LMFDB - Embedded newform 1143.2.a.j.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1143,2,Mod(1,1143)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1143, 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("1143.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1143 = 3^{2} \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1143.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:	$9.12690095103$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 2x^{8} - 14x^{7} + 26x^{6} + 59x^{5} - 99x^{4} - 66x^{3} + 102x^{2} - 24x + 1 $ x^9 - 2x^8 - 14x^7 + 26x^6 + 59x^5 - 99x^4 - 66x^3 + 102x^2 - 24x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 381)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.682747$ of defining polynomial
Character	$\chi$	$=$	1143.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.682747 q^{2} -1.53386 q^{4} -3.92611 q^{5} +1.75230 q^{7} -2.41273 q^{8} +O(q^{10})$	\(q+0.682747 q^{2} -1.53386 q^{4} -3.92611 q^{5} +1.75230 q^{7} -2.41273 q^{8} -2.68054 q^{10} +1.87675 q^{11} -2.72641 q^{13} +1.19638 q^{14} +1.42043 q^{16} +1.56644 q^{17} +0.433563 q^{19} +6.02209 q^{20} +1.28134 q^{22} -3.96274 q^{23} +10.4144 q^{25} -1.86145 q^{26} -2.68778 q^{28} +0.326279 q^{29} +8.28579 q^{31} +5.79525 q^{32} +1.06948 q^{34} -6.87974 q^{35} +2.00914 q^{37} +0.296014 q^{38} +9.47265 q^{40} +10.2464 q^{41} +5.51041 q^{43} -2.87866 q^{44} -2.70555 q^{46} +8.54855 q^{47} -3.92943 q^{49} +7.11038 q^{50} +4.18191 q^{52} -3.63647 q^{53} -7.36832 q^{55} -4.22784 q^{56} +0.222766 q^{58} +8.32957 q^{59} -12.1064 q^{61} +5.65710 q^{62} +1.11584 q^{64} +10.7042 q^{65} +14.1028 q^{67} -2.40269 q^{68} -4.69713 q^{70} -6.95753 q^{71} +1.27642 q^{73} +1.37173 q^{74} -0.665023 q^{76} +3.28863 q^{77} -0.149503 q^{79} -5.57675 q^{80} +6.99574 q^{82} +10.1128 q^{83} -6.15001 q^{85} +3.76222 q^{86} -4.52809 q^{88} +14.2227 q^{89} -4.77749 q^{91} +6.07828 q^{92} +5.83650 q^{94} -1.70222 q^{95} -15.4087 q^{97} -2.68281 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 2 q^{2} + 14 q^{4} + 4 q^{5} + 10 q^{7} + 6 q^{8}+O(q^{10}) $ 9 * q + 2 * q^2 + 14 * q^4 + 4 * q^5 + 10 * q^7 + 6 * q^8	$ 9 q + 2 q^{2} + 14 q^{4} + 4 q^{5} + 10 q^{7} + 6 q^{8} - 4 q^{10} - 8 q^{11} + 14 q^{13} - 4 q^{14} + 32 q^{16} + 6 q^{17} + 12 q^{19} + 28 q^{20} - 18 q^{22} + 4 q^{23} + 21 q^{25} + 14 q^{26} + 8 q^{29} + 4 q^{31} + 29 q^{32} - 3 q^{34} - 6 q^{35} + 22 q^{37} + 7 q^{38} + 2 q^{41} + 6 q^{43} - 17 q^{44} - 10 q^{46} + 2 q^{47} + 23 q^{49} + 20 q^{50} - 9 q^{52} + 12 q^{53} - 22 q^{55} - 18 q^{56} - 28 q^{58} + 6 q^{59} + 2 q^{61} + 15 q^{62} + 24 q^{64} - 4 q^{65} + 18 q^{67} + 24 q^{68} - 72 q^{70} - 24 q^{71} + 14 q^{73} - 3 q^{74} + 4 q^{76} + 18 q^{77} + 12 q^{79} + 86 q^{80} + 4 q^{82} + 20 q^{83} - 24 q^{85} - 16 q^{86} - 55 q^{88} + 30 q^{89} + 14 q^{91} + 46 q^{92} - 66 q^{94} + 32 q^{95} + 12 q^{97} + 62 q^{98}+O(q^{100}) $ 9 * q + 2 * q^2 + 14 * q^4 + 4 * q^5 + 10 * q^7 + 6 * q^8 - 4 * q^10 - 8 * q^11 + 14 * q^13 - 4 * q^14 + 32 * q^16 + 6 * q^17 + 12 * q^19 + 28 * q^20 - 18 * q^22 + 4 * q^23 + 21 * q^25 + 14 * q^26 + 8 * q^29 + 4 * q^31 + 29 * q^32 - 3 * q^34 - 6 * q^35 + 22 * q^37 + 7 * q^38 + 2 * q^41 + 6 * q^43 - 17 * q^44 - 10 * q^46 + 2 * q^47 + 23 * q^49 + 20 * q^50 - 9 * q^52 + 12 * q^53 - 22 * q^55 - 18 * q^56 - 28 * q^58 + 6 * q^59 + 2 * q^61 + 15 * q^62 + 24 * q^64 - 4 * q^65 + 18 * q^67 + 24 * q^68 - 72 * q^70 - 24 * q^71 + 14 * q^73 - 3 * q^74 + 4 * q^76 + 18 * q^77 + 12 * q^79 + 86 * q^80 + 4 * q^82 + 20 * q^83 - 24 * q^85 - 16 * q^86 - 55 * q^88 + 30 * q^89 + 14 * q^91 + 46 * q^92 - 66 * q^94 + 32 * q^95 + 12 * q^97 + 62 * 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 $

$2$	0.682747	0.482775	0.241388	−	0.970429i	$-0.422397\pi$
$2$	0.682747	0.482775	0.241388	+	0.970429i	$0.422397\pi$
$3$	0	0
$3$	0	0
$4$	−1.53386	−0.766928
$5$	−3.92611	−1.75581	−0.877906	−	0.478834i	$-0.841060\pi$
$5$	−3.92611	−1.75581	−0.877906	+	0.478834i	$0.841060\pi$
$6$	0	0
$7$	1.75230	0.662308	0.331154	−	0.943577i	$-0.392562\pi$
$7$	1.75230	0.662308	0.331154	+	0.943577i	$0.392562\pi$
$8$	−2.41273	−0.853029
$9$	0	0
$10$	−2.68054	−0.847662
$11$	1.87675	0.565861	0.282930	−	0.959140i	$-0.408693\pi$
$11$	1.87675	0.565861	0.282930	+	0.959140i	$0.408693\pi$
$12$	0	0
$13$	−2.72641	−0.756169	−0.378084	−	0.925771i	$-0.623417\pi$
$13$	−2.72641	−0.756169	−0.378084	+	0.925771i	$0.623417\pi$
$14$	1.19638	0.319746
$15$	0	0
$16$	1.42043	0.355106
$17$	1.56644	0.379917	0.189958	−	0.981792i	$-0.439165\pi$
$17$	1.56644	0.379917	0.189958	+	0.981792i	$0.439165\pi$
$18$	0	0
$19$	0.433563	0.0994662	0.0497331	−	0.998763i	$-0.484163\pi$
$19$	0.433563	0.0994662	0.0497331	+	0.998763i	$0.484163\pi$
$20$	6.02209	1.34658
$21$	0	0
$22$	1.28134	0.273184
$23$	−3.96274	−0.826289	−0.413145	−	0.910665i	$-0.635570\pi$
$23$	−3.96274	−0.826289	−0.413145	+	0.910665i	$0.635570\pi$
$24$	0	0
$25$	10.4144	2.08287
$26$	−1.86145	−0.365060
$27$	0	0
$28$	−2.68778	−0.507943
$29$	0.326279	0.0605885	0.0302942	−	0.999541i	$-0.490356\pi$
$29$	0.326279	0.0605885	0.0302942	+	0.999541i	$0.490356\pi$
$30$	0	0
$31$	8.28579	1.48817	0.744086	−	0.668084i	$-0.232885\pi$
$31$	8.28579	1.48817	0.744086	+	0.668084i	$0.232885\pi$
$32$	5.79525	1.02447
$33$	0	0
$34$	1.06948	0.183414
$35$	−6.87974	−1.16289
$36$	0	0
