LMFDB - Embedded newform 2151.2.a.j.1.14

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:	$1$
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			1.14
Root			$-0.921026$ 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)

$f(q)$	$=$	$q+0.921026 q^{2} -1.15171 q^{4} -1.07676 q^{5} +3.99910 q^{7} -2.90281 q^{8} +O(q^{10})$	\(q+0.921026 q^{2} -1.15171 q^{4} -1.07676 q^{5} +3.99910 q^{7} -2.90281 q^{8} -0.991725 q^{10} -3.85807 q^{11} +1.10344 q^{13} +3.68328 q^{14} -0.370136 q^{16} -4.98207 q^{17} +6.77486 q^{19} +1.24012 q^{20} -3.55338 q^{22} -4.48583 q^{23} -3.84058 q^{25} +1.01629 q^{26} -4.60582 q^{28} +0.455789 q^{29} -0.141534 q^{31} +5.46471 q^{32} -4.58862 q^{34} -4.30608 q^{35} -5.62791 q^{37} +6.23982 q^{38} +3.12563 q^{40} -3.87964 q^{41} -5.21280 q^{43} +4.44339 q^{44} -4.13156 q^{46} +0.0217628 q^{47} +8.99284 q^{49} -3.53728 q^{50} -1.27084 q^{52} +8.05621 q^{53} +4.15422 q^{55} -11.6086 q^{56} +0.419793 q^{58} -14.3144 q^{59} -13.6860 q^{61} -0.130356 q^{62} +5.77341 q^{64} -1.18814 q^{65} -8.40462 q^{67} +5.73791 q^{68} -3.96601 q^{70} -1.78926 q^{71} +8.35852 q^{73} -5.18345 q^{74} -7.80269 q^{76} -15.4288 q^{77} -16.7789 q^{79} +0.398548 q^{80} -3.57325 q^{82} +3.41314 q^{83} +5.36450 q^{85} -4.80112 q^{86} +11.1992 q^{88} +7.49447 q^{89} +4.41276 q^{91} +5.16638 q^{92} +0.0200441 q^{94} -7.29491 q^{95} -9.46397 q^{97} +8.28263 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.921026	0.651263	0.325632	−	0.945497i	$-0.394423\pi$
$2$	0.921026	0.651263	0.325632	+	0.945497i	$0.394423\pi$
$3$	0	0
$3$	0	0
$4$	−1.15171	−0.575856
$5$	−1.07676	−0.481542	−0.240771	−	0.970582i	$-0.577400\pi$
$5$	−1.07676	−0.481542	−0.240771	+	0.970582i	$0.577400\pi$
$6$	0	0
$7$	3.99910	1.51152	0.755760	−	0.654849i	$-0.227268\pi$
$7$	3.99910	1.51152	0.755760	+	0.654849i	$0.227268\pi$
$8$	−2.90281	−1.02630
$9$	0	0
$10$	−0.991725	−0.313611
$11$	−3.85807	−1.16325	−0.581626	−	0.813456i	$-0.697583\pi$
$11$	−3.85807	−1.16325	−0.581626	+	0.813456i	$0.697583\pi$
$12$	0	0
$13$	1.10344	0.306038	0.153019	−	0.988223i	$-0.451100\pi$
$13$	1.10344	0.306038	0.153019	+	0.988223i	$0.451100\pi$
$14$	3.68328	0.984397
$15$	0	0
$16$	−0.370136	−0.0925340
$17$	−4.98207	−1.20833	−0.604165	−	0.796859i	$-0.706493\pi$
$17$	−4.98207	−1.20833	−0.604165	+	0.796859i	$0.706493\pi$
$18$	0	0
$19$	6.77486	1.55426	0.777130	−	0.629340i	$-0.216675\pi$
$19$	6.77486	1.55426	0.777130	+	0.629340i	$0.216675\pi$
$20$	1.24012	0.277299
$21$	0	0
$22$	−3.55338	−0.757584
$23$	−4.48583	−0.935359	−0.467680	−	0.883898i	$-0.654910\pi$
$23$	−4.48583	−0.935359	−0.467680	+	0.883898i	$0.654910\pi$
$24$	0	0
$25$	−3.84058	−0.768117
$26$	1.01629	0.199311
$27$	0	0
$28$	−4.60582	−0.870417
$29$	0.455789	0.0846379	0.0423189	−	0.999104i	$-0.486525\pi$
$29$	0.455789	0.0846379	0.0423189	+	0.999104i	$0.486525\pi$
$30$	0	0
$31$	−0.141534	−0.0254202	−0.0127101	−	0.999919i	$-0.504046\pi$
$31$	−0.141534	−0.0254202	−0.0127101	+	0.999919i	$0.504046\pi$
$32$	5.46471	0.966033
$33$	0	0
$34$	−4.58862	−0.786941
$35$	−4.30608	−0.727861
$36$	0	0
$37$	−5.62791	−0.925224	−0.462612	−	0.886561i	$-0.653088\pi$
$37$	−5.62791	−0.925224	−0.462612	+	0.886561i	$0.653088\pi$
$38$	6.23982	1.01223
$39$	0	0
$40$	3.12563	0.494206
$41$	−3.87964	−0.605898	−0.302949	−	0.953007i	$-0.597971\pi$
$41$	−3.87964	−0.605898	−0.302949	+	0.953007i	$0.597971\pi$
$42$	0	0
$43$	−5.21280	−0.794945	−0.397472	−	0.917614i	$-0.630113\pi$
$43$	−5.21280	−0.794945	−0.397472	+	0.917614i	$0.630113\pi$
$44$	4.44339	0.669866
$45$	0	0
$46$	−4.13156	−0.609165
$47$	0.0217628	0.00317443	0.00158722	−	0.999999i	$-0.499495\pi$
$47$	0.0217628	0.00317443	0.00158722	+	0.999999i	$0.499495\pi$
$48$	0	0
$49$	8.99284	1.28469
$50$	−3.53728	−0.500246
$51$	0	0
$52$	−1.27084	−0.176234
$53$	8.05621	1.10661	0.553303	−	0.832980i	$-0.313367\pi$
$53$	8.05621	1.10661	0.553303	+	0.832980i	$0.313367\pi$
$54$	0	0
$55$	4.15422	0.560155
$56$	−11.6086	−1.55127
$57$	0	0
$58$	0.419793	0.0551215
$59$	−14.3144	−1.86358	−0.931790	−	0.362998i	$-0.881753\pi$
$59$	−14.3144	−1.86358	−0.931790	+	0.362998i	$0.881753\pi$
$60$	0	0
$61$	−13.6860	−1.75232	−0.876159	−	0.482023i	$-0.839902\pi$
$61$	−13.6860	−1.75232	−0.876159	+	0.482023i	$0.839902\pi$
$62$	−0.130356	−0.0165553
$63$	0	0
$64$	5.77341	0.721676
$65$	−1.18814	−0.147370
$66$	0	0
$67$	−8.40462	−1.02679	−0.513394	−	0.858153i	$-0.671612\pi$
$67$	−8.40462	−1.02679	−0.513394	+	0.858153i	$0.671612\pi$
$68$	5.73791	0.695824
$69$	0	0
$70$	−3.96601	−0.474029
$71$	−1.78926	−0.212346	−0.106173	−	0.994348i	$-0.533860\pi$
$71$	−1.78926	−0.212346	−0.106173	+	0.994348i	$0.533860\pi$
$72$	0	0
$73$	8.35852	0.978291	0.489146	−	0.872202i	$-0.337309\pi$
$73$	8.35852	0.978291	0.489146	+	0.872202i	$0.337309\pi$
$74$	−5.18345	−0.602564
$75$	0	0
$76$	−7.80269	−0.895030
$77$	−15.4288	−1.75828
$78$	0	0
