LMFDB - Embedded newform 4923.2.a.l.1.7

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			1.7
Root			$-0.924759$ 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)

$f(q)$	$=$	$q-0.924759 q^{2} -1.14482 q^{4} +3.95421 q^{5} -3.08028 q^{7} +2.90820 q^{8} +O(q^{10})$	\(q-0.924759 q^{2} -1.14482 q^{4} +3.95421 q^{5} -3.08028 q^{7} +2.90820 q^{8} -3.65669 q^{10} +3.50276 q^{11} +3.47242 q^{13} +2.84852 q^{14} -0.399740 q^{16} +3.04988 q^{17} +7.32400 q^{19} -4.52687 q^{20} -3.23921 q^{22} +6.16987 q^{23} +10.6358 q^{25} -3.21115 q^{26} +3.52637 q^{28} +3.33808 q^{29} +5.01284 q^{31} -5.44674 q^{32} -2.82040 q^{34} -12.1801 q^{35} -9.39929 q^{37} -6.77293 q^{38} +11.4996 q^{40} +2.63973 q^{41} +0.457015 q^{43} -4.01003 q^{44} -5.70564 q^{46} +4.87863 q^{47} +2.48812 q^{49} -9.83553 q^{50} -3.97530 q^{52} -0.519948 q^{53} +13.8506 q^{55} -8.95807 q^{56} -3.08692 q^{58} -10.3773 q^{59} +2.71700 q^{61} -4.63567 q^{62} +5.83640 q^{64} +13.7307 q^{65} -11.2680 q^{67} -3.49157 q^{68} +11.2636 q^{70} -6.36651 q^{71} -1.19812 q^{73} +8.69207 q^{74} -8.38468 q^{76} -10.7895 q^{77} -11.6933 q^{79} -1.58066 q^{80} -2.44111 q^{82} +9.73676 q^{83} +12.0599 q^{85} -0.422629 q^{86} +10.1867 q^{88} -11.4691 q^{89} -10.6960 q^{91} -7.06340 q^{92} -4.51155 q^{94} +28.9607 q^{95} +7.25071 q^{97} -2.30091 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.924759	−0.653903	−0.326952	−	0.945041i	$-0.606021\pi$
$2$	−0.924759	−0.653903	−0.326952	+	0.945041i	$0.606021\pi$
$3$	0	0
$3$	0	0
$4$	−1.14482	−0.572411
$5$	3.95421	1.76838	0.884188	−	0.467131i	$-0.154712\pi$
$5$	3.95421	1.76838	0.884188	+	0.467131i	$0.154712\pi$
$6$	0	0
$7$	−3.08028	−1.16424	−0.582118	−	0.813104i	$-0.697776\pi$
$7$	−3.08028	−1.16424	−0.582118	+	0.813104i	$0.697776\pi$
$8$	2.90820	1.02820
$9$	0	0
$10$	−3.65669	−1.15635
$11$	3.50276	1.05612	0.528061	−	0.849207i	$-0.322919\pi$
$11$	3.50276	1.05612	0.528061	+	0.849207i	$0.322919\pi$
$12$	0	0
$13$	3.47242	0.963075	0.481538	−	0.876425i	$-0.340079\pi$
$13$	3.47242	0.963075	0.481538	+	0.876425i	$0.340079\pi$
$14$	2.84852	0.761298
$15$	0	0
$16$	−0.399740	−0.0999351
$17$	3.04988	0.739705	0.369852	−	0.929091i	$-0.379408\pi$
$17$	3.04988	0.739705	0.369852	+	0.929091i	$0.379408\pi$
$18$	0	0
$19$	7.32400	1.68024	0.840121	−	0.542399i	$-0.182484\pi$
$19$	7.32400	1.68024	0.840121	+	0.542399i	$0.182484\pi$
$20$	−4.52687	−1.01224
$21$	0	0
$22$	−3.23921	−0.690601
$23$	6.16987	1.28651	0.643253	−	0.765653i	$-0.277584\pi$
$23$	6.16987	1.28651	0.643253	+	0.765653i	$0.277584\pi$
$24$	0	0
$25$	10.6358	2.12716
$26$	−3.21115	−0.629758
$27$	0	0
$28$	3.52637	0.666421
$29$	3.33808	0.619867	0.309933	−	0.950758i	$-0.399693\pi$
$29$	3.33808	0.619867	0.309933	+	0.950758i	$0.399693\pi$
$30$	0	0
$31$	5.01284	0.900333	0.450167	−	0.892945i	$-0.351365\pi$
$31$	5.01284	0.900333	0.450167	+	0.892945i	$0.351365\pi$
$32$	−5.44674	−0.962856
$33$	0	0
$34$	−2.82040	−0.483695
$35$	−12.1801	−2.05881
$36$	0	0
$37$	−9.39929	−1.54523	−0.772617	−	0.634873i	$-0.781053\pi$
$37$	−9.39929	−1.54523	−0.772617	+	0.634873i	$0.781053\pi$
$38$	−6.77293	−1.09872
$39$	0	0
$40$	11.4996	1.81825
$41$	2.63973	0.412256	0.206128	−	0.978525i	$-0.433914\pi$
$41$	2.63973	0.412256	0.206128	+	0.978525i	$0.433914\pi$
$42$	0	0
$43$	0.457015	0.0696942	0.0348471	−	0.999393i	$-0.488906\pi$
$43$	0.457015	0.0696942	0.0348471	+	0.999393i	$0.488906\pi$
$44$	−4.01003	−0.604535
$45$	0	0
$46$	−5.70564	−0.841251
$47$	4.87863	0.711621	0.355811	−	0.934558i	$-0.384205\pi$
$47$	4.87863	0.711621	0.355811	+	0.934558i	$0.384205\pi$
$48$	0	0
$49$	2.48812	0.355446
$50$	−9.83553	−1.39095
$51$	0	0
$52$	−3.97530	−0.551275
$53$	−0.519948	−0.0714203	−0.0357101	−	0.999362i	$-0.511369\pi$
$53$	−0.519948	−0.0714203	−0.0357101	+	0.999362i	$0.511369\pi$
$54$	0	0
$55$	13.8506	1.86762
$56$	−8.95807	−1.19707
$57$	0	0
$58$	−3.08692	−0.405333
$59$	−10.3773	−1.35101	−0.675503	−	0.737357i	$-0.736073\pi$
$59$	−10.3773	−1.35101	−0.675503	+	0.737357i	$0.736073\pi$
$60$	0	0
$61$	2.71700	0.347877	0.173938	−	0.984757i	$-0.444351\pi$
$61$	2.71700	0.347877	0.173938	+	0.984757i	$0.444351\pi$
$62$	−4.63567	−0.588731
$63$	0	0
$64$	5.83640	0.729550
$65$	13.7307	1.70308
$66$	0	0
$67$	−11.2680	−1.37661	−0.688304	−	0.725423i	$-0.741644\pi$
$67$	−11.2680	−1.37661	−0.688304	+	0.725423i	$0.741644\pi$
$68$	−3.49157	−0.423415
$69$	0	0
$70$	11.2636	1.34626
$71$	−6.36651	−0.755566	−0.377783	−	0.925894i	$-0.623313\pi$
$71$	−6.36651	−0.755566	−0.377783	+	0.925894i	$0.623313\pi$
$72$	0	0
$73$	−1.19812	−0.140229	−0.0701145	−	0.997539i	$-0.522336\pi$
$73$	−1.19812	−0.140229	−0.0701145	+	0.997539i	$0.522336\pi$
$74$	8.69207	1.01043
$75$	0	0
$76$	−8.38468	−0.961788
$77$	−10.7895	−1.22957
$78$	0	0
$79$	−11.6933	−1.31560	−0.657802	−	0.753191i	$-0.728513\pi$
$79$	−11.6933	−1.31560	−0.657802	+	0.753191i	$0.728513\pi$
