LMFDB - Embedded newform 2169.2.a.h.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2169,2,Mod(1,2169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2169, 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("2169.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2169 = 3^{2} \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2169.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:	$17.3195521984$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 14 x^{10} + 44 x^{9} + 65 x^{8} - 219 x^{7} - 123 x^{6} + 444 x^{5} + 105 x^{4} + \cdots - 1 $ x^12 - 3x^11 - 14x^10 + 44x^9 + 65x^8 - 219x^7 - 123x^6 + 444x^5 + 105x^4 - 328x^3 - 45x^2 + 18*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 241)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.115670$ of defining polynomial
Character	$\chi$	$=$	2169.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.115670 q^{2} -1.98662 q^{4} +1.31091 q^{5} +3.19647 q^{7} +0.461133 q^{8} +O(q^{10})$	\(q-0.115670 q^{2} -1.98662 q^{4} +1.31091 q^{5} +3.19647 q^{7} +0.461133 q^{8} -0.151633 q^{10} -1.38968 q^{11} -5.87704 q^{13} -0.369736 q^{14} +3.91990 q^{16} -5.28927 q^{17} +4.99913 q^{19} -2.60428 q^{20} +0.160744 q^{22} -3.07207 q^{23} -3.28152 q^{25} +0.679798 q^{26} -6.35017 q^{28} -3.28657 q^{29} +0.672296 q^{31} -1.37568 q^{32} +0.611810 q^{34} +4.19028 q^{35} -3.79547 q^{37} -0.578249 q^{38} +0.604503 q^{40} -0.970489 q^{41} +7.93946 q^{43} +2.76077 q^{44} +0.355346 q^{46} -2.82021 q^{47} +3.21740 q^{49} +0.379573 q^{50} +11.6754 q^{52} -8.45419 q^{53} -1.82175 q^{55} +1.47400 q^{56} +0.380158 q^{58} -5.70844 q^{59} +0.717980 q^{61} -0.0777646 q^{62} -7.68068 q^{64} -7.70427 q^{65} +8.81215 q^{67} +10.5078 q^{68} -0.484690 q^{70} -15.8552 q^{71} -8.75018 q^{73} +0.439022 q^{74} -9.93137 q^{76} -4.44207 q^{77} +11.4452 q^{79} +5.13864 q^{80} +0.112256 q^{82} +11.9246 q^{83} -6.93376 q^{85} -0.918358 q^{86} -0.640827 q^{88} +11.9996 q^{89} -18.7858 q^{91} +6.10304 q^{92} +0.326214 q^{94} +6.55340 q^{95} +1.18886 q^{97} -0.372157 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{2} + 13 q^{4} - 6 q^{5} + 3 q^{7} - 9 q^{8}+O(q^{10}) $ 12 * q - 3 * q^2 + 13 * q^4 - 6 * q^5 + 3 * q^7 - 9 * q^8	$ 12 q - 3 q^{2} + 13 q^{4} - 6 q^{5} + 3 q^{7} - 9 q^{8} - 7 q^{10} - 22 q^{11} - 5 q^{13} - 6 q^{14} + 15 q^{16} + 4 q^{17} - 6 q^{19} - 10 q^{20} - 12 q^{22} - 32 q^{23} + 4 q^{25} - 8 q^{26} - 11 q^{28} - 6 q^{29} + 8 q^{31} - q^{32} - 19 q^{34} - 15 q^{35} - 8 q^{37} + 10 q^{38} - 52 q^{40} + q^{41} - 2 q^{43} - 42 q^{44} - 25 q^{46} - 34 q^{47} - 9 q^{49} + 27 q^{50} - 41 q^{52} - 5 q^{53} - 3 q^{55} - q^{56} - 33 q^{58} - 26 q^{59} - 26 q^{61} + 17 q^{62} + 13 q^{64} + 25 q^{65} + 6 q^{67} + 35 q^{68} - 4 q^{70} - 94 q^{71} - 22 q^{73} - 26 q^{74} - 20 q^{76} + 7 q^{77} + 9 q^{79} - 19 q^{80} + 15 q^{82} + 8 q^{83} + 4 q^{85} - 9 q^{86} + 6 q^{88} + 3 q^{89} - 20 q^{91} - 36 q^{92} + 48 q^{94} - 33 q^{95} - 29 q^{97} - 28 q^{98}+O(q^{100}) $ 12 * q - 3 * q^2 + 13 * q^4 - 6 * q^5 + 3 * q^7 - 9 * q^8 - 7 * q^10 - 22 * q^11 - 5 * q^13 - 6 * q^14 + 15 * q^16 + 4 * q^17 - 6 * q^19 - 10 * q^20 - 12 * q^22 - 32 * q^23 + 4 * q^25 - 8 * q^26 - 11 * q^28 - 6 * q^29 + 8 * q^31 - q^32 - 19 * q^34 - 15 * q^35 - 8 * q^37 + 10 * q^38 - 52 * q^40 + q^41 - 2 * q^43 - 42 * q^44 - 25 * q^46 - 34 * q^47 - 9 * q^49 + 27 * q^50 - 41 * q^52 - 5 * q^53 - 3 * q^55 - q^56 - 33 * q^58 - 26 * q^59 - 26 * q^61 + 17 * q^62 + 13 * q^64 + 25 * q^65 + 6 * q^67 + 35 * q^68 - 4 * q^70 - 94 * q^71 - 22 * q^73 - 26 * q^74 - 20 * q^76 + 7 * q^77 + 9 * q^79 - 19 * q^80 + 15 * q^82 + 8 * q^83 + 4 * q^85 - 9 * q^86 + 6 * q^88 + 3 * q^89 - 20 * q^91 - 36 * q^92 + 48 * q^94 - 33 * q^95 - 29 * q^97 - 28 * 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 $

$2$	−0.115670	−0.0817911	−0.0408955	−	0.999163i	$-0.513021\pi$
$2$	−0.115670	−0.0817911	−0.0408955	+	0.999163i	$0.513021\pi$
$3$	0	0
$3$	0	0
$4$	−1.98662	−0.993310
$5$	1.31091	0.586256	0.293128	−	0.956073i	$-0.405304\pi$
$5$	1.31091	0.586256	0.293128	+	0.956073i	$0.405304\pi$
$6$	0	0
$7$	3.19647	1.20815	0.604076	−	0.796927i	$-0.293543\pi$
$7$	3.19647	1.20815	0.604076	+	0.796927i	$0.293543\pi$
$8$	0.461133	0.163035
$9$	0	0
$10$	−0.151633	−0.0479506
$11$	−1.38968	−0.419005	−0.209502	−	0.977808i	$-0.567184\pi$
$11$	−1.38968	−0.419005	−0.209502	+	0.977808i	$0.567184\pi$
$12$	0	0
$13$	−5.87704	−1.63000	−0.814999	−	0.579463i	$-0.803262\pi$
$13$	−5.87704	−1.63000	−0.814999	+	0.579463i	$0.803262\pi$
$14$	−0.369736	−0.0988160
$15$	0	0
$16$	3.91990	0.979975
$17$	−5.28927	−1.28284	−0.641418	−	0.767191i	$-0.721654\pi$
$17$	−5.28927	−1.28284	−0.641418	+	0.767191i	$0.721654\pi$
$18$	0	0
$19$	4.99913	1.14688	0.573439	−	0.819248i	$-0.305609\pi$
$19$	4.99913	1.14688	0.573439	+	0.819248i	$0.305609\pi$
$20$	−2.60428	−0.582335
$21$	0	0
$22$	0.160744	0.0342708
$23$	−3.07207	−0.640571	−0.320285	−	0.947321i	$-0.603779\pi$
$23$	−3.07207	−0.640571	−0.320285	+	0.947321i	$0.603779\pi$
$24$	0	0
$25$	−3.28152	−0.656303
$26$	0.679798	0.133319
$27$	0	0
$28$	−6.35017	−1.20007
$29$	−3.28657	−0.610301	−0.305150	−	0.952304i	$-0.598707\pi$
$29$	−3.28657	−0.610301	−0.305150	+	0.952304i	$0.598707\pi$
$30$	0	0
$31$	0.672296	0.120748	0.0603740	−	0.998176i	$-0.480771\pi$
$31$	0.672296	0.120748	0.0603740	+	0.998176i	$0.480771\pi$
$32$	−1.37568	−0.243188
$33$	0	0
$34$	0.611810	0.104925