$79$	−16.7789	−1.88778	−0.943889	−	0.330264i	$-0.892862\pi$
$79$	−16.7789	−1.88778	−0.943889	+	0.330264i	$0.892862\pi$
$80$	0.398548	0.0445591
$81$	0	0
$82$	−3.57325	−0.394599
$83$	3.41314	0.374641	0.187320	−	0.982299i	$-0.440020\pi$
$83$	3.41314	0.374641	0.187320	+	0.982299i	$0.440020\pi$
$84$	0	0
$85$	5.36450	0.581862
$86$	−4.80112	−0.517719
$87$	0	0
$88$	11.1992	1.19384
$89$	7.49447	0.794413	0.397206	−	0.917729i	$-0.369980\pi$
$89$	7.49447	0.794413	0.397206	+	0.917729i	$0.369980\pi$
$90$	0	0
$91$	4.41276	0.462583
$92$	5.16638	0.538632
$93$	0	0
$94$	0.0200441	0.00206739
$95$	−7.29491	−0.748442
$96$	0	0
$97$	−9.46397	−0.960920	−0.480460	−	0.877017i	$-0.659530\pi$
$97$	−9.46397	−0.960920	−0.480460	+	0.877017i	$0.659530\pi$
$98$	8.28263	0.836672
$99$	0	0
$100$	4.42325	0.442325
$101$	16.8805	1.67968	0.839838	−	0.542837i	$-0.182650\pi$
$101$	16.8805	1.67968	0.839838	+	0.542837i	$0.182650\pi$
$102$	0	0
$103$	−0.103493	−0.0101975	−0.00509875	−	0.999987i	$-0.501623\pi$
$103$	−0.103493	−0.0101975	−0.00509875	+	0.999987i	$0.501623\pi$
$104$	−3.20306	−0.314086
$105$	0	0
$106$	7.41997	0.720691
$107$	−18.4298	−1.78167	−0.890837	−	0.454323i	$-0.849881\pi$
$107$	−18.4298	−1.78167	−0.890837	+	0.454323i	$0.849881\pi$
$108$	0	0
$109$	−3.54270	−0.339329	−0.169664	−	0.985502i	$-0.554268\pi$
$109$	−3.54270	−0.339329	−0.169664	+	0.985502i	$0.554268\pi$
$110$	3.82615	0.364809
$111$	0	0
$112$	−1.48021	−0.139867
$113$	−13.3935	−1.25995	−0.629976	−	0.776614i	$-0.716935\pi$
$113$	−13.3935	−1.25995	−0.629976	+	0.776614i	$0.716935\pi$
$114$	0	0
$115$	4.83016	0.450415
$116$	−0.524937	−0.0487392
$117$	0	0
$118$	−13.1840	−1.21368
$119$	−19.9238	−1.82641
$120$	0	0
$121$	3.88472	0.353157
$122$	−12.6052	−1.14122
$123$	0	0
$124$	0.163006	0.0146384
$125$	9.51920	0.851423
$126$	0	0
$127$	−2.82374	−0.250566	−0.125283	−	0.992121i	$-0.539984\pi$
$127$	−2.82374	−0.250566	−0.125283	+	0.992121i	$0.539984\pi$
$128$	−5.61196	−0.496032
$129$	0	0
$130$	−1.09431	−0.0959769
$131$	7.18818	0.628034	0.314017	−	0.949417i	$-0.398325\pi$
$131$	7.18818	0.628034	0.314017	+	0.949417i	$0.398325\pi$
$132$	0	0
$133$	27.0934	2.34929
$134$	−7.74087	−0.668709
$135$	0	0
$136$	14.4620	1.24011
$137$	9.45738	0.807999	0.403999	−	0.914759i	$-0.367620\pi$
$137$	9.45738	0.807999	0.403999	+	0.914759i	$0.367620\pi$
$138$	0	0
$139$	7.46510	0.633182	0.316591	−	0.948562i	$-0.397462\pi$
$139$	7.46510	0.633182	0.316591	+	0.948562i	$0.397462\pi$
$140$	4.95937	0.419143
$141$	0	0
$142$	−1.64795	−0.138293
$143$	−4.25714	−0.356000
$144$	0	0
$145$	−0.490776	−0.0407567
$146$	7.69841	0.637125
$147$	0	0
$148$	6.48174	0.532795
$149$	−9.92379	−0.812988	−0.406494	−	0.913653i	$-0.633249\pi$
$149$	−9.92379	−0.812988	−0.406494	+	0.913653i	$0.633249\pi$
$150$	0	0
$151$	9.70932	0.790133	0.395067	−	0.918652i	$-0.370722\pi$
$151$	9.70932	0.790133	0.395067	+	0.918652i	$0.370722\pi$
$152$	−19.6661	−1.59513
$153$	0	0
$154$	−14.2104	−1.14510
$155$	0.152398	0.0122409
$156$	0	0
$157$	−16.4632	−1.31391	−0.656955	−	0.753930i	$-0.728156\pi$
$157$	−16.4632	−1.31391	−0.656955	+	0.753930i	$0.728156\pi$
$158$	−15.4538	−1.22944
$159$	0	0
$160$	−5.88419	−0.465186
$161$	−17.9393	−1.41381
$162$	0	0
$163$	20.4963	1.60540	0.802698	−	0.596386i	$-0.203397\pi$
$163$	20.4963	1.60540	0.802698	+	0.596386i	$0.203397\pi$
$164$	4.46823	0.348910
$165$	0	0
$166$	3.14359	0.243990
$167$	2.79371	0.216184	0.108092	−	0.994141i	$-0.465526\pi$
$167$	2.79371	0.216184	0.108092	+	0.994141i	$0.465526\pi$
$168$	0	0
$169$	−11.7824	−0.906341
$170$	4.94085	0.378946
$171$	0	0
$172$	6.00365	0.457774
$173$	−16.5368	−1.25727	−0.628634	−	0.777701i	$-0.716386\pi$
$173$	−16.5368	−1.25727	−0.628634	+	0.777701i	$0.716386\pi$
$174$	0	0
$175$	−15.3589	−1.16102
$176$	1.42801	0.107640
$177$	0	0
$178$	6.90260	0.517372
$179$	−8.54297	−0.638531	−0.319266	−	0.947665i	$-0.603436\pi$
$179$	−8.54297	−0.638531	−0.319266	+	0.947665i	$0.603436\pi$
$180$	0	0
$181$	0.983405	0.0730959	0.0365480	−	0.999332i	$-0.488364\pi$
$181$	0.983405	0.0730959	0.0365480	+	0.999332i	$0.488364\pi$
$182$	4.06426	0.301263
$183$	0	0
$184$	13.0215	0.959957
$185$	6.05992	0.445534
$186$	0	0
$187$	19.2212	1.40559
$188$	−0.0250645	−0.00182802
$189$	0	0
$190$	−6.71880	−0.487433
$191$	−19.7781	−1.43109	−0.715547	−	0.698564i	$-0.753823\pi$
$191$	−19.7781	−1.43109	−0.715547	+	0.698564i	$0.753823\pi$
$192$	0	0
$193$	21.0252	1.51342	0.756712	−	0.653748i	$-0.226805\pi$
$193$	21.0252	1.51342	0.756712	+	0.653748i	$0.226805\pi$
$194$	−8.71656	−0.625812
$195$	0	0
$196$	−10.3572	−0.739797
$197$	18.7152	1.33340	0.666702	−	0.745325i	$-0.267706\pi$
$197$	18.7152	1.33340	0.666702	+	0.745325i	$0.267706\pi$
$198$	0	0
$199$	−4.36925	−0.309728	−0.154864	−	0.987936i	$-0.549494\pi$
$199$	−4.36925	−0.309728	−0.154864	+	0.987936i	$0.549494\pi$
$200$	11.1485	0.788316
$201$	0	0
$202$	15.5474	1.09391
$203$	1.82275	0.127932
$204$	0	0
$205$	4.17745	0.291766
