LMFDB - Embedded newform 2151.2.a.j.1.11

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			1.11
Root			$0.0888996$ 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)

$f(q)$	$=$	$q-0.0888996 q^{2} -1.99210 q^{4} -0.674348 q^{5} +0.687902 q^{7} +0.354896 q^{8} +O(q^{10})$	\(q-0.0888996 q^{2} -1.99210 q^{4} -0.674348 q^{5} +0.687902 q^{7} +0.354896 q^{8} +0.0599493 q^{10} -0.536956 q^{11} -0.628799 q^{13} -0.0611542 q^{14} +3.95264 q^{16} +5.05676 q^{17} -0.724853 q^{19} +1.34337 q^{20} +0.0477352 q^{22} -7.36157 q^{23} -4.54525 q^{25} +0.0559000 q^{26} -1.37037 q^{28} -3.61524 q^{29} +5.74731 q^{31} -1.06118 q^{32} -0.449544 q^{34} -0.463885 q^{35} -0.110957 q^{37} +0.0644391 q^{38} -0.239323 q^{40} +8.05251 q^{41} +12.2197 q^{43} +1.06967 q^{44} +0.654440 q^{46} +1.48719 q^{47} -6.52679 q^{49} +0.404071 q^{50} +1.25263 q^{52} -6.92286 q^{53} +0.362095 q^{55} +0.244133 q^{56} +0.321393 q^{58} -9.81893 q^{59} +6.30376 q^{61} -0.510934 q^{62} -7.81095 q^{64} +0.424030 q^{65} -7.02929 q^{67} -10.0736 q^{68} +0.0412392 q^{70} -9.21843 q^{71} -12.1931 q^{73} +0.00986406 q^{74} +1.44398 q^{76} -0.369373 q^{77} -6.71164 q^{79} -2.66546 q^{80} -0.715865 q^{82} -8.10511 q^{83} -3.41002 q^{85} -1.08633 q^{86} -0.190563 q^{88} -13.1414 q^{89} -0.432552 q^{91} +14.6650 q^{92} -0.132211 q^{94} +0.488803 q^{95} +8.31742 q^{97} +0.580229 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.0888996	−0.0628615	−0.0314308	−	0.999506i	$-0.510006\pi$
$2$	−0.0888996	−0.0628615	−0.0314308	+	0.999506i	$0.510006\pi$
$3$	0	0
$3$	0	0
$4$	−1.99210	−0.996048
$5$	−0.674348	−0.301578	−0.150789	−	0.988566i	$-0.548181\pi$
$5$	−0.674348	−0.301578	−0.150789	+	0.988566i	$0.548181\pi$
$6$	0	0
$7$	0.687902	0.260002	0.130001	−	0.991514i	$-0.458502\pi$
$7$	0.687902	0.260002	0.130001	+	0.991514i	$0.458502\pi$
$8$	0.354896	0.125475
$9$	0	0
$10$	0.0599493	0.0189576
$11$	−0.536956	−0.161898	−0.0809492	−	0.996718i	$-0.525795\pi$
$11$	−0.536956	−0.161898	−0.0809492	+	0.996718i	$0.525795\pi$
$12$	0	0
$13$	−0.628799	−0.174397	−0.0871987	−	0.996191i	$-0.527792\pi$
$13$	−0.628799	−0.174397	−0.0871987	+	0.996191i	$0.527792\pi$
$14$	−0.0611542	−0.0163441
$15$	0	0
$16$	3.95264	0.988161
$17$	5.05676	1.22644	0.613222	−	0.789911i	$-0.289873\pi$
$17$	5.05676	1.22644	0.613222	+	0.789911i	$0.289873\pi$
$18$	0	0
$19$	−0.724853	−0.166293	−0.0831463	−	0.996537i	$-0.526497\pi$
$19$	−0.724853	−0.166293	−0.0831463	+	0.996537i	$0.526497\pi$
$20$	1.34337	0.300386
$21$	0	0
$22$	0.0477352	0.0101772
$23$	−7.36157	−1.53499	−0.767496	−	0.641053i	$-0.778498\pi$
$23$	−7.36157	−1.53499	−0.767496	+	0.641053i	$0.778498\pi$
$24$	0	0
$25$	−4.54525	−0.909051
$26$	0.0559000	0.0109629
$27$	0	0
$28$	−1.37037	−0.258975
$29$	−3.61524	−0.671332	−0.335666	−	0.941981i	$-0.608961\pi$
$29$	−3.61524	−0.671332	−0.335666	+	0.941981i	$0.608961\pi$
$30$	0	0
$31$	5.74731	1.03225	0.516124	−	0.856514i	$-0.327374\pi$
$31$	5.74731	1.03225	0.516124	+	0.856514i	$0.327374\pi$
$32$	−1.06118	−0.187592
$33$	0	0
$34$	−0.449544	−0.0770961
$35$	−0.463885	−0.0784109
$36$	0	0
$37$	−0.110957	−0.0182413	−0.00912063	−	0.999958i	$-0.502903\pi$
$37$	−0.110957	−0.0182413	−0.00912063	+	0.999958i	$0.502903\pi$
$38$	0.0644391	0.0104534
$39$	0	0
$40$	−0.239323	−0.0378403
$41$	8.05251	1.25759	0.628796	−	0.777571i	$-0.283548\pi$
$41$	8.05251	1.25759	0.628796	+	0.777571i	$0.283548\pi$
$42$	0	0
$43$	12.2197	1.86349	0.931746	−	0.363111i	$-0.118285\pi$
$43$	12.2197	1.86349	0.931746	+	0.363111i	$0.118285\pi$
$44$	1.06967	0.161259
$45$	0	0
$46$	0.654440	0.0964919
$47$	1.48719	0.216929	0.108465	−	0.994100i	$-0.465407\pi$
$47$	1.48719	0.216929	0.108465	+	0.994100i	$0.465407\pi$
$48$	0	0
$49$	−6.52679	−0.932399
$50$	0.404071	0.0571443
$51$	0	0
$52$	1.25263	0.173708
$53$	−6.92286	−0.950927	−0.475464	−	0.879735i	$-0.657720\pi$
$53$	−6.92286	−0.950927	−0.475464	+	0.879735i	$0.657720\pi$
$54$	0	0
$55$	0.362095	0.0488249
$56$	0.244133	0.0326237
$57$	0	0
$58$	0.321393	0.0422010
$59$	−9.81893	−1.27832	−0.639158	−	0.769076i	$-0.720717\pi$
$59$	−9.81893	−1.27832	−0.639158	+	0.769076i	$0.720717\pi$
$60$	0	0
$61$	6.30376	0.807113	0.403557	−	0.914955i	$-0.367774\pi$
$61$	6.30376	0.807113	0.403557	+	0.914955i	$0.367774\pi$
$62$	−0.510934	−0.0648886
$63$	0	0
$64$	−7.81095	−0.976369
$65$	0.424030	0.0525944
$66$	0	0
$67$	−7.02929	−0.858764	−0.429382	−	0.903123i	$-0.641269\pi$
$67$	−7.02929	−0.858764	−0.429382	+	0.903123i	$0.641269\pi$
$68$	−10.0736	−1.22160
$69$	0	0
$70$	0.0412392	0.00492903
$71$	−9.21843	−1.09403	−0.547013	−	0.837124i	$-0.684235\pi$
$71$	−9.21843	−1.09403	−0.547013	+	0.837124i	$0.684235\pi$
$72$	0	0
$73$	−12.1931	−1.42710	−0.713548	−	0.700606i	$-0.752913\pi$
$73$	−12.1931	−1.42710	−0.713548	+	0.700606i	$0.752913\pi$
$74$	0.00986406	0.00114667
$75$	0	0
$76$	1.44398	0.165636
$77$	−0.369373	−0.0420940
$78$	0	0
$79$	−6.71164	−0.755118	−0.377559	−	0.925985i	$-0.623236\pi$