$80$	−1.58066	−0.176723
$81$	0	0
$82$	−2.44111	−0.269576
$83$	9.73676	1.06875	0.534374	−	0.845248i	$-0.320547\pi$
$83$	9.73676	1.06875	0.534374	+	0.845248i	$0.320547\pi$
$84$	0	0
$85$	12.0599	1.30808
$86$	−0.422629	−0.0455732
$87$	0	0
$88$	10.1867	1.08591
$89$	−11.4691	−1.21572	−0.607859	−	0.794045i	$-0.707972\pi$
$89$	−11.4691	−1.21572	−0.607859	+	0.794045i	$0.707972\pi$
$90$	0	0
$91$	−10.6960	−1.12125
$92$	−7.06340	−0.736410
$93$	0	0
$94$	−4.51155	−0.465331
$95$	28.9607	2.97130
$96$	0	0
$97$	7.25071	0.736198	0.368099	−	0.929787i	$-0.380009\pi$
$97$	7.25071	0.736198	0.368099	+	0.929787i	$0.380009\pi$
$98$	−2.30091	−0.232427
$99$	0	0
$100$	−12.1761	−1.21761
$101$	15.4716	1.53948	0.769741	−	0.638356i	$-0.220385\pi$
$101$	15.4716	1.53948	0.769741	+	0.638356i	$0.220385\pi$
$102$	0	0
$103$	−11.1594	−1.09957	−0.549785	−	0.835306i	$-0.685290\pi$
$103$	−11.1594	−1.09957	−0.549785	+	0.835306i	$0.685290\pi$
$104$	10.0985	0.990238
$105$	0	0
$106$	0.480826	0.0467019
$107$	15.8859	1.53574	0.767872	−	0.640603i	$-0.221316\pi$
$107$	15.8859	1.53574	0.767872	+	0.640603i	$0.221316\pi$
$108$	0	0
$109$	2.66177	0.254951	0.127476	−	0.991842i	$-0.459313\pi$
$109$	2.66177	0.254951	0.127476	+	0.991842i	$0.459313\pi$
$110$	−12.8085	−1.22124
$111$	0	0
$112$	1.23131	0.116348
$113$	3.96953	0.373422	0.186711	−	0.982415i	$-0.440217\pi$
$113$	3.96953	0.373422	0.186711	+	0.982415i	$0.440217\pi$
$114$	0	0
$115$	24.3970	2.27503
$116$	−3.82151	−0.354818
$117$	0	0
$118$	9.59647	0.883427
$119$	−9.39449	−0.861191
$120$	0	0
$121$	1.26931	0.115392
$122$	−2.51257	−0.227478
$123$	0	0
$124$	−5.73881	−0.515360
$125$	22.2851	1.99324
$126$	0	0
$127$	−14.0151	−1.24364	−0.621822	−	0.783159i	$-0.713607\pi$
$127$	−14.0151	−1.24364	−0.621822	+	0.783159i	$0.713607\pi$
$128$	5.49622	0.485801
$129$	0	0
$130$	−12.6976	−1.11365
$131$	−21.9251	−1.91560	−0.957802	−	0.287430i	$-0.907199\pi$
$131$	−21.9251	−1.91560	−0.957802	+	0.287430i	$0.907199\pi$
$132$	0	0
$133$	−22.5600	−1.95620
$134$	10.4202	0.900168
$135$	0	0
$136$	8.86967	0.760568
$137$	−0.725655	−0.0619969	−0.0309984	−	0.999519i	$-0.509869\pi$
$137$	−0.725655	−0.0619969	−0.0309984	+	0.999519i	$0.509869\pi$
$138$	0	0
$139$	2.05950	0.174685	0.0873423	−	0.996178i	$-0.472163\pi$
$139$	2.05950	0.174685	0.0873423	+	0.996178i	$0.472163\pi$
$140$	13.9440	1.17848
$141$	0	0
$142$	5.88749	0.494067
$143$	12.1630	1.01712
$144$	0	0
$145$	13.1995	1.09616
$146$	1.10797	0.0916961
$147$	0	0
$148$	10.7605	0.884508
$149$	−13.4893	−1.10509	−0.552544	−	0.833484i	$-0.686343\pi$
$149$	−13.4893	−1.10509	−0.552544	+	0.833484i	$0.686343\pi$
$150$	0	0
$151$	0.695017	0.0565597	0.0282798	−	0.999600i	$-0.490997\pi$
$151$	0.695017	0.0565597	0.0282798	+	0.999600i	$0.490997\pi$
$152$	21.2997	1.72763
$153$	0	0
$154$	9.97766	0.804023
$155$	19.8218	1.59213
$156$	0	0
$157$	10.9574	0.874499	0.437249	−	0.899340i	$-0.355953\pi$
$157$	10.9574	0.874499	0.437249	+	0.899340i	$0.355953\pi$
$158$	10.8135	0.860277
$159$	0	0
$160$	−21.5375	−1.70269
$161$	−19.0049	−1.49780
$162$	0	0
$163$	15.9789	1.25156	0.625780	−	0.779999i	$-0.284781\pi$
$163$	15.9789	1.25156	0.625780	+	0.779999i	$0.284781\pi$
$164$	−3.02202	−0.235980
$165$	0	0
$166$	−9.00415	−0.698858
$167$	−13.3872	−1.03594	−0.517968	−	0.855400i	$-0.673311\pi$
$167$	−13.3872	−1.03594	−0.517968	+	0.855400i	$0.673311\pi$
$168$	0	0
$169$	−0.942324	−0.0724865
$170$	−11.1525	−0.855355
$171$	0	0
$172$	−0.523201	−0.0398937
$173$	17.9016	1.36103	0.680515	−	0.732734i	$-0.261756\pi$
$173$	17.9016	1.36103	0.680515	+	0.732734i	$0.261756\pi$
$174$	0	0
$175$	−32.7612	−2.47651
$176$	−1.40019	−0.105544
$177$	0	0
$178$	10.6061	0.794962
$179$	−24.0861	−1.80028	−0.900139	−	0.435602i	$-0.856536\pi$
$179$	−24.0861	−1.80028	−0.900139	+	0.435602i	$0.856536\pi$
$180$	0	0
$181$	4.37140	0.324924	0.162462	−	0.986715i	$-0.448057\pi$
$181$	4.37140	0.324924	0.162462	+	0.986715i	$0.448057\pi$
$182$	9.89123	0.733187
$183$	0	0
$184$	17.9432	1.32279
$185$	−37.1668	−2.73255
$186$	0	0
$187$	10.6830	0.781218
$188$	−5.58516	−0.407340
$189$	0	0
$190$	−26.7816	−1.94294
$191$	14.3169	1.03593	0.517967	−	0.855401i	$-0.326689\pi$
$191$	14.3169	1.03593	0.517967	+	0.855401i	$0.326689\pi$
$192$	0	0
$193$	12.7297	0.916305	0.458152	−	0.888874i	$-0.348511\pi$
$193$	12.7297	0.916305	0.458152	+	0.888874i	$0.348511\pi$
$194$	−6.70516	−0.481402
$195$	0	0
$196$	−2.84846	−0.203461
$197$	−13.4238	−0.956407	−0.478203	−	0.878249i	$-0.658712\pi$
$197$	−13.4238	−0.956407	−0.478203	+	0.878249i	$0.658712\pi$
$198$	0	0
$199$	−1.56808	−0.111159	−0.0555793	−	0.998454i	$-0.517701\pi$
$199$	−1.56808	−0.111159	−0.0555793	+	0.998454i	$0.517701\pi$
$200$	30.9310	2.18715
$201$	0	0
$202$	−14.3075	−1.00667
$203$	−10.2822	−0.721671
$204$	0	0
$205$	10.4380	0.729024
$206$	10.3198	0.719012
$207$	0	0
$208$	−1.38807	−0.0962450
