LMFDB - Embedded newform 4923.2.a.l.1.14

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			1.14
Root			$1.74487$ 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)

$f(q)$	$=$	$q+1.74487 q^{2} +1.04455 q^{4} +3.61409 q^{5} +4.28084 q^{7} -1.66712 q^{8} +O(q^{10})$	\(q+1.74487 q^{2} +1.04455 q^{4} +3.61409 q^{5} +4.28084 q^{7} -1.66712 q^{8} +6.30610 q^{10} +1.82207 q^{11} +1.54763 q^{13} +7.46949 q^{14} -4.99801 q^{16} -5.89804 q^{17} +5.77207 q^{19} +3.77512 q^{20} +3.17927 q^{22} +7.36494 q^{23} +8.06166 q^{25} +2.70041 q^{26} +4.47157 q^{28} +4.20778 q^{29} -8.68625 q^{31} -5.38662 q^{32} -10.2913 q^{34} +15.4714 q^{35} -9.26711 q^{37} +10.0715 q^{38} -6.02514 q^{40} -8.29157 q^{41} -2.50047 q^{43} +1.90326 q^{44} +12.8508 q^{46} +6.07577 q^{47} +11.3256 q^{49} +14.0665 q^{50} +1.61659 q^{52} +4.35174 q^{53} +6.58514 q^{55} -7.13669 q^{56} +7.34201 q^{58} +13.6982 q^{59} -9.67314 q^{61} -15.1563 q^{62} +0.597111 q^{64} +5.59329 q^{65} -3.67729 q^{67} -6.16082 q^{68} +26.9954 q^{70} -6.92649 q^{71} +2.36768 q^{73} -16.1699 q^{74} +6.02925 q^{76} +7.80001 q^{77} +6.27816 q^{79} -18.0633 q^{80} -14.4677 q^{82} +2.57942 q^{83} -21.3161 q^{85} -4.36299 q^{86} -3.03762 q^{88} -8.45581 q^{89} +6.62517 q^{91} +7.69309 q^{92} +10.6014 q^{94} +20.8608 q^{95} -11.0935 q^{97} +19.7617 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.74487	1.23381	0.616903	−	0.787039i	$-0.288387\pi$
$2$	1.74487	1.23381	0.616903	+	0.787039i	$0.288387\pi$
$3$	0	0
$3$	0	0
$4$	1.04455	0.522277
$5$	3.61409	1.61627	0.808135	−	0.588997i	$-0.200477\pi$
$5$	3.61409	1.61627	0.808135	+	0.588997i	$0.200477\pi$
$6$	0	0
$7$	4.28084	1.61801	0.809003	−	0.587804i	$-0.200008\pi$
$7$	4.28084	1.61801	0.809003	+	0.587804i	$0.200008\pi$
$8$	−1.66712	−0.589417
$9$	0	0
$10$	6.30610	1.99416
$11$	1.82207	0.549376	0.274688	−	0.961533i	$-0.411425\pi$
$11$	1.82207	0.549376	0.274688	+	0.961533i	$0.411425\pi$
$12$	0	0
$13$	1.54763	0.429236	0.214618	−	0.976698i	$-0.431149\pi$
$13$	1.54763	0.429236	0.214618	+	0.976698i	$0.431149\pi$
$14$	7.46949	1.99631
$15$	0	0
$16$	−4.99801	−1.24950
$17$	−5.89804	−1.43048	−0.715242	−	0.698877i	$-0.753684\pi$
$17$	−5.89804	−1.43048	−0.715242	+	0.698877i	$0.753684\pi$
$18$	0	0
$19$	5.77207	1.32420	0.662102	−	0.749414i	$-0.269664\pi$
$19$	5.77207	1.32420	0.662102	+	0.749414i	$0.269664\pi$
$20$	3.77512	0.844142
$21$	0	0
$22$	3.17927	0.677824
$23$	7.36494	1.53570	0.767848	−	0.640632i	$-0.221327\pi$
$23$	7.36494	1.53570	0.767848	+	0.640632i	$0.221327\pi$
$24$	0	0
$25$	8.06166	1.61233
$26$	2.70041	0.529594
$27$	0	0
$28$	4.47157	0.845048
$29$	4.20778	0.781365	0.390683	−	0.920525i	$-0.372239\pi$
$29$	4.20778	0.781365	0.390683	+	0.920525i	$0.372239\pi$
$30$	0	0
$31$	−8.68625	−1.56010	−0.780049	−	0.625719i	$-0.784806\pi$
$31$	−8.68625	−1.56010	−0.780049	+	0.625719i	$0.784806\pi$
$32$	−5.38662	−0.952228
$33$	0	0
$34$	−10.2913	−1.76494
$35$	15.4714	2.61514
$36$	0	0
$37$	−9.26711	−1.52350	−0.761752	−	0.647869i	$-0.775660\pi$
$37$	−9.26711	−1.52350	−0.761752	+	0.647869i	$0.775660\pi$
$38$	10.0715	1.63381
$39$	0	0
$40$	−6.02514	−0.952658
$41$	−8.29157	−1.29493	−0.647463	−	0.762097i	$-0.724170\pi$
$41$	−8.29157	−1.29493	−0.647463	+	0.762097i	$0.724170\pi$
$42$	0	0
$43$	−2.50047	−0.381319	−0.190659	−	0.981656i	$-0.561063\pi$
$43$	−2.50047	−0.381319	−0.190659	+	0.981656i	$0.561063\pi$
$44$	1.90326	0.286927
$45$	0	0
$46$	12.8508	1.89475
$47$	6.07577	0.886242	0.443121	−	0.896462i	$-0.353871\pi$
$47$	6.07577	0.886242	0.443121	+	0.896462i	$0.353871\pi$
$48$	0	0
$49$	11.3256	1.61794
$50$	14.0665	1.98930
$51$	0	0
$52$	1.61659	0.224180
$53$	4.35174	0.597757	0.298879	−	0.954291i	$-0.403387\pi$
$53$	4.35174	0.597757	0.298879	+	0.954291i	$0.403387\pi$
$54$	0	0
$55$	6.58514	0.887941
$56$	−7.13669	−0.953680
$57$	0	0
$58$	7.34201	0.964053
$59$	13.6982	1.78336	0.891680	−	0.452667i	$-0.149527\pi$
$59$	13.6982	1.78336	0.891680	+	0.452667i	$0.149527\pi$
$60$	0	0
$61$	−9.67314	−1.23852	−0.619259	−	0.785186i	$-0.712567\pi$
$61$	−9.67314	−1.23852	−0.619259	+	0.785186i	$0.712567\pi$
$62$	−15.1563	−1.92486
$63$	0	0
$64$	0.597111	0.0746389
$65$	5.59329	0.693762
$66$	0	0
$67$	−3.67729	−0.449253	−0.224626	−	0.974445i	$-0.572116\pi$
$67$	−3.67729	−0.449253	−0.224626	+	0.974445i	$0.572116\pi$
$68$	−6.16082	−0.747110
$69$	0	0
$70$	26.9954	3.22657
$71$	−6.92649	−0.822023	−0.411012	−	0.911630i	$-0.634824\pi$
$71$	−6.92649	−0.822023	−0.411012	+	0.911630i	$0.634824\pi$
$72$	0	0
$73$	2.36768	0.277116	0.138558	−	0.990354i	$-0.455753\pi$
$73$	2.36768	0.277116	0.138558	+	0.990354i	$0.455753\pi$
$74$	−16.1699	−1.87971
$75$	0	0
$76$	6.02925	0.691602
$77$	7.80001	0.888894
$78$	0	0
$79$	6.27816	0.706348	0.353174	−	0.935558i	$-0.385102\pi$
$79$	6.27816	0.706348	0.353174	+	0.935558i	$0.385102\pi$
$80$	−18.0633	−2.01954
$81$	0	0
