LMFDB - Embedded newform 4923.2.a.l.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4923,2,Mod(1,4923)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4923, 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("4923.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4923 = 3^{2} \cdot 547 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4923.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:	$39.3103529151$
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} - 4 x^{17} - 18 x^{16} + 84 x^{15} + 116 x^{14} - 708 x^{13} - 282 x^{12} + 3104 x^{11} + \cdots + 328 $ x^18 - 4x^17 - 18x^16 + 84x^15 + 116x^14 - 708x^13 - 282x^12 + 3104x^11 - 137x^10 - 7703x^9 + 2068x^8 + 11068x^7 - 4274x^6 - 9021x^5 + 4048x^4 + 3834x^3 - 1851x^2 - 654*x + 328
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 547)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$0.763493$ of defining polynomial
Character	$\chi$	$=$	4923.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.763493 q^{2} -1.41708 q^{4} +1.51218 q^{5} +1.20167 q^{7} -2.60892 q^{8} +O(q^{10})$	\(q+0.763493 q^{2} -1.41708 q^{4} +1.51218 q^{5} +1.20167 q^{7} -2.60892 q^{8} +1.15454 q^{10} +5.83960 q^{11} -5.40780 q^{13} +0.917468 q^{14} +0.842269 q^{16} -1.92787 q^{17} +0.965947 q^{19} -2.14288 q^{20} +4.45849 q^{22} +0.470051 q^{23} -2.71331 q^{25} -4.12882 q^{26} -1.70286 q^{28} +3.67331 q^{29} +0.675583 q^{31} +5.86090 q^{32} -1.47191 q^{34} +1.81715 q^{35} +5.66915 q^{37} +0.737494 q^{38} -3.94515 q^{40} +5.00667 q^{41} +3.18500 q^{43} -8.27517 q^{44} +0.358881 q^{46} +10.1509 q^{47} -5.55598 q^{49} -2.07159 q^{50} +7.66328 q^{52} -8.25219 q^{53} +8.83053 q^{55} -3.13506 q^{56} +2.80455 q^{58} +5.23947 q^{59} -3.53188 q^{61} +0.515803 q^{62} +2.79021 q^{64} -8.17758 q^{65} +8.70291 q^{67} +2.73194 q^{68} +1.38738 q^{70} +9.50861 q^{71} +9.68659 q^{73} +4.32836 q^{74} -1.36882 q^{76} +7.01728 q^{77} -5.23633 q^{79} +1.27366 q^{80} +3.82256 q^{82} -14.9853 q^{83} -2.91529 q^{85} +2.43172 q^{86} -15.2350 q^{88} -10.2143 q^{89} -6.49841 q^{91} -0.666099 q^{92} +7.75014 q^{94} +1.46069 q^{95} -6.07529 q^{97} -4.24195 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 4 q^{2} + 16 q^{4} + 27 q^{5} - 11 q^{7} + 12 q^{8}+O(q^{10}) $ 18 * q + 4 * q^2 + 16 * q^4 + 27 * q^5 - 11 * q^7 + 12 * q^8	$ 18 q + 4 q^{2} + 16 q^{4} + 27 q^{5} - 11 q^{7} + 12 q^{8} - 5 q^{10} - 2 q^{11} - 25 q^{13} + 7 q^{14} + 8 q^{16} + 30 q^{17} + 4 q^{19} + 41 q^{20} - 24 q^{22} + 26 q^{23} + 31 q^{25} + 18 q^{26} - 16 q^{28} + 18 q^{29} - 5 q^{31} + 28 q^{32} + 5 q^{34} + 9 q^{35} - 18 q^{37} + 45 q^{38} + 7 q^{40} + 17 q^{41} + 8 q^{43} - 12 q^{44} + 30 q^{46} + 52 q^{47} + 29 q^{49} - 13 q^{50} - 14 q^{52} + 60 q^{53} + 11 q^{55} - 7 q^{56} + 14 q^{58} + 8 q^{59} - 26 q^{61} - 4 q^{62} + 44 q^{64} + 6 q^{65} + 12 q^{67} + 61 q^{68} + 35 q^{70} + q^{71} - 2 q^{73} - 16 q^{74} + 66 q^{76} + 73 q^{77} + 18 q^{79} + 32 q^{80} + 44 q^{82} + 43 q^{83} + 51 q^{85} - 4 q^{86} - 17 q^{88} + 28 q^{89} - q^{91} + 68 q^{92} + 78 q^{94} + 18 q^{95} - 34 q^{97} - 34 q^{98}+O(q^{100}) $ 18 * q + 4 * q^2 + 16 * q^4 + 27 * q^5 - 11 * q^7 + 12 * q^8 - 5 * q^10 - 2 * q^11 - 25 * q^13 + 7 * q^14 + 8 * q^16 + 30 * q^17 + 4 * q^19 + 41 * q^20 - 24 * q^22 + 26 * q^23 + 31 * q^25 + 18 * q^26 - 16 * q^28 + 18 * q^29 - 5 * q^31 + 28 * q^32 + 5 * q^34 + 9 * q^35 - 18 * q^37 + 45 * q^38 + 7 * q^40 + 17 * q^41 + 8 * q^43 - 12 * q^44 + 30 * q^46 + 52 * q^47 + 29 * q^49 - 13 * q^50 - 14 * q^52 + 60 * q^53 + 11 * q^55 - 7 * q^56 + 14 * q^58 + 8 * q^59 - 26 * q^61 - 4 * q^62 + 44 * q^64 + 6 * q^65 + 12 * q^67 + 61 * q^68 + 35 * q^70 + q^71 - 2 * q^73 - 16 * q^74 + 66 * q^76 + 73 * q^77 + 18 * q^79 + 32 * q^80 + 44 * q^82 + 43 * q^83 + 51 * q^85 - 4 * q^86 - 17 * q^88 + 28 * q^89 - q^91 + 68 * q^92 + 78 * q^94 + 18 * q^95 - 34 * q^97 - 34 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.763493	0.539871	0.269935	−	0.962878i	$-0.412998\pi$
$2$	0.763493	0.539871	0.269935	+	0.962878i	$0.412998\pi$
$3$	0	0
$3$	0	0
$4$	−1.41708	−0.708539
$5$	1.51218	0.676268	0.338134	−	0.941098i	$-0.390204\pi$
$5$	1.51218	0.676268	0.338134	+	0.941098i	$0.390204\pi$
$6$	0	0
$7$	1.20167	0.454189	0.227095	−	0.973873i	$-0.427077\pi$
$7$	1.20167	0.454189	0.227095	+	0.973873i	$0.427077\pi$
$8$	−2.60892	−0.922391
$9$	0	0
$10$	1.15454	0.365098
$11$	5.83960	1.76070	0.880352	−	0.474321i	$-0.157306\pi$
$11$	5.83960	1.76070	0.880352	+	0.474321i	$0.157306\pi$
$12$	0	0
$13$	−5.40780	−1.49986	−0.749928	−	0.661520i	$-0.769912\pi$
$13$	−5.40780	−1.49986	−0.749928	+	0.661520i	$0.769912\pi$
$14$	0.917468	0.245204
$15$	0	0
$16$	0.842269	0.210567
$17$	−1.92787	−0.467577	−0.233788	−	0.972288i	$-0.575112\pi$
$17$	−1.92787	−0.467577	−0.233788	+	0.972288i	$0.575112\pi$
$18$	0	0
$19$	0.965947	0.221603	0.110802	−	0.993843i	$-0.464658\pi$
$19$	0.965947	0.221603	0.110802	+	0.993843i	$0.464658\pi$
$20$	−2.14288	−0.479163
$21$	0	0
$22$	4.45849	0.950553
$23$	0.470051	0.0980124	0.0490062	−	0.998798i	$-0.484395\pi$
$23$	0.470051	0.0980124	0.0490062	+	0.998798i	$0.484395\pi$
$24$	0	0
$25$	−2.71331	−0.542661
$26$	−4.12882	−0.809728
$27$	0	0
$28$	−1.70286	−0.321811
$29$	3.67331	0.682117	0.341059	−	0.940042i	$-0.389215\pi$
$29$	3.67331	0.682117	0.341059	+	0.940042i	$0.389215\pi$
$30$	0	0
$31$	0.675583	0.121338	0.0606691	−	0.998158i	$-0.480677\pi$