$37$	2.00914	0.330300	0.165150	−	0.986268i	$-0.447189\pi$
$37$	2.00914	0.330300	0.165150	+	0.986268i	$0.447189\pi$
$38$	0.296014	0.0480198
$39$	0	0
$40$	9.47265	1.49776
$41$	10.2464	1.60023	0.800113	−	0.599849i	$-0.204773\pi$
$41$	10.2464	1.60023	0.800113	+	0.599849i	$0.204773\pi$
$42$	0	0
$43$	5.51041	0.840330	0.420165	−	0.907448i	$-0.361972\pi$
$43$	5.51041	0.840330	0.420165	+	0.907448i	$0.361972\pi$
$44$	−2.87866	−0.433974
$45$	0	0
$46$	−2.70555	−0.398912
$47$	8.54855	1.24693	0.623467	−	0.781850i	$-0.285724\pi$
$47$	8.54855	1.24693	0.623467	+	0.781850i	$0.285724\pi$
$48$	0	0
$49$	−3.92943	−0.561348
$50$	7.11038	1.00556
$51$	0	0
$52$	4.18191	0.579927
$53$	−3.63647	−0.499507	−0.249753	−	0.968309i	$-0.580350\pi$
$53$	−3.63647	−0.499507	−0.249753	+	0.968309i	$0.580350\pi$
$54$	0	0
$55$	−7.36832	−0.993545
$56$	−4.22784	−0.564968
$57$	0	0
$58$	0.222766	0.0292506
$59$	8.32957	1.08442	0.542209	−	0.840244i	$-0.317588\pi$
$59$	8.32957	1.08442	0.542209	+	0.840244i	$0.317588\pi$
$60$	0	0
$61$	−12.1064	−1.55007	−0.775036	−	0.631917i	$-0.782268\pi$
$61$	−12.1064	−1.55007	−0.775036	+	0.631917i	$0.782268\pi$
$62$	5.65710	0.718453
$63$	0	0
$64$	1.11584	0.139480
$65$	10.7042	1.32769
$66$	0	0
$67$	14.1028	1.72293	0.861463	−	0.507821i	$-0.169549\pi$
$67$	14.1028	1.72293	0.861463	+	0.507821i	$0.169549\pi$
$68$	−2.40269	−0.291369
$69$	0	0
$70$	−4.69713	−0.561414
$71$	−6.95753	−0.825706	−0.412853	−	0.910798i	$-0.635468\pi$
$71$	−6.95753	−0.825706	−0.412853	+	0.910798i	$0.635468\pi$
$72$	0	0
$73$	1.27642	0.149394	0.0746971	−	0.997206i	$-0.476201\pi$
$73$	1.27642	0.149394	0.0746971	+	0.997206i	$0.476201\pi$
$74$	1.37173	0.159461
$75$	0	0
$76$	−0.665023	−0.0762834
$77$	3.28863	0.374774
$78$	0	0
$79$	−0.149503	−0.0168204	−0.00841018	−	0.999965i	$-0.502677\pi$
$79$	−0.149503	−0.0168204	−0.00841018	+	0.999965i	$0.502677\pi$
$80$	−5.57675	−0.623500
$81$	0	0
$82$	6.99574	0.772550
$83$	10.1128	1.11002	0.555009	−	0.831844i	$-0.312715\pi$
$83$	10.1128	1.11002	0.555009	+	0.831844i	$0.312715\pi$
$84$	0	0
$85$	−6.15001	−0.667062
$86$	3.76222	0.405691
$87$	0	0
$88$	−4.52809	−0.482696
$89$	14.2227	1.50760	0.753800	−	0.657103i	$-0.228219\pi$
$89$	14.2227	1.50760	0.753800	+	0.657103i	$0.228219\pi$
$90$	0	0
$91$	−4.77749	−0.500817
$92$	6.07828	0.633704
$93$	0	0
$94$	5.83650	0.601989
$95$	−1.70222	−0.174644
$96$	0	0
$97$	−15.4087	−1.56451	−0.782257	−	0.622956i	$-0.785932\pi$
$97$	−15.4087	−1.56451	−0.782257	+	0.622956i	$0.785932\pi$
$98$	−2.68281	−0.271005
$99$	0	0
$100$	−15.9741	−1.59741
$101$	−7.45281	−0.741582	−0.370791	−	0.928716i	$-0.620913\pi$
$101$	−7.45281	−0.741582	−0.370791	+	0.928716i	$0.620913\pi$
$102$	0	0
$103$	−1.25470	−0.123629	−0.0618144	−	0.998088i	$-0.519689\pi$
$103$	−1.25470	−0.123629	−0.0618144	+	0.998088i	$0.519689\pi$
$104$	6.57808	0.645034
$105$	0	0
$106$	−2.48279	−0.241150
$107$	8.06189	0.779373	0.389686	−	0.920948i	$-0.372583\pi$
$107$	8.06189	0.779373	0.389686	+	0.920948i	$0.372583\pi$
$108$	0	0
$109$	−9.79205	−0.937908	−0.468954	−	0.883223i	$-0.655369\pi$
$109$	−9.79205	−0.937908	−0.468954	+	0.883223i	$0.655369\pi$
$110$	−5.03070	−0.479659
$111$	0	0
$112$	2.48902	0.235190
$113$	−10.7352	−1.00988	−0.504941	−	0.863154i	$-0.668486\pi$
$113$	−10.7352	−1.00988	−0.504941	+	0.863154i	$0.668486\pi$
$114$	0	0
$115$	15.5582	1.45081
$116$	−0.500465	−0.0464670
$117$	0	0
$118$	5.68699	0.523530
$119$	2.74487	0.251622
$120$	0	0
$121$	−7.47782	−0.679802
$122$	−8.26564	−0.748336
$123$	0	0
$124$	−12.7092	−1.14132
$125$	−21.2574	−1.90132
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−10.8287	−0.957128
$129$	0	0
$130$	7.30825	0.640976
$131$	−3.22588	−0.281846	−0.140923	−	0.990021i	$-0.545007\pi$
$131$	−3.22588	−0.281846	−0.140923	+	0.990021i	$0.545007\pi$
$132$	0	0
$133$	0.759734	0.0658773
$134$	9.62862	0.831786
$135$	0	0
$136$	−3.77939	−0.324080
$137$	20.3748	1.74073	0.870367	−	0.492404i	$-0.163882\pi$
$137$	20.3748	1.74073	0.870367	+	0.492404i	$0.163882\pi$
$138$	0	0
$139$	−9.41243	−0.798352	−0.399176	−	0.916874i	$-0.630704\pi$
$139$	−9.41243	−0.798352	−0.399176	+	0.916874i	$0.630704\pi$
$140$	10.5525	0.891852
$141$	0	0
$142$	−4.75023	−0.398631
$143$	−5.11678	−0.427886
$144$	0	0
$145$	−1.28101	−0.106382
$146$	0.871475	0.0721238
$147$	0	0
$148$	−3.08173	−0.253317
$149$	−22.2532	−1.82306	−0.911528	−	0.411238i	$-0.865097\pi$
$149$	−22.2532	−1.82306	−0.911528	+	0.411238i	$0.865097\pi$
$150$	0	0
$151$	13.1042	1.06641	0.533203	−	0.845987i	$-0.320988\pi$
$151$	13.1042	1.06641	0.533203	+	0.845987i	$0.320988\pi$
$152$	−1.04607	−0.0848475
$153$	0	0
$154$	2.24531	0.180932
$155$	−32.5309	−2.61295
$156$	0	0
$157$	−8.63326	−0.689009	−0.344504	−	0.938785i	$-0.611953\pi$
$157$	−8.63326	−0.689009	−0.344504	+	0.938785i	$0.611953\pi$
$158$	−0.102073	−0.00812045
$159$	0	0
$160$	−22.7528	−1.79877
$161$	−6.94393	−0.547258
$162$	0	0
$163$	−0.874922	−0.0685292	−0.0342646	−	0.999413i	$-0.510909\pi$
$163$	−0.874922	−0.0685292	−0.0342646	+	0.999413i	$0.510909\pi$
$164$	−15.7166	−1.22726
$165$	0	0
$166$	6.90446	0.535890
$167$	11.6126	0.898610	0.449305	−	0.893378i	$-0.351672\pi$
$167$	11.6126	0.898610	0.449305	+	0.893378i	$0.351672\pi$
$168$	0	0
$169$	−5.56671	−0.428209
$170$	−4.19890	−0.322041
$171$	0	0
$172$	−8.45218	−0.644473
$173$	16.2920	1.23866	0.619329	−	0.785131i	$-0.287405\pi$