$209$	25.6542	1.77454
$210$	0	0
$211$	−16.3089	−1.12275	−0.561376	−	0.827561i	$-0.689728\pi$
$211$	−16.3089	−1.12275	−0.561376	+	0.827561i	$0.689728\pi$
$212$	0.595247	0.0408817
$213$	0	0
$214$	−14.6906	−1.00423
$215$	1.80714	0.123246
$216$	0	0
$217$	−15.4410	−1.04820
$218$	−2.46149	−0.166713
$219$	0	0
$220$	−15.8565	−1.06905
$221$	10.5905	0.712391
$222$	0	0
$223$	−6.07966	−0.407124	−0.203562	−	0.979062i	$-0.565252\pi$
$223$	−6.07966	−0.407124	−0.203562	+	0.979062i	$0.565252\pi$
$224$	16.7775	1.12099
$225$	0	0
$226$	−3.67086	−0.244182
$227$	−19.1850	−1.27336	−0.636678	−	0.771130i	$-0.719692\pi$
$227$	−19.1850	−1.27336	−0.636678	+	0.771130i	$0.719692\pi$
$228$	0	0
$229$	9.82576	0.649305	0.324652	−	0.945833i	$-0.394753\pi$
$229$	9.82576	0.649305	0.324652	+	0.945833i	$0.394753\pi$
$230$	−22.5613	−1.48765
$231$	0	0
$232$	9.70782	0.637350
$233$	30.2087	1.97904	0.989519	−	0.144406i	$-0.0461271\pi$
$233$	30.2087	1.97904	0.989519	+	0.144406i	$0.0461271\pi$
$234$	0	0
$235$	19.2911	1.25841
$236$	11.8801	0.773330
$237$	0	0
$238$	8.68763	0.563136
$239$	−20.0414	−1.29637	−0.648186	−	0.761482i	$-0.724472\pi$
$239$	−20.0414	−1.29637	−0.648186	+	0.761482i	$0.724472\pi$
$240$	0	0
$241$	−22.2442	−1.43287	−0.716436	−	0.697653i	$-0.754228\pi$
$241$	−22.2442	−1.43287	−0.716436	+	0.697653i	$0.754228\pi$
$242$	−1.17381	−0.0754551
$243$	0	0
$244$	−3.11048	−0.199128
$245$	9.83857	0.628563
$246$	0	0
$247$	25.4320	1.61820
$248$	14.5784	0.925726
$249$	0	0
$250$	−20.6083	−1.30338
$251$	12.5368	0.791313	0.395656	−	0.918399i	$-0.370517\pi$
$251$	12.5368	0.791313	0.395656	+	0.918399i	$0.370517\pi$
$252$	0	0
$253$	21.6116	1.35871
$254$	12.9606	0.813222
$255$	0	0
$256$	−16.7555	−1.04722
$257$	−13.1786	−0.822055	−0.411028	−	0.911623i	$-0.634830\pi$
$257$	−13.1786	−0.822055	−0.411028	+	0.911623i	$0.634830\pi$
$258$	0	0
$259$	28.9524	1.79902
$260$	−15.7192	−0.974861
$261$	0	0
$262$	20.2754	1.25262
$263$	−1.98374	−0.122322	−0.0611612	−	0.998128i	$-0.519480\pi$
$263$	−1.98374	−0.122322	−0.0611612	+	0.998128i	$0.519480\pi$
$264$	0	0
$265$	−2.05598	−0.126298
$266$	20.8625	1.27916
$267$	0	0
$268$	12.8999	0.787985
$269$	0.592176	0.0361056	0.0180528	−	0.999837i	$-0.494253\pi$
$269$	0.592176	0.0361056	0.0180528	+	0.999837i	$0.494253\pi$
$270$	0	0
$271$	−21.4167	−1.30097	−0.650486	−	0.759518i	$-0.725435\pi$
$271$	−21.4167	−1.30097	−0.650486	+	0.759518i	$0.725435\pi$
$272$	−1.21916	−0.0739225
$273$	0	0
$274$	0.671056	0.0405400
$275$	37.2546	2.24653
$276$	0	0
$277$	−9.10027	−0.546783	−0.273391	−	0.961903i	$-0.588145\pi$
$277$	−9.10027	−0.546783	−0.273391	+	0.961903i	$0.588145\pi$
$278$	−1.90454	−0.114227
$279$	0	0
$280$	−35.4221	−2.11688
$281$	2.45423	0.146407	0.0732036	−	0.997317i	$-0.476678\pi$
$281$	2.45423	0.146407	0.0732036	+	0.997317i	$0.476678\pi$
$282$	0	0
$283$	−7.57111	−0.450056	−0.225028	−	0.974352i	$-0.572247\pi$
$283$	−7.57111	−0.450056	−0.225028	+	0.974352i	$0.572247\pi$
$284$	7.28852	0.432494
$285$	0	0
$286$	−11.2479	−0.665100
$287$	−8.13110	−0.479964
$288$	0	0
$289$	−7.69823	−0.452837
$290$	−12.2063	−0.716781
$291$	0	0
$292$	1.37163	0.0802685
$293$	15.9522	0.931935	0.465967	−	0.884802i	$-0.345706\pi$
$293$	15.9522	0.931935	0.465967	+	0.884802i	$0.345706\pi$
$294$	0	0
$295$	−41.0339	−2.38909
$296$	−27.3350	−1.58882
$297$	0	0
$298$	12.4744	0.722620
$299$	21.4244	1.23900
$300$	0	0
$301$	−1.40774	−0.0811405
$302$	−0.642723	−0.0369846
$303$	0	0
$304$	−2.92770	−0.167915
$305$	10.7436	0.615177
$306$	0	0
$307$	−10.0561	−0.573931	−0.286965	−	0.957941i	$-0.592646\pi$
$307$	−10.0561	−0.573931	−0.286965	+	0.957941i	$0.592646\pi$
$308$	12.3520	0.703822
$309$	0	0
$310$	−18.3304	−1.04110
$311$	17.8541	1.01241	0.506207	−	0.862412i	$-0.331047\pi$
$311$	17.8541	1.01241	0.506207	+	0.862412i	$0.331047\pi$
$312$	0	0
$313$	−11.0835	−0.626475	−0.313238	−	0.949675i	$-0.601414\pi$
$313$	−11.0835	−0.626475	−0.313238	+	0.949675i	$0.601414\pi$
$314$	−10.1330	−0.571837
$315$	0	0
$316$	13.3868	0.753065
$317$	−4.47220	−0.251184	−0.125592	−	0.992082i	$-0.540083\pi$
$317$	−4.47220	−0.251184	−0.125592	+	0.992082i	$0.540083\pi$
$318$	0	0
$319$	11.6925	0.654654
$320$	23.0783	1.29012
$321$	0	0
$322$	17.5750	0.979415
$323$	22.3373	1.24288
$324$	0	0
$325$	36.9319	2.04861
$326$	−14.7766	−0.818399
$327$	0	0
$328$	7.67686	0.423884
$329$	−15.0275	−0.828495
$330$	0	0
$331$	28.5978	1.57187	0.785937	−	0.618306i	$-0.212181\pi$
$331$	28.5978	1.57187	0.785937	+	0.618306i	$0.212181\pi$
$332$	−11.1469	−0.611763
$333$	0	0
$334$	12.3800	0.677402
$335$	−44.5561	−2.43436
$336$	0	0
$337$	34.0126	1.85278	0.926392	−	0.376561i	$-0.122893\pi$
$337$	34.0126	1.85278	0.926392	+	0.376561i	$0.122893\pi$
$338$	0.871422	0.0473991
$339$	0	0
$340$	−13.8064	−0.748757
$341$	17.5588	0.950861
$342$	0	0
$343$	13.8978	0.750413
$344$	1.32909	0.0716599
$345$	0	0
