LMFDB - Embedded newform 2169.2.a.h.1.9

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			1.9
Root			$-1.28632$ 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)

$f(q)$	$=$	$q+1.28632 q^{2} -0.345373 q^{4} -0.612768 q^{5} +1.03110 q^{7} -3.01691 q^{8} +O(q^{10})$	\(q+1.28632 q^{2} -0.345373 q^{4} -0.612768 q^{5} +1.03110 q^{7} -3.01691 q^{8} -0.788217 q^{10} -0.227935 q^{11} +3.38088 q^{13} +1.32633 q^{14} -3.18997 q^{16} -7.12130 q^{17} +3.40112 q^{19} +0.211634 q^{20} -0.293198 q^{22} -6.91488 q^{23} -4.62452 q^{25} +4.34890 q^{26} -0.356115 q^{28} -0.569431 q^{29} +4.93697 q^{31} +1.93048 q^{32} -9.16030 q^{34} -0.631826 q^{35} -5.37832 q^{37} +4.37494 q^{38} +1.84866 q^{40} -10.7559 q^{41} -0.910247 q^{43} +0.0787228 q^{44} -8.89477 q^{46} +8.50333 q^{47} -5.93683 q^{49} -5.94862 q^{50} -1.16766 q^{52} +7.76696 q^{53} +0.139671 q^{55} -3.11074 q^{56} -0.732472 q^{58} -11.2505 q^{59} -3.65450 q^{61} +6.35054 q^{62} +8.86317 q^{64} -2.07169 q^{65} -12.0694 q^{67} +2.45951 q^{68} -0.812732 q^{70} +9.48630 q^{71} -7.10488 q^{73} -6.91825 q^{74} -1.17466 q^{76} -0.235024 q^{77} +0.366201 q^{79} +1.95471 q^{80} -13.8356 q^{82} -17.8030 q^{83} +4.36370 q^{85} -1.17087 q^{86} +0.687660 q^{88} +7.54236 q^{89} +3.48603 q^{91} +2.38822 q^{92} +10.9380 q^{94} -2.08410 q^{95} -7.85505 q^{97} -7.63668 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.28632	0.909568	0.454784	−	0.890602i	$-0.349717\pi$
$2$	1.28632	0.909568	0.454784	+	0.890602i	$0.349717\pi$
$3$	0	0
$3$	0	0
$4$	−0.345373	−0.172687
$5$	−0.612768	−0.274038	−0.137019	−	0.990568i	$-0.543752\pi$
$5$	−0.612768	−0.274038	−0.137019	+	0.990568i	$0.543752\pi$
$6$	0	0
$7$	1.03110	0.389720	0.194860	−	0.980831i	$-0.437575\pi$
$7$	1.03110	0.389720	0.194860	+	0.980831i	$0.437575\pi$
$8$	−3.01691	−1.06664
$9$	0	0
$10$	−0.788217	−0.249256
$11$	−0.227935	−0.0687251	−0.0343625	−	0.999409i	$-0.510940\pi$
$11$	−0.227935	−0.0687251	−0.0343625	+	0.999409i	$0.510940\pi$
$12$	0	0
$13$	3.38088	0.937686	0.468843	−	0.883281i	$-0.344671\pi$
$13$	3.38088	0.937686	0.468843	+	0.883281i	$0.344671\pi$
$14$	1.32633	0.354477
$15$	0	0
$16$	−3.18997	−0.797493
$17$	−7.12130	−1.72717	−0.863585	−	0.504203i	$-0.831786\pi$
$17$	−7.12130	−1.72717	−0.863585	+	0.504203i	$0.831786\pi$
$18$	0	0
$19$	3.40112	0.780270	0.390135	−	0.920758i	$-0.372428\pi$
$19$	3.40112	0.780270	0.390135	+	0.920758i	$0.372428\pi$
$20$	0.211634	0.0473227
$21$	0	0
$22$	−0.293198	−0.0625101
$23$	−6.91488	−1.44185	−0.720926	−	0.693012i	$-0.756283\pi$
$23$	−6.91488	−1.44185	−0.720926	+	0.693012i	$0.756283\pi$
$24$	0	0
$25$	−4.62452	−0.924903
$26$	4.34890	0.852889
$27$	0	0
$28$	−0.356115	−0.0672995
$29$	−0.569431	−0.105741	−0.0528703	−	0.998601i	$-0.516837\pi$
$29$	−0.569431	−0.105741	−0.0528703	+	0.998601i	$0.516837\pi$
$30$	0	0
$31$	4.93697	0.886706	0.443353	−	0.896347i	$-0.353789\pi$
$31$	4.93697	0.886706	0.443353	+	0.896347i	$0.353789\pi$
$32$	1.93048	0.341264
$33$	0	0
$34$	−9.16030	−1.57098
$35$	−0.631826	−0.106798
$36$	0	0
$37$	−5.37832	−0.884190	−0.442095	−	0.896968i	$-0.645765\pi$
$37$	−5.37832	−0.884190	−0.442095	+	0.896968i	$0.645765\pi$
$38$	4.37494	0.709709
$39$	0	0
$40$	1.84866	0.292299
$41$	−10.7559	−1.67979	−0.839897	−	0.542747i	$-0.817384\pi$
$41$	−10.7559	−1.67979	−0.839897	+	0.542747i	$0.817384\pi$
$42$	0	0
$43$	−0.910247	−0.138811	−0.0694057	−	0.997589i	$-0.522110\pi$
$43$	−0.910247	−0.138811	−0.0694057	+	0.997589i	$0.522110\pi$
$44$	0.0787228	0.0118679
$45$	0	0
$46$	−8.89477	−1.31146
$47$	8.50333	1.24034	0.620169	−	0.784468i	$-0.287064\pi$
$47$	8.50333	1.24034	0.620169	+	0.784468i	$0.287064\pi$
$48$	0	0
$49$	−5.93683	−0.848118
$50$	−5.94862	−0.841262
$51$	0	0
$52$	−1.16766	−0.161926
$53$	7.76696	1.06687	0.533437	−	0.845840i	$-0.320900\pi$
$53$	7.76696	1.06687	0.533437	+	0.845840i	$0.320900\pi$
$54$	0	0
$55$	0.139671	0.0188333
$56$	−3.11074	−0.415690
$57$	0	0
$58$	−0.732472	−0.0961783
$59$	−11.2505	−1.46468	−0.732342	−	0.680937i	$-0.761573\pi$
$59$	−11.2505	−1.46468	−0.732342	+	0.680937i	$0.761573\pi$
$60$	0	0
$61$	−3.65450	−0.467910	−0.233955	−	0.972247i	$-0.575167\pi$
$61$	−3.65450	−0.467910	−0.233955	+	0.972247i	$0.575167\pi$
$62$	6.35054	0.806519
$63$	0	0
$64$	8.86317	1.10790
$65$	−2.07169	−0.256962
$66$	0	0
$67$	−12.0694	−1.47451	−0.737257	−	0.675613i	$-0.763879\pi$
$67$	−12.0694	−1.47451	−0.737257	+	0.675613i	$0.763879\pi$
$68$	2.45951	0.298259
$69$	0	0
$70$	−0.812732	−0.0971401
$71$	9.48630	1.12582	0.562908	−	0.826519i	$-0.309682\pi$
$71$	9.48630	1.12582	0.562908	+	0.826519i	$0.309682\pi$
$72$	0	0
$73$	−7.10488	−0.831563	−0.415782	−	0.909464i	$-0.636492\pi$
$73$	−7.10488	−0.831563	−0.415782	+	0.909464i	$0.636492\pi$
$74$	−6.91825	−0.804231
$75$	0	0
$76$	−1.17466	−0.134742
$77$	−0.235024	−0.0267835
$78$	0	0
$79$	0.366201	0.0412008	0.0206004	−	0.999788i	$-0.493442\pi$