$35$	4.19028	0.708286
$36$	0	0
$37$	−3.79547	−0.623971	−0.311986	−	0.950087i	$-0.600994\pi$
$37$	−3.79547	−0.623971	−0.311986	+	0.950087i	$0.600994\pi$
$38$	−0.578249	−0.0938044
$39$	0	0
$40$	0.604503	0.0955803
$41$	−0.970489	−0.151565	−0.0757824	−	0.997124i	$-0.524145\pi$
$41$	−0.970489	−0.151565	−0.0757824	+	0.997124i	$0.524145\pi$
$42$	0	0
$43$	7.93946	1.21076	0.605378	−	0.795938i	$-0.293022\pi$
$43$	7.93946	1.21076	0.605378	+	0.795938i	$0.293022\pi$
$44$	2.76077	0.416201
$45$	0	0
$46$	0.355346	0.0523930
$47$	−2.82021	−0.411370	−0.205685	−	0.978618i	$-0.565942\pi$
$47$	−2.82021	−0.411370	−0.205685	+	0.978618i	$0.565942\pi$
$48$	0	0
$49$	3.21740	0.459629
$50$	0.379573	0.0536798
$51$	0	0
$52$	11.6754	1.61909
$53$	−8.45419	−1.16127	−0.580636	−	0.814163i	$-0.697196\pi$
$53$	−8.45419	−1.16127	−0.580636	+	0.814163i	$0.697196\pi$
$54$	0	0
$55$	−1.82175	−0.245644
$56$	1.47400	0.196971
$57$	0	0
$58$	0.380158	0.0499172
$59$	−5.70844	−0.743176	−0.371588	−	0.928398i	$-0.621187\pi$
$59$	−5.70844	−0.743176	−0.371588	+	0.928398i	$0.621187\pi$
$60$	0	0
$61$	0.717980	0.0919279	0.0459639	−	0.998943i	$-0.485364\pi$
$61$	0.717980	0.0919279	0.0459639	+	0.998943i	$0.485364\pi$
$62$	−0.0777646	−0.00987611
$63$	0	0
$64$	−7.68068	−0.960085
$65$	−7.70427	−0.955597
$66$	0	0
$67$	8.81215	1.07658	0.538288	−	0.842761i	$-0.319071\pi$
$67$	8.81215	1.07658	0.538288	+	0.842761i	$0.319071\pi$
$68$	10.5078	1.27425
$69$	0	0
$70$	−0.484690	−0.0579315
$71$	−15.8552	−1.88166	−0.940832	−	0.338874i	$-0.889954\pi$
$71$	−15.8552	−1.88166	−0.940832	+	0.338874i	$0.889954\pi$
$72$	0	0
$73$	−8.75018	−1.02413	−0.512066	−	0.858946i	$-0.671120\pi$
$73$	−8.75018	−1.02413	−0.512066	+	0.858946i	$0.671120\pi$
$74$	0.439022	0.0510353
$75$	0	0
$76$	−9.93137	−1.13921
$77$	−4.44207	−0.506221
$78$	0	0
$79$	11.4452	1.28769	0.643843	−	0.765158i	$-0.277339\pi$
$79$	11.4452	1.28769	0.643843	+	0.765158i	$0.277339\pi$
$80$	5.13864	0.574517
$81$	0	0
$82$	0.112256	0.0123967
$83$	11.9246	1.30889	0.654446	−	0.756109i	$-0.272902\pi$
$83$	11.9246	1.30889	0.654446	+	0.756109i	$0.272902\pi$
$84$	0	0
$85$	−6.93376	−0.752071
$86$	−0.918358	−0.0990291
$87$	0	0
$88$	−0.640827	−0.0683124
$89$	11.9996	1.27195	0.635975	−	0.771709i	$-0.280598\pi$
$89$	11.9996	1.27195	0.635975	+	0.771709i	$0.280598\pi$
$90$	0	0
$91$	−18.7858	−1.96928
$92$	6.10304	0.636285
$93$	0	0
$94$	0.326214	0.0336464
$95$	6.55340	0.672365
$96$	0	0
$97$	1.18886	0.120710	0.0603550	−	0.998177i	$-0.480777\pi$
$97$	1.18886	0.120710	0.0603550	+	0.998177i	$0.480777\pi$
$98$	−0.372157	−0.0375936
$99$	0	0
$100$	6.51913	0.651913
$101$	−19.9051	−1.98064	−0.990318	−	0.138820i	$-0.955669\pi$
$101$	−19.9051	−1.98064	−0.990318	+	0.138820i	$0.955669\pi$
$102$	0	0
$103$	−2.74129	−0.270108	−0.135054	−	0.990838i	$-0.543121\pi$
$103$	−2.74129	−0.270108	−0.135054	+	0.990838i	$0.543121\pi$
$104$	−2.71010	−0.265747
$105$	0	0
$106$	0.977897	0.0949817
$107$	9.95829	0.962704	0.481352	−	0.876527i	$-0.340146\pi$
$107$	9.95829	0.962704	0.481352	+	0.876527i	$0.340146\pi$
$108$	0	0
$109$	−5.65622	−0.541768	−0.270884	−	0.962612i	$-0.587316\pi$
$109$	−5.65622	−0.541768	−0.270884	+	0.962612i	$0.587316\pi$
$110$	0.210721	0.0200915
$111$	0	0
$112$	12.5298	1.18396
$113$	12.9604	1.21921	0.609604	−	0.792706i	$-0.291329\pi$
$113$	12.9604	1.21921	0.609604	+	0.792706i	$0.291329\pi$
$114$	0	0
$115$	−4.02720	−0.375539
$116$	6.52917	0.606218
$117$	0	0
$118$	0.660296	0.0607852
$119$	−16.9070	−1.54986
$120$	0	0
$121$	−9.06879	−0.824435
$122$	−0.0830488	−0.00751888
$123$	0	0
$124$	−1.33560	−0.119940
$125$	−10.8563	−0.971019
$126$	0	0
$127$	−10.4647	−0.928593	−0.464297	−	0.885680i	$-0.653693\pi$
$127$	−10.4647	−0.928593	−0.464297	+	0.885680i	$0.653693\pi$
$128$	3.63979	0.321715
$129$	0	0
$130$	0.891153	0.0781593
$131$	−12.9607	−1.13238	−0.566190	−	0.824275i	$-0.691583\pi$
$131$	−12.9607	−1.13238	−0.566190	+	0.824275i	$0.691583\pi$
$132$	0	0
$133$	15.9795	1.38560
$134$	−1.01930	−0.0880543
$135$	0	0
$136$	−2.43906	−0.209147
$137$	−14.1420	−1.20824	−0.604118	−	0.796895i	$-0.706474\pi$
$137$	−14.1420	−1.20824	−0.604118	+	0.796895i	$0.706474\pi$
$138$	0	0
$139$	−16.8589	−1.42996	−0.714978	−	0.699147i	$-0.753563\pi$
$139$	−16.8589	−1.42996	−0.714978	+	0.699147i	$0.753563\pi$
$140$	−8.32449	−0.703548
$141$	0	0
$142$	1.83397	0.153903
$143$	8.16721	0.682976
$144$	0	0
$145$	−4.30840	−0.357793
$146$	1.01213	0.0837648
$147$	0	0
$148$	7.54015	0.619797
$149$	−16.2523	−1.33144	−0.665722	−	0.746200i	$-0.731876\pi$
$149$	−16.2523	−1.33144	−0.665722	+	0.746200i	$0.731876\pi$
$150$	0	0
$151$	−10.4085	−0.847035	−0.423517	−	0.905888i	$-0.639205\pi$
$151$	−10.4085	−0.847035	−0.423517	+	0.905888i	$0.639205\pi$
$152$	2.30526	0.186981
$153$	0	0
$154$	0.513814	0.0414043
$155$	0.881320	0.0707893
$156$	0	0
$157$	−18.9894	−1.51552	−0.757758	−	0.652535i	$-0.773705\pi$
$157$	−18.9894	−1.51552	−0.757758	+	0.652535i	$0.773705\pi$
$158$	−1.32387	−0.105321
$159$	0	0
$160$	−1.80339	−0.142571
$161$	−9.81977	−0.773906
$162$	0	0
$163$	7.43947	0.582704	0.291352	−	0.956616i	$-0.405895\pi$
$163$	7.43947	0.582704	0.291352	+	0.956616i	$0.405895\pi$
$164$	1.92799	0.150551
$165$	0	0
$166$	−1.37932	−0.107056
$167$	−13.4159	−1.03815	−0.519076	−	0.854728i	$-0.673724\pi$
$167$	−13.4159	−1.03815	−0.519076	+	0.854728i	$0.673724\pi$
$168$	0	0
$169$	21.5396	1.65689
$170$	0.802028	0.0615127
$171$	0	0
