LMFDB - Embedded newform 279.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [279,2,Mod(1,279)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(279, 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("279.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 279 = 3^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	279.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:	$2.22782621639$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 31)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	279.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.61803 q^{2} +0.618034 q^{4} -1.00000 q^{5} +0.236068 q^{7} +2.23607 q^{8} +O(q^{10})$	\(q-1.61803 q^{2} +0.618034 q^{4} -1.00000 q^{5} +0.236068 q^{7} +2.23607 q^{8} +1.61803 q^{10} -2.00000 q^{11} -3.23607 q^{13} -0.381966 q^{14} -4.85410 q^{16} -0.763932 q^{17} -2.23607 q^{19} -0.618034 q^{20} +3.23607 q^{22} -5.70820 q^{23} -4.00000 q^{25} +5.23607 q^{26} +0.145898 q^{28} -2.76393 q^{29} +1.00000 q^{31} +3.38197 q^{32} +1.23607 q^{34} -0.236068 q^{35} -2.00000 q^{37} +3.61803 q^{38} -2.23607 q^{40} -7.00000 q^{41} +1.23607 q^{43} -1.23607 q^{44} +9.23607 q^{46} -2.47214 q^{47} -6.94427 q^{49} +6.47214 q^{50} -2.00000 q^{52} +10.4721 q^{53} +2.00000 q^{55} +0.527864 q^{56} +4.47214 q^{58} -2.23607 q^{59} +8.18034 q^{61} -1.61803 q^{62} +4.23607 q^{64} +3.23607 q^{65} +8.00000 q^{67} -0.472136 q^{68} +0.381966 q^{70} +9.18034 q^{71} +8.47214 q^{73} +3.23607 q^{74} -1.38197 q^{76} -0.472136 q^{77} -11.7082 q^{79} +4.85410 q^{80} +11.3262 q^{82} +14.9443 q^{83} +0.763932 q^{85} -2.00000 q^{86} -4.47214 q^{88} -11.7082 q^{89} -0.763932 q^{91} -3.52786 q^{92} +4.00000 q^{94} +2.23607 q^{95} -15.9443 q^{97} +11.2361 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - q^{4} - 2 q^{5} - 4 q^{7}+O(q^{10}) $ 2 * q - q^2 - q^4 - 2 * q^5 - 4 * q^7	$ 2 q - q^{2} - q^{4} - 2 q^{5} - 4 q^{7} + q^{10} - 4 q^{11} - 2 q^{13} - 3 q^{14} - 3 q^{16} - 6 q^{17} + q^{20} + 2 q^{22} + 2 q^{23} - 8 q^{25} + 6 q^{26} + 7 q^{28} - 10 q^{29} + 2 q^{31} + 9 q^{32} - 2 q^{34} + 4 q^{35} - 4 q^{37} + 5 q^{38} - 14 q^{41} - 2 q^{43} + 2 q^{44} + 14 q^{46} + 4 q^{47} + 4 q^{49} + 4 q^{50} - 4 q^{52} + 12 q^{53} + 4 q^{55} + 10 q^{56} - 6 q^{61} - q^{62} + 4 q^{64} + 2 q^{65} + 16 q^{67} + 8 q^{68} + 3 q^{70} - 4 q^{71} + 8 q^{73} + 2 q^{74} - 5 q^{76} + 8 q^{77} - 10 q^{79} + 3 q^{80} + 7 q^{82} + 12 q^{83} + 6 q^{85} - 4 q^{86} - 10 q^{89} - 6 q^{91} - 16 q^{92} + 8 q^{94} - 14 q^{97} + 18 q^{98}+O(q^{100}) $ 2 * q - q^2 - q^4 - 2 * q^5 - 4 * q^7 + q^10 - 4 * q^11 - 2 * q^13 - 3 * q^14 - 3 * q^16 - 6 * q^17 + q^20 + 2 * q^22 + 2 * q^23 - 8 * q^25 + 6 * q^26 + 7 * q^28 - 10 * q^29 + 2 * q^31 + 9 * q^32 - 2 * q^34 + 4 * q^35 - 4 * q^37 + 5 * q^38 - 14 * q^41 - 2 * q^43 + 2 * q^44 + 14 * q^46 + 4 * q^47 + 4 * q^49 + 4 * q^50 - 4 * q^52 + 12 * q^53 + 4 * q^55 + 10 * q^56 - 6 * q^61 - q^62 + 4 * q^64 + 2 * q^65 + 16 * q^67 + 8 * q^68 + 3 * q^70 - 4 * q^71 + 8 * q^73 + 2 * q^74 - 5 * q^76 + 8 * q^77 - 10 * q^79 + 3 * q^80 + 7 * q^82 + 12 * q^83 + 6 * q^85 - 4 * q^86 - 10 * q^89 - 6 * q^91 - 16 * q^92 + 8 * q^94 - 14 * q^97 + 18 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.61803	−1.14412	−0.572061	−	0.820211i	$-0.693856\pi$
$2$	−1.61803	−1.14412	−0.572061	+	0.820211i	$0.693856\pi$
$3$	0	0
$3$	0	0
$4$	0.618034	0.309017
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	0.236068	0.0892253	0.0446127	−	0.999004i	$-0.485795\pi$
$7$	0.236068	0.0892253	0.0446127	+	0.999004i	$0.485795\pi$
$8$	2.23607	0.790569
$9$	0	0
$10$	1.61803	0.511667
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	−3.23607	−0.897524	−0.448762	−	0.893651i	$-0.648135\pi$
$13$	−3.23607	−0.897524	−0.448762	+	0.893651i	$0.648135\pi$
$14$	−0.381966	−0.102085
$15$	0	0
$16$	−4.85410	−1.21353
$17$	−0.763932	−0.185281	−0.0926404	−	0.995700i	$-0.529531\pi$
$17$	−0.763932	−0.185281	−0.0926404	+	0.995700i	$0.529531\pi$
$18$	0	0
$19$	−2.23607	−0.512989	−0.256495	−	0.966546i	$-0.582568\pi$
$19$	−2.23607	−0.512989	−0.256495	+	0.966546i	$0.582568\pi$
$20$	−0.618034	−0.138197
$21$	0	0
$22$	3.23607	0.689932
$23$	−5.70820	−1.19024	−0.595121	−	0.803636i	$-0.702896\pi$
$23$	−5.70820	−1.19024	−0.595121	+	0.803636i	$0.702896\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	5.23607	1.02688
$27$	0	0
$28$	0.145898	0.0275721
$29$	−2.76393	−0.513249	−0.256625	−	0.966511i	$-0.582610\pi$
$29$	−2.76393	−0.513249	−0.256625	+	0.966511i	$0.582610\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	3.38197	0.597853
$33$	0	0
$34$	1.23607	0.211984
$35$	−0.236068	−0.0399028
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	3.61803	0.586923
$39$	0	0
$40$	−2.23607	−0.353553
$41$	−7.00000	−1.09322	−0.546608	−	0.837389i	$-0.684081\pi$
$41$	−7.00000	−1.09322	−0.546608	+	0.837389i	$0.684081\pi$
$42$	0	0
$43$	1.23607	0.188499	0.0942493	−	0.995549i	$-0.469955\pi$
$43$	1.23607	0.188499	0.0942493	+	0.995549i	$0.469955\pi$
$44$	−1.23607	−0.186344
$45$	0	0
$46$	9.23607	1.36178
$47$	−2.47214	−0.360598	−0.180299	−	0.983612i	$-0.557707\pi$
$47$	−2.47214	−0.360598	−0.180299	+	0.983612i	$0.557707\pi$
$48$	0	0
$49$	−6.94427	−0.992039
$50$	6.47214	0.915298
$51$	0	0
$52$	−2.00000	−0.277350
$53$	10.4721	1.43846	0.719229	−	0.694773i	$-0.244495\pi$
$53$	10.4721	1.43846	0.719229	+	0.694773i	$0.244495\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	0.527864	0.0705388
$57$	0	0
$58$	4.47214	0.587220
$59$	−2.23607	−0.291111	−0.145556	−	0.989350i	$-0.546497\pi$