$173$	16.2920	1.23866	0.619329	+	0.785131i	$0.287405\pi$
$174$	0	0
$175$	18.2491	1.37950
$176$	2.66578	0.200941
$177$	0	0
$178$	9.71050	0.727833
$179$	−10.3903	−0.776606	−0.388303	−	0.921532i	$-0.626939\pi$
$179$	−10.3903	−0.776606	−0.388303	+	0.921532i	$0.626939\pi$
$180$	0	0
$181$	18.6888	1.38913	0.694564	−	0.719431i	$-0.255597\pi$
$181$	18.6888	1.38913	0.694564	+	0.719431i	$0.255597\pi$
$182$	−3.26182	−0.241782
$183$	0	0
$184$	9.56104	0.704849
$185$	−7.88811	−0.579945
$186$	0	0
$187$	2.93981	0.214980
$188$	−13.1122	−0.956308
$189$	0	0
$190$	−1.16218	−0.0843137
$191$	25.5434	1.84826	0.924129	−	0.382080i	$-0.124792\pi$
$191$	25.5434	1.84826	0.924129	+	0.382080i	$0.124792\pi$
$192$	0	0
$193$	17.0139	1.22469	0.612344	−	0.790591i	$-0.290227\pi$
$193$	17.0139	1.22469	0.612344	+	0.790591i	$0.290227\pi$
$194$	−10.5202	−0.755309
$195$	0	0
$196$	6.02718	0.430513
$197$	0.493580	0.0351661	0.0175830	−	0.999845i	$-0.494403\pi$
$197$	0.493580	0.0351661	0.0175830	+	0.999845i	$0.494403\pi$
$198$	0	0
$199$	19.3177	1.36939	0.684697	−	0.728828i	$-0.259935\pi$
$199$	19.3177	1.36939	0.684697	+	0.728828i	$0.259935\pi$
$200$	−25.1271	−1.77675
$201$	0	0
$202$	−5.08839	−0.358018
$203$	0.571740	0.0401283
$204$	0	0
$205$	−40.2287	−2.80970
$206$	−0.856640	−0.0596849
$207$	0	0
$208$	−3.87266	−0.268520
$209$	0.813688	0.0562840
$210$	0	0
$211$	2.69669	0.185648	0.0928239	−	0.995683i	$-0.470411\pi$
$211$	2.69669	0.185648	0.0928239	+	0.995683i	$0.470411\pi$
$212$	5.57781	0.383086
$213$	0	0
$214$	5.50424	0.376262
$215$	−21.6345	−1.47546
$216$	0	0
$217$	14.5192	0.985629
$218$	−6.68550	−0.452799
$219$	0	0
$220$	11.3019	0.761977
$221$	−4.27074	−0.287281
$222$	0	0
$223$	23.3651	1.56464	0.782321	−	0.622875i	$-0.214035\pi$
$223$	23.3651	1.56464	0.782321	+	0.622875i	$0.214035\pi$
$224$	10.1550	0.678512
$225$	0	0
$226$	−7.32942	−0.487546
$227$	4.78583	0.317647	0.158823	−	0.987307i	$-0.449230\pi$
$227$	4.78583	0.317647	0.158823	+	0.987307i	$0.449230\pi$
$228$	0	0
$229$	24.4752	1.61737	0.808684	−	0.588243i	$-0.200180\pi$
$229$	24.4752	1.61737	0.808684	+	0.588243i	$0.200180\pi$
$230$	10.6223	0.700414
$231$	0	0
$232$	−0.787223	−0.0516837
$233$	−18.7963	−1.23138	−0.615692	−	0.787987i	$-0.711123\pi$
$233$	−18.7963	−1.23138	−0.615692	+	0.787987i	$0.711123\pi$
$234$	0	0
$235$	−33.5626	−2.18938
$236$	−12.7764	−0.831670
$237$	0	0
$238$	1.87406	0.121477
$239$	14.5357	0.940236	0.470118	−	0.882603i	$-0.344211\pi$
$239$	14.5357	0.940236	0.470118	+	0.882603i	$0.344211\pi$
$240$	0	0
$241$	14.1465	0.911259	0.455630	−	0.890169i	$-0.349414\pi$
$241$	14.1465	0.911259	0.455630	+	0.890169i	$0.349414\pi$
$242$	−5.10546	−0.328191
$243$	0	0
$244$	18.5695	1.18879
$245$	15.4274	0.985620
$246$	0	0
$247$	−1.18207	−0.0752132
$248$	−19.9914	−1.26945
$249$	0	0
$250$	−14.5134	−0.917910
$251$	5.28411	0.333530	0.166765	−	0.985997i	$-0.446668\pi$
$251$	5.28411	0.333530	0.166765	+	0.985997i	$0.446668\pi$
$252$	0	0
$253$	−7.43707	−0.467565
$254$	−0.682747	−0.0428394
$255$	0	0
$256$	−9.62493	−0.601558
$257$	12.2257	0.762619	0.381310	−	0.924447i	$-0.375473\pi$
$257$	12.2257	0.762619	0.381310	+	0.924447i	$0.375473\pi$
$258$	0	0
$259$	3.52062	0.218761
$260$	−16.4187	−1.01824
$261$	0	0
$262$	−2.20246	−0.136068
$263$	10.1991	0.628906	0.314453	−	0.949273i	$-0.398179\pi$
$263$	10.1991	0.628906	0.314453	+	0.949273i	$0.398179\pi$
$264$	0	0
$265$	14.2772	0.877040
$266$	0.518706	0.0318039
$267$	0	0
$268$	−21.6316	−1.32136
$269$	15.1445	0.923375	0.461688	−	0.887043i	$-0.347244\pi$
$269$	15.1445	0.923375	0.461688	+	0.887043i	$0.347244\pi$
$270$	0	0
$271$	11.4395	0.694901	0.347451	−	0.937698i	$-0.387047\pi$
$271$	11.4395	0.694901	0.347451	+	0.937698i	$0.387047\pi$
$272$	2.22501	0.134911
$273$	0	0
$274$	13.9108	0.840383
$275$	19.5451	1.17862
$276$	0	0
$277$	−16.6762	−1.00198	−0.500989	−	0.865453i	$-0.667030\pi$
$277$	−16.6762	−1.00198	−0.500989	+	0.865453i	$0.667030\pi$
$278$	−6.42631	−0.385425
$279$	0	0
$280$	16.5990	0.991978
$281$	−8.46118	−0.504751	−0.252376	−	0.967629i	$-0.581212\pi$
$281$	−8.46118	−0.504751	−0.252376	+	0.967629i	$0.581212\pi$
$282$	0	0
$283$	−7.91915	−0.470744	−0.235372	−	0.971905i	$-0.575631\pi$
$283$	−7.91915	−0.470744	−0.235372	+	0.971905i	$0.575631\pi$
$284$	10.6718	0.633257
$285$	0	0
$286$	−3.49347	−0.206573
$287$	17.9549	1.05984
$288$	0	0
$289$	−14.5463	−0.855663
$290$	−0.874605	−0.0513586
$291$	0	0
$292$	−1.95785	−0.114575
$293$	2.58608	0.151080	0.0755401	−	0.997143i	$-0.475932\pi$
$293$	2.58608	0.151080	0.0755401	+	0.997143i	$0.475932\pi$
$294$	0	0
$295$	−32.7028	−1.90403
$296$	−4.84751	−0.281756
$297$	0	0
$298$	−15.1933	−0.880127
$299$	10.8040	0.624814
$300$	0	0
$301$	9.65592	0.556558
$302$	8.94688	0.514835
$303$	0	0
$304$	0.615844	0.0353211
$305$	47.5313	2.72163
$306$	0	0
$307$	−14.7044	−0.839225	−0.419612	−	0.907703i	$-0.637834\pi$
$307$	−14.7044	−0.839225	−0.419612	+	0.907703i	$0.637834\pi$
$308$	−5.04429	−0.287425
$309$	0	0
$310$	−22.2104	−1.26147
$311$	−21.5149	−1.22000	−0.609998	−	0.792403i	$-0.708830\pi$
$311$	−21.5149	−1.22000	−0.609998	+	0.792403i	$0.708830\pi$
$312$	0	0
$313$	27.4314	1.55052	0.775258	−	0.631644i	$-0.217620\pi$
$313$	27.4314	1.55052	0.775258	+	0.631644i	$0.217620\pi$
$314$	−5.89433	−0.332637
$315$	0	0
$316$	0.229315	0.0129000
$317$	18.2137	1.02298	0.511492	−	0.859288i	$-0.329093\pi$