$172$	−15.7727	−1.20266
$173$	3.18442	0.242107	0.121053	−	0.992646i	$-0.461373\pi$
$173$	3.18442	0.242107	0.121053	+	0.992646i	$0.461373\pi$
$174$	0	0
$175$	−10.4893	−0.792914
$176$	−5.44741	−0.410614
$177$	0	0
$178$	−1.38799	−0.104034
$179$	−11.8290	−0.884142	−0.442071	−	0.896980i	$-0.645756\pi$
$179$	−11.8290	−0.884142	−0.442071	+	0.896980i	$0.645756\pi$
$180$	0	0
$181$	−3.92038	−0.291399	−0.145700	−	0.989329i	$-0.546543\pi$
$181$	−3.92038	−0.291399	−0.145700	+	0.989329i	$0.546543\pi$
$182$	2.17295	0.161070
$183$	0	0
$184$	−1.41663	−0.104435
$185$	−4.97551	−0.365807
$186$	0	0
$187$	7.35040	0.537514
$188$	5.60269	0.408618
$189$	0	0
$190$	−0.758032	−0.0549934
$191$	−1.27094	−0.0919617	−0.0459809	−	0.998942i	$-0.514641\pi$
$191$	−1.27094	−0.0919617	−0.0459809	+	0.998942i	$0.514641\pi$
$192$	0	0
$193$	−7.18460	−0.517159	−0.258580	−	0.965990i	$-0.583254\pi$
$193$	−7.18460	−0.517159	−0.258580	+	0.965990i	$0.583254\pi$
$194$	−0.137515	−0.00987300
$195$	0	0
$196$	−6.39176	−0.456554
$197$	−6.64645	−0.473540	−0.236770	−	0.971566i	$-0.576089\pi$
$197$	−6.64645	−0.473540	−0.236770	+	0.971566i	$0.576089\pi$
$198$	0	0
$199$	12.7358	0.902819	0.451410	−	0.892317i	$-0.350921\pi$
$199$	12.7358	0.902819	0.451410	+	0.892317i	$0.350921\pi$
$200$	−1.51321	−0.107000
$201$	0	0
$202$	2.30243	0.161998
$203$	−10.5054	−0.737336
$204$	0	0
$205$	−1.27222	−0.0888559
$206$	0.317085	0.0220924
$207$	0	0
$208$	−23.0374	−1.59736
$209$	−6.94719	−0.480547
$210$	0	0
$211$	5.22527	0.359722	0.179861	−	0.983692i	$-0.442435\pi$
$211$	5.22527	0.359722	0.179861	+	0.983692i	$0.442435\pi$
$212$	16.7953	1.15350
$213$	0	0
$214$	−1.15188	−0.0787406
$215$	10.4079	0.709814
$216$	0	0
$217$	2.14897	0.145882
$218$	0.654256	0.0443118
$219$	0	0
$220$	3.61912	0.244001
$221$	31.0853	2.09102
$222$	0	0
$223$	−1.68503	−0.112838	−0.0564191	−	0.998407i	$-0.517968\pi$
$223$	−1.68503	−0.112838	−0.0564191	+	0.998407i	$0.517968\pi$
$224$	−4.39732	−0.293808
$225$	0	0
$226$	−1.49913	−0.0997203
$227$	1.27020	0.0843064	0.0421532	−	0.999111i	$-0.486578\pi$
$227$	1.27020	0.0843064	0.0421532	+	0.999111i	$0.486578\pi$
$228$	0	0
$229$	24.4833	1.61790	0.808950	−	0.587877i	$-0.200036\pi$
$229$	24.4833	1.61790	0.808950	+	0.587877i	$0.200036\pi$
$230$	0.465827	0.0307157
$231$	0	0
$232$	−1.51554	−0.0995004
$233$	−3.58351	−0.234764	−0.117382	−	0.993087i	$-0.537450\pi$
$233$	−3.58351	−0.234764	−0.117382	+	0.993087i	$0.537450\pi$
$234$	0	0
$235$	−3.69704	−0.241169
$236$	11.3405	0.738204
$237$	0	0
$238$	1.95563	0.126765
$239$	27.0261	1.74817	0.874087	−	0.485769i	$-0.161460\pi$
$239$	27.0261	1.74817	0.874087	+	0.485769i	$0.161460\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	1.04899	0.0674315
$243$	0	0
$244$	−1.42635	−0.0913129
$245$	4.21772	0.269460
$246$	0	0
$247$	−29.3801	−1.86941
$248$	0.310018	0.0196862
$249$	0	0
$250$	1.25575	0.0794207
$251$	19.0710	1.20375	0.601874	−	0.798591i	$-0.294421\pi$
$251$	19.0710	1.20375	0.601874	+	0.798591i	$0.294421\pi$
$252$	0	0
$253$	4.26920	0.268402
$254$	1.21045	0.0759506
$255$	0	0
$256$	14.9403	0.933771
$257$	29.8662	1.86300	0.931502	−	0.363736i	$-0.118499\pi$
$257$	29.8662	1.86300	0.931502	+	0.363736i	$0.118499\pi$
$258$	0	0
$259$	−12.1321	−0.753851
$260$	15.3055	0.949204
$261$	0	0
$262$	1.49916	0.0926186
$263$	14.4912	0.893563	0.446781	−	0.894643i	$-0.352570\pi$
$263$	14.4912	0.893563	0.446781	+	0.894643i	$0.352570\pi$
$264$	0	0
$265$	−11.0827	−0.680803
$266$	−1.84835	−0.113330
$267$	0	0
$268$	−17.5064	−1.06937
$269$	−9.73320	−0.593444	−0.296722	−	0.954964i	$-0.595893\pi$
$269$	−9.73320	−0.593444	−0.296722	+	0.954964i	$0.595893\pi$
$270$	0	0
$271$	17.7984	1.08117	0.540587	−	0.841288i	$-0.318202\pi$
$271$	17.7984	1.08117	0.540587	+	0.841288i	$0.318202\pi$
$272$	−20.7334	−1.25715
$273$	0	0
$274$	1.63581	0.0988229
$275$	4.56026	0.274994
$276$	0	0
$277$	2.28565	0.137332	0.0686658	−	0.997640i	$-0.478126\pi$
$277$	2.28565	0.137332	0.0686658	+	0.997640i	$0.478126\pi$
$278$	1.95007	0.116958
$279$	0	0
$280$	1.93227	0.115475
$281$	−9.34433	−0.557436	−0.278718	−	0.960373i	$-0.589910\pi$
$281$	−9.34433	−0.557436	−0.278718	+	0.960373i	$0.589910\pi$
$282$	0	0
$283$	8.86628	0.527045	0.263523	−	0.964653i	$-0.415116\pi$
$283$	8.86628	0.527045	0.263523	+	0.964653i	$0.415116\pi$
$284$	31.4982	1.86908
$285$	0	0
$286$	−0.944702	−0.0558614
$287$	−3.10214	−0.183113
$288$	0	0
$289$	10.9764	0.645670
$290$	0.498352	0.0292643
$291$	0	0
$292$	17.3833	1.01728
$293$	−17.1228	−1.00032	−0.500161	−	0.865932i	$-0.666726\pi$
$293$	−17.1228	−1.00032	−0.500161	+	0.865932i	$0.666726\pi$
$294$	0	0
$295$	−7.48325	−0.435692
$296$	−1.75021	−0.101729
$297$	0	0
$298$	1.87991	0.108900
$299$	18.0547	1.04413
$300$	0	0
$301$	25.3782	1.46278
$302$	1.20396	0.0692799
$303$	0	0
$304$	19.5961	1.12391
$305$	0.941206	0.0538933
$306$	0	0
$307$	5.03009	0.287082	0.143541	−	0.989644i	$-0.454151\pi$
$307$	5.03009	0.287082	0.143541	+	0.989644i	$0.454151\pi$
$308$	8.82471	0.502834
$309$	0	0
$310$	−0.101942	−0.00578993
$311$	−9.99750	−0.566906	−0.283453	−	0.958986i	$-0.591480\pi$
$311$	−9.99750	−0.566906	−0.283453	+	0.958986i	$0.591480\pi$
$312$	0	0
$313$	28.1466	1.59094	0.795471	−	0.605992i	$-0.207224\pi$
$313$	28.1466	1.59094	0.795471	+	0.605992i	$0.207224\pi$
$314$	2.19650	0.123956
$315$	0	0
$316$	−22.7373	−1.27907
$317$	5.66871	0.318387	0.159193	−	0.987247i	$-0.449111\pi$
$317$	5.66871	0.318387	0.159193	+	0.987247i	$0.449111\pi$