$31$	0.675583	0.121338	0.0606691	+	0.998158i	$0.480677\pi$
$32$	5.86090	1.03607
$33$	0	0
$34$	−1.47191	−0.252431
$35$	1.81715	0.307154
$36$	0	0
$37$	5.66915	0.932003	0.466002	−	0.884784i	$-0.345694\pi$
$37$	5.66915	0.932003	0.466002	+	0.884784i	$0.345694\pi$
$38$	0.737494	0.119637
$39$	0	0
$40$	−3.94515	−0.623784
$41$	5.00667	0.781911	0.390955	−	0.920410i	$-0.372145\pi$
$41$	5.00667	0.781911	0.390955	+	0.920410i	$0.372145\pi$
$42$	0	0
$43$	3.18500	0.485707	0.242854	−	0.970063i	$-0.421917\pi$
$43$	3.18500	0.485707	0.242854	+	0.970063i	$0.421917\pi$
$44$	−8.27517	−1.24753
$45$	0	0
$46$	0.358881	0.0529141
$47$	10.1509	1.48066	0.740331	−	0.672243i	$-0.234669\pi$
$47$	10.1509	1.48066	0.740331	+	0.672243i	$0.234669\pi$
$48$	0	0
$49$	−5.55598	−0.793712
$50$	−2.07159	−0.292967
$51$	0	0
$52$	7.66328	1.06271
$53$	−8.25219	−1.13353	−0.566763	−	0.823881i	$-0.691804\pi$
$53$	−8.25219	−1.13353	−0.566763	+	0.823881i	$0.691804\pi$
$54$	0	0
$55$	8.83053	1.19071
$56$	−3.13506	−0.418940
$57$	0	0
$58$	2.80455	0.368255
$59$	5.23947	0.682121	0.341060	−	0.940041i	$-0.389214\pi$
$59$	5.23947	0.682121	0.341060	+	0.940041i	$0.389214\pi$
$60$	0	0
$61$	−3.53188	−0.452211	−0.226106	−	0.974103i	$-0.572599\pi$
$61$	−3.53188	−0.452211	−0.226106	+	0.974103i	$0.572599\pi$
$62$	0.515803	0.0655070
$63$	0	0
$64$	2.79021	0.348777
$65$	−8.17758	−1.01430
$66$	0	0
$67$	8.70291	1.06323	0.531615	−	0.846986i	$-0.321585\pi$
$67$	8.70291	1.06323	0.531615	+	0.846986i	$0.321585\pi$
$68$	2.73194	0.331296
$69$	0	0
$70$	1.38738	0.165823
$71$	9.50861	1.12846	0.564232	−	0.825616i	$-0.309172\pi$
$71$	9.50861	1.12846	0.564232	+	0.825616i	$0.309172\pi$
$72$	0	0
$73$	9.68659	1.13373	0.566865	−	0.823811i	$-0.308156\pi$
$73$	9.68659	1.13373	0.566865	+	0.823811i	$0.308156\pi$
$74$	4.32836	0.503161
$75$	0	0
$76$	−1.36882	−0.157015
$77$	7.01728	0.799693
$78$	0	0
$79$	−5.23633	−0.589133	−0.294567	−	0.955631i	$-0.595175\pi$
$79$	−5.23633	−0.589133	−0.294567	+	0.955631i	$0.595175\pi$
$80$	1.27366	0.142400
$81$	0	0
$82$	3.82256	0.422131
$83$	−14.9853	−1.64485	−0.822427	−	0.568871i	$-0.807380\pi$
$83$	−14.9853	−1.64485	−0.822427	+	0.568871i	$0.807380\pi$
$84$	0	0
$85$	−2.91529	−0.316207
$86$	2.43172	0.262219
$87$	0	0
$88$	−15.2350	−1.62406
$89$	−10.2143	−1.08271	−0.541357	−	0.840793i	$-0.682089\pi$
$89$	−10.2143	−1.08271	−0.541357	+	0.840793i	$0.682089\pi$
$90$	0	0
$91$	−6.49841	−0.681218
$92$	−0.666099	−0.0694457
$93$	0	0
$94$	7.75014	0.799366
$95$	1.46069	0.149863
$96$	0	0
$97$	−6.07529	−0.616853	−0.308426	−	0.951248i	$-0.599802\pi$
$97$	−6.07529	−0.616853	−0.308426	+	0.951248i	$0.599802\pi$
$98$	−4.24195	−0.428502
$99$	0	0
$100$	3.84497	0.384497
$101$	2.36723	0.235549	0.117774	−	0.993040i	$-0.462424\pi$
$101$	2.36723	0.235549	0.117774	+	0.993040i	$0.462424\pi$
$102$	0	0
$103$	12.4607	1.22779	0.613895	−	0.789388i	$-0.289602\pi$
$103$	12.4607	1.22779	0.613895	+	0.789388i	$0.289602\pi$
$104$	14.1085	1.38345
$105$	0	0
$106$	−6.30049	−0.611958
$107$	17.8981	1.73027	0.865137	−	0.501536i	$-0.167232\pi$
$107$	17.8981	1.73027	0.865137	+	0.501536i	$0.167232\pi$
$108$	0	0
$109$	−2.89181	−0.276985	−0.138492	−	0.990363i	$-0.544226\pi$
$109$	−2.89181	−0.276985	−0.138492	+	0.990363i	$0.544226\pi$
$110$	6.74205	0.642829
$111$	0	0
$112$	1.01213	0.0956374
$113$	−5.34990	−0.503276	−0.251638	−	0.967821i	$-0.580969\pi$
$113$	−5.34990	−0.503276	−0.251638	+	0.967821i	$0.580969\pi$
$114$	0	0
$115$	0.710803	0.0662827
$116$	−5.20537	−0.483307
$117$	0	0
$118$	4.00030	0.368257
$119$	−2.31666	−0.212368
$120$	0	0
$121$	23.1009	2.10008
$122$	−2.69657	−0.244136
$123$	0	0
$124$	−0.957354	−0.0859730
$125$	−11.6639	−1.04325
$126$	0	0
$127$	14.6006	1.29559	0.647795	−	0.761815i	$-0.275691\pi$
$127$	14.6006	1.29559	0.647795	+	0.761815i	$0.275691\pi$
$128$	−9.59148	−0.847775
$129$	0	0
$130$	−6.24353	−0.547594
$131$	19.8795	1.73688	0.868441	−	0.495792i	$-0.165122\pi$
$131$	19.8795	1.73688	0.868441	+	0.495792i	$0.165122\pi$
$132$	0	0
$133$	1.16075	0.100650
$134$	6.64461	0.574007
$135$	0	0
$136$	5.02964	0.431288
$137$	−19.2841	−1.64755	−0.823777	−	0.566914i	$-0.808137\pi$
$137$	−19.2841	−1.64755	−0.823777	+	0.566914i	$0.808137\pi$
$138$	0	0
$139$	11.2982	0.958297	0.479149	−	0.877734i	$-0.340946\pi$
$139$	11.2982	0.958297	0.479149	+	0.877734i	$0.340946\pi$
$140$	−2.57504	−0.217631
$141$	0	0
$142$	7.25976	0.609225
$143$	−31.5794	−2.64080
$144$	0	0
$145$	5.55472	0.461294
$146$	7.39564	0.612067
$147$	0	0
$148$	−8.03364	−0.660361
$149$	−0.559854	−0.0458650	−0.0229325	−	0.999737i	$-0.507300\pi$
$149$	−0.559854	−0.0458650	−0.0229325	+	0.999737i	$0.507300\pi$
$150$	0	0
$151$	−7.77394	−0.632634	−0.316317	−	0.948653i	$-0.602446\pi$
$151$	−7.77394	−0.632634	−0.316317	+	0.948653i	$0.602446\pi$
$152$	−2.52007	−0.204405
$153$	0	0
$154$	5.35764	0.431731
$155$	1.02160	0.0820572
$156$	0	0
$157$	−0.453089	−0.0361604	−0.0180802	−	0.999837i	$-0.505755\pi$
$157$	−0.453089	−0.0361604	−0.0180802	+	0.999837i	$0.505755\pi$
$158$	−3.99790	−0.318056
$159$	0	0
$160$	8.86274	0.700661
$161$	0.564847	0.0445162
$162$	0	0
$163$	−7.12748	−0.558267	−0.279134	−	0.960252i	$-0.590047\pi$
$163$	−7.12748	−0.558267	−0.279134	+	0.960252i	$0.590047\pi$
$164$	−7.09485	−0.554014
$165$	0	0
$166$	−11.4412	−0.888009
$167$	10.5555	0.816808	0.408404	−	0.912801i	$-0.366086\pi$
$167$	10.5555	0.816808	0.408404	+	0.912801i	$0.366086\pi$
