892$	−20.3923	−0.682785
$893$	28.2487	0.945307
$894$	9.94744	0.332692
$895$	0	0
$896$	1.26795	0.0423592
$897$	4.92820	0.164548
$898$	21.9282	0.731754
$899$	−32.5885	−1.08689
$900$	0	0
$901$	−5.19615	−0.173109
$902$	0	0
$903$	8.78461	0.292334
$904$	−6.80385	−0.226293
$905$	0	0
$906$	−8.00000	−0.265782
$907$	−8.39230	−0.278662	−0.139331	−	0.990246i	$-0.544495\pi$
$907$	−8.39230	−0.278662	−0.139331	+	0.990246i	$0.544495\pi$
$908$	−28.3923	−0.942232
$909$	−24.6410	−0.817291
$910$	0	0
$911$	−17.4641	−0.578612	−0.289306	−	0.957237i	$-0.593424\pi$
$911$	−17.4641	−0.578612	−0.289306	+	0.957237i	$0.593424\pi$
$912$	3.07180	0.101717
$913$	0	0
$914$	−15.5885	−0.515620
$915$	0	0
$916$	9.33975	0.308594
$917$	−6.92820	−0.228789
$918$	−6.92820	−0.228665
$919$	−28.1051	−0.927102	−0.463551	−	0.886070i	$-0.653425\pi$
$919$	−28.1051	−0.927102	−0.463551	+	0.886070i	$0.653425\pi$
$920$	0	0
$921$	6.39230	0.210634
$922$	−8.51666	−0.280481
$923$	17.0718	0.561925
$924$	0	0
$925$	0	0
$926$	4.00000	0.131448
$927$	19.7128	0.647454
$928$	3.73205	0.122511
$929$	14.0718	0.461681	0.230840	−	0.972992i	$-0.425852\pi$
$929$	14.0718	0.461681	0.230840	+	0.972992i	$0.425852\pi$
$930$	0	0
$931$	−22.6269	−0.741568
$932$	13.1962	0.432254
$933$	9.07180	0.296997
$934$	−25.1769	−0.823814
$935$	0	0
$936$	6.07180	0.198463
$937$	−16.2679	−0.531451	−0.265725	−	0.964049i	$-0.585611\pi$
$937$	−16.2679	−0.531451	−0.265725	+	0.964049i	$0.585611\pi$
$938$	9.21539	0.300893
$939$	4.58846	0.149739
$940$	0	0
$941$	2.51666	0.0820408	0.0410204	−	0.999158i	$-0.486939\pi$
$941$	2.51666	0.0820408	0.0410204	+	0.999158i	$0.486939\pi$
$942$	−7.32051	−0.238515
$943$	−28.5885	−0.930968
$944$	13.8564	0.450988
$945$	0	0
$946$	0	0
$947$	−45.9090	−1.49184	−0.745920	−	0.666035i	$-0.767990\pi$
$947$	−45.9090	−1.49184	−0.745920	+	0.666035i	$0.767990\pi$
$948$	5.32051	0.172802
$949$	−31.8564	−1.03410
$950$	0	0
$951$	−18.6410	−0.604476
$952$	−2.19615	−0.0711777
$953$	19.3397	0.626476	0.313238	−	0.949675i	$-0.398586\pi$
$953$	19.3397	0.626476	0.313238	+	0.949675i	$0.398586\pi$
$954$	7.39230	0.239335
$955$	0	0
$956$	−16.7321	−0.541153
$957$	0	0
$958$	−2.14359	−0.0692564
$959$	7.60770	0.245665
$960$	0	0
$961$	45.2487	1.45964
$962$	19.5359	0.629863
$963$	28.7321	0.925877
$964$	19.3205	0.622272
$965$	0	0
$966$	2.53590	0.0815912
$967$	53.3731	1.71636	0.858181	−	0.513347i	$-0.171595\pi$
$967$	53.3731	1.71636	0.858181	+	0.513347i	$0.171595\pi$
$968$	0	0