$59$	−2.23607	−0.291111	−0.145556	+	0.989350i	$0.546497\pi$
$60$	0	0
$61$	8.18034	1.04739	0.523693	−	0.851907i	$-0.324554\pi$
$61$	8.18034	1.04739	0.523693	+	0.851907i	$0.324554\pi$
$62$	−1.61803	−0.205491
$63$	0	0
$64$	4.23607	0.529508
$65$	3.23607	0.401385
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	−0.472136	−0.0572549
$69$	0	0
$70$	0.381966	0.0456537
$71$	9.18034	1.08951	0.544753	−	0.838597i	$-0.316623\pi$
$71$	9.18034	1.08951	0.544753	+	0.838597i	$0.316623\pi$
$72$	0	0
$73$	8.47214	0.991589	0.495794	−	0.868440i	$-0.334877\pi$
$73$	8.47214	0.991589	0.495794	+	0.868440i	$0.334877\pi$
$74$	3.23607	0.376185
$75$	0	0
$76$	−1.38197	−0.158522
$77$	−0.472136	−0.0538049
$78$	0	0
$79$	−11.7082	−1.31728	−0.658638	−	0.752460i	$-0.728867\pi$
$79$	−11.7082	−1.31728	−0.658638	+	0.752460i	$0.728867\pi$
$80$	4.85410	0.542705
$81$	0	0
$82$	11.3262	1.25077
$83$	14.9443	1.64035	0.820173	−	0.572115i	$-0.193877\pi$
$83$	14.9443	1.64035	0.820173	+	0.572115i	$0.193877\pi$
$84$	0	0
$85$	0.763932	0.0828601
$86$	−2.00000	−0.215666
$87$	0	0
$88$	−4.47214	−0.476731
$89$	−11.7082	−1.24107	−0.620534	−	0.784180i	$-0.713084\pi$
$89$	−11.7082	−1.24107	−0.620534	+	0.784180i	$0.713084\pi$
$90$	0	0
$91$	−0.763932	−0.0800818
$92$	−3.52786	−0.367805
$93$	0	0
$94$	4.00000	0.412568
$95$	2.23607	0.229416
$96$	0	0
$97$	−15.9443	−1.61890	−0.809448	−	0.587192i	$-0.800233\pi$
$97$	−15.9443	−1.61890	−0.809448	+	0.587192i	$0.800233\pi$
$98$	11.2361	1.13501
$99$	0	0
$100$	−2.47214	−0.247214
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	0	0
$103$	6.23607	0.614458	0.307229	−	0.951636i	$-0.400598\pi$
$103$	6.23607	0.614458	0.307229	+	0.951636i	$0.400598\pi$
$104$	−7.23607	−0.709555
$105$	0	0
$106$	−16.9443	−1.64577
$107$	−5.76393	−0.557220	−0.278610	−	0.960404i	$-0.589874\pi$
$107$	−5.76393	−0.557220	−0.278610	+	0.960404i	$0.589874\pi$
$108$	0	0
$109$	−13.9443	−1.33562	−0.667810	−	0.744332i	$-0.732768\pi$
$109$	−13.9443	−1.33562	−0.667810	+	0.744332i	$0.732768\pi$
$110$	−3.23607	−0.308547
$111$	0	0
$112$	−1.14590	−0.108277
$113$	−3.47214	−0.326631	−0.163316	−	0.986574i	$-0.552219\pi$
$113$	−3.47214	−0.326631	−0.163316	+	0.986574i	$0.552219\pi$
$114$	0	0
$115$	5.70820	0.532293
$116$	−1.70820	−0.158603
$117$	0	0
$118$	3.61803	0.333067
$119$	−0.180340	−0.0165317
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−13.2361	−1.19834
$123$	0	0
$124$	0.618034	0.0555011
$125$	9.00000	0.804984
$126$	0	0
$127$	12.4721	1.10672	0.553362	−	0.832941i	$-0.313345\pi$
$127$	12.4721	1.10672	0.553362	+	0.832941i	$0.313345\pi$
$128$	−13.6180	−1.20368
$129$	0	0
$130$	−5.23607	−0.459234
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	−0.527864	−0.0457716
$134$	−12.9443	−1.11821
$135$	0	0
$136$	−1.70820	−0.146477
$137$	−6.29180	−0.537544	−0.268772	−	0.963204i	$-0.586618\pi$
$137$	−6.29180	−0.537544	−0.268772	+	0.963204i	$0.586618\pi$
$138$	0	0
$139$	13.4164	1.13796	0.568982	−	0.822350i	$-0.307337\pi$
$139$	13.4164	1.13796	0.568982	+	0.822350i	$0.307337\pi$
$140$	−0.145898	−0.0123306
$141$	0	0
$142$	−14.8541	−1.24653
$143$	6.47214	0.541227
$144$	0	0
$145$	2.76393	0.229532
$146$	−13.7082	−1.13450
$147$	0	0
$148$	−1.23607	−0.101604
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	−14.1803	−1.15398	−0.576990	−	0.816751i	$-0.695773\pi$
$151$	−14.1803	−1.15398	−0.576990	+	0.816751i	$0.695773\pi$
$152$	−5.00000	−0.405554
$153$	0	0
$154$	0.763932	0.0615594
$155$	−1.00000	−0.0803219
$156$	0	0
$157$	20.8885	1.66709	0.833544	−	0.552454i	$-0.186308\pi$
$157$	20.8885	1.66709	0.833544	+	0.552454i	$0.186308\pi$
$158$	18.9443	1.50713
$159$	0	0
$160$	−3.38197	−0.267368
$161$	−1.34752	−0.106200
$162$	0	0
$163$	10.7082	0.838731	0.419366	−	0.907817i	$-0.362253\pi$
$163$	10.7082	0.838731	0.419366	+	0.907817i	$0.362253\pi$
$164$	−4.32624	−0.337822
$165$	0	0
$166$	−24.1803	−1.87676
$167$	6.47214	0.500829	0.250414	−	0.968139i	$-0.419433\pi$
$167$	6.47214	0.500829	0.250414	+	0.968139i	$0.419433\pi$
$168$	0	0
$169$	−2.52786	−0.194451
$170$	−1.23607	−0.0948021
$171$	0	0
$172$	0.763932	0.0582493
$173$	−2.94427	−0.223849	−0.111924	−	0.993717i	$-0.535701\pi$
$173$	−2.94427	−0.223849	−0.111924	+	0.993717i	$0.535701\pi$
$174$	0	0
$175$	−0.944272	−0.0713802
$176$	9.70820	0.731783
$177$	0	0
$178$	18.9443	1.41993
$179$	−1.70820	−0.127677	−0.0638386	−	0.997960i	$-0.520334\pi$
$179$	−1.70820	−0.127677	−0.0638386	+	0.997960i	$0.520334\pi$
$180$	0	0
$181$	−4.18034	−0.310722	−0.155361	−	0.987858i	$-0.549654\pi$
$181$	−4.18034	−0.310722	−0.155361	+	0.987858i	$0.549654\pi$
$182$	1.23607	0.0916235
$183$	0	0
$184$	−12.7639	−0.940970
$185$	2.00000	0.147043
$186$	0	0
$187$	1.52786	0.111728
$188$	−1.52786	−0.111431
$189$	0	0
$190$	−3.61803	−0.262480
$191$	19.1803	1.38784	0.693920	−	0.720052i	$-0.255882\pi$
$191$	19.1803	1.38784	0.693920	+	0.720052i	$0.255882\pi$
$192$	0	0
$193$	3.47214	0.249930	0.124965	−	0.992161i	$-0.460118\pi$
$193$	3.47214	0.249930	0.124965	+	0.992161i	$0.460118\pi$
$194$	25.7984	1.85222
$195$	0	0
$196$	−4.29180	−0.306557
$197$	−11.4164	−0.813385	−0.406693	−	0.913565i	$-0.633318\pi$
$197$	−11.4164	−0.813385	−0.406693	+	0.913565i	$0.633318\pi$
$198$	0	0
$199$	−18.9443	−1.34292	−0.671462	−	0.741039i	$-0.734333\pi$
$199$	−18.9443	−1.34292	−0.671462	+	0.741039i	$0.734333\pi$
$200$	−8.94427	−0.632456
$201$	0	0
$202$	−4.85410	−0.341533
$203$	−0.652476	−0.0457948
$204$	0	0
$205$	7.00000	0.488901
$206$	−10.0902	−0.703015
$207$	0	0
$208$	15.7082	1.08917
$209$	4.47214	0.309344
