LMFDB - Embedded newform 4842.2.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4842,2,Mod(1,4842)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4842, 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("4842.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4842 = 2 \cdot 3^{2} \cdot 269 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4842.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:	$38.6635646587$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 15x^{5} + 16x^{4} + 49x^{3} - 53x^{2} - 44x + 48 $ x^7 - x^6 - 15x^5 + 16x^4 + 49x^3 - 53x^2 - 44*x + 48
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 538)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.22542$ of defining polynomial
Character	$\chi$	$=$	4842.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{4} -4.15424 q^{5} +4.87818 q^{7} -1.00000 q^{8} +4.15424 q^{10} +1.62910 q^{11} -2.16682 q^{13} -4.87818 q^{14} +1.00000 q^{16} +6.78448 q^{17} -8.61766 q^{19} -4.15424 q^{20} -1.62910 q^{22} -2.76765 q^{23} +12.2577 q^{25} +2.16682 q^{26} +4.87818 q^{28} -2.10510 q^{29} -3.62414 q^{31} -1.00000 q^{32} -6.78448 q^{34} -20.2651 q^{35} +1.19760 q^{37} +8.61766 q^{38} +4.15424 q^{40} -5.43715 q^{41} -9.63402 q^{43} +1.62910 q^{44} +2.76765 q^{46} +7.07613 q^{47} +16.7967 q^{49} -12.2577 q^{50} -2.16682 q^{52} +3.03890 q^{53} -6.76765 q^{55} -4.87818 q^{56} +2.10510 q^{58} -4.21827 q^{59} +8.74744 q^{61} +3.62414 q^{62} +1.00000 q^{64} +9.00148 q^{65} -1.50761 q^{67} +6.78448 q^{68} +20.2651 q^{70} -11.0270 q^{71} +1.37039 q^{73} -1.19760 q^{74} -8.61766 q^{76} +7.94703 q^{77} -14.9749 q^{79} -4.15424 q^{80} +5.43715 q^{82} +8.08276 q^{83} -28.1843 q^{85} +9.63402 q^{86} -1.62910 q^{88} -3.31299 q^{89} -10.5701 q^{91} -2.76765 q^{92} -7.07613 q^{94} +35.7998 q^{95} +16.8474 q^{97} -16.7967 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 7 q^{2} + 7 q^{4} - 7 q^{5} + 6 q^{7} - 7 q^{8} + 7 q^{10} + 3 q^{11} - 9 q^{13} - 6 q^{14} + 7 q^{16} - 8 q^{17} - 11 q^{19} - 7 q^{20} - 3 q^{22} - 12 q^{23} + 22 q^{25} + 9 q^{26} + 6 q^{28} + 5 q^{29}+ \cdots - 15 q^{98}+O(q^{100}) $ 7 * q - 7 * q^2 + 7 * q^4 - 7 * q^5 + 6 * q^7 - 7 * q^8 + 7 * q^10 + 3 * q^11 - 9 * q^13 - 6 * q^14 + 7 * q^16 - 8 * q^17 - 11 * q^19 - 7 * q^20 - 3 * q^22 - 12 * q^23 + 22 * q^25 + 9 * q^26 + 6 * q^28 + 5 * q^29 + 14 * q^31 - 7 * q^32 + 8 * q^34 + 4 * q^35 + 13 * q^37 + 11 * q^38 + 7 * q^40 - 12 * q^41 - 11 * q^43 + 3 * q^44 + 12 * q^46 - 2 * q^47 + 15 * q^49 - 22 * q^50 - 9 * q^52 - 19 * q^53 - 40 * q^55 - 6 * q^56 - 5 * q^58 + 9 * q^59 - 3 * q^61 - 14 * q^62 + 7 * q^64 + 10 * q^65 - 33 * q^67 - 8 * q^68 - 4 * q^70 - 28 * q^71 - 14 * q^73 - 13 * q^74 - 11 * q^76 - 10 * q^77 + 2 * q^79 - 7 * q^80 + 12 * q^82 + 7 * q^83 - 16 * q^85 + 11 * q^86 - 3 * q^88 - 18 * q^89 - 26 * q^91 - 12 * q^92 + 2 * q^94 + 34 * q^95 - 4 * q^97 - 15 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−4.15424	−1.85783	−0.928916	−	0.370291i	$-0.879258\pi$
$5$	−4.15424	−1.85783	−0.928916	+	0.370291i	$0.879258\pi$
$6$	0	0
$7$	4.87818	1.84378	0.921890	−	0.387452i	$-0.126645\pi$
$7$	4.87818	1.84378	0.921890	+	0.387452i	$0.126645\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	4.15424	1.31369
$11$	1.62910	0.491191	0.245596	−	0.969372i	$-0.421016\pi$
$11$	1.62910	0.491191	0.245596	+	0.969372i	$0.421016\pi$
$12$	0	0
$13$	−2.16682	−0.600968	−0.300484	−	0.953787i	$-0.597148\pi$
$13$	−2.16682	−0.600968	−0.300484	+	0.953787i	$0.597148\pi$
$14$	−4.87818	−1.30375
$15$	0	0
$16$	1.00000	0.250000
$17$	6.78448	1.64548	0.822739	−	0.568420i	$-0.192445\pi$
$17$	6.78448	1.64548	0.822739	+	0.568420i	$0.192445\pi$
$18$	0	0
$19$	−8.61766	−1.97703	−0.988513	−	0.151136i	$-0.951707\pi$
$19$	−8.61766	−1.97703	−0.988513	+	0.151136i	$0.951707\pi$
$20$	−4.15424	−0.928916
$21$	0	0
$22$	−1.62910	−0.347325
$23$	−2.76765	−0.577096	−0.288548	−	0.957465i	$-0.593172\pi$
$23$	−2.76765	−0.577096	−0.288548	+	0.957465i	$0.593172\pi$
$24$	0	0
$25$	12.2577	2.45154
$26$	2.16682	0.424948
$27$	0	0
$28$	4.87818	0.921890
$29$	−2.10510	−0.390907	−0.195454	−	0.980713i	$-0.562618\pi$
$29$	−2.10510	−0.390907	−0.195454	+	0.980713i	$0.562618\pi$
$30$	0	0
$31$	−3.62414	−0.650914	−0.325457	−	0.945557i	$-0.605518\pi$
$31$	−3.62414	−0.650914	−0.325457	+	0.945557i	$0.605518\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−6.78448	−1.16353
$35$	−20.2651	−3.42543
$36$	0	0
$37$	1.19760	0.196885	0.0984424	−	0.995143i	$-0.468614\pi$
$37$	1.19760	0.196885	0.0984424	+	0.995143i	$0.468614\pi$
$38$	8.61766	1.39797
$39$	0	0
$40$	4.15424	0.656843
$41$	−5.43715	−0.849140	−0.424570	−	0.905395i	$-0.639575\pi$
$41$	−5.43715	−0.849140	−0.424570	+	0.905395i	$0.639575\pi$
$42$	0	0
$43$	−9.63402	−1.46917	−0.734587	−	0.678514i	$-0.762624\pi$
$43$	−9.63402	−1.46917	−0.734587	+	0.678514i	$0.762624\pi$
$44$	1.62910	0.245596
$45$	0	0
$46$	2.76765	0.408068
$47$	7.07613	1.03216	0.516080	−	0.856541i	$-0.327391\pi$
$47$	7.07613	1.03216	0.516080	+	0.856541i	$0.327391\pi$
$48$	0	0
$49$	16.7967	2.39952
$50$	−12.2577	−1.73350
$51$	0	0
$52$	−2.16682	−0.300484
$53$	3.03890	0.417425	0.208712	−	0.977977i	$-0.433073\pi$