$79$	0.366201	0.0412008	0.0206004	+	0.999788i	$0.493442\pi$
$80$	1.95471	0.218543
$81$	0	0
$82$	−13.8356	−1.52789
$83$	−17.8030	−1.95413	−0.977067	−	0.212932i	$-0.931699\pi$
$83$	−17.8030	−1.95413	−0.977067	+	0.212932i	$0.931699\pi$
$84$	0	0
$85$	4.36370	0.473310
$86$	−1.17087	−0.126258
$87$	0	0
$88$	0.687660	0.0733048
$89$	7.54236	0.799489	0.399744	−	0.916627i	$-0.369099\pi$
$89$	7.54236	0.799489	0.399744	+	0.916627i	$0.369099\pi$
$90$	0	0
$91$	3.48603	0.365435
$92$	2.38822	0.248989
$93$	0	0
$94$	10.9380	1.12817
$95$	−2.08410	−0.213824
$96$	0	0
$97$	−7.85505	−0.797560	−0.398780	−	0.917047i	$-0.630566\pi$
$97$	−7.85505	−0.797560	−0.398780	+	0.917047i	$0.630566\pi$
$98$	−7.63668	−0.771421
$99$	0	0
$100$	1.59718	0.159718
$101$	−17.4801	−1.73933	−0.869666	−	0.493640i	$-0.835666\pi$
$101$	−17.4801	−1.73933	−0.869666	+	0.493640i	$0.835666\pi$
$102$	0	0
$103$	6.10361	0.601407	0.300703	−	0.953718i	$-0.402779\pi$
$103$	6.10361	0.601407	0.300703	+	0.953718i	$0.402779\pi$
$104$	−10.1998	−1.00017
$105$	0	0
$106$	9.99081	0.970394
$107$	−7.86203	−0.760051	−0.380026	−	0.924976i	$-0.624085\pi$
$107$	−7.86203	−0.760051	−0.380026	+	0.924976i	$0.624085\pi$
$108$	0	0
$109$	−4.17241	−0.399644	−0.199822	−	0.979832i	$-0.564036\pi$
$109$	−4.17241	−0.399644	−0.199822	+	0.979832i	$0.564036\pi$
$110$	0.179662	0.0171301
$111$	0	0
$112$	−3.28919	−0.310799
$113$	−3.62203	−0.340732	−0.170366	−	0.985381i	$-0.554495\pi$
$113$	−3.62203	−0.340732	−0.170366	+	0.985381i	$0.554495\pi$
$114$	0	0
$115$	4.23721	0.395122
$116$	0.196666	0.0182600
$117$	0	0
$118$	−14.4717	−1.33223
$119$	−7.34279	−0.673113
$120$	0	0
$121$	−10.9480	−0.995277
$122$	−4.70086	−0.425596
$123$	0	0
$124$	−1.70510	−0.153122
$125$	5.89759	0.527497
$126$	0	0
$127$	13.5446	1.20189	0.600943	−	0.799292i	$-0.294792\pi$
$127$	13.5446	1.20189	0.600943	+	0.799292i	$0.294792\pi$
$128$	7.53993	0.666442
$129$	0	0
$130$	−2.66486	−0.233724
$131$	−1.03003	−0.0899942	−0.0449971	−	0.998987i	$-0.514328\pi$
$131$	−1.03003	−0.0899942	−0.0449971	+	0.998987i	$0.514328\pi$
$132$	0	0
$133$	3.50690	0.304087
$134$	−15.5252	−1.34117
$135$	0	0
$136$	21.4843	1.84226
$137$	11.5780	0.989180	0.494590	−	0.869127i	$-0.335318\pi$
$137$	11.5780	0.989180	0.494590	+	0.869127i	$0.335318\pi$
$138$	0	0
$139$	0.110960	0.00941151	0.00470575	−	0.999989i	$-0.498502\pi$
$139$	0.110960	0.00941151	0.00470575	+	0.999989i	$0.498502\pi$
$140$	0.218216	0.0184426
$141$	0	0
$142$	12.2024	1.02401
$143$	−0.770621	−0.0644425
$144$	0	0
$145$	0.348929	0.0289769
$146$	−9.13917	−0.756363
$147$	0	0
$148$	1.85753	0.152688
$149$	19.5413	1.60089	0.800443	−	0.599409i	$-0.204598\pi$
$149$	19.5413	1.60089	0.800443	+	0.599409i	$0.204598\pi$
$150$	0	0
$151$	17.7745	1.44647	0.723233	−	0.690604i	$-0.242655\pi$
$151$	17.7745	1.44647	0.723233	+	0.690604i	$0.242655\pi$
$152$	−10.2609	−0.832266
$153$	0	0
$154$	−0.302317	−0.0243614
$155$	−3.02522	−0.242991
$156$	0	0
$157$	6.98884	0.557770	0.278885	−	0.960325i	$-0.410035\pi$
$157$	6.98884	0.557770	0.278885	+	0.960325i	$0.410035\pi$
$158$	0.471053	0.0374749
$159$	0	0
$160$	−1.18294	−0.0935194
$161$	−7.12995	−0.561918
$162$	0	0
$163$	−8.87693	−0.695295	−0.347647	−	0.937625i	$-0.613019\pi$
$163$	−8.87693	−0.695295	−0.347647	+	0.937625i	$0.613019\pi$
$164$	3.71481	0.290078
$165$	0	0
$166$	−22.9004	−1.77742
$167$	−4.85435	−0.375641	−0.187820	−	0.982203i	$-0.560142\pi$
$167$	−4.85435	−0.375641	−0.187820	+	0.982203i	$0.560142\pi$
$168$	0	0
$169$	−1.56968	−0.120745
$170$	5.61313	0.430508
$171$	0	0
$172$	0.314375	0.0239709
$173$	13.5277	1.02849	0.514247	−	0.857642i	$-0.328072\pi$
$173$	13.5277	1.02849	0.514247	+	0.857642i	$0.328072\pi$
$174$	0	0
$175$	−4.76835	−0.360453
$176$	0.727107	0.0548077
$177$	0	0
$178$	9.70191	0.727189
$179$	10.4800	0.783310	0.391655	−	0.920112i	$-0.371903\pi$
$179$	10.4800	0.783310	0.391655	+	0.920112i	$0.371903\pi$
$180$	0	0
$181$	−8.82686	−0.656095	−0.328048	−	0.944661i	$-0.606391\pi$
$181$	−8.82686	−0.656095	−0.328048	+	0.944661i	$0.606391\pi$
$182$	4.48416	0.332388
$183$	0	0
$184$	20.8616	1.53793
$185$	3.29566	0.242302
$186$	0	0
$187$	1.62320	0.118700
$188$	−2.93682	−0.214190
$189$	0	0
$190$	−2.68082	−0.194487
$191$	3.25920	0.235827	0.117914	−	0.993024i	$-0.462379\pi$
$191$	3.25920	0.235827	0.117914	+	0.993024i	$0.462379\pi$
$192$	0	0
$193$	16.9658	1.22123	0.610613	−	0.791929i	$-0.290923\pi$
$193$	16.9658	1.22123	0.610613	+	0.791929i	$0.290923\pi$
$194$	−10.1041	−0.725434
$195$	0	0
$196$	2.05042	0.146459
$197$	18.5572	1.32214	0.661072	−	0.750322i	$-0.270102\pi$
$197$	18.5572	1.32214	0.661072	+	0.750322i	$0.270102\pi$
$198$	0	0
$199$	−19.6484	−1.39284	−0.696420	−	0.717634i	$-0.745225\pi$
$199$	−19.6484	−1.39284	−0.696420	+	0.717634i	$0.745225\pi$
$200$	13.9517	0.986537
$201$	0	0
$202$	−22.4850	−1.58204
$203$	−0.587141	−0.0412092
$204$	0	0
$205$	6.59088	0.460327
$206$	7.85121	0.547020
$207$	0	0