$168$	0	0
$169$	16.2443	1.24957
$170$	−2.22580	−0.170711
$171$	0	0
$172$	−4.51339	−0.344143
$173$	16.7583	1.27411	0.637055	−	0.770818i	$-0.280152\pi$
$173$	16.7583	1.27411	0.637055	+	0.770818i	$0.280152\pi$
$174$	0	0
$175$	−3.26050	−0.246471
$176$	4.91851	0.370747
$177$	0	0
$178$	−7.79855	−0.584526
$179$	−4.75191	−0.355174	−0.177587	−	0.984105i	$-0.556829\pi$
$179$	−4.75191	−0.355174	−0.177587	+	0.984105i	$0.556829\pi$
$180$	0	0
$181$	−9.02008	−0.670457	−0.335229	−	0.942137i	$-0.608814\pi$
$181$	−9.02008	−0.670457	−0.335229	+	0.942137i	$0.608814\pi$
$182$	−4.96149	−0.367770
$183$	0	0
$184$	−1.22632	−0.0904058
$185$	8.57279	0.630284
$186$	0	0
$187$	−11.2580	−0.823264
$188$	−14.3846	−1.04911
$189$	0	0
$190$	1.11522	0.0809069
$191$	24.6001	1.78000	0.890000	−	0.455960i	$-0.150704\pi$
$191$	24.6001	1.78000	0.890000	+	0.455960i	$0.150704\pi$
$192$	0	0
$193$	15.5317	1.11800	0.558998	−	0.829169i	$-0.311186\pi$
$193$	15.5317	1.11800	0.558998	+	0.829169i	$0.311186\pi$
$194$	−4.63844	−0.333021
$195$	0	0
$196$	7.87327	0.562376
$197$	15.1172	1.07705	0.538527	−	0.842608i	$-0.318981\pi$
$197$	15.1172	1.07705	0.538527	+	0.842608i	$0.318981\pi$
$198$	0	0
$199$	13.6303	0.966227	0.483114	−	0.875558i	$-0.339506\pi$
$199$	13.6303	0.966227	0.483114	+	0.875558i	$0.339506\pi$
$200$	7.07879	0.500546
$201$	0	0
$202$	1.80737	0.127166
$203$	4.41412	0.309810
$204$	0	0
$205$	7.57100	0.528781
$206$	9.51366	0.662848
$207$	0	0
$208$	−4.55483	−0.315820
$209$	5.64074	0.390178
$210$	0	0
$211$	11.3226	0.779478	0.389739	−	0.920925i	$-0.372565\pi$
$211$	11.3226	0.779478	0.389739	+	0.920925i	$0.372565\pi$
$212$	11.6940	0.803147
$213$	0	0
$214$	13.6651	0.934124
$215$	4.81629	0.328468
$216$	0	0
$217$	0.811829	0.0551106
$218$	−2.20787	−0.149536
$219$	0	0
$220$	−12.5136	−0.843664
$221$	10.4255	0.701297
$222$	0	0
$223$	5.49894	0.368236	0.184118	−	0.982904i	$-0.441057\pi$
$223$	5.49894	0.368236	0.184118	+	0.982904i	$0.441057\pi$
$224$	7.04288	0.470572
$225$	0	0
$226$	−4.08461	−0.271704
$227$	2.09244	0.138880	0.0694402	−	0.997586i	$-0.477879\pi$
$227$	2.09244	0.138880	0.0694402	+	0.997586i	$0.477879\pi$
$228$	0	0
$229$	−16.4218	−1.08518	−0.542592	−	0.839996i	$-0.682557\pi$
$229$	−16.4218	−1.08518	−0.542592	+	0.839996i	$0.682557\pi$
$230$	0.542693	0.0357841
$231$	0	0
$232$	−9.58336	−0.629179
$233$	10.0033	0.655338	0.327669	−	0.944793i	$-0.393737\pi$
$233$	10.0033	0.655338	0.327669	+	0.944793i	$0.393737\pi$
$234$	0	0
$235$	15.3500	1.00132
$236$	−7.42474	−0.483309
$237$	0	0
$238$	−1.76876	−0.114651
$239$	−12.8814	−0.833227	−0.416614	−	0.909084i	$-0.636783\pi$
$239$	−12.8814	−0.833227	−0.416614	+	0.909084i	$0.636783\pi$
$240$	0	0
$241$	9.58394	0.617356	0.308678	−	0.951167i	$-0.400114\pi$
$241$	9.58394	0.617356	0.308678	+	0.951167i	$0.400114\pi$
$242$	17.6374	1.13377
$243$	0	0
$244$	5.00496	0.320409
$245$	−8.40166	−0.536762
$246$	0	0
$247$	−5.22365	−0.332373
$248$	−1.76254	−0.111921
$249$	0	0
$250$	−8.90532	−0.563222
$251$	7.09126	0.447596	0.223798	−	0.974636i	$-0.428154\pi$
$251$	7.09126	0.447596	0.223798	+	0.974636i	$0.428154\pi$
$252$	0	0
$253$	2.74491	0.172571
$254$	11.1474	0.699452
$255$	0	0
$256$	−12.9035	−0.806466
$257$	7.64221	0.476708	0.238354	−	0.971178i	$-0.423392\pi$
$257$	7.64221	0.476708	0.238354	+	0.971178i	$0.423392\pi$
$258$	0	0
$259$	6.81246	0.423306
$260$	11.5883	0.718675
$261$	0	0
$262$	15.1779	0.937693
$263$	18.7853	1.15835	0.579176	−	0.815202i	$-0.303374\pi$
$263$	18.7853	1.15835	0.579176	+	0.815202i	$0.303374\pi$
$264$	0	0
$265$	−12.4788	−0.766567
$266$	0.886226	0.0543380
$267$	0	0
$268$	−12.3327	−0.753340
$269$	17.1967	1.04850	0.524251	−	0.851564i	$-0.324346\pi$
$269$	17.1967	1.04850	0.524251	+	0.851564i	$0.324346\pi$
$270$	0	0
$271$	−13.7691	−0.836416	−0.418208	−	0.908351i	$-0.637342\pi$
$271$	−13.7691	−0.836416	−0.418208	+	0.908351i	$0.637342\pi$
$272$	−1.62378	−0.0984563
$273$	0	0
$274$	−14.7233	−0.889467
$275$	−15.8446	−0.955466
$276$	0	0
$277$	−18.7759	−1.12813	−0.564066	−	0.825730i	$-0.690764\pi$
$277$	−18.7759	−1.12813	−0.564066	+	0.825730i	$0.690764\pi$
$278$	8.62606	0.517357
$279$	0	0
$280$	−4.74078	−0.283316
$281$	−9.00392	−0.537129	−0.268564	−	0.963262i	$-0.586549\pi$
$281$	−9.00392	−0.537129	−0.268564	+	0.963262i	$0.586549\pi$
$282$	0	0
$283$	14.0938	0.837789	0.418894	−	0.908035i	$-0.362418\pi$
$283$	14.0938	0.837789	0.418894	+	0.908035i	$0.362418\pi$
$284$	−13.4744	−0.799561
$285$	0	0
$286$	−24.1106	−1.42569
$287$	6.01638	0.355135
$288$	0	0
$289$	−13.2833	−0.781372
$290$	4.24099	0.249039
$291$	0	0
$292$	−13.7267	−0.803292
$293$	25.2452	1.47484	0.737421	−	0.675434i	$-0.236044\pi$
$293$	25.2452	1.47484	0.737421	+	0.675434i	$0.236044\pi$
$294$	0	0
$295$	7.92303	0.461297
$296$	−14.7903	−0.859671
$297$	0	0
$298$	−0.427444	−0.0247612
$299$	−2.54194	−0.147004
$300$	0	0
$301$	3.82732	0.220603
$302$	−5.93535	−0.341541
$303$	0	0
$304$	0.813587	0.0466624
$305$	−5.34085	−0.305816
$306$	0	0
$307$	16.6513	0.950342	0.475171	−	0.879893i	$-0.342386\pi$
$307$	16.6513	0.950342	0.475171	+	0.879893i	$0.342386\pi$
$308$	−9.94404	−0.566614
$309$	0	0
$310$	0.779988	0.0443003
$311$	3.26046	0.184884	0.0924418	−	0.995718i	$-0.470533\pi$
$311$	3.26046	0.184884	0.0924418	+	0.995718i	$0.470533\pi$
$312$	0	0
$313$	−5.48703	−0.310145	−0.155073	−	0.987903i	$-0.549561\pi$
$313$	−5.48703	−0.310145	−0.155073	+	0.987903i	$0.549561\pi$