$346$	−16.5546	−0.889982
$347$	5.18016	0.278086	0.139043	−	0.990286i	$-0.455597\pi$
$347$	5.18016	0.278086	0.139043	+	0.990286i	$0.455597\pi$
$348$	0	0
$349$	4.21519	0.225634	0.112817	−	0.993616i	$-0.464013\pi$
$349$	4.21519	0.225634	0.112817	+	0.993616i	$0.464013\pi$
$350$	30.2962	1.61940
$351$	0	0
$352$	−19.0786	−1.01689
$353$	−22.6574	−1.20593	−0.602967	−	0.797766i	$-0.706015\pi$
$353$	−22.6574	−1.20593	−0.602967	+	0.797766i	$0.706015\pi$
$354$	0	0
$355$	−25.1745	−1.33613
$356$	13.1300	0.695891
$357$	0	0
$358$	22.2738	1.17721
$359$	5.93624	0.313303	0.156651	−	0.987654i	$-0.449930\pi$
$359$	5.93624	0.313303	0.156651	+	0.987654i	$0.449930\pi$
$360$	0	0
$361$	34.6410	1.82321
$362$	−4.04249	−0.212469
$363$	0	0
$364$	12.2450	0.641814
$365$	−4.73760	−0.247978
$366$	0	0
$367$	16.1865	0.844930	0.422465	−	0.906379i	$-0.361165\pi$
$367$	16.1865	0.844930	0.422465	+	0.906379i	$0.361165\pi$
$368$	−2.46635	−0.128567
$369$	0	0
$370$	34.3703	1.78683
$371$	1.60158	0.0831501
$372$	0	0
$373$	−8.96744	−0.464316	−0.232158	−	0.972678i	$-0.574579\pi$
$373$	−8.96744	−0.464316	−0.232158	+	0.972678i	$0.574579\pi$
$374$	−9.87919	−0.510841
$375$	0	0
$376$	14.1880	0.731692
$377$	11.5912	0.596978
$378$	0	0
$379$	−32.2073	−1.65438	−0.827189	−	0.561924i	$-0.810061\pi$
$379$	−32.2073	−1.65438	−0.827189	+	0.561924i	$0.810061\pi$
$380$	−33.1548	−1.70080
$381$	0	0
$382$	−13.2397	−0.677400
$383$	−32.7239	−1.67211	−0.836056	−	0.548644i	$-0.815144\pi$
$383$	−32.7239	−1.67211	−0.836056	+	0.548644i	$0.815144\pi$
$384$	0	0
$385$	−42.6639	−2.17435
$386$	−11.7719	−0.599175
$387$	0	0
$388$	−8.30077	−0.421408
$389$	15.2384	0.772616	0.386308	−	0.922370i	$-0.373750\pi$
$389$	15.2384	0.772616	0.386308	+	0.922370i	$0.373750\pi$
$390$	0	0
$391$	18.8174	0.951635
$392$	7.23596	0.365471
$393$	0	0
$394$	12.4138	0.625397
$395$	−46.2379	−2.32648
$396$	0	0
$397$	−31.8678	−1.59940	−0.799699	−	0.600401i	$-0.795008\pi$
$397$	−31.8678	−1.59940	−0.799699	+	0.600401i	$0.795008\pi$
$398$	1.45010	0.0726869
$399$	0	0
$400$	−4.25155	−0.212578
$401$	7.87598	0.393307	0.196654	−	0.980473i	$-0.436993\pi$
$401$	7.87598	0.393307	0.196654	+	0.980473i	$0.436993\pi$
$402$	0	0
$403$	17.4067	0.867088
$404$	−17.7122	−0.881216
$405$	0	0
$406$	9.50858	0.471903
$407$	−32.9234	−1.63195
$408$	0	0
$409$	−21.2120	−1.04886	−0.524432	−	0.851452i	$-0.675722\pi$
$409$	−21.2120	−1.04886	−0.524432	+	0.851452i	$0.675722\pi$
$410$	−9.65267	−0.476711
$411$	0	0
$412$	12.7755	0.629405
$413$	31.9649	1.57289
$414$	0	0
$415$	38.5012	1.88995
$416$	−18.9133	−0.927303
$417$	0	0
$418$	−23.7239	−1.16038
$419$	17.3991	0.850003	0.425001	−	0.905193i	$-0.360274\pi$
$419$	17.3991	0.850003	0.425001	+	0.905193i	$0.360274\pi$
$420$	0	0
$421$	−13.3962	−0.652890	−0.326445	−	0.945216i	$-0.605851\pi$
$421$	−13.3962	−0.652890	−0.326445	+	0.945216i	$0.605851\pi$
$422$	15.0818	0.734171
$423$	0	0
$424$	−1.51211	−0.0734346
$425$	32.4379	1.57347
$426$	0	0
$427$	−8.36913	−0.405011
$428$	−18.1865	−0.879076
$429$	0	0
$430$	−1.67116	−0.0805907
$431$	−32.3565	−1.55856	−0.779279	−	0.626677i	$-0.784414\pi$
$431$	−32.3565	−1.55856	−0.779279	+	0.626677i	$0.784414\pi$
$432$	0	0
$433$	8.08088	0.388342	0.194171	−	0.980968i	$-0.437798\pi$
$433$	8.08088	0.388342	0.194171	+	0.980968i	$0.437798\pi$
$434$	14.2792	0.685422
$435$	0	0
$436$	−3.04725	−0.145937
$437$	45.1881	2.16164
$438$	0	0
$439$	26.4717	1.26342	0.631712	−	0.775203i	$-0.282353\pi$
$439$	26.4717	1.26342	0.631712	+	0.775203i	$0.282353\pi$
$440$	40.2804	1.92029
$441$	0	0
$442$	−9.79362	−0.465835
$443$	38.7060	1.83898	0.919488	−	0.393118i	$-0.128604\pi$
$443$	38.7060	1.83898	0.919488	+	0.393118i	$0.128604\pi$
$444$	0	0
$445$	−45.3511	−2.14985
$446$	5.62222	0.266220
$447$	0	0
$448$	−17.9777	−0.849368
$449$	4.41373	0.208297	0.104148	−	0.994562i	$-0.466788\pi$
$449$	4.41373	0.208297	0.104148	+	0.994562i	$0.466788\pi$
$450$	0	0
$451$	9.24633	0.435393
$452$	−4.54440	−0.213751
$453$	0	0
$454$	17.7415	0.832651
$455$	−42.2943	−1.98279
$456$	0	0
$457$	0.171589	0.00802659	0.00401329	−	0.999992i	$-0.498723\pi$
$457$	0.171589	0.00802659	0.00401329	+	0.999992i	$0.498723\pi$
$458$	−9.08646	−0.424582
$459$	0	0
$460$	−27.9302	−1.30225
$461$	32.6067	1.51864	0.759322	−	0.650715i	$-0.225531\pi$
$461$	32.6067	1.51864	0.759322	+	0.650715i	$0.225531\pi$
$462$	0	0
$463$	25.5148	1.18577	0.592887	−	0.805285i	$-0.297988\pi$
$463$	25.5148	1.18577	0.592887	+	0.805285i	$0.297988\pi$
$464$	−1.33437	−0.0619465
$465$	0	0
$466$	−27.9357	−1.29410
$467$	−3.58271	−0.165788	−0.0828939	−	0.996558i	$-0.526416\pi$
$467$	−3.58271	−0.165788	−0.0828939	+	0.996558i	$0.526416\pi$
$468$	0	0
$469$	34.7086	1.60270
$470$	−17.8396	−0.822881
$471$	0	0
$472$	−30.1792	−1.38911
$473$	1.60081	0.0736055
$474$	0	0
$475$	77.8965	3.57414
$476$	10.7550	0.492955
$477$	0	0
$478$	18.5335	0.847702