$208$	−10.7849	−0.747798
$209$	−0.775235	−0.0536241
$210$	0	0
$211$	−8.18657	−0.563587	−0.281794	−	0.959475i	$-0.590929\pi$
$211$	−8.18657	−0.563587	−0.281794	+	0.959475i	$0.590929\pi$
$212$	−2.68250	−0.184235
$213$	0	0
$214$	−10.1131	−0.691318
$215$	0.557770	0.0380396
$216$	0	0
$217$	5.09052	0.345567
$218$	−5.36706	−0.363503
$219$	0	0
$220$	−0.0482388	−0.00325226
$221$	−24.0762	−1.61954
$222$	0	0
$223$	15.5070	1.03843	0.519213	−	0.854645i	$-0.326225\pi$
$223$	15.5070	1.03843	0.519213	+	0.854645i	$0.326225\pi$
$224$	1.99053	0.132998
$225$	0	0
$226$	−4.65910	−0.309919
$227$	27.1128	1.79954	0.899769	−	0.436366i	$-0.143735\pi$
$227$	27.1128	1.79954	0.899769	+	0.436366i	$0.143735\pi$
$228$	0	0
$229$	8.10947	0.535889	0.267944	−	0.963434i	$-0.413656\pi$
$229$	8.10947	0.535889	0.267944	+	0.963434i	$0.413656\pi$
$230$	5.45042	0.359390
$231$	0	0
$232$	1.71792	0.112787
$233$	4.34697	0.284780	0.142390	−	0.989811i	$-0.454521\pi$
$233$	4.34697	0.284780	0.142390	+	0.989811i	$0.454521\pi$
$234$	0	0
$235$	−5.21056	−0.339900
$236$	3.88561	0.252932
$237$	0	0
$238$	−9.44520	−0.612241
$239$	−18.3619	−1.18773	−0.593867	−	0.804563i	$-0.702399\pi$
$239$	−18.3619	−1.18773	−0.593867	+	0.804563i	$0.702399\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	−14.0827	−0.905272
$243$	0	0
$244$	1.26217	0.0808019
$245$	3.63790	0.232417
$246$	0	0
$247$	11.4988	0.731649
$248$	−14.8944	−0.945795
$249$	0	0
$250$	7.58621	0.479794
$251$	−12.3629	−0.780337	−0.390169	−	0.920743i	$-0.627583\pi$
$251$	−12.3629	−0.780337	−0.390169	+	0.920743i	$0.627583\pi$
$252$	0	0
$253$	1.57614	0.0990914
$254$	17.4227	1.09320
$255$	0	0
$256$	−8.02755	−0.501722
$257$	12.1589	0.758452	0.379226	−	0.925304i	$-0.376190\pi$
$257$	12.1589	0.758452	0.379226	+	0.925304i	$0.376190\pi$
$258$	0	0
$259$	−5.54560	−0.344587
$260$	0.715507	0.0443739
$261$	0	0
$262$	−1.32495	−0.0818558
$263$	5.76048	0.355207	0.177603	−	0.984102i	$-0.443166\pi$
$263$	5.76048	0.355207	0.177603	+	0.984102i	$0.443166\pi$
$264$	0	0
$265$	−4.75934	−0.292364
$266$	4.51101	0.276588
$267$	0	0
$268$	4.16846	0.254629
$269$	−14.4585	−0.881551	−0.440776	−	0.897617i	$-0.645297\pi$
$269$	−14.4585	−0.881551	−0.440776	+	0.897617i	$0.645297\pi$
$270$	0	0
$271$	6.81638	0.414065	0.207033	−	0.978334i	$-0.433619\pi$
$271$	6.81638	0.414065	0.207033	+	0.978334i	$0.433619\pi$
$272$	22.7167	1.37741
$273$	0	0
$274$	14.8931	0.899726
$275$	1.05409	0.0635640
$276$	0	0
$277$	−25.1323	−1.51005	−0.755027	−	0.655693i	$-0.772376\pi$
$277$	−25.1323	−1.51005	−0.755027	+	0.655693i	$0.772376\pi$
$278$	0.142730	0.00856040
$279$	0	0
$280$	1.90616	0.113915
$281$	6.03331	0.359917	0.179959	−	0.983674i	$-0.442404\pi$
$281$	6.03331	0.359917	0.179959	+	0.983674i	$0.442404\pi$
$282$	0	0
$283$	−3.47423	−0.206521	−0.103261	−	0.994654i	$-0.532928\pi$
$283$	−3.47423	−0.206521	−0.103261	+	0.994654i	$0.532928\pi$
$284$	−3.27631	−0.194414
$285$	0	0
$286$	−0.991267	−0.0586148
$287$	−11.0905	−0.654649
$288$	0	0
$289$	33.7130	1.98312
$290$	0.448835	0.0263565
$291$	0	0
$292$	2.45384	0.143600
$293$	19.7300	1.15264	0.576321	−	0.817224i	$-0.304488\pi$
$293$	19.7300	1.15264	0.576321	+	0.817224i	$0.304488\pi$
$294$	0	0
$295$	6.89391	0.401379
$296$	16.2259	0.943111
$297$	0	0
$298$	25.1364	1.45611
$299$	−23.3783	−1.35200
$300$	0	0
$301$	−0.938558	−0.0540976
$302$	22.8637	1.31566
$303$	0	0
$304$	−10.8495	−0.622260
$305$	2.23936	0.128225
$306$	0	0
$307$	−0.0265060	−0.00151278	−0.000756388	−	1.00000i	$-0.500241\pi$
$307$	−0.0265060	−0.00151278	−0.000756388		1.00000i	$0.500241\pi$
$308$	0.0811712	0.00462516
$309$	0	0
$310$	−3.89140	−0.221017
$311$	−21.2822	−1.20680	−0.603400	−	0.797439i	$-0.706188\pi$
$311$	−21.2822	−1.20680	−0.603400	+	0.797439i	$0.706188\pi$
$312$	0	0
$313$	10.6916	0.604323	0.302161	−	0.953257i	$-0.402292\pi$
$313$	10.6916	0.604323	0.302161	+	0.953257i	$0.402292\pi$
$314$	8.98990	0.507329
$315$	0	0
$316$	−0.126476	−0.00711484
$317$	2.07826	0.116727	0.0583633	−	0.998295i	$-0.481412\pi$
$317$	2.07826	0.116727	0.0583633	+	0.998295i	$0.481412\pi$
$318$	0	0
$319$	0.129793	0.00726703
$320$	−5.43106	−0.303605
$321$	0	0
$322$	−9.17141	−0.511103
$323$	−24.2204	−1.34766
$324$	0	0
$325$	−15.6349	−0.867269
$326$	−11.4186	−0.632418
$327$	0	0
$328$	32.4496	1.79173
$329$	8.76780	0.483384
$330$	0	0
$331$	−28.1272	−1.54601	−0.773006	−	0.634399i	$-0.781248\pi$
$331$	−28.1272	−1.54601	−0.773006	+	0.634399i	$0.781248\pi$
$332$	6.14869	0.337453
$333$	0	0
$334$	−6.24426	−0.341671
$335$	7.39575	0.404073
$336$	0	0
$337$	23.5835	1.28468	0.642338	−	0.766422i	$-0.277965\pi$
$337$	23.5835	1.28468	0.642338	+	0.766422i	$0.277965\pi$
$338$	−2.01911	−0.109825
$339$	0	0
$340$	−1.50711	−0.0817344
$341$	−1.12531	−0.0609389
$342$	0	0
$343$	−13.3392	−0.720249
$344$	2.74613	0.148061
$345$	0	0
$346$	17.4010	0.935484
$347$	21.8697	1.17403	0.587013	−	0.809578i	$-0.300304\pi$