$206$	−0.0953200	−0.00664126
$207$	0	0
$208$	−0.408422	−0.0283189
$209$	−26.1379	−1.80800
$210$	0	0
$211$	−14.9305	−1.02786	−0.513928	−	0.857833i	$-0.671810\pi$
$211$	−14.9305	−1.02786	−0.513928	+	0.857833i	$0.671810\pi$
$212$	−9.27843	−0.637245
$213$	0	0
$214$	−16.9743	−1.16034
$215$	5.61295	0.382800
$216$	0	0
$217$	−0.566008	−0.0384231
$218$	−3.26291	−0.220992
$219$	0	0
$220$	−4.78447	−0.322569
$221$	−5.49740	−0.369795
$222$	0	0
$223$	−3.01452	−0.201867	−0.100934	−	0.994893i	$-0.532183\pi$
$223$	−3.01452	−0.201867	−0.100934	+	0.994893i	$0.532183\pi$
$224$	21.8539	1.46018
$225$	0	0
$226$	−12.3357	−0.820561
$227$	27.4158	1.81965	0.909825	−	0.414993i	$-0.136216\pi$
$227$	27.4158	1.81965	0.909825	+	0.414993i	$0.136216\pi$
$228$	0	0
$229$	−23.6259	−1.56124	−0.780621	−	0.625005i	$-0.785097\pi$
$229$	−23.6259	−1.56124	−0.780621	+	0.625005i	$0.785097\pi$
$230$	4.44870	0.293339
$231$	0	0
$232$	−1.32307	−0.0868636
$233$	−7.58457	−0.496882	−0.248441	−	0.968647i	$-0.579918\pi$
$233$	−7.58457	−0.496882	−0.248441	+	0.968647i	$0.579918\pi$
$234$	0	0
$235$	−0.0234334	−0.00152862
$236$	16.4861	1.07315
$237$	0	0
$238$	−18.3504	−1.18948
$239$	−1.00000	−0.0646846
$239$	−1.00000	−0.0646846
$240$	0	0
$241$	29.6913	1.91259	0.956293	−	0.292410i	$-0.0944570\pi$
$241$	29.6913	1.91259	0.956293	+	0.292410i	$0.0944570\pi$
$242$	3.57793	0.229998
$243$	0	0
$244$	15.7624	1.00908
$245$	−9.68314	−0.618633
$246$	0	0
$247$	7.47563	0.475663
$248$	0.410845	0.0260887
$249$	0	0
$250$	8.76743	0.554501
$251$	22.5521	1.42347	0.711737	−	0.702446i	$-0.247909\pi$
$251$	22.5521	1.42347	0.711737	+	0.702446i	$0.247909\pi$
$252$	0	0
$253$	17.3066	1.08806
$254$	−2.60074	−0.163185
$255$	0	0
$256$	−16.7156	−1.04472
$257$	14.9158	0.930420	0.465210	−	0.885200i	$-0.345979\pi$
$257$	14.9158	0.930420	0.465210	+	0.885200i	$0.345979\pi$
$258$	0	0
$259$	−22.5066	−1.39849
$260$	1.36839	0.0848641
$261$	0	0
$262$	6.62050	0.409016
$263$	7.93666	0.489395	0.244698	−	0.969599i	$-0.421311\pi$
$263$	7.93666	0.489395	0.244698	+	0.969599i	$0.421311\pi$
$264$	0	0
$265$	−8.67461	−0.532877
$266$	24.9537	1.53001
$267$	0	0
$268$	9.67970	0.591282
$269$	28.7193	1.75105	0.875524	−	0.483174i	$-0.160516\pi$
$269$	28.7193	1.75105	0.875524	+	0.483174i	$0.160516\pi$
$270$	0	0
$271$	−1.47810	−0.0897884	−0.0448942	−	0.998992i	$-0.514295\pi$
$271$	−1.47810	−0.0897884	−0.0448942	+	0.998992i	$0.514295\pi$
$272$	1.84404	0.111812
$273$	0	0
$274$	8.71049	0.526220
$275$	14.8173	0.893514
$276$	0	0
$277$	29.4540	1.76972	0.884859	−	0.465859i	$-0.154255\pi$
$277$	29.4540	1.76972	0.884859	+	0.465859i	$0.154255\pi$
$278$	6.87555	0.412368
$279$	0	0
$280$	12.4997	0.747001
$281$	−16.0239	−0.955904	−0.477952	−	0.878386i	$-0.658621\pi$
$281$	−16.0239	−0.955904	−0.477952	+	0.878386i	$0.658621\pi$
$282$	0	0
$283$	8.00350	0.475759	0.237879	−	0.971295i	$-0.423548\pi$
$283$	8.00350	0.475759	0.237879	+	0.971295i	$0.423548\pi$
$284$	2.06071	0.122281
$285$	0	0
$286$	−3.92093	−0.231850
$287$	−15.5151	−0.915827
$288$	0	0
$289$	7.82105	0.460062
$290$	−0.452017	−0.0265434
$291$	0	0
$292$	−9.62661	−0.563355
$293$	−12.1458	−0.709567	−0.354784	−	0.934948i	$-0.615445\pi$
$293$	−12.1458	−0.709567	−0.354784	+	0.934948i	$0.615445\pi$
$294$	0	0
$295$	15.4132	0.897393
$296$	16.3368	0.949554
$297$	0	0
$298$	−9.14006	−0.529470
$299$	−4.94982	−0.286256
$300$	0	0
$301$	−20.8465	−1.20157
$302$	8.94253	0.514585
$303$	0	0
$304$	−2.50762	−0.143822
$305$	14.7366	0.843815
$306$	0	0
$307$	18.4095	1.05069	0.525343	−	0.850891i	$-0.323937\pi$
$307$	18.4095	1.05069	0.525343	+	0.850891i	$0.323937\pi$
$308$	17.7696	1.01252
$309$	0	0
$310$	0.140363	0.00797206
$311$	33.7729	1.91509	0.957543	−	0.288292i	$-0.0930874\pi$
$311$	33.7729	1.91509	0.957543	+	0.288292i	$0.0930874\pi$
$312$	0	0
$313$	−13.7399	−0.776625	−0.388312	−	0.921528i	$-0.626942\pi$
$313$	−13.7399	−0.776625	−0.388312	+	0.921528i	$0.626942\pi$
$314$	−15.1631	−0.855702
$315$	0	0
$316$	19.3245	1.08709
$317$	4.88391	0.274308	0.137154	−	0.990550i	$-0.456205\pi$
$317$	4.88391	0.274308	0.137154	+	0.990550i	$0.456205\pi$
$318$	0	0
$319$	−1.75847	−0.0984552
$320$	−6.21659	−0.347518
$321$	0	0
$322$	−16.5225	−0.920765
$323$	−33.7528	−1.87806
$324$	0	0
$325$	−4.23784	−0.235073
$326$	18.8776	1.04554
$327$	0	0
$328$	11.2618	0.621832
$329$	0.0870317	0.00479822
$330$	0	0
$331$	6.51734	0.358225	0.179113	−	0.983829i	$-0.442677\pi$
$331$	6.51734	0.358225	0.179113	+	0.983829i	$0.442677\pi$
$332$	−3.93096	−0.215739
$333$	0	0
$334$	2.57308	0.140793
$335$	9.04977	0.494442
$336$	0	0
$337$	2.02175	0.110132	0.0550659	−	0.998483i	$-0.482463\pi$
$337$	2.02175	0.110132	0.0550659	+	0.998483i	$0.482463\pi$
$338$	−10.8519	−0.590267
$339$	0	0
$340$	−6.17836	−0.335069
$341$	0.546048	0.0295701
$342$	0	0
$343$	7.96956	0.430316
$344$	15.1318	0.815850
$345$	0	0
$346$	−15.2308	−0.818813