$210$	0	0
$211$	23.1803	1.59580	0.797900	−	0.602790i	$-0.205944\pi$
$211$	23.1803	1.59580	0.797900	+	0.602790i	$0.205944\pi$
$212$	6.47214	0.444508
$213$	0	0
$214$	9.32624	0.637528
$215$	−1.23607	−0.0842991
$216$	0	0
$217$	0.236068	0.0160253
$218$	22.5623	1.52811
$219$	0	0
$220$	1.23607	0.0833357
$221$	2.47214	0.166294
$222$	0	0
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	0.798374	0.0533436
$225$	0	0
$226$	5.61803	0.373706
$227$	6.47214	0.429571	0.214785	−	0.976661i	$-0.431095\pi$
$227$	6.47214	0.429571	0.214785	+	0.976661i	$0.431095\pi$
$228$	0	0
$229$	−13.4164	−0.886581	−0.443291	−	0.896378i	$-0.646189\pi$
$229$	−13.4164	−0.886581	−0.443291	+	0.896378i	$0.646189\pi$
$230$	−9.23607	−0.609008
$231$	0	0
$232$	−6.18034	−0.405759
$233$	−17.9443	−1.17557	−0.587784	−	0.809018i	$-0.700000\pi$
$233$	−17.9443	−1.17557	−0.587784	+	0.809018i	$0.700000\pi$
$234$	0	0
$235$	2.47214	0.161264
$236$	−1.38197	−0.0899583
$237$	0	0
$238$	0.291796	0.0189143
$239$	11.7082	0.757341	0.378670	−	0.925532i	$-0.376381\pi$
$239$	11.7082	0.757341	0.378670	+	0.925532i	$0.376381\pi$
$240$	0	0
$241$	14.3607	0.925053	0.462526	−	0.886606i	$-0.346943\pi$
$241$	14.3607	0.925053	0.462526	+	0.886606i	$0.346943\pi$
$242$	11.3262	0.728078
$243$	0	0
$244$	5.05573	0.323660
$245$	6.94427	0.443653
$246$	0	0
$247$	7.23607	0.460420
$248$	2.23607	0.141990
$249$	0	0
$250$	−14.5623	−0.921001
$251$	1.81966	0.114856	0.0574280	−	0.998350i	$-0.481710\pi$
$251$	1.81966	0.114856	0.0574280	+	0.998350i	$0.481710\pi$
$252$	0	0
$253$	11.4164	0.717743
$254$	−20.1803	−1.26623
$255$	0	0
$256$	13.5623	0.847644
$257$	−1.94427	−0.121280	−0.0606402	−	0.998160i	$-0.519314\pi$
$257$	−1.94427	−0.121280	−0.0606402	+	0.998160i	$0.519314\pi$
$258$	0	0
$259$	−0.472136	−0.0293371
$260$	2.00000	0.124035
$261$	0	0
$262$	19.4164	1.19955
$263$	23.2361	1.43280	0.716399	−	0.697691i	$-0.245789\pi$
$263$	23.2361	1.43280	0.716399	+	0.697691i	$0.245789\pi$
$264$	0	0
$265$	−10.4721	−0.643298
$266$	0.854102	0.0523684
$267$	0	0
$268$	4.94427	0.302019
$269$	11.0557	0.674080	0.337040	−	0.941490i	$-0.390574\pi$
$269$	11.0557	0.674080	0.337040	+	0.941490i	$0.390574\pi$
$270$	0	0
$271$	−14.1803	−0.861394	−0.430697	−	0.902497i	$-0.641732\pi$
$271$	−14.1803	−0.861394	−0.430697	+	0.902497i	$0.641732\pi$
$272$	3.70820	0.224843
$273$	0	0
$274$	10.1803	0.615017
$275$	8.00000	0.482418
$276$	0	0
$277$	−12.6525	−0.760214	−0.380107	−	0.924943i	$-0.624113\pi$
$277$	−12.6525	−0.760214	−0.380107	+	0.924943i	$0.624113\pi$
$278$	−21.7082	−1.30197
$279$	0	0
$280$	−0.527864	−0.0315459
$281$	−17.0000	−1.01413	−0.507067	−	0.861906i	$-0.669271\pi$
$281$	−17.0000	−1.01413	−0.507067	+	0.861906i	$0.669271\pi$
$282$	0	0
$283$	−13.8885	−0.825588	−0.412794	−	0.910824i	$-0.635447\pi$
$283$	−13.8885	−0.825588	−0.412794	+	0.910824i	$0.635447\pi$
$284$	5.67376	0.336676
$285$	0	0
$286$	−10.4721	−0.619230
$287$	−1.65248	−0.0975426
$288$	0	0
$289$	−16.4164	−0.965671
$290$	−4.47214	−0.262613
$291$	0	0
$292$	5.23607	0.306418
$293$	0.472136	0.0275825	0.0137912	−	0.999905i	$-0.495610\pi$
$293$	0.472136	0.0275825	0.0137912	+	0.999905i	$0.495610\pi$
$294$	0	0
$295$	2.23607	0.130189
$296$	−4.47214	−0.259938
$297$	0	0
$298$	16.1803	0.937302
$299$	18.4721	1.06827
$300$	0	0
$301$	0.291796	0.0168188
$302$	22.9443	1.32029
$303$	0	0
$304$	10.8541	0.622525
$305$	−8.18034	−0.468405
$306$	0	0
$307$	−28.7082	−1.63846	−0.819232	−	0.573462i	$-0.805600\pi$
$307$	−28.7082	−1.63846	−0.819232	+	0.573462i	$0.805600\pi$
$308$	−0.291796	−0.0166266
$309$	0	0
$310$	1.61803	0.0918982
$311$	29.1803	1.65467	0.827333	−	0.561712i	$-0.189857\pi$
$311$	29.1803	1.65467	0.827333	+	0.561712i	$0.189857\pi$
$312$	0	0
$313$	16.7639	0.947553	0.473777	−	0.880645i	$-0.342890\pi$
$313$	16.7639	0.947553	0.473777	+	0.880645i	$0.342890\pi$
$314$	−33.7984	−1.90735
$315$	0	0
$316$	−7.23607	−0.407061
$317$	−4.05573	−0.227792	−0.113896	−	0.993493i	$-0.536333\pi$
$317$	−4.05573	−0.227792	−0.113896	+	0.993493i	$0.536333\pi$
$318$	0	0
$319$	5.52786	0.309501
$320$	−4.23607	−0.236803
$321$	0	0
$322$	2.18034	0.121506
$323$	1.70820	0.0950470
$324$	0	0
$325$	12.9443	0.718019
$326$	−17.3262	−0.959612
$327$	0	0
$328$	−15.6525	−0.864263
$329$	−0.583592	−0.0321745
$330$	0	0
$331$	2.00000	0.109930	0.0549650	−	0.998488i	$-0.482495\pi$
$331$	2.00000	0.109930	0.0549650	+	0.998488i	$0.482495\pi$
$332$	9.23607	0.506895
$333$	0	0
$334$	−10.4721	−0.573010
$335$	−8.00000	−0.437087
$336$	0	0
$337$	−14.7639	−0.804243	−0.402121	−	0.915586i	$-0.631727\pi$
$337$	−14.7639	−0.804243	−0.402121	+	0.915586i	$0.631727\pi$
$338$	4.09017	0.222476
$339$	0	0
$340$	0.472136	0.0256052
$341$	−2.00000	−0.108306
$342$	0	0
$343$	−3.29180	−0.177740
$344$	2.76393	0.149021
$345$	0	0
$346$	4.76393	0.256111
$347$	−24.1803	−1.29807	−0.649034	−	0.760759i	$-0.724827\pi$
$347$	−24.1803	−1.29807	−0.649034	+	0.760759i	$0.724827\pi$
$348$	0	0
$349$	−7.88854	−0.422264	−0.211132	−	0.977458i	$-0.567715\pi$
$349$	−7.88854	−0.422264	−0.211132	+	0.977458i	$0.567715\pi$
$350$	1.52786	0.0816678
$351$	0	0
$352$	−6.76393	−0.360519
$353$	−7.41641	−0.394736	−0.197368	−	0.980330i	$-0.563239\pi$
$353$	−7.41641	−0.394736	−0.197368	+	0.980330i	$0.563239\pi$
$354$	0	0
$355$	−9.18034	−0.487242
$356$	−7.23607	−0.383511
$357$	0	0
$358$	2.76393	0.146078
$359$	−22.2361	−1.17357	−0.586787	−	0.809741i	$-0.699608\pi$
$359$	−22.2361	−1.17357	−0.586787	+	0.809741i	$0.699608\pi$
$360$	0	0
$361$	−14.0000	−0.736842
$362$	6.76393	0.355504
$363$	0	0