$347$	21.8697	1.17403	0.587013	+	0.809578i	$0.300304\pi$
$348$	0	0
$349$	18.8525	1.00915	0.504575	−	0.863368i	$-0.331649\pi$
$349$	18.8525	1.00915	0.504575	+	0.863368i	$0.331649\pi$
$350$	−6.13364	−0.327857
$351$	0	0
$352$	−0.440025	−0.0234534
$353$	11.3768	0.605526	0.302763	−	0.953066i	$-0.402091\pi$
$353$	11.3768	0.605526	0.302763	+	0.953066i	$0.402091\pi$
$354$	0	0
$355$	−5.81289	−0.308516
$356$	−2.60493	−0.138061
$357$	0	0
$358$	13.4806	0.712473
$359$	0.146567	0.00773548	0.00386774	−	0.999993i	$-0.498769\pi$
$359$	0.146567	0.00773548	0.00386774	+	0.999993i	$0.498769\pi$
$360$	0	0
$361$	−7.43238	−0.391178
$362$	−11.3542	−0.596763
$363$	0	0
$364$	−1.20398	−0.0631058
$365$	4.35364	0.227880
$366$	0	0
$367$	32.6647	1.70508	0.852541	−	0.522660i	$-0.175060\pi$
$367$	32.6647	1.70508	0.852541	+	0.522660i	$0.175060\pi$
$368$	22.0583	1.14987
$369$	0	0
$370$	4.23928	0.220390
$371$	8.00852	0.415782
$372$	0	0
$373$	−17.7530	−0.919218	−0.459609	−	0.888121i	$-0.652010\pi$
$373$	−17.7530	−0.919218	−0.459609	+	0.888121i	$0.652010\pi$
$374$	2.08795	0.107966
$375$	0	0
$376$	−25.6538	−1.32299
$377$	−1.92517	−0.0991515
$378$	0	0
$379$	−4.82435	−0.247810	−0.123905	−	0.992294i	$-0.539542\pi$
$379$	−4.82435	−0.247810	−0.123905	+	0.992294i	$0.539542\pi$
$380$	0.719791	0.0369245
$381$	0	0
$382$	4.19238	0.214501
$383$	−14.7127	−0.751783	−0.375892	−	0.926664i	$-0.622664\pi$
$383$	−14.7127	−0.751783	−0.375892	+	0.926664i	$0.622664\pi$
$384$	0	0
$385$	0.144015	0.00733970
$386$	21.8235	1.11079
$387$	0	0
$388$	2.71293	0.137728
$389$	−4.90322	−0.248603	−0.124301	−	0.992244i	$-0.539669\pi$
$389$	−4.90322	−0.248603	−0.124301	+	0.992244i	$0.539669\pi$
$390$	0	0
$391$	49.2430	2.49032
$392$	17.9109	0.904635
$393$	0	0
$394$	23.8705	1.20258
$395$	−0.224396	−0.0112906
$396$	0	0
$397$	−14.8959	−0.747606	−0.373803	−	0.927508i	$-0.621946\pi$
$397$	−14.8959	−0.747606	−0.373803	+	0.927508i	$0.621946\pi$
$398$	−25.2742	−1.26688
$399$	0	0
$400$	14.7521	0.737603
$401$	−6.72128	−0.335645	−0.167822	−	0.985817i	$-0.553673\pi$
$401$	−6.72128	−0.335645	−0.167822	+	0.985817i	$0.553673\pi$
$402$	0	0
$403$	16.6913	0.831452
$404$	6.03715	0.300360
$405$	0	0
$406$	−0.755253	−0.0374826
$407$	1.22591	0.0607660
$408$	0	0
$409$	−2.59935	−0.128529	−0.0642647	−	0.997933i	$-0.520470\pi$
$409$	−2.59935	−0.128529	−0.0642647	+	0.997933i	$0.520470\pi$
$410$	8.47800	0.418699
$411$	0	0
$412$	−2.10803	−0.103855
$413$	−11.6004	−0.570817
$414$	0	0
$415$	10.9091	0.535507
$416$	6.52672	0.319999
$417$	0	0
$418$	−0.997203	−0.0487748
$419$	13.1622	0.643018	0.321509	−	0.946907i	$-0.395810\pi$
$419$	13.1622	0.643018	0.321509	+	0.946907i	$0.395810\pi$
$420$	0	0
$421$	−6.53715	−0.318601	−0.159301	−	0.987230i	$-0.550924\pi$
$421$	−6.53715	−0.318601	−0.159301	+	0.987230i	$0.550924\pi$
$422$	−10.5306	−0.512620
$423$	0	0
$424$	−23.4322	−1.13797
$425$	32.9326	1.59746
$426$	0	0
$427$	−3.76816	−0.182354
$428$	2.71534	0.131251
$429$	0	0
$430$	0.717472	0.0345996
$431$	−10.8319	−0.521755	−0.260878	−	0.965372i	$-0.584012\pi$
$431$	−10.8319	−0.521755	−0.260878	+	0.965372i	$0.584012\pi$
$432$	0	0
$433$	−25.6037	−1.23044	−0.615218	−	0.788357i	$-0.710932\pi$
$433$	−25.6037	−1.23044	−0.615218	+	0.788357i	$0.710932\pi$
$434$	6.54805	0.314317
$435$	0	0
$436$	1.44104	0.0690132
$437$	−23.5183	−1.12503
$438$	0	0
$439$	14.6842	0.700839	0.350419	−	0.936593i	$-0.386039\pi$
$439$	14.6842	0.700839	0.350419	+	0.936593i	$0.386039\pi$
$440$	−0.421375	−0.0200883
$441$	0	0
$442$	−30.9698	−1.47308
$443$	15.2687	0.725436	0.362718	−	0.931899i	$-0.381849\pi$
$443$	15.2687	0.725436	0.362718	+	0.931899i	$0.381849\pi$
$444$	0	0
$445$	−4.62171	−0.219090
$446$	19.9470	0.944518
$447$	0	0
$448$	9.13883	0.431769
$449$	−31.2160	−1.47317	−0.736587	−	0.676343i	$-0.763564\pi$
$449$	−31.2160	−1.47317	−0.736587	+	0.676343i	$0.763564\pi$
$450$	0	0
$451$	2.45165	0.115444
$452$	1.25095	0.0588399
$453$	0	0
$454$	34.8758	1.63680
$455$	−2.13612	−0.100143
$456$	0	0
$457$	9.43632	0.441412	0.220706	−	0.975340i	$-0.429164\pi$
$457$	9.43632	0.441412	0.220706	+	0.975340i	$0.429164\pi$
$458$	10.4314	0.487427
$459$	0	0
$460$	−1.46342	−0.0682324
$461$	−8.65525	−0.403115	−0.201557	−	0.979477i	$-0.564600\pi$
$461$	−8.65525	−0.403115	−0.201557	+	0.979477i	$0.564600\pi$
$462$	0	0
$463$	−12.5415	−0.582853	−0.291426	−	0.956593i	$-0.594130\pi$
$463$	−12.5415	−0.582853	−0.291426	+	0.956593i	$0.594130\pi$
$464$	1.81647	0.0843274
$465$	0	0
$466$	5.59161	0.259026
$467$	5.28477	0.244550	0.122275	−	0.992496i	$-0.460981\pi$
$467$	5.28477	0.244550	0.122275	+	0.992496i	$0.460981\pi$
$468$	0	0
$469$	−12.4448	−0.574647
$470$	−6.70247	−0.309162
$471$	0	0
$472$	33.9416	1.56229
$473$	0.207477	0.00953982
$474$	0	0
$475$	−15.7285	−0.721675
$476$	2.53601	0.116238
$477$	0	0
$478$	−23.6194	−1.08032
$479$	29.0596	1.32777	0.663884	−	0.747835i	$-0.268907\pi$