$479$	27.2764	1.24629	0.623145	−	0.782106i	$-0.285854\pi$
$479$	27.2764	1.24629	0.623145	+	0.782106i	$0.285854\pi$
$480$	0	0
$481$	−32.6382	−1.48818
$482$	20.5705	0.936959
$483$	0	0
$484$	−1.45313	−0.0660515
$485$	28.6709	1.30188
$486$	0	0
$487$	−32.4550	−1.47068	−0.735339	−	0.677700i	$-0.762977\pi$
$487$	−32.4550	−1.47068	−0.735339	+	0.677700i	$0.762977\pi$
$488$	7.90159	0.357688
$489$	0	0
$490$	−9.09830	−0.411019
$491$	−8.45330	−0.381492	−0.190746	−	0.981639i	$-0.561091\pi$
$491$	−8.45330	−0.381492	−0.190746	+	0.981639i	$0.561091\pi$
$492$	0	0
$493$	10.1808	0.458518
$494$	−23.5185	−1.05815
$495$	0	0
$496$	−2.00384	−0.0899749
$497$	19.6106	0.879657
$498$	0	0
$499$	−28.5526	−1.27819	−0.639094	−	0.769128i	$-0.720691\pi$
$499$	−28.5526	−1.27819	−0.639094	+	0.769128i	$0.720691\pi$
$500$	−25.5124	−1.14095
$501$	0	0
$502$	−11.5935	−0.517442
$503$	−14.5826	−0.650208	−0.325104	−	0.945678i	$-0.605399\pi$
$503$	−14.5826	−0.650208	−0.325104	+	0.945678i	$0.605399\pi$
$504$	0	0
$505$	61.1780	2.72238
$506$	−19.9855	−0.888463
$507$	0	0
$508$	16.0448	0.711875
$509$	16.1161	0.714331	0.357166	−	0.934041i	$-0.383743\pi$
$509$	16.1161	0.714331	0.357166	+	0.934041i	$0.383743\pi$
$510$	0	0
$511$	3.69053	0.163260
$512$	4.50233	0.198977
$513$	0	0
$514$	12.1870	0.537545
$515$	−44.1267	−1.94445
$516$	0	0
$517$	17.0887	0.751558
$518$	−26.7740	−1.17638
$519$	0	0
$520$	39.9315	1.75111
$521$	7.13196	0.312457	0.156228	−	0.987721i	$-0.450066\pi$
$521$	7.13196	0.312457	0.156228	+	0.987721i	$0.450066\pi$
$522$	0	0
$523$	−4.52473	−0.197853	−0.0989264	−	0.995095i	$-0.531541\pi$
$523$	−4.52473	−0.197853	−0.0989264	+	0.995095i	$0.531541\pi$
$524$	25.1003	1.09651
$525$	0	0
$526$	1.83448	0.0799870
$527$	15.2886	0.665981
$528$	0	0
$529$	15.0673	0.655099
$530$	1.90129	0.0825866
$531$	0	0
$532$	25.8272	1.11975
$533$	9.16623	0.397034
$534$	0	0
$535$	62.8160	2.71577
$536$	−32.7697	−1.41543
$537$	0	0
$538$	−0.547619	−0.0236096
$539$	8.71529	0.375394
$540$	0	0
$541$	−6.61816	−0.284537	−0.142268	−	0.989828i	$-0.545440\pi$
$541$	−6.61816	−0.284537	−0.142268	+	0.989828i	$0.545440\pi$
$542$	19.8053	0.850709
$543$	0	0
$544$	−16.6119	−0.712229
$545$	10.5252	0.450849
$546$	0	0
$547$	−1.00000	−0.0427569
$547$	−1.00000	−0.0427569
$548$	0.830745	0.0354877
$549$	0	0
$550$	−34.4515	−1.46902
$551$	24.4481	1.04153
$552$	0	0
$553$	36.0188	1.53167
$554$	8.41556	0.357543
$555$	0	0
$556$	−2.35776	−0.0999914
$557$	9.58172	0.405991	0.202995	−	0.979180i	$-0.434932\pi$
$557$	9.58172	0.405991	0.202995	+	0.979180i	$0.434932\pi$
$558$	0	0
$559$	1.58695	0.0671207
$560$	4.86887	0.205747
$561$	0	0
$562$	−2.26957	−0.0957362
$563$	−44.4878	−1.87494	−0.937468	−	0.348072i	$-0.886836\pi$
$563$	−44.4878	−1.87494	−0.937468	+	0.348072i	$0.886836\pi$
$564$	0	0
$565$	15.6964	0.660351
$566$	7.00145	0.294293
$567$	0	0
$568$	−18.5151	−0.776876
$569$	14.0592	0.589394	0.294697	−	0.955591i	$-0.404781\pi$
$569$	14.0592	0.589394	0.294697	+	0.955591i	$0.404781\pi$
$570$	0	0
$571$	38.0971	1.59431	0.797156	−	0.603773i	$-0.206337\pi$
$571$	38.0971	1.59431	0.797156	+	0.603773i	$0.206337\pi$
$572$	−13.9245	−0.582213
$573$	0	0
$574$	7.51931	0.313850
$575$	65.6214	2.73660
$576$	0	0
$577$	7.80242	0.324819	0.162410	−	0.986723i	$-0.448073\pi$
$577$	7.80242	0.324819	0.162410	+	0.986723i	$0.448073\pi$
$578$	7.11900	0.296111
$579$	0	0
$580$	−15.1111	−0.627453
$581$	−29.9919	−1.24428
$582$	0	0
$583$	−1.82125	−0.0754285
$584$	−3.48436	−0.144184
$585$	0	0
$586$	−14.7519	−0.609395
$587$	26.8275	1.10729	0.553644	−	0.832753i	$-0.313237\pi$
$587$	26.8275	1.10729	0.553644	+	0.832753i	$0.313237\pi$
$588$	0	0
$589$	36.7141	1.51278
$590$	37.9465	1.56223
$591$	0	0
$592$	3.75728	0.154423
$593$	34.0163	1.39688	0.698441	−	0.715667i	$-0.253877\pi$
$593$	34.0163	1.39688	0.698441	+	0.715667i	$0.253877\pi$
$594$	0	0
$595$	−37.1478	−1.52291
$596$	15.4428	0.632564
$597$	0	0
$598$	−19.8124	−0.810187
$599$	−22.3457	−0.913021	−0.456510	−	0.889718i	$-0.650901\pi$
$599$	−22.3457	−0.913021	−0.456510	+	0.889718i	$0.650901\pi$
$600$	0	0
$601$	25.2733	1.03092	0.515460	−	0.856914i	$-0.327621\pi$
$601$	25.2733	1.03092	0.515460	+	0.856914i	$0.327621\pi$
$602$	1.30182	0.0530580
$603$	0	0
$604$	−0.795671	−0.0323754
$605$	5.01912	0.204056
$606$	0	0
$607$	40.1847	1.63105	0.815524	−	0.578723i	$-0.196449\pi$
$607$	40.1847	1.63105	0.815524	+	0.578723i	$0.196449\pi$
$608$	−39.8919	−1.61783
$609$	0	0
$610$	−9.93524	−0.402266
$611$	16.9406	0.685344
$612$	0	0
$613$	44.1657	1.78384	0.891919	−	0.452196i	$-0.149359\pi$
$613$	44.1657	1.78384	0.891919	+	0.452196i	$0.149359\pi$
$614$	9.29945	0.375295
$615$	0	0
$616$	−31.3780	−1.26425
$617$	22.9118	0.922394	0.461197	−	0.887298i	$-0.347420\pi$
$617$	22.9118	0.922394	0.461197	+	0.887298i	$0.347420\pi$
$618$	0	0
$619$	26.6090	1.06951	0.534754	−	0.845008i	$-0.320404\pi$