$79$	−6.71164	−0.755118	−0.377559	+	0.925985i	$0.623236\pi$
$80$	−2.66546	−0.298007
$81$	0	0
$82$	−0.715865	−0.0790541
$83$	−8.10511	−0.889651	−0.444825	−	0.895617i	$-0.646734\pi$
$83$	−8.10511	−0.889651	−0.444825	+	0.895617i	$0.646734\pi$
$84$	0	0
$85$	−3.41002	−0.369868
$86$	−1.08633	−0.117142
$87$	0	0
$88$	−0.190563	−0.0203141
$89$	−13.1414	−1.39298	−0.696492	−	0.717564i	$-0.745257\pi$
$89$	−13.1414	−1.39298	−0.696492	+	0.717564i	$0.745257\pi$
$90$	0	0
$91$	−0.432552	−0.0453438
$92$	14.6650	1.52893
$93$	0	0
$94$	−0.132211	−0.0136365
$95$	0.488803	0.0501502
$96$	0	0
$97$	8.31742	0.844506	0.422253	−	0.906478i	$-0.361239\pi$
$97$	8.31742	0.844506	0.422253	+	0.906478i	$0.361239\pi$
$98$	0.580229	0.0586120
$99$	0	0
$100$	9.05459	0.905459
$101$	−13.3756	−1.33092	−0.665462	−	0.746432i	$-0.731765\pi$
$101$	−13.3756	−1.33092	−0.665462	+	0.746432i	$0.731765\pi$
$102$	0	0
$103$	3.09332	0.304794	0.152397	−	0.988319i	$-0.451301\pi$
$103$	3.09332	0.304794	0.152397	+	0.988319i	$0.451301\pi$
$104$	−0.223158	−0.0218825
$105$	0	0
$106$	0.615439	0.0597767
$107$	−11.5001	−1.11175	−0.555877	−	0.831265i	$-0.687617\pi$
$107$	−11.5001	−1.11175	−0.555877	+	0.831265i	$0.687617\pi$
$108$	0	0
$109$	10.0986	0.967275	0.483637	−	0.875268i	$-0.339315\pi$
$109$	10.0986	0.967275	0.483637	+	0.875268i	$0.339315\pi$
$110$	−0.0321901	−0.00306921
$111$	0	0
$112$	2.71903	0.256924
$113$	−20.8956	−1.96570	−0.982848	−	0.184415i	$-0.940961\pi$
$113$	−20.8956	−1.96570	−0.982848	+	0.184415i	$0.940961\pi$
$114$	0	0
$115$	4.96426	0.462920
$116$	7.20190	0.668679
$117$	0	0
$118$	0.872898	0.0803568
$119$	3.47855	0.318878
$120$	0	0
$121$	−10.7117	−0.973789
$122$	−0.560401	−0.0507364
$123$	0	0
$124$	−11.4492	−1.02817
$125$	6.43683	0.575727
$126$	0	0
$127$	5.25364	0.466185	0.233092	−	0.972455i	$-0.425116\pi$
$127$	5.25364	0.466185	0.233092	+	0.972455i	$0.425116\pi$
$128$	2.81675	0.248968
$129$	0	0
$130$	−0.0376961	−0.00330616
$131$	−7.68069	−0.671065	−0.335532	−	0.942029i	$-0.608916\pi$
$131$	−7.68069	−0.671065	−0.335532	+	0.942029i	$0.608916\pi$
$132$	0	0
$133$	−0.498628	−0.0432365
$134$	0.624901	0.0539832
$135$	0	0
$136$	1.79462	0.153888
$137$	11.5008	0.982578	0.491289	−	0.870996i	$-0.336526\pi$
$137$	11.5008	0.982578	0.491289	+	0.870996i	$0.336526\pi$
$138$	0	0
$139$	17.1077	1.45106	0.725529	−	0.688192i	$-0.241595\pi$
$139$	17.1077	1.45106	0.725529	+	0.688192i	$0.241595\pi$
$140$	0.924105	0.0781011
$141$	0	0
$142$	0.819514	0.0687721
$143$	0.337638	0.0282347
$144$	0	0
$145$	2.43793	0.202459
$146$	1.08396	0.0897094
$147$	0	0
$148$	0.221038	0.0181692
$149$	−23.5946	−1.93294	−0.966470	−	0.256778i	$-0.917339\pi$
$149$	−23.5946	−1.93294	−0.966470	+	0.256778i	$0.917339\pi$
$150$	0	0
$151$	−11.3698	−0.925263	−0.462631	−	0.886551i	$-0.653095\pi$
$151$	−11.3698	−0.925263	−0.462631	+	0.886551i	$0.653095\pi$
$152$	−0.257247	−0.0208655
$153$	0	0
$154$	0.0328371	0.00264609
$155$	−3.87569	−0.311303
$156$	0	0
$157$	−0.914383	−0.0729757	−0.0364878	−	0.999334i	$-0.511617\pi$
$157$	−0.914383	−0.0729757	−0.0364878	+	0.999334i	$0.511617\pi$
$158$	0.596662	0.0474679
$159$	0	0
$160$	0.715605	0.0565735
$161$	−5.06403	−0.399102
$162$	0	0
$163$	−9.90233	−0.775610	−0.387805	−	0.921741i	$-0.626767\pi$
$163$	−9.90233	−0.775610	−0.387805	+	0.921741i	$0.626767\pi$
$164$	−16.0414	−1.25262
$165$	0	0
$166$	0.720541	0.0559248
$167$	−14.3435	−1.10993	−0.554965	−	0.831873i	$-0.687269\pi$
$167$	−14.3435	−1.10993	−0.554965	+	0.831873i	$0.687269\pi$
$168$	0	0
$169$	−12.6046	−0.969586
$170$	0.303149	0.0232505
$171$	0	0
$172$	−24.3429	−1.85613
$173$	7.17674	0.545637	0.272819	−	0.962065i	$-0.412044\pi$
$173$	7.17674	0.545637	0.272819	+	0.962065i	$0.412044\pi$
$174$	0	0
$175$	−3.12669	−0.236355
$176$	−2.12240	−0.159982
$177$	0	0
$178$	1.16826	0.0875651
$179$	18.9923	1.41955	0.709774	−	0.704429i	$-0.248797\pi$
$179$	18.9923	1.41955	0.709774	+	0.704429i	$0.248797\pi$
$180$	0	0
$181$	20.1558	1.49817	0.749086	−	0.662473i	$-0.230493\pi$
$181$	20.1558	1.49817	0.749086	+	0.662473i	$0.230493\pi$
$182$	0.0384537	0.00285038
$183$	0	0
$184$	−2.61259	−0.192603
$185$	0.0748239	0.00550116
$186$	0	0
$187$	−2.71526	−0.198559
$188$	−2.96263	−0.216072
$189$	0	0
$190$	−0.0434544	−0.00315251
$191$	7.35271	0.532023	0.266012	−	0.963970i	$-0.414294\pi$
$191$	7.35271	0.532023	0.266012	+	0.963970i	$0.414294\pi$
$192$	0	0
$193$	5.59970	0.403076	0.201538	−	0.979481i	$-0.435406\pi$
$193$	5.59970	0.403076	0.201538	+	0.979481i	$0.435406\pi$
$194$	−0.739415	−0.0530869
$195$	0	0
$196$	13.0020	0.928714
$197$	−23.4497	−1.67072	−0.835362	−	0.549700i	$-0.814742\pi$
$197$	−23.4497	−1.67072	−0.835362	+	0.549700i	$0.814742\pi$
$198$	0	0
$199$	8.09735	0.574006	0.287003	−	0.957930i	$-0.407341\pi$
$199$	8.09735	0.574006	0.287003	+	0.957930i	$0.407341\pi$
$200$	−1.61309	−0.114063
$201$	0	0
$202$	1.18909	0.0836639
$203$	−2.48693	−0.174548
$204$	0	0
