LMFDB - Embedded newform 4923.2.a.n.1.21

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:	$1$
Dimension:	$25$
Twist minimal:	no (minimal twist has level 547)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.21
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)

$f(q)$	$=$	$q+1.98290 q^{2} +1.93190 q^{4} -0.712381 q^{5} +4.39659 q^{7} -0.135039 q^{8} +O(q^{10})$	\(q+1.98290 q^{2} +1.93190 q^{4} -0.712381 q^{5} +4.39659 q^{7} -0.135039 q^{8} -1.41258 q^{10} -3.35537 q^{11} -2.53115 q^{13} +8.71801 q^{14} -4.13157 q^{16} -4.59046 q^{17} -3.55428 q^{19} -1.37625 q^{20} -6.65336 q^{22} +2.55608 q^{23} -4.49251 q^{25} -5.01902 q^{26} +8.49377 q^{28} -5.14817 q^{29} +3.73387 q^{31} -7.92241 q^{32} -9.10242 q^{34} -3.13205 q^{35} -10.4553 q^{37} -7.04778 q^{38} +0.0961991 q^{40} +6.42906 q^{41} -2.39294 q^{43} -6.48223 q^{44} +5.06845 q^{46} -0.655082 q^{47} +12.3300 q^{49} -8.90821 q^{50} -4.88993 q^{52} +1.90404 q^{53} +2.39030 q^{55} -0.593710 q^{56} -10.2083 q^{58} -11.5584 q^{59} -4.56217 q^{61} +7.40391 q^{62} -7.44623 q^{64} +1.80315 q^{65} +13.5959 q^{67} -8.86829 q^{68} -6.21055 q^{70} -9.44418 q^{71} -8.94150 q^{73} -20.7317 q^{74} -6.86650 q^{76} -14.7522 q^{77} +6.83371 q^{79} +2.94325 q^{80} +12.7482 q^{82} -13.3811 q^{83} +3.27016 q^{85} -4.74497 q^{86} +0.453105 q^{88} +15.4078 q^{89} -11.1284 q^{91} +4.93808 q^{92} -1.29896 q^{94} +2.53200 q^{95} +6.59022 q^{97} +24.4492 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 4 q^{2} + 26 q^{4} - 29 q^{5} + 5 q^{7} - 6 q^{8}+O(q^{10}) $ 25 * q - 4 * q^2 + 26 * q^4 - 29 * q^5 + 5 * q^7 - 6 * q^8	$ 25 q - 4 q^{2} + 26 q^{4} - 29 q^{5} + 5 q^{7} - 6 q^{8} - q^{10} - 10 q^{11} + 19 q^{13} - 9 q^{14} + 16 q^{16} - 40 q^{17} - 33 q^{20} - 10 q^{22} - 26 q^{23} + 36 q^{25} + 8 q^{26} - 8 q^{28} - 30 q^{29} - 5 q^{31} - 6 q^{32} - 7 q^{34} - 11 q^{35} + 26 q^{37} - 25 q^{38} - 25 q^{40} - 9 q^{41} - 10 q^{43} - 34 q^{46} - 28 q^{47} + 20 q^{49} + 9 q^{50} - 2 q^{52} - 80 q^{53} - q^{55} - 7 q^{56} - 24 q^{58} + 2 q^{59} + 22 q^{61} - 36 q^{62} - 28 q^{64} - 30 q^{65} - 16 q^{67} - 59 q^{68} - 61 q^{70} + q^{71} + 2 q^{73} + 8 q^{74} - 46 q^{76} - 67 q^{77} - 34 q^{79} - 30 q^{80} - 4 q^{82} - 15 q^{83} + 15 q^{85} + 44 q^{86} - 55 q^{88} - 38 q^{89} - 41 q^{91} - 40 q^{92} - 46 q^{94} + 46 q^{95} - 2 q^{97} + 14 q^{98}+O(q^{100}) $ 25 * q - 4 * q^2 + 26 * q^4 - 29 * q^5 + 5 * q^7 - 6 * q^8 - q^10 - 10 * q^11 + 19 * q^13 - 9 * q^14 + 