$318$	0	0
$319$	4.56728	0.255719
$320$	−10.0687	−0.562856
$321$	0	0
$322$	1.13585	0.0632986
$323$	−26.4417	−1.47126
$324$	0	0
$325$	19.2856	1.06977
$326$	−0.860524	−0.0476600
$327$	0	0
$328$	−0.447524	−0.0247104
$329$	−9.01472	−0.496998
$330$	0	0
$331$	−14.8950	−0.818701	−0.409350	−	0.912377i	$-0.634245\pi$
$331$	−14.8950	−0.818701	−0.409350	+	0.912377i	$0.634245\pi$
$332$	−23.6896	−1.30014
$333$	0	0
$334$	1.55181	0.0849115
$335$	11.5519	0.631149
$336$	0	0
$337$	21.4134	1.16646	0.583231	−	0.812306i	$-0.301788\pi$
$337$	21.4134	1.16646	0.583231	+	0.812306i	$0.301788\pi$
$338$	−2.49149	−0.135519
$339$	0	0
$340$	13.7747	0.747040
$341$	−0.934278	−0.0505940
$342$	0	0
$343$	−12.0909	−0.652850
$344$	3.66115	0.197396
$345$	0	0
$346$	−0.368342	−0.0198022
$347$	6.12740	0.328936	0.164468	−	0.986382i	$-0.447409\pi$
$347$	6.12740	0.328936	0.164468	+	0.986382i	$0.447409\pi$
$348$	0	0
$349$	−7.15661	−0.383085	−0.191542	−	0.981484i	$-0.561349\pi$
$349$	−7.15661	−0.383085	−0.191542	+	0.981484i	$0.561349\pi$
$350$	1.21329	0.0648533
$351$	0	0
$352$	1.91176	0.101897
$353$	9.47379	0.504239	0.252120	−	0.967696i	$-0.418872\pi$
$353$	9.47379	0.504239	0.252120	+	0.967696i	$0.418872\pi$
$354$	0	0
$355$	−20.7847	−1.10314
$356$	−23.8386	−1.26344
$357$	0	0
$358$	1.36826	0.0723149
$359$	−7.14694	−0.377201	−0.188601	−	0.982054i	$-0.560395\pi$
$359$	−7.14694	−0.377201	−0.188601	+	0.982054i	$0.560395\pi$
$360$	0	0
$361$	5.99126	0.315330
$362$	0.453470	0.0238339
$363$	0	0
$364$	37.3202	1.95611
$365$	−11.4707	−0.600404
$366$	0	0
$367$	2.63925	0.137767	0.0688837	−	0.997625i	$-0.478056\pi$
$367$	2.63925	0.137767	0.0688837	+	0.997625i	$0.478056\pi$
$368$	−12.0422	−0.627744
$369$	0	0
$370$	0.575518	0.0299198
$371$	−27.0235	−1.40299
$372$	0	0
$373$	20.5114	1.06204	0.531020	−	0.847360i	$-0.321809\pi$
$373$	20.5114	1.06204	0.531020	+	0.847360i	$0.321809\pi$
$374$	−0.850221	−0.0439639
$375$	0	0
$376$	−1.30049	−0.0670678
$377$	19.3153	0.994789
$378$	0	0
$379$	−24.7263	−1.27010	−0.635051	−	0.772470i	$-0.719021\pi$
$379$	−24.7263	−1.27010	−0.635051	+	0.772470i	$0.719021\pi$
$380$	−13.0191	−0.667867
$381$	0	0
$382$	0.147009	0.00752165
$383$	33.2690	1.69997	0.849983	−	0.526810i	$-0.176612\pi$
$383$	33.2690	1.69997	0.849983	+	0.526810i	$0.176612\pi$
$384$	0	0
$385$	−5.82315	−0.296775
$386$	0.831044	0.0422990
$387$	0	0
$388$	−2.36181	−0.119902
$389$	11.2024	0.567985	0.283992	−	0.958827i	$-0.408341\pi$
$389$	11.2024	0.567985	0.283992	+	0.958827i	$0.408341\pi$
$390$	0	0
$391$	16.2490	0.821748
$392$	1.48365	0.0749356
$393$	0	0
$394$	0.768796	0.0387314
$395$	15.0036	0.754914
$396$	0	0
$397$	5.20629	0.261296	0.130648	−	0.991429i	$-0.458294\pi$
$397$	5.20629	0.261296	0.130648	+	0.991429i	$0.458294\pi$
$398$	−1.47315	−0.0738426
$399$	0	0
$400$	−12.8632	−0.643161
$401$	2.79646	0.139649	0.0698243	−	0.997559i	$-0.477756\pi$
$401$	2.79646	0.139649	0.0698243	+	0.997559i	$0.477756\pi$
$402$	0	0
$403$	−3.95111	−0.196819
$404$	39.5440	1.96739
$405$	0	0
$406$	1.21516	0.0603075
$407$	5.27449	0.261447
$408$	0	0
$409$	0.743107	0.0367443	0.0183721	−	0.999831i	$-0.494152\pi$
$409$	0.743107	0.0367443	0.0183721	+	0.999831i	$0.494152\pi$
$410$	0.147158	0.00726762
$411$	0	0
$412$	5.44591	0.268301
$413$	−18.2469	−0.897869
$414$	0	0
$415$	15.6320	0.767346
$416$	8.08493	0.396396
$417$	0	0
$418$	0.803582	0.0393045
$419$	11.8016	0.576544	0.288272	−	0.957549i	$-0.406919\pi$
$419$	11.8016	0.576544	0.288272	+	0.957549i	$0.406919\pi$
$420$	0	0
$421$	−9.89956	−0.482475	−0.241237	−	0.970466i	$-0.577553\pi$
$421$	−9.89956	−0.482475	−0.241237	+	0.970466i	$0.577553\pi$
$422$	−0.604407	−0.0294221
$423$	0	0
$424$	−3.89850	−0.189328
$425$	17.3568	0.841930
$426$	0	0
$427$	2.29500	0.111063
$428$	−19.7833	−0.956264
$429$	0	0
$430$	−1.20388	−0.0580565
$431$	−9.64934	−0.464793	−0.232396	−	0.972621i	$-0.574657\pi$
$431$	−9.64934	−0.464793	−0.232396	+	0.972621i	$0.574657\pi$
$432$	0	0
$433$	−38.9496	−1.87180	−0.935900	−	0.352265i	$-0.885412\pi$
$433$	−38.9496	−1.87180	−0.935900	+	0.352265i	$0.885412\pi$
$434$	−0.248572	−0.0119318
$435$	0	0
$436$	11.2368	0.538144
$437$	−15.3577	−0.734657
$438$	0	0
$439$	−16.8089	−0.802243	−0.401121	−	0.916025i	$-0.631379\pi$
$439$	−16.8089	−0.802243	−0.401121	+	0.916025i	$0.631379\pi$
$440$	−0.840066	−0.0400486
$441$	0	0
$442$	−3.59563	−0.171027
$443$	0.438324	0.0208254	0.0104127	−	0.999946i	$-0.496685\pi$
$443$	0.438324	0.0208254	0.0104127	+	0.999946i	$0.496685\pi$
$444$	0	0
$445$	15.7303	0.745689
$446$	0.194908	0.00922917
$447$	0	0
$448$	−24.5510	−1.15993
$449$	31.7722	1.49942	0.749711	−	0.661766i	$-0.230193\pi$
$449$	31.7722	1.49942	0.749711	+	0.661766i	$0.230193\pi$
$450$	0	0
$451$	1.34867	0.0635064
$452$	−25.7473	−1.21105
$453$	0	0
$454$	−0.146925	−0.00689551
$455$	−24.6264	−1.15451
$456$	0	0
$457$	17.1543	0.802444	0.401222	−	0.915981i	$-0.368586\pi$
$457$	17.1543	0.802444	0.401222	+	0.915981i	$0.368586\pi$
$458$	−2.83198	−0.132330
$459$	0	0
$460$	8.00053	0.373026
$461$	30.9242	1.44028	0.720141	−	0.693827i	$-0.244077\pi$
$461$	30.9242	1.44028	0.720141	+	0.693827i	$0.244077\pi$
$462$	0	0
$463$	15.6931	0.729319	0.364659	−	0.931141i	$-0.381185\pi$
$463$	15.6931	0.729319	0.364659	+	0.931141i	$0.381185\pi$
$464$	−12.8830	−0.598080
$465$	0	0
$466$	0.414505	0.0192016
$467$	−21.3617	−0.988501	−0.494250	−	0.869320i	$-0.664557\pi$