$82$	−14.4677	−1.59769
$83$	2.57942	0.283128	0.141564	−	0.989929i	$-0.454787\pi$
$83$	2.57942	0.283128	0.141564	+	0.989929i	$0.454787\pi$
$84$	0	0
$85$	−21.3161	−2.31205
$86$	−4.36299	−0.470473
$87$	0	0
$88$	−3.03762	−0.323812
$89$	−8.45581	−0.896314	−0.448157	−	0.893955i	$-0.647919\pi$
$89$	−8.45581	−0.896314	−0.448157	+	0.893955i	$0.647919\pi$
$90$	0	0
$91$	6.62517	0.694507
$92$	7.69309	0.802060
$93$	0	0
$94$	10.6014	1.09345
$95$	20.8608	2.14027
$96$	0	0
$97$	−11.0935	−1.12637	−0.563187	−	0.826329i	$-0.690425\pi$
$97$	−11.0935	−1.12637	−0.563187	+	0.826329i	$0.690425\pi$
$98$	19.7617	1.99623
$99$	0	0
$100$	8.42084	0.842084
$101$	12.1254	1.20652	0.603259	−	0.797545i	$-0.293869\pi$
$101$	12.1254	1.20652	0.603259	+	0.797545i	$0.293869\pi$
$102$	0	0
$103$	−10.8356	−1.06767	−0.533833	−	0.845590i	$-0.679249\pi$
$103$	−10.8356	−1.06767	−0.533833	+	0.845590i	$0.679249\pi$
$104$	−2.58010	−0.252999
$105$	0	0
$106$	7.59320	0.737517
$107$	−15.7365	−1.52130	−0.760652	−	0.649160i	$-0.775121\pi$
$107$	−15.7365	−1.52130	−0.760652	+	0.649160i	$0.775121\pi$
$108$	0	0
$109$	−7.74111	−0.741464	−0.370732	−	0.928740i	$-0.620893\pi$
$109$	−7.74111	−0.741464	−0.370732	+	0.928740i	$0.620893\pi$
$110$	11.4902	1.09555
$111$	0	0
$112$	−21.3957	−2.02170
$113$	0.0433232	0.00407550	0.00203775	−	0.999998i	$-0.499351\pi$
$113$	0.0433232	0.00407550	0.00203775	+	0.999998i	$0.499351\pi$
$114$	0	0
$115$	26.6176	2.48210
$116$	4.39526	0.408089
$117$	0	0
$118$	23.9016	2.20032
$119$	−25.2486	−2.31453
$120$	0	0
$121$	−7.68004	−0.698186
$122$	−16.8783	−1.52809
$123$	0	0
$124$	−9.07327	−0.814803
$125$	11.0651	0.989694
$126$	0	0
$127$	0.557420	0.0494630	0.0247315	−	0.999694i	$-0.492127\pi$
$127$	0.557420	0.0494630	0.0247315	+	0.999694i	$0.492127\pi$
$128$	11.8151	1.04432
$129$	0	0
$130$	9.75954	0.855968
$131$	4.31192	0.376735	0.188367	−	0.982099i	$-0.439680\pi$
$131$	4.31192	0.376735	0.188367	+	0.982099i	$0.439680\pi$
$132$	0	0
$133$	24.7093	2.14257
$134$	−6.41638	−0.554291
$135$	0	0
$136$	9.83276	0.843152
$137$	9.86688	0.842985	0.421492	−	0.906832i	$-0.361506\pi$
$137$	9.86688	0.842985	0.421492	+	0.906832i	$0.361506\pi$
$138$	0	0
$139$	11.3673	0.964163	0.482082	−	0.876126i	$-0.339881\pi$
$139$	11.3673	0.964163	0.482082	+	0.876126i	$0.339881\pi$
$140$	16.1607	1.36583
$141$	0	0
$142$	−12.0858	−1.01422
$143$	2.81990	0.235812
$144$	0	0
$145$	15.2073	1.26290
$146$	4.13128	0.341907
$147$	0	0
$148$	−9.68001	−0.795692
$149$	−14.2550	−1.16781	−0.583907	−	0.811821i	$-0.698477\pi$
$149$	−14.2550	−1.16781	−0.583907	+	0.811821i	$0.698477\pi$
$150$	0	0
$151$	17.5151	1.42536	0.712680	−	0.701489i	$-0.247481\pi$
$151$	17.5151	1.42536	0.712680	+	0.701489i	$0.247481\pi$
$152$	−9.62276	−0.780509
$153$	0	0
$154$	13.6100	1.09672
$155$	−31.3929	−2.52154
$156$	0	0
$157$	−0.960703	−0.0766725	−0.0383362	−	0.999265i	$-0.512206\pi$
$157$	−0.960703	−0.0766725	−0.0383362	+	0.999265i	$0.512206\pi$
$158$	10.9545	0.871497
$159$	0	0
$160$	−19.4677	−1.53906
$161$	31.5282	2.48477
$162$	0	0
$163$	−8.10300	−0.634676	−0.317338	−	0.948313i	$-0.602789\pi$
$163$	−8.10300	−0.634676	−0.317338	+	0.948313i	$0.602789\pi$
$164$	−8.66100	−0.676311
$165$	0	0
$166$	4.50074	0.349325
$167$	12.7559	0.987083	0.493541	−	0.869722i	$-0.335702\pi$
$167$	12.7559	0.987083	0.493541	+	0.869722i	$0.335702\pi$
$168$	0	0
$169$	−10.6048	−0.815756
$170$	−37.1936	−2.85262
$171$	0	0
$172$	−2.61188	−0.199154
$173$	6.36016	0.483554	0.241777	−	0.970332i	$-0.422270\pi$
$173$	6.36016	0.483554	0.241777	+	0.970332i	$0.422270\pi$
$174$	0	0
$175$	34.5107	2.60876
$176$	−9.10675	−0.686447
$177$	0	0
$178$	−14.7542	−1.10588
$179$	−15.6282	−1.16811	−0.584055	−	0.811714i	$-0.698535\pi$
$179$	−15.6282	−1.16811	−0.584055	+	0.811714i	$0.698535\pi$
$180$	0	0
$181$	−12.0567	−0.896164	−0.448082	−	0.893992i	$-0.647893\pi$
$181$	−12.0567	−0.896164	−0.448082	+	0.893992i	$0.647893\pi$
$182$	11.5600	0.856887
$183$	0	0
$184$	−12.2783	−0.905166
$185$	−33.4922	−2.46240
$186$	0	0
$187$	−10.7467	−0.785874
$188$	6.34648	0.462864
$189$	0	0
$190$	36.3993	2.64068
$191$	15.1622	1.09710	0.548550	−	0.836118i	$-0.315180\pi$
$191$	15.1622	1.09710	0.548550	+	0.836118i	$0.315180\pi$
$192$	0	0
$193$	−6.95522	−0.500648	−0.250324	−	0.968162i	$-0.580537\pi$
$193$	−6.95522	−0.500648	−0.250324	+	0.968162i	$0.580537\pi$
$194$	−19.3567	−1.38973
$195$	0	0
$196$	11.8302	0.845015
$197$	14.5377	1.03577	0.517884	−	0.855451i	$-0.326720\pi$
$197$	14.5377	1.03577	0.517884	+	0.855451i	$0.326720\pi$
$198$	0	0
$199$	−5.94365	−0.421334	−0.210667	−	0.977558i	$-0.567564\pi$
$199$	−5.94365	−0.421334	−0.210667	+	0.977558i	$0.567564\pi$
$200$	−13.4398	−0.950336
$201$	0	0
$202$	21.1571	1.48861
$203$	18.0128	1.26425
$204$	0	0
$205$	−29.9665	−2.09295
$206$	−18.9067	−1.31729
$207$	0	0
$208$	−7.73509	−0.536332
$209$	10.5171	0.727486