$364$	−0.472136	−0.0247466
$365$	−8.47214	−0.443452
$366$	0	0
$367$	18.0000	0.939592	0.469796	−	0.882775i	$-0.344327\pi$
$367$	18.0000	0.939592	0.469796	+	0.882775i	$0.344327\pi$
$368$	27.7082	1.44439
$369$	0	0
$370$	−3.23607	−0.168235
$371$	2.47214	0.128347
$372$	0	0
$373$	19.0000	0.983783	0.491891	−	0.870657i	$-0.336306\pi$
$373$	19.0000	0.983783	0.491891	+	0.870657i	$0.336306\pi$
$374$	−2.47214	−0.127831
$375$	0	0
$376$	−5.52786	−0.285078
$377$	8.94427	0.460653
$378$	0	0
$379$	−2.11146	−0.108458	−0.0542291	−	0.998529i	$-0.517270\pi$
$379$	−2.11146	−0.108458	−0.0542291	+	0.998529i	$0.517270\pi$
$380$	1.38197	0.0708934
$381$	0	0
$382$	−31.0344	−1.58786
$383$	23.8885	1.22065	0.610324	−	0.792152i	$-0.291039\pi$
$383$	23.8885	1.22065	0.610324	+	0.792152i	$0.291039\pi$
$384$	0	0
$385$	0.472136	0.0240623
$386$	−5.61803	−0.285950
$387$	0	0
$388$	−9.85410	−0.500266
$389$	17.8885	0.906985	0.453493	−	0.891260i	$-0.350178\pi$
$389$	17.8885	0.906985	0.453493	+	0.891260i	$0.350178\pi$
$390$	0	0
$391$	4.36068	0.220529
$392$	−15.5279	−0.784276
$393$	0	0
$394$	18.4721	0.930613
$395$	11.7082	0.589104
$396$	0	0
$397$	−7.00000	−0.351320	−0.175660	−	0.984451i	$-0.556206\pi$
$397$	−7.00000	−0.351320	−0.175660	+	0.984451i	$0.556206\pi$
$398$	30.6525	1.53647
$399$	0	0
$400$	19.4164	0.970820
$401$	−38.1803	−1.90664	−0.953318	−	0.301969i	$-0.902356\pi$
$401$	−38.1803	−1.90664	−0.953318	+	0.301969i	$0.902356\pi$
$402$	0	0
$403$	−3.23607	−0.161200
$404$	1.85410	0.0922450
$405$	0	0
$406$	1.05573	0.0523949
$407$	4.00000	0.198273
$408$	0	0
$409$	−3.81966	−0.188870	−0.0944350	−	0.995531i	$-0.530104\pi$
$409$	−3.81966	−0.188870	−0.0944350	+	0.995531i	$0.530104\pi$
$410$	−11.3262	−0.559363
$411$	0	0
$412$	3.85410	0.189878
$413$	−0.527864	−0.0259745
$414$	0	0
$415$	−14.9443	−0.733585
$416$	−10.9443	−0.536587
$417$	0	0
$418$	−7.23607	−0.353928
$419$	10.1246	0.494620	0.247310	−	0.968936i	$-0.420453\pi$
$419$	10.1246	0.494620	0.247310	+	0.968936i	$0.420453\pi$
$420$	0	0
$421$	29.3607	1.43095	0.715476	−	0.698637i	$-0.246210\pi$
$421$	29.3607	1.43095	0.715476	+	0.698637i	$0.246210\pi$
$422$	−37.5066	−1.82579
$423$	0	0
$424$	23.4164	1.13720
$425$	3.05573	0.148225
$426$	0	0
$427$	1.93112	0.0934533
$428$	−3.56231	−0.172191
$429$	0	0
$430$	2.00000	0.0964486
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	0	0
$433$	10.1803	0.489236	0.244618	−	0.969620i	$-0.421337\pi$
$433$	10.1803	0.489236	0.244618	+	0.969620i	$0.421337\pi$
$434$	−0.381966	−0.0183350
$435$	0	0
$436$	−8.61803	−0.412729
$437$	12.7639	0.610582
$438$	0	0
$439$	−1.18034	−0.0563345	−0.0281673	−	0.999603i	$-0.508967\pi$
$439$	−1.18034	−0.0563345	−0.0281673	+	0.999603i	$0.508967\pi$
$440$	4.47214	0.213201
$441$	0	0
$442$	−4.00000	−0.190261
$443$	−30.7082	−1.45899	−0.729495	−	0.683986i	$-0.760245\pi$
$443$	−30.7082	−1.45899	−0.729495	+	0.683986i	$0.760245\pi$
$444$	0	0
$445$	11.7082	0.555022
$446$	−6.47214	−0.306465
$447$	0	0
$448$	1.00000	0.0472456
$449$	31.3050	1.47737	0.738686	−	0.674050i	$-0.235447\pi$
$449$	31.3050	1.47737	0.738686	+	0.674050i	$0.235447\pi$
$450$	0	0
$451$	14.0000	0.659234
$452$	−2.14590	−0.100935
$453$	0	0
$454$	−10.4721	−0.491482
$455$	0.763932	0.0358137
$456$	0	0
$457$	−3.05573	−0.142941	−0.0714705	−	0.997443i	$-0.522769\pi$
$457$	−3.05573	−0.142941	−0.0714705	+	0.997443i	$0.522769\pi$
$458$	21.7082	1.01436
$459$	0	0
$460$	3.52786	0.164488
$461$	−34.3607	−1.60034	−0.800168	−	0.599776i	$-0.795256\pi$
$461$	−34.3607	−1.60034	−0.800168	+	0.599776i	$0.795256\pi$
$462$	0	0
$463$	−2.58359	−0.120070	−0.0600349	−	0.998196i	$-0.519121\pi$
$463$	−2.58359	−0.120070	−0.0600349	+	0.998196i	$0.519121\pi$
$464$	13.4164	0.622841
$465$	0	0
$466$	29.0344	1.34499
$467$	−4.70820	−0.217870	−0.108935	−	0.994049i	$-0.534744\pi$
$467$	−4.70820	−0.217870	−0.108935	+	0.994049i	$0.534744\pi$
$468$	0	0
$469$	1.88854	0.0872049
$470$	−4.00000	−0.184506
$471$	0	0
$472$	−5.00000	−0.230144
$473$	−2.47214	−0.113669
$474$	0	0
$475$	8.94427	0.410391
$476$	−0.111456	−0.00510859
$477$	0	0
$478$	−18.9443	−0.866491
$479$	−23.2918	−1.06423	−0.532115	−	0.846672i	$-0.678602\pi$
$479$	−23.2918	−1.06423	−0.532115	+	0.846672i	$0.678602\pi$
$480$	0	0
$481$	6.47214	0.295104
$482$	−23.2361	−1.05837
$483$	0	0
$484$	−4.32624	−0.196647
$485$	15.9443	0.723992
$486$	0	0
$487$	−19.2361	−0.871669	−0.435835	−	0.900027i	$-0.643547\pi$
$487$	−19.2361	−0.871669	−0.435835	+	0.900027i	$0.643547\pi$
$488$	18.2918	0.828031
$489$	0	0
$490$	−11.2361	−0.507594
$491$	−4.36068	−0.196795	−0.0983974	−	0.995147i	$-0.531372\pi$
$491$	−4.36068	−0.196795	−0.0983974	+	0.995147i	$0.531372\pi$
$492$	0	0
$493$	2.11146	0.0950952
$494$	−11.7082	−0.526777
$495$	0	0
$496$	−4.85410	−0.217956
$497$	2.16718	0.0972115
$498$	0	0
$499$	−6.58359	−0.294722	−0.147361	−	0.989083i	$-0.547078\pi$
$499$	−6.58359	−0.294722	−0.147361	+	0.989083i	$0.547078\pi$
$500$	5.56231	0.248754
$501$	0	0
$502$	−2.94427	−0.131409
$503$	−29.6525	−1.32214	−0.661069	−	0.750325i	$-0.729897\pi$
$503$	−29.6525	−1.32214	−0.661069	+	0.750325i	$0.729897\pi$
$504$	0	0
$505$	−3.00000	−0.133498
$506$	−18.4721	−0.821187
$507$	0	0
$508$	7.70820	0.341996
$509$	−29.5967	−1.31185	−0.655926	−	0.754825i	$-0.727722\pi$
$509$	−29.5967	−1.31185	−0.655926	+	0.754825i	$0.727722\pi$
$510$	0	0
$511$	2.00000	0.0884748
$512$	5.29180	0.233867
$513$	0	0
$514$	3.14590	0.138760
$515$	−6.23607	−0.274794
$516$	0	0
$517$	4.94427	0.217449
$518$	0.763932	0.0335652
$519$	0	0
$520$	7.23607	0.317323
