LMFDB - Embedded newform 2151.2.a.k.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,2,Mod(1,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, 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("2151.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2151.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:	$17.1758214748$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 4 x^{19} - 21 x^{18} + 96 x^{17} + 164 x^{16} - 936 x^{15} - 540 x^{14} + 4804 x^{13} + \cdots + 1 $ x^20 - 4x^19 - 21x^18 + 96x^17 + 164x^16 - 936x^15 - 540x^14 + 4804x^13 + 229x^12 - 14020x^11 + 3356x^10 + 23404x^9 - 9429x^8 - 21252x^7 + 10479x^6 + 9108x^5 - 4844x^4 - 1184x^3 + 640x^2 - 56*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Root			$1.66953$ of defining polynomial
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.66953 q^{2} +0.787326 q^{4} -3.19534 q^{5} -2.48662 q^{7} -2.02459 q^{8} +O(q^{10})$	\(q+1.66953 q^{2} +0.787326 q^{4} -3.19534 q^{5} -2.48662 q^{7} -2.02459 q^{8} -5.33471 q^{10} +4.44538 q^{11} -0.0955662 q^{13} -4.15149 q^{14} -4.95477 q^{16} -0.658610 q^{17} +5.40875 q^{19} -2.51577 q^{20} +7.42169 q^{22} -0.635524 q^{23} +5.21017 q^{25} -0.159551 q^{26} -1.95778 q^{28} +9.38370 q^{29} +7.21262 q^{31} -4.22294 q^{32} -1.09957 q^{34} +7.94559 q^{35} -11.9620 q^{37} +9.03006 q^{38} +6.46926 q^{40} +6.90694 q^{41} +7.01304 q^{43} +3.49996 q^{44} -1.06103 q^{46} +7.53884 q^{47} -0.816716 q^{49} +8.69853 q^{50} -0.0752418 q^{52} +0.341267 q^{53} -14.2045 q^{55} +5.03440 q^{56} +15.6664 q^{58} +1.57718 q^{59} -1.38928 q^{61} +12.0417 q^{62} +2.85921 q^{64} +0.305366 q^{65} -0.142775 q^{67} -0.518541 q^{68} +13.2654 q^{70} -3.28506 q^{71} +0.402150 q^{73} -19.9709 q^{74} +4.25845 q^{76} -11.0540 q^{77} +6.98905 q^{79} +15.8322 q^{80} +11.5313 q^{82} +0.0483201 q^{83} +2.10448 q^{85} +11.7085 q^{86} -9.00009 q^{88} +5.95912 q^{89} +0.237637 q^{91} -0.500365 q^{92} +12.5863 q^{94} -17.2828 q^{95} -11.1375 q^{97} -1.36353 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 4 q^{2} + 18 q^{4} + 16 q^{5} - 4 q^{7} + 12 q^{8}+O(q^{10}) $ 20 * q + 4 * q^2 + 18 * q^4 + 16 * q^5 - 4 * q^7 + 12 * q^8	$ 20 q + 4 q^{2} + 18 q^{4} + 16 q^{5} - 4 q^{7} + 12 q^{8} + 4 q^{10} + 12 q^{11} - 4 q^{13} + 20 q^{14} + 22 q^{16} + 24 q^{17} - 4 q^{19} + 40 q^{20} - 6 q^{22} + 12 q^{23} + 22 q^{25} + 30 q^{26} - 12 q^{28} + 24 q^{29} - 4 q^{31} + 28 q^{32} + 8 q^{34} + 20 q^{35} - 10 q^{37} + 26 q^{38} + 6 q^{40} + 66 q^{41} + 8 q^{43} + 36 q^{44} - 12 q^{46} + 28 q^{47} + 18 q^{49} + 28 q^{50} - 18 q^{52} + 28 q^{53} - 4 q^{55} + 60 q^{56} + 54 q^{59} - 4 q^{61} + 20 q^{62} + 22 q^{64} + 42 q^{65} + 12 q^{67} + 12 q^{68} + 20 q^{70} + 36 q^{71} + 14 q^{73} - 50 q^{76} + 8 q^{77} - 12 q^{79} + 88 q^{80} - 8 q^{82} + 20 q^{83} + 4 q^{85} + 18 q^{86} - 10 q^{88} + 130 q^{89} - 6 q^{91} - 46 q^{92} - 26 q^{94} - 2 q^{97} + 12 q^{98}+O(q^{100}) $ 20 * q + 4 * q^2 + 18 * q^4 + 16 * q^5 - 4 * q^7 + 12 * q^8 + 4 * q^10 + 12 * q^11 - 4 * q^13 + 20 * q^14 + 22 * q^16 + 24 * q^17 - 4 * q^19 + 40 * q^20 - 6 * q^22 + 12 * q^23 + 22 * q^25 + 30 * q^26 - 12 * q^28 + 24 * q^29 - 4 * q^31 + 28 * q^32 + 8 * q^34 + 20 * q^35 - 10 * q^37 + 26 * q^38 + 6 * q^40 + 66 * q^41 + 8 * q^43 + 36 * q^44 - 12 * q^46 + 28 * q^47 + 18 * q^49 + 28 * q^50 - 18 * q^52 + 28 * q^53 - 4 * q^55 + 60 * q^56 + 54 * q^59 - 4 * q^61 + 20 * q^62 + 22 * q^64 + 42 * q^65 + 12 * q^67 + 12 * q^68 + 20 * q^70 + 36 * q^71 + 14 * q^73 - 50 * q^76 + 8 * q^77 - 12 * q^79 + 88 * q^80 - 8 * q^82 + 20 * q^83 + 4 * q^85 + 18 * q^86 - 10 * q^88 + 130 * q^89 - 6 * q^91 - 46 * q^92 - 26 * q^94 - 2 * q^97 + 12 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.66953	1.18054	0.590268	−	0.807208i	$-0.299022\pi$
$2$	1.66953	1.18054	0.590268	+	0.807208i	$0.299022\pi$
$3$	0	0
$3$	0	0
$4$	0.787326	0.393663
$5$	−3.19534	−1.42900	−0.714499	−	0.699637i	$-0.753345\pi$
$5$	−3.19534	−1.42900	−0.714499	+	0.699637i	$0.753345\pi$
$6$	0	0
$7$	−2.48662	−0.939854	−0.469927	−	0.882705i	$-0.655720\pi$
$7$	−2.48662	−0.939854	−0.469927	+	0.882705i	$0.655720\pi$
$8$	−2.02459	−0.715802
$9$	0	0
$10$	−5.33471	−1.68698
$11$	4.44538	1.34033	0.670166	−	0.742211i	$-0.266223\pi$
$11$	4.44538	1.34033	0.670166	+	0.742211i	$0.266223\pi$
$12$	0	0
$13$	−0.0955662	−0.0265053	−0.0132526	−	0.999912i	$-0.504219\pi$
$13$	−0.0955662	−0.0265053	−0.0132526	+	0.999912i	$0.504219\pi$
$14$	−4.15149	−1.10953
$15$	0	0
$16$	−4.95477	−1.23869
$17$	−0.658610	−0.159736	−0.0798682	−	0.996805i	$-0.525450\pi$
$17$	−0.658610	−0.159736	−0.0798682	+	0.996805i	$0.525450\pi$
$18$	0	0
$19$	5.40875	1.24085	0.620426	−	0.784265i	$-0.286960\pi$
$19$	5.40875	1.24085	0.620426	+	0.784265i	$0.286960\pi$
$20$	−2.51577	−0.562544
$21$	0	0
$22$	7.42169	1.58231
$23$	−0.635524	−0.132516	−0.0662580	−	0.997803i	$-0.521106\pi$
$23$	−0.635524	−0.132516	−0.0662580	+	0.997803i	$0.521106\pi$
$24$	0	0
$25$	5.21017	1.04203
$26$	−0.159551	−0.0312904
$27$	0	0
$28$	−1.95778	−0.369986
$29$	9.38370	1.74251	0.871255	−	0.490831i	$-0.163307\pi$
$29$	9.38370	1.74251	0.871255	+	0.490831i	$0.163307\pi$
$30$	0	0
$31$	7.21262	1.29542	0.647712	−	0.761885i	$-0.275726\pi$
$31$	7.21262	1.29542	0.647712	+	0.761885i	$0.275726\pi$
$32$	−4.22294	−0.746518
$33$	0	0
$34$	−1.09957	−0.188574
$35$	7.94559	1.34305
$36$	0	0