$479$	29.0596	1.32777	0.663884	+	0.747835i	$0.268907\pi$
$480$	0	0
$481$	−18.1834	−0.829093
$482$	1.28632	0.0585904
$483$	0	0
$484$	3.78116	0.171871
$485$	4.81332	0.218562
$486$	0	0
$487$	−36.2854	−1.64425	−0.822124	−	0.569309i	$-0.807211\pi$
$487$	−36.2854	−1.64425	−0.822124	+	0.569309i	$0.807211\pi$
$488$	11.0253	0.499091
$489$	0	0
$490$	4.67951	0.211399
$491$	3.99282	0.180193	0.0900967	−	0.995933i	$-0.471282\pi$
$491$	3.99282	0.180193	0.0900967	+	0.995933i	$0.471282\pi$
$492$	0	0
$493$	4.05509	0.182632
$494$	14.7911	0.665484
$495$	0	0
$496$	−15.7488	−0.707142
$497$	9.78134	0.438753
$498$	0	0
$499$	9.25262	0.414204	0.207102	−	0.978319i	$-0.433597\pi$
$499$	9.25262	0.414204	0.207102	+	0.978319i	$0.433597\pi$
$500$	−2.03687	−0.0910916
$501$	0	0
$502$	−15.9026	−0.709769
$503$	−30.0225	−1.33864	−0.669319	−	0.742975i	$-0.733414\pi$
$503$	−30.0225	−1.33864	−0.669319	+	0.742975i	$0.733414\pi$
$504$	0	0
$505$	10.7112	0.476643
$506$	2.02743	0.0901303
$507$	0	0
$508$	−4.67793	−0.207550
$509$	−8.01192	−0.355122	−0.177561	−	0.984110i	$-0.556821\pi$
$509$	−8.01192	−0.355122	−0.177561	+	0.984110i	$0.556821\pi$
$510$	0	0
$511$	−7.32586	−0.324077
$512$	−25.4059	−1.12279
$513$	0	0
$514$	15.6403	0.689864
$515$	−3.74009	−0.164808
$516$	0	0
$517$	−1.93821	−0.0852423
$518$	−7.13343	−0.313425
$519$	0	0
$520$	6.25010	0.274085
$521$	10.1940	0.446608	0.223304	−	0.974749i	$-0.428316\pi$
$521$	10.1940	0.446608	0.223304	+	0.974749i	$0.428316\pi$
$522$	0	0
$523$	−41.0217	−1.79375	−0.896877	−	0.442281i	$-0.854170\pi$
$523$	−41.0217	−1.79375	−0.896877	+	0.442281i	$0.854170\pi$
$524$	0.355745	0.0155408
$525$	0	0
$526$	7.40984	0.323084
$527$	−35.1577	−1.53149
$528$	0	0
$529$	24.8156	1.07894
$530$	−6.12205	−0.265925
$531$	0	0
$532$	−1.21119	−0.0525118
$533$	−36.3644	−1.57512
$534$	0	0
$535$	4.81760	0.208283
$536$	36.4123	1.57277
$537$	0	0
$538$	−18.5983	−0.801831
$539$	1.35321	0.0582870
$540$	0	0
$541$	−25.9044	−1.11372	−0.556858	−	0.830608i	$-0.687993\pi$
$541$	−25.9044	−1.11372	−0.556858	+	0.830608i	$0.687993\pi$
$542$	8.76806	0.376620
$543$	0	0
$544$	−13.7476	−0.589422
$545$	2.55671	0.109518
$546$	0	0
$547$	−43.6488	−1.86629	−0.933145	−	0.359501i	$-0.882947\pi$
$547$	−43.6488	−1.86629	−0.933145	+	0.359501i	$0.882947\pi$
$548$	−3.99875	−0.170818
$549$	0	0
$550$	1.35590	0.0578158
$551$	−1.93670	−0.0825063
$552$	0	0
$553$	0.377591	0.0160568
$554$	−32.3283	−1.37350
$555$	0	0
$556$	−0.0383227	−0.00162524
$557$	−22.1122	−0.936925	−0.468463	−	0.883483i	$-0.655192\pi$
$557$	−22.1122	−0.936925	−0.468463	+	0.883483i	$0.655192\pi$
$558$	0	0
$559$	−3.07743	−0.130162
$560$	2.01551	0.0851707
$561$	0	0
$562$	7.76078	0.327369
$563$	−19.0461	−0.802697	−0.401348	−	0.915925i	$-0.631458\pi$
$563$	−19.0461	−0.802697	−0.401348	+	0.915925i	$0.631458\pi$
$564$	0	0
$565$	2.21946	0.0933735
$566$	−4.46898	−0.187845
$567$	0	0
$568$	−28.6193	−1.20084
$569$	24.1421	1.01209	0.506045	−	0.862507i	$-0.331107\pi$
$569$	24.1421	1.01209	0.506045	+	0.862507i	$0.331107\pi$
$570$	0	0
$571$	−37.2649	−1.55949	−0.779745	−	0.626098i	$-0.784651\pi$
$571$	−37.2649	−1.55949	−0.779745	+	0.626098i	$0.784651\pi$
$572$	0.266152	0.0111284
$573$	0	0
$574$	−14.2659	−0.595447
$575$	31.9780	1.33357
$576$	0	0
$577$	−6.94471	−0.289112	−0.144556	−	0.989497i	$-0.546175\pi$
$577$	−6.94471	−0.289112	−0.144556	+	0.989497i	$0.546175\pi$
$578$	43.3658	1.80378
$579$	0	0
$580$	−0.120511	−0.00500393
$581$	−18.3567	−0.761565
$582$	0	0
$583$	−1.77036	−0.0733209
$584$	21.4348	0.886977
$585$	0	0
$586$	25.3792	1.04841
$587$	33.6812	1.39017	0.695085	−	0.718927i	$-0.255367\pi$
$587$	33.6812	1.39017	0.695085	+	0.718927i	$0.255367\pi$
$588$	0	0
$589$	16.7912	0.691871
$590$	8.86780	0.365081
$591$	0	0
$592$	17.1567	0.705135
$593$	25.1034	1.03087	0.515437	−	0.856927i	$-0.327630\pi$
$593$	25.1034	1.03087	0.515437	+	0.856927i	$0.327630\pi$
$594$	0	0
$595$	4.49942	0.184458
$596$	−6.74905	−0.276452
$597$	0	0
$598$	−30.0721	−1.22974
$599$	5.93752	0.242601	0.121300	−	0.992616i	$-0.461294\pi$
$599$	5.93752	0.242601	0.121300	+	0.992616i	$0.461294\pi$
$600$	0	0
$601$	−29.7235	−1.21245	−0.606223	−	0.795294i	$-0.707316\pi$
$601$	−29.7235	−1.21245	−0.606223	+	0.795294i	$0.707316\pi$
$602$	−1.20729	−0.0492054
$603$	0	0
$604$	−6.13883	−0.249785
$605$	6.70861	0.272744
$606$	0	0
$607$	16.5792	0.672929	0.336464	−	0.941696i	$-0.390769\pi$
$607$	16.5792	0.672929	0.336464	+	0.941696i	$0.390769\pi$
$608$	6.56580	0.266279
$609$	0	0
$610$	2.88054	0.116629
$611$	28.7487	1.16305
$612$	0	0
$613$	39.4836	1.59473	0.797364	−	0.603499i	$-0.206227\pi$
$613$	39.4836	1.59473	0.797364	+	0.603499i	$0.206227\pi$
$614$	−0.0340952	−0.00137597
$615$	0	0
$616$	0.709047	0.0285683
$617$	−38.4323	−1.54723	−0.773613	−	0.633658i	$-0.781553\pi$
$617$	−38.4323	−1.54723	−0.773613	+	0.633658i	$0.781553\pi$
$618$	0	0
$619$	14.9244	0.599861	0.299931	−	0.953961i	$-0.403036\pi$