$317$	18.2137	1.02298	0.511492	+	0.859288i	$0.329093\pi$
$318$	0	0
$319$	0.612343	0.0342846
$320$	−4.38093	−0.244901
$321$	0	0
$322$	−4.74095	−0.264203
$323$	0.679149	0.0377889
$324$	0	0
$325$	−28.3938	−1.57500
$326$	−0.597351	−0.0330842
$327$	0	0
$328$	−24.7219	−1.36504
$329$	14.9797	0.825855
$330$	0	0
$331$	8.68098	0.477150	0.238575	−	0.971124i	$-0.423320\pi$
$331$	8.68098	0.477150	0.238575	+	0.971124i	$0.423320\pi$
$332$	−15.5115	−0.851305
$333$	0	0
$334$	7.92847	0.433827
$335$	−55.3690	−3.02513
$336$	0	0
$337$	19.1300	1.04208	0.521038	−	0.853533i	$-0.325545\pi$
$337$	19.1300	1.04208	0.521038	+	0.853533i	$0.325545\pi$
$338$	−3.80066	−0.206729
$339$	0	0
$340$	9.43323	0.511589
$341$	15.5503	0.842098
$342$	0	0
$343$	−19.1517	−1.03409
$344$	−13.2951	−0.716826
$345$	0	0
$346$	11.1233	0.597994
$347$	−25.5227	−1.37013	−0.685066	−	0.728481i	$-0.740227\pi$
$347$	−25.5227	−1.37013	−0.685066	+	0.728481i	$0.740227\pi$
$348$	0	0
$349$	−29.5263	−1.58051	−0.790253	−	0.612780i	$-0.790051\pi$
$349$	−29.5263	−1.58051	−0.790253	+	0.612780i	$0.790051\pi$
$350$	12.4595	0.665991
$351$	0	0
$352$	10.8762	0.579705
$353$	14.1469	0.752964	0.376482	−	0.926424i	$-0.377134\pi$
$353$	14.1469	0.752964	0.376482	+	0.926424i	$0.377134\pi$
$354$	0	0
$355$	27.3160	1.44978
$356$	−21.8155	−1.15622
$357$	0	0
$358$	−7.09394	−0.374926
$359$	−27.5293	−1.45294	−0.726470	−	0.687198i	$-0.758840\pi$
$359$	−27.5293	−1.45294	−0.726470	+	0.687198i	$0.758840\pi$
$360$	0	0
$361$	−18.8120	−0.990106
$362$	12.7597	0.670637
$363$	0	0
$364$	7.32798	0.384091
$365$	−5.01138	−0.262308
$366$	0	0
$367$	−27.6340	−1.44248	−0.721241	−	0.692684i	$-0.756428\pi$
$367$	−27.6340	−1.44248	−0.721241	+	0.692684i	$0.756428\pi$
$368$	−5.62878	−0.293421
$369$	0	0
$370$	−5.38558	−0.279983
$371$	−6.37219	−0.330828
$372$	0	0
$373$	0.571631	0.0295979	0.0147990	−	0.999890i	$-0.495289\pi$
$373$	0.571631	0.0295979	0.0147990	+	0.999890i	$0.495289\pi$
$374$	2.00715	0.103787
$375$	0	0
$376$	−20.6253	−1.06367
$377$	−0.889569	−0.0458151
$378$	0	0
$379$	32.8952	1.68971	0.844857	−	0.534992i	$-0.179686\pi$
$379$	32.8952	1.68971	0.844857	+	0.534992i	$0.179686\pi$
$380$	2.61096	0.133939
$381$	0	0
$382$	17.4397	0.892294
$383$	−13.6608	−0.698032	−0.349016	−	0.937117i	$-0.613484\pi$
$383$	−13.6608	−0.698032	−0.349016	+	0.937117i	$0.613484\pi$
$384$	0	0
$385$	−12.9115	−0.658033
$386$	11.6162	0.591249
$387$	0	0
$388$	23.6347	1.19987
$389$	11.8028	0.598426	0.299213	−	0.954186i	$-0.403276\pi$
$389$	11.8028	0.598426	0.299213	+	0.954186i	$0.403276\pi$
$390$	0	0
$391$	−6.20739	−0.313921
$392$	9.48066	0.478846
$393$	0	0
$394$	0.336990	0.0169773
$395$	0.586964	0.0295334
$396$	0	0
$397$	16.9669	0.851547	0.425773	−	0.904830i	$-0.360002\pi$
$397$	16.9669	0.851547	0.425773	+	0.904830i	$0.360002\pi$
$398$	13.1891	0.661109
$399$	0	0
$400$	14.7928	0.739642
$401$	−14.6188	−0.730026	−0.365013	−	0.931003i	$-0.618935\pi$
$401$	−14.6188	−0.730026	−0.365013	+	0.931003i	$0.618935\pi$
$402$	0	0
$403$	−22.5904	−1.12531
$404$	11.4315	0.568740
$405$	0	0
$406$	0.390354	0.0193729
$407$	3.77065	0.186904
$408$	0	0
$409$	16.6845	0.824994	0.412497	−	0.910959i	$-0.364657\pi$
$409$	16.6845	0.824994	0.412497	+	0.910959i	$0.364657\pi$
$410$	−27.4661	−1.35645
$411$	0	0
$412$	1.92452	0.0948144
$413$	14.5959	0.718219
$414$	0	0
$415$	−39.7038	−1.94898
$416$	−15.8002	−0.774669
$417$	0	0
$418$	0.555544	0.0271725
$419$	−35.3686	−1.72787	−0.863934	−	0.503606i	$-0.832006\pi$
$419$	−35.3686	−1.72787	−0.863934	+	0.503606i	$0.832006\pi$
$420$	0	0
$421$	6.13005	0.298760	0.149380	−	0.988780i	$-0.452272\pi$
$421$	6.13005	0.298760	0.149380	+	0.988780i	$0.452272\pi$
$422$	1.84116	0.0896262
$423$	0	0
$424$	8.77381	0.426094
$425$	16.3134	0.791318
$426$	0	0
$427$	−21.2142	−1.02663
$428$	−12.3658	−0.597723
$429$	0	0
$430$	−14.7709	−0.712316
$431$	−21.9152	−1.05562	−0.527810	−	0.849363i	$-0.676987\pi$
$431$	−21.9152	−1.05562	−0.527810	+	0.849363i	$0.676987\pi$
$432$	0	0
$433$	16.5552	0.795594	0.397797	−	0.917473i	$-0.369775\pi$
$433$	16.5552	0.795594	0.397797	+	0.917473i	$0.369775\pi$
$434$	9.91296	0.475837
$435$	0	0
$436$	15.0196	0.719308
$437$	−1.71810	−0.0821878
$438$	0	0
$439$	18.8319	0.898798	0.449399	−	0.893331i	$-0.351638\pi$
$439$	18.8319	0.898798	0.449399	+	0.893331i	$0.351638\pi$
$440$	17.7778	0.847523
$441$	0	0
$442$	−2.91584	−0.138692
$443$	−9.44120	−0.448565	−0.224282	−	0.974524i	$-0.572004\pi$
$443$	−9.44120	−0.448565	−0.224282	+	0.974524i	$0.572004\pi$
$444$	0	0
$445$	−55.8398	−2.64706
$446$	15.9525	0.755371
$447$	0	0
$448$	1.95530	0.0923791
$449$	8.48798	0.400572	0.200286	−	0.979737i	$-0.435813\pi$
$449$	8.48798	0.400572	0.200286	+	0.979737i	$0.435813\pi$
$450$	0	0
$451$	19.2300	0.905505
$452$	16.4662	0.774506
$453$	0	0
$454$	3.26752	0.153352
$455$	18.7570	0.879340
$456$	0	0
$457$	−13.3986	−0.626759	−0.313379	−	0.949628i	$-0.601461\pi$
$457$	−13.3986	−0.626759	−0.313379	+	0.949628i	$0.601461\pi$
$458$	16.7104	0.780826
$459$	0	0
$460$	−23.8640	−1.11267
$461$	−33.9696	−1.58212	−0.791062	−	0.611737i	$-0.790471\pi$
$461$	−33.9696	−1.58212	−0.791062	+	0.611737i	$0.790471\pi$
$462$	0	0
$463$	1.37759	0.0640218	0.0320109	−	0.999488i	$-0.489809\pi$
$463$	1.37759	0.0640218	0.0320109	+	0.999488i	$0.489809\pi$
$464$	0.463455	0.0215154
$465$	0	0
$466$	−12.8331	−0.594482
$467$	29.8471	1.38116	0.690579	−	0.723257i	$-0.257356\pi$