$969$	−5.32051	−0.170919
$970$	0	0
$971$	18.7321	0.601140	0.300570	−	0.953760i	$-0.402823\pi$
$971$	18.7321	0.601140	0.300570	+	0.953760i	$0.402823\pi$
$972$	15.2679	0.489720
$973$	11.7513	0.376729
$974$	−38.8372	−1.24442
$975$	0	0
$976$	−8.92820	−0.285785
$977$	41.0526	1.31339	0.656694	−	0.754157i	$-0.271954\pi$
$977$	41.0526	1.31339	0.656694	+	0.754157i	$0.271954\pi$
$978$	6.67949	0.213587
$979$	0	0
$980$	0	0
$981$	−20.3731	−0.650462
$982$	−35.9090	−1.14590
$983$	44.0000	1.40338	0.701691	−	0.712481i	$-0.252429\pi$
$983$	44.0000	1.40338	0.701691	+	0.712481i	$0.252429\pi$
$984$	7.66025	0.244200
$985$	0	0
$986$	−6.46410	−0.205859
$987$	−6.24871	−0.198899
$988$	10.3397	0.328951
$989$	−25.8564	−0.822186
$990$	0	0
$991$	−38.2487	−1.21501	−0.607505	−	0.794316i	$-0.707830\pi$
$991$	−38.2487	−1.21501	−0.607505	+	0.794316i	$0.707830\pi$
$992$	−8.73205	−0.277243
$993$	−6.92820	−0.219860
$994$	8.78461	0.278631
$995$	0	0
$996$	2.39230	0.0758031
$997$	21.1436	0.669624	0.334812	−	0.942285i	$-0.391327\pi$
$997$	21.1436	0.669624	0.334812	+	0.942285i	$0.391327\pi$
$998$	10.9282	0.345926
$999$	31.7128	1.00335

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6050.2.a.bt.1.2		2
5.2	odd	4		1210.2.b.j.969.1	yes	4
5.3	odd	4		1210.2.b.j.969.4	yes	4
5.4	even	2		6050.2.a.cu.1.1		2
11.10	odd	2		6050.2.a.cn.1.2		2
55.32	even	4		1210.2.b.i.969.3	yes	4
55.43	even	4		1210.2.b.i.969.2	✓	4
55.54	odd	2		6050.2.a.cd.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1210.2.b.i.969.2	✓	4	55.43	even	4
1210.2.b.i.969.3	yes	4	55.32	even	4
1210.2.b.j.969.1	yes	4	5.2	odd	4
1210.2.b.j.969.4	yes	4	5.3	odd	4
6050.2.a.bt.1.2		2	1.1	even	1	trivial
6050.2.a.cd.1.1		2	55.54	odd	2
6050.2.a.cn.1.2		2	11.10	odd	2
6050.2.a.cu.1.1		2	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6050 = 2 \cdot 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6050.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +0.732051 q^{3} +1.00000 q^{4} -0.732051 q^{6} -1.26795 q^{7} -1.00000 q^{8} -2.46410 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +0.732051 q^{3} +1.00000 q^{4} -0.732051 q^{6} -1.26795 q^{7} -1.00000 q^{8} -2.46410 q^{9} +0.732051 q^{12} +2.46410 q^{13} +1.26795 q^{14} +1.00000 q^{16} -1.73205 q^{17} +2.46410 q^{18} +4.19615 q^{19} -0.928203 q^{21} +2.73205 q^{23} -0.732051 q^{24} -2.46410 q^{26} -4.00000 q^{27} -1.26795 q^{28} -3.73205 q^{29} +8.73205 q^{31} -1.00000 q^{32} +1.73205 q^{34} -2.46410 