$619$	26.6090	1.06951	0.534754	+	0.845008i	$0.320404\pi$
$620$	−22.6925	−0.911351
$621$	0	0
$622$	−16.5107	−0.662020
$623$	35.3279	1.41538
$624$	0	0
$625$	34.9409	1.39764
$626$	10.2495	0.409654
$627$	0	0
$628$	−12.5443	−0.500572
$629$	−28.6667	−1.14302
$630$	0	0
$631$	−10.4893	−0.417573	−0.208787	−	0.977961i	$-0.566951\pi$
$631$	−10.4893	−0.417573	−0.208787	+	0.977961i	$0.566951\pi$
$632$	−34.0066	−1.35271
$633$	0	0
$634$	4.13570	0.164250
$635$	−55.4189	−2.19923
$636$	0	0
$637$	8.63980	0.342321
$638$	−10.8127	−0.428080
$639$	0	0
$640$	21.7332	0.859080
$641$	−11.4048	−0.450463	−0.225232	−	0.974305i	$-0.572314\pi$
$641$	−11.4048	−0.450463	−0.225232	+	0.974305i	$0.572314\pi$
$642$	0	0
$643$	−24.0029	−0.946581	−0.473291	−	0.880906i	$-0.656934\pi$
$643$	−24.0029	−0.946581	−0.473291	+	0.880906i	$0.656934\pi$
$644$	21.7572	0.857356
$645$	0	0
$646$	−20.6566	−0.812725
$647$	−39.8641	−1.56722	−0.783609	−	0.621254i	$-0.786623\pi$
$647$	−39.8641	−1.56722	−0.783609	+	0.621254i	$0.786623\pi$
$648$	0	0
$649$	−36.3491	−1.42683
$650$	−34.1531	−1.33959
$651$	0	0
$652$	−18.2929	−0.716407
$653$	35.3964	1.38517	0.692584	−	0.721337i	$-0.256472\pi$
$653$	35.3964	1.38517	0.692584	+	0.721337i	$0.256472\pi$
$654$	0	0
$655$	−86.6964	−3.38751
$656$	−1.05521	−0.0411989
$657$	0	0
$658$	13.8968	0.541755
$659$	−24.7728	−0.965010	−0.482505	−	0.875893i	$-0.660273\pi$
$659$	−24.7728	−0.965010	−0.482505	+	0.875893i	$0.660273\pi$
$660$	0	0
$661$	−23.3350	−0.907626	−0.453813	−	0.891097i	$-0.649937\pi$
$661$	−23.3350	−0.907626	−0.453813	+	0.891097i	$0.649937\pi$
$662$	−26.4460	−1.02785
$663$	0	0
$664$	28.3164	1.09889
$665$	−89.2069	−3.45930
$666$	0	0
$667$	20.5955	0.797463
$668$	15.3260	0.592981
$669$	0	0
$670$	41.2036	1.59184
$671$	9.51700	0.367400
$672$	0	0
$673$	−5.98598	−0.230742	−0.115371	−	0.993322i	$-0.536806\pi$
$673$	−5.98598	−0.230742	−0.115371	+	0.993322i	$0.536806\pi$
$674$	−31.4534	−1.21154
$675$	0	0
$676$	1.07879	0.0414920
$677$	27.5508	1.05886	0.529431	−	0.848353i	$-0.322405\pi$
$677$	27.5508	1.05886	0.529431	+	0.848353i	$0.322405\pi$
$678$	0	0
$679$	−22.3342	−0.857109
$680$	35.0725	1.34497
$681$	0	0
$682$	−16.2376	−0.621771
$683$	−41.1849	−1.57590	−0.787948	−	0.615741i	$-0.788857\pi$
$683$	−41.1849	−1.57590	−0.787948	+	0.615741i	$0.788857\pi$
$684$	0	0
$685$	−2.86939	−0.109634
$686$	−12.8521	−0.490697
$687$	0	0
$688$	−0.182688	−0.00696490
$689$	−1.80547	−0.0687831
$690$	0	0
$691$	−6.87493	−0.261535	−0.130767	−	0.991413i	$-0.541744\pi$
$691$	−6.87493	−0.261535	−0.130767	+	0.991413i	$0.541744\pi$
$692$	−20.4941	−0.779069
$693$	0	0
$694$	−4.79040	−0.181841
$695$	8.14370	0.308908
$696$	0	0
$697$	8.05086	0.304948
$698$	−3.89803	−0.147543
$699$	0	0
$700$	37.5057	1.41758
$701$	−15.9268	−0.601548	−0.300774	−	0.953695i	$-0.597245\pi$
$701$	−15.9268	−0.601548	−0.300774	+	0.953695i	$0.597245\pi$
$702$	0	0
$703$	−68.8404	−2.59637
$704$	20.4435	0.770493
$705$	0	0
$706$	20.9526	0.788563
$707$	−47.6569	−1.79232
$708$	0	0
$709$	−8.19363	−0.307718	−0.153859	−	0.988093i	$-0.549170\pi$
$709$	−8.19363	−0.307718	−0.153859	+	0.988093i	$0.549170\pi$
$710$	23.2804	0.873696
$711$	0	0
$712$	−33.3544	−1.25001
$713$	30.9286	1.15828
$714$	0	0
$715$	48.0952	1.79866
$716$	27.5743	1.03050
$717$	0	0
$718$	−5.48959	−0.204870
$719$	−13.0692	−0.487397	−0.243699	−	0.969851i	$-0.578361\pi$
$719$	−13.0692	−0.487397	−0.243699	+	0.969851i	$0.578361\pi$
$720$	0	0
$721$	34.3741	1.28016
$722$	−32.0346	−1.19220
$723$	0	0
$724$	−5.00447	−0.185990
$725$	35.5031	1.31855
$726$	0	0
$727$	−18.2107	−0.675397	−0.337699	−	0.941254i	$-0.609648\pi$
$727$	−18.2107	−0.675397	−0.337699	+	0.941254i	$0.609648\pi$
$728$	−31.1062	−1.15287
$729$	0	0
$730$	4.38114	0.162153
$731$	1.39384	0.0515531
$732$	0	0
$733$	−7.19358	−0.265701	−0.132851	−	0.991136i	$-0.542413\pi$
$733$	−7.19358	−0.265701	−0.132851	+	0.991136i	$0.542413\pi$
$734$	−14.9686	−0.552502
$735$	0	0
$736$	−33.6057	−1.23872
$737$	−39.4691	−1.45386
$738$	0	0
$739$	−0.780351	−0.0287057	−0.0143528	−	0.999897i	$-0.504569\pi$
$739$	−0.780351	−0.0287057	−0.0143528	+	0.999897i	$0.504569\pi$
$740$	42.5493	1.56414
$741$	0	0
$742$	−1.48108	−0.0543721
$743$	13.0511	0.478797	0.239399	−	0.970921i	$-0.423050\pi$
$743$	13.0511	0.478797	0.239399	+	0.970921i	$0.423050\pi$
$744$	0	0
$745$	−53.3396	−1.95421
$746$	8.29272	0.303618
$747$	0	0
$748$	−12.2301	−0.447178
$749$	−48.9329	−1.78797
$750$	0	0
$751$	−28.3735	−1.03536	−0.517681	−	0.855573i	$-0.673205\pi$
$751$	−28.3735	−1.03536	−0.517681	+	0.855573i	$0.673205\pi$
$752$	−1.95019	−0.0711159
$753$	0	0
$754$	−10.7191	−0.390366
$755$	2.74824	0.100019
$756$	0	0
$757$	−11.4037	−0.414476	−0.207238	−	0.978291i	$-0.566447\pi$
$757$	−11.4037	−0.414476	−0.207238	+	0.978291i	$0.566447\pi$
$758$	29.7840	1.08180
$759$	0	0
$760$	84.2234	3.05510
$761$	−19.8526	−0.719658	−0.359829	−	0.933018i	$-0.617165\pi$