$619$	14.9244	0.599861	0.299931	+	0.953961i	$0.403036\pi$
$620$	1.04483	0.0419614
$621$	0	0
$622$	−27.3757	−1.09767
$623$	7.77695	0.311577
$624$	0	0
$625$	19.5087	0.780349
$626$	13.7528	0.549672
$627$	0	0
$628$	−2.41376	−0.0963195
$629$	38.3006	1.52715
$630$	0	0
$631$	30.8867	1.22958	0.614791	−	0.788690i	$-0.289241\pi$
$631$	30.8867	1.22958	0.614791	+	0.788690i	$0.289241\pi$
$632$	−1.10479	−0.0439464
$633$	0	0
$634$	2.67331	0.106171
$635$	−8.29967	−0.329362
$636$	0	0
$637$	−20.0717	−0.795269
$638$	0.166956	0.00660986
$639$	0	0
$640$	−4.62022	−0.182630
$641$	−31.6649	−1.25069	−0.625344	−	0.780349i	$-0.715041\pi$
$641$	−31.6649	−1.25069	−0.625344	+	0.780349i	$0.715041\pi$
$642$	0	0
$643$	−17.2630	−0.680786	−0.340393	−	0.940283i	$-0.610560\pi$
$643$	−17.2630	−0.680786	−0.340393	+	0.940283i	$0.610560\pi$
$644$	2.46249	0.0970359
$645$	0	0
$646$	−31.1553	−1.22579
$647$	−35.2693	−1.38658	−0.693289	−	0.720659i	$-0.743839\pi$
$647$	−35.2693	−1.38658	−0.693289	+	0.720659i	$0.743839\pi$
$648$	0	0
$649$	2.56437	0.100661
$650$	−20.1115	−0.788840
$651$	0	0
$652$	3.06586	0.120068
$653$	38.0371	1.48851	0.744253	−	0.667898i	$-0.232806\pi$
$653$	38.0371	1.48851	0.744253	+	0.667898i	$0.232806\pi$
$654$	0	0
$655$	0.631169	0.0246618
$656$	34.3111	1.33962
$657$	0	0
$658$	11.2782	0.439671
$659$	7.14619	0.278376	0.139188	−	0.990266i	$-0.455551\pi$
$659$	7.14619	0.278376	0.139188	+	0.990266i	$0.455551\pi$
$660$	0	0
$661$	33.6588	1.30917	0.654587	−	0.755986i	$-0.272842\pi$
$661$	33.6588	1.30917	0.654587	+	0.755986i	$0.272842\pi$
$662$	−36.1807	−1.40620
$663$	0	0
$664$	53.7100	2.08435
$665$	−2.14892	−0.0833314
$666$	0	0
$667$	3.93754	0.152462
$668$	1.67656	0.0648682
$669$	0	0
$670$	9.51332	0.367531
$671$	0.832989	0.0321572
$672$	0	0
$673$	−4.10894	−0.158388	−0.0791939	−	0.996859i	$-0.525235\pi$
$673$	−4.10894	−0.158388	−0.0791939	+	0.996859i	$0.525235\pi$
$674$	30.3360	1.16850
$675$	0	0
$676$	0.542126	0.0208510
$677$	26.3822	1.01395	0.506976	−	0.861960i	$-0.330763\pi$
$677$	26.3822	1.01395	0.506976	+	0.861960i	$0.330763\pi$
$678$	0	0
$679$	−8.09936	−0.310825
$680$	−13.1649	−0.504850
$681$	0	0
$682$	−1.44751	−0.0554281
$683$	18.1548	0.694676	0.347338	−	0.937740i	$-0.387086\pi$
$683$	18.1548	0.694676	0.347338	+	0.937740i	$0.387086\pi$
$684$	0	0
$685$	−7.09465	−0.271073
$686$	−17.1585	−0.655115
$687$	0	0
$688$	2.90366	0.110701
$689$	26.2591	1.00039
$690$	0	0
$691$	28.1899	1.07239	0.536196	−	0.844093i	$-0.319861\pi$
$691$	28.1899	1.07239	0.536196	+	0.844093i	$0.319861\pi$
$692$	−4.67212	−0.177607
$693$	0	0
$694$	28.1315	1.06786
$695$	−0.0679927	−0.00257911
$696$	0	0
$697$	76.5962	2.90129
$698$	24.2504	0.917890
$699$	0	0
$700$	1.64686	0.0622455
$701$	−39.3669	−1.48687	−0.743434	−	0.668809i	$-0.766804\pi$
$701$	−39.3669	−1.48687	−0.743434	+	0.668809i	$0.766804\pi$
$702$	0	0
$703$	−18.2923	−0.689907
$704$	−2.02023	−0.0761402
$705$	0	0
$706$	14.6342	0.550766
$707$	−18.0237	−0.677853
$708$	0	0
$709$	48.4112	1.81812	0.909060	−	0.416664i	$-0.136801\pi$
$709$	48.4112	1.81812	0.909060	+	0.416664i	$0.136801\pi$
$710$	−7.47726	−0.280617
$711$	0	0
$712$	−22.7546	−0.852765
$713$	−34.1386	−1.27850
$714$	0	0
$715$	0.472211	0.0176597
$716$	−3.61950	−0.135267
$717$	0	0
$718$	0.188532	0.00703595
$719$	10.8564	0.404877	0.202438	−	0.979295i	$-0.435113\pi$
$719$	10.8564	0.404877	0.202438	+	0.979295i	$0.435113\pi$
$720$	0	0
$721$	6.29345	0.234380
$722$	−9.56045	−0.355803
$723$	0	0
$724$	3.04856	0.113299
$725$	2.63334	0.0977999
$726$	0	0
$727$	−37.5110	−1.39121	−0.695604	−	0.718426i	$-0.744863\pi$
$727$	−37.5110	−1.39121	−0.695604	+	0.718426i	$0.744863\pi$
$728$	−10.5170	−0.389787
$729$	0	0
$730$	5.60019	0.207272
$731$	6.48215	0.239751
$732$	0	0
$733$	−1.31177	−0.0484512	−0.0242256	−	0.999707i	$-0.507712\pi$
$733$	−1.31177	−0.0484512	−0.0242256	+	0.999707i	$0.507712\pi$
$734$	42.0173	1.55089
$735$	0	0
$736$	−13.3491	−0.492053
$737$	2.75105	0.101336
$738$	0	0
$739$	52.6696	1.93748	0.968742	−	0.248072i	$-0.0797969\pi$
$739$	52.6696	1.93748	0.968742	+	0.248072i	$0.0797969\pi$
$740$	−1.13823	−0.0418423
$741$	0	0
$742$	10.3015	0.378182
$743$	−44.6512	−1.63809	−0.819046	−	0.573728i	$-0.805497\pi$
$743$	−44.6512	−1.63809	−0.819046	+	0.573728i	$0.805497\pi$
$744$	0	0
$745$	−11.9743	−0.438704
$746$	−22.8362	−0.836091
$747$	0	0
$748$	−0.560609	−0.0204979
$749$	−8.10656	−0.296207
$750$	0	0
$751$	−30.0001	−1.09472	−0.547360	−	0.836897i	$-0.684367\pi$
$751$	−30.0001	−1.09472	−0.547360	+	0.836897i	$0.684367\pi$
$752$	−27.1254	−0.989160
$753$	0	0
$754$	−2.47640	−0.0901850
$755$	−10.8916	−0.396386
$756$	0	0
$757$	10.8998	0.396160	0.198080	−	0.980186i	$-0.436529\pi$
$757$	10.8998	0.396160	0.198080	+	0.980186i	$0.436529\pi$
$758$	−6.20568	−0.225400
$759$	0	0
$760$	6.28752	0.228072
$761$	−54.3944	−1.97180	−0.985898	−	0.167348i	$-0.946480\pi$
$761$	−54.3944	−1.97180	−0.985898	+	0.167348i	$0.946480\pi$