$37$	−11.9620	−1.96655	−0.983273	−	0.182140i	$-0.941698\pi$
$37$	−11.9620	−1.96655	−0.983273	+	0.182140i	$0.941698\pi$
$38$	9.03006	1.46487
$39$	0	0
$40$	6.46926	1.02288
$41$	6.90694	1.07868	0.539342	−	0.842087i	$-0.318673\pi$
$41$	6.90694	1.07868	0.539342	+	0.842087i	$0.318673\pi$
$42$	0	0
$43$	7.01304	1.06948	0.534739	−	0.845017i	$-0.320410\pi$
$43$	7.01304	1.06948	0.534739	+	0.845017i	$0.320410\pi$
$44$	3.49996	0.527639
$45$	0	0
$46$	−1.06103	−0.156440
$47$	7.53884	1.09965	0.549827	−	0.835279i	$-0.314694\pi$
$47$	7.53884	1.09965	0.549827	+	0.835279i	$0.314694\pi$
$48$	0	0
$49$	−0.816716	−0.116674
$50$	8.69853	1.23016
$51$	0	0
$52$	−0.0752418	−0.0104342
$53$	0.341267	0.0468766	0.0234383	−	0.999725i	$-0.492539\pi$
$53$	0.341267	0.0468766	0.0234383	+	0.999725i	$0.492539\pi$
$54$	0	0
$55$	−14.2045	−1.91533
$56$	5.03440	0.672750
$57$	0	0
$58$	15.6664	2.05709
$59$	1.57718	0.205332	0.102666	−	0.994716i	$-0.467263\pi$
$59$	1.57718	0.205332	0.102666	+	0.994716i	$0.467263\pi$
$60$	0	0
$61$	−1.38928	−0.177879	−0.0889396	−	0.996037i	$-0.528348\pi$
$61$	−1.38928	−0.177879	−0.0889396	+	0.996037i	$0.528348\pi$
$62$	12.0417	1.52929
$63$	0	0
$64$	2.85921	0.357402
$65$	0.305366	0.0378760
$66$	0	0
$67$	−0.142775	−0.0174427	−0.00872136	−	0.999962i	$-0.502776\pi$
$67$	−0.142775	−0.0174427	−0.00872136	+	0.999962i	$0.502776\pi$
$68$	−0.518541	−0.0628823
$69$	0	0
$70$	13.2654	1.58552
$71$	−3.28506	−0.389865	−0.194933	−	0.980817i	$-0.562449\pi$
$71$	−3.28506	−0.389865	−0.194933	+	0.980817i	$0.562449\pi$
$72$	0	0
$73$	0.402150	0.0470681	0.0235340	−	0.999723i	$-0.492508\pi$
$73$	0.402150	0.0470681	0.0235340	+	0.999723i	$0.492508\pi$
$74$	−19.9709	−2.32158
$75$	0	0
$76$	4.25845	0.488478
$77$	−11.0540	−1.25972
$78$	0	0
$79$	6.98905	0.786330	0.393165	−	0.919468i	$-0.371380\pi$
$79$	6.98905	0.786330	0.393165	+	0.919468i	$0.371380\pi$
$80$	15.8322	1.77009
$81$	0	0
$82$	11.5313	1.27342
$83$	0.0483201	0.00530382	0.00265191	−	0.999996i	$-0.499156\pi$
$83$	0.0483201	0.00530382	0.00265191	+	0.999996i	$0.499156\pi$
$84$	0	0
$85$	2.10448	0.228263
$86$	11.7085	1.26256
$87$	0	0
$88$	−9.00009	−0.959412
$89$	5.95912	0.631665	0.315833	−	0.948815i	$-0.397716\pi$
$89$	5.95912	0.631665	0.315833	+	0.948815i	$0.397716\pi$
$90$	0	0
$91$	0.237637	0.0249111
$92$	−0.500365	−0.0521667
$93$	0	0
$94$	12.5863	1.29818
$95$	−17.2828	−1.77317
$96$	0	0
$97$	−11.1375	−1.13084	−0.565421	−	0.824802i	$-0.691286\pi$
$97$	−11.1375	−1.13084	−0.565421	+	0.824802i	$0.691286\pi$
$98$	−1.36353	−0.137737
$99$	0	0
$100$	4.10211	0.410211
$101$	−0.600219	−0.0597240	−0.0298620	−	0.999554i	$-0.509507\pi$
$101$	−0.600219	−0.0597240	−0.0298620	+	0.999554i	$0.509507\pi$
$102$	0	0
$103$	0.225700	0.0222389	0.0111195	−	0.999938i	$-0.496460\pi$
$103$	0.225700	0.0222389	0.0111195	+	0.999938i	$0.496460\pi$
$104$	0.193483	0.0189725
$105$	0	0
$106$	0.569755	0.0553395
$107$	8.77148	0.847971	0.423986	−	0.905669i	$-0.360631\pi$
$107$	8.77148	0.847971	0.423986	+	0.905669i	$0.360631\pi$
$108$	0	0
$109$	4.72372	0.452450	0.226225	−	0.974075i	$-0.427361\pi$
$109$	4.72372	0.452450	0.226225	+	0.974075i	$0.427361\pi$
$110$	−23.7148	−2.26112
$111$	0	0
$112$	12.3206	1.16419
$113$	−11.7376	−1.10418	−0.552089	−	0.833785i	$-0.686169\pi$
$113$	−11.7376	−1.10418	−0.552089	+	0.833785i	$0.686169\pi$
$114$	0	0
$115$	2.03071	0.189365
$116$	7.38803	0.685962
$117$	0	0
$118$	2.63315	0.242401
$119$	1.63771	0.150129
$120$	0	0
$121$	8.76140	0.796491
$122$	−2.31944	−0.209993
$123$	0	0
$124$	5.67869	0.509961
$125$	−0.671570	−0.0600671
$126$	0	0
$127$	−0.436274	−0.0387130	−0.0193565	−	0.999813i	$-0.506162\pi$
$127$	−0.436274	−0.0387130	−0.0193565	+	0.999813i	$0.506162\pi$
$128$	13.2194	1.16844
$129$	0	0
$130$	0.509817	0.0447139
$131$	11.9696	1.04579	0.522894	−	0.852398i	$-0.324853\pi$
$131$	11.9696	1.04579	0.522894	+	0.852398i	$0.324853\pi$
$132$	0	0
$133$	−13.4495	−1.16622
$134$	−0.238367	−0.0205917
$135$	0	0
$136$	1.33342	0.114340
$137$	18.5405	1.58402	0.792009	−	0.610509i	$-0.209035\pi$
$137$	18.5405	1.58402	0.792009	+	0.610509i	$0.209035\pi$
$138$	0	0
$139$	0.00950510	0.000806212 0	0.000403106	−	1.00000i	$-0.499872\pi$
$139$	0.00950510	0.000806212 0	0.000403106		1.00000i	$0.499872\pi$
$140$	6.25577	0.528709
$141$	0	0
$142$	−5.48451	−0.460250
$143$	−0.424828	−0.0355259
$144$	0	0
$145$	−29.9841	−2.49004
$146$	0.671401	0.0555655
$147$	0	0
$148$	−9.41802	−0.774156
$149$	18.0008	1.47468	0.737342	−	0.675519i	$-0.236081\pi$
$149$	18.0008	1.47468	0.737342	+	0.675519i	$0.236081\pi$
$150$	0	0
$151$	−6.75284	−0.549538	−0.274769	−	0.961510i	$-0.588601\pi$
$151$	−6.75284	−0.549538	−0.274769	+	0.961510i	$0.588601\pi$
$152$	−10.9505	−0.888204
$153$	0	0
$154$	−18.4549	−1.48714
$155$	−23.0467	−1.85116
$156$	0	0
$157$	−3.73808	−0.298331	−0.149166	−	0.988812i	$-0.547659\pi$
$157$	−3.73808	−0.298331	−0.149166	+	0.988812i	$0.547659\pi$
$158$	11.6684	0.928290
$159$	0	0
$160$	13.4937	1.06677
$161$	1.58031	0.124546
$162$	0	0
$163$	14.3516	1.12410	0.562052	−	0.827102i	$-0.310012\pi$
$163$	14.3516	1.12410	0.562052	+	0.827102i	$0.310012\pi$
$164$	5.43802	0.424638
$165$	0	0
$166$	0.0806718	0.00626135
$167$	0.336460	0.0260360	0.0130180	−	0.999915i	$-0.495856\pi$
$167$	0.336460	0.0260360	0.0130180	+	0.999915i	$0.495856\pi$
$168$	0	0
$169$	−12.9909	−0.999297
$170$	3.51349	0.269472
$171$	0	0