$210$	0	0
$211$	−27.1667	−1.87024	−0.935118	−	0.354337i	$-0.884707\pi$
$211$	−27.1667	−1.87024	−0.935118	+	0.354337i	$0.884707\pi$
$212$	4.54563	0.312195
$213$	0	0
$214$	−27.4581	−1.87699
$215$	−9.03694	−0.616314
$216$	0	0
$217$	−37.1845	−2.52425
$218$	−13.5072	−0.914823
$219$	0	0
$220$	6.87854	0.463751
$221$	−9.12800	−0.614016
$222$	0	0
$223$	3.41548	0.228717	0.114359	−	0.993440i	$-0.463519\pi$
$223$	3.41548	0.228717	0.114359	+	0.993440i	$0.463519\pi$
$224$	−23.0592	−1.54071
$225$	0	0
$226$	0.0755931	0.00502838
$227$	22.1539	1.47041	0.735204	−	0.677846i	$-0.237086\pi$
$227$	22.1539	1.47041	0.735204	+	0.677846i	$0.237086\pi$
$228$	0	0
$229$	1.65658	0.109470	0.0547350	−	0.998501i	$-0.482569\pi$
$229$	1.65658	0.109470	0.0547350	+	0.998501i	$0.482569\pi$
$230$	46.4441	3.06243
$231$	0	0
$232$	−7.01489	−0.460550
$233$	−9.45846	−0.619644	−0.309822	−	0.950795i	$-0.600269\pi$
$233$	−9.45846	−0.619644	−0.309822	+	0.950795i	$0.600269\pi$
$234$	0	0
$235$	21.9584	1.43241
$236$	14.3086	0.931408
$237$	0	0
$238$	−44.0554	−2.85568
$239$	20.2498	1.30985	0.654926	−	0.755693i	$-0.272700\pi$
$239$	20.2498	1.30985	0.654926	+	0.755693i	$0.272700\pi$
$240$	0	0
$241$	12.5021	0.805328	0.402664	−	0.915348i	$-0.368084\pi$
$241$	12.5021	0.805328	0.402664	+	0.915348i	$0.368084\pi$
$242$	−13.4006	−0.861426
$243$	0	0
$244$	−10.1041	−0.646850
$245$	40.9318	2.61504
$246$	0	0
$247$	8.93305	0.568397
$248$	14.4811	0.919548
$249$	0	0
$250$	19.3071	1.22109
$251$	−15.4087	−0.972590	−0.486295	−	0.873795i	$-0.661652\pi$
$251$	−15.4087	−0.972590	−0.486295	+	0.873795i	$0.661652\pi$
$252$	0	0
$253$	13.4195	0.843675
$254$	0.972622	0.0610277
$255$	0	0
$256$	19.4216	1.21385
$257$	5.96426	0.372040	0.186020	−	0.982546i	$-0.440441\pi$
$257$	5.96426	0.372040	0.186020	+	0.982546i	$0.440441\pi$
$258$	0	0
$259$	−39.6710	−2.46504
$260$	5.84250	0.362336
$261$	0	0
$262$	7.52373	0.464817
$263$	−22.1233	−1.36418	−0.682089	−	0.731269i	$-0.738928\pi$
$263$	−22.1233	−1.36418	−0.682089	+	0.731269i	$0.738928\pi$
$264$	0	0
$265$	15.7276	0.966138
$266$	43.1144	2.64352
$267$	0	0
$268$	−3.84113	−0.234635
$269$	8.96149	0.546391	0.273196	−	0.961958i	$-0.411919\pi$
$269$	8.96149	0.546391	0.273196	+	0.961958i	$0.411919\pi$
$270$	0	0
$271$	2.25602	0.137043	0.0685217	−	0.997650i	$-0.478172\pi$
$271$	2.25602	0.137043	0.0685217	+	0.997650i	$0.478172\pi$
$272$	29.4785	1.78740
$273$	0	0
$274$	17.2164	1.04008
$275$	14.6889	0.885777
$276$	0	0
$277$	−12.2849	−0.738128	−0.369064	−	0.929404i	$-0.620322\pi$
$277$	−12.2849	−0.738128	−0.369064	+	0.929404i	$0.620322\pi$
$278$	19.8344	1.18959
$279$	0	0
$280$	−25.7927	−1.54141
$281$	31.3296	1.86897	0.934484	−	0.356005i	$-0.115861\pi$
$281$	31.3296	1.86897	0.934484	+	0.356005i	$0.115861\pi$
$282$	0	0
$283$	26.8312	1.59495	0.797474	−	0.603353i	$-0.206169\pi$
$283$	26.8312	1.59495	0.797474	+	0.603353i	$0.206169\pi$
$284$	−7.23510	−0.429324
$285$	0	0
$286$	4.92035	0.290946
$287$	−35.4949	−2.09520
$288$	0	0
$289$	17.7869	1.04629
$290$	26.5347	1.55817
$291$	0	0
$292$	2.47317	0.144731
$293$	−8.91567	−0.520859	−0.260430	−	0.965493i	$-0.583864\pi$
$293$	−8.91567	−0.520859	−0.260430	+	0.965493i	$0.583864\pi$
$294$	0	0
$295$	49.5067	2.88239
$296$	15.4494	0.897979
$297$	0	0
$298$	−24.8730	−1.44086
$299$	11.3982	0.659177
$300$	0	0
$301$	−10.7041	−0.616976
$302$	30.5615	1.75862
$303$	0	0
$304$	−28.8489	−1.65460
$305$	−34.9596	−2.00178
$306$	0	0
$307$	−20.6821	−1.18039	−0.590195	−	0.807260i	$-0.700949\pi$
$307$	−20.6821	−1.18039	−0.590195	+	0.807260i	$0.700949\pi$
$308$	8.14754	0.464249
$309$	0	0
$310$	−54.7764	−3.11109
$311$	−9.54165	−0.541057	−0.270529	−	0.962712i	$-0.587198\pi$
$311$	−9.54165	−0.541057	−0.270529	+	0.962712i	$0.587198\pi$
$312$	0	0
$313$	−19.8618	−1.12265	−0.561327	−	0.827594i	$-0.689709\pi$
$313$	−19.8618	−1.12265	−0.561327	+	0.827594i	$0.689709\pi$
$314$	−1.67630	−0.0945989
$315$	0	0
$316$	6.55788	0.368910
$317$	12.6386	0.709857	0.354929	−	0.934893i	$-0.384505\pi$
$317$	12.6386	0.709857	0.354929	+	0.934893i	$0.384505\pi$
$318$	0	0
$319$	7.66689	0.429263
$320$	2.15802	0.120637
$321$	0	0
$322$	55.0124	3.06572
$323$	−34.0439	−1.89425
$324$	0	0
$325$	12.4765	0.692071
$326$	−14.1386	−0.783067
$327$	0	0
$328$	13.8231	0.763252
$329$	26.0094	1.43395
$330$	0	0
$331$	17.8621	0.981790	0.490895	−	0.871219i	$-0.336670\pi$
$331$	17.8621	0.981790	0.490895	+	0.871219i	$0.336670\pi$
$332$	2.69434	0.147871
$333$	0	0
$334$	22.2574	1.21787
$335$	−13.2901	−0.726114
$336$	0	0
$337$	11.1406	0.606865	0.303432	−	0.952853i	$-0.401867\pi$
$337$	11.1406	0.606865	0.303432	+	0.952853i	$0.401867\pi$
$338$	−18.5040	−1.00648
$339$	0	0
$340$	−22.2658	−1.20753
$341$	−15.8270	−0.857080
$342$	0	0
$343$	18.5172	0.999836
$344$	4.16860	0.224756
$345$	0	0
$346$	11.0976	0.596612
$347$	0.533400	0.0286344	0.0143172	−	0.999898i	$-0.495443\pi$