$762$	0	0
$763$	−4.30218	−0.155749
$764$	−1.12564	−0.0407242
$765$	0	0
$766$	−18.9253	−0.683798
$767$	−38.0364	−1.37341
$768$	0	0
$769$	−16.9359	−0.610723	−0.305361	−	0.952237i	$-0.598777\pi$
$769$	−16.9359	−0.610723	−0.305361	+	0.952237i	$0.598777\pi$
$770$	0.185250	0.00667596
$771$	0	0
$772$	−5.85954	−0.210890
$773$	29.5026	1.06113	0.530567	−	0.847643i	$-0.321979\pi$
$773$	29.5026	1.06113	0.530567	+	0.847643i	$0.321979\pi$
$774$	0	0
$775$	−22.8311	−0.820117
$776$	23.6980	0.850707
$777$	0	0
$778$	−6.30712	−0.226121
$779$	−36.5822	−1.31069
$780$	0	0
$781$	−2.16226	−0.0773718
$782$	63.3423	2.26512
$783$	0	0
$784$	18.9383	0.676368
$785$	−4.28253	−0.152850
$786$	0	0
$787$	−5.96992	−0.212805	−0.106402	−	0.994323i	$-0.533933\pi$
$787$	−5.96992	−0.212805	−0.106402	+	0.994323i	$0.533933\pi$
$788$	−6.40916	−0.228317
$789$	0	0
$790$	−0.288646	−0.0102696
$791$	−3.73468	−0.132790
$792$	0	0
$793$	−12.3554	−0.438753
$794$	−19.1610	−0.679998
$795$	0	0
$796$	6.78605	0.240525
$797$	−19.1226	−0.677357	−0.338679	−	0.940902i	$-0.609980\pi$
$797$	−19.1226	−0.677357	−0.338679	+	0.940902i	$0.609980\pi$
$798$	0	0
$799$	−60.5548	−2.14227
$800$	−8.92755	−0.315637
$801$	0	0
$802$	−8.64573	−0.305291
$803$	1.61945	0.0571492
$804$	0	0
$805$	4.36900	0.153987
$806$	21.4704	0.756262
$807$	0	0
$808$	52.7358	1.85524
$809$	−34.3768	−1.20862	−0.604312	−	0.796748i	$-0.706552\pi$
$809$	−34.3768	−1.20862	−0.604312	+	0.796748i	$0.706552\pi$
$810$	0	0
$811$	18.2250	0.639967	0.319984	−	0.947423i	$-0.396323\pi$
$811$	18.2250	0.639967	0.319984	+	0.947423i	$0.396323\pi$
$812$	0.202783	0.00711629
$813$	0	0
$814$	1.57691	0.0552708
$815$	5.43949	0.190537
$816$	0	0
$817$	−3.09586	−0.108310
$818$	−3.34360	−0.116906
$819$	0	0
$820$	−2.27632	−0.0794924
$821$	−36.2130	−1.26384	−0.631920	−	0.775033i	$-0.717733\pi$
$821$	−36.2130	−1.26384	−0.631920	+	0.775033i	$0.717733\pi$
$822$	0	0
$823$	−42.2212	−1.47174	−0.735869	−	0.677124i	$-0.763226\pi$
$823$	−42.2212	−1.47174	−0.735869	+	0.677124i	$0.763226\pi$
$824$	−18.4140	−0.641483
$825$	0	0
$826$	−14.9218	−0.519196
$827$	−15.1566	−0.527048	−0.263524	−	0.964653i	$-0.584885\pi$
$827$	−15.1566	−0.527048	−0.263524	+	0.964653i	$0.584885\pi$
$828$	0	0
$829$	−3.49294	−0.121315	−0.0606575	−	0.998159i	$-0.519320\pi$
$829$	−3.49294	−0.121315	−0.0606575	+	0.998159i	$0.519320\pi$
$830$	14.0326	0.487080
$831$	0	0
$832$	29.9653	1.03886
$833$	42.2780	1.46484
$834$	0	0
$835$	2.97459	0.102940
$836$	0.267746	0.00926017
$837$	0	0
$838$	16.9309	0.584868
$839$	28.5020	0.983999	0.491999	−	0.870596i	$-0.336266\pi$
$839$	28.5020	0.983999	0.491999	+	0.870596i	$0.336266\pi$
$840$	0	0
$841$	−28.6757	−0.988819
$842$	−8.40889	−0.289789
$843$	0	0
$844$	2.82743	0.0973240
$845$	0.961849	0.0330886
$846$	0	0
$847$	−11.2886	−0.387879
$848$	−24.7764	−0.850824
$849$	0	0
$850$	42.3619	1.45300
$851$	37.1904	1.27487
$852$	0	0
$853$	5.52236	0.189082	0.0945410	−	0.995521i	$-0.469862\pi$
$853$	5.52236	0.189082	0.0945410	+	0.995521i	$0.469862\pi$
$854$	−4.84707	−0.165863
$855$	0	0
$856$	23.7190	0.810700
$857$	17.0172	0.581298	0.290649	−	0.956830i	$-0.406129\pi$
$857$	17.0172	0.581298	0.290649	+	0.956830i	$0.406129\pi$
$858$	0	0
$859$	−1.16642	−0.0397979	−0.0198989	−	0.999802i	$-0.506334\pi$
$859$	−1.16642	−0.0397979	−0.0198989	+	0.999802i	$0.506334\pi$
$860$	−0.192639	−0.00656893
$861$	0	0
$862$	−13.9333	−0.474572
$863$	8.50057	0.289363	0.144681	−	0.989478i	$-0.453784\pi$
$863$	8.50057	0.289363	0.144681	+	0.989478i	$0.453784\pi$
$864$	0	0
$865$	−8.28935	−0.281846
$866$	−32.9346	−1.11916
$867$	0	0
$868$	−1.75813	−0.0596749
$869$	−0.0834701	−0.00283153
$870$	0	0
$871$	−40.8052	−1.38263
$872$	12.5878	0.426275
$873$	0	0
$874$	−30.2522	−1.02329
$875$	6.08102	0.205576
$876$	0	0
$877$	7.37310	0.248972	0.124486	−	0.992221i	$-0.460272\pi$
$877$	7.37310	0.248972	0.124486	+	0.992221i	$0.460272\pi$
$878$	18.8886	0.637460
$879$	0	0
$880$	−0.445547	−0.0150194
$881$	48.8726	1.64656	0.823279	−	0.567637i	$-0.192142\pi$
$881$	48.8726	1.64656	0.823279	+	0.567637i	$0.192142\pi$
$882$	0	0
$883$	−26.8006	−0.901911	−0.450956	−	0.892546i	$-0.648917\pi$
$883$	−26.8006	−0.901911	−0.450956	+	0.892546i	$0.648917\pi$
$884$	8.31529	0.279674
$885$	0	0
$886$	19.6404	0.659833
$887$	3.15346	0.105883	0.0529414	−	0.998598i	$-0.483140\pi$
$887$	3.15346	0.105883	0.0529414	+	0.998598i	$0.483140\pi$
$888$	0	0
$889$	13.9658	0.468399
$890$	−5.94502	−0.199277
$891$	0	0
$892$	−5.35571	−0.179322
$893$	28.9208	0.967799
$894$	0	0
$895$	−6.42179	−0.214657
$896$	7.77443	0.259726
$897$	0	0
$898$	−40.1538	−1.33995
$899$	−2.81126	−0.0937609
$900$	0	0
$901$	−55.3109	−1.84267
$902$	3.15362	0.105004
$903$	0	0
$904$	10.9273	0.363438
$905$	5.40881	0.179795
$906$	0	0
$907$	−25.7319	−0.854414	−0.427207	−	0.904154i	$-0.640502\pi$
$907$	−25.7319	−0.854414	−0.427207	+	0.904154i	$0.640502\pi$
$908$	−9.36404	−0.310756