$467$	29.8471	1.38116	0.690579	+	0.723257i	$0.257356\pi$
$468$	0	0
$469$	24.7123	1.14111
$470$	−22.9148	−1.05698
$471$	0	0
$472$	−20.0970	−0.925040
$473$	10.3417	0.475510
$474$	0	0
$475$	4.51528	0.207175
$476$	−4.21024	−0.192976
$477$	0	0
$478$	9.92421	0.453923
$479$	21.0999	0.964080	0.482040	−	0.876149i	$-0.339896\pi$
$479$	21.0999	0.964080	0.482040	+	0.876149i	$0.339896\pi$
$480$	0	0
$481$	−5.47773	−0.249763
$482$	9.65852	0.439933
$483$	0	0
$484$	11.4699	0.521359
$485$	60.4962	2.74699
$486$	0	0
$487$	23.1093	1.04718	0.523592	−	0.851969i	$-0.324592\pi$
$487$	23.1093	1.04718	0.523592	+	0.851969i	$0.324592\pi$
$488$	29.2096	1.32226
$489$	0	0
$490$	10.5330	0.475833
$491$	−17.9363	−0.809453	−0.404726	−	0.914438i	$-0.632633\pi$
$491$	−17.9363	−0.809453	−0.404726	+	0.914438i	$0.632633\pi$
$492$	0	0
$493$	0.511095	0.0230186
$494$	−0.807054	−0.0363111
$495$	0	0
$496$	11.7693	0.528459
$497$	−12.1917	−0.546872
$498$	0	0
$499$	33.1405	1.48357	0.741786	−	0.670637i	$-0.233979\pi$
$499$	33.1405	1.48357	0.741786	+	0.670637i	$0.233979\pi$
$500$	32.6058	1.45818
$501$	0	0
$502$	3.60771	0.161020
$503$	30.0952	1.34188	0.670938	−	0.741513i	$-0.265892\pi$
$503$	30.0952	1.34188	0.670938	+	0.741513i	$0.265892\pi$
$504$	0	0
$505$	29.2606	1.30208
$506$	−5.07764	−0.225729
$507$	0	0
$508$	1.53386	0.0680539
$509$	−12.1085	−0.536698	−0.268349	−	0.963322i	$-0.586478\pi$
$509$	−12.1085	−0.536698	−0.268349	+	0.963322i	$0.586478\pi$
$510$	0	0
$511$	2.23668	0.0989450
$512$	15.0859	0.666711
$513$	0	0
$514$	8.34708	0.368174
$515$	4.92607	0.217069
$516$	0	0
$517$	16.0435	0.705591
$518$	2.40369	0.105612
$519$	0	0
$520$	−25.8263	−1.13256
$521$	12.7736	0.559620	0.279810	−	0.960055i	$-0.409728\pi$
$521$	12.7736	0.559620	0.279810	+	0.960055i	$0.409728\pi$
$522$	0	0
$523$	−6.25406	−0.273471	−0.136735	−	0.990608i	$-0.543661\pi$
$523$	−6.25406	−0.273471	−0.136735	+	0.990608i	$0.543661\pi$
$524$	4.94803	0.216156
$525$	0	0
$526$	6.96343	0.303620
$527$	12.9792	0.565381
$528$	0	0
$529$	−7.29666	−0.317246
$530$	9.74770	0.423413
$531$	0	0
$532$	−1.16532	−0.0505231
$533$	−27.9360	−1.21004
$534$	0	0
$535$	−31.6519	−1.36843
$536$	−34.0261	−1.46971
$537$	0	0
$538$	10.3399	0.445783
$539$	−7.37455	−0.317645
$540$	0	0
$541$	−8.28320	−0.356122	−0.178061	−	0.984019i	$-0.556983\pi$
$541$	−8.28320	−0.356122	−0.178061	+	0.984019i	$0.556983\pi$
$542$	7.81030	0.335481
$543$	0	0
$544$	9.07790	0.389212
$545$	38.4447	1.64679
$546$	0	0
$547$	24.5582	1.05003	0.525016	−	0.851092i	$-0.324059\pi$
$547$	24.5582	1.05003	0.525016	+	0.851092i	$0.324059\pi$
$548$	−31.2519	−1.33502
$549$	0	0
$550$	13.3444	0.569007
$551$	0.141462	0.00602650
$552$	0	0
$553$	−0.261974	−0.0111403
$554$	−11.3857	−0.483731
$555$	0	0
$556$	14.4373	0.612278
$557$	−28.8486	−1.22235	−0.611177	−	0.791494i	$-0.709304\pi$
$557$	−28.8486	−1.22235	−0.611177	+	0.791494i	$0.709304\pi$
$558$	0	0
$559$	−15.0236	−0.635432
$560$	−9.77216	−0.412949
$561$	0	0
$562$	−5.77685	−0.243682
$563$	−40.8627	−1.72216	−0.861078	−	0.508473i	$-0.830210\pi$
$563$	−40.8627	−1.72216	−0.861078	+	0.508473i	$0.830210\pi$
$564$	0	0
$565$	42.1476	1.77316
$566$	−5.40678	−0.227264
$567$	0	0
$568$	16.7866	0.704352
$569$	12.8588	0.539069	0.269535	−	0.962991i	$-0.413130\pi$
$569$	12.8588	0.539069	0.269535	+	0.962991i	$0.413130\pi$
$570$	0	0
$571$	44.9051	1.87922	0.939609	−	0.342250i	$-0.111189\pi$
$571$	44.9051	1.87922	0.939609	+	0.342250i	$0.111189\pi$
$572$	7.84840	0.328158
$573$	0	0
$574$	12.2587	0.511666
$575$	−41.2695	−1.72106
$576$	0	0
$577$	6.38498	0.265810	0.132905	−	0.991129i	$-0.457569\pi$
$577$	6.38498	0.265810	0.132905	+	0.991129i	$0.457569\pi$
$578$	−9.93143	−0.413093
$579$	0	0
$580$	1.96488	0.0815873
$581$	17.7206	0.735175
$582$	0	0
$583$	−6.82473	−0.282651
$584$	−3.07967	−0.127438
$585$	0	0
$586$	1.76564	0.0729378
$587$	−6.96680	−0.287551	−0.143775	−	0.989610i	$-0.545924\pi$
$587$	−6.96680	−0.287551	−0.143775	+	0.989610i	$0.545924\pi$
$588$	0	0
$589$	3.59241	0.148023
$590$	−22.3278	−0.919220
$591$	0	0
$592$	2.85383	0.117292
$593$	9.39119	0.385650	0.192825	−	0.981233i	$-0.438235\pi$
$593$	9.39119	0.385650	0.192825	+	0.981233i	$0.438235\pi$
$594$	0	0
$595$	−10.7767	−0.441801
$596$	34.1333	1.39815
$597$	0	0
$598$	7.37644	0.301645
$599$	−39.9692	−1.63310	−0.816548	−	0.577278i	$-0.804115\pi$
$599$	−39.9692	−1.63310	−0.816548	+	0.577278i	$0.804115\pi$
$600$	0	0
$601$	8.06140	0.328831	0.164416	−	0.986391i	$-0.447426\pi$
$601$	8.06140	0.328831	0.164416	+	0.986391i	$0.447426\pi$
$602$	6.59255	0.268692
$603$	0	0
$604$	−20.1000	−0.817857
$605$	29.3588	1.19360
$606$	0	0
$607$	−23.3283	−0.946866	−0.473433	−	0.880830i	$-0.656985\pi$
$607$	−23.3283	−0.946866	−0.473433	+	0.880830i	$0.656985\pi$
$608$	2.51261	0.101900
$609$	0	0
$610$	32.4519	1.31394
$611$	−23.3068	−0.942893
$612$	0	0
$613$	−32.3773	−1.30771	−0.653853	−	0.756621i	$-0.726849\pi$
$613$	−32.3773	−1.30771	−0.653853	+	0.756621i	$0.726849\pi$
$614$	−10.0394	−0.405157
$615$	0	0
$616$	−7.93458	−0.319694
$617$	−28.6970	−1.15530	−0.577650	−	0.816285i	$-0.696030\pi$
$617$	−28.6970	−1.15530	−0.577650	+	0.816285i	$0.696030\pi$
$618$	0	0
$619$	−18.8070	−0.755919	−0.377960	−	0.925822i	$-0.623374\pi$
$619$	−18.8070	−0.755919	−0.377960	+	0.925822i	$0.623374\pi$
$620$	49.8978	2.00394
$621$	0	0
$622$	−14.6892	−0.588984
$623$	24.9224	0.998497
