LMFDB - Embedded newform 6093.2.a.l.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6093,2,Mod(1,6093)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6093, 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("6093.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6093 = 3^{2} \cdot 677 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6093.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:	$48.6528499516$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 7 x^{17} + 99 x^{15} - 148 x^{14} - 514 x^{13} + 1204 x^{12} + 1143 x^{11} - 4191 x^{10} + \cdots + 48 $ x^18 - 7x^17 + 99x^15 - 148x^14 - 514x^13 + 1204x^12 + 1143x^11 - 4191x^10 - 668x^9 + 7531x^8 - 1457x^7 - 7108x^6 + 2569x^5 + 3238x^4 - 1325x^3 - 571x^2 + 160x + 48
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2031)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.46392$ of defining polynomial
Character	$\chi$	$=$	6093.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.46392 q^{2} +0.143048 q^{4} -0.387918 q^{5} -2.28879 q^{7} +2.71842 q^{8} +0.567880 q^{10} +1.78909 q^{11} +0.983519 q^{13} +3.35060 q^{14} -4.26563 q^{16} +1.60300 q^{17} -4.74512 q^{19} -0.0554910 q^{20} -2.61907 q^{22} +3.45899 q^{23} -4.84952 q^{25} -1.43979 q^{26} -0.327407 q^{28} +9.04599 q^{29} +2.62121 q^{31} +0.807685 q^{32} -2.34665 q^{34} +0.887865 q^{35} -0.148493 q^{37} +6.94645 q^{38} -1.05453 q^{40} -5.04274 q^{41} +6.65939 q^{43} +0.255925 q^{44} -5.06368 q^{46} -7.92475 q^{47} -1.76143 q^{49} +7.09929 q^{50} +0.140690 q^{52} +1.72374 q^{53} -0.694020 q^{55} -6.22190 q^{56} -13.2426 q^{58} +0.0905729 q^{59} +7.66859 q^{61} -3.83723 q^{62} +7.34888 q^{64} -0.381525 q^{65} -9.42400 q^{67} +0.229306 q^{68} -1.29976 q^{70} +12.4803 q^{71} -4.58440 q^{73} +0.217382 q^{74} -0.678780 q^{76} -4.09485 q^{77} -3.84230 q^{79} +1.65472 q^{80} +7.38214 q^{82} +9.85407 q^{83} -0.621833 q^{85} -9.74879 q^{86} +4.86349 q^{88} -5.42797 q^{89} -2.25107 q^{91} +0.494802 q^{92} +11.6012 q^{94} +1.84072 q^{95} +0.0506965 q^{97} +2.57858 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 7 q^{2} + 13 q^{4} + 13 q^{5} - 3 q^{7} + 18 q^{8} - 6 q^{10} + 26 q^{11} - 11 q^{13} + 11 q^{14} - q^{16} + 34 q^{17} - 2 q^{19} + 10 q^{20} + 8 q^{22} + 35 q^{23} - 5 q^{25} + 17 q^{26} - 5 q^{28}+ \cdots - 12 q^{98}+O(q^{100}) $ 18 * q + 7 * q^2 + 13 * q^4 + 13 * q^5 - 3 * q^7 + 18 * q^8 - 6 * q^10 + 26 * q^11 - 11 * q^13 + 11 * q^14 - q^16 + 34 * q^17 - 2 * q^19 + 10 * q^20 + 8 * q^22 + 35 * q^23 - 5 * q^25 + 17 * q^26 - 5 * q^28 + 25 * q^29 - 11 * q^31 + 18 * q^32 - 3 * q^34 + 27 * q^35 - q^37 + 18 * q^38 - 8 * q^40 + 44 * q^41 + 5 * q^43 + 46 * q^44 + 20 * q^46 + 51 * q^47 - 33 * q^49 - 11 * q^50 - 2 * q^52 + 28 * q^53 + q^55 + q^56 + 22 * q^58 + 44 * q^59 - 7 * q^61 - q^62 + 8 * q^64 + 11 * q^65 + 2 * q^67 + 11 * q^68 + 16 * q^70 + 23 * q^71 + q^73 + 9 * q^74 + 3 * q^76 + 23 * q^77 + 9 * q^79 + 15 * q^80 + 19 * q^82 + 45 * q^83 + 9 * q^85 + 4 * q^86 + 51 * q^88 + 90 * q^89 - 24 * q^91 + 57 * q^92 + 24 * q^94 + 9 * q^95 - 17 * q^97 - 12 * q^98

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 $

$2$	−1.46392	−1.03514	−0.517572	−	0.855640i	$-0.673164\pi$
$2$	−1.46392	−1.03514	−0.517572	+	0.855640i	$0.673164\pi$
$3$	0	0
$3$	0	0
$4$	0.143048	0.0715240
$5$	−0.387918	−0.173482	−0.0867412	−	0.996231i	$-0.527645\pi$
$5$	−0.387918	−0.173482	−0.0867412	+	0.996231i	$0.527645\pi$
$6$	0	0
$7$	−2.28879	−0.865082	−0.432541	−	0.901614i	$-0.642383\pi$
$7$	−2.28879	−0.865082	−0.432541	+	0.901614i	$0.642383\pi$
$8$	2.71842	0.961107
$9$	0	0
$10$	0.567880	0.179579
$11$	1.78909	0.539430	0.269715	−	0.962940i	$-0.413070\pi$
$11$	1.78909	0.539430	0.269715	+	0.962940i	$0.413070\pi$
$12$	0	0
$13$	0.983519	0.272779	0.136389	−	0.990655i	$-0.456450\pi$
$13$	0.983519	0.272779	0.136389	+	0.990655i	$0.456450\pi$
$14$	3.35060	0.895485
$15$	0	0
$16$	−4.26563	−1.06641
$17$	1.60300	0.388784	0.194392	−	0.980924i	$-0.437727\pi$
$17$	1.60300	0.388784	0.194392	+	0.980924i	$0.437727\pi$
$18$	0	0
$19$	−4.74512	−1.08860	−0.544302	−	0.838889i	$-0.683206\pi$
$19$	−4.74512	−1.08860	−0.544302	+	0.838889i	$0.683206\pi$
$20$	−0.0554910	−0.0124082
$21$	0	0
$22$	−2.61907	−0.558388
$23$	3.45899	0.721250	0.360625	−	0.932711i	$-0.382563\pi$
$23$	3.45899	0.721250	0.360625	+	0.932711i	$0.382563\pi$
$24$	0	0
$25$	−4.84952	−0.969904
$26$	−1.43979	−0.282366
$27$	0	0
$28$	−0.327407	−0.0618742
$29$	9.04599	1.67980	0.839899	−	0.542743i	$-0.182614\pi$
$29$	9.04599	1.67980	0.839899	+	0.542743i	$0.182614\pi$
$30$	0	0
$31$	2.62121	0.470783	0.235391	−	0.971901i	$-0.424363\pi$
$31$	2.62121	0.470783	0.235391	+	0.971901i	$0.424363\pi$
$32$	0.807685	0.142780
$33$	0	0
$34$	−2.34665	−0.402448
$35$	0.887865	0.150077
$36$	0	0
$37$	−0.148493	−0.0244122	−0.0122061	−	0.999926i	$-0.503885\pi$
$37$	−0.148493	−0.0244122	−0.0122061	+	0.999926i	$0.503885\pi$
$38$	6.94645	1.12686
$39$	0	0
$40$	−1.05453	−0.166735
$41$	−5.04274	−0.787543	−0.393772	−	0.919208i	$-0.628830\pi$
$41$	−5.04274	−0.787543	−0.393772	+	0.919208i	$0.628830\pi$
$42$	0	0
$43$	6.65939	1.01555	0.507774	−	0.861490i	$-0.330468\pi$
$43$	6.65939	1.01555	0.507774	+	0.861490i	$0.330468\pi$