$53$	3.03890	0.417425	0.208712	+	0.977977i	$0.433073\pi$
$54$	0	0
$55$	−6.76765	−0.912550
$56$	−4.87818	−0.651875
$57$	0	0
$58$	2.10510	0.276413
$59$	−4.21827	−0.549172	−0.274586	−	0.961563i	$-0.588541\pi$
$59$	−4.21827	−0.549172	−0.274586	+	0.961563i	$0.588541\pi$
$60$	0	0
$61$	8.74744	1.11999	0.559997	−	0.828494i	$-0.310802\pi$
$61$	8.74744	1.11999	0.559997	+	0.828494i	$0.310802\pi$
$62$	3.62414	0.460266
$63$	0	0
$64$	1.00000	0.125000
$65$	9.00148	1.11650
$66$	0	0
$67$	−1.50761	−0.184184	−0.0920921	−	0.995750i	$-0.529355\pi$
$67$	−1.50761	−0.184184	−0.0920921	+	0.995750i	$0.529355\pi$
$68$	6.78448	0.822739
$69$	0	0
$70$	20.2651	2.42215
$71$	−11.0270	−1.30866	−0.654331	−	0.756208i	$-0.727050\pi$
$71$	−11.0270	−1.30866	−0.654331	+	0.756208i	$0.727050\pi$
$72$	0	0
$73$	1.37039	0.160391	0.0801957	−	0.996779i	$-0.474445\pi$
$73$	1.37039	0.160391	0.0801957	+	0.996779i	$0.474445\pi$
$74$	−1.19760	−0.139219
$75$	0	0
$76$	−8.61766	−0.988513
$77$	7.94703	0.905648
$78$	0	0
$79$	−14.9749	−1.68480	−0.842401	−	0.538850i	$-0.818859\pi$
$79$	−14.9749	−1.68480	−0.842401	+	0.538850i	$0.818859\pi$
$80$	−4.15424	−0.464458
$81$	0	0
$82$	5.43715	0.600432
$83$	8.08276	0.887198	0.443599	−	0.896225i	$-0.353701\pi$
$83$	8.08276	0.887198	0.443599	+	0.896225i	$0.353701\pi$
$84$	0	0
$85$	−28.1843	−3.05702
$86$	9.63402	1.03886
$87$	0	0
$88$	−1.62910	−0.173662
$89$	−3.31299	−0.351176	−0.175588	−	0.984464i	$-0.556183\pi$
$89$	−3.31299	−0.351176	−0.175588	+	0.984464i	$0.556183\pi$
$90$	0	0
$91$	−10.5701	−1.10805
$92$	−2.76765	−0.288548
$93$	0	0
$94$	−7.07613	−0.729847
$95$	35.7998	3.67298
$96$	0	0
$97$	16.8474	1.71059	0.855296	−	0.518139i	$-0.173375\pi$
$97$	16.8474	1.71059	0.855296	+	0.518139i	$0.173375\pi$
$98$	−16.7967	−1.69672
$99$	0	0
$100$	12.2577	1.22577
$101$	−2.93034	−0.291580	−0.145790	−	0.989316i	$-0.546572\pi$
$101$	−2.93034	−0.291580	−0.145790	+	0.989316i	$0.546572\pi$
$102$	0	0
$103$	1.25819	0.123973	0.0619867	−	0.998077i	$-0.480256\pi$
$103$	1.25819	0.123973	0.0619867	+	0.998077i	$0.480256\pi$
$104$	2.16682	0.212474
$105$	0	0
$106$	−3.03890	−0.295164
$107$	−9.35946	−0.904813	−0.452407	−	0.891812i	$-0.649434\pi$
$107$	−9.35946	−0.904813	−0.452407	+	0.891812i	$0.649434\pi$
$108$	0	0
$109$	11.4192	1.09376	0.546882	−	0.837210i	$-0.315815\pi$
$109$	11.4192	1.09376	0.546882	+	0.837210i	$0.315815\pi$
$110$	6.76765	0.645270
$111$	0	0
$112$	4.87818	0.460945
$113$	−0.316816	−0.0298036	−0.0149018	−	0.999889i	$-0.504744\pi$
$113$	−0.316816	−0.0298036	−0.0149018	+	0.999889i	$0.504744\pi$
$114$	0	0
$115$	11.4975	1.07215
$116$	−2.10510	−0.195454
$117$	0	0
$118$	4.21827	0.388323
$119$	33.0959	3.03390
$120$	0	0
$121$	−8.34604	−0.758731
$122$	−8.74744	−0.791956
$123$	0	0
$124$	−3.62414	−0.325457
$125$	−30.1501	−2.69671
$126$	0	0
$127$	−2.08772	−0.185255	−0.0926276	−	0.995701i	$-0.529527\pi$
$127$	−2.08772	−0.185255	−0.0926276	+	0.995701i	$0.529527\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−9.00148	−0.789482
$131$	5.17397	0.452052	0.226026	−	0.974121i	$-0.427427\pi$
$131$	5.17397	0.452052	0.226026	+	0.974121i	$0.427427\pi$
$132$	0	0
$133$	−42.0385	−3.64520
$134$	1.50761	0.130238
$135$	0	0
$136$	−6.78448	−0.581764
$137$	−6.35949	−0.543328	−0.271664	−	0.962392i	$-0.587574\pi$
$137$	−6.35949	−0.543328	−0.271664	+	0.962392i	$0.587574\pi$
$138$	0	0
$139$	−10.6786	−0.905746	−0.452873	−	0.891575i	$-0.649601\pi$
$139$	−10.6786	−0.905746	−0.452873	+	0.891575i	$0.649601\pi$
$140$	−20.2651	−1.71272
$141$	0	0
$142$	11.0270	0.925364
$143$	−3.52996	−0.295190
$144$	0	0
$145$	8.74508	0.726239
$146$	−1.37039	−0.113414
$147$	0	0
$148$	1.19760	0.0984424
$149$	3.47426	0.284622	0.142311	−	0.989822i	$-0.454547\pi$
$149$	3.47426	0.284622	0.142311	+	0.989822i	$0.454547\pi$
$150$	0	0
$151$	11.2407	0.914752	0.457376	−	0.889273i	$-0.348789\pi$
$151$	11.2407	0.914752	0.457376	+	0.889273i	$0.348789\pi$
$152$	8.61766	0.698984
$153$	0	0
$154$	−7.94703	−0.640390
$155$	15.0555	1.20929
$156$	0	0
$157$	8.59533	0.685982	0.342991	−	0.939339i	$-0.388560\pi$
$157$	8.59533	0.685982	0.342991	+	0.939339i	$0.388560\pi$
$158$	14.9749	1.19134
$159$	0	0
$160$	4.15424	0.328421
$161$	−13.5011	−1.06404
$162$	0	0
$163$	10.1530	0.795241	0.397621	−	0.917550i	$-0.369836\pi$
$163$	10.1530	0.795241	0.397621	+	0.917550i	$0.369836\pi$
$164$	−5.43715	−0.424570
$165$	0	0
$166$	−8.08276	−0.627344
$167$	−4.37572	−0.338604	−0.169302	−	0.985564i	$-0.554151\pi$
$167$	−4.37572	−0.338604	−0.169302	+	0.985564i	$0.554151\pi$
$168$	0	0
$169$	−8.30489	−0.638838
$170$	28.1843	2.16164
$171$	0	0
$172$	−9.63402	−0.734587
$173$	22.7651	1.73080	0.865398	−	0.501085i	$-0.167066\pi$
$173$	22.7651	1.73080	0.865398	+	0.501085i	$0.167066\pi$
$174$	0	0
$175$	59.7952	4.52009
$176$	1.62910	0.122798
$177$	0	0
$178$	3.31299	0.248319
$179$	−12.1634	−0.909139	−0.454569	−	0.890711i	$-0.650207\pi$
$179$	−12.1634	−0.909139	−0.454569	+	0.890711i	$0.650207\pi$
$180$	0	0
$181$	−4.61048	−0.342694	−0.171347	−	0.985211i	$-0.554812\pi$
$181$	−4.61048	−0.342694	−0.171347	+	0.985211i	$0.554812\pi$
$182$	10.5701	0.783511
$183$	0	0
$184$	2.76765	0.204034