$172$	5.52155	0.421014
$173$	−11.3461	−0.862628	−0.431314	−	0.902202i	$-0.641950\pi$
$173$	−11.3461	−0.862628	−0.431314	+	0.902202i	$0.641950\pi$
$174$	0	0
$175$	−12.9557	−0.979361
$176$	−22.0258	−1.66026
$177$	0	0
$178$	9.94892	0.745703
$179$	4.38379	0.327660	0.163830	−	0.986489i	$-0.447615\pi$
$179$	4.38379	0.327660	0.163830	+	0.986489i	$0.447615\pi$
$180$	0	0
$181$	−8.91623	−0.662738	−0.331369	−	0.943501i	$-0.607511\pi$
$181$	−8.91623	−0.662738	−0.331369	+	0.943501i	$0.607511\pi$
$182$	0.396742	0.0294084
$183$	0	0
$184$	1.28668	0.0948552
$185$	38.2227	2.81019
$186$	0	0
$187$	−2.92777	−0.214100
$188$	5.93553	0.432893
$189$	0	0
$190$	−28.8541	−2.09329
$191$	9.60112	0.694713	0.347356	−	0.937733i	$-0.387079\pi$
$191$	9.60112	0.694713	0.347356	+	0.937733i	$0.387079\pi$
$192$	0	0
$193$	−20.1533	−1.45067	−0.725333	−	0.688398i	$-0.758314\pi$
$193$	−20.1533	−1.45067	−0.725333	+	0.688398i	$0.758314\pi$
$194$	−18.5944	−1.33500
$195$	0	0
$196$	−0.643022	−0.0459302
$197$	−21.4790	−1.53032	−0.765158	−	0.643842i	$-0.777339\pi$
$197$	−21.4790	−1.53032	−0.765158	+	0.643842i	$0.777339\pi$
$198$	0	0
$199$	0.0533765	0.00378376	0.00189188	−	0.999998i	$-0.499398\pi$
$199$	0.0533765	0.00378376	0.00189188	+	0.999998i	$0.499398\pi$
$200$	−10.5485	−0.745890
$201$	0	0
$202$	−1.00208	−0.0705063
$203$	−23.3337	−1.63771
$204$	0	0
$205$	−22.0700	−1.54144
$206$	0.376813	0.0262538
$207$	0	0
$208$	0.473508	0.0328319
$209$	24.0439	1.66315
$210$	0	0
$211$	−3.28901	−0.226425	−0.113212	−	0.993571i	$-0.536114\pi$
$211$	−3.28901	−0.226425	−0.113212	+	0.993571i	$0.536114\pi$
$212$	0.268688	0.0184536
$213$	0	0
$214$	14.6442	1.00106
$215$	−22.4090	−1.52828
$216$	0	0
$217$	−17.9351	−1.21751
$218$	7.88639	0.534134
$219$	0	0
$220$	−11.1836	−0.753996
$221$	0.0629408	0.00423386
$222$	0	0
$223$	−4.32252	−0.289458	−0.144729	−	0.989471i	$-0.546231\pi$
$223$	−4.32252	−0.289458	−0.144729	+	0.989471i	$0.546231\pi$
$224$	10.5009	0.701618
$225$	0	0
$226$	−19.5962	−1.30352
$227$	22.3311	1.48217	0.741085	−	0.671411i	$-0.234312\pi$
$227$	22.3311	1.48217	0.741085	+	0.671411i	$0.234312\pi$
$228$	0	0
$229$	19.6537	1.29875	0.649376	−	0.760468i	$-0.275030\pi$
$229$	19.6537	1.29875	0.649376	+	0.760468i	$0.275030\pi$
$230$	3.39033	0.223552
$231$	0	0
$232$	−18.9982	−1.24729
$233$	3.43729	0.225185	0.112592	−	0.993641i	$-0.464085\pi$
$233$	3.43729	0.225185	0.112592	+	0.993641i	$0.464085\pi$
$234$	0	0
$235$	−24.0891	−1.57140
$236$	1.24176	0.0808315
$237$	0	0
$238$	2.73421	0.177232
$239$	1.00000	0.0646846
$239$	1.00000	0.0646846
$240$	0	0
$241$	8.22484	0.529808	0.264904	−	0.964275i	$-0.414660\pi$
$241$	8.22484	0.529808	0.264904	+	0.964275i	$0.414660\pi$
$242$	14.6274	0.940285
$243$	0	0
$244$	−1.09382	−0.0700245
$245$	2.60968	0.166727
$246$	0	0
$247$	−0.516893	−0.0328891
$248$	−14.6026	−0.927268
$249$	0	0
$250$	−1.12121	−0.0709113
$251$	−15.3444	−0.968531	−0.484265	−	0.874921i	$-0.660913\pi$
$251$	−15.3444	−0.968531	−0.484265	+	0.874921i	$0.660913\pi$
$252$	0	0
$253$	−2.82515	−0.177615
$254$	−0.728371	−0.0457021
$255$	0	0
$256$	16.3518	1.02199
$257$	27.1321	1.69245	0.846225	−	0.532825i	$-0.178870\pi$
$257$	27.1321	1.69245	0.846225	+	0.532825i	$0.178870\pi$
$258$	0	0
$259$	29.7450	1.84827
$260$	0.240423	0.0149104
$261$	0	0
$262$	19.9836	1.23459
$263$	−15.0146	−0.925840	−0.462920	−	0.886400i	$-0.653198\pi$
$263$	−15.0146	−0.925840	−0.462920	+	0.886400i	$0.653198\pi$
$264$	0	0
$265$	−1.09046	−0.0669865
$266$	−22.4543	−1.37676
$267$	0	0
$268$	−0.112410	−0.00686655
$269$	24.3072	1.48204	0.741018	−	0.671485i	$-0.234343\pi$
$269$	24.3072	1.48204	0.741018	+	0.671485i	$0.234343\pi$
$270$	0	0
$271$	−27.2074	−1.65273	−0.826366	−	0.563133i	$-0.809596\pi$
$271$	−27.2074	−1.65273	−0.826366	+	0.563133i	$0.809596\pi$
$272$	3.26326	0.197864
$273$	0	0
$274$	30.9538	1.86999
$275$	23.1612	1.39667
$276$	0	0
$277$	−31.1450	−1.87132	−0.935660	−	0.352902i	$-0.885195\pi$
$277$	−31.1450	−1.87132	−0.935660	+	0.352902i	$0.885195\pi$
$278$	0.0158690	0.000951761 0
$279$	0	0
$280$	−16.0866	−0.961358
$281$	21.5282	1.28426	0.642132	−	0.766594i	$-0.278050\pi$
$281$	21.5282	1.28426	0.642132	+	0.766594i	$0.278050\pi$
$282$	0	0
$283$	29.2986	1.74162	0.870809	−	0.491621i	$-0.163596\pi$
$283$	29.2986	1.74162	0.870809	+	0.491621i	$0.163596\pi$
$284$	−2.58642	−0.153476
$285$	0	0
$286$	−0.709263	−0.0419396
$287$	−17.1750	−1.01381
$288$	0	0
$289$	−16.5662	−0.974484
$290$	−50.0593	−2.93958
$291$	0	0
$292$	0.316623	0.0185290
$293$	10.7054	0.625417	0.312709	−	0.949849i	$-0.398764\pi$
$293$	10.7054	0.625417	0.312709	+	0.949849i	$0.398764\pi$
$294$	0	0
$295$	−5.03962	−0.293418
$296$	24.2182	1.40766
$297$	0	0
$298$	30.0529	1.74092
$299$	0.0607346	0.00351237
$300$	0	0
$301$	−17.4388	−1.00515
$302$	−11.2741	−0.648749
$303$	0	0
$304$	−26.7991	−1.53703
$305$	4.43922	0.254189
$306$	0	0
$307$	21.3838	1.22044	0.610219	−	0.792233i	$-0.291081\pi$
$307$	21.3838	1.22044	0.610219	+	0.792233i	$0.291081\pi$
$308$	−8.70308	−0.495904
$309$	0	0
$310$	−38.4772	−2.18536
$311$	2.67787	0.151848	0.0759240	−	0.997114i	$-0.475809\pi$
$311$	2.67787	0.151848	0.0759240	+	0.997114i	$0.475809\pi$
$312$	0	0
$313$	0.666770	0.0376881	0.0188440	−	0.999822i	$-0.494001\pi$
$313$	0.666770	0.0376881	0.0188440	+	0.999822i	$0.494001\pi$
$314$	−6.24084	−0.352191
$315$	0	0
$316$	5.50266	0.309549