$624$	0	0
$625$	31.3871	1.25549
$626$	18.7287	0.748551
$627$	0	0
$628$	13.2422	0.528420
$629$	3.14719	0.125487
$630$	0	0
$631$	−36.6630	−1.45953	−0.729766	−	0.683697i	$-0.760371\pi$
$631$	−36.6630	−1.45953	−0.729766	+	0.683697i	$0.760371\pi$
$632$	0.360710	0.0143483
$633$	0	0
$634$	12.4354	0.493871
$635$	3.92611	0.155803
$636$	0	0
$637$	10.7132	0.424474
$638$	0.418076	0.0165518
$639$	0	0
$640$	42.5146	1.68054
$641$	7.77164	0.306961	0.153481	−	0.988152i	$-0.450952\pi$
$641$	7.77164	0.306961	0.153481	+	0.988152i	$0.450952\pi$
$642$	0	0
$643$	−29.9605	−1.18153	−0.590763	−	0.806845i	$-0.701173\pi$
$643$	−29.9605	−1.18153	−0.590763	+	0.806845i	$0.701173\pi$
$644$	10.6510	0.419708
$645$	0	0
$646$	0.463687	0.0182435
$647$	6.96319	0.273751	0.136876	−	0.990588i	$-0.456294\pi$
$647$	6.96319	0.273751	0.136876	+	0.990588i	$0.456294\pi$
$648$	0	0
$649$	15.6325	0.613629
$650$	−19.3858	−0.760373
$651$	0	0
$652$	1.34200	0.0525570
$653$	36.5263	1.42938	0.714692	−	0.699439i	$-0.246567\pi$
$653$	36.5263	1.42938	0.714692	+	0.699439i	$0.246567\pi$
$654$	0	0
$655$	12.6652	0.494869
$656$	14.5543	0.568251
$657$	0	0
$658$	10.2273	0.398702
$659$	28.1336	1.09593	0.547965	−	0.836502i	$-0.315403\pi$
$659$	28.1336	1.09593	0.547965	+	0.836502i	$0.315403\pi$
$660$	0	0
$661$	15.4945	0.602665	0.301332	−	0.953519i	$-0.402569\pi$
$661$	15.4945	0.602665	0.301332	+	0.953519i	$0.402569\pi$
$662$	5.92692	0.230356
$663$	0	0
$664$	−24.3994	−0.946879
$665$	−2.98280	−0.115668
$666$	0	0
$667$	−1.29296	−0.0500636
$668$	−17.8121	−0.689169
$669$	0	0
$670$	−37.8030	−1.46046
$671$	−22.7207	−0.877125
$672$	0	0
$673$	−24.7447	−0.953838	−0.476919	−	0.878947i	$-0.658246\pi$
$673$	−24.7447	−0.953838	−0.476919	+	0.878947i	$0.658246\pi$
$674$	13.0609	0.503089
$675$	0	0
$676$	8.53853	0.328405
$677$	−0.929389	−0.0357193	−0.0178597	−	0.999841i	$-0.505685\pi$
$677$	−0.929389	−0.0357193	−0.0178597	+	0.999841i	$0.505685\pi$
$678$	0	0
$679$	−27.0007	−1.03619
$680$	14.8383	0.569024
$681$	0	0
$682$	10.6170	0.406544
$683$	−27.2913	−1.04427	−0.522136	−	0.852862i	$-0.674865\pi$
$683$	−27.2913	−1.04427	−0.522136	+	0.852862i	$0.674865\pi$
$684$	0	0
$685$	−79.9936	−3.05640
$686$	−13.0758	−0.499235
$687$	0	0
$688$	7.82713	0.298407
$689$	9.91448	0.377712
$690$	0	0
$691$	−7.95653	−0.302681	−0.151340	−	0.988482i	$-0.548359\pi$
$691$	−7.95653	−0.302681	−0.151340	+	0.988482i	$0.548359\pi$
$692$	−24.9896	−0.949962
$693$	0	0
$694$	−17.4256	−0.661466
$695$	36.9543	1.40176
$696$	0	0
$697$	16.0504	0.607953
$698$	−20.1590	−0.763030
$699$	0	0
$700$	−27.9915	−1.05798
$701$	−35.9759	−1.35879	−0.679396	−	0.733771i	$-0.737758\pi$
$701$	−35.9759	−1.35879	−0.679396	+	0.733771i	$0.737758\pi$
$702$	0	0
$703$	0.871088	0.0328537
$704$	2.09416	0.0789265
$705$	0	0
$706$	9.65876	0.363512
$707$	−13.0596	−0.491156
$708$	0	0
$709$	−31.1972	−1.17164	−0.585818	−	0.810442i	$-0.699227\pi$
$709$	−31.1972	−1.17164	−0.585818	+	0.810442i	$0.699227\pi$
$710$	18.6500	0.699920
$711$	0	0
$712$	−34.3155	−1.28603
$713$	−32.8345	−1.22966
$714$	0	0
$715$	20.0890	0.751288
$716$	15.9372	0.595601
$717$	0	0
$718$	−18.7955	−0.701444
$719$	9.28322	0.346206	0.173103	−	0.984904i	$-0.444621\pi$
$719$	9.28322	0.346206	0.173103	+	0.984904i	$0.444621\pi$
$720$	0	0
$721$	−2.19861	−0.0818804
$722$	−12.8439	−0.477999
$723$	0	0
$724$	−28.6659	−1.06536
$725$	3.39799	0.126198
$726$	0	0
$727$	8.01820	0.297379	0.148689	−	0.988884i	$-0.452495\pi$
$727$	8.01820	0.297379	0.148689	+	0.988884i	$0.452495\pi$
$728$	11.5268	0.427212
$729$	0	0
$730$	−3.42151	−0.126636
$731$	8.63172	0.319256
$732$	0	0
$733$	−12.4333	−0.459235	−0.229618	−	0.973281i	$-0.573748\pi$
$733$	−12.4333	−0.459235	−0.229618	+	0.973281i	$0.573748\pi$
$734$	−18.8670	−0.696395
$735$	0	0
$736$	−22.9651	−0.846505
$737$	26.4673	0.974936
$738$	0	0
$739$	12.8565	0.472934	0.236467	−	0.971640i	$-0.424011\pi$
$739$	12.8565	0.472934	0.236467	+	0.971640i	$0.424011\pi$
$740$	12.0992	0.444776
$741$	0	0
$742$	−4.35060	−0.159715
$743$	32.6724	1.19863	0.599317	−	0.800512i	$-0.295439\pi$
$743$	32.6724	1.19863	0.599317	+	0.800512i	$0.295439\pi$
$744$	0	0
$745$	87.3687	3.20094
$746$	0.390280	0.0142892
$747$	0	0
$748$	−4.50924	−0.164874
$749$	14.1269	0.516185
$750$	0	0
$751$	−3.62185	−0.132163	−0.0660817	−	0.997814i	$-0.521050\pi$
$751$	−3.62185	−0.132163	−0.0660817	+	0.997814i	$0.521050\pi$
$752$	12.1426	0.442794
$753$	0	0
$754$	−0.607351	−0.0221184
$755$	−51.4487	−1.87241
$756$	0	0
$757$	49.6247	1.80364	0.901821	−	0.432111i	$-0.142231\pi$
$757$	49.6247	1.80364	0.901821	+	0.432111i	$0.142231\pi$
$758$	22.4591	0.815752
$759$	0	0
$760$	4.10699	0.148976
$761$	39.6727	1.43813	0.719067	−	0.694941i	$-0.244569\pi$
$761$	39.6727	1.43813	0.719067	+	0.694941i	$0.244569\pi$
$762$	0	0
$763$	−17.1586	−0.621184
$764$	−39.1800	−1.41748
$765$	0	0
$766$	−9.32685	−0.336993
$767$	−22.7098	−0.820003
$768$	0	0
$769$	20.7114	0.746871	0.373436	−	0.927656i	$-0.378180\pi$
$769$	20.7114	0.746871	0.373436	+	0.927656i	$0.378180\pi$
$770$	−8.81532	−0.317682
$771$	0	0
$772$	−26.0969	−0.939247
$773$	−9.48210	−0.341048	−0.170524	−	0.985354i	$-0.554546\pi$
$773$	−9.48210	−0.341048	−0.170524	+	0.985354i	$0.554546\pi$
$774$	0	0
$775$	86.2912	3.09967
$776$	37.1770	1.33458
$777$	0	0
$778$	8.05834	0.288905
$779$	4.44248	0.159168
$780$	0	0
$781$	−13.0575	−0.467235