$521$	−2.00000	−0.0876216	−0.0438108	−	0.999040i	$-0.513950\pi$
$521$	−2.00000	−0.0876216	−0.0438108	+	0.999040i	$0.513950\pi$
$522$	0	0
$523$	−17.7082	−0.774326	−0.387163	−	0.922011i	$-0.626545\pi$
$523$	−17.7082	−0.774326	−0.387163	+	0.922011i	$0.626545\pi$
$524$	−7.41641	−0.323987
$525$	0	0
$526$	−37.5967	−1.63930
$527$	−0.763932	−0.0332774
$528$	0	0
$529$	9.58359	0.416678
$530$	16.9443	0.736012
$531$	0	0
$532$	−0.326238	−0.0141442
$533$	22.6525	0.981188
$534$	0	0
$535$	5.76393	0.249197
$536$	17.8885	0.772667
$537$	0	0
$538$	−17.8885	−0.771230
$539$	13.8885	0.598222
$540$	0	0
$541$	−25.3607	−1.09034	−0.545170	−	0.838325i	$-0.683535\pi$
$541$	−25.3607	−1.09034	−0.545170	+	0.838325i	$0.683535\pi$
$542$	22.9443	0.985541
$543$	0	0
$544$	−2.58359	−0.110771
$545$	13.9443	0.597307
$546$	0	0
$547$	−12.1246	−0.518411	−0.259205	−	0.965822i	$-0.583461\pi$
$547$	−12.1246	−0.518411	−0.259205	+	0.965822i	$0.583461\pi$
$548$	−3.88854	−0.166110
$549$	0	0
$550$	−12.9443	−0.551946
$551$	6.18034	0.263291
$552$	0	0
$553$	−2.76393	−0.117534
$554$	20.4721	0.869778
$555$	0	0
$556$	8.29180	0.351650
$557$	12.0000	0.508456	0.254228	−	0.967144i	$-0.418179\pi$
$557$	12.0000	0.508456	0.254228	+	0.967144i	$0.418179\pi$
$558$	0	0
$559$	−4.00000	−0.169182
$560$	1.14590	0.0484230
$561$	0	0
$562$	27.5066	1.16029
$563$	−27.5410	−1.16072	−0.580358	−	0.814362i	$-0.697087\pi$
$563$	−27.5410	−1.16072	−0.580358	+	0.814362i	$0.697087\pi$
$564$	0	0
$565$	3.47214	0.146074
$566$	22.4721	0.944574
$567$	0	0
$568$	20.5279	0.861330
$569$	−5.52786	−0.231740	−0.115870	−	0.993264i	$-0.536966\pi$
$569$	−5.52786	−0.231740	−0.115870	+	0.993264i	$0.536966\pi$
$570$	0	0
$571$	28.1803	1.17931	0.589655	−	0.807655i	$-0.299264\pi$
$571$	28.1803	1.17931	0.589655	+	0.807655i	$0.299264\pi$
$572$	4.00000	0.167248
$573$	0	0
$574$	2.67376	0.111601
$575$	22.8328	0.952194
$576$	0	0
$577$	−28.8328	−1.20033	−0.600163	−	0.799878i	$-0.704898\pi$
$577$	−28.8328	−1.20033	−0.600163	+	0.799878i	$0.704898\pi$
$578$	26.5623	1.10485
$579$	0	0
$580$	1.70820	0.0709293
$581$	3.52786	0.146360
$582$	0	0
$583$	−20.9443	−0.867423
$584$	18.9443	0.783920
$585$	0	0
$586$	−0.763932	−0.0315577
$587$	6.47214	0.267134	0.133567	−	0.991040i	$-0.457357\pi$
$587$	6.47214	0.267134	0.133567	+	0.991040i	$0.457357\pi$
$588$	0	0
$589$	−2.23607	−0.0921356
$590$	−3.61803	−0.148952
$591$	0	0
$592$	9.70820	0.399005
$593$	6.52786	0.268067	0.134034	−	0.990977i	$-0.457207\pi$
$593$	6.52786	0.268067	0.134034	+	0.990977i	$0.457207\pi$
$594$	0	0
$595$	0.180340	0.00739321
$596$	−6.18034	−0.253157
$597$	0	0
$598$	−29.8885	−1.22223
$599$	14.5967	0.596407	0.298203	−	0.954502i	$-0.403613\pi$
$599$	14.5967	0.596407	0.298203	+	0.954502i	$0.403613\pi$
$600$	0	0
$601$	30.5410	1.24579	0.622897	−	0.782304i	$-0.285956\pi$
$601$	30.5410	1.24579	0.622897	+	0.782304i	$0.285956\pi$
$602$	−0.472136	−0.0192428
$603$	0	0
$604$	−8.76393	−0.356599
$605$	7.00000	0.284590
$606$	0	0
$607$	22.4721	0.912116	0.456058	−	0.889950i	$-0.349261\pi$
$607$	22.4721	0.912116	0.456058	+	0.889950i	$0.349261\pi$
$608$	−7.56231	−0.306692
$609$	0	0
$610$	13.2361	0.535913
$611$	8.00000	0.323645
$612$	0	0
$613$	−43.8885	−1.77264	−0.886321	−	0.463072i	$-0.846747\pi$
$613$	−43.8885	−1.77264	−0.886321	+	0.463072i	$0.846747\pi$
$614$	46.4508	1.87460
$615$	0	0
$616$	−1.05573	−0.0425365
$617$	−32.4721	−1.30728	−0.653639	−	0.756806i	$-0.726759\pi$
$617$	−32.4721	−1.30728	−0.653639	+	0.756806i	$0.726759\pi$
$618$	0	0
$619$	6.18034	0.248409	0.124204	−	0.992257i	$-0.460362\pi$
$619$	6.18034	0.248409	0.124204	+	0.992257i	$0.460362\pi$
$620$	−0.618034	−0.0248208
$621$	0	0
$622$	−47.2148	−1.89314
$623$	−2.76393	−0.110735
$624$	0	0
$625$	11.0000	0.440000
$626$	−27.1246	−1.08412
$627$	0	0
$628$	12.9098	0.515158
$629$	1.52786	0.0609199
$630$	0	0
$631$	34.3607	1.36788	0.683939	−	0.729540i	$-0.260266\pi$
$631$	34.3607	1.36788	0.683939	+	0.729540i	$0.260266\pi$
$632$	−26.1803	−1.04140
$633$	0	0
$634$	6.56231	0.260622
$635$	−12.4721	−0.494942
$636$	0	0
$637$	22.4721	0.890378
$638$	−8.94427	−0.354107
$639$	0	0
$640$	13.6180	0.538300
$641$	−12.0000	−0.473972	−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000	−0.473972	−0.236986	+	0.971513i	$0.576159\pi$
$642$	0	0
$643$	19.5279	0.770104	0.385052	−	0.922895i	$-0.374184\pi$
$643$	19.5279	0.770104	0.385052	+	0.922895i	$0.374184\pi$
$644$	−0.832816	−0.0328175
$645$	0	0
$646$	−2.76393	−0.108745
$647$	0.944272	0.0371232	0.0185616	−	0.999828i	$-0.494091\pi$
$647$	0.944272	0.0371232	0.0185616	+	0.999828i	$0.494091\pi$
$648$	0	0
$649$	4.47214	0.175547
$650$	−20.9443	−0.821502
$651$	0	0
$652$	6.61803	0.259182
$653$	47.3050	1.85119	0.925593	−	0.378521i	$-0.123567\pi$
$653$	47.3050	1.85119	0.925593	+	0.378521i	$0.123567\pi$
$654$	0	0
$655$	12.0000	0.468879
$656$	33.9787	1.32665
$657$	0	0
$658$	0.944272	0.0368116
$659$	−25.6525	−0.999279	−0.499639	−	0.866234i	$-0.666534\pi$
$659$	−25.6525	−0.999279	−0.499639	+	0.866234i	$0.666534\pi$
$660$	0	0
$661$	−0.639320	−0.0248667	−0.0124333	−	0.999923i	$-0.503958\pi$
$661$	−0.639320	−0.0248667	−0.0124333	+	0.999923i	$0.503958\pi$
$662$	−3.23607	−0.125773
$663$	0	0
$664$	33.4164	1.29681
$665$	0.527864	0.0204697
$666$	0	0
$667$	15.7771	0.610891
$668$	4.00000	0.154765
$669$	0	0
$670$	12.9443	0.500081
$671$	−16.3607	−0.631597
$672$	0	0
$673$	−29.0132	−1.11837	−0.559187	−	0.829041i	$-0.688887\pi$
$673$	−29.0132	−1.11837	−0.559187	+	0.829041i	$0.688887\pi$
$674$	23.8885	0.920152
$675$	0	0
$676$	−1.56231	−0.0600887