$205$	−5.43020	−0.379261
$206$	−0.274995	−0.0191598
$207$	0	0
$208$	−2.48542	−0.172333
$209$	0.389214	0.0269225
$210$	0	0
$211$	−26.2444	−1.80674	−0.903371	−	0.428860i	$-0.858915\pi$
$211$	−26.2444	−1.80674	−0.903371	+	0.428860i	$0.858915\pi$
$212$	13.7910	0.947170
$213$	0	0
$214$	1.02235	0.0698865
$215$	−8.24036	−0.561988
$216$	0	0
$217$	3.95359	0.268387
$218$	−0.897765	−0.0608044
$219$	0	0
$220$	−0.721329	−0.0486320
$221$	−3.17968	−0.213889
$222$	0	0
$223$	5.41208	0.362420	0.181210	−	0.983444i	$-0.441999\pi$
$223$	5.41208	0.362420	0.181210	+	0.983444i	$0.441999\pi$
$224$	−0.729987	−0.0487743
$225$	0	0
$226$	1.85761	0.123567
$227$	23.2617	1.54394	0.771968	−	0.635661i	$-0.219272\pi$
$227$	23.2617	1.54394	0.771968	+	0.635661i	$0.219272\pi$
$228$	0	0
$229$	21.9483	1.45039	0.725193	−	0.688546i	$-0.241751\pi$
$229$	21.9483	1.45039	0.725193	+	0.688546i	$0.241751\pi$
$230$	−0.441321	−0.0290998
$231$	0	0
$232$	−1.28303	−0.0842352
$233$	4.41858	0.289471	0.144736	−	0.989470i	$-0.453767\pi$
$233$	4.41858	0.289471	0.144736	+	0.989470i	$0.453767\pi$
$234$	0	0
$235$	−1.00289	−0.0654210
$236$	19.5603	1.27326
$237$	0	0
$238$	−0.309242	−0.0200452
$239$	−1.00000	−0.0646846
$239$	−1.00000	−0.0646846
$240$	0	0
$241$	−11.2098	−0.722087	−0.361044	−	0.932549i	$-0.617579\pi$
$241$	−11.2098	−0.722087	−0.361044	+	0.932549i	$0.617579\pi$
$242$	0.952264	0.0612138
$243$	0	0
$244$	−12.5577	−0.803924
$245$	4.40133	0.281191
$246$	0	0
$247$	0.455787	0.0290010
$248$	2.03970	0.129521
$249$	0	0
$250$	−0.572231	−0.0361911
$251$	24.1478	1.52419	0.762096	−	0.647464i	$-0.224170\pi$
$251$	24.1478	1.52419	0.762096	+	0.647464i	$0.224170\pi$
$252$	0	0
$253$	3.95284	0.248513
$254$	−0.467046	−0.0293051
$255$	0	0
$256$	15.3715	0.960718
$257$	−23.7037	−1.47859	−0.739297	−	0.673379i	$-0.764842\pi$
$257$	−23.7037	−1.47859	−0.739297	+	0.673379i	$0.764842\pi$
$258$	0	0
$259$	−0.0763277	−0.00474277
$260$	−0.844708	−0.0523866
$261$	0	0
$262$	0.682810	0.0421841
$263$	19.8888	1.22640	0.613198	−	0.789929i	$-0.289883\pi$
$263$	19.8888	1.22640	0.613198	+	0.789929i	$0.289883\pi$
$264$	0	0
$265$	4.66842	0.286779
$266$	0.0443278	0.00271791
$267$	0	0
$268$	14.0030	0.855371
$269$	−16.7074	−1.01867	−0.509335	−	0.860568i	$-0.670109\pi$
$269$	−16.7074	−1.01867	−0.509335	+	0.860568i	$0.670109\pi$
$270$	0	0
$271$	18.0253	1.09496	0.547481	−	0.836818i	$-0.315587\pi$
$271$	18.0253	1.09496	0.547481	+	0.836818i	$0.315587\pi$
$272$	19.9876	1.21192
$273$	0	0
$274$	−1.02242	−0.0617664
$275$	2.44060	0.147174
$276$	0	0
$277$	−23.7188	−1.42513	−0.712563	−	0.701608i	$-0.752466\pi$
$277$	−23.7188	−1.42513	−0.712563	+	0.701608i	$0.752466\pi$
$278$	−1.52087	−0.0912157
$279$	0	0
$280$	−0.164631	−0.00983858
$281$	8.96828	0.535003	0.267501	−	0.963557i	$-0.413802\pi$
$281$	8.96828	0.535003	0.267501	+	0.963557i	$0.413802\pi$
$282$	0	0
$283$	2.74613	0.163240	0.0816201	−	0.996664i	$-0.473991\pi$
$283$	2.74613	0.163240	0.0816201	+	0.996664i	$0.473991\pi$
$284$	18.3640	1.08970
$285$	0	0
$286$	−0.0300158	−0.00177487
$287$	5.53934	0.326977
$288$	0	0
$289$	8.57079	0.504164
$290$	−0.216731	−0.0127269
$291$	0	0
$292$	24.2899	1.42146
$293$	26.5055	1.54847	0.774235	−	0.632898i	$-0.218135\pi$
$293$	26.5055	1.54847	0.774235	+	0.632898i	$0.218135\pi$
$294$	0	0
$295$	6.62138	0.385511
$296$	−0.0393783	−0.00228882
$297$	0	0
$298$	2.09755	0.121508
$299$	4.62895	0.267699
$300$	0	0
$301$	8.40598	0.484512
$302$	1.01077	0.0581634
$303$	0	0
$304$	−2.86509	−0.164324
$305$	−4.25093	−0.243407
$306$	0	0
$307$	−27.6982	−1.58082	−0.790410	−	0.612578i	$-0.790133\pi$
$307$	−27.6982	−1.58082	−0.790410	+	0.612578i	$0.790133\pi$
$308$	0.735827	0.0419276
$309$	0	0
$310$	0.344547	0.0195690
$311$	9.87576	0.560003	0.280001	−	0.960000i	$-0.409665\pi$
$311$	9.87576	0.560003	0.280001	+	0.960000i	$0.409665\pi$
$312$	0	0
$313$	−14.8397	−0.838792	−0.419396	−	0.907803i	$-0.637758\pi$
$313$	−14.8397	−0.838792	−0.419396	+	0.907803i	$0.637758\pi$
$314$	0.0812883	0.00458736
$315$	0	0
$316$	13.3702	0.752134
$317$	19.8717	1.11610	0.558052	−	0.829806i	$-0.311549\pi$
$317$	19.8717	1.11610	0.558052	+	0.829806i	$0.311549\pi$
$318$	0	0
$319$	1.94122	0.108688
$320$	5.26730	0.294451
$321$	0	0
$322$	0.450190	0.0250881
$323$	−3.66541	−0.203949
$324$	0	0
$325$	2.85805	0.158536
$326$	0.880313	0.0487560
$327$	0	0
$328$	2.85780	0.157796
$329$	1.02304	0.0564021
$330$	0	0
$331$	−15.2217	−0.836658	−0.418329	−	0.908296i	$-0.637384\pi$
$331$	−15.2217	−0.836658	−0.418329	+	0.908296i	$0.637384\pi$
$332$	16.1462	0.886135
$333$	0	0
$334$	1.27513	0.0697719
$335$	4.74019	0.258984
$336$	0	0
$337$	−31.2836	−1.70413	−0.852063	−	0.523440i	$-0.824648\pi$
$337$	−31.2836	−1.70413	−0.852063	+	0.523440i	$0.824648\pi$
$338$	1.12054	0.0609496
$339$	0	0
$340$	6.79308	0.368407
$341$	−3.08605	−0.167119
$342$	0	0
$343$	−9.30510	−0.502428
$344$	4.33673	0.233821
$345$	0	0
$346$	−0.638009	−0.0342996