$317$	−24.4769	−1.37476	−0.687379	−	0.726299i	$-0.741239\pi$
$317$	−24.4769	−1.37476	−0.687379	+	0.726299i	$0.741239\pi$
$318$	0	0
$319$	41.7141	2.33554
$320$	−9.13615	−0.510726
$321$	0	0
$322$	2.63837	0.147031
$323$	−3.56225	−0.198209
$324$	0	0
$325$	−0.497916	−0.0276194
$326$	23.9604	1.32704
$327$	0	0
$328$	−13.9838	−0.772124
$329$	−18.7462	−1.03351
$330$	0	0
$331$	−28.3306	−1.55719	−0.778595	−	0.627526i	$-0.784068\pi$
$331$	−28.3306	−1.55719	−0.778595	+	0.627526i	$0.784068\pi$
$332$	0.0380437	0.00208792
$333$	0	0
$334$	0.561729	0.0307364
$335$	0.456213	0.0249256
$336$	0	0
$337$	28.2985	1.54152	0.770758	−	0.637127i	$-0.219878\pi$
$337$	28.2985	1.54152	0.770758	+	0.637127i	$0.219878\pi$
$338$	−21.6886	−1.17971
$339$	0	0
$340$	1.65691	0.0898587
$341$	32.0628	1.73630
$342$	0	0
$343$	19.4372	1.04951
$344$	−14.1985	−0.765534
$345$	0	0
$346$	−18.9426	−1.01836
$347$	36.1955	1.94307	0.971537	−	0.236887i	$-0.0761270\pi$
$347$	36.1955	1.94307	0.971537	+	0.236887i	$0.0761270\pi$
$348$	0	0
$349$	35.8094	1.91683	0.958416	−	0.285375i	$-0.0921180\pi$
$349$	35.8094	1.91683	0.958416	+	0.285375i	$0.0921180\pi$
$350$	−21.6300	−1.15617
$351$	0	0
$352$	−18.7726	−1.00058
$353$	−27.6089	−1.46948	−0.734738	−	0.678351i	$-0.762695\pi$
$353$	−27.6089	−1.46948	−0.734738	+	0.678351i	$0.762695\pi$
$354$	0	0
$355$	10.4969	0.557117
$356$	4.69177	0.248663
$357$	0	0
$358$	7.31887	0.386814
$359$	16.6008	0.876154	0.438077	−	0.898937i	$-0.355660\pi$
$359$	16.6008	0.876154	0.438077	+	0.898937i	$0.355660\pi$
$360$	0	0
$361$	10.2545	0.539713
$362$	−14.8859	−0.782386
$363$	0	0
$364$	0.187098	0.00980659
$365$	−1.28500	−0.0672602
$366$	0	0
$367$	21.8482	1.14047	0.570234	−	0.821483i	$-0.306853\pi$
$367$	21.8482	1.14047	0.570234	+	0.821483i	$0.306853\pi$
$368$	3.14888	0.164147
$369$	0	0
$370$	63.8139	3.31753
$371$	−0.848601	−0.0440572
$372$	0	0
$373$	−26.2374	−1.35852	−0.679261	−	0.733897i	$-0.737699\pi$
$373$	−26.2374	−1.35852	−0.679261	+	0.733897i	$0.737699\pi$
$374$	−4.88800	−0.252752
$375$	0	0
$376$	−15.2631	−0.787134
$377$	−0.896765	−0.0461857
$378$	0	0
$379$	−23.8736	−1.22630	−0.613151	−	0.789966i	$-0.710098\pi$
$379$	−23.8736	−1.22630	−0.613151	+	0.789966i	$0.710098\pi$
$380$	−13.6072	−0.698033
$381$	0	0
$382$	16.0294	0.820133
$383$	1.53922	0.0786507	0.0393254	−	0.999226i	$-0.487479\pi$
$383$	1.53922	0.0786507	0.0393254	+	0.999226i	$0.487479\pi$
$384$	0	0
$385$	35.3212	1.80013
$386$	−33.6465	−1.71256
$387$	0	0
$388$	−8.76885	−0.445171
$389$	20.7468	1.05191	0.525953	−	0.850514i	$-0.323709\pi$
$389$	20.7468	1.05191	0.525953	+	0.850514i	$0.323709\pi$
$390$	0	0
$391$	0.418563	0.0211676
$392$	1.65352	0.0835153
$393$	0	0
$394$	−35.8598	−1.80659
$395$	−22.3324	−1.12366
$396$	0	0
$397$	0.280543	0.0140800	0.00704002	−	0.999975i	$-0.497759\pi$
$397$	0.280543	0.0140800	0.00704002	+	0.999975i	$0.497759\pi$
$398$	0.0891135	0.00446686
$399$	0	0
$400$	−25.8152	−1.29076
$401$	14.9965	0.748888	0.374444	−	0.927250i	$-0.377834\pi$
$401$	14.9965	0.748888	0.374444	+	0.927250i	$0.377834\pi$
$402$	0	0
$403$	−0.689283	−0.0343356
$404$	−0.472568	−0.0235112
$405$	0	0
$406$	−38.9563	−1.93337
$407$	−53.1757	−2.63582
$408$	0	0
$409$	−2.64336	−0.130706	−0.0653528	−	0.997862i	$-0.520817\pi$
$409$	−2.64336	−0.130706	−0.0653528	+	0.997862i	$0.520817\pi$
$410$	−36.8465	−1.81972
$411$	0	0
$412$	0.177700	0.00875464
$413$	−3.92185	−0.192982
$414$	0	0
$415$	−0.154399	−0.00757915
$416$	0.403571	0.0197867
$417$	0	0
$418$	40.1420	1.96341
$419$	−12.1813	−0.595094	−0.297547	−	0.954707i	$-0.596169\pi$
$419$	−12.1813	−0.595094	−0.297547	+	0.954707i	$0.596169\pi$
$420$	0	0
$421$	6.22710	0.303490	0.151745	−	0.988420i	$-0.451511\pi$
$421$	6.22710	0.303490	0.151745	+	0.988420i	$0.451511\pi$
$422$	−5.49110	−0.267302
$423$	0	0
$424$	−0.690926	−0.0335544
$425$	−3.43147	−0.166451
$426$	0	0
$427$	3.45461	0.167180
$428$	6.90602	0.333815
$429$	0	0
$430$	−37.4125	−1.80419
$431$	−22.4753	−1.08260	−0.541299	−	0.840830i	$-0.682067\pi$
$431$	−22.4753	−1.08260	−0.541299	+	0.840830i	$0.682067\pi$
$432$	0	0
$433$	11.0267	0.529908	0.264954	−	0.964261i	$-0.414643\pi$
$433$	11.0267	0.529908	0.264954	+	0.964261i	$0.414643\pi$
$434$	−29.9431	−1.43731
$435$	0	0
$436$	3.71911	0.178113
$437$	−3.43739	−0.164433
$438$	0	0
$439$	37.6999	1.79932	0.899659	−	0.436593i	$-0.143815\pi$
$439$	37.6999	1.79932	0.899659	+	0.436593i	$0.143815\pi$
$440$	28.7583	1.37100
$441$	0	0
$442$	0.105082	0.00499822
$443$	12.1550	0.577503	0.288751	−	0.957404i	$-0.406760\pi$
$443$	12.1550	0.577503	0.288751	+	0.957404i	$0.406760\pi$
$444$	0	0
$445$	−19.0414	−0.902648
$446$	−7.21658	−0.341715
$447$	0	0
$448$	−7.10978	−0.335906
$449$	−4.49986	−0.212362	−0.106181	−	0.994347i	$-0.533862\pi$
$449$	−4.49986	−0.212362	−0.106181	+	0.994347i	$0.533862\pi$
$450$	0	0
$451$	30.7040	1.44579
$452$	−9.24130	−0.434674
$453$	0	0
$454$	37.2825	1.74975
$455$	−0.759330	−0.0355979
$456$	0	0
$457$	3.25466	0.152246	0.0761232	−	0.997098i	$-0.475746\pi$
$457$	3.25466	0.152246	0.0761232	+	0.997098i	$0.475746\pi$
$458$	32.8124	1.53322
$459$	0	0
$460$	1.59883	0.0745460
$461$	−38.0611	−1.77268	−0.886341	−	0.463033i	$-0.846761\pi$
$461$	−38.0611	−1.77268	−0.886341	+	0.463033i	$0.846761\pi$
$462$	0	0
$463$	−9.16202	−0.425795	−0.212898	−	0.977075i	$-0.568290\pi$
$463$	−9.16202	−0.425795	−0.212898	+	0.977075i	$0.568290\pi$
$464$	−46.4941	−2.15843