$347$	−18.7227	−1.00508	−0.502542	−	0.864553i	$-0.667602\pi$
$347$	−18.7227	−1.00508	−0.502542	+	0.864553i	$0.667602\pi$
$348$	0	0
$349$	14.1099	0.755283	0.377642	−	0.925952i	$-0.376735\pi$
$349$	14.1099	0.755283	0.377642	+	0.925952i	$0.376735\pi$
$350$	−14.1459	−0.756132
$351$	0	0
$352$	−21.0832	−1.12374
$353$	20.7618	1.10504	0.552519	−	0.833500i	$-0.313667\pi$
$353$	20.7618	1.10504	0.552519	+	0.833500i	$0.313667\pi$
$354$	0	0
$355$	1.92661	0.102254
$356$	−8.63147	−0.457467
$357$	0	0
$358$	−7.86829	−0.415852
$359$	−3.96492	−0.209261	−0.104630	−	0.994511i	$-0.533366\pi$
$359$	−3.96492	−0.209261	−0.104630	+	0.994511i	$0.533366\pi$
$360$	0	0
$361$	26.8987	1.41572
$362$	0.905742	0.0476047
$363$	0	0
$364$	−5.08222	−0.266381
$365$	−9.00014	−0.471089
$366$	0	0
$367$	−19.4443	−1.01499	−0.507493	−	0.861656i	$-0.669428\pi$
$367$	−19.4443	−1.01499	−0.507493	+	0.861656i	$0.669428\pi$
$368$	1.66037	0.0865525
$369$	0	0
$370$	5.58134	0.290160
$371$	32.2176	1.67266
$372$	0	0
$373$	−3.77156	−0.195284	−0.0976419	−	0.995222i	$-0.531130\pi$
$373$	−3.77156	−0.195284	−0.0976419	+	0.995222i	$0.531130\pi$
$374$	17.7032	0.915411
$375$	0	0
$376$	−0.0631732	−0.00325791
$377$	0.502934	0.0259024
$378$	0	0
$379$	15.1994	0.780740	0.390370	−	0.920658i	$-0.372347\pi$
$379$	15.1994	0.780740	0.390370	+	0.920658i	$0.372347\pi$
$380$	8.40163	0.430995
$381$	0	0
$382$	−18.2162	−0.932019
$383$	7.52457	0.384488	0.192244	−	0.981347i	$-0.438424\pi$
$383$	7.52457	0.384488	0.192244	+	0.981347i	$0.438424\pi$
$384$	0	0
$385$	16.6132	0.846686
$386$	19.3647	0.985638
$387$	0	0
$388$	10.8998	0.553352
$389$	30.3583	1.53923	0.769613	−	0.638510i	$-0.220449\pi$
$389$	30.3583	1.53923	0.769613	+	0.638510i	$0.220449\pi$
$390$	0	0
$391$	22.3487	1.13022
$392$	−26.1045	−1.31847
$393$	0	0
$394$	17.2372	0.868397
$395$	18.0669	0.909045
$396$	0	0
$397$	4.62399	0.232071	0.116036	−	0.993245i	$-0.462981\pi$
$397$	4.62399	0.232071	0.116036	+	0.993245i	$0.462981\pi$
$398$	−4.02419	−0.201715
$399$	0	0
$400$	1.42154	0.0710769
$401$	−17.7060	−0.884194	−0.442097	−	0.896967i	$-0.645765\pi$
$401$	−17.7060	−0.884194	−0.442097	+	0.896967i	$0.645765\pi$
$402$	0	0
$403$	−0.156173	−0.00777956
$404$	−19.4415	−0.967251
$405$	0	0
$406$	1.67880	0.0833173
$407$	21.7129	1.07627
$408$	0	0
$409$	4.47676	0.221362	0.110681	−	0.993856i	$-0.464697\pi$
$409$	4.47676	0.221362	0.110681	+	0.993856i	$0.464697\pi$
$410$	3.84754	0.190016
$411$	0	0
$412$	0.119194	0.00587229
$413$	−57.2449	−2.81684
$414$	0	0
$415$	−3.67514	−0.180405
$416$	6.02996	0.295643
$417$	0	0
$418$	−24.0737	−1.17748
$419$	22.5224	1.10029	0.550146	−	0.835069i	$-0.314572\pi$
$419$	22.5224	1.10029	0.550146	+	0.835069i	$0.314572\pi$
$420$	0	0
$421$	−31.4367	−1.53213	−0.766067	−	0.642761i	$-0.777789\pi$
$421$	−31.4367	−1.53213	−0.766067	+	0.642761i	$0.777789\pi$
$422$	−13.7514	−0.669405
$423$	0	0
$424$	−23.3856	−1.13571
$425$	19.1341	0.928139
$426$	0	0
$427$	−54.7319	−2.64866
$428$	21.2258	1.02599
$429$	0	0
$430$	5.16967	0.249303
$431$	−7.61874	−0.366982	−0.183491	−	0.983021i	$-0.558740\pi$
$431$	−7.61874	−0.366982	−0.183491	+	0.983021i	$0.558740\pi$
$432$	0	0
$433$	−6.89352	−0.331281	−0.165641	−	0.986186i	$-0.552969\pi$
$433$	−6.89352	−0.331281	−0.165641	+	0.986186i	$0.552969\pi$
$434$	−0.521308	−0.0250236
$435$	0	0
$436$	4.08016	0.195404
$437$	−30.3908	−1.45379
$438$	0	0
$439$	19.2553	0.919003	0.459502	−	0.888177i	$-0.348028\pi$
$439$	19.2553	0.919003	0.459502	+	0.888177i	$0.348028\pi$
$440$	−12.0589	−0.574886
$441$	0	0
$442$	−5.06325	−0.240834
$443$	−6.89520	−0.327601	−0.163800	−	0.986494i	$-0.552375\pi$
$443$	−6.89520	−0.327601	−0.163800	+	0.986494i	$0.552375\pi$
$444$	0	0
$445$	−8.06976	−0.382543
$446$	−2.77645	−0.131469
$447$	0	0
$448$	23.0885	1.09083
$449$	6.70630	0.316490	0.158245	−	0.987400i	$-0.449416\pi$
$449$	6.70630	0.316490	0.158245	+	0.987400i	$0.449416\pi$
$450$	0	0
$451$	14.9679	0.704812
$452$	15.4254	0.725551
$453$	0	0
$454$	25.2506	1.18507
$455$	−4.75149	−0.222753
$456$	0	0
$457$	−30.4544	−1.42460	−0.712298	−	0.701877i	$-0.752345\pi$
$457$	−30.4544	−1.42460	−0.712298	+	0.701877i	$0.752345\pi$
$458$	−21.7600	−1.01678
$459$	0	0
$460$	−5.56296	−0.259374
$461$	−4.43558	−0.206585	−0.103293	−	0.994651i	$-0.532938\pi$
$461$	−4.43558	−0.206585	−0.103293	+	0.994651i	$0.532938\pi$
$462$	0	0
$463$	−29.3900	−1.36587	−0.682935	−	0.730480i	$-0.739297\pi$
$463$	−29.3900	−1.36587	−0.682935	+	0.730480i	$0.739297\pi$
$464$	−0.168704	−0.00783188
$465$	0	0
$466$	−6.98558	−0.323601
$467$	−7.49113	−0.346648	−0.173324	−	0.984865i	$-0.555451\pi$
$467$	−7.49113	−0.346648	−0.173324	+	0.984865i	$0.555451\pi$
$468$	0	0
$469$	−33.6109	−1.55201
$470$	−0.0215827	−0.000995537 0
$471$	0	0
$472$	41.5520	1.91259
$473$	20.1114	0.924722
$474$	0	0
$475$	−26.0194	−1.19385
$476$	22.9465	1.05175
$477$	0	0
$478$	−0.921026	−0.0421267
$479$	5.20065	0.237624	0.118812	−	0.992917i	$-0.462091\pi$