$347$	0.533400	0.0286344	0.0143172	+	0.999898i	$0.495443\pi$
$348$	0	0
$349$	26.6600	1.42708	0.713539	−	0.700616i	$-0.247091\pi$
$349$	26.6600	1.42708	0.713539	+	0.700616i	$0.247091\pi$
$350$	60.2165	3.21871
$351$	0	0
$352$	−9.81481	−0.523131
$353$	2.12890	0.113310	0.0566548	−	0.998394i	$-0.481957\pi$
$353$	2.12890	0.113310	0.0566548	+	0.998394i	$0.481957\pi$
$354$	0	0
$355$	−25.0330	−1.32861
$356$	−8.83255	−0.468124
$357$	0	0
$358$	−27.2692	−1.44122
$359$	−1.54866	−0.0817353	−0.0408677	−	0.999165i	$-0.513012\pi$
$359$	−1.54866	−0.0817353	−0.0408677	+	0.999165i	$0.513012\pi$
$360$	0	0
$361$	14.3168	0.753517
$362$	−21.0372	−1.10569
$363$	0	0
$364$	6.92036	0.362725
$365$	8.55700	0.447894
$366$	0	0
$367$	8.10473	0.423064	0.211532	−	0.977371i	$-0.432155\pi$
$367$	8.10473	0.423064	0.211532	+	0.977371i	$0.432155\pi$
$368$	−36.8101	−1.91886
$369$	0	0
$370$	−58.4394	−3.03812
$371$	18.6291	0.967175
$372$	0	0
$373$	36.4059	1.88503	0.942514	−	0.334167i	$-0.108455\pi$
$373$	36.4059	1.88503	0.942514	+	0.334167i	$0.108455\pi$
$374$	−18.7515	−0.969616
$375$	0	0
$376$	−10.1291	−0.522366
$377$	6.51210	0.335390
$378$	0	0
$379$	−1.17841	−0.0605309	−0.0302655	−	0.999542i	$-0.509635\pi$
$379$	−1.17841	−0.0605309	−0.0302655	+	0.999542i	$0.509635\pi$
$380$	21.7902	1.11782
$381$	0	0
$382$	26.4560	1.35361
$383$	−3.86230	−0.197355	−0.0986773	−	0.995119i	$-0.531461\pi$
$383$	−3.86230	−0.197355	−0.0986773	+	0.995119i	$0.531461\pi$
$384$	0	0
$385$	28.1900	1.43669
$386$	−12.1359	−0.617702
$387$	0	0
$388$	−11.5878	−0.588280
$389$	9.33318	0.473211	0.236605	−	0.971606i	$-0.423965\pi$
$389$	9.33318	0.473211	0.236605	+	0.971606i	$0.423965\pi$
$390$	0	0
$391$	−43.4387	−2.19679
$392$	−18.8812	−0.953644
$393$	0	0
$394$	25.3663	1.27794
$395$	22.6898	1.14165
$396$	0	0
$397$	−29.7948	−1.49536	−0.747680	−	0.664059i	$-0.768832\pi$
$397$	−29.7948	−1.49536	−0.747680	+	0.664059i	$0.768832\pi$
$398$	−10.3709	−0.519844
$399$	0	0
$400$	−40.2923	−2.01461
$401$	−5.34757	−0.267045	−0.133522	−	0.991046i	$-0.542629\pi$
$401$	−5.34757	−0.267045	−0.133522	+	0.991046i	$0.542629\pi$
$402$	0	0
$403$	−13.4431	−0.669650
$404$	12.6656	0.630137
$405$	0	0
$406$	31.4300	1.55984
$407$	−16.8854	−0.836977
$408$	0	0
$409$	−6.05618	−0.299459	−0.149729	−	0.988727i	$-0.547840\pi$
$409$	−6.05618	−0.299459	−0.149729	+	0.988727i	$0.547840\pi$
$410$	−52.2875	−2.58230
$411$	0	0
$412$	−11.3184	−0.557617
$413$	58.6400	2.88549
$414$	0	0
$415$	9.32226	0.457612
$416$	−8.33651	−0.408731
$417$	0	0
$418$	18.3510	0.897577
$419$	−8.84901	−0.432303	−0.216151	−	0.976360i	$-0.569350\pi$
$419$	−8.84901	−0.432303	−0.216151	+	0.976360i	$0.569350\pi$
$420$	0	0
$421$	−18.4589	−0.899631	−0.449816	−	0.893121i	$-0.648510\pi$
$421$	−18.4589	−0.899631	−0.449816	+	0.893121i	$0.648510\pi$
$422$	−47.4023	−2.30751
$423$	0	0
$424$	−7.25489	−0.352328
$425$	−47.5480	−2.30642
$426$	0	0
$427$	−41.4092	−2.00393
$428$	−16.4376	−0.794543
$429$	0	0
$430$	−15.7682	−0.760413
$431$	−17.9207	−0.863212	−0.431606	−	0.902062i	$-0.642053\pi$
$431$	−17.9207	−0.863212	−0.431606	+	0.902062i	$0.642053\pi$
$432$	0	0
$433$	−39.6184	−1.90394	−0.951969	−	0.306193i	$-0.900945\pi$
$433$	−39.6184	−1.90394	−0.951969	+	0.306193i	$0.900945\pi$
$434$	−64.8819	−3.11443
$435$	0	0
$436$	−8.08602	−0.387250
$437$	42.5110	2.03358
$438$	0	0
$439$	14.0915	0.672553	0.336276	−	0.941763i	$-0.390832\pi$
$439$	14.0915	0.672553	0.336276	+	0.941763i	$0.390832\pi$
$440$	−10.9782	−0.523367
$441$	0	0
$442$	−15.9271	−0.757576
$443$	24.4089	1.15970	0.579850	−	0.814723i	$-0.303111\pi$
$443$	24.4089	1.15970	0.579850	+	0.814723i	$0.303111\pi$
$444$	0	0
$445$	−30.5601	−1.44869
$446$	5.95955	0.282193
$447$	0	0
$448$	2.55614	0.120766
$449$	−25.2207	−1.19024	−0.595120	−	0.803637i	$-0.702895\pi$
$449$	−25.2207	−1.19024	−0.595120	+	0.803637i	$0.702895\pi$
$450$	0	0
$451$	−15.1079	−0.711401
$452$	0.0452534	0.00212854
$453$	0	0
$454$	38.6556	1.81420
$455$	23.9440	1.12251
$456$	0	0
$457$	33.5255	1.56826	0.784128	−	0.620599i	$-0.213111\pi$
$457$	33.5255	1.56826	0.784128	+	0.620599i	$0.213111\pi$
$458$	2.89051	0.135065
$459$	0	0
$460$	27.8035	1.29635
$461$	−8.70385	−0.405379	−0.202689	−	0.979243i	$-0.564968\pi$
$461$	−8.70385	−0.405379	−0.202689	+	0.979243i	$0.564968\pi$
$462$	0	0
$463$	16.2456	0.754997	0.377498	−	0.926010i	$-0.376784\pi$
$463$	16.2456	0.754997	0.377498	+	0.926010i	$0.376784\pi$
$464$	−21.0306	−0.976319
$465$	0	0
$466$	−16.5037	−0.764521
$467$	−13.7642	−0.636932	−0.318466	−	0.947934i	$-0.603168\pi$
$467$	−13.7642	−0.636932	−0.318466	+	0.947934i	$0.603168\pi$
$468$	0	0
$469$	−15.7419	−0.726894
$470$	38.3144	1.76731
$471$	0	0
$472$	−22.8367	−1.05114
$473$	−4.55605	−0.209487
$474$	0	0
$475$	46.5325	2.13506
$476$	−26.3735	−1.20883
$477$	0	0
$478$	35.3332	1.61610
$479$	−5.69599	−0.260256	−0.130128	−	0.991497i	$-0.541539\pi$