$782$	−4.23808	−0.151553
$783$	0	0
$784$	−5.58147	−0.199338
$785$	33.8951	1.20977
$786$	0	0
$787$	−28.6501	−1.02127	−0.510633	−	0.859799i	$-0.670589\pi$
$787$	−28.6501	−1.02127	−0.510633	+	0.859799i	$0.670589\pi$
$788$	−0.757080	−0.0269699
$789$	0	0
$790$	0.400748	0.0142580
$791$	−18.8113	−0.668853
$792$	0	0
$793$	33.0071	1.17212
$794$	11.5841	0.411106
$795$	0	0
$796$	−29.6305	−1.05023
$797$	1.70421	0.0603662	0.0301831	−	0.999544i	$-0.490391\pi$
$797$	1.70421	0.0603662	0.0301831	+	0.999544i	$0.490391\pi$
$798$	0	0
$799$	13.3908	0.473731
$800$	60.3539	2.13383
$801$	0	0
$802$	−9.98091	−0.352438
$803$	2.39553	0.0845363
$804$	0	0
$805$	27.2627	0.960882
$806$	−15.4236	−0.543271
$807$	0	0
$808$	17.9816	0.632592
$809$	−2.17207	−0.0763659	−0.0381829	−	0.999271i	$-0.512157\pi$
$809$	−2.17207	−0.0763659	−0.0381829	+	0.999271i	$0.512157\pi$
$810$	0	0
$811$	23.0485	0.809341	0.404671	−	0.914462i	$-0.367386\pi$
$811$	23.0485	0.809341	0.404671	+	0.914462i	$0.367386\pi$
$812$	−0.876966	−0.0307755
$813$	0	0
$814$	2.57440	0.0902327
$815$	3.43504	0.120324
$816$	0	0
$817$	2.38911	0.0835844
$818$	11.3913	0.398287
$819$	0	0
$820$	61.7051	2.15483
$821$	−16.4692	−0.574780	−0.287390	−	0.957814i	$-0.592788\pi$
$821$	−16.4692	−0.574780	−0.287390	+	0.957814i	$0.592788\pi$
$822$	0	0
$823$	45.9664	1.60229	0.801143	−	0.598473i	$-0.204226\pi$
$823$	45.9664	1.60229	0.801143	+	0.598473i	$0.204226\pi$
$824$	3.02724	0.105459
$825$	0	0
$826$	9.96533	0.346738
$827$	10.6911	0.371767	0.185883	−	0.982572i	$-0.440485\pi$
$827$	10.6911	0.371767	0.185883	+	0.982572i	$0.440485\pi$
$828$	0	0
$829$	50.9761	1.77047	0.885237	−	0.465141i	$-0.153996\pi$
$829$	50.9761	1.77047	0.885237	+	0.465141i	$0.153996\pi$
$830$	−27.1077	−0.940921
$831$	0	0
$832$	−3.04224	−0.105471
$833$	−6.15521	−0.213265
$834$	0	0
$835$	−45.5924	−1.57779
$836$	−1.24808	−0.0431658
$837$	0	0
$838$	−24.1478	−0.834172
$839$	−33.3331	−1.15079	−0.575394	−	0.817876i	$-0.695151\pi$
$839$	−33.3331	−1.15079	−0.575394	+	0.817876i	$0.695151\pi$
$840$	0	0
$841$	−28.8935	−0.996329
$842$	4.18528	0.144234
$843$	0	0
$844$	−4.13634	−0.142379
$845$	21.8555	0.751853
$846$	0	0
$847$	−13.1034	−0.450238
$848$	−5.16533	−0.177378
$849$	0	0
$850$	11.1380	0.382029
$851$	−7.96170	−0.272924
$852$	0	0
$853$	−53.4607	−1.83046	−0.915229	−	0.402934i	$-0.867991\pi$
$853$	−53.4607	−1.83046	−0.915229	+	0.402934i	$0.867991\pi$
$854$	−14.4839	−0.495629
$855$	0	0
$856$	−19.4512	−0.664828
$857$	−7.79499	−0.266272	−0.133136	−	0.991098i	$-0.542505\pi$
$857$	−7.79499	−0.266272	−0.133136	+	0.991098i	$0.542505\pi$
$858$	0	0
$859$	29.2342	0.997459	0.498730	−	0.866758i	$-0.333800\pi$
$859$	29.2342	0.997459	0.498730	+	0.866758i	$0.333800\pi$
$860$	33.1842	1.13157
$861$	0	0
$862$	−14.9626	−0.509627
$863$	28.5823	0.972954	0.486477	−	0.873693i	$-0.338282\pi$
$863$	28.5823	0.972954	0.486477	+	0.873693i	$0.338282\pi$
$864$	0	0
$865$	−63.9643	−2.17485
$866$	11.3030	0.384093
$867$	0	0
$868$	−22.2704	−0.755906
$869$	−0.280579	−0.00951798
$870$	0	0
$871$	−38.4498	−1.30282
$872$	23.6256	0.800063
$873$	0	0
$874$	−1.17303	−0.0396783
$875$	−37.2494	−1.25926
$876$	0	0
$877$	17.2120	0.581208	0.290604	−	0.956843i	$-0.406144\pi$
$877$	17.2120	0.581208	0.290604	+	0.956843i	$0.406144\pi$
$878$	12.8574	0.433918
$879$	0	0
$880$	−10.4662	−0.352814
$881$	−6.11779	−0.206114	−0.103057	−	0.994675i	$-0.532862\pi$
$881$	−6.11779	−0.206114	−0.103057	+	0.994675i	$0.532862\pi$
$882$	0	0
$883$	25.0363	0.842540	0.421270	−	0.906935i	$-0.361585\pi$
$883$	25.0363	0.842540	0.421270	+	0.906935i	$0.361585\pi$
$884$	6.55070	0.220324
$885$	0	0
$886$	−6.44595	−0.216556
$887$	44.2916	1.48717	0.743584	−	0.668643i	$-0.233124\pi$
$887$	44.2916	1.48717	0.743584	+	0.668643i	$0.233124\pi$
$888$	0	0
$889$	−1.75230	−0.0587704
$890$	−38.1245	−1.27794
$891$	0	0
$892$	−35.8387	−1.19997
$893$	3.70633	0.124028
$894$	0	0
$895$	40.7934	1.36357
$896$	−18.9751	−0.633914
$897$	0	0
$898$	5.79515	0.193387
$899$	2.70348	0.0901661
$900$	0	0
$901$	−5.69629	−0.189771
$902$	13.1292	0.437156
$903$	0	0
$904$	25.9011	0.861459
$905$	−73.3744	−2.43905
$906$	0	0
$907$	9.26205	0.307541	0.153771	−	0.988107i	$-0.450858\pi$
$907$	9.26205	0.307541	0.153771	+	0.988107i	$0.450858\pi$
$908$	−7.34078	−0.243612
$909$	0	0
$910$	12.8063	0.424524
$911$	0.383797	0.0127158	0.00635788	−	0.999980i	$-0.497976\pi$
$911$	0.383797	0.0127158	0.00635788	+	0.999980i	$0.497976\pi$
$912$	0	0
$913$	18.9791	0.628116
$914$	−9.14784	−0.302584
$915$	0	0
$916$	−37.5415	−1.24041
$917$	−5.65272	−0.186669
$918$	0	0
$919$	−38.1271	−1.25770	−0.628848	−	0.777528i	$-0.716473\pi$
$919$	−38.1271	−1.25770	−0.628848	+	0.777528i	$0.716473\pi$
$920$	−37.5377	−1.23758
$921$	0	0
$922$	−23.1927	−0.763810
$923$	18.9690	0.624374
$924$	0	0
$925$	20.9239	0.687974
$926$	0.940543	0.0309082
$927$	0	0
$928$	1.89087	0.0620708
$929$	−30.7660	−1.00940	−0.504700	−	0.863295i	$-0.668397\pi$
$929$	−30.7660	−1.00940	−0.504700	+	0.863295i	$0.668397\pi$
$930$	0	0
$931$	−1.70366	−0.0558351
$932$	28.8308	0.944383
$933$	0	0
$934$	20.3780	0.666789
$935$	−11.5420	−0.377464
$936$	0	0
$937$	42.0990	1.37531	0.687657	−	0.726036i	$-0.258639\pi$
$937$	42.0990	1.37531	0.687657	+	0.726036i	$0.258639\pi$
$938$	16.8723	0.550899
$939$	0	0
$940$	51.4801	1.67910
$941$	38.0545	1.24054	0.620271	−	0.784388i	$-0.287023\pi$