$761$	−19.8526	−0.719658	−0.359829	+	0.933018i	$0.617165\pi$
$762$	0	0
$763$	−8.19899	−0.296823
$764$	−16.3903	−0.592980
$765$	0	0
$766$	30.2617	1.09340
$767$	−36.0342	−1.30112
$768$	0	0
$769$	−50.3300	−1.81495	−0.907474	−	0.420109i	$-0.861992\pi$
$769$	−50.3300	−1.81495	−0.907474	+	0.420109i	$0.861992\pi$
$770$	39.4538	1.42181
$771$	0	0
$772$	−14.5733	−0.524503
$773$	−37.6442	−1.35397	−0.676984	−	0.735998i	$-0.736713\pi$
$773$	−37.6442	−1.35397	−0.676984	+	0.735998i	$0.736713\pi$
$774$	0	0
$775$	53.3155	1.91515
$776$	21.0865	0.756962
$777$	0	0
$778$	−14.0918	−0.505216
$779$	19.3334	0.692690
$780$	0	0
$781$	−22.3003	−0.797969
$782$	−17.4015	−0.622277
$783$	0	0
$784$	−0.994604	−0.0355216
$785$	43.3280	1.54644
$786$	0	0
$787$	−39.1131	−1.39423	−0.697116	−	0.716958i	$-0.745534\pi$
$787$	−39.1131	−1.39423	−0.697116	+	0.716958i	$0.745534\pi$
$788$	15.3679	0.547458
$789$	0	0
$790$	42.7589	1.52129
$791$	−12.2273	−0.434751
$792$	0	0
$793$	9.43456	0.335031
$794$	29.4700	1.04585
$795$	0	0
$796$	1.79518	0.0636284
$797$	41.6279	1.47454	0.737269	−	0.675600i	$-0.236115\pi$
$797$	41.6279	1.47454	0.737269	+	0.675600i	$0.236115\pi$
$798$	0	0
$799$	14.8792	0.526389
$800$	−57.9303	−2.04815
$801$	0	0
$802$	−7.28338	−0.257185
$803$	−4.19671	−0.148099
$804$	0	0
$805$	−75.1495	−2.64867
$806$	−16.0970	−0.566992
$807$	0	0
$808$	44.9945	1.58290
$809$	1.32763	0.0466770	0.0233385	−	0.999728i	$-0.492570\pi$
$809$	1.32763	0.0466770	0.0233385	+	0.999728i	$0.492570\pi$
$810$	0	0
$811$	−47.7210	−1.67571	−0.837856	−	0.545892i	$-0.816191\pi$
$811$	−47.7210	−1.67571	−0.837856	+	0.545892i	$0.816191\pi$
$812$	11.7713	0.413092
$813$	0	0
$814$	30.4462	1.06714
$815$	63.1838	2.21323
$816$	0	0
$817$	3.34718	0.117103
$818$	19.6159	0.685855
$819$	0	0
$820$	−11.9497	−0.417301
$821$	−18.2322	−0.636307	−0.318153	−	0.948039i	$-0.603063\pi$
$821$	−18.2322	−0.636307	−0.318153	+	0.948039i	$0.603063\pi$
$822$	0	0
$823$	10.6665	0.371810	0.185905	−	0.982568i	$-0.440478\pi$
$823$	10.6665	0.371810	0.185905	+	0.982568i	$0.440478\pi$
$824$	−32.4538	−1.13058
$825$	0	0
$826$	−29.5598	−1.02852
$827$	3.91954	0.136296	0.0681478	−	0.997675i	$-0.478291\pi$
$827$	3.91954	0.136296	0.0681478	+	0.997675i	$0.478291\pi$
$828$	0	0
$829$	34.1483	1.18602	0.593009	−	0.805196i	$-0.297940\pi$
$829$	34.1483	1.18602	0.593009	+	0.805196i	$0.297940\pi$
$830$	−35.6043	−1.23584
$831$	0	0
$832$	20.2664	0.702611
$833$	7.58848	0.262925
$834$	0	0
$835$	−52.9360	−1.83193
$836$	−29.3695	−1.01577
$837$	0	0
$838$	−16.0900	−0.555819
$839$	−18.3406	−0.633188	−0.316594	−	0.948561i	$-0.602539\pi$
$839$	−18.3406	−0.633188	−0.316594	+	0.948561i	$0.602539\pi$
$840$	0	0
$841$	−17.8572	−0.615765
$842$	12.3882	0.426927
$843$	0	0
$844$	18.6708	0.642675
$845$	−3.72615	−0.128183
$846$	0	0
$847$	−3.90983	−0.134343
$848$	0.207844	0.00713740
$849$	0	0
$850$	−29.9972	−1.02890
$851$	−57.9924	−1.98795
$852$	0	0
$853$	−24.8775	−0.851790	−0.425895	−	0.904773i	$-0.640041\pi$
$853$	−24.8775	−0.851790	−0.425895	+	0.904773i	$0.640041\pi$
$854$	7.73942	0.264838
$855$	0	0
$856$	46.1993	1.57906
$857$	−7.01498	−0.239627	−0.119814	−	0.992796i	$-0.538230\pi$
$857$	−7.01498	−0.239627	−0.119814	+	0.992796i	$0.538230\pi$
$858$	0	0
$859$	29.1462	0.994456	0.497228	−	0.867620i	$-0.334351\pi$
$859$	29.1462	0.994456	0.497228	+	0.867620i	$0.334351\pi$
$860$	−2.06885	−0.0705471
$861$	0	0
$862$	29.9220	1.01915
$863$	21.2381	0.722952	0.361476	−	0.932381i	$-0.382273\pi$
$863$	21.2381	0.722952	0.361476	+	0.932381i	$0.382273\pi$
$864$	0	0
$865$	70.7866	2.40682
$866$	−7.47287	−0.253938
$867$	0	0
$868$	17.6771	0.600001
$869$	−40.9589	−1.38944
$870$	0	0
$871$	−39.1272	−1.32578
$872$	7.74095	0.262142
$873$	0	0
$874$	−41.7881	−1.41350
$875$	−68.6442	−2.32060
$876$	0	0
$877$	−0.350498	−0.0118355	−0.00591773	−	0.999982i	$-0.501884\pi$
$877$	−0.350498	−0.0118355	−0.00591773	+	0.999982i	$0.501884\pi$
$878$	−24.4799	−0.826157
$879$	0	0
$880$	−5.53666	−0.186641
$881$	−20.5547	−0.692504	−0.346252	−	0.938142i	$-0.612546\pi$
$881$	−20.5547	−0.692504	−0.346252	+	0.938142i	$0.612546\pi$
$882$	0	0
$883$	38.4958	1.29549	0.647744	−	0.761858i	$-0.275713\pi$
$883$	38.4958	1.29549	0.647744	+	0.761858i	$0.275713\pi$
$884$	−12.1242	−0.407780
$885$	0	0
$886$	−35.7937	−1.20251
$887$	32.5923	1.09434	0.547171	−	0.837021i	$-0.315705\pi$
$887$	32.5923	1.09434	0.547171	+	0.837021i	$0.315705\pi$
$888$	0	0
$889$	43.1706	1.44789
$890$	41.9388	1.40579
$891$	0	0
$892$	6.96012	0.233042
$893$	35.7311	1.19570
$894$	0	0
$895$	−95.2415	−3.18357
$896$	−16.9299	−0.565588
$897$	0	0
$898$	−4.08164	−0.136206
$899$	16.7333	0.558087
$900$	0	0
$901$	−1.58578	−0.0528299
$902$	−8.55062	−0.284705
$903$	0	0
$904$	11.5442	0.383954
$905$	17.2854	0.574588
$906$	0	0
$907$	−18.8155	−0.624757	−0.312379	−	0.949958i	$-0.601126\pi$