$465$	0	0
$466$	5.73866	0.265838
$467$	4.39258	0.203265	0.101632	−	0.994822i	$-0.467593\pi$
$467$	4.39258	0.203265	0.101632	+	0.994822i	$0.467593\pi$
$468$	0	0
$469$	0.355027	0.0163936
$470$	−40.2175	−1.85509
$471$	0	0
$472$	−3.19315	−0.146977
$473$	31.1756	1.43346
$474$	0	0
$475$	28.1805	1.29301
$476$	1.28941	0.0591002
$477$	0	0
$478$	1.66953	0.0763625
$479$	24.3906	1.11444	0.557219	−	0.830366i	$-0.311869\pi$
$479$	24.3906	1.11444	0.557219	+	0.830366i	$0.311869\pi$
$480$	0	0
$481$	1.14317	0.0521239
$482$	13.7316	0.625458
$483$	0	0
$484$	6.89808	0.313549
$485$	35.5881	1.61597
$486$	0	0
$487$	−33.7875	−1.53106	−0.765530	−	0.643400i	$-0.777523\pi$
$487$	−33.7875	−1.53106	−0.765530	+	0.643400i	$0.777523\pi$
$488$	2.81273	0.127326
$489$	0	0
$490$	4.35694	0.196827
$491$	6.60713	0.298175	0.149088	−	0.988824i	$-0.452366\pi$
$491$	6.60713	0.298175	0.149088	+	0.988824i	$0.452366\pi$
$492$	0	0
$493$	−6.18020	−0.278342
$494$	−0.862968	−0.0388268
$495$	0	0
$496$	−35.7369	−1.60463
$497$	8.16871	0.366417
$498$	0	0
$499$	−7.45431	−0.333701	−0.166850	−	0.985982i	$-0.553360\pi$
$499$	−7.45431	−0.333701	−0.166850	+	0.985982i	$0.553360\pi$
$500$	−0.528745	−0.0236462
$501$	0	0
$502$	−25.6179	−1.14338
$503$	−1.72248	−0.0768015	−0.0384008	−	0.999262i	$-0.512226\pi$
$503$	−1.72248	−0.0768015	−0.0384008	+	0.999262i	$0.512226\pi$
$504$	0	0
$505$	1.91790	0.0853455
$506$	−4.71666	−0.209681
$507$	0	0
$508$	−0.343490	−0.0152399
$509$	−6.17725	−0.273802	−0.136901	−	0.990585i	$-0.543714\pi$
$509$	−6.17725	−0.273802	−0.136901	+	0.990585i	$0.543714\pi$
$510$	0	0
$511$	−0.999994	−0.0442371
$512$	0.860923	0.0380478
$513$	0	0
$514$	45.2977	1.99800
$515$	−0.721189	−0.0317794
$516$	0	0
$517$	33.5130	1.47390
$518$	49.6602	2.18194
$519$	0	0
$520$	−0.618242	−0.0271117
$521$	26.4121	1.15714	0.578568	−	0.815634i	$-0.303612\pi$
$521$	26.4121	1.15714	0.578568	+	0.815634i	$0.303612\pi$
$522$	0	0
$523$	30.6730	1.34124	0.670618	−	0.741803i	$-0.266029\pi$
$523$	30.6730	1.34124	0.670618	+	0.741803i	$0.266029\pi$
$524$	9.42397	0.411688
$525$	0	0
$526$	−25.0673	−1.09299
$527$	−4.75030	−0.206926
$528$	0	0
$529$	−22.5961	−0.982440
$530$	−1.82056	−0.0790800
$531$	0	0
$532$	−10.5891	−0.459098
$533$	−0.660070	−0.0285908
$534$	0	0
$535$	−28.0278	−1.21175
$536$	0.289061	0.0124855
$537$	0	0
$538$	40.5816	1.74960
$539$	−3.63061	−0.156382
$540$	0	0
$541$	−8.59747	−0.369634	−0.184817	−	0.982773i	$-0.559169\pi$
$541$	−8.59747	−0.369634	−0.184817	+	0.982773i	$0.559169\pi$
$542$	−45.4236	−1.95111
$543$	0	0
$544$	2.78127	0.119246
$545$	−15.0939	−0.646551
$546$	0	0
$547$	20.3050	0.868178	0.434089	−	0.900870i	$-0.357070\pi$
$547$	20.3050	0.868178	0.434089	+	0.900870i	$0.357070\pi$
$548$	14.5974	0.623570
$549$	0	0
$550$	38.6683	1.64882
$551$	50.7541	2.16220
$552$	0	0
$553$	−17.3791	−0.739035
$554$	−51.9974	−2.20916
$555$	0	0
$556$	0.00748361	0.000317376 0
$557$	−25.7890	−1.09272	−0.546358	−	0.837552i	$-0.683986\pi$
$557$	−25.7890	−1.09272	−0.546358	+	0.837552i	$0.683986\pi$
$558$	0	0
$559$	−0.670209	−0.0283468
$560$	−39.3686	−1.66363
$561$	0	0
$562$	35.9419	1.51612
$563$	10.6066	0.447016	0.223508	−	0.974702i	$-0.428249\pi$
$563$	10.6066	0.447016	0.223508	+	0.974702i	$0.428249\pi$
$564$	0	0
$565$	37.5055	1.57787
$566$	48.9148	2.05604
$567$	0	0
$568$	6.65092	0.279066
$569$	−31.2790	−1.31128	−0.655641	−	0.755072i	$-0.727602\pi$
$569$	−31.2790	−1.31128	−0.655641	+	0.755072i	$0.727602\pi$
$570$	0	0
$571$	−3.31115	−0.138567	−0.0692836	−	0.997597i	$-0.522071\pi$
$571$	−3.31115	−0.138567	−0.0692836	+	0.997597i	$0.522071\pi$
$572$	−0.334478	−0.0139852
$573$	0	0
$574$	−28.6741	−1.19683
$575$	−3.31119	−0.138086
$576$	0	0
$577$	43.3143	1.80320	0.901599	−	0.432573i	$-0.142394\pi$
$577$	43.3143	1.80320	0.901599	+	0.432573i	$0.142394\pi$
$578$	−27.6578	−1.15041
$579$	0	0
$580$	−23.6073	−0.980238
$581$	−0.120154	−0.00498482
$582$	0	0
$583$	1.51706	0.0628302
$584$	−0.814190	−0.0336914
$585$	0	0
$586$	17.8730	0.738327
$587$	−36.3210	−1.49913	−0.749565	−	0.661931i	$-0.769737\pi$
$587$	−36.3210	−1.49913	−0.749565	+	0.661931i	$0.769737\pi$
$588$	0	0
$589$	39.0112	1.60743
$590$	−8.41380	−0.346391
$591$	0	0
$592$	59.2691	2.43594
$593$	−25.2745	−1.03790	−0.518949	−	0.854805i	$-0.673677\pi$
$593$	−25.2745	−1.03790	−0.518949	+	0.854805i	$0.673677\pi$
$594$	0	0
$595$	−5.23304	−0.214534
$596$	14.1725	0.580529
$597$	0	0
$598$	0.101398	0.00414648
$599$	28.3772	1.15946	0.579731	−	0.814808i	$-0.303158\pi$
$599$	28.3772	1.15946	0.579731	+	0.814808i	$0.303158\pi$
$600$	0	0
$601$	22.8258	0.931084	0.465542	−	0.885026i	$-0.345859\pi$
$601$	22.8258	0.931084	0.465542	+	0.885026i	$0.345859\pi$
$602$	−29.1145	−1.18662
$603$	0	0
$604$	−5.31669	−0.216333
$605$	−27.9956	−1.13818
$606$	0	0
$607$	27.1407	1.10161	0.550803	−	0.834635i	$-0.314321\pi$
$607$	27.1407	1.10161	0.550803	+	0.834635i	$0.314321\pi$
$608$	−22.8408	−0.926318
$609$	0	0
$610$	7.41140	0.300079
$611$	−0.720458	−0.0291466
$612$	0	0
$613$	14.9443	0.603595	0.301797	−	0.953372i	$-0.402413\pi$
$613$	14.9443	0.603595	0.301797	+	0.953372i	$0.402413\pi$
$614$	35.7009	1.44077
$615$	0	0
$616$	22.3798	0.901708
$617$	−6.00090	−0.241587	−0.120794	−	0.992678i	$-0.538544\pi$
$617$	−6.00090	−0.241587	−0.120794	+	0.992678i	$0.538544\pi$
$618$	0	0
$619$	−28.7394	−1.15513	−0.577567	−	0.816343i	$-0.695998\pi$