$479$	5.20065	0.237624	0.118812	+	0.992917i	$0.462091\pi$
$480$	0	0
$481$	−6.21005	−0.283154
$482$	27.3465	1.24560
$483$	0	0
$484$	−4.47408	−0.203367
$485$	10.1904	0.462724
$486$	0	0
$487$	−26.9614	−1.22174	−0.610868	−	0.791732i	$-0.709180\pi$
$487$	−26.9614	−1.22174	−0.610868	+	0.791732i	$0.709180\pi$
$488$	39.7279	1.79840
$489$	0	0
$490$	−8.91842	−0.402893
$491$	5.89472	0.266025	0.133013	−	0.991114i	$-0.457535\pi$
$491$	5.89472	0.266025	0.133013	+	0.991114i	$0.457535\pi$
$492$	0	0
$493$	−2.27077	−0.102270
$494$	6.88524	0.309782
$495$	0	0
$496$	0.0523868	0.00235223
$497$	−7.15544	−0.320965
$498$	0	0
$499$	13.9968	0.626585	0.313293	−	0.949657i	$-0.398568\pi$
$499$	13.9968	0.626585	0.313293	+	0.949657i	$0.398568\pi$
$500$	−10.9634	−0.490297
$501$	0	0
$502$	20.7710	0.927056
$503$	−21.0763	−0.939747	−0.469873	−	0.882734i	$-0.655700\pi$
$503$	−21.0763	−0.939747	−0.469873	+	0.882734i	$0.655700\pi$
$504$	0	0
$505$	−18.1763	−0.808835
$506$	15.9399	0.708613
$507$	0	0
$508$	3.25213	0.144290
$509$	−26.8372	−1.18954	−0.594768	−	0.803897i	$-0.702756\pi$
$509$	−26.8372	−1.18954	−0.594768	+	0.803897i	$0.702756\pi$
$510$	0	0
$511$	33.4266	1.47871
$512$	−4.17155	−0.184358
$513$	0	0
$514$	13.7378	0.605949
$515$	0.111438	0.00491053
$516$	0	0
$517$	−0.0839625	−0.00369267
$518$	−20.7292	−0.910788
$519$	0	0
$520$	3.44893	0.151246
$521$	−38.1749	−1.67247	−0.836237	−	0.548368i	$-0.815249\pi$
$521$	−38.1749	−1.67247	−0.836237	+	0.548368i	$0.815249\pi$
$522$	0	0
$523$	−27.7790	−1.21469	−0.607344	−	0.794439i	$-0.707765\pi$
$523$	−27.7790	−1.21469	−0.607344	+	0.794439i	$0.707765\pi$
$524$	−8.27871	−0.361657
$525$	0	0
$526$	7.30986	0.318725
$527$	0.705132	0.0307160
$528$	0	0
$529$	−2.87737	−0.125103
$530$	−7.98954	−0.347043
$531$	0	0
$532$	−31.2038	−1.35285
$533$	−4.28094	−0.185428
$534$	0	0
$535$	19.8445	0.857952
$536$	24.3970	1.05379
$537$	0	0
$538$	26.4512	1.14039
$539$	−34.6950	−1.49442
$540$	0	0
$541$	27.8323	1.19660	0.598302	−	0.801270i	$-0.295842\pi$
$541$	27.8323	1.19660	0.598302	+	0.801270i	$0.295842\pi$
$542$	−1.36137	−0.0584759
$543$	0	0
$544$	−27.2256	−1.16729
$545$	3.81464	0.163401
$546$	0	0
$547$	−19.9472	−0.852880	−0.426440	−	0.904516i	$-0.640232\pi$
$547$	−19.9472	−0.852880	−0.426440	+	0.904516i	$0.640232\pi$
$548$	−10.8922	−0.465291
$549$	0	0
$550$	13.6471	0.581913
$551$	3.08791	0.131549
$552$	0	0
$553$	−67.1007	−2.85341
$554$	27.1279	1.15255
$555$	0	0
$556$	−8.59765	−0.364621
$557$	−13.5520	−0.574215	−0.287107	−	0.957898i	$-0.592694\pi$
$557$	−13.5520	−0.574215	−0.287107	+	0.957898i	$0.592694\pi$
$558$	0	0
$559$	−5.75200	−0.243283
$560$	1.59384	0.0673519
$561$	0	0
$562$	−14.7584	−0.622546
$563$	−28.1055	−1.18451	−0.592253	−	0.805752i	$-0.701761\pi$
$563$	−28.1055	−1.18451	−0.592253	+	0.805752i	$0.701761\pi$
$564$	0	0
$565$	14.4216	0.606721
$566$	7.37143	0.309844
$567$	0	0
$568$	5.19388	0.217930
$569$	−5.43746	−0.227950	−0.113975	−	0.993484i	$-0.536358\pi$
$569$	−5.43746	−0.227950	−0.113975	+	0.993484i	$0.536358\pi$
$570$	0	0
$571$	14.9824	0.626993	0.313496	−	0.949589i	$-0.398500\pi$
$571$	14.9824	0.626993	0.313496	+	0.949589i	$0.398500\pi$
$572$	4.90299	0.205005
$573$	0	0
$574$	−14.2898	−0.596444
$575$	17.2282	0.718465
$576$	0	0
$577$	21.1328	0.879768	0.439884	−	0.898055i	$-0.355020\pi$
$577$	21.1328	0.879768	0.439884	+	0.898055i	$0.355020\pi$
$578$	7.20339	0.299621
$579$	0	0
$580$	0.565232	0.0234700
$581$	13.6495	0.566277
$582$	0	0
$583$	−31.0814	−1.28726
$584$	−24.2632	−1.00402
$585$	0	0
$586$	−11.1866	−0.462115
$587$	26.9642	1.11293	0.556465	−	0.830871i	$-0.312157\pi$
$587$	26.9642	1.11293	0.556465	+	0.830871i	$0.312157\pi$
$588$	0	0
$589$	−0.958872	−0.0395096
$590$	14.1960	0.584439
$591$	0	0
$592$	2.08309	0.0856146
$593$	1.66422	0.0683415	0.0341708	−	0.999416i	$-0.489121\pi$
$593$	1.66422	0.0683415	0.0341708	+	0.999416i	$0.489121\pi$
$594$	0	0
$595$	21.4532	0.879496
$596$	11.4293	0.468164
$597$	0	0
$598$	−4.55891	−0.186428
$599$	−19.8878	−0.812592	−0.406296	−	0.913741i	$-0.633180\pi$
$599$	−19.8878	−0.812592	−0.406296	+	0.913741i	$0.633180\pi$
$600$	0	0
$601$	−13.8715	−0.565832	−0.282916	−	0.959145i	$-0.591302\pi$
$601$	−13.8715	−0.565832	−0.282916	+	0.959145i	$0.591302\pi$
$602$	−19.2002	−0.782542
$603$	0	0
$604$	−11.1823	−0.455003
$605$	−4.18292	−0.170060
$606$	0	0
$607$	−21.6595	−0.879131	−0.439565	−	0.898211i	$-0.644868\pi$
$607$	−21.6595	−0.879131	−0.439565	+	0.898211i	$0.644868\pi$
$608$	37.0226	1.50147
$609$	0	0
$610$	13.5728	0.549546
$611$	0.0240139	0.000971497 0
$612$	0	0
$613$	36.5608	1.47668	0.738339	−	0.674430i	$-0.235610\pi$
$613$	36.5608	1.47668	0.738339	+	0.674430i	$0.235610\pi$
$614$	16.9556	0.684273
$615$	0	0
$616$	44.7869	1.80452
$617$	20.3869	0.820746	0.410373	−	0.911918i	$-0.365399\pi$
$617$	20.3869	0.820746	0.410373	+	0.911918i	$0.365399\pi$
$618$	0	0
$619$	10.3450	0.415802	0.207901	−	0.978150i	$-0.433337\pi$