$907$	−18.8155	−0.624757	−0.312379	+	0.949958i	$0.601126\pi$
$908$	21.9634	0.728882
$909$	0	0
$910$	39.1120	1.29655
$911$	6.57999	0.218005	0.109002	−	0.994041i	$-0.465234\pi$
$911$	6.57999	0.218005	0.109002	+	0.994041i	$0.465234\pi$
$912$	0	0
$913$	34.1055	1.12873
$914$	−0.158678	−0.00524861
$915$	0	0
$916$	−11.2487	−0.371669
$917$	67.5354	2.23021
$918$	0	0
$919$	26.0037	0.857782	0.428891	−	0.903356i	$-0.358904\pi$
$919$	26.0037	0.857782	0.428891	+	0.903356i	$0.358904\pi$
$920$	70.9513	2.33919
$921$	0	0
$922$	−30.1533	−0.993046
$923$	−22.1072	−0.727667
$924$	0	0
$925$	−99.9688	−3.28695
$926$	−23.5951	−0.775382
$927$	0	0
$928$	−18.1817	−0.596843
$929$	−8.06188	−0.264502	−0.132251	−	0.991216i	$-0.542220\pi$
$929$	−8.06188	−0.264502	−0.132251	+	0.991216i	$0.542220\pi$
$930$	0	0
$931$	18.2230	0.597236
$932$	−34.5836	−1.13282
$933$	0	0
$934$	3.31314	0.108409
$935$	42.2428	1.38149
$936$	0	0
$937$	23.0454	0.752862	0.376431	−	0.926445i	$-0.377151\pi$
$937$	23.0454	0.752862	0.376431	+	0.926445i	$0.377151\pi$
$938$	−32.0971	−1.04801
$939$	0	0
$940$	−22.0849	−0.720330
$941$	−37.3296	−1.21691	−0.608454	−	0.793589i	$-0.708210\pi$
$941$	−37.3296	−1.21691	−0.608454	+	0.793589i	$0.708210\pi$
$942$	0	0
$943$	16.2868	0.530370
$944$	4.14822	0.135013
$945$	0	0
$946$	−1.48037	−0.0481309
$947$	29.3980	0.955306	0.477653	−	0.878549i	$-0.341488\pi$
$947$	29.3980	0.955306	0.477653	+	0.878549i	$0.341488\pi$
$948$	0	0
$949$	−4.16036	−0.135051
$950$	−72.0355	−2.33714
$951$	0	0
$952$	−27.3211	−0.885480
$953$	−17.1464	−0.555426	−0.277713	−	0.960664i	$-0.589576\pi$
$953$	−17.1464	−0.555426	−0.277713	+	0.960664i	$0.589576\pi$
$954$	0	0
$955$	56.6120	1.83192
$956$	22.9439	0.742058
$957$	0	0
$958$	−25.2241	−0.814953
$959$	2.23522	0.0721790
$960$	0	0
$961$	−5.87140	−0.189400
$962$	30.1825	0.973123
$963$	0	0
$964$	25.4656	0.820191
$965$	50.3360	1.62037
$966$	0	0
$967$	−34.1166	−1.09711	−0.548557	−	0.836113i	$-0.684823\pi$
$967$	−34.1166	−1.09711	−0.548557	+	0.836113i	$0.684823\pi$
$968$	3.69141	0.118646
$969$	0	0
$970$	−26.5136	−0.851301
$971$	6.71553	0.215512	0.107756	−	0.994177i	$-0.465634\pi$
$971$	6.71553	0.215512	0.107756	+	0.994177i	$0.465634\pi$
$972$	0	0
$973$	−6.34384	−0.203374
$974$	30.0131	0.961680
$975$	0	0
$976$	−1.08610	−0.0347651
$977$	12.8585	0.411380	0.205690	−	0.978617i	$-0.434056\pi$
$977$	12.8585	0.411380	0.205690	+	0.978617i	$0.434056\pi$
$978$	0	0
$979$	−40.1734	−1.28395
$980$	−11.2634	−0.359796
$981$	0	0
$982$	7.81726	0.249459
$983$	−44.7830	−1.42836	−0.714178	−	0.699964i	$-0.753199\pi$
$983$	−44.7830	−1.42836	−0.714178	+	0.699964i	$0.753199\pi$
$984$	0	0
$985$	−53.0806	−1.69129
$986$	−9.41475	−0.299827
$987$	0	0
$988$	−29.1151	−0.926274
$989$	2.81973	0.0896620
$990$	0	0
$991$	38.3443	1.21805	0.609024	−	0.793152i	$-0.291561\pi$
$991$	38.3443	1.21805	0.609024	+	0.793152i	$0.291561\pi$
$992$	−27.3036	−0.866892
$993$	0	0
$994$	−18.1351	−0.575211
$995$	−6.20054	−0.196570
$996$	0	0
$997$	−50.7982	−1.60880	−0.804398	−	0.594090i	$-0.797512\pi$
$997$	−50.7982	−1.60880	−0.804398	+	0.594090i	$0.797512\pi$
$998$	26.4042	0.835811
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4923.2.a.l.1.7		18
3.2	odd	2		547.2.a.b.1.12	✓	18
12.11	even	2		8752.2.a.s.1.2		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.a.b.1.12	✓	18	3.2	odd	2
4923.2.a.l.1.7		18	1.1	even	1	trivial
8752.2.a.s.1.2		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)

\(f(q)\)	\(=\)	\(q-0.924759 q^{2} -1.14482 q^{4} +3.95421 q^{5} -3.08028 q^{7} +2.90820 q^{8} +O(q^{10})\)	\(q-0.924759 q^{2} -1.14482 q^{4} +3.95421 q^{5} -3.08028 q^{7} +2.90820 q^{8} -3.65669 q^{10} +3.50276 q^{11} +3.47242 q^{13} +2.84852 q^{14} -0.399740 q^{16} +3.04988 q^{17} +7.32400 q^{19} -4.52687 q^{20} -3.23921 q^{22} +6.16987 q^{23} +10.6358 q^{25} -3.21115 q^{26} +3.52637 q^{28} +3.33808 q^{29} +5.01284 q^{31} -5.44674 q^{32} -2.82040 q^{34} -12.1801 q^{35} -9.39929 q^{37} -6.77293 q^{38} +11.4996 q^{40} +2.63973 q^{41} +0.457015 q^{43} -4.01003 q^{44} -5.70564 q^{46} +4.87863 q^{47} +2.48812 q^{49} -9.83553 q^{50} -3.97530 q^{52} -0.519948 q^{53} +13.8506 q^{55} -8.95807 q^{56} -3.08692 q^{58} -10.3773 q^{59} +2.71700 q^{61} -4.63567 q^{62} +5.83640 q^{64} +13.7307 q^{65} -11.2680 q^{67} -3.49157 q^{68} +11.2636 q^{70} -6.36651 q^{71} -1.19812 q^{73} +8.69207 q^{74} -8.38468 q^{76} -10.7895 q^{77} -11.6933 q^{79} -1.58066 q^{80} -2.44111 q^{82} +9.73676 q^{83} +12.0599 q^{85} -0.422629 q^{86} +10.1867 q^{88} -11.4691 q^{89} -10.6960 q^{91} -7.06340 q^{92} -4.51155 q^{94} +28.9607 q^{95} +7.25071 q^{97} -2.30091 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 4 q^{2} + 16 q^{4} + 27 q^{5} - 11 q^{7} + 12 q^{8}+O(q^{10}) \) 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

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