$619$	−28.7394	−1.15513	−0.577567	+	0.816343i	$0.695998\pi$
$620$	−18.1453	−0.728733
$621$	0	0
$622$	4.47078	0.179262
$623$	−14.8181	−0.593673
$624$	0	0
$625$	−23.9050	−0.956199
$626$	1.11319	0.0444921
$627$	0	0
$628$	−2.94309	−0.117442
$629$	7.87831	0.314129
$630$	0	0
$631$	−24.2406	−0.965003	−0.482501	−	0.875895i	$-0.660272\pi$
$631$	−24.2406	−0.965003	−0.482501	+	0.875895i	$0.660272\pi$
$632$	−14.1500	−0.562856
$633$	0	0
$634$	−40.8649	−1.62295
$635$	1.39404	0.0553208
$636$	0	0
$637$	0.0780505	0.00309247
$638$	69.6429	2.75719
$639$	0	0
$640$	−42.2405	−1.66970
$641$	−47.5182	−1.87686	−0.938428	−	0.345476i	$-0.887717\pi$
$641$	−47.5182	−1.87686	−0.938428	+	0.345476i	$0.887717\pi$
$642$	0	0
$643$	−24.9865	−0.985370	−0.492685	−	0.870208i	$-0.663984\pi$
$643$	−24.9865	−0.985370	−0.492685	+	0.870208i	$0.663984\pi$
$644$	1.24422	0.0490291
$645$	0	0
$646$	−5.94729	−0.233993
$647$	−31.5472	−1.24025	−0.620125	−	0.784503i	$-0.712918\pi$
$647$	−31.5472	−1.24025	−0.620125	+	0.784503i	$0.712918\pi$
$648$	0	0
$649$	7.01117	0.275212
$650$	−0.831286	−0.0326057
$651$	0	0
$652$	11.2994	0.442518
$653$	7.47349	0.292460	0.146230	−	0.989251i	$-0.453286\pi$
$653$	7.47349	0.292460	0.146230	+	0.989251i	$0.453286\pi$
$654$	0	0
$655$	−38.2468	−1.49443
$656$	−34.2223	−1.33616
$657$	0	0
$658$	−31.2974	−1.22010
$659$	−20.6225	−0.803340	−0.401670	−	0.915785i	$-0.631570\pi$
$659$	−20.6225	−0.803340	−0.401670	+	0.915785i	$0.631570\pi$
$660$	0	0
$661$	38.4744	1.49648	0.748240	−	0.663428i	$-0.230899\pi$
$661$	38.4744	1.49648	0.748240	+	0.663428i	$0.230899\pi$
$662$	−47.2988	−1.83832
$663$	0	0
$664$	−0.0978286	−0.00379649
$665$	42.9757	1.66653
$666$	0	0
$667$	−5.96357	−0.230910
$668$	0.264903	0.0102494
$669$	0	0
$670$	0.761661	0.0294255
$671$	−6.17588	−0.238417
$672$	0	0
$673$	23.5232	0.906754	0.453377	−	0.891319i	$-0.350219\pi$
$673$	23.5232	0.906754	0.453377	+	0.891319i	$0.350219\pi$
$674$	47.2451	1.81981
$675$	0	0
$676$	−10.2281	−0.393387
$677$	−8.45475	−0.324942	−0.162471	−	0.986713i	$-0.551946\pi$
$677$	−8.45475	−0.324942	−0.162471	+	0.986713i	$0.551946\pi$
$678$	0	0
$679$	27.6948	1.06283
$680$	−4.26072	−0.163391
$681$	0	0
$682$	53.5298	2.04976
$683$	−21.6625	−0.828894	−0.414447	−	0.910074i	$-0.636025\pi$
$683$	−21.6625	−0.828894	−0.414447	+	0.910074i	$0.636025\pi$
$684$	0	0
$685$	−59.2430	−2.26356
$686$	32.4510	1.23898
$687$	0	0
$688$	−34.7480	−1.32475
$689$	−0.0326136	−0.00124248
$690$	0	0
$691$	2.61127	0.0993375	0.0496688	−	0.998766i	$-0.484183\pi$
$691$	2.61127	0.0993375	0.0496688	+	0.998766i	$0.484183\pi$
$692$	−8.93308	−0.339585
$693$	0	0
$694$	60.4294	2.29387
$695$	−0.0303720	−0.00115207
$696$	0	0
$697$	−4.54898	−0.172305
$698$	59.7848	2.26289
$699$	0	0
$700$	−10.2004	−0.385538
$701$	34.6693	1.30944	0.654721	−	0.755870i	$-0.272786\pi$
$701$	34.6693	1.30944	0.654721	+	0.755870i	$0.272786\pi$
$702$	0	0
$703$	−64.6996	−2.44019
$704$	12.7103	0.479037
$705$	0	0
$706$	−46.0939	−1.73477
$707$	1.49252	0.0561319
$708$	0	0
$709$	27.0398	1.01550	0.507750	−	0.861504i	$-0.330477\pi$
$709$	27.0398	1.01550	0.507750	+	0.861504i	$0.330477\pi$
$710$	17.5249	0.657696
$711$	0	0
$712$	−12.0648	−0.452147
$713$	−4.58380	−0.171664
$714$	0	0
$715$	1.35747	0.0507664
$716$	3.45148	0.128988
$717$	0	0
$718$	27.7154	1.03433
$719$	−30.6086	−1.14151	−0.570753	−	0.821122i	$-0.693349\pi$
$719$	−30.6086	−1.14151	−0.570753	+	0.821122i	$0.693349\pi$
$720$	0	0
$721$	−0.561231	−0.0209013
$722$	17.1203	0.637150
$723$	0	0
$724$	−7.01998	−0.260896
$725$	48.8907	1.81575
$726$	0	0
$727$	−45.7934	−1.69838	−0.849192	−	0.528085i	$-0.822910\pi$
$727$	−45.7934	−1.69838	−0.849192	+	0.528085i	$0.822910\pi$
$728$	−0.481118	−0.0178314
$729$	0	0
$730$	−2.14535	−0.0794030
$731$	−4.61885	−0.170834
$732$	0	0
$733$	−23.1892	−0.856511	−0.428256	−	0.903658i	$-0.640872\pi$
$733$	−23.1892	−0.856511	−0.428256	+	0.903658i	$0.640872\pi$
$734$	36.4762	1.34636
$735$	0	0
$736$	2.68378	0.0989256
$737$	−0.634688	−0.0233790
$738$	0	0
$739$	4.00843	0.147452	0.0737262	−	0.997279i	$-0.476511\pi$
$739$	4.00843	0.147452	0.0737262	+	0.997279i	$0.476511\pi$
$740$	30.0937	1.10627
$741$	0	0
$742$	−1.41676	−0.0520110
$743$	−35.7259	−1.31065	−0.655327	−	0.755345i	$-0.727469\pi$
$743$	−35.7259	−1.31065	−0.655327	+	0.755345i	$0.727469\pi$
$744$	0	0
$745$	−57.5187	−2.10732
$746$	−43.8041	−1.60378
$747$	0	0
$748$	−2.30511	−0.0842832
$749$	−21.8114	−0.796970
$750$	0	0
$751$	−31.8260	−1.16135	−0.580674	−	0.814136i	$-0.697211\pi$
$751$	−31.8260	−1.16135	−0.580674	+	0.814136i	$0.697211\pi$
$752$	−37.3532	−1.36213
$753$	0	0
$754$	−1.49717	−0.0545239
$755$	21.5776	0.785288
$756$	0	0
$757$	−36.5025	−1.32671	−0.663353	−	0.748306i	$-0.730867\pi$
$757$	−36.5025	−1.32671	−0.663353	+	0.748306i	$0.730867\pi$
$758$	−39.8576	−1.44769
$759$	0	0
$760$	34.9906	1.26924
$761$	47.8246	1.73364	0.866819	−	0.498622i	$-0.166161\pi$
$761$	47.8246	1.73364	0.866819	+	0.498622i	$0.166161\pi$
$762$	0	0
$763$	−11.7461	−0.425238
$764$	7.55922	0.273483
$765$	0	0
$766$	2.56978	0.0928499
$767$	−0.150725	−0.00544237
$768$	0	0
$769$	23.5915	0.850732	0.425366	−	0.905021i	$-0.360145\pi$
$769$	23.5915	0.850732	0.425366	+	0.905021i	$0.360145\pi$
$770$	58.9697	2.12512
$771$	0	0
$772$	−15.8672	−0.571074
$773$	−22.1221	−0.795675	−0.397838	−	0.917456i	$-0.630239\pi$
$773$	−22.1221	−0.795675	−0.397838	+	0.917456i	$0.630239\pi$