$677$	46.7214	1.79565	0.897824	−	0.440355i	$-0.145147\pi$
$677$	46.7214	1.79565	0.897824	+	0.440355i	$0.145147\pi$
$678$	0	0
$679$	−3.76393	−0.144446
$680$	1.70820	0.0655066
$681$	0	0
$682$	3.23607	0.123915
$683$	−5.18034	−0.198220	−0.0991101	−	0.995076i	$-0.531600\pi$
$683$	−5.18034	−0.198220	−0.0991101	+	0.995076i	$0.531600\pi$
$684$	0	0
$685$	6.29180	0.240397
$686$	5.32624	0.203357
$687$	0	0
$688$	−6.00000	−0.228748
$689$	−33.8885	−1.29105
$690$	0	0
$691$	3.18034	0.120986	0.0604929	−	0.998169i	$-0.480733\pi$
$691$	3.18034	0.120986	0.0604929	+	0.998169i	$0.480733\pi$
$692$	−1.81966	−0.0691731
$693$	0	0
$694$	39.1246	1.48515
$695$	−13.4164	−0.508913
$696$	0	0
$697$	5.34752	0.202552
$698$	12.7639	0.483122
$699$	0	0
$700$	−0.583592	−0.0220577
$701$	−7.00000	−0.264386	−0.132193	−	0.991224i	$-0.542202\pi$
$701$	−7.00000	−0.264386	−0.132193	+	0.991224i	$0.542202\pi$
$702$	0	0
$703$	4.47214	0.168670
$704$	−8.47214	−0.319306
$705$	0	0
$706$	12.0000	0.451626
$707$	0.708204	0.0266348
$708$	0	0
$709$	25.5279	0.958719	0.479360	−	0.877619i	$-0.340869\pi$
$709$	25.5279	0.958719	0.479360	+	0.877619i	$0.340869\pi$
$710$	14.8541	0.557465
$711$	0	0
$712$	−26.1803	−0.981150
$713$	−5.70820	−0.213774
$714$	0	0
$715$	−6.47214	−0.242044
$716$	−1.05573	−0.0394544
$717$	0	0
$718$	35.9787	1.34271
$719$	13.8197	0.515386	0.257693	−	0.966227i	$-0.417038\pi$
$719$	13.8197	0.515386	0.257693	+	0.966227i	$0.417038\pi$
$720$	0	0
$721$	1.47214	0.0548252
$722$	22.6525	0.843038
$723$	0	0
$724$	−2.58359	−0.0960184
$725$	11.0557	0.410599
$726$	0	0
$727$	−44.2361	−1.64062	−0.820312	−	0.571916i	$-0.806200\pi$
$727$	−44.2361	−1.64062	−0.820312	+	0.571916i	$0.806200\pi$
$728$	−1.70820	−0.0633102
$729$	0	0
$730$	13.7082	0.507363
$731$	−0.944272	−0.0349252
$732$	0	0
$733$	3.47214	0.128246	0.0641231	−	0.997942i	$-0.479575\pi$
$733$	3.47214	0.128246	0.0641231	+	0.997942i	$0.479575\pi$
$734$	−29.1246	−1.07501
$735$	0	0
$736$	−19.3050	−0.711590
$737$	−16.0000	−0.589368
$738$	0	0
$739$	6.18034	0.227347	0.113674	−	0.993518i	$-0.463738\pi$
$739$	6.18034	0.227347	0.113674	+	0.993518i	$0.463738\pi$
$740$	1.23607	0.0454388
$741$	0	0
$742$	−4.00000	−0.146845
$743$	−50.1803	−1.84094	−0.920469	−	0.390815i	$-0.872193\pi$
$743$	−50.1803	−1.84094	−0.920469	+	0.390815i	$0.872193\pi$
$744$	0	0
$745$	10.0000	0.366372
$746$	−30.7426	−1.12557
$747$	0	0
$748$	0.944272	0.0345260
$749$	−1.36068	−0.0497182
$750$	0	0
$751$	−21.5410	−0.786043	−0.393021	−	0.919529i	$-0.628570\pi$
$751$	−21.5410	−0.786043	−0.393021	+	0.919529i	$0.628570\pi$
$752$	12.0000	0.437595
$753$	0	0
$754$	−14.4721	−0.527044
$755$	14.1803	0.516075
$756$	0	0
$757$	8.65248	0.314480	0.157240	−	0.987560i	$-0.449740\pi$
$757$	8.65248	0.314480	0.157240	+	0.987560i	$0.449740\pi$
$758$	3.41641	0.124090
$759$	0	0
$760$	5.00000	0.181369
$761$	−2.00000	−0.0724999	−0.0362500	−	0.999343i	$-0.511541\pi$
$761$	−2.00000	−0.0724999	−0.0362500	+	0.999343i	$0.511541\pi$
$762$	0	0
$763$	−3.29180	−0.119171
$764$	11.8541	0.428866
$765$	0	0
$766$	−38.6525	−1.39657
$767$	7.23607	0.261279
$768$	0	0
$769$	−47.3607	−1.70787	−0.853935	−	0.520380i	$-0.825790\pi$
$769$	−47.3607	−1.70787	−0.853935	+	0.520380i	$0.825790\pi$
$770$	−0.763932	−0.0275302
$771$	0	0
$772$	2.14590	0.0772326
$773$	11.1246	0.400124	0.200062	−	0.979783i	$-0.435886\pi$
$773$	11.1246	0.400124	0.200062	+	0.979783i	$0.435886\pi$
$774$	0	0
$775$	−4.00000	−0.143684
$776$	−35.6525	−1.27985
$777$	0	0
$778$	−28.9443	−1.03770
$779$	15.6525	0.560808
$780$	0	0
$781$	−18.3607	−0.656997
$782$	−7.05573	−0.252312
$783$	0	0
$784$	33.7082	1.20386
$785$	−20.8885	−0.745544
$786$	0	0
$787$	7.34752	0.261911	0.130955	−	0.991388i	$-0.458196\pi$
$787$	7.34752	0.261911	0.130955	+	0.991388i	$0.458196\pi$
$788$	−7.05573	−0.251350
$789$	0	0
$790$	−18.9443	−0.674007
$791$	−0.819660	−0.0291438
$792$	0	0
$793$	−26.4721	−0.940053
$794$	11.3262	0.401953
$795$	0	0
$796$	−11.7082	−0.414986
$797$	55.4164	1.96295	0.981475	−	0.191591i	$-0.0613646\pi$
$797$	55.4164	1.96295	0.981475	+	0.191591i	$0.0613646\pi$
$798$	0	0
$799$	1.88854	0.0668119
$800$	−13.5279	−0.478282
$801$	0	0
$802$	61.7771	2.18142
$803$	−16.9443	−0.597950
$804$	0	0
$805$	1.34752	0.0474940
$806$	5.23607	0.184433
$807$	0	0
$808$	6.70820	0.235994
$809$	−23.4164	−0.823277	−0.411639	−	0.911347i	$-0.635043\pi$
$809$	−23.4164	−0.823277	−0.411639	+	0.911347i	$0.635043\pi$
$810$	0	0
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	−0.403252	−0.0141514
$813$	0	0
$814$	−6.47214	−0.226848
$815$	−10.7082	−0.375092
$816$	0	0
$817$	−2.76393	−0.0966977
$818$	6.18034	0.216091
$819$	0	0
$820$	4.32624	0.151079
$821$	−30.5410	−1.06589	−0.532944	−	0.846150i	$-0.678915\pi$
$821$	−30.5410	−1.06589	−0.532944	+	0.846150i	$0.678915\pi$
$822$	0	0
$823$	−14.2918	−0.498181	−0.249090	−	0.968480i	$-0.580132\pi$
$823$	−14.2918	−0.498181	−0.249090	+	0.968480i	$0.580132\pi$
$824$	13.9443	0.485772
$825$	0	0
$826$	0.854102	0.0297180
$827$	−17.3475	−0.603233	−0.301616	−	0.953429i	$-0.597526\pi$
$827$	−17.3475	−0.603233	−0.301616	+	0.953429i	$0.597526\pi$
$828$	0	0
$829$	16.8328	0.584628	0.292314	−	0.956322i	$-0.405575\pi$
$829$	16.8328	0.584628	0.292314	+	0.956322i	$0.405575\pi$
$830$	24.1803	0.839312
$831$	0	0
$832$	−13.7082	−0.475246
$833$	5.30495	0.183806
$834$	0	0
$835$	−6.47214	−0.223978
$836$	2.76393	0.0955926
$837$	0	0
$838$	−16.3820	−0.565906
$839$	−28.9443	−0.999267	−0.499634	−	0.866237i	$-0.666532\pi$
$839$	−28.9443	−0.999267	−0.499634	+	0.866237i	$0.666532\pi$
$840$	0	0