$467$	−21.3617	−0.988501	−0.494250	+	0.869320i	$0.664557\pi$
$468$	0	0
$469$	28.1678	1.30067
$470$	0.427637	0.0197254
$471$	0	0
$472$	−2.63235	−0.121164
$473$	−11.0333	−0.507313
$474$	0	0
$475$	−16.4047	−0.752700
$476$	33.5878	1.53949
$477$	0	0
$478$	−3.12611	−0.142985
$479$	−27.3633	−1.25026	−0.625130	−	0.780521i	$-0.714954\pi$
$479$	−27.3633	−1.25026	−0.625130	+	0.780521i	$0.714954\pi$
$480$	0	0
$481$	22.3061	1.01707
$482$	−0.115670	−0.00526863
$483$	0	0
$484$	18.0162	0.818920
$485$	1.55848	0.0707670
$486$	0	0
$487$	−9.12015	−0.413274	−0.206637	−	0.978418i	$-0.566252\pi$
$487$	−9.12015	−0.413274	−0.206637	+	0.978418i	$0.566252\pi$
$488$	0.331084	0.0149875
$489$	0	0
$490$	−0.487864	−0.0220395
$491$	7.29684	0.329302	0.164651	−	0.986352i	$-0.447350\pi$
$491$	7.29684	0.329302	0.164651	+	0.986352i	$0.447350\pi$
$492$	0	0
$493$	17.3836	0.782916
$494$	3.39839	0.152901
$495$	0	0
$496$	2.63534	0.118330
$497$	−50.6806	−2.27333
$498$	0	0
$499$	1.98278	0.0887613	0.0443807	−	0.999015i	$-0.485869\pi$
$499$	1.98278	0.0887613	0.0443807	+	0.999015i	$0.485869\pi$
$500$	21.5674	0.964523
$501$	0	0
$502$	−2.20594	−0.0984559
$503$	−7.01318	−0.312702	−0.156351	−	0.987702i	$-0.549973\pi$
$503$	−7.01318	−0.312702	−0.156351	+	0.987702i	$0.549973\pi$
$504$	0	0
$505$	−26.0938	−1.16116
$506$	−0.493818	−0.0219529
$507$	0	0
$508$	20.7894	0.922381
$509$	28.5738	1.26651	0.633256	−	0.773943i	$-0.281718\pi$
$509$	28.5738	1.26651	0.633256	+	0.773943i	$0.281718\pi$
$510$	0	0
$511$	−27.9697	−1.23731
$512$	−9.00772	−0.398089
$513$	0	0
$514$	−3.45463	−0.152377
$515$	−3.59359	−0.158352
$516$	0	0
$517$	3.91920	0.172366
$518$	1.40332	0.0616583
$519$	0	0
$520$	−3.55269	−0.155796
$521$	13.1589	0.576501	0.288250	−	0.957555i	$-0.406926\pi$
$521$	13.1589	0.576501	0.288250	+	0.957555i	$0.406926\pi$
$522$	0	0
$523$	23.8792	1.04416	0.522082	−	0.852895i	$-0.325155\pi$
$523$	23.8792	1.04416	0.522082	+	0.852895i	$0.325155\pi$
$524$	25.7480	1.12480
$525$	0	0
$526$	−1.67619	−0.0730855
$527$	−3.55596	−0.154900
$528$	0	0
$529$	−13.5624	−0.589669
$530$	1.28193	0.0556836
$531$	0	0
$532$	−31.7453	−1.37633
$533$	5.70360	0.247050
$534$	0	0
$535$	13.0544	0.564391
$536$	4.06357	0.175520
$537$	0	0
$538$	1.12584	0.0485384
$539$	−4.47116	−0.192587
$540$	0	0
$541$	−7.11389	−0.305850	−0.152925	−	0.988238i	$-0.548869\pi$
$541$	−7.11389	−0.305850	−0.152925	+	0.988238i	$0.548869\pi$
$542$	−2.05874	−0.0884304
$543$	0	0
$544$	7.27635	0.311971
$545$	−7.41480	−0.317615
$546$	0	0
$547$	2.10736	0.0901044	0.0450522	−	0.998985i	$-0.485655\pi$
$547$	2.10736	0.0901044	0.0450522	+	0.998985i	$0.485655\pi$
$548$	28.0949	1.20015
$549$	0	0
$550$	−0.527486	−0.0224921
$551$	−16.4300	−0.699941
$552$	0	0
$553$	36.5842	1.55572
$554$	−0.264382	−0.0112325
$555$	0	0
$556$	33.4923	1.42039
$557$	9.69468	0.410777	0.205388	−	0.978681i	$-0.434154\pi$
$557$	9.69468	0.410777	0.205388	+	0.978681i	$0.434154\pi$
$558$	0	0
$559$	−46.6605	−1.97353
$560$	16.4255	0.694103
$561$	0	0
$562$	1.08086	0.0455933
$563$	12.6385	0.532647	0.266324	−	0.963884i	$-0.414191\pi$
$563$	12.6385	0.532647	0.266324	+	0.963884i	$0.414191\pi$
$564$	0	0
$565$	16.9899	0.714768
$566$	−1.02556	−0.0431076
$567$	0	0
$568$	−7.31134	−0.306777
$569$	−17.6815	−0.741245	−0.370622	−	0.928784i	$-0.620856\pi$
$569$	−17.6815	−0.741245	−0.370622	+	0.928784i	$0.620856\pi$
$570$	0	0
$571$	−24.5248	−1.02633	−0.513166	−	0.858289i	$-0.671527\pi$
$571$	−24.5248	−1.02633	−0.513166	+	0.858289i	$0.671527\pi$
$572$	−16.2251	−0.678407
$573$	0	0
$574$	0.358824	0.0149770
$575$	10.0810	0.420409
$576$	0	0
$577$	−35.7883	−1.48988	−0.744942	−	0.667129i	$-0.767523\pi$
$577$	−35.7883	−1.48988	−0.744942	+	0.667129i	$0.767523\pi$
$578$	−1.26964	−0.0528101
$579$	0	0
$580$	8.55915	0.355399
$581$	38.1165	1.58134
$582$	0	0
$583$	11.7486	0.486578
$584$	−4.03500	−0.166969
$585$	0	0
$586$	1.98059	0.0818175
$587$	42.4486	1.75204	0.876021	−	0.482273i	$-0.160189\pi$
$587$	42.4486	1.75204	0.876021	+	0.482273i	$0.160189\pi$
$588$	0	0
$589$	3.36089	0.138483
$590$	0.865588	0.0356357
$591$	0	0
$592$	−14.8779	−0.611476
$593$	−17.8382	−0.732527	−0.366263	−	0.930511i	$-0.619363\pi$
$593$	−17.8382	−0.732527	−0.366263	+	0.930511i	$0.619363\pi$
$594$	0	0
$595$	−22.1635	−0.908616
$596$	32.2872	1.32254
$597$	0	0
$598$	−2.08839	−0.0854004
$599$	−21.3734	−0.873291	−0.436646	−	0.899634i	$-0.643834\pi$
$599$	−21.3734	−0.873291	−0.436646	+	0.899634i	$0.643834\pi$
$600$	0	0
$601$	−42.3095	−1.72584	−0.862920	−	0.505341i	$-0.831367\pi$
$601$	−42.3095	−1.72584	−0.862920	+	0.505341i	$0.831367\pi$
$602$	−2.93550	−0.119642
$603$	0	0
$604$	20.6778	0.841368
$605$	−11.8884	−0.483330
$606$	0	0
$607$	1.51410	0.0614552	0.0307276	−	0.999528i	$-0.490218\pi$
$607$	1.51410	0.0614552	0.0307276	+	0.999528i	$0.490218\pi$
$608$	−6.87720	−0.278907
$609$	0	0
$610$	−0.108869	−0.00440799
$611$	16.5745	0.670533
$612$	0	0
$613$	−15.4991	−0.626001	−0.313000	−	0.949753i	$-0.601334\pi$
$613$	−15.4991	−0.626001	−0.313000	+	0.949753i	$0.601334\pi$
$614$	−0.581830	−0.0234808
$615$	0	0
$616$	−2.04838	−0.0825317
$617$	−22.9695	−0.924718	−0.462359	−	0.886693i	$-0.652997\pi$
$617$	−22.9695	−0.924718	−0.462359	+	0.886693i	$0.652997\pi$
$618$	0	0
$619$	36.7503	1.47712	0.738559	−	0.674189i	$-0.235507\pi$
$619$	36.7503	1.47712	0.738559	+	0.674189i	$0.235507\pi$
$620$	−1.75085	−0.0703157
$621$	0	0
$622$	1.15641	0.0463679
$623$	38.3562	1.53671
$624$	0	0
$625$	2.17594	0.0870375