$314$	−0.345930	−0.0195220
$315$	0	0
$316$	7.42029	0.417424
$317$	22.3722	1.25655	0.628275	−	0.777992i	$-0.283761\pi$
$317$	22.3722	1.25655	0.628275	+	0.777992i	$0.283761\pi$
$318$	0	0
$319$	21.4507	1.20101
$320$	4.21931	0.235867
$321$	0	0
$322$	0.431257	0.0240330
$323$	−1.86222	−0.103617
$324$	0	0
$325$	14.6730	0.813913
$326$	−5.44178	−0.301392
$327$	0	0
$328$	−13.0620	−0.721227
$329$	12.1981	0.672501
$330$	0	0
$331$	−33.2750	−1.82896	−0.914479	−	0.404634i	$-0.867399\pi$
$331$	−33.2750	−1.82896	−0.914479	+	0.404634i	$0.867399\pi$
$332$	21.2354	1.16544
$333$	0	0
$334$	8.05903	0.440971
$335$	13.1604	0.719028
$336$	0	0
$337$	4.21753	0.229743	0.114872	−	0.993380i	$-0.463354\pi$
$337$	4.21753	0.229743	0.114872	+	0.993380i	$0.463354\pi$
$338$	12.4024	0.674604
$339$	0	0
$340$	4.13119	0.224045
$341$	3.94513	0.213641
$342$	0	0
$343$	−15.0882	−0.814685
$344$	−8.30939	−0.448012
$345$	0	0
$346$	12.7948	0.687855
$347$	17.8458	0.958013	0.479007	−	0.877811i	$-0.340997\pi$
$347$	17.8458	0.958013	0.479007	+	0.877811i	$0.340997\pi$
$348$	0	0
$349$	−4.70118	−0.251648	−0.125824	−	0.992053i	$-0.540157\pi$
$349$	−4.70118	−0.251648	−0.125824	+	0.992053i	$0.540157\pi$
$350$	−2.48937	−0.133063
$351$	0	0
$352$	34.2253	1.82421
$353$	25.5243	1.35852	0.679261	−	0.733896i	$-0.262300\pi$
$353$	25.5243	1.35852	0.679261	+	0.733896i	$0.262300\pi$
$354$	0	0
$355$	14.3787	0.763145
$356$	14.4745	0.767145
$357$	0	0
$358$	−3.62805	−0.191748
$359$	−32.0017	−1.68898	−0.844492	−	0.535568i	$-0.820097\pi$
$359$	−32.0017	−1.68898	−0.844492	+	0.535568i	$0.820097\pi$
$360$	0	0
$361$	−18.0669	−0.950892
$362$	−6.88676	−0.361960
$363$	0	0
$364$	9.20875	0.482670
$365$	14.6479	0.766705
$366$	0	0
$367$	14.7807	0.771546	0.385773	−	0.922594i	$-0.373935\pi$
$367$	14.7807	0.771546	0.385773	+	0.922594i	$0.373935\pi$
$368$	0.395909	0.0206382
$369$	0	0
$370$	6.54526	0.340272
$371$	−9.91643	−0.514835
$372$	0	0
$373$	−30.5239	−1.58047	−0.790235	−	0.612804i	$-0.790042\pi$
$373$	−30.5239	−1.58047	−0.790235	+	0.612804i	$0.790042\pi$
$374$	−8.59538	−0.444456
$375$	0	0
$376$	−26.4828	−1.36575
$377$	−19.8646	−1.02308
$378$	0	0
$379$	−13.3030	−0.683329	−0.341665	−	0.939822i	$-0.610991\pi$
$379$	−13.3030	−0.683329	−0.341665	+	0.939822i	$0.610991\pi$
$380$	−2.06991	−0.106184
$381$	0	0
$382$	18.7820	0.960971
$383$	−20.6416	−1.05474	−0.527369	−	0.849636i	$-0.676821\pi$
$383$	−20.6416	−1.05474	−0.527369	+	0.849636i	$0.676821\pi$
$384$	0	0
$385$	10.6114	0.540807
$386$	11.8583	0.603573
$387$	0	0
$388$	8.60917	0.437064
$389$	−18.0385	−0.914590	−0.457295	−	0.889315i	$-0.651182\pi$
$389$	−18.0385	−0.914590	−0.457295	+	0.889315i	$0.651182\pi$
$390$	0	0
$391$	−0.906196	−0.0458283
$392$	14.4951	0.732113
$393$	0	0
$394$	11.5419	0.581470
$395$	−7.91828	−0.398412
$396$	0	0
$397$	−24.2893	−1.21904	−0.609521	−	0.792770i	$-0.708638\pi$
$397$	−24.2893	−1.21904	−0.609521	+	0.792770i	$0.708638\pi$
$398$	10.4066	0.521638
$399$	0	0
$400$	−2.28533	−0.114267
$401$	−11.4436	−0.571464	−0.285732	−	0.958310i	$-0.592237\pi$
$401$	−11.4436	−0.571464	−0.285732	+	0.958310i	$0.592237\pi$
$402$	0	0
$403$	−3.65342	−0.181990
$404$	−3.35456	−0.166895
$405$	0	0
$406$	3.37015	0.167258
$407$	33.1056	1.64098
$408$	0	0
$409$	4.34200	0.214698	0.107349	−	0.994221i	$-0.465764\pi$
$409$	4.34200	0.214698	0.107349	+	0.994221i	$0.465764\pi$
$410$	5.78040	0.285474
$411$	0	0
$412$	−17.6578	−0.869938
$413$	6.29613	0.309812
$414$	0	0
$415$	−22.6605	−1.11236
$416$	−31.6946	−1.55395
$417$	0	0
$418$	4.30667	0.210646
$419$	35.0918	1.71434	0.857172	−	0.515030i	$-0.172219\pi$
$419$	35.0918	1.71434	0.857172	+	0.515030i	$0.172219\pi$
$420$	0	0
$421$	−36.2929	−1.76881	−0.884405	−	0.466720i	$-0.845436\pi$
$421$	−36.2929	−1.76881	−0.884405	+	0.466720i	$0.845436\pi$
$422$	8.64471	0.420818
$423$	0	0
$424$	21.5293	1.04555
$425$	5.23089	0.253736
$426$	0	0
$427$	−4.24417	−0.205390
$428$	−25.3630	−1.22597
$429$	0	0
$430$	3.67721	0.177331
$431$	−8.92616	−0.429958	−0.214979	−	0.976619i	$-0.568968\pi$
$431$	−8.92616	−0.429958	−0.214979	+	0.976619i	$0.568968\pi$
$432$	0	0
$433$	40.1476	1.92937	0.964684	−	0.263408i	$-0.0848466\pi$
$433$	40.1476	1.92937	0.964684	+	0.263408i	$0.0848466\pi$
$434$	0.619826	0.0297526
$435$	0	0
$436$	4.09792	0.196255
$437$	0.454045	0.0217199
$438$	0	0
$439$	−16.4516	−0.785194	−0.392597	−	0.919711i	$-0.628423\pi$
$439$	−16.4516	−0.785194	−0.392597	+	0.919711i	$0.628423\pi$
$440$	−23.0381	−1.09830
$441$	0	0
$442$	7.95982	0.378610
$443$	−14.1442	−0.672012	−0.336006	−	0.941860i	$-0.609076\pi$
$443$	−14.1442	−0.672012	−0.336006	+	0.941860i	$0.609076\pi$
$444$	0	0
$445$	−15.4459	−0.732205
$446$	4.19840	0.198800
$447$	0	0
$448$	3.35292	0.158411
$449$	−6.96741	−0.328813	−0.164406	−	0.986393i	$-0.552571\pi$
$449$	−6.96741	−0.328813	−0.164406	+	0.986393i	$0.552571\pi$
$450$	0	0
$451$	29.2369	1.37671
$452$	7.58122	0.356591
$453$	0	0
$454$	1.59757	0.0749775
$455$	−9.82677	−0.460686
$456$	0	0
$457$	−16.6490	−0.778809	−0.389404	−	0.921067i	$-0.627319\pi$
$457$	−16.6490	−0.778809	−0.389404	+	0.921067i	$0.627319\pi$
$458$	−12.5379	−0.585859
$459$	0	0
$460$	−1.00726	−0.0469639
$461$	−18.1015	−0.843072	−0.421536	−	0.906812i	$-0.638509\pi$
$461$	−18.1015	−0.843072	−0.421536	+	0.906812i	$0.638509\pi$
$462$	0	0
$463$	33.1519	1.54070	0.770350	−	0.637621i	$-0.220081\pi$
$463$	33.1519	1.54070	0.770350	+	0.637621i	$0.220081\pi$