16 * q^16 - 40 * q^17 - 33 * q^20 - 10 * q^22 - 26 * q^23 + 36 * q^25 + 8 * q^26 - 8 * q^28 - 30 * q^29 - 5 * q^31 - 6 * q^32 - 7 * q^34 - 11 * q^35 + 26 * q^37 - 25 * q^38 - 25 * q^40 - 9 * q^41 - 10 * q^43 - 34 * q^46 - 28 * q^47 + 20 * q^49 + 9 * q^50 - 2 * q^52 - 80 * q^53 - q^55 - 7 * q^56 - 24 * q^58 + 2 * q^59 + 22 * q^61 - 36 * q^62 - 28 * q^64 - 30 * q^65 - 16 * q^67 - 59 * q^68 - 61 * q^70 + q^71 + 2 * q^73 + 8 * q^74 - 46 * q^76 - 67 * q^77 - 34 * q^79 - 30 * q^80 - 4 * q^82 - 15 * q^83 + 15 * q^85 + 44 * q^86 - 55 * q^88 - 38 * q^89 - 41 * q^91 - 40 * q^92 - 46 * q^94 + 46 * q^95 - 2 * q^97 + 14 * 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 $

$2$	1.98290	1.40212	0.701062	−	0.713101i	$-0.252710\pi$
$2$	1.98290	1.40212	0.701062	+	0.713101i	$0.252710\pi$
$3$	0	0
$3$	0	0
$4$	1.93190	0.965949
$5$	−0.712381	−0.318587	−0.159293	−	0.987231i	$-0.550922\pi$
$5$	−0.712381	−0.318587	−0.159293	+	0.987231i	$0.550922\pi$
$6$	0	0
$7$	4.39659	1.66176	0.830878	−	0.556455i	$-0.187839\pi$
$7$	4.39659	1.66176	0.830878	+	0.556455i	$0.187839\pi$
$8$	−0.135039	−0.0477434
$9$	0	0
$10$	−1.41258	−0.446698
$11$	−3.35537	−1.01168	−0.505841	−	0.862627i	$-0.668818\pi$
$11$	−3.35537	−1.01168	−0.505841	+	0.862627i	$0.668818\pi$
$12$	0	0
$13$	−2.53115	−0.702015	−0.351008	−	0.936373i	$-0.614161\pi$
$13$	−2.53115	−0.702015	−0.351008	+	0.936373i	$0.614161\pi$
$14$	8.71801	2.32999
$15$	0	0
$16$	−4.13157	−1.03289
$17$	−4.59046	−1.11335	−0.556675	−	0.830731i	$-0.687923\pi$
$17$	−4.59046	−1.11335	−0.556675	+	0.830731i	$0.687923\pi$
$18$	0	0
$19$	−3.55428	−0.815407	−0.407704	−	0.913114i	$-0.633670\pi$
$19$	−3.55428	−0.815407	−0.407704	+	0.913114i	$0.633670\pi$
$20$	−1.37625	−0.307739
$21$	0	0
$22$	−6.65336	−1.41850
$23$	2.55608	0.532979	0.266489	−	0.963838i	$-0.414136\pi$
$23$	2.55608	0.532979	0.266489	+	0.963838i	$0.414136\pi$
$24$	0	0
$25$	−4.49251	−0.898503
$26$	−5.01902	−0.984312
$27$	0	0
$28$	8.49377	1.60517
$29$	−5.14817	−0.955990	−0.477995	−	0.878362i	$-0.658636\pi$
$29$	−5.14817	−0.955990	−0.477995	+	0.878362i	$0.658636\pi$
$30$	0	0
$31$	3.73387	0.670624	0.335312	−	0.942107i	$-0.391158\pi$
$31$	3.73387	0.670624	0.335312	+	0.942107i	$0.391158\pi$
$32$	−7.92241	−1.40050
$33$	0	0
$34$	−9.10242	−1.56105
$35$	−3.13205	−0.529413
$36$	0	0
$37$	−10.4553	−1.71883	−0.859417	−	0.511276i	$-0.829173\pi$
$37$	−10.4553	−1.71883	−0.859417	+	0.511276i	$0.829173\pi$
$38$	−7.04778	−1.14330
$39$	0	0
$40$	0.0961991	0.0152104