$185$	−4.97513	−0.365779
$186$	0	0
$187$	11.0526	0.808244
$188$	7.07613	0.516080
$189$	0	0
$190$	−35.7998	−2.59719
$191$	−3.30553	−0.239180	−0.119590	−	0.992823i	$-0.538158\pi$
$191$	−3.30553	−0.239180	−0.119590	+	0.992823i	$0.538158\pi$
$192$	0	0
$193$	−18.5157	−1.33279	−0.666394	−	0.745600i	$-0.732163\pi$
$193$	−18.5157	−1.33279	−0.666394	+	0.745600i	$0.732163\pi$
$194$	−16.8474	−1.20957
$195$	0	0
$196$	16.7967	1.19976
$197$	−8.78109	−0.625627	−0.312814	−	0.949815i	$-0.601271\pi$
$197$	−8.78109	−0.625627	−0.312814	+	0.949815i	$0.601271\pi$
$198$	0	0
$199$	−12.4370	−0.881634	−0.440817	−	0.897597i	$-0.645311\pi$
$199$	−12.4370	−0.881634	−0.440817	+	0.897597i	$0.645311\pi$
$200$	−12.2577	−0.866749
$201$	0	0
$202$	2.93034	0.206178
$203$	−10.2691	−0.720746
$204$	0	0
$205$	22.5872	1.57756
$206$	−1.25819	−0.0876625
$207$	0	0
$208$	−2.16682	−0.150242
$209$	−14.0390	−0.971098
$210$	0	0
$211$	−10.9290	−0.752381	−0.376191	−	0.926542i	$-0.622766\pi$
$211$	−10.9290	−0.752381	−0.376191	+	0.926542i	$0.622766\pi$
$212$	3.03890	0.208712
$213$	0	0
$214$	9.35946	0.639800
$215$	40.0220	2.72948
$216$	0	0
$217$	−17.6792	−1.20014
$218$	−11.4192	−0.773408
$219$	0	0
$220$	−6.76765	−0.456275
$221$	−14.7007	−0.988879
$222$	0	0
$223$	8.79372	0.588871	0.294436	−	0.955671i	$-0.404868\pi$
$223$	8.79372	0.588871	0.294436	+	0.955671i	$0.404868\pi$
$224$	−4.87818	−0.325937
$225$	0	0
$226$	0.316816	0.0210743
$227$	−6.74871	−0.447928	−0.223964	−	0.974597i	$-0.571900\pi$
$227$	−6.74871	−0.447928	−0.223964	+	0.974597i	$0.571900\pi$
$228$	0	0
$229$	−23.2606	−1.53711	−0.768553	−	0.639787i	$-0.779023\pi$
$229$	−23.2606	−1.53711	−0.768553	+	0.639787i	$0.779023\pi$
$230$	−11.4975	−0.758122
$231$	0	0
$232$	2.10510	0.138206
$233$	−11.0721	−0.725357	−0.362679	−	0.931914i	$-0.618138\pi$
$233$	−11.0721	−0.725357	−0.362679	+	0.931914i	$0.618138\pi$
$234$	0	0
$235$	−29.3959	−1.91758
$236$	−4.21827	−0.274586
$237$	0	0
$238$	−33.0959	−2.14529
$239$	3.99072	0.258138	0.129069	−	0.991636i	$-0.458801\pi$
$239$	3.99072	0.258138	0.129069	+	0.991636i	$0.458801\pi$
$240$	0	0
$241$	−16.5408	−1.06549	−0.532745	−	0.846276i	$-0.678839\pi$
$241$	−16.5408	−1.06549	−0.532745	+	0.846276i	$0.678839\pi$
$242$	8.34604	0.536504
$243$	0	0
$244$	8.74744	0.559997
$245$	−69.7773	−4.45791
$246$	0	0
$247$	18.6729	1.18813
$248$	3.62414	0.230133
$249$	0	0
$250$	30.1501	1.90686
$251$	−24.0533	−1.51823	−0.759117	−	0.650955i	$-0.774369\pi$
$251$	−24.0533	−1.51823	−0.759117	+	0.650955i	$0.774369\pi$
$252$	0	0
$253$	−4.50877	−0.283464
$254$	2.08772	0.130995
$255$	0	0
$256$	1.00000	0.0625000
$257$	13.6822	0.853470	0.426735	−	0.904377i	$-0.359664\pi$
$257$	13.6822	0.853470	0.426735	+	0.904377i	$0.359664\pi$
$258$	0	0
$259$	5.84213	0.363012
$260$	9.00148	0.558248
$261$	0	0
$262$	−5.17397	−0.319649
$263$	−11.3942	−0.702596	−0.351298	−	0.936264i	$-0.614260\pi$
$263$	−11.3942	−0.702596	−0.351298	+	0.936264i	$0.614260\pi$
$264$	0	0
$265$	−12.6243	−0.775505
$266$	42.0385	2.57755
$267$	0	0
$268$	−1.50761	−0.0920921
$269$	1.00000	0.0609711
$269$	1.00000	0.0609711
$270$	0	0
$271$	−24.6167	−1.49536	−0.747679	−	0.664061i	$-0.768832\pi$
$271$	−24.6167	−1.49536	−0.747679	+	0.664061i	$0.768832\pi$
$272$	6.78448	0.411369
$273$	0	0
$274$	6.35949	0.384191
$275$	19.9690	1.20417
$276$	0	0
$277$	4.70762	0.282853	0.141427	−	0.989949i	$-0.454831\pi$
$277$	4.70762	0.282853	0.141427	+	0.989949i	$0.454831\pi$
$278$	10.6786	0.640459
$279$	0	0
$280$	20.2651	1.21107
$281$	−6.51540	−0.388676	−0.194338	−	0.980935i	$-0.562256\pi$
$281$	−6.51540	−0.388676	−0.194338	+	0.980935i	$0.562256\pi$
$282$	0	0
$283$	−8.67417	−0.515626	−0.257813	−	0.966195i	$-0.583002\pi$
$283$	−8.67417	−0.515626	−0.257813	+	0.966195i	$0.583002\pi$
$284$	−11.0270	−0.654331
$285$	0	0
$286$	3.52996	0.208731
$287$	−26.5234	−1.56563
$288$	0	0
$289$	29.0291	1.70760
$290$	−8.74508	−0.513529
$291$	0	0
$292$	1.37039	0.0801957
$293$	−20.3798	−1.19060	−0.595300	−	0.803504i	$-0.702967\pi$
$293$	−20.3798	−1.19060	−0.595300	+	0.803504i	$0.702967\pi$
$294$	0	0
$295$	17.5237	1.02027
$296$	−1.19760	−0.0696093
$297$	0	0
$298$	−3.47426	−0.201258
$299$	5.99701	0.346816
$300$	0	0
$301$	−46.9965	−2.70883
$302$	−11.2407	−0.646827
$303$	0	0
$304$	−8.61766	−0.494256
$305$	−36.3389	−2.08076
$306$	0	0
$307$	−17.5028	−0.998939	−0.499470	−	0.866331i	$-0.666472\pi$
$307$	−17.5028	−0.998939	−0.499470	+	0.866331i	$0.666472\pi$
$308$	7.94703	0.452824
$309$	0	0
$310$	−15.0555	−0.855096
$311$	−22.4715	−1.27424	−0.637120	−	0.770764i	$-0.719875\pi$
$311$	−22.4715	−1.27424	−0.637120	+	0.770764i	$0.719875\pi$
$312$	0	0
$313$	−32.6321	−1.84447	−0.922236	−	0.386626i	$-0.873640\pi$
$313$	−32.6321	−1.84447	−0.922236	+	0.386626i	$0.873640\pi$
$314$	−8.59533	−0.485063
$315$	0	0
$316$	−14.9749	−0.842401
$317$	19.5532	1.09822	0.549110	−	0.835750i	$-0.314967\pi$
$317$	19.5532	1.09822	0.549110	+	0.835750i	$0.314967\pi$
$318$	0	0
$319$	−3.42941	−0.192010
$320$	−4.15424	−0.232229
$321$	0	0
$322$	13.5011	0.752388
$323$	−58.4663	−3.25315
$324$	0	0
$325$	−26.5602	−1.47329
$326$	−10.1530	−0.562321
$327$	0	0
$328$	5.43715	0.300216
$329$	34.5186	1.90307