$909$	0	0
$910$	−2.74775	−0.0910869
$911$	−10.3710	−0.343606	−0.171803	−	0.985131i	$-0.554959\pi$
$911$	−10.3710	−0.343606	−0.171803	+	0.985131i	$0.554959\pi$
$912$	0	0
$913$	4.05793	0.134298
$914$	12.1381	0.401494
$915$	0	0
$916$	−2.80079	−0.0925409
$917$	−1.06207	−0.0350725
$918$	0	0
$919$	−33.5553	−1.10689	−0.553444	−	0.832887i	$-0.686687\pi$
$919$	−33.5553	−1.10689	−0.553444	+	0.832887i	$0.686687\pi$
$920$	−12.7833	−0.421452
$921$	0	0
$922$	−11.1334	−0.366660
$923$	32.0720	1.05566
$924$	0	0
$925$	24.8721	0.817790
$926$	−16.1324	−0.530144
$927$	0	0
$928$	−1.09928	−0.0360855
$929$	−11.6222	−0.381314	−0.190657	−	0.981657i	$-0.561062\pi$
$929$	−11.6222	−0.381314	−0.190657	+	0.981657i	$0.561062\pi$
$930$	0	0
$931$	−20.1919	−0.661762
$932$	−1.50133	−0.0491777
$933$	0	0
$934$	6.79792	0.222435
$935$	−0.994642	−0.0325283
$936$	0	0
$937$	46.3339	1.51366	0.756831	−	0.653611i	$-0.226747\pi$
$937$	46.3339	1.51366	0.756831	+	0.653611i	$0.226747\pi$
$938$	−16.0080	−0.522681
$939$	0	0
$940$	1.79959	0.0586961
$941$	7.22995	0.235690	0.117845	−	0.993032i	$-0.462401\pi$
$941$	7.22995	0.235690	0.117845	+	0.993032i	$0.462401\pi$
$942$	0	0
$943$	74.3759	2.42201
$944$	35.8886	1.16807
$945$	0	0
$946$	0.266883	0.00867711
$947$	−30.3453	−0.986091	−0.493045	−	0.870004i	$-0.664116\pi$
$947$	−30.3453	−0.986091	−0.493045	+	0.870004i	$0.664116\pi$
$948$	0	0
$949$	−24.0207	−0.779745
$950$	−20.2320	−0.656412
$951$	0	0
$952$	22.1525	0.717967
$953$	−4.89634	−0.158608	−0.0793040	−	0.996850i	$-0.525270\pi$
$953$	−4.89634	−0.158608	−0.0793040	+	0.996850i	$0.525270\pi$
$954$	0	0
$955$	−1.99713	−0.0646256
$956$	6.34172	0.205106
$957$	0	0
$958$	37.3801	1.20770
$959$	11.9382	0.385503
$960$	0	0
$961$	−6.62631	−0.213752
$962$	−23.3898	−0.754116
$963$	0	0
$964$	−0.345373	−0.0111237
$965$	−10.3961	−0.334662
$966$	0	0
$967$	25.6038	0.823363	0.411681	−	0.911328i	$-0.364942\pi$
$967$	25.6038	0.823363	0.411681	+	0.911328i	$0.364942\pi$
$968$	33.0292	1.06160
$969$	0	0
$970$	6.19148	0.198797
$971$	−19.2440	−0.617570	−0.308785	−	0.951132i	$-0.599922\pi$
$971$	−19.2440	−0.617570	−0.308785	+	0.951132i	$0.599922\pi$
$972$	0	0
$973$	0.114411	0.00366785
$974$	−46.6747	−1.49555
$975$	0	0
$976$	11.6577	0.373155
$977$	−14.0956	−0.450957	−0.225479	−	0.974248i	$-0.572395\pi$
$977$	−14.0956	−0.450957	−0.225479	+	0.974248i	$0.572395\pi$
$978$	0	0
$979$	−1.71917	−0.0549449
$980$	−1.25643	−0.0401353
$981$	0	0
$982$	5.13606	0.163898
$983$	50.6189	1.61449	0.807246	−	0.590216i	$-0.200957\pi$
$983$	50.6189	1.61449	0.807246	+	0.590216i	$0.200957\pi$
$984$	0	0
$985$	−11.3712	−0.362318
$986$	5.21615	0.166116
$987$	0	0
$988$	−3.97137	−0.126346
$989$	6.29425	0.200145
$990$	0	0
$991$	15.7226	0.499444	0.249722	−	0.968318i	$-0.419661\pi$
$991$	15.7226	0.499444	0.249722	+	0.968318i	$0.419661\pi$
$992$	9.53074	0.302601
$993$	0	0
$994$	12.5820	0.399076
$995$	12.0399	0.381691
$996$	0	0
$997$	−9.08507	−0.287727	−0.143864	−	0.989598i	$-0.545953\pi$
$997$	−9.08507	−0.287727	−0.143864	+	0.989598i	$0.545953\pi$
$998$	11.9019	0.376747
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2169.2.a.h.1.9		12
3.2	odd	2		241.2.a.b.1.4	✓	12
12.11	even	2		3856.2.a.n.1.7		12
15.14	odd	2		6025.2.a.h.1.9		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
241.2.a.b.1.4	✓	12	3.2	odd	2
2169.2.a.h.1.9		12	1.1	even	1	trivial
3856.2.a.n.1.7		12	12.11	even	2
6025.2.a.h.1.9		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)

\(f(q)\)	\(=\)	\(q+1.28632 q^{2} -0.345373 q^{4} -0.612768 q^{5} +1.03110 q^{7} -3.01691 q^{8} +O(q^{10})\)	\(q+1.28632 q^{2} -0.345373 q^{4} -0.612768 q^{5} +1.03110 q^{7} -3.01691 q^{8} -0.788217 q^{10} -0.227935 q^{11} +3.38088 q^{13} +1.32633 q^{14} -3.18997 q^{16} -7.12130 q^{17} +3.40112 q^{19} +0.211634 q^{20} -0.293198 q^{22} -6.91488 q^{23} -4.62452 q^{25} +4.34890 q^{26} -0.356115 q^{28} -0.569431 q^{29} +4.93697 q^{31} +1.93048 q^{32} -9.16030 q^{34} -0.631826 q^{35} -5.37832 q^{37} +4.37494 q^{38} +1.84866 q^{40} -10.7559 q^{41} -0.910247 q^{43} +0.0787228 q^{44} -8.89477 q^{46} +8.50333 q^{47} -5.93683 q^{49} -5.94862 q^{50} -1.16766 q^{52} +7.76696 q^{53} +0.139671 q^{55} -3.11074 q^{56} -0.732472 q^{58} -11.2505 q^{59} -3.65450 q^{61} +6.35054 q^{62} +8.86317 q^{64} -2.07169 q^{65} -12.0694 q^{67} +2.45951 q^{68} -0.812732 q^{70} +9.48630 q^{71} -7.10488 q^{73} -6.91825 q^{74} -1.17466 q^{76} -0.235024 q^{77} +0.366201 q^{79} +1.95471 q^{80} -13.8356 q^{82} -17.8030 q^{83} +4.36370 q^{85} -1.17087 q^{86} +0.687660 q^{88} +7.54236 q^{89} +3.48603 q^{91} +2.38822 q^{92} +10.9380 q^{94} -2.08410 q^{95} -7.85505 q^{97} -7.63668 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{2} + 13 q^{4} - 6 q^{5} + 3 q^{7} - 9 q^{8}+O(q^{10}) \) 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

Properties

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