$44$	0.255925	0.0385822
$45$	0	0
$46$	−5.06368	−0.746598
$47$	−7.92475	−1.15594	−0.577972	−	0.816057i	$-0.696156\pi$
$47$	−7.92475	−1.15594	−0.577972	+	0.816057i	$0.696156\pi$
$48$	0	0
$49$	−1.76143	−0.251633
$50$	7.09929	1.00399
$51$	0	0
$52$	0.140690	0.0195102
$53$	1.72374	0.236774	0.118387	−	0.992968i	$-0.462228\pi$
$53$	1.72374	0.236774	0.118387	+	0.992968i	$0.462228\pi$
$54$	0	0
$55$	−0.694020	−0.0935817
$56$	−6.22190	−0.831436
$57$	0	0
$58$	−13.2426	−1.73883
$59$	0.0905729	0.0117916	0.00589579	−	0.999983i	$-0.498123\pi$
$59$	0.0905729	0.0117916	0.00589579	+	0.999983i	$0.498123\pi$
$60$	0	0
$61$	7.66859	0.981863	0.490931	−	0.871198i	$-0.336657\pi$
$61$	7.66859	0.981863	0.490931	+	0.871198i	$0.336657\pi$
$62$	−3.83723	−0.487328
$63$	0	0
$64$	7.34888	0.918611
$65$	−0.381525	−0.0473224
$66$	0	0
$67$	−9.42400	−1.15132	−0.575662	−	0.817688i	$-0.695256\pi$
$67$	−9.42400	−1.15132	−0.575662	+	0.817688i	$0.695256\pi$
$68$	0.229306	0.0278074
$69$	0	0
$70$	−1.29976	−0.155351
$71$	12.4803	1.48114	0.740572	−	0.671977i	$-0.234555\pi$
$71$	12.4803	1.48114	0.740572	+	0.671977i	$0.234555\pi$
$72$	0	0
$73$	−4.58440	−0.536563	−0.268282	−	0.963341i	$-0.586456\pi$
$73$	−4.58440	−0.536563	−0.268282	+	0.963341i	$0.586456\pi$
$74$	0.217382	0.0252701
$75$	0	0
$76$	−0.678780	−0.0778614
$77$	−4.09485	−0.466651
$78$	0	0
$79$	−3.84230	−0.432293	−0.216146	−	0.976361i	$-0.569349\pi$
$79$	−3.84230	−0.432293	−0.216146	+	0.976361i	$0.569349\pi$
$80$	1.65472	0.185003
$81$	0	0
$82$	7.38214	0.815221
$83$	9.85407	1.08162	0.540812	−	0.841143i	$-0.318117\pi$
$83$	9.85407	1.08162	0.540812	+	0.841143i	$0.318117\pi$
$84$	0	0
$85$	−0.621833	−0.0674472
$86$	−9.74879	−1.05124
$87$	0	0
$88$	4.86349	0.518450
$89$	−5.42797	−0.575364	−0.287682	−	0.957726i	$-0.592885\pi$
$89$	−5.42797	−0.575364	−0.287682	+	0.957726i	$0.592885\pi$
$90$	0	0
$91$	−2.25107	−0.235976
$92$	0.494802	0.0515867
$93$	0	0
$94$	11.6012	1.19657
$95$	1.84072	0.188854
$96$	0	0
$97$	0.0506965	0.00514745	0.00257372	−	0.999997i	$-0.499181\pi$
$97$	0.0506965	0.00514745	0.00257372	+	0.999997i	$0.499181\pi$
$98$	2.57858	0.260476
$99$	0	0
$100$	−0.693714	−0.0693714
$101$	8.03677	0.799689	0.399844	−	0.916583i	$-0.369064\pi$
$101$	8.03677	0.799689	0.399844	+	0.916583i	$0.369064\pi$
$102$	0	0
$103$	9.35634	0.921907	0.460954	−	0.887424i	$-0.347507\pi$
$103$	9.35634	0.921907	0.460954	+	0.887424i	$0.347507\pi$
$104$	2.67362	0.262170
$105$	0	0
$106$	−2.52341	−0.245096
$107$	−9.29217	−0.898308	−0.449154	−	0.893454i	$-0.648275\pi$
$107$	−9.29217	−0.898308	−0.449154	+	0.893454i	$0.648275\pi$
$108$	0	0
$109$	−17.3534	−1.66216	−0.831079	−	0.556154i	$-0.812277\pi$
$109$	−17.3534	−1.66216	−0.831079	+	0.556154i	$0.812277\pi$
$110$	1.01599	0.0968705
$111$	0	0
$112$	9.76315	0.922531
$113$	10.7244	1.00886	0.504432	−	0.863452i	$-0.331702\pi$
$113$	10.7244	1.00886	0.504432	+	0.863452i	$0.331702\pi$
$114$	0	0
$115$	−1.34181	−0.125124
$116$	1.29401	0.120146
$117$	0	0
$118$	−0.132591	−0.0122060
$119$	−3.66893	−0.336330
$120$	0	0
$121$	−7.79917	−0.709015
$122$	−11.2262	−1.01637
$123$	0	0
$124$	0.374959	0.0336723
$125$	3.82081	0.341744
$126$	0	0
$127$	0.558833	0.0495884	0.0247942	−	0.999693i	$-0.492107\pi$
$127$	0.558833	0.0495884	0.0247942	+	0.999693i	$0.492107\pi$
$128$	−12.3735	−1.09367
$129$	0	0
$130$	0.558520	0.0489855
$131$	−21.6051	−1.88764	−0.943822	−	0.330453i	$-0.892798\pi$
$131$	−21.6051	−1.88764	−0.943822	+	0.330453i	$0.892798\pi$
$132$	0	0
$133$	10.8606	0.941733
$134$	13.7959	1.19179
$135$	0	0
$136$	4.35762	0.373663
$137$	−9.15886	−0.782494	−0.391247	−	0.920286i	$-0.627956\pi$
$137$	−9.15886	−0.782494	−0.391247	+	0.920286i	$0.627956\pi$
$138$	0	0
$139$	9.19295	0.779736	0.389868	−	0.920871i	$-0.372521\pi$
$139$	9.19295	0.779736	0.389868	+	0.920871i	$0.372521\pi$
$140$	0.127007	0.0107341
$141$	0	0
$142$	−18.2702	−1.53320
$143$	1.75960	0.147145
$144$	0	0
$145$	−3.50910	−0.291415
$146$	6.71117	0.555421
$147$	0	0
$148$	−0.0212417	−0.00174606
$149$	−15.8387	−1.29756	−0.648778	−	0.760978i	$-0.724720\pi$
$149$	−15.8387	−1.29756	−0.648778	+	0.760978i	$0.724720\pi$
$150$	0	0
$151$	9.02723	0.734625	0.367313	−	0.930098i	$-0.380278\pi$
$151$	9.02723	0.734625	0.367313	+	0.930098i	$0.380278\pi$
$152$	−12.8992	−1.04627
$153$	0	0
$154$	5.99451	0.483052
$155$	−1.01681	−0.0816725
$156$	0	0
$157$	−4.12933	−0.329557	−0.164778	−	0.986331i	$-0.552691\pi$
$157$	−4.12933	−0.329557	−0.164778	+	0.986331i	$0.552691\pi$
$158$	5.62480	0.447485
$159$	0	0
$160$	−0.313316	−0.0247698
$161$	−7.91692	−0.623941
$162$	0	0
$163$	−8.78697	−0.688249	−0.344124	−	0.938924i	$-0.611824\pi$
$163$	−8.78697	−0.688249	−0.344124	+	0.938924i	$0.611824\pi$
$164$	−0.721354	−0.0563283
$165$	0	0
$166$	−14.4255	−1.11964
$167$	−5.89653	−0.456287	−0.228144	−	0.973627i	$-0.573266\pi$
$167$	−5.89653	−0.456287	−0.228144	+	0.973627i	$0.573266\pi$
$168$	0	0
$169$	−12.0327	−0.925592
$170$	0.910310	0.0698176
$171$	0	0
$172$	0.952613	0.0726361
$173$	23.0978	1.75609	0.878045	−	0.478577i	$-0.158847\pi$
$173$	23.0978	1.75609	0.878045	+	0.478577i	$0.158847\pi$
$174$	0	0
$175$	11.0995	0.839047
$176$	−7.63159	−0.575253
$177$	0	0
$178$	7.94609	0.595585
$179$	−21.6632	−1.61918	−0.809591	−	0.586995i	$-0.800311\pi$