$347$	−2.32374	−0.124745	−0.0623726	−	0.998053i	$-0.519867\pi$
$347$	−2.32374	−0.124745	−0.0623726	+	0.998053i	$0.519867\pi$
$348$	0	0
$349$	17.1644	0.918788	0.459394	−	0.888233i	$-0.348067\pi$
$349$	17.1644	0.918788	0.459394	+	0.888233i	$0.348067\pi$
$350$	0.277961	0.0148577
$351$	0	0
$352$	0.569807	0.0303708
$353$	−21.0255	−1.11907	−0.559537	−	0.828805i	$-0.689021\pi$
$353$	−21.0255	−1.11907	−0.559537	+	0.828805i	$0.689021\pi$
$354$	0	0
$355$	6.21643	0.329934
$356$	26.1789	1.38748
$357$	0	0
$358$	−1.68840	−0.0892350
$359$	−31.0325	−1.63783	−0.818917	−	0.573912i	$-0.805425\pi$
$359$	−31.0325	−1.63783	−0.818917	+	0.573912i	$0.805425\pi$
$360$	0	0
$361$	−18.4746	−0.972347
$362$	−1.79185	−0.0941773
$363$	0	0
$364$	0.861685	0.0451646
$365$	8.22240	0.430380
$366$	0	0
$367$	−0.316706	−0.0165319	−0.00826595	−	0.999966i	$-0.502631\pi$
$367$	−0.316706	−0.0165319	−0.00826595	+	0.999966i	$0.502631\pi$
$368$	−29.0976	−1.51682
$369$	0	0
$370$	−0.00665181	−0.000345811 0
$371$	−4.76224	−0.247243
$372$	0	0
$373$	−2.67698	−0.138609	−0.0693045	−	0.997596i	$-0.522078\pi$
$373$	−2.67698	−0.138609	−0.0693045	+	0.997596i	$0.522078\pi$
$374$	0.241385	0.0124817
$375$	0	0
$376$	0.527798	0.0272191
$377$	2.27326	0.117079
$378$	0	0
$379$	−17.6822	−0.908275	−0.454137	−	0.890932i	$-0.650052\pi$
$379$	−17.6822	−0.908275	−0.454137	+	0.890932i	$0.650052\pi$
$380$	−0.973744	−0.0499520
$381$	0	0
$382$	−0.653653	−0.0334438
$383$	−10.1403	−0.518145	−0.259072	−	0.965858i	$-0.583417\pi$
$383$	−10.1403	−0.518145	−0.259072	+	0.965858i	$0.583417\pi$
$384$	0	0
$385$	0.249086	0.0126946
$386$	−0.497811	−0.0253379
$387$	0	0
$388$	−16.5691	−0.841169
$389$	9.21319	0.467127	0.233564	−	0.972341i	$-0.424961\pi$
$389$	9.21319	0.467127	0.233564	+	0.972341i	$0.424961\pi$
$390$	0	0
$391$	−37.2256	−1.88258
$392$	−2.31633	−0.116992
$393$	0	0
$394$	2.08467	0.105024
$395$	4.52598	0.227727
$396$	0	0
$397$	30.4727	1.52938	0.764689	−	0.644399i	$-0.222892\pi$
$397$	30.4727	1.52938	0.764689	+	0.644399i	$0.222892\pi$
$398$	−0.719851	−0.0360829
$399$	0	0
$400$	−17.9658	−0.898289
$401$	34.4093	1.71832	0.859160	−	0.511707i	$-0.170987\pi$
$401$	34.4093	1.71832	0.859160	+	0.511707i	$0.170987\pi$
$402$	0	0
$403$	−3.61391	−0.180021
$404$	26.6455	1.32566
$405$	0	0
$406$	0.221087	0.0109724
$407$	0.0595792	0.00295323
$408$	0	0
$409$	−19.0633	−0.942618	−0.471309	−	0.881968i	$-0.656218\pi$
$409$	−19.0633	−0.942618	−0.471309	+	0.881968i	$0.656218\pi$
$410$	0.482742	0.0238409
$411$	0	0
$412$	−6.16219	−0.303589
$413$	−6.75446	−0.332365
$414$	0	0
$415$	5.46566	0.268299
$416$	0.667269	0.0327156
$417$	0	0
$418$	−0.0346010	−0.00169239
$419$	−3.95215	−0.193075	−0.0965375	−	0.995329i	$-0.530777\pi$
$419$	−3.95215	−0.193075	−0.0965375	+	0.995329i	$0.530777\pi$
$420$	0	0
$421$	27.8805	1.35881	0.679406	−	0.733762i	$-0.262237\pi$
$421$	27.8805	1.35881	0.679406	+	0.733762i	$0.262237\pi$
$422$	2.33312	0.113574
$423$	0	0
$424$	−2.45689	−0.119317
$425$	−22.9842	−1.11490
$426$	0	0
$427$	4.33637	0.209851
$428$	22.9093	1.10736
$429$	0	0
$430$	0.732564	0.0353274
$431$	−8.80087	−0.423923	−0.211961	−	0.977278i	$-0.567985\pi$
$431$	−8.80087	−0.423923	−0.211961	+	0.977278i	$0.567985\pi$
$432$	0	0
$433$	38.8304	1.86607	0.933034	−	0.359788i	$-0.117151\pi$
$433$	38.8304	1.86607	0.933034	+	0.359788i	$0.117151\pi$
$434$	−0.351472	−0.0168712
$435$	0	0
$436$	−20.1175	−0.963453
$437$	5.33605	0.255258
$438$	0	0
$439$	26.9487	1.28619	0.643095	−	0.765786i	$-0.277650\pi$
$439$	26.9487	1.28619	0.643095	+	0.765786i	$0.277650\pi$
$440$	0.128506	0.00612629
$441$	0	0
$442$	0.282673	0.0134454
$443$	14.5012	0.688972	0.344486	−	0.938792i	$-0.388053\pi$
$443$	14.5012	0.688972	0.344486	+	0.938792i	$0.388053\pi$
$444$	0	0
$445$	8.86187	0.420093
$446$	−0.481132	−0.0227823
$447$	0	0
$448$	−5.37316	−0.253858
$449$	−8.58911	−0.405345	−0.202673	−	0.979247i	$-0.564963\pi$
$449$	−8.58911	−0.405345	−0.202673	+	0.979247i	$0.564963\pi$
$450$	0	0
$451$	−4.32385	−0.203602
$452$	41.6261	1.95793
$453$	0	0
$454$	−2.06796	−0.0970541
$455$	0.291691	0.0136747
$456$	0	0
$457$	13.4901	0.631040	0.315520	−	0.948919i	$-0.397821\pi$
$457$	13.4901	0.631040	0.315520	+	0.948919i	$0.397821\pi$
$458$	−1.95120	−0.0911734
$459$	0	0
$460$	−9.88929	−0.461090
$461$	20.1254	0.937332	0.468666	−	0.883375i	$-0.344735\pi$
$461$	20.1254	0.937332	0.468666	+	0.883375i	$0.344735\pi$
$462$	0	0
$463$	19.2303	0.893709	0.446855	−	0.894607i	$-0.352544\pi$
$463$	19.2303	0.893709	0.446855	+	0.894607i	$0.352544\pi$
$464$	−14.2897	−0.663384
$465$	0	0
$466$	−0.392810	−0.0181966
$467$	17.7598	0.821828	0.410914	−	0.911674i	$-0.365210\pi$
$467$	17.7598	0.821828	0.410914	+	0.911674i	$0.365210\pi$
$468$	0	0
$469$	−4.83546	−0.223281
$470$	0.0891561	0.00411246
$471$	0	0
$472$	−3.48470	−0.160396
$473$	−6.56146	−0.301696
$474$	0	0
$475$	3.29464	0.151168
$476$	−6.92961	−0.317618
$477$	0	0
$478$	0.0888996	0.00406617
$479$	28.8979	1.32038	0.660189	−	0.751100i	$-0.270476\pi$