$841$	−21.3607	−0.736575
$842$	−47.5066	−1.63718
$843$	0	0
$844$	14.3262	0.493129
$845$	2.52786	0.0869612
$846$	0	0
$847$	−1.65248	−0.0567797
$848$	−50.8328	−1.74561
$849$	0	0
$850$	−4.94427	−0.169587
$851$	11.4164	0.391349
$852$	0	0
$853$	10.5836	0.362375	0.181188	−	0.983449i	$-0.442006\pi$
$853$	10.5836	0.362375	0.181188	+	0.983449i	$0.442006\pi$
$854$	−3.12461	−0.106922
$855$	0	0
$856$	−12.8885	−0.440521
$857$	55.6656	1.90150	0.950751	−	0.309956i	$-0.100314\pi$
$857$	55.6656	1.90150	0.950751	+	0.309956i	$0.100314\pi$
$858$	0	0
$859$	2.11146	0.0720420	0.0360210	−	0.999351i	$-0.488532\pi$
$859$	2.11146	0.0720420	0.0360210	+	0.999351i	$0.488532\pi$
$860$	−0.763932	−0.0260499
$861$	0	0
$862$	19.4164	0.661325
$863$	9.81966	0.334265	0.167133	−	0.985934i	$-0.446549\pi$
$863$	9.81966	0.334265	0.167133	+	0.985934i	$0.446549\pi$
$864$	0	0
$865$	2.94427	0.100108
$866$	−16.4721	−0.559746
$867$	0	0
$868$	0.145898	0.00495210
$869$	23.4164	0.794347
$870$	0	0
$871$	−25.8885	−0.877200
$872$	−31.1803	−1.05590
$873$	0	0
$874$	−20.6525	−0.698580
$875$	2.12461	0.0718250
$876$	0	0
$877$	−18.0557	−0.609699	−0.304849	−	0.952401i	$-0.598606\pi$
$877$	−18.0557	−0.609699	−0.304849	+	0.952401i	$0.598606\pi$
$878$	1.90983	0.0644536
$879$	0	0
$880$	−9.70820	−0.327263
$881$	20.3607	0.685969	0.342984	−	0.939341i	$-0.388562\pi$
$881$	20.3607	0.685969	0.342984	+	0.939341i	$0.388562\pi$
$882$	0	0
$883$	−31.7771	−1.06938	−0.534692	−	0.845047i	$-0.679572\pi$
$883$	−31.7771	−1.06938	−0.534692	+	0.845047i	$0.679572\pi$
$884$	1.52786	0.0513876
$885$	0	0
$886$	49.6869	1.66926
$887$	−27.0689	−0.908884	−0.454442	−	0.890776i	$-0.650161\pi$
$887$	−27.0689	−0.908884	−0.454442	+	0.890776i	$0.650161\pi$
$888$	0	0
$889$	2.94427	0.0987477
$890$	−18.9443	−0.635013
$891$	0	0
$892$	2.47214	0.0827732
$893$	5.52786	0.184983
$894$	0	0
$895$	1.70820	0.0570990
$896$	−3.21478	−0.107398
$897$	0	0
$898$	−50.6525	−1.69030
$899$	−2.76393	−0.0921823
$900$	0	0
$901$	−8.00000	−0.266519
$902$	−22.6525	−0.754245
$903$	0	0
$904$	−7.76393	−0.258225
$905$	4.18034	0.138959
$906$	0	0
$907$	−24.2361	−0.804745	−0.402373	−	0.915476i	$-0.631814\pi$
$907$	−24.2361	−0.804745	−0.402373	+	0.915476i	$0.631814\pi$
$908$	4.00000	0.132745
$909$	0	0
$910$	−1.23607	−0.0409753
$911$	−18.1803	−0.602342	−0.301171	−	0.953570i	$-0.597377\pi$
$911$	−18.1803	−0.602342	−0.301171	+	0.953570i	$0.597377\pi$
$912$	0	0
$913$	−29.8885	−0.989166
$914$	4.94427	0.163542
$915$	0	0
$916$	−8.29180	−0.273969
$917$	−2.83282	−0.0935478
$918$	0	0
$919$	−14.4721	−0.477392	−0.238696	−	0.971094i	$-0.576720\pi$
$919$	−14.4721	−0.477392	−0.238696	+	0.971094i	$0.576720\pi$
$920$	12.7639	0.420814
$921$	0	0
$922$	55.5967	1.83098
$923$	−29.7082	−0.977857
$924$	0	0
$925$	8.00000	0.263038
$926$	4.18034	0.137374
$927$	0	0
$928$	−9.34752	−0.306848
$929$	20.0000	0.656179	0.328089	−	0.944647i	$-0.393595\pi$
$929$	20.0000	0.656179	0.328089	+	0.944647i	$0.393595\pi$
$930$	0	0
$931$	15.5279	0.508905
$932$	−11.0902	−0.363271
$933$	0	0
$934$	7.61803	0.249270
$935$	−1.52786	−0.0499665
$936$	0	0
$937$	9.05573	0.295838	0.147919	−	0.988999i	$-0.452743\pi$
$937$	9.05573	0.295838	0.147919	+	0.988999i	$0.452743\pi$
$938$	−3.05573	−0.0997731
$939$	0	0
$940$	1.52786	0.0498334
$941$	38.0000	1.23876	0.619382	−	0.785090i	$-0.287383\pi$
$941$	38.0000	1.23876	0.619382	+	0.785090i	$0.287383\pi$
$942$	0	0
$943$	39.9574	1.30119
$944$	10.8541	0.353271
$945$	0	0
$946$	4.00000	0.130051
$947$	13.0557	0.424254	0.212127	−	0.977242i	$-0.431961\pi$
$947$	13.0557	0.424254	0.212127	+	0.977242i	$0.431961\pi$
$948$	0	0
$949$	−27.4164	−0.889974
$950$	−14.4721	−0.469538
$951$	0	0
$952$	−0.403252	−0.0130695
$953$	−45.7082	−1.48063	−0.740317	−	0.672258i	$-0.765325\pi$
$953$	−45.7082	−1.48063	−0.740317	+	0.672258i	$0.765325\pi$
$954$	0	0
$955$	−19.1803	−0.620661
$956$	7.23607	0.234031
$957$	0	0
$958$	37.6869	1.21761
$959$	−1.48529	−0.0479626
$960$	0	0
$961$	1.00000	0.0322581
$962$	−10.4721	−0.337635
$963$	0	0
$964$	8.87539	0.285857
$965$	−3.47214	−0.111772
$966$	0	0
$967$	60.3607	1.94107	0.970534	−	0.240963i	$-0.0774632\pi$
$967$	60.3607	1.94107	0.970534	+	0.240963i	$0.0774632\pi$
$968$	−15.6525	−0.503090
$969$	0	0
$970$	−25.7984	−0.828336
$971$	28.0000	0.898563	0.449281	−	0.893390i	$-0.351680\pi$
$971$	28.0000	0.898563	0.449281	+	0.893390i	$0.351680\pi$
$972$	0	0
$973$	3.16718	0.101535
$974$	31.1246	0.997297
$975$	0	0
$976$	−39.7082	−1.27103
$977$	47.2492	1.51164	0.755818	−	0.654781i	$-0.227239\pi$
$977$	47.2492	1.51164	0.755818	+	0.654781i	$0.227239\pi$
$978$	0	0
$979$	23.4164	0.748392
$980$	4.29180	0.137096
$981$	0	0
$982$	7.05573	0.225157
$983$	−39.5279	−1.26074	−0.630372	−	0.776294i	$-0.717097\pi$
$983$	−39.5279	−1.26074	−0.630372	+	0.776294i	$0.717097\pi$
$984$	0	0
$985$	11.4164	0.363757
$986$	−3.41641	−0.108801
$987$	0	0
$988$	4.47214	0.142278
$989$	−7.05573	−0.224359
$990$	0	0
$991$	−16.5410	−0.525443	−0.262721	−	0.964872i	$-0.584620\pi$
$991$	−16.5410	−0.525443	−0.262721	+	0.964872i	$0.584620\pi$
$992$	3.38197	0.107378
$993$	0	0
$994$	−3.50658	−0.111222
$995$	18.9443	0.600574
$996$	0	0
$997$	−29.3607	−0.929862	−0.464931	−	0.885347i	$-0.653921\pi$
$997$	−29.3607	−0.929862	−0.464931	+	0.885347i	$0.653921\pi$
$998$	10.6525	0.337198
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	279.2.a.a.1.1		2
3.2	odd	2		31.2.a.a.1.2	✓	2
4.3	odd	2		4464.2.a.bf.1.1		2
5.4	even	2		6975.2.a.y.1.2		2
12.11	even	2		496.2.a.i.1.2		2
15.2	even	4		775.2.b.d.249.4		4
15.8	even	4		775.2.b.d.249.1		4