$479$	−5.69599	−0.260256	−0.130128	+	0.991497i	$0.541539\pi$
$480$	0	0
$481$	−14.3421	−0.653943
$482$	21.8144	0.993619
$483$	0	0
$484$	−8.02223	−0.364647
$485$	−40.0930	−1.82053
$486$	0	0
$487$	22.3309	1.01191	0.505954	−	0.862560i	$-0.331140\pi$
$487$	22.3309	1.01191	0.505954	+	0.862560i	$0.331140\pi$
$488$	16.1263	0.730004
$489$	0	0
$490$	71.4204	3.22645
$491$	−27.1853	−1.22686	−0.613429	−	0.789750i	$-0.710210\pi$
$491$	−27.1853	−1.22686	−0.613429	+	0.789750i	$0.710210\pi$
$492$	0	0
$493$	−24.8177	−1.11773
$494$	15.5870	0.701291
$495$	0	0
$496$	43.4140	1.94935
$497$	−29.6512	−1.33004
$498$	0	0
$499$	19.0533	0.852941	0.426470	−	0.904502i	$-0.359757\pi$
$499$	19.0533	0.852941	0.426470	+	0.904502i	$0.359757\pi$
$500$	11.5581	0.516895
$501$	0	0
$502$	−26.8861	−1.19999
$503$	8.63613	0.385066	0.192533	−	0.981290i	$-0.438330\pi$
$503$	8.63613	0.385066	0.192533	+	0.981290i	$0.438330\pi$
$504$	0	0
$505$	43.8222	1.95006
$506$	23.4152	1.04093
$507$	0	0
$508$	0.582255	0.0258334
$509$	−25.5400	−1.13204	−0.566020	−	0.824392i	$-0.691517\pi$
$509$	−25.5400	−1.13204	−0.566020	+	0.824392i	$0.691517\pi$
$510$	0	0
$511$	10.1356	0.448375
$512$	10.2578	0.453334
$513$	0	0
$514$	10.4068	0.459025
$515$	−39.1609	−1.72564
$516$	0	0
$517$	11.0705	0.486880
$518$	−69.2206	−3.04138
$519$	0	0
$520$	−9.32470	−0.408915
$521$	39.4866	1.72994	0.864970	−	0.501824i	$-0.167337\pi$
$521$	39.4866	1.72994	0.864970	+	0.501824i	$0.167337\pi$
$522$	0	0
$523$	−17.3341	−0.757967	−0.378984	−	0.925403i	$-0.623726\pi$
$523$	−17.3341	−0.757967	−0.378984	+	0.925403i	$0.623726\pi$
$524$	4.50404	0.196760
$525$	0	0
$526$	−38.6021	−1.68313
$527$	51.2319	2.23169
$528$	0	0
$529$	31.2424	1.35837
$530$	27.4425	1.19203
$531$	0	0
$532$	25.8102	1.11902
$533$	−12.8323	−0.555829
$534$	0	0
$535$	−56.8731	−2.45884
$536$	6.13050	0.264797
$537$	0	0
$538$	15.6366	0.674141
$539$	20.6361	0.888859
$540$	0	0
$541$	−26.4300	−1.13632	−0.568158	−	0.822919i	$-0.692344\pi$
$541$	−26.4300	−1.13632	−0.568158	+	0.822919i	$0.692344\pi$
$542$	3.93645	0.169085
$543$	0	0
$544$	31.7705	1.36215
$545$	−27.9771	−1.19841
$546$	0	0
$547$	−1.00000	−0.0427569
$547$	−1.00000	−0.0427569
$548$	10.3065	0.440272
$549$	0	0
$550$	25.6302	1.09288
$551$	24.2876	1.03469
$552$	0	0
$553$	26.8758	1.14288
$554$	−21.4355	−0.910706
$555$	0	0
$556$	11.8738	0.503561
$557$	−1.27168	−0.0538829	−0.0269414	−	0.999637i	$-0.508577\pi$
$557$	−1.27168	−0.0538829	−0.0269414	+	0.999637i	$0.508577\pi$
$558$	0	0
$559$	−3.86982	−0.163676
$560$	−77.3261	−3.26762
$561$	0	0
$562$	54.6660	2.30594
$563$	3.96838	0.167247	0.0836237	−	0.996497i	$-0.473351\pi$
$563$	3.96838	0.167247	0.0836237	+	0.996497i	$0.473351\pi$
$564$	0	0
$565$	0.156574	0.00658711
$566$	46.8168	1.96786
$567$	0	0
$568$	11.5473	0.484515
$569$	−16.1671	−0.677761	−0.338881	−	0.940829i	$-0.610048\pi$
$569$	−16.1671	−0.677761	−0.338881	+	0.940829i	$0.610048\pi$
$570$	0	0
$571$	17.9457	0.751005	0.375503	−	0.926821i	$-0.377470\pi$
$571$	17.9457	0.751005	0.375503	+	0.926821i	$0.377470\pi$
$572$	2.94554	0.123159
$573$	0	0
$574$	−61.9338	−2.58507
$575$	59.3737	2.47605
$576$	0	0
$577$	16.6268	0.692184	0.346092	−	0.938200i	$-0.387508\pi$
$577$	16.6268	0.692184	0.346092	+	0.938200i	$0.387508\pi$
$578$	31.0357	1.29091
$579$	0	0
$580$	15.8849	0.659583
$581$	11.0421	0.458103
$582$	0	0
$583$	7.92919	0.328394
$584$	−3.94721	−0.163337
$585$	0	0
$586$	−15.5566	−0.642639
$587$	−33.9183	−1.39996	−0.699979	−	0.714163i	$-0.746807\pi$
$587$	−33.9183	−1.39996	−0.699979	+	0.714163i	$0.746807\pi$
$588$	0	0
$589$	−50.1377	−2.06589
$590$	86.3825	3.55631
$591$	0	0
$592$	46.3172	1.90362
$593$	−23.7489	−0.975252	−0.487626	−	0.873053i	$-0.662137\pi$
$593$	−23.7489	−0.975252	−0.487626	+	0.873053i	$0.662137\pi$
$594$	0	0
$595$	−91.2506	−3.74091
$596$	−14.8901	−0.609922
$597$	0	0
$598$	19.8884	0.813296
$599$	−35.6788	−1.45779	−0.728897	−	0.684623i	$-0.759967\pi$
$599$	−35.6788	−1.45779	−0.728897	+	0.684623i	$0.759967\pi$
$600$	0	0
$601$	−7.46392	−0.304460	−0.152230	−	0.988345i	$-0.548645\pi$
$601$	−7.46392	−0.304460	−0.152230	+	0.988345i	$0.548645\pi$
$602$	−18.6773	−0.761229
$603$	0	0
$604$	18.2955	0.744433
$605$	−27.7564	−1.12846
$606$	0	0
$607$	−3.18262	−0.129179	−0.0645893	−	0.997912i	$-0.520574\pi$
$607$	−3.18262	−0.129179	−0.0645893	+	0.997912i	$0.520574\pi$
$608$	−31.0919	−1.26094
$609$	0	0
$610$	−60.9998	−2.46981
$611$	9.40307	0.380407
$612$	0	0
$613$	−21.7245	−0.877446	−0.438723	−	0.898622i	$-0.644569\pi$
$613$	−21.7245	−0.877446	−0.438723	+	0.898622i	$0.644569\pi$
$614$	−36.0875	−1.45637
$615$	0	0
$616$	−13.0036	−0.523929
$617$	6.04399	0.243322	0.121661	−	0.992572i	$-0.461178\pi$
$617$	6.04399	0.243322	0.121661	+	0.992572i	$0.461178\pi$
$618$	0	0
$619$	−10.9041	−0.438272	−0.219136	−	0.975694i	$-0.570324\pi$