$479$	28.8979	1.32038	0.660189	+	0.751100i	$0.270476\pi$
$480$	0	0
$481$	0.0697698	0.00318123
$482$	0.996548	0.0453915
$483$	0	0
$484$	21.3387	0.969941
$485$	−5.60884	−0.254684
$486$	0	0
$487$	−8.62899	−0.391017	−0.195508	−	0.980702i	$-0.562636\pi$
$487$	−8.62899	−0.391017	−0.195508	+	0.980702i	$0.562636\pi$
$488$	2.23718	0.101272
$489$	0	0
$490$	−0.391276	−0.0176761
$491$	−24.6724	−1.11345	−0.556725	−	0.830697i	$-0.687942\pi$
$491$	−24.6724	−1.11345	−0.556725	+	0.830697i	$0.687942\pi$
$492$	0	0
$493$	−18.2814	−0.823351
$494$	−0.0405193	−0.00182305
$495$	0	0
$496$	22.7171	1.02003
$497$	−6.34137	−0.284449
$498$	0	0
$499$	−0.336524	−0.0150649	−0.00753244	−	0.999972i	$-0.502398\pi$
$499$	−0.336524	−0.0150649	−0.00753244	+	0.999972i	$0.502398\pi$
$500$	−12.8228	−0.573452
$501$	0	0
$502$	−2.14673	−0.0958131
$503$	−33.4007	−1.48926	−0.744632	−	0.667476i	$-0.767375\pi$
$503$	−33.4007	−1.48926	−0.744632	+	0.667476i	$0.767375\pi$
$504$	0	0
$505$	9.01983	0.401377
$506$	−0.351406	−0.0156219
$507$	0	0
$508$	−10.4658	−0.464343
$509$	−35.3725	−1.56786	−0.783929	−	0.620851i	$-0.786787\pi$
$509$	−35.3725	−1.56786	−0.783929	+	0.620851i	$0.786787\pi$
$510$	0	0
$511$	−8.38766	−0.371048
$512$	−7.00002	−0.309360
$513$	0	0
$514$	2.10725	0.0929466
$515$	−2.08598	−0.0919191
$516$	0	0
$517$	−0.798557	−0.0351205
$518$	0.00678550	0.000298138 0
$519$	0	0
$520$	0.150486	0.00659926
$521$	11.3345	0.496572	0.248286	−	0.968687i	$-0.420133\pi$
$521$	11.3345	0.496572	0.248286	+	0.968687i	$0.420133\pi$
$522$	0	0
$523$	−40.7825	−1.78329	−0.891646	−	0.452733i	$-0.850449\pi$
$523$	−40.7825	−1.78329	−0.891646	+	0.452733i	$0.850449\pi$
$524$	15.3007	0.668413
$525$	0	0
$526$	−1.76811	−0.0770931
$527$	29.0628	1.26599
$528$	0	0
$529$	31.1926	1.35620
$530$	−0.415020	−0.0180273
$531$	0	0
$532$	0.993314	0.0430656
$533$	−5.06341	−0.219321
$534$	0	0
$535$	7.75505	0.335280
$536$	−2.49466	−0.107753
$537$	0	0
$538$	1.48528	0.0640352
$539$	3.50460	0.150954
$540$	0	0
$541$	15.1438	0.651082	0.325541	−	0.945528i	$-0.394454\pi$
$541$	15.1438	0.651082	0.325541	+	0.945528i	$0.394454\pi$
$542$	−1.60245	−0.0688310
$543$	0	0
$544$	−5.36613	−0.230071
$545$	−6.81000	−0.291709
$546$	0	0
$547$	28.0014	1.19725	0.598627	−	0.801028i	$-0.295713\pi$
$547$	28.0014	1.19725	0.598627	+	0.801028i	$0.295713\pi$
$548$	−22.9107	−0.978696
$549$	0	0
$550$	−0.216969	−0.00925157
$551$	2.62051	0.111638
$552$	0	0
$553$	−4.61695	−0.196333
$554$	2.10859	0.0895855
$555$	0	0
$556$	−34.0802	−1.44532
$557$	39.4275	1.67060	0.835298	−	0.549798i	$-0.185295\pi$
$557$	39.4275	1.67060	0.835298	+	0.549798i	$0.185295\pi$
$558$	0	0
$559$	−7.68376	−0.324988
$560$	−1.83357	−0.0774826
$561$	0	0
$562$	−0.797276	−0.0336311
$563$	−11.7452	−0.495003	−0.247502	−	0.968887i	$-0.579610\pi$
$563$	−11.7452	−0.495003	−0.247502	+	0.968887i	$0.579610\pi$
$564$	0	0
$565$	14.0909	0.592810
$566$	−0.244129	−0.0102615
$567$	0	0
$568$	−3.27158	−0.137272
$569$	16.2860	0.682743	0.341371	−	0.939928i	$-0.389109\pi$
$569$	16.2860	0.682743	0.341371	+	0.939928i	$0.389109\pi$
$570$	0	0
$571$	3.21609	0.134589	0.0672946	−	0.997733i	$-0.478563\pi$
$571$	3.21609	0.134589	0.0672946	+	0.997733i	$0.478563\pi$
$572$	−0.672607	−0.0281231
$573$	0	0
$574$	−0.492445	−0.0205542
$575$	33.4602	1.39539
$576$	0	0
$577$	22.9258	0.954412	0.477206	−	0.878792i	$-0.341650\pi$
$577$	22.9258	0.954412	0.477206	+	0.878792i	$0.341650\pi$
$578$	−0.761940	−0.0316925
$579$	0	0
$580$	−4.85659	−0.201659
$581$	−5.57552	−0.231311
$582$	0	0
$583$	3.71727	0.153954
$584$	−4.32728	−0.179064
$585$	0	0
$586$	−2.35633	−0.0973392
$587$	7.86672	0.324694	0.162347	−	0.986734i	$-0.448094\pi$
$587$	7.86672	0.324694	0.162347	+	0.986734i	$0.448094\pi$
$588$	0	0
$589$	−4.16596	−0.171655
$590$	−0.588638	−0.0242338
$591$	0	0
$592$	−0.438575	−0.0180253
$593$	−6.59271	−0.270730	−0.135365	−	0.990796i	$-0.543221\pi$
$593$	−6.59271	−0.270730	−0.135365	+	0.990796i	$0.543221\pi$
$594$	0	0
$595$	−2.34576	−0.0961666
$596$	47.0026	1.92530
$597$	0	0
$598$	−0.411511	−0.0168280
$599$	32.3184	1.32050	0.660248	−	0.751048i	$-0.270451\pi$
$599$	32.3184	1.32050	0.660248	+	0.751048i	$0.270451\pi$
$600$	0	0
$601$	−34.1450	−1.39280	−0.696402	−	0.717652i	$-0.745217\pi$
$601$	−34.1450	−1.39280	−0.696402	+	0.717652i	$0.745217\pi$
$602$	−0.747288	−0.0304572
$603$	0	0
$604$	22.6498	0.921606
$605$	7.22340	0.293673
$606$	0	0
$607$	−19.5667	−0.794186	−0.397093	−	0.917778i	$-0.629981\pi$
$607$	−19.5667	−0.794186	−0.397093	+	0.917778i	$0.629981\pi$
$608$	0.769199	0.0311952
$609$	0	0
$610$	0.377906	0.0153010
$611$	−0.935145	−0.0378319
$612$	0	0
$613$	−46.3571	−1.87235	−0.936174	−	0.351538i	$-0.885659\pi$
$613$	−46.3571	−1.87235	−0.936174	+	0.351538i	$0.885659\pi$
$614$	2.46236	0.0993728
$615$	0	0
$616$	−0.131089	−0.00528172
$617$	−33.5775	−1.35178	−0.675889	−	0.737004i	$-0.736240\pi$
$617$	−33.5775	−1.35178	−0.675889	+	0.737004i	$0.736240\pi$