$179$	−21.6632	−1.61918	−0.809591	+	0.586995i	$0.800311\pi$
$180$	0	0
$181$	23.3481	1.73545	0.867726	−	0.497044i	$-0.165581\pi$
$181$	23.3481	1.73545	0.867726	+	0.497044i	$0.165581\pi$
$182$	3.29538	0.244270
$183$	0	0
$184$	9.40300	0.693198
$185$	0.0576033	0.00423508
$186$	0	0
$187$	2.86790	0.209722
$188$	−1.13362	−0.0826777
$189$	0	0
$190$	−2.69466	−0.195491
$191$	24.6992	1.78717	0.893587	−	0.448890i	$-0.148181\pi$
$191$	24.6992	1.78717	0.893587	+	0.448890i	$0.148181\pi$
$192$	0	0
$193$	−11.8669	−0.854197	−0.427098	−	0.904205i	$-0.640464\pi$
$193$	−11.8669	−0.854197	−0.427098	+	0.904205i	$0.640464\pi$
$194$	−0.0742154	−0.00532835
$195$	0	0
$196$	−0.251969	−0.0179978
$197$	2.04200	0.145486	0.0727432	−	0.997351i	$-0.476825\pi$
$197$	2.04200	0.145486	0.0727432	+	0.997351i	$0.476825\pi$
$198$	0	0
$199$	21.1109	1.49651	0.748256	−	0.663410i	$-0.230891\pi$
$199$	21.1109	1.49651	0.748256	+	0.663410i	$0.230891\pi$
$200$	−13.1830	−0.932181
$201$	0	0
$202$	−11.7652	−0.827793
$203$	−20.7044	−1.45316
$204$	0	0
$205$	1.95617	0.136625
$206$	−13.6969	−0.954307
$207$	0	0
$208$	−4.19533	−0.290894
$209$	−8.48943	−0.587226
$210$	0	0
$211$	18.5988	1.28040	0.640198	−	0.768210i	$-0.278852\pi$
$211$	18.5988	1.28040	0.640198	+	0.768210i	$0.278852\pi$
$212$	0.246578	0.0169351
$213$	0	0
$214$	13.6030	0.929879
$215$	−2.58330	−0.176180
$216$	0	0
$217$	−5.99940	−0.407266
$218$	25.4040	1.72057
$219$	0	0
$220$	−0.0992782	−0.00669334
$221$	1.57658	0.106052
$222$	0	0
$223$	−0.939854	−0.0629372	−0.0314686	−	0.999505i	$-0.510018\pi$
$223$	−0.939854	−0.0629372	−0.0314686	+	0.999505i	$0.510018\pi$
$224$	−1.84862	−0.123516
$225$	0	0
$226$	−15.6996	−1.04432
$227$	8.16364	0.541840	0.270920	−	0.962602i	$-0.412672\pi$
$227$	8.16364	0.541840	0.270920	+	0.962602i	$0.412672\pi$
$228$	0	0
$229$	−11.5793	−0.765179	−0.382589	−	0.923918i	$-0.624968\pi$
$229$	−11.5793	−0.765179	−0.382589	+	0.923918i	$0.624968\pi$
$230$	1.96429	0.129522
$231$	0	0
$232$	24.5908	1.61446
$233$	7.90222	0.517692	0.258846	−	0.965919i	$-0.416658\pi$
$233$	7.90222	0.517692	0.258846	+	0.965919i	$0.416658\pi$
$234$	0	0
$235$	3.07416	0.200536
$236$	0.0129563	0.000843382 0
$237$	0	0
$238$	5.37100	0.348150
$239$	−16.0618	−1.03895	−0.519476	−	0.854485i	$-0.673873\pi$
$239$	−16.0618	−1.03895	−0.519476	+	0.854485i	$0.673873\pi$
$240$	0	0
$241$	9.31539	0.600057	0.300029	−	0.953930i	$-0.403004\pi$
$241$	9.31539	0.600057	0.300029	+	0.953930i	$0.403004\pi$
$242$	11.4173	0.733933
$243$	0	0
$244$	1.09698	0.0702268
$245$	0.683291	0.0436539
$246$	0	0
$247$	−4.66691	−0.296948
$248$	7.12555	0.452473
$249$	0	0
$250$	−5.59334	−0.353754
$251$	20.5549	1.29742	0.648708	−	0.761038i	$-0.275310\pi$
$251$	20.5549	1.29742	0.648708	+	0.761038i	$0.275310\pi$
$252$	0	0
$253$	6.18844	0.389064
$254$	−0.818084	−0.0513312
$255$	0	0
$256$	3.41601	0.213501
$257$	28.7839	1.79549	0.897747	−	0.440512i	$-0.145203\pi$
$257$	28.7839	1.79549	0.897747	+	0.440512i	$0.145203\pi$
$258$	0	0
$259$	0.339871	0.0211185
$260$	−0.0545764	−0.00338468
$261$	0	0
$262$	31.6280	1.95398
$263$	14.3614	0.885565	0.442782	−	0.896629i	$-0.353991\pi$
$263$	14.3614	0.885565	0.442782	+	0.896629i	$0.353991\pi$
$264$	0	0
$265$	−0.668672	−0.0410762
$266$	−15.8990	−0.974829
$267$	0	0
$268$	−1.34808	−0.0823474
$269$	−0.286312	−0.0174568	−0.00872839	−	0.999962i	$-0.502778\pi$
$269$	−0.286312	−0.0174568	−0.00872839	+	0.999962i	$0.502778\pi$
$270$	0	0
$271$	23.1392	1.40561	0.702804	−	0.711384i	$-0.251931\pi$
$271$	23.1392	1.40561	0.702804	+	0.711384i	$0.251931\pi$
$272$	−6.83780	−0.414603
$273$	0	0
$274$	13.4078	0.809994
$275$	−8.67621	−0.523195
$276$	0	0
$277$	−12.8797	−0.773865	−0.386932	−	0.922108i	$-0.626465\pi$
$277$	−12.8797	−0.773865	−0.386932	+	0.922108i	$0.626465\pi$
$278$	−13.4577	−0.807139
$279$	0	0
$280$	2.41359	0.144240
$281$	16.5883	0.989578	0.494789	−	0.869013i	$-0.335245\pi$
$281$	16.5883	0.989578	0.494789	+	0.869013i	$0.335245\pi$
$282$	0	0
$283$	−0.328687	−0.0195384	−0.00976922	−	0.999952i	$-0.503110\pi$
$283$	−0.328687	−0.0195384	−0.00976922	+	0.999952i	$0.503110\pi$
$284$	1.78529	0.105937
$285$	0	0
$286$	−2.57591	−0.152317
$287$	11.5418	0.681290
$288$	0	0
$289$	−14.4304	−0.848847
$290$	5.13703	0.301657
$291$	0	0
$292$	−0.655789	−0.0383772
$293$	−10.3647	−0.605510	−0.302755	−	0.953068i	$-0.597906\pi$
$293$	−10.3647	−0.605510	−0.302755	+	0.953068i	$0.597906\pi$
$294$	0	0
$295$	−0.0351349	−0.00204563
$296$	−0.403667	−0.0234627
$297$	0	0
$298$	23.1865	1.34316
$299$	3.40199	0.196742
$300$	0	0
$301$	−15.2420	−0.878533
$302$	−13.2151	−0.760443
$303$	0	0
$304$	20.2409	1.16090
$305$	−2.97479	−0.170336
$306$	0	0
$307$	−5.02595	−0.286846	−0.143423	−	0.989661i	$-0.545811\pi$
$307$	−5.02595	−0.286846	−0.143423	+	0.989661i	$0.545811\pi$
$308$	−0.585760	−0.0333768
$309$	0	0
$310$	1.48853	0.0845429
$311$	−3.63333	−0.206027	−0.103014	−	0.994680i	$-0.532849\pi$
$311$	−3.63333	−0.206027	−0.103014	+	0.994680i	$0.532849\pi$
$312$	0	0
$313$	23.0916	1.30521	0.652606	−	0.757697i	$-0.273676\pi$
$313$	23.0916	1.30521	0.652606	+	0.757697i	$0.273676\pi$
$314$	6.04499	0.341139
$315$	0	0
$316$	−0.549634	−0.0309193
$317$	4.51754	0.253730	0.126865	−	0.991920i	$-0.459508\pi$