$41$	6.42906	1.00405	0.502025	−	0.864853i	$-0.332589\pi$
$41$	6.42906	1.00405	0.502025	+	0.864853i	$0.332589\pi$
$42$	0	0
$43$	−2.39294	−0.364920	−0.182460	−	0.983213i	$-0.558406\pi$
$43$	−2.39294	−0.364920	−0.182460	+	0.983213i	$0.558406\pi$
$44$	−6.48223	−0.977233
$45$	0	0
$46$	5.06845	0.747302
$47$	−0.655082	−0.0955535	−0.0477768	−	0.998858i	$-0.515214\pi$
$47$	−0.655082	−0.0955535	−0.0477768	+	0.998858i	$0.515214\pi$
$48$	0	0
$49$	12.3300	1.76143
$50$	−8.90821	−1.25981
$51$	0	0
$52$	−4.88993	−0.678111
$53$	1.90404	0.261540	0.130770	−	0.991413i	$-0.458255\pi$
$53$	1.90404	0.261540	0.130770	+	0.991413i	$0.458255\pi$
$54$	0	0
$55$	2.39030	0.322308
$56$	−0.593710	−0.0793379
$57$	0	0
$58$	−10.2083	−1.34042
$59$	−11.5584	−1.50478	−0.752391	−	0.658717i	$-0.771099\pi$
$59$	−11.5584	−1.50478	−0.752391	+	0.658717i	$0.771099\pi$
$60$	0	0
$61$	−4.56217	−0.584126	−0.292063	−	0.956399i	$-0.594342\pi$
$61$	−4.56217	−0.584126	−0.292063	+	0.956399i	$0.594342\pi$
$62$	7.40391	0.940297
$63$	0	0
$64$	−7.44623	−0.930778
$65$	1.80315	0.223653
$66$	0	0
$67$	13.5959	1.66100	0.830499	−	0.557020i	$-0.188056\pi$
$67$	13.5959	1.66100	0.830499	+	0.557020i	$0.188056\pi$
$68$	−8.86829	−1.07544
$69$	0	0
$70$	−6.21055	−0.742302
$71$	−9.44418	−1.12082	−0.560409	−	0.828216i	$-0.689356\pi$
$71$	−9.44418	−1.12082	−0.560409	+	0.828216i	$0.689356\pi$
$72$	0	0
$73$	−8.94150	−1.04652	−0.523262	−	0.852172i	$-0.675285\pi$
$73$	−8.94150	−1.04652	−0.523262	+	0.852172i	$0.675285\pi$
$74$	−20.7317	−2.41002
$75$	0	0
$76$	−6.86650	−0.787642
$77$	−14.7522	−1.68117
$78$	0	0
$79$	6.83371	0.768852	0.384426	−	0.923156i	$-0.374399\pi$
$79$	6.83371	0.768852	0.384426	+	0.923156i	$0.374399\pi$
$80$	2.94325	0.329065
$81$	0	0
$82$	12.7482	1.40780
$83$	−13.3811	−1.46877	−0.734385	−	0.678733i	$-0.762530\pi$
$83$	−13.3811	−1.46877	−0.734385	+	0.678733i	$0.762530\pi$
$84$	0	0
$85$	3.27016	0.354698
$86$	−4.74497	−0.511663
$87$	0	0
$88$	0.453105	0.0483011
$89$	15.4078	1.63323	0.816614	−	0.577185i	$-0.195849\pi$
$89$	15.4078	1.63323	0.816614	+	0.577185i	$0.195849\pi$
$90$	0	0
$91$	−11.1284	−1.16658
$92$	4.93808	0.514830
$93$	0	0
$94$	−1.29896	−0.133978
$95$	2.53200	0.259778
$96$	0	0
$97$	6.59022	0.669135	0.334568	−	0.942372i	$-0.391410\pi$
$97$	6.59022	0.669135	0.334568	+	0.942372i	$0.391410\pi$
$98$	24.4492	2.46974
$99$	0	0
$100$	−8.67908	−0.867908
$101$	8.18518	0.814455	0.407228	−	0.913327i	$-0.366496\pi$
$101$	8.18518	0.814455	0.407228	+	0.913327i	$0.366496\pi$