$464$	3.09392	0.143632
$465$	0	0
$466$	7.63745	0.353798
$467$	−3.70462	−0.171430	−0.0857148	−	0.996320i	$-0.527317\pi$
$467$	−3.70462	−0.171430	−0.0857148	+	0.996320i	$0.527317\pi$
$468$	0	0
$469$	10.4580	0.482907
$470$	11.7196	0.540586
$471$	0	0
$472$	−13.6693	−0.629182
$473$	18.5991	0.855187
$474$	0	0
$475$	−2.62091	−0.120256
$476$	3.28290	0.150471
$477$	0	0
$478$	−9.83484	−0.449835
$479$	−13.6667	−0.624449	−0.312225	−	0.950008i	$-0.601074\pi$
$479$	−13.6667	−0.624449	−0.312225	+	0.950008i	$0.601074\pi$
$480$	0	0
$481$	−30.6577	−1.39787
$482$	7.31727	0.333292
$483$	0	0
$484$	−32.7358	−1.48799
$485$	−9.18695	−0.417158
$486$	0	0
$487$	26.6571	1.20795	0.603974	−	0.797004i	$-0.293583\pi$
$487$	26.6571	1.20795	0.603974	+	0.797004i	$0.293583\pi$
$488$	9.21438	0.417116
$489$	0	0
$490$	−6.41461	−0.289782
$491$	−8.27214	−0.373317	−0.186658	−	0.982425i	$-0.559766\pi$
$491$	−8.27214	−0.373317	−0.186658	+	0.982425i	$0.559766\pi$
$492$	0	0
$493$	−7.08166	−0.318942
$494$	−3.98822	−0.179439
$495$	0	0
$496$	0.569023	0.0255499
$497$	11.4262	0.512537
$498$	0	0
$499$	−32.1496	−1.43921	−0.719607	−	0.694382i	$-0.755678\pi$
$499$	−32.1496	−1.43921	−0.719607	+	0.694382i	$0.755678\pi$
$500$	16.5287	0.739186
$501$	0	0
$502$	5.41412	0.241644
$503$	4.95942	0.221130	0.110565	−	0.993869i	$-0.464734\pi$
$503$	4.95942	0.221130	0.110565	+	0.993869i	$0.464734\pi$
$504$	0	0
$505$	3.57969	0.159294
$506$	2.09572	0.0931660
$507$	0	0
$508$	−20.6901	−0.917977
$509$	−13.2112	−0.585577	−0.292788	−	0.956177i	$-0.594583\pi$
$509$	−13.2112	−0.585577	−0.292788	+	0.956177i	$0.594583\pi$
$510$	0	0
$511$	11.6401	0.514928
$512$	9.33127	0.412388
$513$	0	0
$514$	5.83477	0.257361
$515$	18.8429	0.830315
$516$	0	0
$517$	59.2772	2.60701
$518$	5.20127	0.228531
$519$	0	0
$520$	21.3346	0.935585
$521$	7.65030	0.335166	0.167583	−	0.985858i	$-0.446404\pi$
$521$	7.65030	0.335166	0.167583	+	0.985858i	$0.446404\pi$
$522$	0	0
$523$	38.3821	1.67833	0.839165	−	0.543876i	$-0.183044\pi$
$523$	38.3821	1.67833	0.839165	+	0.543876i	$0.183044\pi$
$524$	−28.1709	−1.23065
$525$	0	0
$526$	14.3425	0.625361
$527$	−1.30243	−0.0567349
$528$	0	0
$529$	−22.7791	−0.990394
$530$	−9.52748	−0.413847
$531$	0	0
$532$	−1.64488	−0.0713144
$533$	−27.0751	−1.17275
$534$	0	0
$535$	27.0652	1.17013
$536$	−22.7051	−0.980713
$537$	0	0
$538$	13.1296	0.566055
$539$	−32.4447	−1.39749
$540$	0	0
$541$	−15.6994	−0.674969	−0.337484	−	0.941331i	$-0.609576\pi$
$541$	−15.6994	−0.674969	−0.337484	+	0.941331i	$0.609576\pi$
$542$	−10.5126	−0.451557
$543$	0	0
$544$	−11.2990	−0.484442
$545$	−4.37294	−0.187316
$546$	0	0
$547$	−1.00000	−0.0427569
$547$	−1.00000	−0.0427569
$548$	27.3271	1.16736
$549$	0	0
$550$	−12.0972	−0.515828
$551$	3.54823	0.151160
$552$	0	0
$553$	−6.29235	−0.267578
$554$	−14.3352	−0.609046
$555$	0	0
$556$	−16.0104	−0.678991
$557$	25.9854	1.10104	0.550518	−	0.834823i	$-0.314430\pi$
$557$	25.9854	1.10104	0.550518	+	0.834823i	$0.314430\pi$
$558$	0	0
$559$	−17.2238	−0.728491
$560$	1.53053	0.0646765
$561$	0	0
$562$	−6.87443	−0.289980
$563$	−11.0040	−0.463762	−0.231881	−	0.972744i	$-0.574488\pi$
$563$	−11.0040	−0.463762	−0.231881	+	0.972744i	$0.574488\pi$
$564$	0	0
$565$	−8.09002	−0.340350
$566$	10.7605	0.452298
$567$	0	0
$568$	−24.8072	−1.04089
$569$	2.32372	0.0974152	0.0487076	−	0.998813i	$-0.484490\pi$
$569$	2.32372	0.0974152	0.0487076	+	0.998813i	$0.484490\pi$
$570$	0	0
$571$	−6.08147	−0.254502	−0.127251	−	0.991871i	$-0.540615\pi$
$571$	−6.08147	−0.254502	−0.127251	+	0.991871i	$0.540615\pi$
$572$	44.7505	1.87111
$573$	0	0
$574$	4.59346	0.191727
$575$	−1.27539	−0.0531875
$576$	0	0
$577$	−43.2965	−1.80246	−0.901228	−	0.433346i	$-0.857333\pi$
$577$	−43.2965	−1.80246	−0.901228	+	0.433346i	$0.857333\pi$
$578$	−10.1417	−0.421840
$579$	0	0
$580$	−7.87147	−0.326845
$581$	−18.0075	−0.747075
$582$	0	0
$583$	−48.1895	−1.99580
$584$	−25.2715	−1.04574
$585$	0	0
$586$	19.2745	0.796224
$587$	−34.3433	−1.41750	−0.708749	−	0.705460i	$-0.750740\pi$
$587$	−34.3433	−1.41750	−0.708749	+	0.705460i	$0.750740\pi$
$588$	0	0
$589$	0.652578	0.0268890
$590$	6.04918	0.249041
$591$	0	0
$592$	4.77495	0.196249
$593$	−12.4576	−0.511572	−0.255786	−	0.966733i	$-0.582334\pi$
$593$	−12.4576	−0.511572	−0.255786	+	0.966733i	$0.582334\pi$
$594$	0	0
$595$	−3.50322	−0.143618
$596$	0.793357	0.0324972
$597$	0	0
$598$	−1.94076	−0.0793634
$599$	−41.2358	−1.68485	−0.842424	−	0.538815i	$-0.818872\pi$
$599$	−41.2358	−1.68485	−0.842424	+	0.538815i	$0.818872\pi$
$600$	0	0
$601$	22.9872	0.937668	0.468834	−	0.883286i	$-0.344674\pi$
$601$	22.9872	0.937668	0.468834	+	0.883286i	$0.344674\pi$
$602$	2.92213	0.119097
$603$	0	0
$604$	11.0163	0.448246
$605$	34.9327	1.42022
$606$	0	0
$607$	−44.6075	−1.81056	−0.905282	−	0.424812i	$-0.860340\pi$
$607$	−44.6075	−1.81056	−0.905282	+	0.424812i	$0.860340\pi$
$608$	5.66132	0.229597
$609$	0	0
$610$	−4.07770	−0.165101
$611$	−54.8941	−2.22078
$612$	0	0
$613$	−21.3775	−0.863428	−0.431714	−	0.902010i	$-0.642091\pi$
$613$	−21.3775	−0.863428	−0.431714	+	0.902010i	$0.642091\pi$
$614$	12.7132	0.513062
$615$	0	0
$616$	−18.3075	−0.737630
$617$	−12.0119	−0.483583	−0.241791	−	0.970328i	$-0.577735\pi$
$617$	−12.0119	−0.483583	−0.241791	+	0.970328i	$0.577735\pi$
$618$	0	0
$619$	19.5487	0.785727	0.392863	−	0.919597i	$-0.371484\pi$
$619$	19.5487	0.785727	0.392863	+	0.919597i	$0.371484\pi$
$620$	−1.44769	−0.0581408