$317$	4.51754	0.253730	0.126865	+	0.991920i	$0.459508\pi$
$318$	0	0
$319$	16.1841	0.906133
$320$	−2.85077	−0.159363
$321$	0	0
$322$	11.5897	0.645869
$323$	−7.60642	−0.423232
$324$	0	0
$325$	−4.76959	−0.264569
$326$	12.8634	0.712437
$327$	0	0
$328$	−13.7083	−0.756913
$329$	18.1381	0.999986
$330$	0	0
$331$	6.57073	0.361160	0.180580	−	0.983560i	$-0.442203\pi$
$331$	6.57073	0.361160	0.180580	+	0.983560i	$0.442203\pi$
$332$	1.40961	0.0773622
$333$	0	0
$334$	8.63202	0.472323
$335$	3.65574	0.199735
$336$	0	0
$337$	21.6310	1.17831	0.589157	−	0.808019i	$-0.299460\pi$
$337$	21.6310	1.17831	0.589157	+	0.808019i	$0.299460\pi$
$338$	17.6148	0.958121
$339$	0	0
$340$	−0.0889519	−0.00482410
$341$	4.68957	0.253954
$342$	0	0
$343$	20.0531	1.08277
$344$	18.1030	0.976050
$345$	0	0
$346$	−33.8132	−1.81781
$347$	1.99669	0.107188	0.0535941	−	0.998563i	$-0.482932\pi$
$347$	1.99669	0.107188	0.0535941	+	0.998563i	$0.482932\pi$
$348$	0	0
$349$	−17.5651	−0.940240	−0.470120	−	0.882602i	$-0.655789\pi$
$349$	−17.5651	−0.940240	−0.470120	+	0.882602i	$0.655789\pi$
$350$	−16.2488	−0.868534
$351$	0	0
$352$	1.44502	0.0770198
$353$	−20.0470	−1.06700	−0.533498	−	0.845802i	$-0.679123\pi$
$353$	−20.0470	−1.06700	−0.533498	+	0.845802i	$0.679123\pi$
$354$	0	0
$355$	−4.84136	−0.256952
$356$	−0.776461	−0.0411523
$357$	0	0
$358$	31.7131	1.67609
$359$	33.8830	1.78828	0.894138	−	0.447791i	$-0.147789\pi$
$359$	33.8830	1.78828	0.894138	+	0.447791i	$0.147789\pi$
$360$	0	0
$361$	3.51614	0.185060
$362$	−34.1797	−1.79644
$363$	0	0
$364$	−0.322011	−0.0168780
$365$	1.77837	0.0930843
$366$	0	0
$367$	−1.55099	−0.0809610	−0.0404805	−	0.999180i	$-0.512889\pi$
$367$	−1.55099	−0.0809610	−0.0404805	+	0.999180i	$0.512889\pi$
$368$	−14.7548	−0.769147
$369$	0	0
$370$	−0.0843264	−0.00438392
$371$	−3.94529	−0.204829
$372$	0	0
$373$	5.59834	0.289871	0.144936	−	0.989441i	$-0.453702\pi$
$373$	5.59834	0.289871	0.144936	+	0.989441i	$0.453702\pi$
$374$	−4.19837	−0.217092
$375$	0	0
$376$	−21.5428	−1.11099
$377$	8.89690	0.458213
$378$	0	0
$379$	−3.49594	−0.179574	−0.0897872	−	0.995961i	$-0.528619\pi$
$379$	−3.49594	−0.179574	−0.0897872	+	0.995961i	$0.528619\pi$
$380$	0.263311	0.0135076
$381$	0	0
$382$	−36.1576	−1.84998
$383$	−23.6240	−1.20713	−0.603565	−	0.797314i	$-0.706253\pi$
$383$	−23.6240	−1.20713	−0.603565	+	0.797314i	$0.706253\pi$
$384$	0	0
$385$	1.58847	0.0809558
$386$	17.3721	0.884217
$387$	0	0
$388$	0.00725203	0.000368166 0
$389$	−15.2090	−0.771126	−0.385563	−	0.922681i	$-0.625993\pi$
$389$	−15.2090	−0.771126	−0.385563	+	0.922681i	$0.625993\pi$
$390$	0	0
$391$	5.54476	0.280411
$392$	−4.78831	−0.241846
$393$	0	0
$394$	−2.98931	−0.150599
$395$	1.49050	0.0749952
$396$	0	0
$397$	12.0872	0.606637	0.303318	−	0.952889i	$-0.401905\pi$
$397$	12.0872	0.606637	0.303318	+	0.952889i	$0.401905\pi$
$398$	−30.9046	−1.54911
$399$	0	0
$400$	20.6863	1.03431
$401$	−6.11032	−0.305135	−0.152567	−	0.988293i	$-0.548754\pi$
$401$	−6.11032	−0.305135	−0.152567	+	0.988293i	$0.548754\pi$
$402$	0	0
$403$	2.57801	0.128420
$404$	1.14964	0.0571969
$405$	0	0
$406$	30.3095	1.50423
$407$	−0.265668	−0.0131687
$408$	0	0
$409$	13.3281	0.659033	0.329516	−	0.944150i	$-0.393114\pi$
$409$	13.3281	0.659033	0.329516	+	0.944150i	$0.393114\pi$
$410$	−2.86367	−0.141427
$411$	0	0
$412$	1.33841	0.0659385
$413$	−0.207303	−0.0102007
$414$	0	0
$415$	−3.82258	−0.187643
$416$	0.794373	0.0389474
$417$	0	0
$418$	12.4278	0.607864
$419$	18.0114	0.879917	0.439958	−	0.898018i	$-0.354993\pi$
$419$	18.0114	0.879917	0.439958	+	0.898018i	$0.354993\pi$
$420$	0	0
$421$	14.0404	0.684290	0.342145	−	0.939647i	$-0.388847\pi$
$421$	14.0404	0.684290	0.342145	+	0.939647i	$0.388847\pi$
$422$	−27.2271	−1.32539
$423$	0	0
$424$	4.68586	0.227565
$425$	−7.77377	−0.377083
$426$	0	0
$427$	−17.5518	−0.849392
$428$	−1.32923	−0.0642506
$429$	0	0
$430$	3.78174	0.182371
$431$	38.7591	1.86696	0.933479	−	0.358632i	$-0.116757\pi$
$431$	38.7591	1.86696	0.933479	+	0.358632i	$0.116757\pi$
$432$	0	0
$433$	−29.1583	−1.40126	−0.700629	−	0.713526i	$-0.747097\pi$
$433$	−29.1583	−1.40126	−0.700629	+	0.713526i	$0.747097\pi$
$434$	8.78261	0.421579
$435$	0	0
$436$	−2.48238	−0.118884
$437$	−16.4133	−0.785156
$438$	0	0
$439$	−9.37853	−0.447613	−0.223807	−	0.974634i	$-0.571848\pi$
$439$	−9.37853	−0.447613	−0.223807	+	0.974634i	$0.571848\pi$
$440$	−1.88664	−0.0899420
$441$	0	0
$442$	−2.30798	−0.109779
$443$	10.6502	0.506009	0.253004	−	0.967465i	$-0.418581\pi$
$443$	10.6502	0.506009	0.253004	+	0.967465i	$0.418581\pi$
$444$	0	0
$445$	2.10561	0.0998155
$446$	1.37587	0.0651491
$447$	0	0
$448$	−16.8201	−0.794674
$449$	22.5384	1.06365	0.531826	−	0.846853i	$-0.321506\pi$
$449$	22.5384	1.06365	0.531826	+	0.846853i	$0.321506\pi$
$450$	0	0
$451$	−9.02190	−0.424825
$452$	1.53410	0.0721580
$453$	0	0
$454$	−11.9509	−0.560882
$455$	0.873232	0.0409377
$456$	0	0
$457$	34.9180	1.63340	0.816698	−	0.577066i	$-0.195802\pi$
$457$	34.9180	1.63340	0.816698	+	0.577066i	$0.195802\pi$
$458$	16.9511	0.792071
$459$	0	0
$460$	−0.191943	−0.00894939
$461$	−28.2073	−1.31374	−0.656872	−	0.754002i	$-0.728121\pi$
$461$	−28.2073	−1.31374	−0.656872	+	0.754002i	$0.728121\pi$
$462$	0	0
$463$	−17.6647	−0.820946	−0.410473	−	0.911873i	$-0.634636\pi$