$619$	10.3450	0.415802	0.207901	+	0.978150i	$0.433337\pi$
$620$	−0.175519	−0.00704900
$621$	0	0
$622$	31.1057	1.24722
$623$	29.9712	1.20077
$624$	0	0
$625$	8.95301	0.358121
$626$	−12.6548	−0.505787
$627$	0	0
$628$	18.9609	0.756623
$629$	28.0387	1.11798
$630$	0	0
$631$	−10.0342	−0.399454	−0.199727	−	0.979852i	$-0.564005\pi$
$631$	−10.0342	−0.399454	−0.199727	+	0.979852i	$0.564005\pi$
$632$	48.7060	1.93742
$633$	0	0
$634$	4.49820	0.178647
$635$	3.04049	0.120658
$636$	0	0
$637$	9.92302	0.393164
$638$	−1.61959	−0.0641203
$639$	0	0
$640$	6.04274	0.238860
$641$	32.5006	1.28370	0.641849	−	0.766831i	$-0.278168\pi$
$641$	32.5006	1.28370	0.641849	+	0.766831i	$0.278168\pi$
$642$	0	0
$643$	−1.29949	−0.0512467	−0.0256234	−	0.999672i	$-0.508157\pi$
$643$	−1.29949	−0.0512467	−0.0256234	+	0.999672i	$0.508157\pi$
$644$	20.6609	0.814153
$645$	0	0
$646$	−31.0872	−1.22311
$647$	11.1792	0.439499	0.219749	−	0.975556i	$-0.429476\pi$
$647$	11.1792	0.439499	0.219749	+	0.975556i	$0.429476\pi$
$648$	0	0
$649$	55.2261	2.16781
$650$	−3.90316	−0.153095
$651$	0	0
$652$	−23.6059	−0.924477
$653$	−16.3420	−0.639512	−0.319756	−	0.947500i	$-0.603601\pi$
$653$	−16.3420	−0.639512	−0.319756	+	0.947500i	$0.603601\pi$
$654$	0	0
$655$	−7.73995	−0.302425
$656$	1.43599	0.0560662
$657$	0	0
$658$	0.0801585	0.00312490
$659$	−40.9312	−1.59445	−0.797227	−	0.603680i	$-0.793701\pi$
$659$	−40.9312	−1.59445	−0.797227	+	0.603680i	$0.793701\pi$
$660$	0	0
$661$	24.6581	0.959089	0.479544	−	0.877518i	$-0.340802\pi$
$661$	24.6581	0.959089	0.479544	+	0.877518i	$0.340802\pi$
$662$	6.00263	0.233299
$663$	0	0
$664$	−9.90769	−0.384493
$665$	−29.1731	−1.13128
$666$	0	0
$667$	−2.04459	−0.0791668
$668$	−3.21755	−0.124491
$669$	0	0
$670$	8.33507	0.322012
$671$	52.8017	2.03839
$672$	0	0
$673$	24.1505	0.930933	0.465467	−	0.885065i	$-0.345887\pi$
$673$	24.1505	0.930933	0.465467	+	0.885065i	$0.345887\pi$
$674$	1.86208	0.0717248
$675$	0	0
$676$	13.5700	0.521922
$677$	−15.9466	−0.612876	−0.306438	−	0.951891i	$-0.599137\pi$
$677$	−15.9466	−0.612876	−0.306438	+	0.951891i	$0.599137\pi$
$678$	0	0
$679$	−37.8474	−1.45245
$680$	−15.5721	−0.597164
$681$	0	0
$682$	0.502924	0.0192579
$683$	41.1186	1.57336	0.786680	−	0.617361i	$-0.211798\pi$
$683$	41.1186	1.57336	0.786680	+	0.617361i	$0.211798\pi$
$684$	0	0
$685$	−10.1833	−0.389086
$686$	7.34017	0.280249
$687$	0	0
$688$	1.92945	0.0735595
$689$	8.88951	0.338663
$690$	0	0
$691$	−6.74938	−0.256759	−0.128379	−	0.991725i	$-0.540977\pi$
$691$	−6.74938	−0.256759	−0.128379	+	0.991725i	$0.540977\pi$
$692$	19.0456	0.724006
$693$	0	0
$694$	−17.2440	−0.654575
$695$	−8.03814	−0.304904
$696$	0	0
$697$	19.3287	0.732125
$698$	12.9955	0.491888
$699$	0	0
$700$	17.6890	0.668582
$701$	−22.1034	−0.834833	−0.417417	−	0.908715i	$-0.637064\pi$
$701$	−22.1034	−0.834833	−0.417417	+	0.908715i	$0.637064\pi$
$702$	0	0
$703$	−38.1283	−1.43804
$704$	−22.2742	−0.839492
$705$	0	0
$706$	19.1221	0.719671
$707$	67.5070	2.53886
$708$	0	0
$709$	−23.1814	−0.870594	−0.435297	−	0.900287i	$-0.643357\pi$
$709$	−23.1814	−0.870594	−0.435297	+	0.900287i	$0.643357\pi$
$710$	1.77445	0.0665941
$711$	0	0
$712$	−21.7550	−0.815304
$713$	0.634896	0.0237770
$714$	0	0
$715$	4.58392	0.171429
$716$	9.83904	0.367702
$717$	0	0
$718$	−3.65179	−0.136284
$719$	−14.6424	−0.546068	−0.273034	−	0.962004i	$-0.588027\pi$
$719$	−14.6424	−0.546068	−0.273034	+	0.962004i	$0.588027\pi$
$720$	0	0
$721$	−0.413880	−0.0154137
$722$	24.7744	0.922008
$723$	0	0
$724$	−1.13260	−0.0420927
$725$	−1.75050	−0.0650118
$726$	0	0
$727$	−9.78781	−0.363010	−0.181505	−	0.983390i	$-0.558097\pi$
$727$	−9.78781	−0.363010	−0.181505	+	0.983390i	$0.558097\pi$
$728$	−12.8094	−0.474747
$729$	0	0
$730$	−8.28936	−0.306803
$731$	25.9706	0.960556
$732$	0	0
$733$	−51.7138	−1.91009	−0.955047	−	0.296456i	$-0.904195\pi$
$733$	−51.7138	−1.91009	−0.955047	+	0.296456i	$0.904195\pi$
$734$	−17.9087	−0.661024
$735$	0	0
$736$	−24.5137	−0.903588
$737$	32.4256	1.19441
$738$	0	0
$739$	−40.5262	−1.49078	−0.745390	−	0.666628i	$-0.767737\pi$
$739$	−40.5262	−1.49078	−0.745390	+	0.666628i	$0.767737\pi$
$740$	−6.97928	−0.256564
$741$	0	0
$742$	29.6732	1.08934
$743$	4.45345	0.163381	0.0816906	−	0.996658i	$-0.473968\pi$
$743$	4.45345	0.163381	0.0816906	+	0.996658i	$0.473968\pi$
$744$	0	0
$745$	10.6856	0.391488
$746$	−3.47370	−0.127181
$747$	0	0
$748$	−22.1373	−0.809419
$749$	−73.7026	−2.69304
$750$	0	0
$751$	53.0922	1.93736	0.968681	−	0.248309i	$-0.0798747\pi$
$751$	53.0922	1.93736	0.968681	+	0.248309i	$0.0798747\pi$
$752$	−0.00805520	−0.000293743 0
$753$	0	0
$754$	0.463215	0.0168693
$755$	−10.4546	−0.380483
$756$	0	0
$757$	−8.84449	−0.321458	−0.160729	−	0.986999i	$-0.551385\pi$
$757$	−8.84449	−0.321458	−0.160729	+	0.986999i	$0.551385\pi$
$758$	13.9990	0.508467
$759$	0	0
$760$	21.1757	0.768124
$761$	20.5342	0.744364	0.372182	−	0.928160i	$-0.378610\pi$