$621$	0	0
$622$	2.48934	0.0998133
$623$	−12.2742	−0.491757
$624$	0	0
$625$	−4.07144	−0.162857
$626$	−4.18931	−0.167438
$627$	0	0
$628$	0.642063	0.0256211
$629$	−10.9294	−0.435783
$630$	0	0
$631$	40.4106	1.60872	0.804360	−	0.594142i	$-0.202508\pi$
$631$	40.4106	1.60872	0.804360	+	0.594142i	$0.202508\pi$
$632$	13.6611	0.543411
$633$	0	0
$634$	17.0810	0.678375
$635$	22.0787	0.876166
$636$	0	0
$637$	30.0457	1.19045
$638$	16.3774	0.648389
$639$	0	0
$640$	−14.5041	−0.573324
$641$	−22.6243	−0.893607	−0.446803	−	0.894632i	$-0.647438\pi$
$641$	−22.6243	−0.893607	−0.446803	+	0.894632i	$0.647438\pi$
$642$	0	0
$643$	40.2337	1.58666	0.793330	−	0.608792i	$-0.208345\pi$
$643$	40.2337	1.58666	0.793330	+	0.608792i	$0.208345\pi$
$644$	−0.800433	−0.0315415
$645$	0	0
$646$	−1.42179	−0.0559396
$647$	26.0029	1.02228	0.511139	−	0.859498i	$-0.329224\pi$
$647$	26.0029	1.02228	0.511139	+	0.859498i	$0.329224\pi$
$648$	0	0
$649$	30.5964	1.20101
$650$	11.2028	0.439408
$651$	0	0
$652$	10.1002	0.395554
$653$	20.6624	0.808584	0.404292	−	0.914630i	$-0.367518\pi$
$653$	20.6624	0.808584	0.404292	+	0.914630i	$0.367518\pi$
$654$	0	0
$655$	30.0615	1.17460
$656$	4.21696	0.164645
$657$	0	0
$658$	9.31313	0.363064
$659$	−5.80002	−0.225937	−0.112968	−	0.993599i	$-0.536036\pi$
$659$	−5.80002	−0.225937	−0.112968	+	0.993599i	$0.536036\pi$
$660$	0	0
$661$	2.81113	0.109340	0.0546702	−	0.998504i	$-0.482589\pi$
$661$	2.81113	0.109340	0.0546702	+	0.998504i	$0.482589\pi$
$662$	−25.4052	−0.987401
$663$	0	0
$664$	39.0955	1.51720
$665$	1.75527	0.0680664
$666$	0	0
$667$	1.72665	0.0668560
$668$	−14.9579	−0.578740
$669$	0	0
$670$	10.0479	0.388183
$671$	−20.6248	−0.796210
$672$	0	0
$673$	−4.40770	−0.169904	−0.0849521	−	0.996385i	$-0.527074\pi$
$673$	−4.40770	−0.169904	−0.0849521	+	0.996385i	$0.527074\pi$
$674$	3.22005	0.124032
$675$	0	0
$676$	−23.0195	−0.885366
$677$	10.0686	0.386969	0.193484	−	0.981103i	$-0.438021\pi$
$677$	10.0686	0.386969	0.193484	+	0.981103i	$0.438021\pi$
$678$	0	0
$679$	−7.30051	−0.280168
$680$	7.60573	0.291667
$681$	0	0
$682$	3.01208	0.115339
$683$	−43.3757	−1.65973	−0.829863	−	0.557968i	$-0.811581\pi$
$683$	−43.3757	−1.65973	−0.829863	+	0.557968i	$0.811581\pi$
$684$	0	0
$685$	−29.1611	−1.11419
$686$	−11.5197	−0.439825
$687$	0	0
$688$	2.68262	0.102274
$689$	44.6262	1.70012
$690$	0	0
$691$	11.2784	0.429052	0.214526	−	0.976718i	$-0.431179\pi$
$691$	11.2784	0.429052	0.214526	+	0.976718i	$0.431179\pi$
$692$	−23.7478	−0.902757
$693$	0	0
$694$	13.6252	0.517204
$695$	17.0849	0.648066
$696$	0	0
$697$	−9.65220	−0.365603
$698$	−3.58931	−0.135858
$699$	0	0
$700$	4.62039	0.174634
$701$	−24.4031	−0.921691	−0.460845	−	0.887480i	$-0.652454\pi$
$701$	−24.4031	−0.921691	−0.460845	+	0.887480i	$0.652454\pi$
$702$	0	0
$703$	5.47610	0.206535
$704$	16.2937	0.614093
$705$	0	0
$706$	19.4876	0.733427
$707$	2.84464	0.106984
$708$	0	0
$709$	20.1044	0.755037	0.377518	−	0.926002i	$-0.376777\pi$
$709$	20.1044	0.755037	0.377518	+	0.926002i	$0.376777\pi$
$710$	10.9781	0.412000
$711$	0	0
$712$	26.6483	0.998685
$713$	0.317559	0.0118927
$714$	0	0
$715$	−47.7538	−1.78589
$716$	6.73383	0.251655
$717$	0	0
$718$	−24.4331	−0.911833
$719$	26.8520	1.00141	0.500705	−	0.865618i	$-0.333074\pi$
$719$	26.8520	1.00141	0.500705	+	0.865618i	$0.333074\pi$
$720$	0	0
$721$	14.9737	0.557649
$722$	−13.7940	−0.513359
$723$	0	0
$724$	12.7822	0.475045
$725$	−9.96683	−0.370159
$726$	0	0
$727$	43.5422	1.61489	0.807446	−	0.589941i	$-0.200849\pi$
$727$	43.5422	1.61489	0.807446	+	0.589941i	$0.200849\pi$
$728$	16.9538	0.628349
$729$	0	0
$730$	11.1836	0.413922
$731$	−6.14025	−0.227105
$732$	0	0
$733$	−29.4062	−1.08614	−0.543072	−	0.839686i	$-0.682739\pi$
$733$	−29.4062	−1.08614	−0.543072	+	0.839686i	$0.682739\pi$
$734$	11.2850	0.416535
$735$	0	0
$736$	2.75492	0.101548
$737$	50.8215	1.87203
$738$	0	0
$739$	−10.0339	−0.369103	−0.184552	−	0.982823i	$-0.559083\pi$
$739$	−10.0339	−0.369103	−0.184552	+	0.982823i	$0.559083\pi$
$740$	−12.1483	−0.446581
$741$	0	0
$742$	−7.57112	−0.277945
$743$	−32.7364	−1.20098	−0.600492	−	0.799631i	$-0.705028\pi$
$743$	−32.7364	−1.20098	−0.600492	+	0.799631i	$0.705028\pi$
$744$	0	0
$745$	−0.846601	−0.0310171
$746$	−23.3048	−0.853250
$747$	0	0
$748$	15.9534	0.583315
$749$	21.5076	0.785872
$750$	0	0
$751$	−29.0656	−1.06062	−0.530309	−	0.847804i	$-0.677924\pi$
$751$	−29.0656	−1.06062	−0.530309	+	0.847804i	$0.677924\pi$
$752$	8.54979	0.311779
$753$	0	0
$754$	−15.1665	−0.552330
$755$	−11.7556	−0.427831
$756$	0	0
$757$	−49.0178	−1.78158	−0.890791	−	0.454414i	$-0.849849\pi$
$757$	−49.0178	−1.78158	−0.890791	+	0.454414i	$0.849849\pi$
$758$	−10.1567	−0.368910
$759$	0	0
$760$	−3.81081	−0.138233
$761$	−42.6524	−1.54615	−0.773073	−	0.634317i	$-0.781282\pi$
$761$	−42.6524	−1.54615	−0.773073	+	0.634317i	$0.781282\pi$
$762$	0	0
$763$	−3.47500	−0.125804
$764$	−34.8603	−1.26120
$765$	0	0
$766$	−15.7597	−0.569423
$767$	−28.3340	−1.02308
$768$	0	0
$769$	23.5755	0.850156	0.425078	−	0.905157i	$-0.360247\pi$
$769$	23.5755	0.850156	0.425078	+	0.905157i	$0.360247\pi$
$770$	8.10173	0.291966
$771$	0	0
$772$	−22.0096	−0.792144
$773$	9.61811	0.345939	0.172970	−	0.984927i	$-0.444664\pi$
$773$	9.61811	0.345939	0.172970	+	0.984927i	$0.444664\pi$
$774$	0	0
$775$	−1.83306	−0.0658456
$776$	15.8499	0.568979
$777$	0	0
$778$	−13.7723	−0.493761
$779$	4.83618	0.173274
$780$	0	0
$781$	55.5264	1.98689