$102$	0	0
$103$	−0.986016	−0.0971550	−0.0485775	−	0.998819i	$-0.515469\pi$
$103$	−0.986016	−0.0971550	−0.0485775	+	0.998819i	$0.515469\pi$
$104$	0.341804	0.0335166
$105$	0	0
$106$	3.77552	0.366711
$107$	13.5258	1.30758	0.653792	−	0.756674i	$-0.273177\pi$
$107$	13.5258	1.30758	0.653792	+	0.756674i	$0.273177\pi$
$108$	0	0
$109$	12.4417	1.19169	0.595847	−	0.803098i	$-0.296816\pi$
$109$	12.4417	1.19169	0.595847	+	0.803098i	$0.296816\pi$
$110$	4.73973	0.451916
$111$	0	0
$112$	−18.1648	−1.71641
$113$	6.83528	0.643009	0.321505	−	0.946908i	$-0.395811\pi$
$113$	6.83528	0.643009	0.321505	+	0.946908i	$0.395811\pi$
$114$	0	0
$115$	−1.82090	−0.169800
$116$	−9.94573	−0.923438
$117$	0	0
$118$	−22.9193	−2.10989
$119$	−20.1824	−1.85011
$120$	0	0
$121$	0.258495	0.0234995
$122$	−9.04633	−0.819016
$123$	0	0
$124$	7.21347	0.647788
$125$	6.76229	0.604838
$126$	0	0
$127$	−8.02668	−0.712253	−0.356126	−	0.934438i	$-0.615903\pi$
$127$	−8.02668	−0.712253	−0.356126	+	0.934438i	$0.615903\pi$
$128$	1.07968	0.0954315
$129$	0	0
$130$	3.57546	0.313589
$131$	7.45795	0.651604	0.325802	−	0.945438i	$-0.394366\pi$
$131$	7.45795	0.651604	0.325802	+	0.945438i	$0.394366\pi$
$132$	0	0
$133$	−15.6267	−1.35501
$134$	26.9592	2.32892
$135$	0	0
$136$	0.619890	0.0531551
$137$	−5.26530	−0.449845	−0.224922	−	0.974377i	$-0.572213\pi$
$137$	−5.26530	−0.449845	−0.224922	+	0.974377i	$0.572213\pi$
$138$	0	0
$139$	−16.6957	−1.41611	−0.708056	−	0.706157i	$-0.750427\pi$
$139$	−16.6957	−1.41611	−0.708056	+	0.706157i	$0.750427\pi$
$140$	−6.05080	−0.511386
$141$	0	0
$142$	−18.7269	−1.57153
$143$	8.49294	0.710216
$144$	0	0
$145$	3.66746	0.304566
$146$	−17.7301	−1.46735
$147$	0	0
$148$	−20.1985	−1.66031
$149$	7.73867	0.633976	0.316988	−	0.948430i	$-0.397328\pi$
$149$	7.73867	0.633976	0.316988	+	0.948430i	$0.397328\pi$
$150$	0	0
$151$	−20.1661	−1.64109	−0.820547	−	0.571579i	$-0.806331\pi$
$151$	−20.1661	−1.64109	−0.820547	+	0.571579i	$0.806331\pi$
$152$	0.479965	0.0389303
$153$	0	0
$154$	−29.2521	−2.35720
$155$	−2.65994	−0.213652
$156$	0	0
$157$	23.0377	1.83861	0.919304	−	0.393548i	$-0.128752\pi$
$157$	23.0377	1.83861	0.919304	+	0.393548i	$0.128752\pi$
$158$	13.5506	1.07803
$159$	0	0
$160$	5.64378	0.446180
$161$	11.2380	0.885680
$162$	0	0
$163$	−4.87315	−0.381694	−0.190847	−	0.981620i	$-0.561124\pi$
$163$	−4.87315	−0.381694	−0.190847	+	0.981620i	$0.561124\pi$
$164$	12.4203	0.969861
$165$	0	0
$166$	−26.5335	−2.05940
$167$	−4.67914	−0.362083	−0.181041	−	0.983475i	$-0.557947\pi$