$626$	−3.25572	−0.130125
$627$	0	0
$628$	37.7247	1.50538
$629$	20.0753	0.800453
$630$	0	0
$631$	−16.5412	−0.658494	−0.329247	−	0.944244i	$-0.606795\pi$
$631$	−16.5412	−0.658494	−0.329247	+	0.944244i	$0.606795\pi$
$632$	5.27775	0.209938
$633$	0	0
$634$	−0.655700	−0.0260412
$635$	−13.7183	−0.544394
$636$	0	0
$637$	−18.9088	−0.749194
$638$	−0.528298	−0.0209155
$639$	0	0
$640$	4.77143	0.188607
$641$	−1.70611	−0.0673873	−0.0336937	−	0.999432i	$-0.510727\pi$
$641$	−1.70611	−0.0673873	−0.0336937	+	0.999432i	$0.510727\pi$
$642$	0	0
$643$	−12.2046	−0.481302	−0.240651	−	0.970612i	$-0.577361\pi$
$643$	−12.2046	−0.481302	−0.240651	+	0.970612i	$0.577361\pi$
$644$	19.5082	0.768729
$645$	0	0
$646$	3.05852	0.120336
$647$	−28.0548	−1.10295	−0.551473	−	0.834193i	$-0.685934\pi$
$647$	−28.0548	−1.10295	−0.551473	+	0.834193i	$0.685934\pi$
$648$	0	0
$649$	7.93291	0.311394
$650$	−2.23077	−0.0874979
$651$	0	0
$652$	−14.7794	−0.578806
$653$	−5.68400	−0.222432	−0.111216	−	0.993796i	$-0.535475\pi$
$653$	−5.68400	−0.222432	−0.111216	+	0.993796i	$0.535475\pi$
$654$	0	0
$655$	−16.9903	−0.663865
$656$	−3.80422	−0.148530
$657$	0	0
$658$	1.04273	0.0406500
$659$	30.6811	1.19517	0.597584	−	0.801806i	$-0.296127\pi$
$659$	30.6811	1.19517	0.597584	+	0.801806i	$0.296127\pi$
$660$	0	0
$661$	−13.8208	−0.537568	−0.268784	−	0.963200i	$-0.586622\pi$
$661$	−13.8208	−0.537568	−0.268784	+	0.963200i	$0.586622\pi$
$662$	1.72290	0.0669624
$663$	0	0
$664$	5.49881	0.213395
$665$	20.9477	0.812318
$666$	0	0
$667$	10.0966	0.390941
$668$	26.6522	1.03121
$669$	0	0
$670$	−1.33621	−0.0516224
$671$	−0.997763	−0.0385182
$672$	0	0
$673$	19.2294	0.741240	0.370620	−	0.928785i	$-0.379145\pi$
$673$	19.2294	0.741240	0.370620	+	0.928785i	$0.379145\pi$
$674$	−2.47689	−0.0954062
$675$	0	0
$676$	−42.7910	−1.64581
$677$	−34.7093	−1.33399	−0.666994	−	0.745063i	$-0.732419\pi$
$677$	−34.7093	−1.33399	−0.666994	+	0.745063i	$0.732419\pi$
$678$	0	0
$679$	3.80014	0.145836
$680$	−3.19738	−0.122614
$681$	0	0
$682$	0.108068	0.00413814
$683$	−2.16428	−0.0828139	−0.0414069	−	0.999142i	$-0.513184\pi$
$683$	−2.16428	−0.0828139	−0.0414069	+	0.999142i	$0.513184\pi$
$684$	0	0
$685$	−18.5389	−0.708336
$686$	1.39856	0.0533973
$687$	0	0
$688$	31.1219	1.18651
$689$	49.6856	1.89287
$690$	0	0
$691$	22.7388	0.865025	0.432513	−	0.901628i	$-0.357627\pi$
$691$	22.7388	0.865025	0.432513	+	0.901628i	$0.357627\pi$
$692$	−6.32623	−0.240487
$693$	0	0
$694$	−0.708756	−0.0269040
$695$	−22.1005	−0.838321
$696$	0	0
$697$	5.13318	0.194433
$698$	0.827806	0.0313329
$699$	0	0
$700$	20.8382	0.787609
$701$	21.4836	0.811424	0.405712	−	0.914001i	$-0.367024\pi$
$701$	21.4836	0.811424	0.405712	+	0.914001i	$0.367024\pi$
$702$	0	0
$703$	−18.9740	−0.715619
$704$	10.6737	0.402280
$705$	0	0
$706$	−1.09583	−0.0412423
$707$	−63.6261	−2.39291
$708$	0	0
$709$	−4.43026	−0.166382	−0.0831910	−	0.996534i	$-0.526511\pi$
$709$	−4.43026	−0.166382	−0.0831910	+	0.996534i	$0.526511\pi$
$710$	2.40417	0.0902268
$711$	0	0
$712$	5.53339	0.207373
$713$	−2.06534	−0.0773476
$714$	0	0
$715$	10.7065	0.400399
$716$	23.4998	0.878227
$717$	0	0
$718$	0.826688	0.0308517
$719$	−37.8736	−1.41245	−0.706224	−	0.707989i	$-0.749603\pi$
$719$	−37.8736	−1.41245	−0.706224	+	0.707989i	$0.749603\pi$
$720$	0	0
$721$	−8.76245	−0.326331
$722$	−0.693010	−0.0257912
$723$	0	0
$724$	7.78830	0.289450
$725$	10.7849	0.400542
$726$	0	0
$727$	−0.346692	−0.0128581	−0.00642904	−	0.999979i	$-0.502046\pi$
$727$	−0.346692	−0.0128581	−0.00642904	+	0.999979i	$0.502046\pi$
$728$	−8.66273	−0.321062
$729$	0	0
$730$	1.32682	0.0491077
$731$	−41.9940	−1.55320
$732$	0	0
$733$	4.28244	0.158176	0.0790878	−	0.996868i	$-0.474799\pi$
$733$	4.28244	0.158176	0.0790878	+	0.996868i	$0.474799\pi$
$734$	−0.305282	−0.0112682
$735$	0	0
$736$	4.22619	0.155779
$737$	−12.2461	−0.451090
$738$	0	0
$739$	4.52351	0.166400	0.0832000	−	0.996533i	$-0.473486\pi$
$739$	4.52351	0.166400	0.0832000	+	0.996533i	$0.473486\pi$
$740$	9.88446	0.363360
$741$	0	0
$742$	3.12581	0.114752
$743$	−14.4360	−0.529606	−0.264803	−	0.964303i	$-0.585307\pi$
$743$	−14.4360	−0.529606	−0.264803	+	0.964303i	$0.585307\pi$
$744$	0	0
$745$	−21.3054	−0.780568
$746$	−2.37255	−0.0868653
$747$	0	0
$748$	−14.6025	−0.533919
$749$	31.8313	1.16309
$750$	0	0
$751$	−18.6022	−0.678804	−0.339402	−	0.940641i	$-0.610225\pi$
$751$	−18.6022	−0.678804	−0.339402	+	0.940641i	$0.610225\pi$
$752$	−11.0550	−0.403133
$753$	0	0
$754$	−2.23420	−0.0813649
$755$	−13.6446	−0.496580
$756$	0	0
$757$	1.36472	0.0496015	0.0248007	−	0.999692i	$-0.492105\pi$
$757$	1.36472	0.0496015	0.0248007	+	0.999692i	$0.492105\pi$
$758$	2.86009	0.103883
$759$	0	0
$760$	3.02199	0.109619
$761$	32.0391	1.16142	0.580709	−	0.814111i	$-0.302776\pi$
$761$	32.0391	1.16142	0.580709	+	0.814111i	$0.302776\pi$
$762$	0	0
$763$	−18.0799	−0.654538
$764$	2.52487	0.0913465
$765$	0	0
$766$	−3.84823	−0.139042
$767$	33.5487	1.21138
$768$	0	0
$769$	−7.53070	−0.271564	−0.135782	−	0.990739i	$-0.543355\pi$
$769$	−7.53070	−0.271564	−0.135782	+	0.990739i	$0.543355\pi$
$770$	0.673564	0.0242736
$771$	0	0
$772$	14.2731	0.513699
$773$	23.1144	0.831368	0.415684	−	0.909509i	$-0.363542\pi$
$773$	23.1144	0.831368	0.415684	+	0.909509i	$0.363542\pi$
$774$	0	0
$775$	−2.20615	−0.0792473
$776$	0.548220	0.0196800
$777$	0	0
$778$	−1.29578	−0.0464561
$779$	−4.85160	−0.173826
$780$	0	0
$781$	22.0336	0.788426
$782$	−1.87952	−0.0672116