$619$	−10.9041	−0.438272	−0.219136	+	0.975694i	$0.570324\pi$
$620$	−32.7916	−1.31694
$621$	0	0
$622$	−16.6489	−0.667559
$623$	−36.1980	−1.45024
$624$	0	0
$625$	−0.317943	−0.0127177
$626$	−34.6561	−1.38514
$627$	0	0
$628$	−1.00351	−0.0400443
$629$	54.6578	2.17935
$630$	0	0
$631$	−6.17161	−0.245688	−0.122844	−	0.992426i	$-0.539201\pi$
$631$	−6.17161	−0.245688	−0.122844	+	0.992426i	$0.539201\pi$
$632$	−10.4665	−0.416334
$633$	0	0
$634$	22.0527	0.875826
$635$	2.01457	0.0799456
$636$	0	0
$637$	17.5279	0.694480
$638$	13.3777	0.529628
$639$	0	0
$640$	42.7009	1.68790
$641$	4.63006	0.182876	0.0914382	−	0.995811i	$-0.470854\pi$
$641$	4.63006	0.182876	0.0914382	+	0.995811i	$0.470854\pi$
$642$	0	0
$643$	−25.9357	−1.02281	−0.511403	−	0.859341i	$-0.670874\pi$
$643$	−25.9357	−1.02281	−0.511403	+	0.859341i	$0.670874\pi$
$644$	32.9329	1.29774
$645$	0	0
$646$	−59.4020	−2.33714
$647$	18.9502	0.745009	0.372505	−	0.928030i	$-0.378499\pi$
$647$	18.9502	0.745009	0.372505	+	0.928030i	$0.378499\pi$
$648$	0	0
$649$	24.9592	0.979735
$650$	21.7698	0.853882
$651$	0	0
$652$	−8.46403	−0.331477
$653$	21.2753	0.832568	0.416284	−	0.909235i	$-0.363332\pi$
$653$	21.2753	0.832568	0.416284	+	0.909235i	$0.363332\pi$
$654$	0	0
$655$	15.5837	0.608905
$656$	41.4414	1.61802
$657$	0	0
$658$	45.3829	1.76921
$659$	−2.22303	−0.0865968	−0.0432984	−	0.999062i	$-0.513787\pi$
$659$	−2.22303	−0.0865968	−0.0432984	+	0.999062i	$0.513787\pi$
$660$	0	0
$661$	−29.9111	−1.16341	−0.581704	−	0.813401i	$-0.697614\pi$
$661$	−29.9111	−1.16341	−0.581704	+	0.813401i	$0.697614\pi$
$662$	31.1670	1.21134
$663$	0	0
$664$	−4.30021	−0.166881
$665$	89.3018	3.46297
$666$	0	0
$667$	30.9901	1.19994
$668$	13.3243	0.515531
$669$	0	0
$670$	−23.1894	−0.895884
$671$	−17.6252	−0.680413
$672$	0	0
$673$	−2.14161	−0.0825530	−0.0412765	−	0.999148i	$-0.513142\pi$
$673$	−2.14161	−0.0825530	−0.0412765	+	0.999148i	$0.513142\pi$
$674$	19.4388	0.748754
$675$	0	0
$676$	−11.0773	−0.426051
$677$	−46.7475	−1.79665	−0.898326	−	0.439329i	$-0.855216\pi$
$677$	−46.7475	−1.79665	−0.898326	+	0.439329i	$0.855216\pi$
$678$	0	0
$679$	−47.4895	−1.82248
$680$	35.5365	1.36276
$681$	0	0
$682$	−27.6160	−1.05747
$683$	−43.9641	−1.68224	−0.841119	−	0.540850i	$-0.818103\pi$
$683$	−43.9641	−1.68224	−0.841119	+	0.540850i	$0.818103\pi$
$684$	0	0
$685$	35.6598	1.36249
$686$	32.3101	1.23360
$687$	0	0
$688$	12.4974	0.476459
$689$	6.73490	0.256579
$690$	0	0
$691$	14.2320	0.541412	0.270706	−	0.962662i	$-0.412743\pi$
$691$	14.2320	0.541412	0.270706	+	0.962662i	$0.412743\pi$
$692$	6.64354	0.252549
$693$	0	0
$694$	0.930711	0.0353293
$695$	41.0825	1.55835
$696$	0	0
$697$	48.9040	1.85237
$698$	46.5181	1.76074
$699$	0	0
$700$	36.0483	1.36250
$701$	−25.9017	−0.978293	−0.489147	−	0.872202i	$-0.662692\pi$
$701$	−25.9017	−0.978293	−0.489147	+	0.872202i	$0.662692\pi$
$702$	0	0
$703$	−53.4904	−2.01743
$704$	1.08798	0.0410048
$705$	0	0
$706$	3.71464	0.139802
$707$	51.9067	1.95215
$708$	0	0
$709$	29.1752	1.09570	0.547849	−	0.836577i	$-0.315447\pi$
$709$	29.1752	1.09570	0.547849	+	0.836577i	$0.315447\pi$
$710$	−43.6792	−1.63925
$711$	0	0
$712$	14.0969	0.528303
$713$	−63.9738	−2.39584
$714$	0	0
$715$	10.1914	0.381136
$716$	−16.3246	−0.610077
$717$	0	0
$718$	−2.70221	−0.100846
$719$	29.1591	1.08745	0.543725	−	0.839263i	$-0.317013\pi$
$719$	29.1591	1.08745	0.543725	+	0.839263i	$0.317013\pi$
$720$	0	0
$721$	−46.3856	−1.72749
$722$	24.9809	0.929694
$723$	0	0
$724$	−12.5938	−0.468046
$725$	33.9217	1.25982
$726$	0	0
$727$	−24.6294	−0.913456	−0.456728	−	0.889606i	$-0.650979\pi$
$727$	−24.6294	−0.913456	−0.456728	+	0.889606i	$0.650979\pi$
$728$	−11.0450	−0.409354
$729$	0	0
$730$	14.9308	0.552614
$731$	14.7479	0.545471
$732$	0	0
$733$	−22.1022	−0.816362	−0.408181	−	0.912901i	$-0.633837\pi$
$733$	−22.1022	−0.816362	−0.408181	+	0.912901i	$0.633837\pi$
$734$	14.1417	0.521978
$735$	0	0
$736$	−39.6721	−1.46233
$737$	−6.70030	−0.246809
$738$	0	0
$739$	−9.16477	−0.337131	−0.168566	−	0.985690i	$-0.553914\pi$
$739$	−9.16477	−0.337131	−0.168566	+	0.985690i	$0.553914\pi$
$740$	−34.9844	−1.28605
$741$	0	0
$742$	32.5053	1.19331
$743$	−32.0802	−1.17691	−0.588454	−	0.808531i	$-0.700263\pi$
$743$	−32.0802	−1.17691	−0.588454	+	0.808531i	$0.700263\pi$
$744$	0	0
$745$	−51.5188	−1.88750
$746$	63.5234	2.32576
$747$	0	0
$748$	−11.2255	−0.410444
$749$	−67.3654	−2.46148
$750$	0	0
$751$	−19.3094	−0.704611	−0.352305	−	0.935885i	$-0.614602\pi$
$751$	−19.3094	−0.704611	−0.352305	+	0.935885i	$0.614602\pi$
$752$	−30.3668	−1.10736
$753$	0	0
$754$	11.3627	0.413807
$755$	63.3012	2.30377
$756$	0	0
$757$	9.52491	0.346189	0.173094	−	0.984905i	$-0.444623\pi$
$757$	9.52491	0.346189	0.173094	+	0.984905i	$0.444623\pi$
$758$	−2.05617	−0.0746834
$759$	0	0
$760$	−34.7775	−1.26151
$761$	−26.1744	−0.948823	−0.474411	−	0.880303i	$-0.657339\pi$