$463$	−17.6647	−0.820946	−0.410473	+	0.911873i	$0.634636\pi$
$464$	−38.5869	−1.79135
$465$	0	0
$466$	−11.5682	−0.535886
$467$	18.6212	0.861688	0.430844	−	0.902426i	$-0.358216\pi$
$467$	18.6212	0.861688	0.430844	+	0.902426i	$0.358216\pi$
$468$	0	0
$469$	21.5696	0.995990
$470$	−4.50031	−0.207584
$471$	0	0
$472$	0.246215	0.0113330
$473$	11.9142	0.547817
$474$	0	0
$475$	23.0115	1.05584
$476$	−0.524833	−0.0240557
$477$	0	0
$478$	23.5131	1.07547
$479$	−2.54315	−0.116200	−0.0580998	−	0.998311i	$-0.518504\pi$
$479$	−2.54315	−0.116200	−0.0580998	+	0.998311i	$0.518504\pi$
$480$	0	0
$481$	−0.146046	−0.00665913
$482$	−13.6369	−0.621146
$483$	0	0
$484$	−1.11566	−0.0507116
$485$	−0.0196661	−0.000892992 0
$486$	0	0
$487$	−1.14047	−0.0516797	−0.0258398	−	0.999666i	$-0.508226\pi$
$487$	−1.14047	−0.0516797	−0.0258398	+	0.999666i	$0.508226\pi$
$488$	20.8465	0.943675
$489$	0	0
$490$	−1.00028	−0.0451881
$491$	27.6703	1.24874	0.624372	−	0.781127i	$-0.285355\pi$
$491$	27.6703	1.24874	0.624372	+	0.781127i	$0.285355\pi$
$492$	0	0
$493$	14.5007	0.653079
$494$	6.83196	0.307385
$495$	0	0
$496$	−11.1811	−0.502047
$497$	−28.5649	−1.28131
$498$	0	0
$499$	−10.2687	−0.459689	−0.229844	−	0.973227i	$-0.573822\pi$
$499$	−10.2687	−0.459689	−0.229844	+	0.973227i	$0.573822\pi$
$500$	0.546559	0.0244429
$501$	0	0
$502$	−30.0907	−1.34301
$503$	1.81344	0.0808574	0.0404287	−	0.999182i	$-0.487128\pi$
$503$	1.81344	0.0808574	0.0404287	+	0.999182i	$0.487128\pi$
$504$	0	0
$505$	−3.11761	−0.138732
$506$	−9.05936	−0.402738
$507$	0	0
$508$	0.0799400	0.00354676
$509$	22.5991	1.00169	0.500844	−	0.865538i	$-0.333023\pi$
$509$	22.5991	1.00169	0.500844	+	0.865538i	$0.333023\pi$
$510$	0	0
$511$	10.4927	0.464171
$512$	19.7463	0.872671
$513$	0	0
$514$	−42.1373	−1.85859
$515$	−3.62950	−0.159935
$516$	0	0
$517$	−14.1781	−0.623551
$518$	−0.497542	−0.0218607
$519$	0	0
$520$	−1.03715	−0.0454818
$521$	7.28990	0.319377	0.159688	−	0.987167i	$-0.448951\pi$
$521$	7.28990	0.319377	0.159688	+	0.987167i	$0.448951\pi$
$522$	0	0
$523$	24.5365	1.07291	0.536453	−	0.843930i	$-0.319764\pi$
$523$	24.5365	1.07291	0.536453	+	0.843930i	$0.319764\pi$
$524$	−3.09056	−0.135012
$525$	0	0
$526$	−21.0239	−0.916688
$527$	4.20179	0.183033
$528$	0	0
$529$	−11.0354	−0.479798
$530$	0.978879	0.0425198
$531$	0	0
$532$	1.55359	0.0673565
$533$	−4.95963	−0.214825
$534$	0	0
$535$	3.60460	0.155841
$536$	−25.6184	−1.10655
$537$	0	0
$538$	0.419137	0.0180703
$539$	−3.15135	−0.135738
$540$	0	0
$541$	−27.8290	−1.19646	−0.598231	−	0.801324i	$-0.704130\pi$
$541$	−27.8290	−1.19646	−0.598231	+	0.801324i	$0.704130\pi$
$542$	−33.8738	−1.45501
$543$	0	0
$544$	1.29472	0.0555106
$545$	6.73172	0.288355
$546$	0	0
$547$	33.7277	1.44209	0.721046	−	0.692887i	$-0.243662\pi$
$547$	33.7277	1.44209	0.721046	+	0.692887i	$0.243662\pi$
$548$	−1.31016	−0.0559671
$549$	0	0
$550$	12.7012	0.541583
$551$	−42.9243	−1.82864
$552$	0	0
$553$	8.79423	0.373969
$554$	18.8548	0.801062
$555$	0	0
$556$	1.31503	0.0557698
$557$	31.3138	1.32681	0.663404	−	0.748261i	$-0.269111\pi$
$557$	31.3138	1.32681	0.663404	+	0.748261i	$0.269111\pi$
$558$	0	0
$559$	6.54964	0.277020
$560$	−3.78731	−0.160043
$561$	0	0
$562$	−24.2839	−1.02436
$563$	32.2048	1.35727	0.678635	−	0.734475i	$-0.262572\pi$
$563$	32.2048	1.35727	0.678635	+	0.734475i	$0.262572\pi$
$564$	0	0
$565$	−4.16018	−0.175020
$566$	0.481171	0.0202251
$567$	0	0
$568$	33.9268	1.42354
$569$	45.0406	1.88820	0.944100	−	0.329658i	$-0.106934\pi$
$569$	45.0406	1.88820	0.944100	+	0.329658i	$0.106934\pi$
$570$	0	0
$571$	−27.9975	−1.17166	−0.585829	−	0.810435i	$-0.699231\pi$
$571$	−27.9975	−1.17166	−0.585829	+	0.810435i	$0.699231\pi$
$572$	0.251707	0.0105244
$573$	0	0
$574$	−16.8962	−0.705233
$575$	−16.7745	−0.699543
$576$	0	0
$577$	−11.4961	−0.478591	−0.239295	−	0.970947i	$-0.576916\pi$
$577$	−11.4961	−0.478591	−0.239295	+	0.970947i	$0.576916\pi$
$578$	21.1249	0.878679
$579$	0	0
$580$	−0.501971	−0.0208432
$581$	−22.5539	−0.935694
$582$	0	0
$583$	3.08393	0.127723
$584$	−12.4623	−0.515695
$585$	0	0
$586$	15.1730	0.626790
$587$	30.6640	1.26564	0.632819	−	0.774300i	$-0.281898\pi$
$587$	30.6640	1.26564	0.632819	+	0.774300i	$0.281898\pi$
$588$	0	0
$589$	−12.4379	−0.512496
$590$	0.0514345	0.00211753
$591$	0	0
$592$	0.633418	0.0260333
$593$	41.7369	1.71393	0.856965	−	0.515374i	$-0.172347\pi$
$593$	41.7369	1.71393	0.856965	+	0.515374i	$0.172347\pi$
$594$	0	0
$595$	1.42325	0.0583474
$596$	−2.26569	−0.0928065
$597$	0	0
$598$	−4.98022	−0.203656
$599$	22.8597	0.934021	0.467010	−	0.884252i	$-0.345331\pi$
$599$	22.8597	0.934021	0.467010	+	0.884252i	$0.345331\pi$
$600$	0	0
$601$	−16.8840	−0.688711	−0.344356	−	0.938839i	$-0.611903\pi$
$601$	−16.8840	−0.688711	−0.344356	+	0.938839i	$0.611903\pi$
$602$	22.3130	0.909408
$603$	0	0
$604$	1.29133	0.0525433
$605$	3.02544	0.123002
$606$	0	0
$607$	2.63279	0.106862	0.0534309	−	0.998572i	$-0.482984\pi$
$607$	2.63279	0.106862	0.0534309	+	0.998572i	$0.482984\pi$
$608$	−3.83256	−0.155431
$609$	0	0
$610$	4.35484	0.176322
$611$	−7.79414	−0.315317
$612$	0	0
$613$	32.8318	1.32606	0.663032	−	0.748591i	$-0.269269\pi$
$613$	32.8318	1.32606	0.663032	+	0.748591i	$0.269269\pi$