$783$	0	0
$784$	12.6119	0.450425
$785$	−24.8933	−0.888481
$786$	0	0
$787$	14.4891	0.516479	0.258240	−	0.966081i	$-0.416858\pi$
$787$	14.4891	0.516479	0.258240	+	0.966081i	$0.416858\pi$
$788$	13.2040	0.470372
$789$	0	0
$790$	−1.73547	−0.0617452
$791$	41.4274	1.47299
$792$	0	0
$793$	−4.21960	−0.149842
$794$	−0.602212	−0.0213717
$795$	0	0
$796$	−25.3013	−0.896780
$797$	13.1633	0.466267	0.233133	−	0.972445i	$-0.425102\pi$
$797$	13.1633	0.466267	0.233133	+	0.972445i	$0.425102\pi$
$798$	0	0
$799$	14.9169	0.527721
$800$	4.51432	0.159605
$801$	0	0
$802$	−0.323467	−0.0114220
$803$	12.1600	0.429116
$804$	0	0
$805$	−12.8728	−0.453707
$806$	0.457026	0.0160980
$807$	0	0
$808$	−9.17891	−0.322913
$809$	33.0845	1.16319	0.581595	−	0.813479i	$-0.302429\pi$
$809$	33.0845	1.16319	0.581595	+	0.813479i	$0.302429\pi$
$810$	0	0
$811$	−4.74569	−0.166644	−0.0833219	−	0.996523i	$-0.526553\pi$
$811$	−4.74569	−0.166644	−0.0833219	+	0.996523i	$0.526553\pi$
$812$	20.8703	0.732403
$813$	0	0
$814$	−0.610101	−0.0213840
$815$	9.75247	0.341614
$816$	0	0
$817$	39.6904	1.38859
$818$	−0.0859553	−0.00300536
$819$	0	0
$820$	2.52742	0.0882615
$821$	33.6149	1.17317	0.586584	−	0.809888i	$-0.300472\pi$
$821$	33.6149	1.17317	0.586584	+	0.809888i	$0.300472\pi$
$822$	0	0
$823$	45.7636	1.59522	0.797609	−	0.603175i	$-0.206098\pi$
$823$	45.7636	1.59522	0.797609	+	0.603175i	$0.206098\pi$
$824$	−1.26410	−0.0440370
$825$	0	0
$826$	2.11061	0.0734377
$827$	−29.9837	−1.04264	−0.521318	−	0.853363i	$-0.674559\pi$
$827$	−29.9837	−1.04264	−0.521318	+	0.853363i	$0.674559\pi$
$828$	0	0
$829$	−2.23933	−0.0777753	−0.0388876	−	0.999244i	$-0.512381\pi$
$829$	−2.23933	−0.0777753	−0.0388876	+	0.999244i	$0.512381\pi$
$830$	−1.80816	−0.0627621
$831$	0	0
$832$	45.1397	1.56494
$833$	−17.0177	−0.589629
$834$	0	0
$835$	−17.5870	−0.608623
$836$	13.8014	0.477332
$837$	0	0
$838$	−1.36509	−0.0471561
$839$	31.0203	1.07094	0.535469	−	0.844555i	$-0.320135\pi$
$839$	31.0203	1.07094	0.535469	+	0.844555i	$0.320135\pi$
$840$	0	0
$841$	−18.1985	−0.627533
$842$	1.14508	0.0394621
$843$	0	0
$844$	−10.3806	−0.357316
$845$	28.2365	0.971364
$846$	0	0
$847$	−28.9881	−0.996042
$848$	−33.1396	−1.13802
$849$	0	0
$850$	−2.00767	−0.0688624
$851$	11.6599	0.399698
$852$	0	0
$853$	−13.7603	−0.471142	−0.235571	−	0.971857i	$-0.575696\pi$
$853$	−13.7603	−0.471142	−0.235571	+	0.971857i	$0.575696\pi$
$854$	−0.265463	−0.00908395
$855$	0	0
$856$	4.59209	0.156954
$857$	−31.2228	−1.06655	−0.533276	−	0.845941i	$-0.679039\pi$
$857$	−31.2228	−1.06655	−0.533276	+	0.845941i	$0.679039\pi$
$858$	0	0
$859$	−55.8137	−1.90434	−0.952170	−	0.305569i	$-0.901153\pi$
$859$	−55.8137	−1.90434	−0.952170	+	0.305569i	$0.901153\pi$
$860$	−20.6766	−0.705065
$861$	0	0
$862$	1.11614	0.0380159
$863$	0.720056	0.0245110	0.0122555	−	0.999925i	$-0.496099\pi$
$863$	0.720056	0.0245110	0.0122555	+	0.999925i	$0.496099\pi$
$864$	0	0
$865$	4.17448	0.141937
$866$	4.50531	0.153097
$867$	0	0
$868$	−4.26920	−0.144906
$869$	−15.9052	−0.539546
$870$	0	0
$871$	−51.7894	−1.75482
$872$	−2.60827	−0.0883272
$873$	0	0
$874$	1.77642	0.0600884
$875$	−34.7019	−1.17314
$876$	0	0
$877$	48.3619	1.63307	0.816533	−	0.577298i	$-0.195893\pi$
$877$	48.3619	1.63307	0.816533	+	0.577298i	$0.195893\pi$
$878$	1.94428	0.0656163
$879$	0	0
$880$	−7.14106	−0.240725
$881$	−36.2946	−1.22279	−0.611397	−	0.791324i	$-0.709392\pi$
$881$	−36.2946	−1.22279	−0.611397	+	0.791324i	$0.709392\pi$
$882$	0	0
$883$	−26.8840	−0.904718	−0.452359	−	0.891836i	$-0.649417\pi$
$883$	−26.8840	−0.904718	−0.452359	+	0.891836i	$0.649417\pi$
$884$	−61.7546	−2.07703
$885$	0	0
$886$	−0.0507010	−0.00170333
$887$	−13.4296	−0.450923	−0.225461	−	0.974252i	$-0.572389\pi$
$887$	−13.4296	−0.450923	−0.225461	+	0.974252i	$0.572389\pi$
$888$	0	0
$889$	−33.4501	−1.12188
$890$	−1.81953	−0.0609908
$891$	0	0
$892$	3.34752	0.112083
$893$	−14.0986	−0.471792
$894$	0	0
$895$	−15.5068	−0.518334
$896$	11.6345	0.388680
$897$	0	0
$898$	−3.67509	−0.122639
$899$	−2.20955	−0.0736926
$900$	0	0
$901$	44.7165	1.48972
$902$	−0.156001	−0.00519426
$903$	0	0
$904$	5.97644	0.198774
$905$	−5.13926	−0.170835
$906$	0	0
$907$	7.62089	0.253047	0.126524	−	0.991964i	$-0.459618\pi$
$907$	7.62089	0.253047	0.126524	+	0.991964i	$0.459618\pi$
$908$	−2.52341	−0.0837424
$909$	0	0
$910$	2.84854	0.0944282
$911$	14.0113	0.464214	0.232107	−	0.972690i	$-0.425438\pi$
$911$	14.0113	0.464214	0.232107	+	0.972690i	$0.425438\pi$
$912$	0	0
$913$	−16.5714	−0.548432
$914$	−1.98424	−0.0656327
$915$	0	0
$916$	−48.6390	−1.60708
$917$	−41.4284	−1.36809
$918$	0	0
$919$	12.5233	0.413107	0.206553	−	0.978435i	$-0.433775\pi$
$919$	12.5233	0.413107	0.206553	+	0.978435i	$0.433775\pi$
$920$	−1.85708	−0.0612260
$921$	0	0
$922$	−3.57700	−0.117802
$923$	93.1815	3.06711
$924$	0	0
$925$	12.4549	0.409514
$926$	−1.81522	−0.0596518
$927$	0	0
$928$	4.52127	0.148418
$929$	−14.3668	−0.471359	−0.235680	−	0.971831i	$-0.575732\pi$
$929$	−14.3668	−0.471359	−0.235680	+	0.971831i	$0.575732\pi$
$930$	0	0
$931$	16.0842	0.527138
$932$	7.11908	0.233193
$933$	0	0
$934$	2.47091	0.0808505
$935$	9.63571	0.315121
$936$	0	0
$937$	57.2739	1.87106	0.935528	−	0.353253i	$-0.114925\pi$
$937$	57.2739	1.87106	0.935528	+	0.353253i	$0.114925\pi$
$938$	−3.25817	−0.106383
$939$	0	0
$940$	7.34462	0.239555
$941$	2.50683	0.0817202	0.0408601	−	0.999165i	$-0.486990\pi$
$941$	2.50683	0.0817202	0.0408601	+	0.999165i	$0.486990\pi$