$761$	−26.1744	−0.948823	−0.474411	+	0.880303i	$0.657339\pi$
$762$	0	0
$763$	−33.1385	−1.19969
$764$	15.8378	0.572991
$765$	0	0
$766$	−6.73920	−0.243497
$767$	21.1999	0.765483
$768$	0	0
$769$	28.8856	1.04164	0.520820	−	0.853666i	$-0.325626\pi$
$769$	28.8856	1.04164	0.520820	+	0.853666i	$0.325626\pi$
$770$	49.1877	1.77260
$771$	0	0
$772$	−7.26511	−0.261477
$773$	4.07993	0.146745	0.0733725	−	0.997305i	$-0.476624\pi$
$773$	4.07993	0.146745	0.0733725	+	0.997305i	$0.476624\pi$
$774$	0	0
$775$	−70.0256	−2.51539
$776$	18.4942	0.663905
$777$	0	0
$778$	16.2851	0.583851
$779$	−47.8596	−1.71475
$780$	0	0
$781$	−12.6206	−0.451600
$782$	−75.7947	−2.71041
$783$	0	0
$784$	−56.6055	−2.02163
$785$	−3.47207	−0.123923
$786$	0	0
$787$	−11.4800	−0.409219	−0.204609	−	0.978844i	$-0.565592\pi$
$787$	−11.4800	−0.409219	−0.204609	+	0.978844i	$0.565592\pi$
$788$	15.1854	0.540958
$789$	0	0
$790$	39.5907	1.40857
$791$	0.185460	0.00659418
$792$	0	0
$793$	−14.9705	−0.531617
$794$	−51.9880	−1.84498
$795$	0	0
$796$	−6.20846	−0.220053
$797$	22.6786	0.803316	0.401658	−	0.915790i	$-0.368434\pi$
$797$	22.6786	0.803316	0.401658	+	0.915790i	$0.368434\pi$
$798$	0	0
$799$	−35.8351	−1.26776
$800$	−43.4251	−1.53531
$801$	0	0
$802$	−9.33079	−0.329482
$803$	4.31408	0.152241
$804$	0	0
$805$	113.946	4.01606
$806$	−23.4565	−0.826219
$807$	0	0
$808$	−20.2145	−0.711142
$809$	8.43197	0.296452	0.148226	−	0.988953i	$-0.452644\pi$
$809$	8.43197	0.296452	0.148226	+	0.988953i	$0.452644\pi$
$810$	0	0
$811$	−25.0503	−0.879637	−0.439818	−	0.898087i	$-0.644957\pi$
$811$	−25.0503	−0.879637	−0.439818	+	0.898087i	$0.644957\pi$
$812$	18.8154	0.660291
$813$	0	0
$814$	−29.4627	−1.03267
$815$	−29.2850	−1.02581
$816$	0	0
$817$	−14.4329	−0.504944
$818$	−10.5672	−0.369474
$819$	0	0
$820$	−31.3017	−1.09310
$821$	−3.90486	−0.136280	−0.0681402	−	0.997676i	$-0.521707\pi$
$821$	−3.90486	−0.136280	−0.0681402	+	0.997676i	$0.521707\pi$
$822$	0	0
$823$	−19.1872	−0.668825	−0.334412	−	0.942427i	$-0.608538\pi$
$823$	−19.1872	−0.668825	−0.334412	+	0.942427i	$0.608538\pi$
$824$	18.0643	0.629300
$825$	0	0
$826$	102.319	3.56013
$827$	32.9961	1.14739	0.573694	−	0.819070i	$-0.305510\pi$
$827$	32.9961	1.14739	0.573694	+	0.819070i	$0.305510\pi$
$828$	0	0
$829$	−9.64456	−0.334969	−0.167485	−	0.985875i	$-0.553564\pi$
$829$	−9.64456	−0.334969	−0.167485	+	0.985875i	$0.553564\pi$
$830$	16.2661	0.564604
$831$	0	0
$832$	0.924109	0.0320377
$833$	−66.7989	−2.31444
$834$	0	0
$835$	46.1011	1.59539
$836$	10.9857	0.379950
$837$	0	0
$838$	−15.4403	−0.533378
$839$	−49.6031	−1.71249	−0.856244	−	0.516572i	$-0.827208\pi$
$839$	−49.6031	−1.71249	−0.856244	+	0.516572i	$0.827208\pi$
$840$	0	0
$841$	−11.2946	−0.389468
$842$	−32.2083	−1.10997
$843$	0	0
$844$	−28.3771	−0.976782
$845$	−38.3268	−1.31848
$846$	0	0
$847$	−32.8771	−1.12967
$848$	−21.7501	−0.746900
$849$	0	0
$850$	−82.9648	−2.84567
$851$	−68.2518	−2.33964
$852$	0	0
$853$	17.9174	0.613480	0.306740	−	0.951793i	$-0.400762\pi$
$853$	17.9174	0.613480	0.306740	+	0.951793i	$0.400762\pi$
$854$	−72.2535	−2.47246
$855$	0	0
$856$	26.2347	0.896683
$857$	2.42086	0.0826950	0.0413475	−	0.999145i	$-0.486835\pi$
$857$	2.42086	0.0826950	0.0413475	+	0.999145i	$0.486835\pi$
$858$	0	0
$859$	−16.6676	−0.568692	−0.284346	−	0.958722i	$-0.591776\pi$
$859$	−16.6676	−0.568692	−0.284346	+	0.958722i	$0.591776\pi$
$860$	−9.43958	−0.321887
$861$	0	0
$862$	−31.2693	−1.06504
$863$	−34.8647	−1.18681	−0.593404	−	0.804905i	$-0.702216\pi$
$863$	−34.8647	−1.18681	−0.593404	+	0.804905i	$0.702216\pi$
$864$	0	0
$865$	22.9862	0.781554
$866$	−69.1288	−2.34909
$867$	0	0
$868$	−38.8412	−1.31836
$869$	11.4393	0.388051
$870$	0	0
$871$	−5.69110	−0.192836
$872$	12.9054	0.437032
$873$	0	0
$874$	74.1760	2.50904
$875$	47.3680	1.60133
$876$	0	0
$877$	−6.48932	−0.219129	−0.109564	−	0.993980i	$-0.534946\pi$
$877$	−6.48932	−0.219129	−0.109564	+	0.993980i	$0.534946\pi$
$878$	24.5879	0.829800
$879$	0	0
$880$	−32.9126	−1.10949
$881$	57.2243	1.92794	0.963968	−	0.266020i	$-0.0857087\pi$
$881$	57.2243	1.92794	0.963968	+	0.266020i	$0.0857087\pi$
$882$	0	0
$883$	−24.1168	−0.811595	−0.405797	−	0.913963i	$-0.633006\pi$
$883$	−24.1168	−0.811595	−0.405797	+	0.913963i	$0.633006\pi$
$884$	−9.53470	−0.320687
$885$	0	0
$886$	42.5902	1.43084
$887$	−42.1176	−1.41417	−0.707086	−	0.707128i	$-0.749990\pi$
$887$	−42.1176	−1.41417	−0.707086	+	0.707128i	$0.749990\pi$
$888$	0	0
$889$	2.38622	0.0800314
$890$	−53.3232	−1.78740
$891$	0	0
$892$	3.56765	0.119454
$893$	35.0698	1.17357
$894$	0	0
$895$	−56.4819	−1.88798
$896$	50.5786	1.68971
$897$	0	0
$898$	−44.0068	−1.46852
$899$	−36.5499	−1.21901
$900$	0	0
$901$	−25.6667	−0.855083
$902$	−26.3612	−0.877731
$903$	0	0
$904$	−0.0722250	−0.00240217
$905$	−43.5739	−1.44844
$906$	0	0
$907$	25.8210	0.857371	0.428686	−	0.903454i	$-0.358977\pi$