$774$	0	0
$775$	37.5790	1.34988
$776$	22.5489	0.809459
$777$	0	0
$778$	34.6374	1.24181
$779$	37.3579	1.33849
$780$	0	0
$781$	−14.6034	−0.522549
$782$	0.698802	0.0249891
$783$	0	0
$784$	4.04664	0.144523
$785$	11.9444	0.426315
$786$	0	0
$787$	40.1991	1.43294	0.716472	−	0.697615i	$-0.245756\pi$
$787$	40.1991	1.43294	0.716472	+	0.697615i	$0.245756\pi$
$788$	−16.9110	−0.602429
$789$	0	0
$790$	−37.2845	−1.32652
$791$	29.1869	1.03777
$792$	0	0
$793$	0.132768	0.00471474
$794$	0.468374	0.0166220
$795$	0	0
$796$	0.0420247	0.00148953
$797$	18.4680	0.654169	0.327084	−	0.944995i	$-0.393934\pi$
$797$	18.4680	0.654169	0.327084	+	0.944995i	$0.393934\pi$
$798$	0	0
$799$	−4.96516	−0.175655
$800$	−22.0023	−0.777897
$801$	0	0
$802$	25.0370	0.884089
$803$	1.78771	0.0630869
$804$	0	0
$805$	−5.04962	−0.177976
$806$	−1.15078	−0.0405344
$807$	0	0
$808$	1.21520	0.0427506
$809$	21.2475	0.747021	0.373510	−	0.927626i	$-0.378154\pi$
$809$	21.2475	0.747021	0.373510	+	0.927626i	$0.378154\pi$
$810$	0	0
$811$	50.7012	1.78036	0.890180	−	0.455609i	$-0.150579\pi$
$811$	50.7012	1.78036	0.890180	+	0.455609i	$0.150579\pi$
$812$	−18.3712	−0.644704
$813$	0	0
$814$	−88.7784	−3.11168
$815$	−45.8582	−1.60634
$816$	0	0
$817$	37.9317	1.32706
$818$	−4.41316	−0.154302
$819$	0	0
$820$	−17.3763	−0.606807
$821$	27.4463	0.957883	0.478941	−	0.877847i	$-0.341021\pi$
$821$	27.4463	0.957883	0.478941	+	0.877847i	$0.341021\pi$
$822$	0	0
$823$	26.9936	0.940938	0.470469	−	0.882416i	$-0.344085\pi$
$823$	26.9936	0.940938	0.470469	+	0.882416i	$0.344085\pi$
$824$	−0.456952	−0.0159187
$825$	0	0
$826$	−6.54765	−0.227822
$827$	−15.3186	−0.532682	−0.266341	−	0.963879i	$-0.585815\pi$
$827$	−15.3186	−0.532682	−0.266341	+	0.963879i	$0.585815\pi$
$828$	0	0
$829$	−39.1591	−1.36005	−0.680026	−	0.733188i	$-0.738031\pi$
$829$	−39.1591	−1.36005	−0.680026	+	0.733188i	$0.738031\pi$
$830$	−0.257774	−0.00894745
$831$	0	0
$832$	−0.273244	−0.00947304
$833$	0.537897	0.0186370
$834$	0	0
$835$	−1.07510	−0.0372054
$836$	18.9304	0.654722
$837$	0	0
$838$	−20.3370	−0.702530
$839$	27.9558	0.965143	0.482571	−	0.875857i	$-0.339703\pi$
$839$	27.9558	0.965143	0.482571	+	0.875857i	$0.339703\pi$
$840$	0	0
$841$	59.0538	2.03634
$842$	10.3963	0.358281
$843$	0	0
$844$	−2.58952	−0.0891351
$845$	41.5102	1.42799
$846$	0	0
$847$	−21.7863	−0.748585
$848$	−1.69090	−0.0580657
$849$	0	0
$850$	−5.72894	−0.196501
$851$	7.60216	0.260599
$852$	0	0
$853$	−27.0275	−0.925404	−0.462702	−	0.886514i	$-0.653120\pi$
$853$	−27.0275	−0.925404	−0.462702	+	0.886514i	$0.653120\pi$
$854$	5.76758	0.197362
$855$	0	0
$856$	−17.7587	−0.606980
$857$	−44.6176	−1.52411	−0.762054	−	0.647514i	$-0.775809\pi$
$857$	−44.6176	−1.52411	−0.762054	+	0.647514i	$0.775809\pi$
$858$	0	0
$859$	−26.4804	−0.903499	−0.451749	−	0.892145i	$-0.649200\pi$
$859$	−26.4804	−0.903499	−0.451749	+	0.892145i	$0.649200\pi$
$860$	−17.6432	−0.601628
$861$	0	0
$862$	−37.5232	−1.27804
$863$	−13.0078	−0.442791	−0.221396	−	0.975184i	$-0.571061\pi$
$863$	−13.0078	−0.442791	−0.221396	+	0.975184i	$0.571061\pi$
$864$	0	0
$865$	36.2546	1.23269
$866$	18.4094	0.625575
$867$	0	0
$868$	−14.1207	−0.479289
$869$	31.0690	1.05394
$870$	0	0
$871$	0.0136444	0.000462324 0
$872$	−9.56362	−0.323865
$873$	0	0
$874$	−5.73882	−0.194119
$875$	1.66994	0.0564543
$876$	0	0
$877$	42.0717	1.42066	0.710330	−	0.703869i	$-0.248546\pi$
$877$	42.0717	1.42066	0.710330	+	0.703869i	$0.248546\pi$
$878$	62.9411	2.12416
$879$	0	0
$880$	70.3799	2.37251
$881$	−35.8562	−1.20802	−0.604012	−	0.796975i	$-0.706432\pi$
$881$	−35.8562	−1.20802	−0.604012	+	0.796975i	$0.706432\pi$
$882$	0	0
$883$	−7.97760	−0.268468	−0.134234	−	0.990950i	$-0.542857\pi$
$883$	−7.97760	−0.268468	−0.134234	+	0.990950i	$0.542857\pi$
$884$	0.0495550	0.00166671
$885$	0	0
$886$	20.2932	0.681762
$887$	−14.7497	−0.495246	−0.247623	−	0.968856i	$-0.579649\pi$
$887$	−14.7497	−0.495246	−0.247623	+	0.968856i	$0.579649\pi$
$888$	0	0
$889$	1.08485	0.0363846
$890$	−31.7901	−1.06561
$891$	0	0
$892$	−3.40324	−0.113949
$893$	40.7757	1.36451
$894$	0	0
$895$	−14.0077	−0.468226
$896$	−32.8717	−1.09817
$897$	0	0
$898$	−7.51265	−0.250700
$899$	67.6811	2.25729
$900$	0	0
$901$	−0.224762	−0.00748789
$902$	51.2612	1.70681
$903$	0	0
$904$	23.7638	0.790373
$905$	28.4903	0.947051
$906$	0	0
$907$	24.0644	0.799045	0.399522	−	0.916723i	$-0.369176\pi$
$907$	24.0644	0.799045	0.399522	+	0.916723i	$0.369176\pi$
$908$	17.5819	0.583476
$909$	0	0
$910$	−1.26772	−0.0420246
$911$	−12.7089	−0.421065	−0.210532	−	0.977587i	$-0.567520\pi$
$911$	−12.7089	−0.421065	−0.210532	+	0.977587i	$0.567520\pi$
$912$	0	0
$913$	0.214801	0.00710888
$914$	5.43375	0.179732
$915$	0	0
$916$	15.4739	0.511271
$917$	−29.7638	−0.982888
$918$	0	0
$919$	−14.4294	−0.475981	−0.237991	−	0.971267i	$-0.576489\pi$
$919$	−14.4294	−0.475981	−0.237991	+	0.971267i	$0.576489\pi$
$920$	−4.11137	−0.135548
$921$	0	0
$922$	−63.5441	−2.09271
$923$	0.313941	0.0103335
$924$	0	0
$925$	−62.3242	−2.04921
$926$	−15.2963	−0.502666
$927$	0	0
$928$	−39.6268	−1.30081
$929$	43.1956	1.41720	0.708601	−	0.705609i	$-0.249326\pi$
$929$	43.1956	1.41720	0.708601	+	0.705609i	$0.249326\pi$
$930$	0	0
$931$	−4.41741	−0.144775
$932$	2.70627	0.0886468
$933$	0	0
$934$	7.33354	0.239961
$935$	9.35521	0.305948
$936$	0	0
$937$	41.4858	1.35528	0.677642	−	0.735392i	$-0.263002\pi$