$618$	0	0
$619$	−9.59081	−0.385487	−0.192744	−	0.981249i	$-0.561739\pi$
$619$	−9.59081	−0.385487	−0.192744	+	0.981249i	$0.561739\pi$
$620$	7.72075	0.310073
$621$	0	0
$622$	−0.877951	−0.0352026
$623$	−9.03998	−0.362179
$624$	0	0
$625$	18.3856	0.735424
$626$	1.31925	0.0527277
$627$	0	0
$628$	1.82154	0.0726873
$629$	−0.561084	−0.0223719
$630$	0	0
$631$	−30.4228	−1.21111	−0.605556	−	0.795803i	$-0.707049\pi$
$631$	−30.4228	−1.21111	−0.605556	+	0.795803i	$0.707049\pi$
$632$	−2.38193	−0.0947482
$633$	0	0
$634$	−1.76658	−0.0701600
$635$	−3.54278	−0.140591
$636$	0	0
$637$	4.10404	0.162608
$638$	−0.172574	−0.00683227
$639$	0	0
$640$	−1.89947	−0.0750832
$641$	−19.5297	−0.771376	−0.385688	−	0.922629i	$-0.626036\pi$
$641$	−19.5297	−0.771376	−0.385688	+	0.922629i	$0.626036\pi$
$642$	0	0
$643$	41.7587	1.64680	0.823401	−	0.567460i	$-0.192074\pi$
$643$	41.7587	1.64680	0.823401	+	0.567460i	$0.192074\pi$
$644$	10.0880	0.397525
$645$	0	0
$646$	0.325853	0.0128205
$647$	−25.7021	−1.01046	−0.505228	−	0.862986i	$-0.668591\pi$
$647$	−25.7021	−1.01046	−0.505228	+	0.862986i	$0.668591\pi$
$648$	0	0
$649$	5.27233	0.206957
$650$	−0.254080	−0.00996582
$651$	0	0
$652$	19.7264	0.772545
$653$	−5.47922	−0.214418	−0.107209	−	0.994236i	$-0.534191\pi$
$653$	−5.47922	−0.214418	−0.107209	+	0.994236i	$0.534191\pi$
$654$	0	0
$655$	5.17946	0.202378
$656$	31.8287	1.24270
$657$	0	0
$658$	−0.0909480	−0.00354552
$659$	41.0759	1.60009	0.800044	−	0.599941i	$-0.204809\pi$
$659$	41.0759	1.60009	0.800044	+	0.599941i	$0.204809\pi$
$660$	0	0
$661$	0.166915	0.00649225	0.00324612	−	0.999995i	$-0.498967\pi$
$661$	0.166915	0.00649225	0.00324612	+	0.999995i	$0.498967\pi$
$662$	1.35320	0.0525936
$663$	0	0
$664$	−2.87647	−0.111629
$665$	0.336249	0.0130392
$666$	0	0
$667$	26.6138	1.03049
$668$	28.5736	1.10554
$669$	0	0
$670$	−0.421401	−0.0162801
$671$	−3.38484	−0.130670
$672$	0	0
$673$	−31.5591	−1.21651	−0.608257	−	0.793741i	$-0.708131\pi$
$673$	−31.5591	−1.21651	−0.608257	+	0.793741i	$0.708131\pi$
$674$	2.78110	0.107124
$675$	0	0
$676$	25.1096	0.965754
$677$	−28.9919	−1.11425	−0.557124	−	0.830429i	$-0.688095\pi$
$677$	−28.9919	−1.11425	−0.557124	+	0.830429i	$0.688095\pi$
$678$	0	0
$679$	5.72157	0.219574
$680$	−1.21020	−0.0464091
$681$	0	0
$682$	0.274349	0.0105054
$683$	−29.7746	−1.13929	−0.569646	−	0.821890i	$-0.692920\pi$
$683$	−29.7746	−1.13929	−0.569646	+	0.821890i	$0.692920\pi$
$684$	0	0
$685$	−7.75554	−0.296324
$686$	0.827220	0.0315834
$687$	0	0
$688$	48.3003	1.84143
$689$	4.35309	0.165839
$690$	0	0
$691$	−21.5776	−0.820852	−0.410426	−	0.911894i	$-0.634620\pi$
$691$	−21.5776	−0.820852	−0.410426	+	0.911894i	$0.634620\pi$
$692$	−14.2968	−0.543481
$693$	0	0
$694$	0.206580	0.00784167
$695$	−11.5366	−0.437607
$696$	0	0
$697$	40.7196	1.54236
$698$	−1.52591	−0.0577564
$699$	0	0
$700$	6.22867	0.235421
$701$	−2.79934	−0.105730	−0.0528649	−	0.998602i	$-0.516835\pi$
$701$	−2.79934	−0.105730	−0.0528649	+	0.998602i	$0.516835\pi$
$702$	0	0
$703$	0.0804277	0.00303339
$704$	4.19414	0.158072
$705$	0	0
$706$	1.86916	0.0703467
$707$	−9.20111	−0.346043
$708$	0	0
$709$	−46.7987	−1.75756	−0.878781	−	0.477225i	$-0.841643\pi$
$709$	−46.7987	−1.75756	−0.878781	+	0.477225i	$0.841643\pi$
$710$	−0.552638	−0.0207401
$711$	0	0
$712$	−4.66382	−0.174784
$713$	−42.3092	−1.58449
$714$	0	0
$715$	−0.227685	−0.00851495
$716$	−37.8344	−1.41394
$717$	0	0
$718$	2.75878	0.102957
$719$	−6.28186	−0.234274	−0.117137	−	0.993116i	$-0.537372\pi$
$719$	−6.28186	−0.234274	−0.117137	+	0.993116i	$0.537372\pi$
$720$	0	0
$721$	2.12790	0.0792471
$722$	1.64238	0.0611232
$723$	0	0
$724$	−40.1524	−1.49225
$725$	16.4322	0.610275
$726$	0	0
$727$	5.34229	0.198135	0.0990673	−	0.995081i	$-0.468414\pi$
$727$	5.34229	0.198135	0.0990673	+	0.995081i	$0.468414\pi$
$728$	−0.153511	−0.00568949
$729$	0	0
$730$	−0.730968	−0.0270544
$731$	61.7922	2.28547
$732$	0	0
$733$	−3.43108	−0.126730	−0.0633650	−	0.997990i	$-0.520183\pi$
$733$	−3.43108	−0.126730	−0.0633650	+	0.997990i	$0.520183\pi$
$734$	0.0281550	0.00103922
$735$	0	0
$736$	7.81195	0.287952
$737$	3.77442	0.139033
$738$	0	0
$739$	45.0227	1.65619	0.828093	−	0.560591i	$-0.189426\pi$
$739$	45.0227	1.65619	0.828093	+	0.560591i	$0.189426\pi$
$740$	−0.149056	−0.00547942
$741$	0	0
$742$	0.423362	0.0155421
$743$	−0.484953	−0.0177912	−0.00889560	−	0.999960i	$-0.502832\pi$
$743$	−0.484953	−0.0177912	−0.00889560	+	0.999960i	$0.502832\pi$
$744$	0	0
$745$	15.9109	0.582932
$746$	0.237983	0.00871317
$747$	0	0
$748$	5.40905	0.197775
$749$	−7.91092	−0.289059
$750$	0	0
$751$	3.09423	0.112910	0.0564550	−	0.998405i	$-0.482020\pi$
$751$	3.09423	0.112910	0.0564550	+	0.998405i	$0.482020\pi$
$752$	5.87834	0.214361
$753$	0	0
$754$	−0.202092	−0.00735974
$755$	7.66722	0.279039
$756$	0	0
$757$	−21.9908	−0.799271	−0.399635	−	0.916674i	$-0.630863\pi$
$757$	−21.9908	−0.799271	−0.399635	+	0.916674i	$0.630863\pi$