$330$	0	0
$331$	−20.1680	−1.10853	−0.554266	−	0.832340i	$-0.687001\pi$
$331$	−20.1680	−1.10853	−0.554266	+	0.832340i	$0.687001\pi$
$332$	8.08276	0.443599
$333$	0	0
$334$	4.37572	0.239429
$335$	6.26298	0.342183
$336$	0	0
$337$	29.1409	1.58741	0.793704	−	0.608305i	$-0.208150\pi$
$337$	29.1409	1.58741	0.793704	+	0.608305i	$0.208150\pi$
$338$	8.30489	0.451727
$339$	0	0
$340$	−28.1843	−1.52851
$341$	−5.90407	−0.319723
$342$	0	0
$343$	47.7899	2.58041
$344$	9.63402	0.519432
$345$	0	0
$346$	−22.7651	−1.22386
$347$	−12.2141	−0.655686	−0.327843	−	0.944732i	$-0.606322\pi$
$347$	−12.2141	−0.655686	−0.327843	+	0.944732i	$0.606322\pi$
$348$	0	0
$349$	15.3346	0.820843	0.410421	−	0.911896i	$-0.365382\pi$
$349$	15.3346	0.820843	0.410421	+	0.911896i	$0.365382\pi$
$350$	−59.7952	−3.19619
$351$	0	0
$352$	−1.62910	−0.0868311
$353$	−30.6859	−1.63324	−0.816622	−	0.577173i	$-0.804156\pi$
$353$	−30.6859	−1.63324	−0.816622	+	0.577173i	$0.804156\pi$
$354$	0	0
$355$	45.8087	2.43127
$356$	−3.31299	−0.175588
$357$	0	0
$358$	12.1634	0.642858
$359$	−0.344824	−0.0181991	−0.00909956	−	0.999959i	$-0.502897\pi$
$359$	−0.344824	−0.0181991	−0.00909956	+	0.999959i	$0.502897\pi$
$360$	0	0
$361$	55.2640	2.90863
$362$	4.61048	0.242322
$363$	0	0
$364$	−10.5701	−0.554026
$365$	−5.69291	−0.297980
$366$	0	0
$367$	22.3072	1.16443	0.582214	−	0.813035i	$-0.302186\pi$
$367$	22.3072	1.16443	0.582214	+	0.813035i	$0.302186\pi$
$368$	−2.76765	−0.144274
$369$	0	0
$370$	4.97513	0.258645
$371$	14.8243	0.769640
$372$	0	0
$373$	−5.92843	−0.306962	−0.153481	−	0.988152i	$-0.549048\pi$
$373$	−5.92843	−0.306962	−0.153481	+	0.988152i	$0.549048\pi$
$374$	−11.0526	−0.571515
$375$	0	0
$376$	−7.07613	−0.364923
$377$	4.56137	0.234922
$378$	0	0
$379$	−2.01119	−0.103308	−0.0516540	−	0.998665i	$-0.516449\pi$
$379$	−2.01119	−0.103308	−0.0516540	+	0.998665i	$0.516449\pi$
$380$	35.7998	1.83649
$381$	0	0
$382$	3.30553	0.169126
$383$	−4.21696	−0.215477	−0.107738	−	0.994179i	$-0.534361\pi$
$383$	−4.21696	−0.215477	−0.107738	+	0.994179i	$0.534361\pi$
$384$	0	0
$385$	−33.0138	−1.68254
$386$	18.5157	0.942423
$387$	0	0
$388$	16.8474	0.855296
$389$	−4.54939	−0.230663	−0.115332	−	0.993327i	$-0.536793\pi$
$389$	−4.54939	−0.230663	−0.115332	+	0.993327i	$0.536793\pi$
$390$	0	0
$391$	−18.7771	−0.949598
$392$	−16.7967	−0.848360
$393$	0	0
$394$	8.78109	0.442385
$395$	62.2091	3.13008
$396$	0	0
$397$	−15.2162	−0.763677	−0.381839	−	0.924229i	$-0.624709\pi$
$397$	−15.2162	−0.763677	−0.381839	+	0.924229i	$0.624709\pi$
$398$	12.4370	0.623409
$399$	0	0
$400$	12.2577	0.612884
$401$	−18.4280	−0.920250	−0.460125	−	0.887854i	$-0.652195\pi$
$401$	−18.4280	−0.920250	−0.460125	+	0.887854i	$0.652195\pi$
$402$	0	0
$403$	7.85285	0.391178
$404$	−2.93034	−0.145790
$405$	0	0
$406$	10.2691	0.509645
$407$	1.95101	0.0967081
$408$	0	0
$409$	36.3293	1.79637	0.898184	−	0.439619i	$-0.144887\pi$
$409$	36.3293	1.79637	0.898184	+	0.439619i	$0.144887\pi$
$410$	−22.5872	−1.11550
$411$	0	0
$412$	1.25819	0.0619867
$413$	−20.5775	−1.01255
$414$	0	0
$415$	−33.5777	−1.64826
$416$	2.16682	0.106237
$417$	0	0
$418$	14.0390	0.686670
$419$	6.63242	0.324015	0.162007	−	0.986790i	$-0.448203\pi$
$419$	6.63242	0.324015	0.162007	+	0.986790i	$0.448203\pi$
$420$	0	0
$421$	−33.1090	−1.61363	−0.806817	−	0.590801i	$-0.798812\pi$
$421$	−33.1090	−1.61363	−0.806817	+	0.590801i	$0.798812\pi$
$422$	10.9290	0.532014
$423$	0	0
$424$	−3.03890	−0.147582
$425$	83.1620	4.03395
$426$	0	0
$427$	42.6716	2.06502
$428$	−9.35946	−0.452407
$429$	0	0
$430$	−40.0220	−1.93003
$431$	16.6498	0.801995	0.400997	−	0.916079i	$-0.368664\pi$
$431$	16.6498	0.801995	0.400997	+	0.916079i	$0.368664\pi$
$432$	0	0
$433$	5.36131	0.257648	0.128824	−	0.991667i	$-0.458880\pi$
$433$	5.36131	0.257648	0.128824	+	0.991667i	$0.458880\pi$
$434$	17.6792	0.848629
$435$	0	0
$436$	11.4192	0.546882
$437$	23.8507	1.14093
$438$	0	0
$439$	−5.42961	−0.259141	−0.129571	−	0.991570i	$-0.541360\pi$
$439$	−5.42961	−0.259141	−0.129571	+	0.991570i	$0.541360\pi$
$440$	6.76765	0.322635
$441$	0	0
$442$	14.7007	0.699243
$443$	−3.66946	−0.174341	−0.0871706	−	0.996193i	$-0.527783\pi$
$443$	−3.66946	−0.174341	−0.0871706	+	0.996193i	$0.527783\pi$
$444$	0	0
$445$	13.7629	0.652426
$446$	−8.79372	−0.416395
$447$	0	0
$448$	4.87818	0.230472
$449$	−6.43513	−0.303693	−0.151846	−	0.988404i	$-0.548522\pi$
$449$	−6.43513	−0.303693	−0.151846	+	0.988404i	$0.548522\pi$
$450$	0	0
$451$	−8.85764	−0.417090
$452$	−0.316816	−0.0149018
$453$	0	0
$454$	6.74871	0.316733
$455$	43.9109	2.05857
$456$	0	0
$457$	2.97606	0.139214	0.0696071	−	0.997574i	$-0.477825\pi$
$457$	2.97606	0.139214	0.0696071	+	0.997574i	$0.477825\pi$
$458$	23.2606	1.08690
$459$	0	0
$460$	11.4975	0.536073
$461$	−9.73958	−0.453617	−0.226809	−	0.973939i	$-0.572829\pi$
$461$	−9.73958	−0.453617	−0.226809	+	0.973939i	$0.572829\pi$
$462$	0	0
$463$	−19.9810	−0.928598	−0.464299	−	0.885679i	$-0.653694\pi$
$463$	−19.9810	−0.928598	−0.464299	+	0.885679i	$0.653694\pi$
$464$	−2.10510	−0.0977268
$465$	0	0
$466$	11.0721	0.512905
$467$	14.3949	0.666115	0.333057	−	0.942907i	$-0.391920\pi$
$467$	14.3949	0.666115	0.333057	+	0.942907i	$0.391920\pi$