$941$	38.0545	1.24054	0.620271	+	0.784388i	$0.287023\pi$
$942$	0	0
$943$	−40.6041	−1.32225
$944$	11.8315	0.385084
$945$	0	0
$946$	7.06074	0.229564
$947$	19.7845	0.642910	0.321455	−	0.946925i	$-0.395828\pi$
$947$	19.7845	0.642910	0.321455	+	0.946925i	$0.395828\pi$
$948$	0	0
$949$	−3.48005	−0.112967
$950$	3.08280	0.100019
$951$	0	0
$952$	−6.62264	−0.214641
$953$	−3.11778	−0.100995	−0.0504974	−	0.998724i	$-0.516081\pi$
$953$	−3.11778	−0.100995	−0.0504974	+	0.998724i	$0.516081\pi$
$954$	0	0
$955$	−100.286	−3.24519
$956$	−22.2957	−0.721094
$957$	0	0
$958$	14.4059	0.465434
$959$	35.7028	1.15290
$960$	0	0
$961$	37.6543	1.21465
$962$	−3.73990	−0.120579
$963$	0	0
$964$	−21.6988	−0.698870
$965$	−66.7985	−2.15032
$966$	0	0
$967$	−55.5899	−1.78765	−0.893826	−	0.448414i	$-0.851989\pi$
$967$	−55.5899	−1.78765	−0.893826	+	0.448414i	$0.851989\pi$
$968$	18.0420	0.579891
$969$	0	0
$970$	41.3036	1.32618
$971$	−14.2495	−0.457289	−0.228644	−	0.973510i	$-0.573429\pi$
$971$	−14.2495	−0.457289	−0.228644	+	0.973510i	$0.573429\pi$
$972$	0	0
$973$	−16.4934	−0.528755
$974$	15.7778	0.505554
$975$	0	0
$976$	−17.1963	−0.550440
$977$	−1.69184	−0.0541268	−0.0270634	−	0.999634i	$-0.508616\pi$
$977$	−1.69184	−0.0541268	−0.0270634	+	0.999634i	$0.508616\pi$
$978$	0	0
$979$	26.6924	0.853092
$980$	−23.6634	−0.755900
$981$	0	0
$982$	−12.2459	−0.390784
$983$	−20.7765	−0.662667	−0.331333	−	0.943514i	$-0.607498\pi$
$983$	−20.7765	−0.662667	−0.331333	+	0.943514i	$0.607498\pi$
$984$	0	0
$985$	−1.93785	−0.0617450
$986$	0.348949	0.0111128
$987$	0	0
$988$	1.81312	0.0576831
$989$	−21.8364	−0.694356
$990$	0	0
$991$	−9.71028	−0.308457	−0.154229	−	0.988035i	$-0.549289\pi$
$991$	−9.71028	−0.308457	−0.154229	+	0.988035i	$0.549289\pi$
$992$	48.0183	1.52458
$993$	0	0
$994$	−8.32385	−0.264016
$995$	−75.8434	−2.40440
$996$	0	0
$997$	−17.9524	−0.568558	−0.284279	−	0.958742i	$-0.591754\pi$
$997$	−17.9524	−0.568558	−0.284279	+	0.958742i	$0.591754\pi$
$998$	22.6266	0.716232
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1143.2.a.j.1.6		9
3.2	odd	2		381.2.a.e.1.4	✓	9
12.11	even	2		6096.2.a.bk.1.9		9
15.14	odd	2		9525.2.a.p.1.6		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
381.2.a.e.1.4	✓	9	3.2	odd	2
1143.2.a.j.1.6		9	1.1	even	1	trivial
6096.2.a.bk.1.9		9	12.11	even	2
9525.2.a.p.1.6		9	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1143 = 3^{2} \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1143.a (trivial)

\(f(q)\)	\(=\)	\(q+0.682747 q^{2} -1.53386 q^{4} -3.92611 q^{5} +1.75230 q^{7} -2.41273 q^{8} +O(q^{10})\)	\(q+0.682747 q^{2} -1.53386 q^{4} -3.92611 q^{5} +1.75230 q^{7} -2.41273 q^{8} -2.68054 q^{10} +1.87675 q^{11} -2.72641 q^{13} +1.19638 q^{14} +1.42043 q^{16} +1.56644 q^{17} +0.433563 q^{19} +6.02209 q^{20} +1.28134 q^{22} -3.96274 q^{23} +10.4144 q^{25} -1.86145 q^{26} -2.68778 q^{28} +0.326279 q^{29} +8.28579 q^{31} +5.79525 q^{32} +1.06948 q^{34} -6.87974 q^{35} +2.00914 q^{37} +0.296014 q^{38} +9.47265 q^{40} +10.2464 q^{41} +5.51041 q^{43} -2.87866 q^{44} -2.70555 q^{46} +8.54855 q^{47} -3.92943 q^{49} +7.11038 q^{50} +4.18191 q^{52} -3.63647 q^{53} -7.36832 q^{55} -4.22784 q^{56} +0.222766 q^{58} +8.32957 q^{59} -12.1064 q^{61} +5.65710 q^{62} +1.11584 q^{64} +10.7042 q^{65} +14.1028 q^{67} -2.40269 q^{68} -4.69713 q^{70} -6.95753 q^{71} +1.27642 q^{73} +1.37173 q^{74} -0.665023 q^{76} +3.28863 q^{77} -0.149503 q^{79} -5.57675 q^{80} +6.99574 q^{82} +10.1128 q^{83} -6.15001 q^{85} +3.76222 q^{86} -4.52809 q^{88} +14.2227 q^{89} -4.77749 q^{91} +6.07828 q^{92} +5.83650 q^{94} -1.70222 q^{95} -15.4087 q^{97} -2.68281 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 2 q^{2} + 14 q^{4} + 4 q^{5} + 10 q^{7} + 6 q^{8}+O(q^{10}) \) 9 * q + 2 * q^2 + 14 * q^4 + 4 * q^5 + 10 * q^7 + 6 * q^8	\( 9 q + 2 q^{2} + 14 q^{4} + 4 q^{5} + 10 q^{7} + 6 q^{8} - 4 q^{10} - 8 q^{11} + 14 q^{13} - 4 q^{14} + 32 q^{16} + 6 q^{17} + 12 q^{19} + 28 q^{20} - 18 q^{22} + 4 q^{23} + 21 q^{25} + 14 q^{26} + 8 q^{29} + 4 q^{31} + 29 q^{32} - 3 q^{34} - 6 q^{35} + 22 q^{37} + 7 q^{38} + 2 q^{41} + 6 q^{43} - 17 q^{44} - 10 q^{46} + 2 q^{47} + 23 q^{49} + 20 q^{50} - 9 q^{52} + 12 q^{53} - 22 q^{55} - 18 q^{56} - 28 q^{58} + 6 q^{59} + 2 q^{61} + 15 q^{62} + 24 q^{64} - 4 q^{65} + 18 q^{67} + 24 q^{68} - 72 q^{70} - 24 q^{71} + 14 q^{73} - 3 q^{74} + 4 q^{76} + 18 q^{77} + 12 q^{79} + 86 q^{80} + 4 q^{82} + 20 q^{83} - 24 q^{85} - 16 q^{86} - 55 q^{88} + 30 q^{89} + 14 q^{91} + 46 q^{92} - 66 q^{94} + 32 q^{95} + 12 q^{97} + 62 q^{98}+O(q^{100}) \) 9 * q + 2 * q^2 + 14 * q^4 + 4 * q^5 + 10 * q^7 + 6 * q^8 - 4 * q^10 - 8 * q^11 + 14 * q^13 - 4 * q^14 + 32 * q^16 + 6 * q^17 + 12 * q^19 + 28 * q^20 - 18 * q^22 + 4 * q^23 + 21 * q^25 + 14 * q^26 + 8 * q^29 + 4 * q^31 + 29 * q^32 - 3 * q^34 - 6 * q^35 + 22 * q^37 + 7 * q^38 + 2 * q^41 + 6 * q^43 - 17 * q^44 - 10 * q^46 + 2 * q^47 + 23 * q^49 + 20 * q^50 - 9 * q^52 + 12 * q^53 - 22 * q^55 - 18 * q^56 - 28 * q^58 + 6 * q^59 + 2 * q^61 + 15 * q^62 + 24 * q^64 - 4 * q^65 + 18 * q^67 + 24 * q^68 - 72 * q^70 - 24 * q^71 + 14 * q^73 - 3 * q^74 + 4 * q^76 + 18 * q^77 + 12 * q^79 + 86 * q^80 + 4 * q^82 + 20 * q^83 - 24 * q^85 - 16 * q^86 - 55 * q^88 + 30 * q^89 + 14 * q^91 + 46 * q^92 - 66 * q^94 + 32 * q^95 + 12 * q^97 + 62 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more