$614$	7.35756	0.296927
$615$	0	0
$616$	−11.1315	−0.448502
$617$	−9.15201	−0.368446	−0.184223	−	0.982884i	$-0.558977\pi$
$617$	−9.15201	−0.368446	−0.184223	+	0.982884i	$0.558977\pi$
$618$	0	0
$619$	11.0552	0.444345	0.222172	−	0.975007i	$-0.428685\pi$
$619$	11.0552	0.444345	0.222172	+	0.975007i	$0.428685\pi$
$620$	−0.145453	−0.00584155
$621$	0	0
$622$	5.31889	0.213268
$623$	12.4235	0.497737
$624$	0	0
$625$	22.7654	0.910617
$626$	−33.8041	−1.35108
$627$	0	0
$628$	−0.590693	−0.0235712
$629$	−0.238035	−0.00949106
$630$	0	0
$631$	31.4339	1.25136	0.625681	−	0.780079i	$-0.284821\pi$
$631$	31.4339	1.25136	0.625681	+	0.780079i	$0.284821\pi$
$632$	−10.4450	−0.415479
$633$	0	0
$634$	−6.61329	−0.262647
$635$	−0.216782	−0.00860272
$636$	0	0
$637$	−1.73240	−0.0686401
$638$	−23.6921	−0.937979
$639$	0	0
$640$	4.79991	0.189733
$641$	35.0632	1.38491	0.692457	−	0.721459i	$-0.256528\pi$
$641$	35.0632	1.38491	0.692457	+	0.721459i	$0.256528\pi$
$642$	0	0
$643$	−28.5069	−1.12420	−0.562101	−	0.827069i	$-0.690007\pi$
$643$	−28.5069	−1.12420	−0.562101	+	0.827069i	$0.690007\pi$
$644$	−1.13250	−0.0446267
$645$	0	0
$646$	11.1351	0.438107
$647$	−5.49463	−0.216016	−0.108008	−	0.994150i	$-0.534447\pi$
$647$	−5.49463	−0.216016	−0.108008	+	0.994150i	$0.534447\pi$
$648$	0	0
$649$	0.162043	0.00636074
$650$	6.98228	0.273868
$651$	0	0
$652$	−1.25696	−0.0492263
$653$	37.8811	1.48240	0.741202	−	0.671283i	$-0.234256\pi$
$653$	37.8811	1.48240	0.741202	+	0.671283i	$0.234256\pi$
$654$	0	0
$655$	8.38101	0.327473
$656$	21.5105	0.839843
$657$	0	0
$658$	−26.5527	−1.03513
$659$	−32.9824	−1.28481	−0.642406	−	0.766364i	$-0.722064\pi$
$659$	−32.9824	−1.28481	−0.642406	+	0.766364i	$0.722064\pi$
$660$	0	0
$661$	3.70894	0.144261	0.0721306	−	0.997395i	$-0.477020\pi$
$661$	3.70894	0.144261	0.0721306	+	0.997395i	$0.477020\pi$
$662$	−9.61899	−0.373853
$663$	0	0
$664$	26.7875	1.03956
$665$	−4.21302	−0.163374
$666$	0	0
$667$	31.2900	1.21155
$668$	−0.843487	−0.0326355
$669$	0	0
$670$	−5.35170	−0.206754
$671$	13.7198	0.529646
$672$	0	0
$673$	−13.4252	−0.517502	−0.258751	−	0.965944i	$-0.583311\pi$
$673$	−13.4252	−0.517502	−0.258751	+	0.965944i	$0.583311\pi$
$674$	−31.6659	−1.21972
$675$	0	0
$676$	−1.72125	−0.0662020
$677$	−1.00000	−0.0384331
$677$	−1.00000	−0.0384331
$678$	0	0
$679$	−0.116034	−0.00445297
$680$	−1.69040	−0.0648240
$681$	0	0
$682$	−6.86513	−0.262880
$683$	−11.9448	−0.457054	−0.228527	−	0.973538i	$-0.573391\pi$
$683$	−11.9448	−0.457054	−0.228527	+	0.973538i	$0.573391\pi$
$684$	0	0
$685$	3.55289	0.135749
$686$	−29.3560	−1.12082
$687$	0	0
$688$	−28.4065	−1.08299
$689$	1.69533	0.0645871
$690$	0	0
$691$	41.1418	1.56511	0.782554	−	0.622583i	$-0.213917\pi$
$691$	41.1418	1.56511	0.782554	+	0.622583i	$0.213917\pi$
$692$	3.30409	0.125603
$693$	0	0
$694$	−2.92299	−0.110955
$695$	−3.56611	−0.135270
$696$	0	0
$697$	−8.08350	−0.306184
$698$	25.7139	0.973285
$699$	0	0
$700$	1.58777	0.0600120
$701$	34.8834	1.31753	0.658765	−	0.752349i	$-0.271079\pi$
$701$	34.8834	1.31753	0.658765	+	0.752349i	$0.271079\pi$
$702$	0	0
$703$	0.704619	0.0265752
$704$	13.1478	0.495526
$705$	0	0
$706$	29.3471	1.10449
$707$	−18.3945	−0.691796
$708$	0	0
$709$	−9.10734	−0.342033	−0.171017	−	0.985268i	$-0.554705\pi$
$709$	−9.10734	−0.342033	−0.171017	+	0.985268i	$0.554705\pi$
$710$	7.08733	0.265983
$711$	0	0
$712$	−14.7555	−0.552986
$713$	9.06674	0.339552
$714$	0	0
$715$	−0.682582	−0.0255271
$716$	−3.09887	−0.115810
$717$	0	0
$718$	−49.6018	−1.85112
$719$	−8.80298	−0.328296	−0.164148	−	0.986436i	$-0.552487\pi$
$719$	−8.80298	−0.328296	−0.164148	+	0.986436i	$0.552487\pi$
$720$	0	0
$721$	−21.4147	−0.797526
$722$	−5.14733	−0.191564
$723$	0	0
$724$	3.33990	0.124126
$725$	−43.8687	−1.62924
$726$	0	0
$727$	41.2941	1.53151	0.765757	−	0.643130i	$-0.222364\pi$
$727$	41.2941	1.53151	0.765757	+	0.643130i	$0.222364\pi$
$728$	−6.11935	−0.226798
$729$	0	0
$730$	−2.60339	−0.0963557
$731$	10.6750	0.394829
$732$	0	0
$733$	19.0753	0.704564	0.352282	−	0.935894i	$-0.385406\pi$
$733$	19.0753	0.704564	0.352282	+	0.935894i	$0.385406\pi$
$734$	2.27052	0.0838063
$735$	0	0
$736$	2.79378	0.102980
$737$	−16.8604	−0.621059
$738$	0	0
$739$	34.9890	1.28709	0.643546	−	0.765407i	$-0.277463\pi$
$739$	34.9890	1.28709	0.643546	+	0.765407i	$0.277463\pi$
$740$	0.00824004	0.000302910 0
$741$	0	0
$742$	5.77557	0.212028
$743$	9.84477	0.361170	0.180585	−	0.983559i	$-0.442201\pi$
$743$	9.84477	0.361170	0.180585	+	0.983559i	$0.442201\pi$
$744$	0	0
$745$	6.14412	0.225103
$746$	−8.19550	−0.300059
$747$	0	0
$748$	0.410248	0.0150002
$749$	21.2679	0.777110
$750$	0	0
$751$	2.32732	0.0849250	0.0424625	−	0.999098i	$-0.486480\pi$
$751$	2.32732	0.0849250	0.0424625	+	0.999098i	$0.486480\pi$
$752$	33.8041	1.23271
$753$	0	0
$754$	−13.0243	−0.474317
$755$	−3.50183	−0.127445
$756$	0	0
$757$	−17.2815	−0.628108	−0.314054	−	0.949405i	$-0.601687\pi$
$757$	−17.2815	−0.628108	−0.314054	+	0.949405i	$0.601687\pi$
$758$	5.11776	0.185885
$759$	0	0
$760$	5.00385	0.181509
$761$	−15.8739	−0.575428	−0.287714	−	0.957716i	$-0.592895\pi$
$761$	−15.8739	−0.575428	−0.287714	+	0.957716i	$0.592895\pi$
$762$	0	0
$763$	39.7184	1.43790
$764$	3.53318	0.127826
$765$	0	0
$766$	34.5835	1.24955
$767$	0.0890801	0.00321650