$761$	20.5342	0.744364	0.372182	+	0.928160i	$0.378610\pi$
$762$	0	0
$763$	−14.1676	−0.512902
$764$	22.7787	0.824104
$765$	0	0
$766$	6.93032	0.250403
$767$	−15.7951	−0.570327
$768$	0	0
$769$	−11.3547	−0.409462	−0.204731	−	0.978818i	$-0.565632\pi$
$769$	−11.3547	−0.409462	−0.204731	+	0.978818i	$0.565632\pi$
$770$	15.3012	0.551416
$771$	0	0
$772$	−24.2149	−0.871514
$773$	−40.7990	−1.46744	−0.733720	−	0.679452i	$-0.762217\pi$
$773$	−40.7990	−1.46744	−0.733720	+	0.679452i	$0.762217\pi$
$774$	0	0
$775$	0.543572	0.0195257
$776$	27.4721	0.986190
$777$	0	0
$778$	27.9608	1.00244
$779$	−26.2840	−0.941723
$780$	0	0
$781$	6.90310	0.247012
$782$	20.5837	0.736073
$783$	0	0
$784$	−3.32857	−0.118878
$785$	17.7270	0.632703
$786$	0	0
$787$	2.97511	0.106051	0.0530257	−	0.998593i	$-0.483113\pi$
$787$	2.97511	0.106051	0.0530257	+	0.998593i	$0.483113\pi$
$788$	−21.5545	−0.767848
$789$	0	0
$790$	16.6401	0.592028
$791$	−53.5619	−1.90444
$792$	0	0
$793$	−15.1017	−0.536276
$794$	4.25881	0.151140
$795$	0	0
$796$	5.03212	0.178359
$797$	−15.2875	−0.541512	−0.270756	−	0.962648i	$-0.587274\pi$
$797$	−15.2875	−0.541512	−0.270756	+	0.962648i	$0.587274\pi$
$798$	0	0
$799$	−0.108424	−0.00383576
$800$	−20.9877	−0.742027
$801$	0	0
$802$	−16.3076	−0.575843
$803$	−32.2478	−1.13800
$804$	0	0
$805$	19.3163	0.680811
$806$	−0.143840	−0.00506654
$807$	0	0
$808$	−49.0009	−1.72385
$809$	−33.2118	−1.16766	−0.583832	−	0.811874i	$-0.698447\pi$
$809$	−33.2118	−1.16766	−0.583832	+	0.811874i	$0.698447\pi$
$810$	0	0
$811$	41.9154	1.47185	0.735925	−	0.677063i	$-0.236748\pi$
$811$	41.9154	1.47185	0.735925	+	0.677063i	$0.236748\pi$
$812$	−2.09928	−0.0736703
$813$	0	0
$814$	19.9981	0.700934
$815$	−22.0697	−0.773066
$816$	0	0
$817$	−35.3160	−1.23555
$818$	4.12321	0.144165
$819$	0	0
$820$	−4.81122	−0.168015
$821$	1.45745	0.0508654	0.0254327	−	0.999677i	$-0.491904\pi$
$821$	1.45745	0.0508654	0.0254327	+	0.999677i	$0.491904\pi$
$822$	0	0
$823$	20.2247	0.704990	0.352495	−	0.935814i	$-0.385333\pi$
$823$	20.2247	0.704990	0.352495	+	0.935814i	$0.385333\pi$
$824$	0.300421	0.0104657
$825$	0	0
$826$	−52.7240	−1.83450
$827$	29.5159	1.02637	0.513184	−	0.858278i	$-0.328466\pi$
$827$	29.5159	1.02637	0.513184	+	0.858278i	$0.328466\pi$
$828$	0	0
$829$	6.24184	0.216788	0.108394	−	0.994108i	$-0.465429\pi$
$829$	6.24184	0.216788	0.108394	+	0.994108i	$0.465429\pi$
$830$	−3.38490	−0.117491
$831$	0	0
$832$	6.37059	0.220860
$833$	−44.8030	−1.55233
$834$	0	0
$835$	−3.00816	−0.104102
$836$	30.1033	1.04115
$837$	0	0
$838$	20.7437	0.716579
$839$	11.6981	0.403863	0.201931	−	0.979400i	$-0.435278\pi$
$839$	11.6981	0.403863	0.201931	+	0.979400i	$0.435278\pi$
$840$	0	0
$841$	−28.7923	−0.992836
$842$	−28.9540	−0.997822
$843$	0	0
$844$	17.1956	0.591897
$845$	12.6869	0.436441
$846$	0	0
$847$	15.5354	0.533803
$848$	−2.98189	−0.102399
$849$	0	0
$850$	17.6230	0.604463
$851$	25.2458	0.865416
$852$	0	0
$853$	24.8809	0.851905	0.425953	−	0.904745i	$-0.359939\pi$
$853$	24.8809	0.851905	0.425953	+	0.904745i	$0.359939\pi$
$854$	−50.4095	−1.72498
$855$	0	0
$856$	53.4981	1.82853
$857$	−37.5138	−1.28145	−0.640724	−	0.767771i	$-0.721366\pi$
$857$	−37.5138	−1.28145	−0.640724	+	0.767771i	$0.721366\pi$
$858$	0	0
$859$	−28.9804	−0.988799	−0.494399	−	0.869235i	$-0.664612\pi$
$859$	−28.9804	−0.988799	−0.494399	+	0.869235i	$0.664612\pi$
$860$	−6.46450	−0.220437
$861$	0	0
$862$	−7.01705	−0.239002
$863$	−15.2759	−0.519999	−0.260000	−	0.965609i	$-0.583722\pi$
$863$	−15.2759	−0.519999	−0.260000	+	0.965609i	$0.583722\pi$
$864$	0	0
$865$	17.8062	0.605428
$866$	−6.34911	−0.215751
$867$	0	0
$868$	0.651878	0.0221262
$869$	64.7343	2.19596
$870$	0	0
$871$	−9.27396	−0.314236
$872$	10.2838	0.348252
$873$	0	0
$874$	−27.9907	−0.946801
$875$	38.0683	1.28694
$876$	0	0
$877$	27.6463	0.933550	0.466775	−	0.884376i	$-0.345416\pi$
$877$	27.6463	0.933550	0.466775	+	0.884376i	$0.345416\pi$
$878$	17.7346	0.598513
$879$	0	0
$880$	−1.53763	−0.0518334
$881$	−38.9625	−1.31268	−0.656341	−	0.754465i	$-0.727897\pi$
$881$	−38.9625	−1.31268	−0.656341	+	0.754465i	$0.727897\pi$
$882$	0	0
$883$	−3.02423	−0.101773	−0.0508867	−	0.998704i	$-0.516205\pi$
$883$	−3.02423	−0.101773	−0.0508867	+	0.998704i	$0.516205\pi$
$884$	6.33142	0.212949
$885$	0	0
$886$	−6.35065	−0.213354
$887$	−52.8701	−1.77521	−0.887603	−	0.460610i	$-0.847631\pi$
$887$	−52.8701	−1.77521	−0.887603	+	0.460610i	$0.847631\pi$
$888$	0	0
$889$	−11.2924	−0.378736
$890$	−7.43246	−0.249136
$891$	0	0
$892$	3.47186	0.116247
$893$	0.147440	0.00493389
$894$	0	0
$895$	9.19874	0.307480
$896$	−22.4428	−0.749762
$897$	0	0
$898$	6.17667	0.206118
$899$	−0.0645095	−0.00215151
$900$	0	0
$901$	−40.1366	−1.33714
$902$	13.7858	0.459019
$903$	0	0
$904$	38.8787	1.29309
$905$	−1.05889	−0.0351988
$906$	0	0
$907$	−8.11733	−0.269531	−0.134766	−	0.990877i	$-0.543028\pi$
$907$	−8.11733	−0.269531	−0.134766	+	0.990877i	$0.543028\pi$