$782$	−0.691874	−0.0247414
$783$	0	0
$784$	−4.67963	−0.167130
$785$	−0.685153	−0.0244541
$786$	0	0
$787$	−27.3656	−0.975479	−0.487740	−	0.872989i	$-0.662178\pi$
$787$	−27.3656	−0.975479	−0.487740	+	0.872989i	$0.662178\pi$
$788$	−21.4222	−0.763135
$789$	0	0
$790$	−6.04555	−0.215091
$791$	−6.42882	−0.228583
$792$	0	0
$793$	19.0997	0.678251
$794$	−18.5447	−0.658126
$795$	0	0
$796$	−19.3152	−0.684610
$797$	10.5153	0.372471	0.186236	−	0.982505i	$-0.440371\pi$
$797$	10.5153	0.372471	0.186236	+	0.982505i	$0.440371\pi$
$798$	0	0
$799$	−19.5696	−0.692322
$800$	−15.9024	−0.562235
$801$	0	0
$802$	−8.73707	−0.308517
$803$	56.5657	1.99616
$804$	0	0
$805$	0.854152	0.0301049
$806$	−2.78936	−0.0982510
$807$	0	0
$808$	−6.17591	−0.217268
$809$	−9.73359	−0.342215	−0.171107	−	0.985252i	$-0.554734\pi$
$809$	−9.73359	−0.342215	−0.171107	+	0.985252i	$0.554734\pi$
$810$	0	0
$811$	11.8190	0.415020	0.207510	−	0.978233i	$-0.433464\pi$
$811$	11.8190	0.415020	0.207510	+	0.978233i	$0.433464\pi$
$812$	−6.25515	−0.219513
$813$	0	0
$814$	25.2759	0.885919
$815$	−10.7780	−0.377538
$816$	0	0
$817$	3.07654	0.107634
$818$	3.31509	0.115909
$819$	0	0
$820$	−10.7287	−0.374662
$821$	−20.8740	−0.728507	−0.364253	−	0.931300i	$-0.618676\pi$
$821$	−20.8740	−0.728507	−0.364253	+	0.931300i	$0.618676\pi$
$822$	0	0
$823$	19.2428	0.670762	0.335381	−	0.942083i	$-0.391135\pi$
$823$	19.2428	0.670762	0.335381	+	0.942083i	$0.391135\pi$
$824$	−32.5089	−1.13250
$825$	0	0
$826$	4.80705	0.167259
$827$	31.8820	1.10864	0.554322	−	0.832302i	$-0.312978\pi$
$827$	31.8820	1.10864	0.554322	+	0.832302i	$0.312978\pi$
$828$	0	0
$829$	−40.5150	−1.40714	−0.703572	−	0.710624i	$-0.748413\pi$
$829$	−40.5150	−1.40714	−0.703572	+	0.710624i	$0.748413\pi$
$830$	−17.3012	−0.600532
$831$	0	0
$832$	−15.0889	−0.523115
$833$	10.7112	0.371121
$834$	0	0
$835$	15.9618	0.552381
$836$	−7.99337	−0.276457
$837$	0	0
$838$	26.7923	0.925525
$839$	30.1260	1.04006	0.520032	−	0.854147i	$-0.325920\pi$
$839$	30.1260	1.04006	0.520032	+	0.854147i	$0.325920\pi$
$840$	0	0
$841$	−15.5068	−0.534716
$842$	−27.7094	−0.954929
$843$	0	0
$844$	−16.0450	−0.552291
$845$	24.5644	0.845041
$846$	0	0
$847$	27.7597	0.953834
$848$	−6.95057	−0.238683
$849$	0	0
$850$	3.99375	0.136985
$851$	2.66479	0.0913479
$852$	0	0
$853$	9.30329	0.318539	0.159269	−	0.987235i	$-0.449086\pi$
$853$	9.30329	0.318539	0.159269	+	0.987235i	$0.449086\pi$
$854$	−3.24039	−0.110884
$855$	0	0
$856$	−46.6946	−1.59599
$857$	−46.7828	−1.59807	−0.799035	−	0.601284i	$-0.794656\pi$
$857$	−46.7828	−1.59807	−0.799035	+	0.601284i	$0.794656\pi$
$858$	0	0
$859$	−11.8562	−0.404528	−0.202264	−	0.979331i	$-0.564830\pi$
$859$	−11.8562	−0.404528	−0.202264	+	0.979331i	$0.564830\pi$
$860$	−6.82507	−0.232733
$861$	0	0
$862$	−6.81506	−0.232122
$863$	34.0665	1.15964	0.579819	−	0.814746i	$-0.303123\pi$
$863$	34.0665	1.15964	0.579819	+	0.814746i	$0.303123\pi$
$864$	0	0
$865$	25.3416	0.861640
$866$	30.6524	1.04161
$867$	0	0
$868$	−1.15043	−0.0390480
$869$	−30.5781	−1.03729
$870$	0	0
$871$	−47.0636	−1.59469
$872$	7.54448	0.255488
$873$	0	0
$874$	0.346660	0.0117259
$875$	−14.0162	−0.473834
$876$	0	0
$877$	6.26125	0.211427	0.105714	−	0.994397i	$-0.466287\pi$
$877$	6.26125	0.211427	0.105714	+	0.994397i	$0.466287\pi$
$878$	−12.5607	−0.423903
$879$	0	0
$880$	7.43768	0.250724
$881$	−3.16036	−0.106475	−0.0532377	−	0.998582i	$-0.516954\pi$
$881$	−3.16036	−0.106475	−0.0532377	+	0.998582i	$0.516954\pi$
$882$	0	0
$883$	−31.3346	−1.05449	−0.527247	−	0.849712i	$-0.676776\pi$
$883$	−31.3346	−1.05449	−0.527247	+	0.849712i	$0.676776\pi$
$884$	−14.7738	−0.496897
$885$	0	0
$886$	−10.7990	−0.362800
$887$	12.6168	0.423632	0.211816	−	0.977310i	$-0.432062\pi$
$887$	12.6168	0.423632	0.211816	+	0.977310i	$0.432062\pi$
$888$	0	0
$889$	17.5451	0.588443
$890$	−11.7928	−0.395296
$891$	0	0
$892$	−7.79243	−0.260910
$893$	9.80523	0.328120
$894$	0	0
$895$	−7.18575	−0.240193
$896$	−11.5258	−0.385051
$897$	0	0
$898$	−5.31957	−0.177516
$899$	2.48163	0.0827669
$900$	0	0
$901$	15.9091	0.530010
$902$	22.3222	0.743248
$903$	0	0
$904$	13.9574	0.464217
$905$	−13.6400	−0.453409
$906$	0	0
$907$	−34.0345	−1.13010	−0.565049	−	0.825057i	$-0.691143\pi$
$907$	−34.0345	−1.13010	−0.565049	+	0.825057i	$0.691143\pi$
$908$	−2.96516	−0.0984022
$909$	0	0
$910$	−7.50267	−0.248711
$911$	41.1212	1.36241	0.681203	−	0.732095i	$-0.261457\pi$
$911$	41.1212	1.36241	0.681203	+	0.732095i	$0.261457\pi$
$912$	0	0
$913$	−87.5083	−2.89610
$914$	−12.7114	−0.420456
$915$	0	0
$916$	23.2710	0.768896
$917$	23.8887	0.788874
$918$	0	0
$919$	−0.391473	−0.0129135	−0.00645675	−	0.999979i	$-0.502055\pi$
$919$	−0.391473	−0.0129135	−0.00645675	+	0.999979i	$0.502055\pi$
$920$	−1.85442	−0.0611385
$921$	0	0
$922$	−13.8204	−0.455150
$923$	−51.4207	−1.69253
$924$	0	0
$925$	−15.3822	−0.505762
$926$	25.3113	0.831779
$927$	0	0
$928$	21.5289	0.706721
$929$	−20.6457	−0.677365	−0.338682	−	0.940901i	$-0.609981\pi$
$929$	−20.6457	−0.677365	−0.338682	+	0.940901i	$0.609981\pi$
$930$	0	0
$931$	−5.36679	−0.175889
$932$	−14.1755	−0.464333
$933$	0	0
$934$	−2.82845	−0.0925499
$935$	−17.0241	−0.556747
$936$	0	0
$937$	−44.5315	−1.45478	−0.727390	−	0.686225i	$-0.759267\pi$
$937$	−44.5315	−1.45478	−0.727390	+	0.686225i	$0.759267\pi$
$938$	7.98464	0.260708
$939$	0	0
$940$	−21.7522	−0.709477
$941$	40.1815	1.30988	0.654940	−	0.755681i	$-0.272694\pi$