$468$	0	0
$469$	−7.35441	−0.339595
$470$	29.3959	1.35593
$471$	0	0
$472$	4.21827	0.194161
$473$	−15.6947	−0.721645
$474$	0	0
$475$	−105.633	−4.84675
$476$	33.0959	1.51695
$477$	0	0
$478$	−3.99072	−0.182531
$479$	22.4679	1.02659	0.513293	−	0.858213i	$-0.328425\pi$
$479$	22.4679	1.02659	0.513293	+	0.858213i	$0.328425\pi$
$480$	0	0
$481$	−2.59499	−0.118321
$482$	16.5408	0.753415
$483$	0	0
$484$	−8.34604	−0.379366
$485$	−69.9880	−3.17799
$486$	0	0
$487$	−15.3289	−0.694618	−0.347309	−	0.937751i	$-0.612904\pi$
$487$	−15.3289	−0.694618	−0.347309	+	0.937751i	$0.612904\pi$
$488$	−8.74744	−0.395978
$489$	0	0
$490$	69.7773	3.15222
$491$	23.8945	1.07834	0.539172	−	0.842196i	$-0.318737\pi$
$491$	23.8945	1.07834	0.539172	+	0.842196i	$0.318737\pi$
$492$	0	0
$493$	−14.2820	−0.643229
$494$	−18.6729	−0.840134
$495$	0	0
$496$	−3.62414	−0.162729
$497$	−53.7917	−2.41289
$498$	0	0
$499$	−15.7627	−0.705635	−0.352818	−	0.935692i	$-0.614776\pi$
$499$	−15.7627	−0.705635	−0.352818	+	0.935692i	$0.614776\pi$
$500$	−30.1501	−1.34836
$501$	0	0
$502$	24.0533	1.07355
$503$	−0.635388	−0.0283305	−0.0141653	−	0.999900i	$-0.504509\pi$
$503$	−0.635388	−0.0283305	−0.0141653	+	0.999900i	$0.504509\pi$
$504$	0	0
$505$	12.1733	0.541706
$506$	4.50877	0.200439
$507$	0	0
$508$	−2.08772	−0.0926276
$509$	−1.74035	−0.0771399	−0.0385699	−	0.999256i	$-0.512280\pi$
$509$	−1.74035	−0.0771399	−0.0385699	+	0.999256i	$0.512280\pi$
$510$	0	0
$511$	6.68499	0.295727
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−13.6822	−0.603495
$515$	−5.22683	−0.230322
$516$	0	0
$517$	11.5277	0.506987
$518$	−5.84213	−0.256688
$519$	0	0
$520$	−9.00148	−0.394741
$521$	−10.5683	−0.463004	−0.231502	−	0.972834i	$-0.574364\pi$
$521$	−10.5683	−0.463004	−0.231502	+	0.972834i	$0.574364\pi$
$522$	0	0
$523$	10.5329	0.460572	0.230286	−	0.973123i	$-0.426034\pi$
$523$	10.5329	0.460572	0.230286	+	0.973123i	$0.426034\pi$
$524$	5.17397	0.226026
$525$	0	0
$526$	11.3942	0.496810
$527$	−24.5879	−1.07106
$528$	0	0
$529$	−15.3401	−0.666961
$530$	12.6243	0.548365
$531$	0	0
$532$	−42.0385	−1.82260
$533$	11.7813	0.510305
$534$	0	0
$535$	38.8814	1.68099
$536$	1.50761	0.0651189
$537$	0	0
$538$	−1.00000	−0.0431131
$539$	27.3634	1.17862
$540$	0	0
$541$	−0.0839881	−0.00361093	−0.00180546	−	0.999998i	$-0.500575\pi$
$541$	−0.0839881	−0.00361093	−0.00180546	+	0.999998i	$0.500575\pi$
$542$	24.6167	1.05738
$543$	0	0
$544$	−6.78448	−0.290882
$545$	−47.4382	−2.03203
$546$	0	0
$547$	1.38614	0.0592669	0.0296334	−	0.999561i	$-0.490566\pi$
$547$	1.38614	0.0592669	0.0296334	+	0.999561i	$0.490566\pi$
$548$	−6.35949	−0.271664
$549$	0	0
$550$	−19.9690	−0.851479
$551$	18.1410	0.772833
$552$	0	0
$553$	−73.0501	−3.10641
$554$	−4.70762	−0.200008
$555$	0	0
$556$	−10.6786	−0.452873
$557$	−17.1840	−0.728111	−0.364056	−	0.931377i	$-0.618608\pi$
$557$	−17.1840	−0.728111	−0.364056	+	0.931377i	$0.618608\pi$
$558$	0	0
$559$	20.8752	0.882926
$560$	−20.2651	−0.856358
$561$	0	0
$562$	6.51540	0.274836
$563$	12.1968	0.514033	0.257017	−	0.966407i	$-0.417260\pi$
$563$	12.1968	0.514033	0.257017	+	0.966407i	$0.417260\pi$
$564$	0	0
$565$	1.31613	0.0553700
$566$	8.67417	0.364602
$567$	0	0
$568$	11.0270	0.462682
$569$	−23.4816	−0.984401	−0.492201	−	0.870482i	$-0.663807\pi$
$569$	−23.4816	−0.984401	−0.492201	+	0.870482i	$0.663807\pi$
$570$	0	0
$571$	−2.61940	−0.109619	−0.0548093	−	0.998497i	$-0.517455\pi$
$571$	−2.61940	−0.109619	−0.0548093	+	0.998497i	$0.517455\pi$
$572$	−3.52996	−0.147595
$573$	0	0
$574$	26.5234	1.10707
$575$	−33.9250	−1.41477
$576$	0	0
$577$	12.1035	0.503874	0.251937	−	0.967744i	$-0.418932\pi$
$577$	12.1035	0.503874	0.251937	+	0.967744i	$0.418932\pi$
$578$	−29.0291	−1.20745
$579$	0	0
$580$	8.74508	0.363120
$581$	39.4292	1.63580
$582$	0	0
$583$	4.95066	0.205035
$584$	−1.37039	−0.0567070
$585$	0	0
$586$	20.3798	0.841881
$587$	3.09578	0.127776	0.0638882	−	0.997957i	$-0.479650\pi$
$587$	3.09578	0.127776	0.0638882	+	0.997957i	$0.479650\pi$
$588$	0	0
$589$	31.2316	1.28687
$590$	−17.5237	−0.721439
$591$	0	0
$592$	1.19760	0.0492212
$593$	24.1730	0.992666	0.496333	−	0.868132i	$-0.334679\pi$
$593$	24.1730	0.992666	0.496333	+	0.868132i	$0.334679\pi$
$594$	0	0
$595$	−137.488	−5.63647
$596$	3.47426	0.142311
$597$	0	0
$598$	−5.99701	−0.245236
$599$	−39.0984	−1.59752	−0.798759	−	0.601651i	$-0.794510\pi$
$599$	−39.0984	−1.59752	−0.798759	+	0.601651i	$0.794510\pi$
$600$	0	0
$601$	−29.5069	−1.20361	−0.601807	−	0.798642i	$-0.705552\pi$
$601$	−29.5069	−1.20361	−0.601807	+	0.798642i	$0.705552\pi$
$602$	46.9965	1.91543
$603$	0	0
$604$	11.2407	0.457376
$605$	34.6714	1.40959
$606$	0	0
$607$	17.2674	0.700861	0.350431	−	0.936589i	$-0.386035\pi$
$607$	17.2674	0.700861	0.350431	+	0.936589i	$0.386035\pi$
$608$	8.61766	0.349492
$609$	0	0
$610$	36.3389	1.47132
$611$	−15.3327	−0.620294
$612$	0	0
$613$	−7.94254	−0.320796	−0.160398	−	0.987052i	$-0.551278\pi$
$613$	−7.94254	−0.320796	−0.160398	+	0.987052i	$0.551278\pi$
$614$	17.5028	0.706357
$615$	0	0
$616$	−7.94703	−0.320195
$617$	−18.0886	−0.728218	−0.364109	−	0.931356i	$-0.618626\pi$
$617$	−18.0886	−0.728218	−0.364109	+	0.931356i	$0.618626\pi$