$908$	−31.5751	−1.04786
$909$	0	0
$910$	−4.37624	−0.145071
$911$	2.82293	0.0935278	0.0467639	−	0.998906i	$-0.485109\pi$
$911$	2.82293	0.0935278	0.0467639	+	0.998906i	$0.485109\pi$
$912$	0	0
$913$	−13.1681	−0.435802
$914$	−28.0493	−0.927787
$915$	0	0
$916$	27.2102	0.899050
$917$	28.7463	0.949286
$918$	0	0
$919$	36.0591	1.18948	0.594739	−	0.803919i	$-0.297255\pi$
$919$	36.0591	1.18948	0.594739	+	0.803919i	$0.297255\pi$
$920$	−14.0210	−0.462260
$921$	0	0
$922$	−4.08528	−0.134542
$923$	−1.97434	−0.0649860
$924$	0	0
$925$	21.6145	0.710680
$926$	−27.0689	−0.889541
$927$	0	0
$928$	2.49075	0.0817630
$929$	−41.9724	−1.37707	−0.688535	−	0.725203i	$-0.741746\pi$
$929$	−41.9724	−1.37707	−0.688535	+	0.725203i	$0.741746\pi$
$930$	0	0
$931$	60.9252	1.99674
$932$	8.73524	0.286132
$933$	0	0
$934$	−6.89953	−0.225759
$935$	−20.6966	−0.676853
$936$	0	0
$937$	−9.17845	−0.299847	−0.149923	−	0.988698i	$-0.547903\pi$
$937$	−9.17845	−0.299847	−0.149923	+	0.988698i	$0.547903\pi$
$938$	−30.9565	−1.01077
$939$	0	0
$940$	0.0269885	0.000880267 0
$941$	18.9045	0.616270	0.308135	−	0.951343i	$-0.400295\pi$
$941$	18.9045	0.616270	0.308135	+	0.951343i	$0.400295\pi$
$942$	0	0
$943$	17.4034	0.566732
$944$	5.29829	0.172445
$945$	0	0
$946$	18.5231	0.602238
$947$	59.9874	1.94933	0.974665	−	0.223670i	$-0.0718038\pi$
$947$	59.9874	1.94933	0.974665	+	0.223670i	$0.0718038\pi$
$948$	0	0
$949$	9.22310	0.299394
$950$	−23.9646	−0.777513
$951$	0	0
$952$	57.8350	1.87444
$953$	−32.9608	−1.06770	−0.533852	−	0.845578i	$-0.679256\pi$
$953$	−32.9608	−1.06770	−0.533852	+	0.845578i	$0.679256\pi$
$954$	0	0
$955$	21.2963	0.689133
$956$	1.15171	0.0372490
$957$	0	0
$958$	4.78993	0.154756
$959$	37.8211	1.22131
$960$	0	0
$961$	−30.9800	−0.999354
$962$	−5.71961	−0.184408
$963$	0	0
$964$	−34.1959	−1.10137
$965$	−22.6391	−0.728778
$966$	0	0
$967$	−12.7497	−0.410001	−0.205001	−	0.978762i	$-0.565720\pi$
$967$	−12.7497	−0.410001	−0.205001	+	0.978762i	$0.565720\pi$
$968$	−11.2766	−0.362444
$969$	0	0
$970$	9.38565	0.301355
$971$	−51.1458	−1.64135	−0.820673	−	0.571398i	$-0.806401\pi$
$971$	−51.1458	−1.64135	−0.820673	+	0.571398i	$0.806401\pi$
$972$	0	0
$973$	29.8537	0.957066
$974$	−24.8321	−0.795673
$975$	0	0
$976$	5.06570	0.162149
$977$	−51.3095	−1.64154	−0.820768	−	0.571262i	$-0.806454\pi$
$977$	−51.3095	−1.64154	−0.820768	+	0.571262i	$0.806454\pi$
$978$	0	0
$979$	−28.9142	−0.924102
$980$	11.1522	0.356244
$981$	0	0
$982$	5.42919	0.173252
$983$	−45.7359	−1.45875	−0.729375	−	0.684114i	$-0.760189\pi$
$983$	−45.7359	−1.45875	−0.729375	+	0.684114i	$0.760189\pi$
$984$	0	0
$985$	−20.1518	−0.642090
$986$	−2.09144	−0.0666050
$987$	0	0
$988$	−8.60977	−0.273913
$989$	23.3837	0.743559
$990$	0	0
$991$	36.0372	1.14476	0.572379	−	0.819989i	$-0.306021\pi$
$991$	36.0372	1.14476	0.572379	+	0.819989i	$0.306021\pi$
$992$	−0.773441	−0.0245568
$993$	0	0
$994$	−6.59034	−0.209033
$995$	4.70464	0.149147
$996$	0	0
$997$	33.6291	1.06504	0.532522	−	0.846416i	$-0.321244\pi$
$997$	33.6291	1.06504	0.532522	+	0.846416i	$0.321244\pi$
$998$	12.8915	0.408072
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.a.j.1.14	✓	20
3.2	odd	2		2151.2.a.k.1.7	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.2.a.j.1.14	✓	20	1.1	even	1	trivial
2151.2.a.k.1.7	yes	20	3.2	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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q+0.921026 q^{2} -1.15171 q^{4} -1.07676 q^{5} +3.99910 q^{7} -2.90281 q^{8} +O(q^{10})\)	\(q+0.921026 q^{2} -1.15171 q^{4} -1.07676 q^{5} +3.99910 q^{7} -2.90281 q^{8} -0.991725 q^{10} -3.85807 q^{11} +1.10344 q^{13} +3.68328 q^{14} -0.370136 q^{16} -4.98207 q^{17} +6.77486 q^{19} +1.24012 q^{20} -3.55338 q^{22} -4.48583 q^{23} -3.84058 q^{25} +1.01629 q^{26} -4.60582 q^{28} +0.455789 q^{29} -0.141534 q^{31} +5.46471 q^{32} -4.58862 q^{34} -4.30608 q^{35} -5.62791 q^{37} +6.23982 q^{38} +3.12563 q^{40} -3.87964 q^{41} -5.21280 q^{43} +4.44339 q^{44} -4.13156 q^{46} +0.0217628 q^{47} +8.99284 q^{49} -3.53728 q^{50} -1.27084 q^{52} +8.05621 q^{53} +4.15422 q^{55} -11.6086 q^{56} +0.419793 q^{58} -14.3144 q^{59} -13.6860 q^{61} -0.130356 q^{62} +5.77341 q^{64} -1.18814 q^{65} -8.40462 q^{67} +5.73791 q^{68} -3.96601 q^{70} -1.78926 q^{71} +8.35852 q^{73} -5.18345 q^{74} -7.80269 q^{76} -15.4288 q^{77} -16.7789 q^{79} +0.398548 q^{80} -3.57325 q^{82} +3.41314 q^{83} +5.36450 q^{85} -4.80112 q^{86} +11.1992 q^{88} +7.49447 q^{89} +4.41276 q^{91} +5.16638 q^{92} +0.0200441 q^{94} -7.29491 q^{95} -9.46397 q^{97} +8.28263 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{2} + 18 q^{4} - 16 q^{5} - 4 q^{7} - 12 q^{8}+O(q^{10}) \) 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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Representations

Groups

Database

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