$942$	0	0
$943$	2.98141	0.0970880
$944$	−22.3765	−0.728294
$945$	0	0
$946$	1.27622	0.0414936
$947$	53.8904	1.75120	0.875601	−	0.483036i	$-0.160466\pi$
$947$	53.8904	1.75120	0.875601	+	0.483036i	$0.160466\pi$
$948$	0	0
$949$	51.4252	1.66933
$950$	1.89753	0.0615642
$951$	0	0
$952$	−7.79636	−0.252682
$953$	20.7256	0.671369	0.335684	−	0.941975i	$-0.391032\pi$
$953$	20.7256	0.671369	0.335684	+	0.941975i	$0.391032\pi$
$954$	0	0
$955$	−1.66608	−0.0539131
$956$	−53.6907	−1.73648
$957$	0	0
$958$	3.16511	0.102260
$959$	−45.2046	−1.45973
$960$	0	0
$961$	−30.5480	−0.985420
$962$	−2.58015	−0.0831874
$963$	0	0
$964$	−1.98662	−0.0639847
$965$	−9.41836	−0.303188
$966$	0	0
$967$	12.2723	0.394652	0.197326	−	0.980338i	$-0.436774\pi$
$967$	12.2723	0.394652	0.197326	+	0.980338i	$0.436774\pi$
$968$	−4.18191	−0.134412
$969$	0	0
$970$	−0.180270	−0.00578811
$971$	−48.8599	−1.56799	−0.783994	−	0.620768i	$-0.786821\pi$
$971$	−48.8599	−1.56799	−0.783994	+	0.620768i	$0.786821\pi$
$972$	0	0
$973$	−53.8890	−1.72760
$974$	1.05493	0.0338021
$975$	0	0
$976$	2.81441	0.0900871
$977$	29.7335	0.951260	0.475630	−	0.879645i	$-0.342220\pi$
$977$	29.7335	0.951260	0.475630	+	0.879645i	$0.342220\pi$
$978$	0	0
$979$	−16.6756	−0.532953
$980$	−8.37902	−0.267658
$981$	0	0
$982$	−0.844026	−0.0269339
$983$	−26.3255	−0.839655	−0.419827	−	0.907604i	$-0.637909\pi$
$983$	−26.3255	−0.839655	−0.419827	+	0.907604i	$0.637909\pi$
$984$	0	0
$985$	−8.71290	−0.277616
$986$	−2.01076	−0.0640356
$987$	0	0
$988$	58.3670	1.85690
$989$	−24.3906	−0.775575
$990$	0	0
$991$	44.7423	1.42129	0.710643	−	0.703552i	$-0.248404\pi$
$991$	44.7423	1.42129	0.710643	+	0.703552i	$0.248404\pi$
$992$	−0.924865	−0.0293645
$993$	0	0
$994$	5.86222	0.185938
$995$	16.6955	0.529284
$996$	0	0
$997$	−39.5818	−1.25357	−0.626784	−	0.779193i	$-0.715629\pi$
$997$	−39.5818	−1.25357	−0.626784	+	0.779193i	$0.715629\pi$
$998$	−0.229348	−0.00725988
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2169.2.a.h.1.6		12
3.2	odd	2		241.2.a.b.1.7	✓	12
12.11	even	2		3856.2.a.n.1.12		12
15.14	odd	2		6025.2.a.h.1.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
241.2.a.b.1.7	✓	12	3.2	odd	2
2169.2.a.h.1.6		12	1.1	even	1	trivial
3856.2.a.n.1.12		12	12.11	even	2
6025.2.a.h.1.6		12	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:	\( N \)	\(=\)	\( 2169 = 3^{2} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2169.a (trivial)

\(f(q)\)	\(=\)	\(q-0.115670 q^{2} -1.98662 q^{4} +1.31091 q^{5} +3.19647 q^{7} +0.461133 q^{8} +O(q^{10})\)	\(q-0.115670 q^{2} -1.98662 q^{4} +1.31091 q^{5} +3.19647 q^{7} +0.461133 q^{8} -0.151633 q^{10} -1.38968 q^{11} -5.87704 q^{13} -0.369736 q^{14} +3.91990 q^{16} -5.28927 q^{17} +4.99913 q^{19} -2.60428 q^{20} +0.160744 q^{22} -3.07207 q^{23} -3.28152 q^{25} +0.679798 q^{26} -6.35017 q^{28} -3.28657 q^{29} +0.672296 q^{31} -1.37568 q^{32} +0.611810 q^{34} +4.19028 q^{35} -3.79547 q^{37} -0.578249 q^{38} +0.604503 q^{40} -0.970489 q^{41} +7.93946 q^{43} +2.76077 q^{44} +0.355346 q^{46} -2.82021 q^{47} +3.21740 q^{49} +0.379573 q^{50} +11.6754 q^{52} -8.45419 q^{53} -1.82175 q^{55} +1.47400 q^{56} +0.380158 q^{58} -5.70844 q^{59} +0.717980 q^{61} -0.0777646 q^{62} -7.68068 q^{64} -7.70427 q^{65} +8.81215 q^{67} +10.5078 q^{68} -0.484690 q^{70} -15.8552 q^{71} -8.75018 q^{73} +0.439022 q^{74} -9.93137 q^{76} -4.44207 q^{77} +11.4452 q^{79} +5.13864 q^{80} +0.112256 q^{82} +11.9246 q^{83} -6.93376 q^{85} -0.918358 q^{86} -0.640827 q^{88} +11.9996 q^{89} -18.7858 q^{91} +6.10304 q^{92} +0.326214 q^{94} +6.55340 q^{95} +1.18886 q^{97} -0.372157 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{2} + 13 q^{4} - 6 q^{5} + 3 q^{7} - 9 q^{8}+O(q^{10}) \) 12 * q - 3 * q^2 + 13 * q^4 - 6 * q^5 + 3 * q^7 - 9 * q^8	\( 12 q - 3 q^{2} + 13 q^{4} - 6 q^{5} + 3 q^{7} - 9 q^{8} - 7 q^{10} - 22 q^{11} - 5 q^{13} - 6 q^{14} + 15 q^{16} + 4 q^{17} - 6 q^{19} - 10 q^{20} - 12 q^{22} - 32 q^{23} + 4 q^{25} - 8 q^{26} - 11 q^{28} - 6 q^{29} + 8 q^{31} - q^{32} - 19 q^{34} - 15 q^{35} - 8 q^{37} + 10 q^{38} - 52 q^{40} + q^{41} - 2 q^{43} - 42 q^{44} - 25 q^{46} - 34 q^{47} - 9 q^{49} + 27 q^{50} - 41 q^{52} - 5 q^{53} - 3 q^{55} - q^{56} - 33 q^{58} - 26 q^{59} - 26 q^{61} + 17 q^{62} + 13 q^{64} + 25 q^{65} + 6 q^{67} + 35 q^{68} - 4 q^{70} - 94 q^{71} - 22 q^{73} - 26 q^{74} - 20 q^{76} + 7 q^{77} + 9 q^{79} - 19 q^{80} + 15 q^{82} + 8 q^{83} + 4 q^{85} - 9 q^{86} + 6 q^{88} + 3 q^{89} - 20 q^{91} - 36 q^{92} + 48 q^{94} - 33 q^{95} - 29 q^{97} - 28 q^{98}+O(q^{100}) \) 12 * q - 3 * q^2 + 13 * q^4 - 6 * q^5 + 3 * q^7 - 9 * q^8 - 7 * q^10 - 22 * q^11 - 5 * q^13 - 6 * q^14 + 15 * q^16 + 4 * q^17 - 6 * q^19 - 10 * q^20 - 12 * q^22 - 32 * q^23 + 4 * q^25 - 8 * q^26 - 11 * q^28 - 6 * q^29 + 8 * q^31 - q^32 - 19 * q^34 - 15 * q^35 - 8 * q^37 + 10 * q^38 - 52 * q^40 + q^41 - 2 * q^43 - 42 * q^44 - 25 * q^46 - 34 * q^47 - 9 * q^49 + 27 * q^50 - 41 * q^52 - 5 * q^53 - 3 * q^55 - q^56 - 33 * q^58 - 26 * q^59 - 26 * q^61 + 17 * q^62 + 13 * q^64 + 25 * q^65 + 6 * q^67 + 35 * q^68 - 4 * q^70 - 94 * q^71 - 22 * q^73 - 26 * q^74 - 20 * q^76 + 7 * q^77 + 9 * q^79 - 19 * q^80 + 15 * q^82 + 8 * q^83 + 4 * q^85 - 9 * q^86 + 6 * q^88 + 3 * q^89 - 20 * q^91 - 36 * q^92 + 48 * q^94 - 33 * q^95 - 29 * q^97 - 28 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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