$618$	0	0
$619$	−33.4949	−1.34627	−0.673136	−	0.739518i	$-0.735053\pi$
$619$	−33.4949	−1.34627	−0.673136	+	0.739518i	$0.735053\pi$
$620$	15.0555	0.604644
$621$	0	0
$622$	22.4715	0.901024
$623$	−16.1614	−0.647491
$624$	0	0
$625$	63.9624	2.55850
$626$	32.6321	1.30424
$627$	0	0
$628$	8.59533	0.342991
$629$	8.12512	0.323970
$630$	0	0
$631$	37.3162	1.48554	0.742768	−	0.669549i	$-0.233512\pi$
$631$	37.3162	1.48554	0.742768	+	0.669549i	$0.233512\pi$
$632$	14.9749	0.595668
$633$	0	0
$634$	−19.5532	−0.776559
$635$	8.67288	0.344173
$636$	0	0
$637$	−36.3954	−1.44204
$638$	3.42941	0.135772
$639$	0	0
$640$	4.15424	0.164211
$641$	18.0393	0.712509	0.356254	−	0.934389i	$-0.384054\pi$
$641$	18.0393	0.712509	0.356254	+	0.934389i	$0.384054\pi$
$642$	0	0
$643$	−33.0496	−1.30335	−0.651675	−	0.758498i	$-0.725933\pi$
$643$	−33.0496	−1.30335	−0.651675	+	0.758498i	$0.725933\pi$
$644$	−13.5011	−0.532019
$645$	0	0
$646$	58.4663	2.30033
$647$	40.3067	1.58462	0.792310	−	0.610118i	$-0.208878\pi$
$647$	40.3067	1.58462	0.792310	+	0.610118i	$0.208878\pi$
$648$	0	0
$649$	−6.87196	−0.269748
$650$	26.5602	1.04178
$651$	0	0
$652$	10.1530	0.397621
$653$	21.3784	0.836601	0.418300	−	0.908309i	$-0.362626\pi$
$653$	21.3784	0.836601	0.418300	+	0.908309i	$0.362626\pi$
$654$	0	0
$655$	−21.4939	−0.839836
$656$	−5.43715	−0.212285
$657$	0	0
$658$	−34.5186	−1.34568
$659$	−23.1777	−0.902875	−0.451438	−	0.892303i	$-0.649089\pi$
$659$	−23.1777	−0.902875	−0.451438	+	0.892303i	$0.649089\pi$
$660$	0	0
$661$	−3.96791	−0.154334	−0.0771669	−	0.997018i	$-0.524587\pi$
$661$	−3.96791	−0.154334	−0.0771669	+	0.997018i	$0.524587\pi$
$662$	20.1680	0.783850
$663$	0	0
$664$	−8.08276	−0.313672
$665$	174.638	6.77217
$666$	0	0
$667$	5.82618	0.225591
$668$	−4.37572	−0.169302
$669$	0	0
$670$	−6.26298	−0.241960
$671$	14.2504	0.550131
$672$	0	0
$673$	32.9589	1.27047	0.635235	−	0.772319i	$-0.280903\pi$
$673$	32.9589	1.27047	0.635235	+	0.772319i	$0.280903\pi$
$674$	−29.1409	−1.12247
$675$	0	0
$676$	−8.30489	−0.319419
$677$	−17.1159	−0.657816	−0.328908	−	0.944362i	$-0.606681\pi$
$677$	−17.1159	−0.657816	−0.328908	+	0.944362i	$0.606681\pi$
$678$	0	0
$679$	82.1846	3.15396
$680$	28.1843	1.08082
$681$	0	0
$682$	5.90407	0.226078
$683$	−33.2813	−1.27347	−0.636736	−	0.771082i	$-0.719716\pi$
$683$	−33.2813	−1.27347	−0.636736	+	0.771082i	$0.719716\pi$
$684$	0	0
$685$	26.4188	1.00941
$686$	−47.7899	−1.82463
$687$	0	0
$688$	−9.63402	−0.367294
$689$	−6.58475	−0.250859
$690$	0	0
$691$	−21.3236	−0.811189	−0.405594	−	0.914053i	$-0.632935\pi$
$691$	−21.3236	−0.811189	−0.405594	+	0.914053i	$0.632935\pi$
$692$	22.7651	0.865398
$693$	0	0
$694$	12.2141	0.463640
$695$	44.3614	1.68272
$696$	0	0
$697$	−36.8882	−1.39724
$698$	−15.3346	−0.580423
$699$	0	0
$700$	59.7952	2.26005
$701$	28.5333	1.07769	0.538844	−	0.842406i	$-0.318861\pi$
$701$	28.5333	1.07769	0.538844	+	0.842406i	$0.318861\pi$
$702$	0	0
$703$	−10.3205	−0.389246
$704$	1.62910	0.0613989
$705$	0	0
$706$	30.6859	1.15488
$707$	−14.2947	−0.537609
$708$	0	0
$709$	4.38294	0.164605	0.0823024	−	0.996607i	$-0.473773\pi$
$709$	4.38294	0.164605	0.0823024	+	0.996607i	$0.473773\pi$
$710$	−45.8087	−1.71917
$711$	0	0
$712$	3.31299	0.124159
$713$	10.0304	0.375640
$714$	0	0
$715$	14.6643	0.548413
$716$	−12.1634	−0.454569
$717$	0	0
$718$	0.344824	0.0128687
$719$	36.1805	1.34930	0.674652	−	0.738136i	$-0.264294\pi$
$719$	36.1805	1.34930	0.674652	+	0.738136i	$0.264294\pi$
$720$	0	0
$721$	6.13770	0.228580
$722$	−55.2640	−2.05671
$723$	0	0
$724$	−4.61048	−0.171347
$725$	−25.8036	−0.958323
$726$	0	0
$727$	−23.0348	−0.854312	−0.427156	−	0.904178i	$-0.640485\pi$
$727$	−23.0348	−0.854312	−0.427156	+	0.904178i	$0.640485\pi$
$728$	10.5701	0.391756
$729$	0	0
$730$	5.69291	0.210704
$731$	−65.3618	−2.41749
$732$	0	0
$733$	−41.3032	−1.52557	−0.762785	−	0.646652i	$-0.776169\pi$
$733$	−41.3032	−1.52557	−0.762785	+	0.646652i	$0.776169\pi$
$734$	−22.3072	−0.823375
$735$	0	0
$736$	2.76765	0.102017
$737$	−2.45605	−0.0904696
$738$	0	0
$739$	−15.2014	−0.559191	−0.279596	−	0.960118i	$-0.590200\pi$
$739$	−15.2014	−0.559191	−0.279596	+	0.960118i	$0.590200\pi$
$740$	−4.97513	−0.182889
$741$	0	0
$742$	−14.8243	−0.544218
$743$	−51.4130	−1.88616	−0.943079	−	0.332568i	$-0.892085\pi$
$743$	−51.4130	−1.88616	−0.943079	+	0.332568i	$0.892085\pi$
$744$	0	0
$745$	−14.4329	−0.528780
$746$	5.92843	0.217055
$747$	0	0
$748$	11.0526	0.404122
$749$	−45.6572	−1.66828
$750$	0	0
$751$	37.2197	1.35816	0.679082	−	0.734062i	$-0.262378\pi$
$751$	37.2197	1.35816	0.679082	+	0.734062i	$0.262378\pi$
$752$	7.07613	0.258040
$753$	0	0
$754$	−4.56137	−0.166115
$755$	−46.6964	−1.69945
$756$	0	0
$757$	51.7191	1.87976	0.939882	−	0.341499i	$-0.110935\pi$
$757$	51.7191	1.87976	0.939882	+	0.341499i	$0.110935\pi$
$758$	2.01119	0.0730499
$759$	0	0
$760$	−35.7998	−1.29859
$761$	−16.8448	−0.610623	−0.305312	−	0.952253i	$-0.598761\pi$
$761$	−16.8448	−0.610623	−0.305312	+	0.952253i	$0.598761\pi$
$762$	0	0
$763$	55.7051	2.01666
$764$	−3.30553	−0.119590
$765$	0	0
$766$	4.21696	0.152365
$767$	9.14022	0.330034
$768$	0	0
$769$	10.2220	0.368614	0.184307	−	0.982869i	$-0.440996\pi$