$167$	−4.67914	−0.362083	−0.181041	+	0.983475i	$0.557947\pi$
$168$	0	0
$169$	−6.59327	−0.507175
$170$	6.48440	0.497330
$171$	0	0
$172$	−4.62292	−0.352494
$173$	8.50196	0.646392	0.323196	−	0.946332i	$-0.395243\pi$
$173$	8.50196	0.646392	0.323196	+	0.946332i	$0.395243\pi$
$174$	0	0
$175$	−19.7517	−1.49309
$176$	13.8629	1.04496
$177$	0	0
$178$	30.5522	2.28999
$179$	−8.80118	−0.657831	−0.328916	−	0.944359i	$-0.606683\pi$
$179$	−8.80118	−0.657831	−0.328916	+	0.944359i	$0.606683\pi$
$180$	0	0
$181$	5.64971	0.419940	0.209970	−	0.977708i	$-0.432663\pi$
$181$	5.64971	0.419940	0.209970	+	0.977708i	$0.432663\pi$
$182$	−22.0666	−1.63568
$183$	0	0
$184$	−0.345169	−0.0254462
$185$	7.44813	0.547597
$186$	0	0
$187$	15.4027	1.12635
$188$	−1.26555	−0.0922999
$189$	0	0
$190$	5.02071	0.364241
$191$	−2.05404	−0.148625	−0.0743124	−	0.997235i	$-0.523676\pi$
$191$	−2.05404	−0.148625	−0.0743124	+	0.997235i	$0.523676\pi$
$192$	0	0
$193$	−18.3204	−1.31873	−0.659366	−	0.751822i	$-0.729175\pi$
$193$	−18.3204	−1.31873	−0.659366	+	0.751822i	$0.729175\pi$
$194$	13.0677	0.938210
$195$	0	0
$196$	23.8203	1.70145
$197$	−22.2367	−1.58430	−0.792149	−	0.610328i	$-0.791038\pi$
$197$	−22.2367	−1.58430	−0.792149	+	0.610328i	$0.791038\pi$
$198$	0	0
$199$	3.69903	0.262217	0.131109	−	0.991368i	$-0.458146\pi$
$199$	3.69903	0.262217	0.131109	+	0.991368i	$0.458146\pi$
$200$	0.606663	0.0428976
$201$	0	0
$202$	16.2304	1.14197
$203$	−22.6344	−1.58862
$204$	0	0
$205$	−4.57994	−0.319877
$206$	−1.95517	−0.136223
$207$	0	0
$208$	10.4576	0.725105
$209$	11.9259	0.824933
$210$	0	0
$211$	17.8259	1.22719	0.613593	−	0.789623i	$-0.289724\pi$
$211$	17.8259	1.22719	0.613593	+	0.789623i	$0.289724\pi$
$212$	3.67841	0.252634
$213$	0	0
$214$	26.8202	1.83339
$215$	1.70469	0.116259
$216$	0	0
$217$	16.4163	1.11441
$218$	24.6706	1.67090
$219$	0	0
$220$	4.61782	0.311333
$221$	11.6191	0.781588
$222$	0	0
$223$	−18.5405	−1.24157	−0.620783	−	0.783982i	$-0.713185\pi$
$223$	−18.5405	−1.24157	−0.620783	+	0.783982i	$0.713185\pi$
$224$	−34.8316	−2.32728
$225$	0	0
$226$	13.5537	0.901578
$227$	1.75864	0.116725	0.0583625	−	0.998295i	$-0.481412\pi$
$227$	1.75864	0.116725	0.0583625	+	0.998295i	$0.481412\pi$
$228$	0	0
$229$	−7.73042	−0.510840	−0.255420	−	0.966830i	$-0.582214\pi$
$229$	−7.73042	−0.510840	−0.255420	+	0.966830i	$0.582214\pi$
$230$	−3.61067	−0.238080
$231$	0	0
$232$	0.695202	0.0456423
$233$	−22.8019	−1.49380	−0.746902	−	0.664934i	$-0.768460\pi$