$758$	1.57194	0.0570955
$759$	0	0
$760$	0.173474	0.00629257
$761$	14.2000	0.514748	0.257374	−	0.966312i	$-0.417143\pi$
$761$	14.2000	0.514748	0.257374	+	0.966312i	$0.417143\pi$
$762$	0	0
$763$	6.94688	0.251494
$764$	−14.6473	−0.529921
$765$	0	0
$766$	0.901468	0.0325713
$767$	6.17413	0.222935
$768$	0	0
$769$	−5.78312	−0.208545	−0.104272	−	0.994549i	$-0.533251\pi$
$769$	−5.78312	−0.208545	−0.104272	+	0.994549i	$0.533251\pi$
$770$	−0.0221437	−0.000798002 0
$771$	0	0
$772$	−11.1552	−0.401483
$773$	40.3814	1.45242	0.726210	−	0.687473i	$-0.241280\pi$
$773$	40.3814	1.45242	0.726210	+	0.687473i	$0.241280\pi$
$774$	0	0
$775$	−26.1230	−0.938366
$776$	2.95182	0.105964
$777$	0	0
$778$	−0.819049	−0.0293643
$779$	−5.83689	−0.209128
$780$	0	0
$781$	4.94989	0.177121
$782$	3.30935	0.118342
$783$	0	0
$784$	−25.7981	−0.921360
$785$	0.616612	0.0220078
$786$	0	0
$787$	30.2097	1.07686	0.538430	−	0.842670i	$-0.319017\pi$
$787$	30.2097	1.07686	0.538430	+	0.842670i	$0.319017\pi$
$788$	46.7141	1.66412
$789$	0	0
$790$	−0.402358	−0.0143153
$791$	−14.3741	−0.511086
$792$	0	0
$793$	−3.96380	−0.140759
$794$	−2.70901	−0.0961390
$795$	0	0
$796$	−16.1307	−0.571738
$797$	28.5813	1.01240	0.506200	−	0.862416i	$-0.331050\pi$
$797$	28.5813	1.01240	0.506200	+	0.862416i	$0.331050\pi$
$798$	0	0
$799$	7.52037	0.266052
$800$	4.82333	0.170531
$801$	0	0
$802$	−3.05898	−0.108016
$803$	6.54717	0.231044
$804$	0	0
$805$	3.41492	0.120360
$806$	0.321275	0.0113164
$807$	0	0
$808$	−4.74695	−0.166997
$809$	−42.6781	−1.50048	−0.750241	−	0.661164i	$-0.770063\pi$
$809$	−42.6781	−1.50048	−0.750241	+	0.661164i	$0.770063\pi$
$810$	0	0
$811$	−14.9127	−0.523655	−0.261828	−	0.965115i	$-0.584325\pi$
$811$	−14.9127	−0.523655	−0.261828	+	0.965115i	$0.584325\pi$
$812$	4.95420	0.173858
$813$	0	0
$814$	−0.00529657	−0.000185645 0
$815$	6.67762	0.233907
$816$	0	0
$817$	−8.85751	−0.309885
$818$	1.69472	0.0592544
$819$	0	0
$820$	10.8175	0.377763
$821$	−25.8710	−0.902905	−0.451452	−	0.892295i	$-0.649094\pi$
$821$	−25.8710	−0.902905	−0.451452	+	0.892295i	$0.649094\pi$
$822$	0	0
$823$	19.4859	0.679236	0.339618	−	0.940563i	$-0.389702\pi$
$823$	19.4859	0.679236	0.339618	+	0.940563i	$0.389702\pi$
$824$	1.09781	0.0382439
$825$	0	0
$826$	0.600468	0.0208930
$827$	42.7539	1.48670	0.743349	−	0.668903i	$-0.233236\pi$
$827$	42.7539	1.48670	0.743349	+	0.668903i	$0.233236\pi$
$828$	0	0
$829$	−21.6962	−0.753542	−0.376771	−	0.926307i	$-0.622966\pi$
$829$	−21.6962	−0.753542	−0.376771	+	0.926307i	$0.622966\pi$
$830$	−0.485895	−0.0168657
$831$	0	0
$832$	4.91152	0.170276
$833$	−33.0044	−1.14353
$834$	0	0
$835$	9.67249	0.334730
$836$	−0.775352	−0.0268161
$837$	0	0
$838$	0.351344	0.0121370
$839$	53.7944	1.85719	0.928595	−	0.371095i	$-0.121017\pi$
$839$	53.7944	1.85719	0.928595	+	0.371095i	$0.121017\pi$
$840$	0	0
$841$	−15.9301	−0.549313
$842$	−2.47857	−0.0854170
$843$	0	0
$844$	52.2815	1.79960
$845$	8.49990	0.292405
$846$	0	0
$847$	−7.36858	−0.253187
$848$	−27.3636	−0.939669
$849$	0	0
$850$	2.04329	0.0700843
$851$	0.816819	0.0280002
$852$	0	0
$853$	−0.311946	−0.0106808	−0.00534041	−	0.999986i	$-0.501700\pi$
$853$	−0.311946	−0.0106808	−0.00534041	+	0.999986i	$0.501700\pi$
$854$	−0.385501	−0.0131916
$855$	0	0
$856$	−4.08133	−0.139497
$857$	−12.6856	−0.433330	−0.216665	−	0.976246i	$-0.569518\pi$
$857$	−12.6856	−0.433330	−0.216665	+	0.976246i	$0.569518\pi$
$858$	0	0
$859$	−1.87231	−0.0638823	−0.0319411	−	0.999490i	$-0.510169\pi$
$859$	−1.87231	−0.0638823	−0.0319411	+	0.999490i	$0.510169\pi$
$860$	16.4156	0.559767
$861$	0	0
$862$	0.782394	0.0266484
$863$	−27.4289	−0.933692	−0.466846	−	0.884339i	$-0.654610\pi$
$863$	−27.4289	−0.933692	−0.466846	+	0.884339i	$0.654610\pi$
$864$	0	0
$865$	−4.83962	−0.164552
$866$	−3.45200	−0.117304
$867$	0	0
$868$	−7.87593	−0.267326
$869$	3.60385	0.122252
$870$	0	0
$871$	4.42001	0.149766
$872$	3.58397	0.121368
$873$	0	0
$874$	−0.474373	−0.0160459
$875$	4.42790	0.149690
$876$	0	0
$877$	−20.8318	−0.703441	−0.351720	−	0.936105i	$-0.614403\pi$
$877$	−20.8318	−0.703441	−0.351720	+	0.936105i	$0.614403\pi$
$878$	−2.39573	−0.0808519
$879$	0	0
$880$	1.43123	0.0482469
$881$	−11.4448	−0.385584	−0.192792	−	0.981240i	$-0.561754\pi$
$881$	−11.4448	−0.385584	−0.192792	+	0.981240i	$0.561754\pi$
$882$	0	0
$883$	2.46065	0.0828076	0.0414038	−	0.999142i	$-0.486817\pi$
$883$	2.46065	0.0828076	0.0414038	+	0.999142i	$0.486817\pi$
$884$	6.33424	0.213044
$885$	0	0
$886$	−1.28915	−0.0433098
$887$	43.7435	1.46876	0.734382	−	0.678737i	$-0.237472\pi$
$887$	43.7435	1.46876	0.734382	+	0.678737i	$0.237472\pi$
$888$	0	0
$889$	3.61399	0.121209
$890$	−0.787817	−0.0264077
$891$	0	0
$892$	−10.7814	−0.360988
$893$	−1.07800	−0.0360737
$894$	0	0
$895$	−12.8074	−0.428104
$896$	1.93765	0.0647322
$897$	0	0
$898$	0.763568	0.0254806
$899$	−20.7779	−0.692981
$900$	0	0
$901$	−35.0072	−1.16626
$902$	0.384388	0.0127987
$903$	0	0
$904$	−7.41578	−0.246645
$905$	−13.5921	−0.451815