$769$	10.2220	0.368614	0.184307	+	0.982869i	$0.440996\pi$
$770$	33.0138	1.18974
$771$	0	0
$772$	−18.5157	−0.666394
$773$	31.6879	1.13973	0.569867	−	0.821737i	$-0.306995\pi$
$773$	31.6879	1.13973	0.569867	+	0.821737i	$0.306995\pi$
$774$	0	0
$775$	−44.4235	−1.59574
$776$	−16.8474	−0.604786
$777$	0	0
$778$	4.54939	0.163103
$779$	46.8555	1.67877
$780$	0	0
$781$	−17.9640	−0.642803
$782$	18.7771	0.671467
$783$	0	0
$784$	16.7967	0.599881
$785$	−35.7070	−1.27444
$786$	0	0
$787$	21.1252	0.753033	0.376516	−	0.926410i	$-0.377122\pi$
$787$	21.1252	0.753033	0.376516	+	0.926410i	$0.377122\pi$
$788$	−8.78109	−0.312814
$789$	0	0
$790$	−62.2091	−2.21330
$791$	−1.54549	−0.0549512
$792$	0	0
$793$	−18.9541	−0.673080
$794$	15.2162	0.540001
$795$	0	0
$796$	−12.4370	−0.440817
$797$	4.27630	0.151474	0.0757371	−	0.997128i	$-0.475869\pi$
$797$	4.27630	0.151474	0.0757371	+	0.997128i	$0.475869\pi$
$798$	0	0
$799$	48.0078	1.69839
$800$	−12.2577	−0.433375
$801$	0	0
$802$	18.4280	0.650715
$803$	2.23249	0.0787829
$804$	0	0
$805$	56.0868	1.97680
$806$	−7.85285	−0.276605
$807$	0	0
$808$	2.93034	0.103089
$809$	−4.78386	−0.168192	−0.0840958	−	0.996458i	$-0.526800\pi$
$809$	−4.78386	−0.168192	−0.0840958	+	0.996458i	$0.526800\pi$
$810$	0	0
$811$	−34.4127	−1.20839	−0.604197	−	0.796835i	$-0.706506\pi$
$811$	−34.4127	−1.20839	−0.604197	+	0.796835i	$0.706506\pi$
$812$	−10.2691	−0.360373
$813$	0	0
$814$	−1.95101	−0.0683829
$815$	−42.1778	−1.47742
$816$	0	0
$817$	83.0227	2.90460
$818$	−36.3293	−1.27022
$819$	0	0
$820$	22.5872	0.788779
$821$	38.7973	1.35404	0.677018	−	0.735967i	$-0.263272\pi$
$821$	38.7973	1.35404	0.677018	+	0.735967i	$0.263272\pi$
$822$	0	0
$823$	−8.52567	−0.297186	−0.148593	−	0.988898i	$-0.547474\pi$
$823$	−8.52567	−0.297186	−0.148593	+	0.988898i	$0.547474\pi$
$824$	−1.25819	−0.0438312
$825$	0	0
$826$	20.5775	0.715982
$827$	−3.84932	−0.133854	−0.0669271	−	0.997758i	$-0.521319\pi$
$827$	−3.84932	−0.133854	−0.0669271	+	0.997758i	$0.521319\pi$
$828$	0	0
$829$	1.76719	0.0613771	0.0306886	−	0.999529i	$-0.490230\pi$
$829$	1.76719	0.0613771	0.0306886	+	0.999529i	$0.490230\pi$
$830$	33.5777	1.16550
$831$	0	0
$832$	−2.16682	−0.0751210
$833$	113.957	3.94836
$834$	0	0
$835$	18.1778	0.629068
$836$	−14.0390	−0.485549
$837$	0	0
$838$	−6.63242	−0.229113
$839$	−48.4527	−1.67277	−0.836386	−	0.548141i	$-0.815336\pi$
$839$	−48.4527	−1.67277	−0.836386	+	0.548141i	$0.815336\pi$
$840$	0	0
$841$	−24.5686	−0.847192
$842$	33.1090	1.14101
$843$	0	0
$844$	−10.9290	−0.376191
$845$	34.5005	1.18685
$846$	0	0
$847$	−40.7135	−1.39893
$848$	3.03890	0.104356
$849$	0	0
$850$	−83.1620	−2.85243
$851$	−3.31455	−0.113621
$852$	0	0
$853$	35.2024	1.20531	0.602654	−	0.798002i	$-0.294110\pi$
$853$	35.2024	1.20531	0.602654	+	0.798002i	$0.294110\pi$
$854$	−42.6716	−1.46019
$855$	0	0
$856$	9.35946	0.319900
$857$	−8.30685	−0.283757	−0.141878	−	0.989884i	$-0.545314\pi$
$857$	−8.30685	−0.283757	−0.141878	+	0.989884i	$0.545314\pi$
$858$	0	0
$859$	48.5886	1.65782	0.828911	−	0.559381i	$-0.188961\pi$
$859$	48.5886	1.65782	0.828911	+	0.559381i	$0.188961\pi$
$860$	40.0220	1.36474
$861$	0	0
$862$	−16.6498	−0.567096
$863$	−50.0516	−1.70378	−0.851888	−	0.523723i	$-0.824543\pi$
$863$	−50.0516	−1.70378	−0.851888	+	0.523723i	$0.824543\pi$
$864$	0	0
$865$	−94.5715	−3.21553
$866$	−5.36131	−0.182185
$867$	0	0
$868$	−17.6792	−0.600071
$869$	−24.3955	−0.827560
$870$	0	0
$871$	3.26672	0.110689
$872$	−11.4192	−0.386704
$873$	0	0
$874$	−23.8507	−0.806761
$875$	−147.078	−4.97214
$876$	0	0
$877$	−35.1921	−1.18835	−0.594177	−	0.804334i	$-0.702522\pi$
$877$	−35.1921	−1.18835	−0.594177	+	0.804334i	$0.702522\pi$
$878$	5.42961	0.183240
$879$	0	0
$880$	−6.76765	−0.228138
$881$	29.4895	0.993527	0.496764	−	0.867886i	$-0.334522\pi$
$881$	29.4895	0.993527	0.496764	+	0.867886i	$0.334522\pi$
$882$	0	0
$883$	−28.1353	−0.946829	−0.473415	−	0.880840i	$-0.656979\pi$
$883$	−28.1353	−0.946829	−0.473415	+	0.880840i	$0.656979\pi$
$884$	−14.7007	−0.494439
$885$	0	0
$886$	3.66946	0.123278
$887$	12.3917	0.416072	0.208036	−	0.978121i	$-0.433293\pi$
$887$	12.3917	0.416072	0.208036	+	0.978121i	$0.433293\pi$
$888$	0	0
$889$	−10.1843	−0.341570
$890$	−13.7629	−0.461335
$891$	0	0
$892$	8.79372	0.294436
$893$	−60.9796	−2.04061
$894$	0	0
$895$	50.5298	1.68903
$896$	−4.87818	−0.162969
$897$	0	0
$898$	6.43513	0.214743
$899$	7.62916	0.254447
$900$	0	0
$901$	20.6173	0.686863
$902$	8.85764	0.294927
$903$	0	0
$904$	0.316816	0.0105372
$905$	19.1530	0.636669
$906$	0	0
$907$	−14.9432	−0.496181	−0.248090	−	0.968737i	$-0.579803\pi$
$907$	−14.9432	−0.496181	−0.248090	+	0.968737i	$0.579803\pi$
$908$	−6.74871	−0.223964
$909$	0	0
$910$	−43.9109	−1.45563
$911$	−29.3458	−0.972269	−0.486134	−	0.873884i	$-0.661593\pi$
$911$	−29.3458	−0.972269	−0.486134	+	0.873884i	$0.661593\pi$
$912$	0	0
$913$	13.1676	0.435784
$914$	−2.97606	−0.0984393
$915$	0	0
$916$	−23.2606	−0.768553
$917$	25.2396	0.833485
$918$	0	0
$919$	24.7551	0.816595	0.408297	−	0.912849i	$-0.366123\pi$
$919$	24.7551	0.816595	0.408297	+	0.912849i	$0.366123\pi$
$920$	−11.4975	−0.379061
$921$	0	0
$922$	9.73958	0.320756
$923$	23.8935	0.786464
$924$	0	0