$937$	41.4858	1.35528	0.677642	+	0.735392i	$0.263002\pi$
$938$	0.592727	0.0193532
$939$	0	0
$940$	−18.9660	−0.618603
$941$	−40.5176	−1.32084	−0.660418	−	0.750898i	$-0.729621\pi$
$941$	−40.5176	−1.32084	−0.660418	+	0.750898i	$0.729621\pi$
$942$	0	0
$943$	−4.38953	−0.142943
$944$	−7.81457	−0.254343
$945$	0	0
$946$	52.0486	1.69224
$947$	−40.6797	−1.32191	−0.660956	−	0.750424i	$-0.729849\pi$
$947$	−40.6797	−1.32191	−0.660956	+	0.750424i	$0.729849\pi$
$948$	0	0
$949$	−0.0384319	−0.00124755
$950$	47.0482	1.52644
$951$	0	0
$952$	−3.31570	−0.107463
$953$	28.0671	0.909183	0.454591	−	0.890700i	$-0.349785\pi$
$953$	28.0671	0.909183	0.454591	+	0.890700i	$0.349785\pi$
$954$	0	0
$955$	−30.6788	−0.992743
$956$	0.787326	0.0254640
$957$	0	0
$958$	40.7209	1.31563
$959$	−46.1031	−1.48875
$960$	0	0
$961$	21.0219	0.678125
$962$	1.90855	0.0615340
$963$	0	0
$964$	6.47563	0.208566
$965$	64.3966	2.07300
$966$	0	0
$967$	37.2824	1.19892	0.599461	−	0.800404i	$-0.295382\pi$
$967$	37.2824	1.19892	0.599461	+	0.800404i	$0.295382\pi$
$968$	−17.7383	−0.570130
$969$	0	0
$970$	59.4153	1.90771
$971$	−27.9856	−0.898100	−0.449050	−	0.893507i	$-0.648238\pi$
$971$	−27.9856	−0.898100	−0.449050	+	0.893507i	$0.648238\pi$
$972$	0	0
$973$	−0.0236356	−0.000757722 0
$974$	−56.4093	−1.80747
$975$	0	0
$976$	6.88357	0.220338
$977$	21.6395	0.692309	0.346154	−	0.938178i	$-0.387487\pi$
$977$	21.6395	0.692309	0.346154	+	0.938178i	$0.387487\pi$
$978$	0	0
$979$	26.4905	0.846641
$980$	2.05467	0.0656341
$981$	0	0
$982$	11.0308	0.352007
$983$	36.9976	1.18004	0.590021	−	0.807388i	$-0.299120\pi$
$983$	36.9976	1.18004	0.590021	+	0.807388i	$0.299120\pi$
$984$	0	0
$985$	68.6327	2.18682
$986$	−10.3180	−0.328593
$987$	0	0
$988$	−0.406964	−0.0129472
$989$	−4.45695	−0.141723
$990$	0	0
$991$	−32.9780	−1.04758	−0.523791	−	0.851847i	$-0.675483\pi$
$991$	−32.9780	−1.04758	−0.523791	+	0.851847i	$0.675483\pi$
$992$	−30.4585	−0.967058
$993$	0	0
$994$	13.6379	0.432568
$995$	−0.170556	−0.00540698
$996$	0	0
$997$	−6.23842	−0.197573	−0.0987863	−	0.995109i	$-0.531496\pi$
$997$	−6.23842	−0.197573	−0.0987863	+	0.995109i	$0.531496\pi$
$998$	−12.4452	−0.393945
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.a.k.1.16	yes	20
3.2	odd	2		2151.2.a.j.1.5	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.2.a.j.1.5	✓	20	3.2	odd	2
2151.2.a.k.1.16	yes	20	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q+1.66953 q^{2} +0.787326 q^{4} -3.19534 q^{5} -2.48662 q^{7} -2.02459 q^{8} +O(q^{10})\)	\(q+1.66953 q^{2} +0.787326 q^{4} -3.19534 q^{5} -2.48662 q^{7} -2.02459 q^{8} -5.33471 q^{10} +4.44538 q^{11} -0.0955662 q^{13} -4.15149 q^{14} -4.95477 q^{16} -0.658610 q^{17} +5.40875 q^{19} -2.51577 q^{20} +7.42169 q^{22} -0.635524 q^{23} +5.21017 q^{25} -0.159551 q^{26} -1.95778 q^{28} +9.38370 q^{29} +7.21262 q^{31} -4.22294 q^{32} -1.09957 q^{34} +7.94559 q^{35} -11.9620 q^{37} +9.03006 q^{38} +6.46926 q^{40} +6.90694 q^{41} +7.01304 q^{43} +3.49996 q^{44} -1.06103 q^{46} +7.53884 q^{47} -0.816716 q^{49} +8.69853 q^{50} -0.0752418 q^{52} +0.341267 q^{53} -14.2045 q^{55} +5.03440 q^{56} +15.6664 q^{58} +1.57718 q^{59} -1.38928 q^{61} +12.0417 q^{62} +2.85921 q^{64} +0.305366 q^{65} -0.142775 q^{67} -0.518541 q^{68} +13.2654 q^{70} -3.28506 q^{71} +0.402150 q^{73} -19.9709 q^{74} +4.25845 q^{76} -11.0540 q^{77} +6.98905 q^{79} +15.8322 q^{80} +11.5313 q^{82} +0.0483201 q^{83} +2.10448 q^{85} +11.7085 q^{86} -9.00009 q^{88} +5.95912 q^{89} +0.237637 q^{91} -0.500365 q^{92} +12.5863 q^{94} -17.2828 q^{95} -11.1375 q^{97} -1.36353 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 4 q^{2} + 18 q^{4} + 16 q^{5} - 4 q^{7} + 12 q^{8}+O(q^{10}) \) 20 * q + 4 * q^2 + 18 * q^4 + 16 * q^5 - 4 * q^7 + 12 * q^8	\( 20 q + 4 q^{2} + 18 q^{4} + 16 q^{5} - 4 q^{7} + 12 q^{8} + 4 q^{10} + 12 q^{11} - 4 q^{13} + 20 q^{14} + 22 q^{16} + 24 q^{17} - 4 q^{19} + 40 q^{20} - 6 q^{22} + 12 q^{23} + 22 q^{25} + 30 q^{26} - 12 q^{28} + 24 q^{29} - 4 q^{31} + 28 q^{32} + 8 q^{34} + 20 q^{35} - 10 q^{37} + 26 q^{38} + 6 q^{40} + 66 q^{41} + 8 q^{43} + 36 q^{44} - 12 q^{46} + 28 q^{47} + 18 q^{49} + 28 q^{50} - 18 q^{52} + 28 q^{53} - 4 q^{55} + 60 q^{56} + 54 q^{59} - 4 q^{61} + 20 q^{62} + 22 q^{64} + 42 q^{65} + 12 q^{67} + 12 q^{68} + 20 q^{70} + 36 q^{71} + 14 q^{73} - 50 q^{76} + 8 q^{77} - 12 q^{79} + 88 q^{80} - 8 q^{82} + 20 q^{83} + 4 q^{85} + 18 q^{86} - 10 q^{88} + 130 q^{89} - 6 q^{91} - 46 q^{92} - 26 q^{94} - 2 q^{97} + 12 q^{98}+O(q^{100}) \) 20 * q + 4 * q^2 + 18 * q^4 + 16 * q^5 - 4 * q^7 + 12 * q^8 + 4 * q^10 + 12 * q^11 - 4 * q^13 + 20 * q^14 + 22 * q^16 + 24 * q^17 - 4 * q^19 + 40 * q^20 - 6 * q^22 + 12 * q^23 + 22 * q^25 + 30 * q^26 - 12 * q^28 + 24 * q^29 - 4 * q^31 + 28 * q^32 + 8 * q^34 + 20 * q^35 - 10 * q^37 + 26 * q^38 + 6 * q^40 + 66 * q^41 + 8 * q^43 + 36 * q^44 - 12 * q^46 + 28 * q^47 + 18 * q^49 + 28 * q^50 - 18 * q^52 + 28 * q^53 - 4 * q^55 + 60 * q^56 + 54 * q^59 - 4 * q^61 + 20 * q^62 + 22 * q^64 + 42 * q^65 + 12 * q^67 + 12 * q^68 + 20 * q^70 + 36 * q^71 + 14 * q^73 - 50 * q^76 + 8 * q^77 - 12 * q^79 + 88 * q^80 - 8 * q^82 + 20 * q^83 + 4 * q^85 + 18 * q^86 - 10 * q^88 + 130 * q^89 - 6 * q^91 - 46 * q^92 - 26 * q^94 - 2 * q^97 + 12 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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