$768$	0	0
$769$	15.3085	0.552040	0.276020	−	0.961152i	$-0.410984\pi$
$769$	15.3085	0.552040	0.276020	+	0.961152i	$0.410984\pi$
$770$	−2.32538	−0.0838010
$771$	0	0
$772$	−1.69753	−0.0610956
$773$	−15.3167	−0.550905	−0.275453	−	0.961315i	$-0.588828\pi$
$773$	−15.3167	−0.550905	−0.275453	+	0.961315i	$0.588828\pi$
$774$	0	0
$775$	−12.7116	−0.456614
$776$	0.137814	0.00494725
$777$	0	0
$778$	22.2647	0.798227
$779$	23.9284	0.857323
$780$	0	0
$781$	22.3284	0.798974
$782$	−8.11706	−0.290266
$783$	0	0
$784$	7.51361	0.268343
$785$	1.60184	0.0571723
$786$	0	0
$787$	−55.7092	−1.98582	−0.992909	−	0.118875i	$-0.962071\pi$
$787$	−55.7092	−1.98582	−0.992909	+	0.118875i	$0.962071\pi$
$788$	0.292104	0.0104058
$789$	0	0
$790$	−2.18197	−0.0776308
$791$	−24.5459	−0.872750
$792$	0	0
$793$	7.54220	0.267832
$794$	−17.6946	−0.627957
$795$	0	0
$796$	3.01987	0.107037
$797$	−2.22098	−0.0786710	−0.0393355	−	0.999226i	$-0.512524\pi$
$797$	−2.22098	−0.0786710	−0.0393355	+	0.999226i	$0.512524\pi$
$798$	0	0
$799$	−12.7034	−0.449413
$800$	−3.91688	−0.138483
$801$	0	0
$802$	8.94500	0.315859
$803$	−8.20189	−0.289438
$804$	0	0
$805$	3.07112	0.108243
$806$	−3.77398	−0.132933
$807$	0	0
$808$	21.8473	0.768586
$809$	−4.24804	−0.149353	−0.0746765	−	0.997208i	$-0.523792\pi$
$809$	−4.24804	−0.149353	−0.0746765	+	0.997208i	$0.523792\pi$
$810$	0	0
$811$	−33.5125	−1.17678	−0.588392	−	0.808576i	$-0.700239\pi$
$811$	−33.5125	−1.17678	−0.588392	+	0.808576i	$0.700239\pi$
$812$	−2.96172	−0.103936
$813$	0	0
$814$	0.388915	0.0136315
$815$	3.40863	0.119399
$816$	0	0
$817$	−31.5996	−1.10553
$818$	−19.5112	−0.682194
$819$	0	0
$820$	0.279826	0.00977196
$821$	−18.8922	−0.659341	−0.329670	−	0.944096i	$-0.606938\pi$
$821$	−18.8922	−0.659341	−0.329670	+	0.944096i	$0.606938\pi$
$822$	0	0
$823$	−0.253189	−0.00882561	−0.00441280	−	0.999990i	$-0.501405\pi$
$823$	−0.253189	−0.00882561	−0.00441280	+	0.999990i	$0.501405\pi$
$824$	25.4345	0.886051
$825$	0	0
$826$	0.303473	0.0105592
$827$	26.9985	0.938830	0.469415	−	0.882978i	$-0.344465\pi$
$827$	26.9985	0.938830	0.469415	+	0.882978i	$0.344465\pi$
$828$	0	0
$829$	−57.0561	−1.98164	−0.990821	−	0.135183i	$-0.956838\pi$
$829$	−57.0561	−1.98164	−0.990821	+	0.135183i	$0.956838\pi$
$830$	5.59593	0.194237
$831$	0	0
$832$	7.22776	0.250578
$833$	−2.82357	−0.0978308
$834$	0	0
$835$	2.28737	0.0791578
$836$	−1.21440	−0.0420008
$837$	0	0
$838$	−26.3672	−0.910841
$839$	38.4221	1.32648	0.663240	−	0.748407i	$-0.269181\pi$
$839$	38.4221	1.32648	0.663240	+	0.748407i	$0.269181\pi$
$840$	0	0
$841$	52.8299	1.82172
$842$	−20.5540	−0.708338
$843$	0	0
$844$	2.66052	0.0915790
$845$	4.66770	0.160574
$846$	0	0
$847$	17.8507	0.613356
$848$	−7.35286	−0.252498
$849$	0	0
$850$	11.3801	0.390336
$851$	−0.513638	−0.0176073
$852$	0	0
$853$	−10.3269	−0.353587	−0.176793	−	0.984248i	$-0.556572\pi$
$853$	−10.3269	−0.353587	−0.176793	+	0.984248i	$0.556572\pi$
$854$	25.6944	0.879243
$855$	0	0
$856$	−25.2600	−0.863370
$857$	−11.3429	−0.387467	−0.193734	−	0.981054i	$-0.562060\pi$
$857$	−11.3429	−0.387467	−0.193734	+	0.981054i	$0.562060\pi$
$858$	0	0
$859$	−8.77050	−0.299245	−0.149623	−	0.988743i	$-0.547806\pi$
$859$	−8.77050	−0.299245	−0.149623	+	0.988743i	$0.547806\pi$
$860$	−0.369536	−0.0126011
$861$	0	0
$862$	−56.7400	−1.93257
$863$	32.5706	1.10872	0.554359	−	0.832278i	$-0.312964\pi$
$863$	32.5706	1.10872	0.554359	+	0.832278i	$0.312964\pi$
$864$	0	0
$865$	−8.96005	−0.304651
$866$	42.6853	1.45050
$867$	0	0
$868$	−0.858202	−0.0291293
$869$	−6.87421	−0.233192
$870$	0	0
$871$	−9.26868	−0.314057
$872$	−47.1740	−1.59751
$873$	0	0
$874$	24.0277	0.812750
$875$	−8.74504	−0.295636
$876$	0	0
$877$	52.5225	1.77356	0.886780	−	0.462192i	$-0.152937\pi$
$877$	52.5225	1.77356	0.886780	+	0.462192i	$0.152937\pi$
$878$	13.7294	0.463344
$879$	0	0
$880$	2.96044	0.0997963
$881$	−39.2658	−1.32290	−0.661450	−	0.749989i	$-0.730059\pi$
$881$	−39.2658	−1.32290	−0.661450	+	0.749989i	$0.730059\pi$
$882$	0	0
$883$	−23.1714	−0.779780	−0.389890	−	0.920862i	$-0.627487\pi$
$883$	−23.1714	−0.779780	−0.389890	+	0.920862i	$0.627487\pi$
$884$	0.225526	0.00758528
$885$	0	0
$886$	−15.5911	−0.523792
$887$	−32.1376	−1.07907	−0.539537	−	0.841962i	$-0.681401\pi$
$887$	−32.1376	−1.07907	−0.539537	+	0.841962i	$0.681401\pi$
$888$	0	0
$889$	−1.27905	−0.0428980
$890$	−3.08244	−0.103323
$891$	0	0
$892$	−0.134444	−0.00450152
$893$	37.6039	1.25837
$894$	0	0
$895$	8.40354	0.280900
$896$	28.3204	0.946118
$897$	0	0
$898$	−32.9943	−1.10103
$899$	23.7114	0.790820
$900$	0	0
$901$	2.76316	0.0920541
$902$	13.2073	0.439755
$903$	0	0
$904$	29.1533	0.969626
$905$	−9.05716	−0.301070
$906$	0	0
$907$	14.0634	0.466967	0.233483	−	0.972361i	$-0.424988\pi$
$907$	14.0634	0.466967	0.233483	+	0.972361i	$0.424988\pi$
$908$	1.16779	0.0387545
$909$	0	0
$910$	−1.27834	−0.0423765
$911$	5.55457	0.184031	0.0920155	−	0.995758i	$-0.470669\pi$
$911$	5.55457	0.184031	0.0920155	+	0.995758i	$0.470669\pi$
$912$	0	0
$913$	17.6298	0.583461
$914$	−51.1170	−1.69080
$915$	0	0
$916$	−1.65639	−0.0547287
$917$	49.4495	1.63297
$918$	0	0
$919$	21.0669	0.694933	0.347466	−	0.937692i	$-0.387042\pi$
$919$	21.0669	0.694933	0.347466	+	0.937692i	$0.387042\pi$