15.14	odd	2		775.2.a.d.1.1		2
21.20	even	2		1519.2.a.a.1.2		2
24.5	odd	2		1984.2.a.r.1.2		2
24.11	even	2		1984.2.a.n.1.1		2
31.30	odd	2		8649.2.a.c.1.1		2
33.32	even	2		3751.2.a.b.1.1		2
39.38	odd	2		5239.2.a.f.1.1		2
51.50	odd	2		8959.2.a.b.1.2		2
93.2	odd	10		961.2.d.d.531.1		4
93.5	odd	6		961.2.c.e.521.2		4
93.8	odd	10		961.2.d.c.374.1		4
93.11	even	30		961.2.g.d.338.1		8
93.14	odd	30		961.2.g.a.816.1		8
93.17	even	30		961.2.g.d.816.1		8
93.20	odd	30		961.2.g.a.338.1		8
93.23	even	10		961.2.d.a.374.1		4
93.26	even	6		961.2.c.c.521.2		4
93.29	even	10		961.2.d.g.531.1		4
93.35	odd	10		961.2.d.c.388.1		4
93.38	odd	30		961.2.g.a.235.1		8
93.41	odd	30		961.2.g.h.844.1		8
93.44	even	30		961.2.g.e.448.1		8
93.47	odd	10		961.2.d.d.628.1		4
93.50	odd	30		961.2.g.h.547.1		8
93.53	even	30		961.2.g.d.732.1		8
93.56	odd	6		961.2.c.e.439.2		4
93.59	odd	30		961.2.g.h.846.1		8
93.65	even	30		961.2.g.e.846.1		8
93.68	even	6		961.2.c.c.439.2		4
93.71	odd	30		961.2.g.a.732.1		8
93.74	even	30		961.2.g.e.547.1		8
93.77	even	10		961.2.d.g.628.1		4
93.80	odd	30		961.2.g.h.448.1		8
93.83	even	30		961.2.g.e.844.1		8
93.86	even	30		961.2.g.d.235.1		8
93.89	even	10		961.2.d.a.388.1		4
93.92	even	2		961.2.a.f.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
31.2.a.a.1.2	✓	2	3.2	odd	2
279.2.a.a.1.1		2	1.1	even	1	trivial
496.2.a.i.1.2		2	12.11	even	2
775.2.a.d.1.1		2	15.14	odd	2
775.2.b.d.249.1		4	15.8	even	4
775.2.b.d.249.4		4	15.2	even	4
961.2.a.f.1.2		2	93.92	even	2
961.2.c.c.439.2		4	93.68	even	6
961.2.c.c.521.2		4	93.26	even	6
961.2.c.e.439.2		4	93.56	odd	6
961.2.c.e.521.2		4	93.5	odd	6
961.2.d.a.374.1		4	93.23	even	10
961.2.d.a.388.1		4	93.89	even	10
961.2.d.c.374.1		4	93.8	odd	10
961.2.d.c.388.1		4	93.35	odd	10
961.2.d.d.531.1		4	93.2	odd	10
961.2.d.d.628.1		4	93.47	odd	10
961.2.d.g.531.1		4	93.29	even	10
961.2.d.g.628.1		4	93.77	even	10
961.2.g.a.235.1		8	93.38	odd	30
961.2.g.a.338.1		8	93.20	odd	30
961.2.g.a.732.1		8	93.71	odd	30
961.2.g.a.816.1		8	93.14	odd	30
961.2.g.d.235.1		8	93.86	even	30
961.2.g.d.338.1		8	93.11	even	30
961.2.g.d.732.1		8	93.53	even	30
961.2.g.d.816.1		8	93.17	even	30
961.2.g.e.448.1		8	93.44	even	30
961.2.g.e.547.1		8	93.74	even	30
961.2.g.e.844.1		8	93.83	even	30
961.2.g.e.846.1		8	93.65	even	30
961.2.g.h.448.1		8	93.80	odd	30
961.2.g.h.547.1		8	93.50	odd	30
961.2.g.h.844.1		8	93.41	odd	30
961.2.g.h.846.1		8	93.59	odd	30
1519.2.a.a.1.2		2	21.20	even	2
1984.2.a.n.1.1		2	24.11	even	2
1984.2.a.r.1.2		2	24.5	odd	2
3751.2.a.b.1.1		2	33.32	even	2
4464.2.a.bf.1.1		2	4.3	odd	2
5239.2.a.f.1.1		2	39.38	odd	2
6975.2.a.y.1.2		2	5.4	even	2
8649.2.a.c.1.1		2	31.30	odd	2
8959.2.a.b.1.2		2	51.50	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:	\( N \)	\(=\)	\( 279 = 3^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	279.a (trivial)

\(f(q)\)	\(=\)	\(q-1.61803 q^{2} +0.618034 q^{4} -1.00000 q^{5} +0.236068 q^{7} +2.23607 q^{8} +O(q^{10})\)	\(q-1.61803 q^{2} +0.618034 q^{4} -1.00000 q^{5} +0.236068 q^{7} +2.23607 q^{8} +1.61803 q^{10} -2.00000 q^{11} -3.23607 q^{13} -0.381966 q^{14} -4.85410 q^{16} -0.763932 q^{17} -2.23607 q^{19} -0.618034 q^{20} +3.23607 q^{22} -5.70820 q^{23} -4.00000 q^{25} +5.23607 q^{26} +0.145898 q^{28} -2.76393 q^{29} +1.00000 q^{31} +3.38197 q^{32} +1.23607 q^{34} -0.236068 q^{35} -2.00000 q^{37} +3.61803 q^{38} -2.23607 q^{40} -7.00000 q^{41} +1.23607 q^{43} -1.23607 q^{44} +9.23607 q^{46} -2.47214 q^{47} -6.94427 q^{49} +6.47214 q^{50} -2.00000 q^{52} +10.4721 q^{53} +2.00000 q^{55} +0.527864 q^{56} +4.47214 q^{58} -2.23607 q^{59} +8.18034 q^{61} -1.61803 q^{62} +4.23607 q^{64} +3.23607 q^{65} +8.00000 q^{67} -0.472136 q^{68} +0.381966 q^{70} +9.18034 q^{71} +8.47214 q^{73} +3.23607 q^{74} -1.38197 q^{76} -0.472136 q^{77} -11.7082 q^{79} +4.85410 q^{80} +11.3262 q^{82} +14.9443 q^{83} +0.763932 q^{85} -2.00000 q^{86} -4.47214 q^{88} -11.7082 q^{89} -0.763932 q^{91} -3.52786 q^{92} +4.00000 q^{94} +2.23607 q^{95} -15.9443 q^{97} +11.2361 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} - 2 q^{5} - 4 q^{7}+O(q^{10}) \) 2 * q - q^2 - q^4 - 2 * q^5 - 4 * q^7	\( 2 q - q^{2} - q^{4} - 2 q^{5} - 4 q^{7} + q^{10} - 4 q^{11} - 2 q^{13} - 3 q^{14} - 3 q^{16} - 6 q^{17} + q^{20} + 2 q^{22} + 2 q^{23} - 8 q^{25} + 6 q^{26} + 7 q^{28} - 10 q^{29} + 2 q^{31} + 9 q^{32} - 2 q^{34} + 4 q^{35} - 4 q^{37} + 5 q^{38} - 14 q^{41} - 2 q^{43} + 2 q^{44} + 14 q^{46} + 4 q^{47} + 4 q^{49} + 4 q^{50} - 4 q^{52} + 12 q^{53} + 4 q^{55} + 10 q^{56} - 6 q^{61} - q^{62} + 4 q^{64} + 2 q^{65} + 16 q^{67} + 8 q^{68} + 3 q^{70} - 4 q^{71} + 8 q^{73} + 2 q^{74} - 5 q^{76} + 8 q^{77} - 10 q^{79} + 3 q^{80} + 7 q^{82} + 12 q^{83} + 6 q^{85} - 4 q^{86} - 10 q^{89} - 6 q^{91} - 16 q^{92} + 8 q^{94} - 14 q^{97} + 18 q^{98}+O(q^{100}) \) 2 * q - q^2 - q^4 - 2 * q^5 - 4 * q^7 + q^10 - 4 * q^11 - 2 * q^13 - 3 * q^14 - 3 * q^16 - 6 * q^17 + q^20 + 2 * q^22 + 2 * q^23 - 8 * q^25 + 6 * q^26 + 7 * q^28 - 10 * q^29 + 2 * q^31 + 9 * q^32 - 2 * q^34 + 4 * q^35 - 4 * q^37 + 5 * q^38 - 14 * q^41 - 2 * q^43 + 2 * q^44 + 14 * q^46 + 4 * q^47 + 4 * q^49 + 4 * q^50 - 4 * q^52 + 12 * q^53 + 4 * q^55 + 10 * q^56 - 6 * q^61 - q^62 + 4 * q^64 + 2 * q^65 + 16 * q^67 + 8 * q^68 + 3 * q^70 - 4 * q^71 + 8 * q^73 + 2 * q^74 - 5 * q^76 + 8 * q^77 - 10 * q^79 + 3 * q^80 + 7 * q^82 + 12 * q^83 + 6 * q^85 - 4 * q^86 - 10 * q^89 - 6 * q^91 - 16 * q^92 + 8 * q^94 - 14 * q^97 + 18 * q^98

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