q^{36} -7.92820 q^{37} -4.19615 q^{38} +1.80385 q^{39} -10.4641 q^{41} +0.928203 q^{42} -9.46410 q^{43} -2.73205 q^{46} +6.73205 q^{47} +0.732051 q^{48} -5.39230 q^{49} -1.26795 q^{51} +2.46410 q^{52} +3.00000 q^{53} +4.00000 q^{54} +1.26795 q^{56} +3.07180 q^{57} +3.73205 q^{58} +13.8564 q^{59} -8.92820 q^{61} -8.73205 q^{62} +3.12436 q^{63} +1.00000 q^{64} +7.26795 q^{67} -1.73205 q^{68} +2.00000 q^{69} +6.92820 q^{71} +2.46410 q^{72} -12.9282 q^{73} +7.92820 q^{74} +4.19615 q^{76} -1.80385 q^{78} +7.26795 q^{79} +4.46410 q^{81} +10.4641 q^{82} +3.26795 q^{83} -0.928203 q^{84} +9.46410 q^{86} -2.73205 q^{87} +0.464102 q^{89} -3.12436 q^{91} +2.73205 q^{92} +6.39230 q^{93} -6.73205 q^{94} -0.732051 q^{96} -17.1962 q^{97} +5.39230 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 6 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 6 * q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 6 q^{7} - 2 q^{8} + 2 q^{9} - 2 q^{12} - 2 q^{13} + 6 q^{14} + 2 q^{16} - 2 q^{18} - 2 q^{19} + 12 q^{21} + 2 q^{23} + 2 q^{24} + 2 q^{26} - 8 q^{27} - 6 q^{28} - 4 q^{29} + 14 q^{31} - 2 q^{32} + 2 q^{36} - 2 q^{37} + 2 q^{38} + 14 q^{39} - 14 q^{41} - 12 q^{42} - 12 q^{43} - 2 q^{46} + 10 q^{47} - 2 q^{48} + 10 q^{49} - 6 q^{51} - 2 q^{52} + 6 q^{53} + 8 q^{54} + 6 q^{56} + 20 q^{57} + 4 q^{58} - 4 q^{61} - 14 q^{62} - 18 q^{63} + 2 q^{64} + 18 q^{67} + 4 q^{69} - 2 q^{72} - 12 q^{73} + 2 q^{74} - 2 q^{76} - 14 q^{78} + 18 q^{79} + 2 q^{81} + 14 q^{82} + 10 q^{83} + 12 q^{84} + 12 q^{86} - 2 q^{87} - 6 q^{89} + 18 q^{91} + 2 q^{92} - 8 q^{93} - 10 q^{94} + 2 q^{96} - 24 q^{97} - 10 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 6 * q^7 - 2 * q^8 + 2 * q^9 - 2 * q^12 - 2 * q^13 + 6 * q^14 + 2 * q^16 - 2 * q^18 - 2 * q^19 + 12 * q^21 + 2 * q^23 + 2 * q^24 + 2 * q^26 - 8 * q^27 - 6 * q^28 - 4 * q^29 + 14 * q^31 - 2 * q^32 + 2 * q^36 - 2 * q^37 + 2 * q^38 + 14 * q^39 - 14 * q^41 - 12 * q^42 - 12 * q^43 - 2 * q^46 + 10 * q^47 - 2 * q^48 + 10 * q^49 - 6 * q^51 - 2 * q^52 + 6 * q^53 + 8 * q^54 + 6 * q^56 + 20 * q^57 + 4 * q^58 - 4 * q^61 - 14 * q^62 - 18 * q^63 + 2 * q^64 + 18 * q^67 + 4 * q^69 - 2 * q^72 - 12 * q^73 + 2 * q^74 - 2 * q^76 - 14 * q^78 + 18 * q^79 + 2 * q^81 + 14 * q^82 + 10 * q^83 + 12 * q^84 + 12 * q^86 - 2 * q^87 - 6 * q^89 + 18 * q^91 + 2 * q^92 - 8 * q^93 - 10 * q^94 + 2 * q^96 - 24 * q^97 - 10 * q^98

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