$925$	14.6799	0.482671
$926$	19.9810	0.656618
$927$	0	0
$928$	2.10510	0.0691032
$929$	27.1303	0.890117	0.445059	−	0.895501i	$-0.353183\pi$
$929$	27.1303	0.890117	0.445059	+	0.895501i	$0.353183\pi$
$930$	0	0
$931$	−144.748	−4.74392
$932$	−11.0721	−0.362679
$933$	0	0
$934$	−14.3949	−0.471014
$935$	−45.9150	−1.50158
$936$	0	0
$937$	44.2931	1.44699	0.723496	−	0.690328i	$-0.242534\pi$
$937$	44.2931	1.44699	0.723496	+	0.690328i	$0.242534\pi$
$938$	7.35441	0.240130
$939$	0	0
$940$	−29.3959	−0.958789
$941$	27.0894	0.883090	0.441545	−	0.897239i	$-0.354431\pi$
$941$	27.0894	0.883090	0.441545	+	0.897239i	$0.354431\pi$
$942$	0	0
$943$	15.0481	0.490035
$944$	−4.21827	−0.137293
$945$	0	0
$946$	15.6947	0.510280
$947$	−44.9196	−1.45969	−0.729845	−	0.683613i	$-0.760408\pi$
$947$	−44.9196	−1.45969	−0.729845	+	0.683613i	$0.760408\pi$
$948$	0	0
$949$	−2.96938	−0.0963901
$950$	105.633	3.42717
$951$	0	0
$952$	−33.0959	−1.07264
$953$	21.0858	0.683036	0.341518	−	0.939875i	$-0.389059\pi$
$953$	21.0858	0.683036	0.341518	+	0.939875i	$0.389059\pi$
$954$	0	0
$955$	13.7320	0.444356
$956$	3.99072	0.129069
$957$	0	0
$958$	−22.4679	−0.725906
$959$	−31.0227	−1.00178
$960$	0	0
$961$	−17.8656	−0.576311
$962$	2.59499	0.0836659
$963$	0	0
$964$	−16.5408	−0.532745
$965$	76.9185	2.47609
$966$	0	0
$967$	25.2415	0.811711	0.405855	−	0.913937i	$-0.366974\pi$
$967$	25.2415	0.811711	0.405855	+	0.913937i	$0.366974\pi$
$968$	8.34604	0.268252
$969$	0	0
$970$	69.9880	2.24718
$971$	34.5382	1.10838	0.554192	−	0.832389i	$-0.313027\pi$
$971$	34.5382	1.10838	0.554192	+	0.832389i	$0.313027\pi$
$972$	0	0
$973$	−52.0921	−1.67000
$974$	15.3289	0.491169
$975$	0	0
$976$	8.74744	0.279999
$977$	−61.0213	−1.95224	−0.976122	−	0.217225i	$-0.930300\pi$
$977$	−61.0213	−1.95224	−0.976122	+	0.217225i	$0.930300\pi$
$978$	0	0
$979$	−5.39718	−0.172495
$980$	−69.7773	−2.22896
$981$	0	0
$982$	−23.8945	−0.762504
$983$	47.5260	1.51585	0.757923	−	0.652345i	$-0.226214\pi$
$983$	47.5260	1.51585	0.757923	+	0.652345i	$0.226214\pi$
$984$	0	0
$985$	36.4787	1.16231
$986$	14.2820	0.454831
$987$	0	0
$988$	18.6729	0.594064
$989$	26.6636	0.847854
$990$	0	0
$991$	−35.6855	−1.13359	−0.566794	−	0.823860i	$-0.691816\pi$
$991$	−35.6855	−1.13359	−0.566794	+	0.823860i	$0.691816\pi$
$992$	3.62414	0.115066
$993$	0	0
$994$	53.7917	1.70617
$995$	51.6662	1.63793
$996$	0	0
$997$	12.2019	0.386438	0.193219	−	0.981156i	$-0.438107\pi$
$997$	12.2019	0.386438	0.193219	+	0.981156i	$0.438107\pi$
$998$	15.7627	0.498960
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4842.2.a.n.1.1		7
3.2	odd	2		538.2.a.e.1.3	✓	7
12.11	even	2		4304.2.a.h.1.5		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
538.2.a.e.1.3	✓	7	3.2	odd	2
4304.2.a.h.1.5		7	12.11	even	2
4842.2.a.n.1.1		7	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:	\( N \)	\(=\)	\( 4842 = 2 \cdot 3^{2} \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4842.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -4.15424 q^{5} +4.87818 q^{7} -1.00000 q^{8} +4.15424 q^{10} +1.62910 q^{11} -2.16682 q^{13} -4.87818 q^{14} +1.00000 q^{16} +6.78448 q^{17} -8.61766 q^{19} -4.15424 q^{20} -1.62910 q^{22} -2.76765 q^{23} +12.2577 q^{25} +2.16682 q^{26} +4.87818 q^{28} -2.10510 q^{29} -3.62414 q^{31} -1.00000 q^{32} -6.78448 q^{34} -20.2651 q^{35} +1.19760 q^{37} +8.61766 q^{38} +4.15424 q^{40} -5.43715 q^{41} -9.63402 q^{43} +1.62910 q^{44} +2.76765 q^{46} +7.07613 q^{47} +16.7967 q^{49} -12.2577 q^{50} -2.16682 q^{52} +3.03890 q^{53} -6.76765 q^{55} -4.87818 q^{56} +2.10510 q^{58} -4.21827 q^{59} +8.74744 q^{61} +3.62414 q^{62} +1.00000 q^{64} +9.00148 q^{65} -1.50761 q^{67} +6.78448 q^{68} +20.2651 q^{70} -11.0270 q^{71} +1.37039 q^{73} -1.19760 q^{74} -8.61766 q^{76} +7.94703 q^{77} -14.9749 q^{79} -4.15424 q^{80} +5.43715 q^{82} +8.08276 q^{83} -28.1843 q^{85} +9.63402 q^{86} -1.62910 q^{88} -3.31299 q^{89} -10.5701 q^{91} -2.76765 q^{92} -7.07613 q^{94} +35.7998 q^{95} +16.8474 q^{97} -16.7967 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 7 q^{2} + 7 q^{4} - 7 q^{5} + 6 q^{7} - 7 q^{8} + 7 q^{10} + 3 q^{11} - 9 q^{13} - 6 q^{14} + 7 q^{16} - 8 q^{17} - 11 q^{19} - 7 q^{20} - 3 q^{22} - 12 q^{23} + 22 q^{25} + 9 q^{26} + 6 q^{28} + 5 q^{29}+ \cdots - 15 q^{98}+O(q^{100}) \) 7 * q - 7 * q^2 + 7 * q^4 - 7 * q^5 + 6 * q^7 - 7 * q^8 + 7 * q^10 + 3 * q^11 - 9 * q^13 - 6 * q^14 + 7 * q^16 - 8 * q^17 - 11 * q^19 - 7 * q^20 - 3 * q^22 - 12 * q^23 + 22 * q^25 + 9 * q^26 + 6 * q^28 + 5 * q^29 + 14 * q^31 - 7 * q^32 + 8 * q^34 + 4 * q^35 + 13 * q^37 + 11 * q^38 + 7 * q^40 - 12 * q^41 - 11 * q^43 + 3 * q^44 + 12 * q^46 - 2 * q^47 + 15 * q^49 - 22 * q^50 - 9 * q^52 - 19 * q^53 - 40 * q^55 - 6 * q^56 - 5 * q^58 + 9 * q^59 - 3 * q^61 - 14 * q^62 + 7 * q^64 + 10 * q^65 - 33 * q^67 - 8 * q^68 - 4 * q^70 - 28 * q^71 - 14 * q^73 - 13 * q^74 - 11 * q^76 - 10 * q^77 + 2 * q^79 - 7 * q^80 + 12 * q^82 + 7 * q^83 - 16 * q^85 + 11 * q^86 - 3 * q^88 - 18 * q^89 - 26 * q^91 - 12 * q^92 + 2 * q^94 + 34 * q^95 - 4 * q^97 - 15 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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