$906$	0	0
$907$	−29.9761	−0.995339	−0.497669	−	0.867367i	$-0.665811\pi$
$907$	−29.9761	−0.995339	−0.497669	+	0.867367i	$0.665811\pi$
$908$	−46.3396	−1.53784
$909$	0	0
$910$	−0.0259312	−0.000859610 0
$911$	−50.0272	−1.65748	−0.828738	−	0.559637i	$-0.810941\pi$
$911$	−50.0272	−1.65748	−0.828738	+	0.559637i	$0.810941\pi$
$912$	0	0
$913$	4.35209	0.144033
$914$	−1.19926	−0.0396682
$915$	0	0
$916$	−43.7232	−1.44465
$917$	−5.28356	−0.174478
$918$	0	0
$919$	24.1329	0.796070	0.398035	−	0.917370i	$-0.369692\pi$
$919$	24.1329	0.796070	0.398035	+	0.917370i	$0.369692\pi$
$920$	1.76179	0.0580846
$921$	0	0
$922$	−1.78914	−0.0589221
$923$	5.79654	0.190795
$924$	0	0
$925$	0.504329	0.0165822
$926$	−1.70957	−0.0561799
$927$	0	0
$928$	3.83641	0.125936
$929$	−49.0310	−1.60865	−0.804327	−	0.594186i	$-0.797474\pi$
$929$	−49.0310	−1.60865	−0.804327	+	0.594186i	$0.797474\pi$
$930$	0	0
$931$	4.73096	0.155051
$932$	−8.80225	−0.288327
$933$	0	0
$934$	−1.57884	−0.0516613
$935$	1.83103	0.0598810
$936$	0	0
$937$	0.755623	0.0246851	0.0123426	−	0.999924i	$-0.496071\pi$
$937$	0.755623	0.0246851	0.0123426	+	0.999924i	$0.496071\pi$
$938$	0.429870	0.0140358
$939$	0	0
$940$	1.99784	0.0651625
$941$	12.4403	0.405542	0.202771	−	0.979226i	$-0.435005\pi$
$941$	12.4403	0.405542	0.202771	+	0.979226i	$0.435005\pi$
$942$	0	0
$943$	−59.2791	−1.93039
$944$	−38.8107	−1.26318
$945$	0	0
$946$	0.583311	0.0189651
$947$	37.7555	1.22689	0.613445	−	0.789738i	$-0.289783\pi$
$947$	37.7555	1.22689	0.613445	+	0.789738i	$0.289783\pi$
$948$	0	0
$949$	7.66702	0.248882
$950$	−0.292892	−0.00950268
$951$	0	0
$952$	1.23452	0.0400111
$953$	6.27937	0.203409	0.101704	−	0.994815i	$-0.467570\pi$
$953$	6.27937	0.203409	0.101704	+	0.994815i	$0.467570\pi$
$954$	0	0
$955$	−4.95829	−0.160446
$956$	1.99210	0.0644290
$957$	0	0
$958$	−2.56901	−0.0830009
$959$	7.91141	0.255473
$960$	0	0
$961$	2.03160	0.0655355
$962$	−0.00620251	−0.000199977 0
$963$	0	0
$964$	22.3310	0.719234
$965$	−3.77615	−0.121559
$966$	0	0
$967$	41.8287	1.34512	0.672560	−	0.740042i	$-0.265195\pi$
$967$	41.8287	1.34512	0.672560	+	0.740042i	$0.265195\pi$
$968$	−3.80153	−0.122186
$969$	0	0
$970$	0.498623	0.0160098
$971$	23.0751	0.740515	0.370258	−	0.928929i	$-0.379269\pi$
$971$	23.0751	0.740515	0.370258	+	0.928929i	$0.379269\pi$
$972$	0	0
$973$	11.7684	0.377278
$974$	0.767114	0.0245799
$975$	0	0
$976$	24.9165	0.797558
$977$	15.2960	0.489363	0.244682	−	0.969603i	$-0.421317\pi$
$977$	15.2960	0.489363	0.244682	+	0.969603i	$0.421317\pi$
$978$	0	0
$979$	7.05635	0.225522
$980$	−8.76788	−0.280080
$981$	0	0
$982$	2.19337	0.0699931
$983$	−43.5763	−1.38987	−0.694935	−	0.719073i	$-0.744567\pi$
$983$	−43.5763	−1.38987	−0.694935	+	0.719073i	$0.744567\pi$
$984$	0	0
$985$	15.8133	0.503853
$986$	1.62521	0.0517571
$987$	0	0
$988$	−0.907972	−0.0288864
$989$	−89.9564	−2.86045
$990$	0	0
$991$	33.3453	1.05925	0.529624	−	0.848232i	$-0.322333\pi$
$991$	33.3453	1.05925	0.529624	+	0.848232i	$0.322333\pi$
$992$	−6.09893	−0.193641
$993$	0	0
$994$	0.563745	0.0178809
$995$	−5.46044	−0.173107
$996$	0	0
$997$	42.7971	1.35540	0.677699	−	0.735340i	$-0.262977\pi$
$997$	42.7971	1.35540	0.677699	+	0.735340i	$0.262977\pi$
$998$	0.0299168	0.000947001 0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.a.j.1.11	✓	20
3.2	odd	2		2151.2.a.k.1.10	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.2.a.j.1.11	✓	20	1.1	even	1	trivial
2151.2.a.k.1.10	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)

\(f(q)\)	\(=\)	\(q-0.0888996 q^{2} -1.99210 q^{4} -0.674348 q^{5} +0.687902 q^{7} +0.354896 q^{8} +O(q^{10})\)	\(q-0.0888996 q^{2} -1.99210 q^{4} -0.674348 q^{5} +0.687902 q^{7} +0.354896 q^{8} +0.0599493 q^{10} -0.536956 q^{11} -0.628799 q^{13} -0.0611542 q^{14} +3.95264 q^{16} +5.05676 q^{17} -0.724853 q^{19} +1.34337 q^{20} +0.0477352 q^{22} -7.36157 q^{23} -4.54525 q^{25} +0.0559000 q^{26} -1.37037 q^{28} -3.61524 q^{29} +5.74731 q^{31} -1.06118 q^{32} -0.449544 q^{34} -0.463885 q^{35} -0.110957 q^{37} +0.0644391 q^{38} -0.239323 q^{40} +8.05251 q^{41} +12.2197 q^{43} +1.06967 q^{44} +0.654440 q^{46} +1.48719 q^{47} -6.52679 q^{49} +0.404071 q^{50} +1.25263 q^{52} -6.92286 q^{53} +0.362095 q^{55} +0.244133 q^{56} +0.321393 q^{58} -9.81893 q^{59} +6.30376 q^{61} -0.510934 q^{62} -7.81095 q^{64} +0.424030 q^{65} -7.02929 q^{67} -10.0736 q^{68} +0.0412392 q^{70} -9.21843 q^{71} -12.1931 q^{73} +0.00986406 q^{74} +1.44398 q^{76} -0.369373 q^{77} -6.71164 q^{79} -2.66546 q^{80} -0.715865 q^{82} -8.10511 q^{83} -3.41002 q^{85} -1.08633 q^{86} -0.190563 q^{88} -13.1414 q^{89} -0.432552 q^{91} +14.6650 q^{92} -0.132211 q^{94} +0.488803 q^{95} +8.31742 q^{97} +0.580229 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{2} + 18 q^{4} - 16 q^{5} - 4 q^{7} - 12 q^{8}+O(q^{10}) \) 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

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