$920$	−3.64760	−0.120258
$921$	0	0
$922$	41.2931	1.35992
$923$	12.2747	0.404025
$924$	0	0
$925$	0.720122	0.0236775
$926$	25.8596	0.849798
$927$	0	0
$928$	7.30631	0.239841
$929$	9.97010	0.327108	0.163554	−	0.986534i	$-0.447704\pi$
$929$	9.97010	0.327108	0.163554	+	0.986534i	$0.447704\pi$
$930$	0	0
$931$	8.35819	0.273929
$932$	1.13040	0.0370274
$933$	0	0
$934$	−27.2599	−0.891972
$935$	−1.11251	−0.0363831
$936$	0	0
$937$	24.7208	0.807594	0.403797	−	0.914849i	$-0.367690\pi$
$937$	24.7208	0.807594	0.403797	+	0.914849i	$0.367690\pi$
$938$	−31.5760	−1.03099
$939$	0	0
$940$	0.439752	0.0143431
$941$	−40.7105	−1.32712	−0.663562	−	0.748122i	$-0.730956\pi$
$941$	−40.7105	−1.32712	−0.663562	+	0.748122i	$0.730956\pi$
$942$	0	0
$943$	−17.4428	−0.568016
$944$	−0.386351	−0.0125746
$945$	0	0
$946$	−17.4414	−0.567070
$947$	−18.8055	−0.611096	−0.305548	−	0.952177i	$-0.598840\pi$
$947$	−18.8055	−0.611096	−0.305548	+	0.952177i	$0.598840\pi$
$948$	0	0
$949$	−4.50884	−0.146363
$950$	−33.6869	−1.09295
$951$	0	0
$952$	−9.97369	−0.323249
$953$	14.7613	0.478166	0.239083	−	0.970999i	$-0.423153\pi$
$953$	14.7613	0.478166	0.239083	+	0.970999i	$0.423153\pi$
$954$	0	0
$955$	−9.58129	−0.310043
$956$	−2.29761	−0.0743101
$957$	0	0
$958$	3.72296	0.120283
$959$	20.9627	0.676922
$960$	0	0
$961$	−24.1293	−0.778364
$962$	0.213799	0.00689316
$963$	0	0
$964$	1.33255	0.0429185
$965$	4.60338	0.148188
$966$	0	0
$967$	34.4322	1.10727	0.553633	−	0.832761i	$-0.313241\pi$
$967$	34.4322	1.10727	0.553633	+	0.832761i	$0.313241\pi$
$968$	−21.2014	−0.681439
$969$	0	0
$970$	0.0287895	0.000924375 0
$971$	−28.1893	−0.904639	−0.452319	−	0.891856i	$-0.649403\pi$
$971$	−28.1893	−0.904639	−0.452319	+	0.891856i	$0.649403\pi$
$972$	0	0
$973$	−21.0407	−0.674535
$974$	1.66955	0.0534959
$975$	0	0
$976$	−32.7114	−1.04707
$977$	31.4468	1.00607	0.503036	−	0.864265i	$-0.332216\pi$
$977$	31.4468	1.00607	0.503036	+	0.864265i	$0.332216\pi$
$978$	0	0
$979$	−9.71112	−0.310369
$980$	0.0977434	0.00312230
$981$	0	0
$982$	−40.5070	−1.29263
$983$	6.88201	0.219502	0.109751	−	0.993959i	$-0.464995\pi$
$983$	6.88201	0.219502	0.109751	+	0.993959i	$0.464995\pi$
$984$	0	0
$985$	−0.792129	−0.0252393
$986$	−21.2278	−0.676031
$987$	0	0
$988$	−0.667593	−0.0212389
$989$	23.0348	0.732464
$990$	0	0
$991$	21.3698	0.678833	0.339416	−	0.940636i	$-0.389770\pi$
$991$	21.3698	0.678833	0.339416	+	0.940636i	$0.389770\pi$
$992$	2.11711	0.0672183
$993$	0	0
$994$	41.8166	1.32634
$995$	−8.18931	−0.259619
$996$	0	0
$997$	15.2851	0.484083	0.242041	−	0.970266i	$-0.422183\pi$
$997$	15.2851	0.484083	0.242041	+	0.970266i	$0.422183\pi$
$998$	15.0325	0.475844
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6093.2.a.l.1.3		18
3.2	odd	2		2031.2.a.g.1.16	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2031.2.a.g.1.16	✓	18	3.2	odd	2
6093.2.a.l.1.3		18	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6093 = 3^{2} \cdot 677 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6093.a (trivial)

\(f(q)\)	\(=\)	\(q-1.46392 q^{2} +0.143048 q^{4} -0.387918 q^{5} -2.28879 q^{7} +2.71842 q^{8} +0.567880 q^{10} +1.78909 q^{11} +0.983519 q^{13} +3.35060 q^{14} -4.26563 q^{16} +1.60300 q^{17} -4.74512 q^{19} -0.0554910 q^{20} -2.61907 q^{22} +3.45899 q^{23} -4.84952 q^{25} -1.43979 q^{26} -0.327407 q^{28} +9.04599 q^{29} +2.62121 q^{31} +0.807685 q^{32} -2.34665 q^{34} +0.887865 q^{35} -0.148493 q^{37} +6.94645 q^{38} -1.05453 q^{40} -5.04274 q^{41} +6.65939 q^{43} +0.255925 q^{44} -5.06368 q^{46} -7.92475 q^{47} -1.76143 q^{49} +7.09929 q^{50} +0.140690 q^{52} +1.72374 q^{53} -0.694020 q^{55} -6.22190 q^{56} -13.2426 q^{58} +0.0905729 q^{59} +7.66859 q^{61} -3.83723 q^{62} +7.34888 q^{64} -0.381525 q^{65} -9.42400 q^{67} +0.229306 q^{68} -1.29976 q^{70} +12.4803 q^{71} -4.58440 q^{73} +0.217382 q^{74} -0.678780 q^{76} -4.09485 q^{77} -3.84230 q^{79} +1.65472 q^{80} +7.38214 q^{82} +9.85407 q^{83} -0.621833 q^{85} -9.74879 q^{86} +4.86349 q^{88} -5.42797 q^{89} -2.25107 q^{91} +0.494802 q^{92} +11.6012 q^{94} +1.84072 q^{95} +0.0506965 q^{97} +2.57858 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 7 q^{2} + 13 q^{4} + 13 q^{5} - 3 q^{7} + 18 q^{8} - 6 q^{10} + 26 q^{11} - 11 q^{13} + 11 q^{14} - q^{16} + 34 q^{17} - 2 q^{19} + 10 q^{20} + 8 q^{22} + 35 q^{23} - 5 q^{25} + 17 q^{26} - 5 q^{28}+ \cdots - 12 q^{98}+O(q^{100}) \) 18 * q + 7 * q^2 + 13 * q^4 + 13 * q^5 - 3 * q^7 + 18 * q^8 - 6 * q^10 + 26 * q^11 - 11 * q^13 + 11 * q^14 - q^16 + 34 * q^17 - 2 * q^19 + 10 * q^20 + 8 * q^22 + 35 * q^23 - 5 * q^25 + 17 * q^26 - 5 * q^28 + 25 * q^29 - 11 * q^31 + 18 * q^32 - 3 * q^34 + 27 * q^35 - q^37 + 18 * q^38 - 8 * q^40 + 44 * q^41 + 5 * q^43 + 46 * q^44 + 20 * q^46 + 51 * q^47 - 33 * q^49 - 11 * q^50 - 2 * q^52 + 28 * q^53 + q^55 + q^56 + 22 * q^58 + 44 * q^59 - 7 * q^61 - q^62 + 8 * q^64 + 11 * q^65 + 2 * q^67 + 11 * q^68 + 16 * q^70 + 23 * q^71 + q^73 + 9 * q^74 + 3 * q^76 + 23 * q^77 + 9 * q^79 + 15 * q^80 + 19 * q^82 + 45 * q^83 + 9 * q^85 + 4 * q^86 + 51 * q^88 + 90 * q^89 - 24 * q^91 + 57 * q^92 + 24 * q^94 + 9 * q^95 - 17 * q^97 - 12 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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