$907$	25.8210	0.857371	0.428686	+	0.903454i	$0.358977\pi$
$908$	23.1410	0.767960
$909$	0	0
$910$	41.7790	1.38496
$911$	20.3911	0.675587	0.337794	−	0.941220i	$-0.390319\pi$
$911$	20.3911	0.675587	0.337794	+	0.941220i	$0.390319\pi$
$912$	0	0
$913$	4.69989	0.155544
$914$	58.4974	1.93492
$915$	0	0
$916$	1.73039	0.0571737
$917$	18.4587	0.609559
$918$	0	0
$919$	−23.0410	−0.760054	−0.380027	−	0.924975i	$-0.624085\pi$
$919$	−23.0410	−0.760054	−0.380027	+	0.924975i	$0.624085\pi$
$920$	−44.3748	−1.46299
$921$	0	0
$922$	−15.1870	−0.500159
$923$	−10.7197	−0.352842
$924$	0	0
$925$	−74.7083	−2.45639
$926$	28.3464	0.931520
$927$	0	0
$928$	−22.6657	−0.744038
$929$	24.2154	0.794483	0.397241	−	0.917714i	$-0.369968\pi$
$929$	24.2154	0.794483	0.397241	+	0.917714i	$0.369968\pi$
$930$	0	0
$931$	65.3722	2.14249
$932$	−9.87987	−0.323626
$933$	0	0
$934$	−24.0167	−0.785850
$935$	−38.8394	−1.27019
$936$	0	0
$937$	28.7273	0.938479	0.469239	−	0.883071i	$-0.344528\pi$
$937$	28.7273	0.938479	0.469239	+	0.883071i	$0.344528\pi$
$938$	−27.4675	−0.896846
$939$	0	0
$940$	22.9367	0.748114
$941$	36.9396	1.20420	0.602098	−	0.798422i	$-0.294331\pi$
$941$	36.9396	1.20420	0.602098	+	0.798422i	$0.294331\pi$
$942$	0	0
$943$	−61.0670	−1.98861
$944$	−68.4640	−2.22831
$945$	0	0
$946$	−7.94969	−0.258467
$947$	56.3251	1.83032	0.915160	−	0.403090i	$-0.132064\pi$
$947$	56.3251	1.83032	0.915160	+	0.403090i	$0.132064\pi$
$948$	0	0
$949$	3.66430	0.118948
$950$	81.1929	2.63425
$951$	0	0
$952$	42.0925	1.36423
$953$	10.9066	0.353298	0.176649	−	0.984274i	$-0.443474\pi$
$953$	10.9066	0.353298	0.176649	+	0.984274i	$0.443474\pi$
$954$	0	0
$955$	54.7977	1.77321
$956$	21.1520	0.684106
$957$	0	0
$958$	−9.93873	−0.321106
$959$	42.2386	1.36395
$960$	0	0
$961$	44.4510	1.43390
$962$	−25.0250	−0.806839
$963$	0	0
$964$	13.0591	0.420605
$965$	−25.1368	−0.809182
$966$	0	0
$967$	16.0179	0.515100	0.257550	−	0.966265i	$-0.417085\pi$
$967$	16.0179	0.515100	0.257550	+	0.966265i	$0.417085\pi$
$968$	12.8036	0.411523
$969$	0	0
$970$	−69.9568	−2.24618
$971$	−24.3509	−0.781457	−0.390728	−	0.920506i	$-0.627777\pi$
$971$	−24.3509	−0.781457	−0.390728	+	0.920506i	$0.627777\pi$
$972$	0	0
$973$	48.6617	1.56002
$974$	38.9644	1.24850
$975$	0	0
$976$	48.3465	1.54753
$977$	9.25929	0.296231	0.148115	−	0.988970i	$-0.452679\pi$
$977$	9.25929	0.296231	0.148115	+	0.988970i	$0.452679\pi$
$978$	0	0
$979$	−15.4071	−0.492413
$980$	42.7555	1.36577
$981$	0	0
$982$	−47.4348	−1.51370
$983$	50.2659	1.60323	0.801617	−	0.597839i	$-0.203974\pi$
$983$	50.2659	1.60323	0.801617	+	0.597839i	$0.203974\pi$
$984$	0	0
$985$	52.5405	1.67408
$986$	−43.3035	−1.37906
$987$	0	0
$988$	9.33106	0.296861
$989$	−18.4159	−0.585590
$990$	0	0
$991$	−37.3702	−1.18711	−0.593553	−	0.804795i	$-0.702275\pi$
$991$	−37.3702	−1.18711	−0.593553	+	0.804795i	$0.702275\pi$
$992$	46.7895	1.48557
$993$	0	0
$994$	−51.7374	−1.64101
$995$	−21.4809	−0.680990
$996$	0	0
$997$	−38.1667	−1.20875	−0.604375	−	0.796700i	$-0.706577\pi$
$997$	−38.1667	−1.20875	−0.604375	+	0.796700i	$0.706577\pi$
$998$	33.2454	1.05236
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4923.2.a.l.1.14		18
3.2	odd	2		547.2.a.b.1.5	✓	18
12.11	even	2		8752.2.a.s.1.12		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.a.b.1.5	✓	18	3.2	odd	2
4923.2.a.l.1.14		18	1.1	even	1	trivial
8752.2.a.s.1.12		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)

\(f(q)\)	\(=\)	\(q+1.74487 q^{2} +1.04455 q^{4} +3.61409 q^{5} +4.28084 q^{7} -1.66712 q^{8} +O(q^{10})\)	\(q+1.74487 q^{2} +1.04455 q^{4} +3.61409 q^{5} +4.28084 q^{7} -1.66712 q^{8} +6.30610 q^{10} +1.82207 q^{11} +1.54763 q^{13} +7.46949 q^{14} -4.99801 q^{16} -5.89804 q^{17} +5.77207 q^{19} +3.77512 q^{20} +3.17927 q^{22} +7.36494 q^{23} +8.06166 q^{25} +2.70041 q^{26} +4.47157 q^{28} +4.20778 q^{29} -8.68625 q^{31} -5.38662 q^{32} -10.2913 q^{34} +15.4714 q^{35} -9.26711 q^{37} +10.0715 q^{38} -6.02514 q^{40} -8.29157 q^{41} -2.50047 q^{43} +1.90326 q^{44} +12.8508 q^{46} +6.07577 q^{47} +11.3256 q^{49} +14.0665 q^{50} +1.61659 q^{52} +4.35174 q^{53} +6.58514 q^{55} -7.13669 q^{56} +7.34201 q^{58} +13.6982 q^{59} -9.67314 q^{61} -15.1563 q^{62} +0.597111 q^{64} +5.59329 q^{65} -3.67729 q^{67} -6.16082 q^{68} +26.9954 q^{70} -6.92649 q^{71} +2.36768 q^{73} -16.1699 q^{74} +6.02925 q^{76} +7.80001 q^{77} +6.27816 q^{79} -18.0633 q^{80} -14.4677 q^{82} +2.57942 q^{83} -21.3161 q^{85} -4.36299 q^{86} -3.03762 q^{88} -8.45581 q^{89} +6.62517 q^{91} +7.69309 q^{92} +10.6014 q^{94} +20.8608 q^{95} -11.0935 q^{97} +19.7617 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 4 q^{2} + 16 q^{4} + 27 q^{5} - 11 q^{7} + 12 q^{8}+O(q^{10}) \) 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

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Database

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