$941$	40.1815	1.30988	0.654940	+	0.755681i	$0.272694\pi$
$942$	0	0
$943$	2.35339	0.0766369
$944$	4.41304	0.143632
$945$	0	0
$946$	14.2003	0.461691
$947$	54.0978	1.75794	0.878972	−	0.476874i	$-0.158230\pi$
$947$	54.0978	1.75794	0.878972	+	0.476874i	$0.158230\pi$
$948$	0	0
$949$	−52.3832	−1.70043
$950$	−2.00105	−0.0649225
$951$	0	0
$952$	6.04398	0.195887
$953$	18.8291	0.609933	0.304967	−	0.952363i	$-0.401355\pi$
$953$	18.8291	0.609933	0.304967	+	0.952363i	$0.401355\pi$
$954$	0	0
$955$	37.1998	1.20376
$956$	18.2539	0.590374
$957$	0	0
$958$	−10.4345	−0.337122
$959$	−23.1732	−0.748302
$960$	0	0
$961$	−30.5436	−0.985277
$962$	−23.4069	−0.754669
$963$	0	0
$964$	−13.5812	−0.437421
$965$	23.4867	0.756065
$966$	0	0
$967$	53.0693	1.70659	0.853296	−	0.521426i	$-0.174600\pi$
$967$	53.0693	1.70659	0.853296	+	0.521426i	$0.174600\pi$
$968$	−60.2682	−1.93709
$969$	0	0
$970$	−7.01417	−0.225211
$971$	−40.4523	−1.29818	−0.649088	−	0.760713i	$-0.724849\pi$
$971$	−40.4523	−1.29818	−0.649088	+	0.760713i	$0.724849\pi$
$972$	0	0
$973$	13.5767	0.435248
$974$	20.3525	0.652136
$975$	0	0
$976$	−2.97480	−0.0952209
$977$	10.6610	0.341075	0.170537	−	0.985351i	$-0.445450\pi$
$977$	10.6610	0.341075	0.170537	+	0.985351i	$0.445450\pi$
$978$	0	0
$979$	−59.6474	−1.90634
$980$	11.9058	0.380317
$981$	0	0
$982$	−6.31572	−0.201543
$983$	−50.7688	−1.61927	−0.809637	−	0.586931i	$-0.800336\pi$
$983$	−50.7688	−1.61927	−0.809637	+	0.586931i	$0.800336\pi$
$984$	0	0
$985$	22.8599	0.728377
$986$	−5.40680	−0.172188
$987$	0	0
$988$	7.40233	0.235499
$989$	1.49711	0.0476054
$990$	0	0
$991$	−31.6640	−1.00584	−0.502919	−	0.864333i	$-0.667741\pi$
$991$	−31.6640	−1.00584	−0.502919	+	0.864333i	$0.667741\pi$
$992$	3.95952	0.125715
$993$	0	0
$994$	8.72385	0.276704
$995$	20.6115	0.653429
$996$	0	0
$997$	46.8252	1.48297	0.741485	−	0.670970i	$-0.234122\pi$
$997$	46.8252	1.48297	0.741485	+	0.670970i	$0.234122\pi$
$998$	−24.5460	−0.776990
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4923.2.a.l.1.10		18
3.2	odd	2		547.2.a.b.1.9	✓	18
12.11	even	2		8752.2.a.s.1.4		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.a.b.1.9	✓	18	3.2	odd	2
4923.2.a.l.1.10		18	1.1	even	1	trivial
8752.2.a.s.1.4		18	12.11	even	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:	\( N \)	\(=\)	\( 4923 = 3^{2} \cdot 547 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4923.a (trivial)

\(f(q)\)	\(=\)	\(q+0.763493 q^{2} -1.41708 q^{4} +1.51218 q^{5} +1.20167 q^{7} -2.60892 q^{8} +O(q^{10})\)	\(q+0.763493 q^{2} -1.41708 q^{4} +1.51218 q^{5} +1.20167 q^{7} -2.60892 q^{8} +1.15454 q^{10} +5.83960 q^{11} -5.40780 q^{13} +0.917468 q^{14} +0.842269 q^{16} -1.92787 q^{17} +0.965947 q^{19} -2.14288 q^{20} +4.45849 q^{22} +0.470051 q^{23} -2.71331 q^{25} -4.12882 q^{26} -1.70286 q^{28} +3.67331 q^{29} +0.675583 q^{31} +5.86090 q^{32} -1.47191 q^{34} +1.81715 q^{35} +5.66915 q^{37} +0.737494 q^{38} -3.94515 q^{40} +5.00667 q^{41} +3.18500 q^{43} -8.27517 q^{44} +0.358881 q^{46} +10.1509 q^{47} -5.55598 q^{49} -2.07159 q^{50} +7.66328 q^{52} -8.25219 q^{53} +8.83053 q^{55} -3.13506 q^{56} +2.80455 q^{58} +5.23947 q^{59} -3.53188 q^{61} +0.515803 q^{62} +2.79021 q^{64} -8.17758 q^{65} +8.70291 q^{67} +2.73194 q^{68} +1.38738 q^{70} +9.50861 q^{71} +9.68659 q^{73} +4.32836 q^{74} -1.36882 q^{76} +7.01728 q^{77} -5.23633 q^{79} +1.27366 q^{80} +3.82256 q^{82} -14.9853 q^{83} -2.91529 q^{85} +2.43172 q^{86} -15.2350 q^{88} -10.2143 q^{89} -6.49841 q^{91} -0.666099 q^{92} +7.75014 q^{94} +1.46069 q^{95} -6.07529 q^{97} -4.24195 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 4 q^{2} + 16 q^{4} + 27 q^{5} - 11 q^{7} + 12 q^{8}+O(q^{10}) \) 18 * q + 4 * q^2 + 16 * q^4 + 27 * q^5 - 11 * q^7 + 12 * q^8	\( 18 q + 4 q^{2} + 16 q^{4} + 27 q^{5} - 11 q^{7} + 12 q^{8} - 5 q^{10} - 2 q^{11} - 25 q^{13} + 7 q^{14} + 8 q^{16} + 30 q^{17} + 4 q^{19} + 41 q^{20} - 24 q^{22} + 26 q^{23} + 31 q^{25} + 18 q^{26} - 16 q^{28} + 18 q^{29} - 5 q^{31} + 28 q^{32} + 5 q^{34} + 9 q^{35} - 18 q^{37} + 45 q^{38} + 7 q^{40} + 17 q^{41} + 8 q^{43} - 12 q^{44} + 30 q^{46} + 52 q^{47} + 29 q^{49} - 13 q^{50} - 14 q^{52} + 60 q^{53} + 11 q^{55} - 7 q^{56} + 14 q^{58} + 8 q^{59} - 26 q^{61} - 4 q^{62} + 44 q^{64} + 6 q^{65} + 12 q^{67} + 61 q^{68} + 35 q^{70} + q^{71} - 2 q^{73} - 16 q^{74} + 66 q^{76} + 73 q^{77} + 18 q^{79} + 32 q^{80} + 44 q^{82} + 43 q^{83} + 51 q^{85} - 4 q^{86} - 17 q^{88} + 28 q^{89} - q^{91} + 68 q^{92} + 78 q^{94} + 18 q^{95} - 34 q^{97} - 34 q^{98}+O(q^{100}) \) 18 * q + 4 * q^2 + 16 * q^4 + 27 * q^5 - 11 * q^7 + 12 * q^8 - 5 * q^10 - 2 * q^11 - 25 * q^13 + 7 * q^14 + 8 * q^16 + 30 * q^17 + 4 * q^19 + 41 * q^20 - 24 * q^22 + 26 * q^23 + 31 * q^25 + 18 * q^26 - 16 * q^28 + 18 * q^29 - 5 * q^31 + 28 * q^32 + 5 * q^34 + 9 * q^35 - 18 * q^37 + 45 * q^38 + 7 * q^40 + 17 * q^41 + 8 * q^43 - 12 * q^44 + 30 * q^46 + 52 * q^47 + 29 * q^49 - 13 * q^50 - 14 * q^52 + 60 * q^53 + 11 * q^55 - 7 * q^56 + 14 * q^58 + 8 * q^59 - 26 * q^61 - 4 * q^62 + 44 * q^64 + 6 * q^65 + 12 * q^67 + 61 * q^68 + 35 * q^70 + q^71 - 2 * q^73 - 16 * q^74 + 66 * q^76 + 73 * q^77 + 18 * q^79 + 32 * q^80 + 44 * q^82 + 43 * q^83 + 51 * q^85 - 4 * q^86 - 17 * q^88 + 28 * q^89 - q^91 + 68 * q^92 + 78 * q^94 + 18 * q^95 - 34 * q^97 - 34 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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