$233$	−22.8019	−1.49380	−0.746902	+	0.664934i	$0.768460\pi$
$234$	0	0
$235$	0.466668	0.0304421
$236$	−22.3297	−1.45354
$237$	0	0
$238$	−40.0196	−2.59409
$239$	14.1169	0.913149	0.456575	−	0.889685i	$-0.349076\pi$
$239$	14.1169	0.913149	0.456575	+	0.889685i	$0.349076\pi$
$240$	0	0
$241$	−4.96966	−0.320124	−0.160062	−	0.987107i	$-0.551169\pi$
$241$	−4.96966	−0.320124	−0.160062	+	0.987107i	$0.551169\pi$
$242$	0.512570	0.0329492
$243$	0	0
$244$	−8.81364	−0.564236
$245$	−8.78367	−0.561168
$246$	0	0
$247$	8.99642	0.572428
$248$	−0.504218	−0.0320179
$249$	0	0
$250$	13.4090	0.848057
$251$	−6.31919	−0.398864	−0.199432	−	0.979912i	$-0.563910\pi$
$251$	−6.31919	−0.398864	−0.199432	+	0.979912i	$0.563910\pi$
$252$	0	0
$253$	−8.57657	−0.539205
$254$	−15.9161	−0.998666
$255$	0	0
$256$	17.0334	1.06459
$257$	−16.1829	−1.00946	−0.504730	−	0.863277i	$-0.668408\pi$
$257$	−16.1829	−1.00946	−0.504730	+	0.863277i	$0.668408\pi$
$258$	0	0
$259$	−45.9675	−2.85628
$260$	3.48349	0.216037
$261$	0	0
$262$	14.7884	0.913630
$263$	17.0403	1.05075	0.525375	−	0.850871i	$-0.323925\pi$
$263$	17.0403	1.05075	0.525375	+	0.850871i	$0.323925\pi$
$264$	0	0
$265$	−1.35640	−0.0833230
$266$	−30.9862	−1.89989
$267$	0	0
$268$	26.2658	1.60444
$269$	−14.8919	−0.907975	−0.453988	−	0.891008i	$-0.649999\pi$
$269$	−14.8919	−0.907975	−0.453988	+	0.891008i	$0.649999\pi$
$270$	0	0
$271$	−0.717096	−0.0435605	−0.0217802	−	0.999763i	$-0.506933\pi$
$271$	−0.717096	−0.0435605	−0.0217802	+	0.999763i	$0.506933\pi$
$272$	18.9658	1.14997
$273$	0	0
$274$	−10.4406	−0.630737
$275$	15.0740	0.908998
$276$	0	0
$277$	−9.12809	−0.548454	−0.274227	−	0.961665i	$-0.588422\pi$
$277$	−9.12809	−0.548454	−0.274227	+	0.961665i	$0.588422\pi$
$278$	−33.1059	−1.98556
$279$	0	0
$280$	0.422948	0.0252760
$281$	26.4024	1.57504	0.787519	−	0.616291i	$-0.211365\pi$
$281$	26.4024	1.57504	0.787519	+	0.616291i	$0.211365\pi$
$282$	0	0
$283$	3.99826	0.237672	0.118836	−	0.992914i	$-0.462084\pi$
$283$	3.99826	0.237672	0.118836	+	0.992914i	$0.462084\pi$
$284$	−18.2452	−1.08265
$285$	0	0
$286$	16.8407	0.995810
$287$	28.2659	1.66849
$288$	0	0
$289$	4.07228	0.239546
$290$	7.27221	0.427039
$291$	0	0
$292$	−17.2741	−1.01089
$293$	−9.63049	−0.562619	−0.281309	−	0.959617i	$-0.590769\pi$
$293$	−9.63049	−0.562619	−0.281309	+	0.959617i	$0.590769\pi$
$294$	0	0
$295$	8.23402	0.479403
$296$	1.41187	0.0820630
$297$	0	0
$298$	15.3450	0.888913
$299$	−6.46981	−0.374159
$300$	0	0
$301$	−10.5208	−0.606408
$302$	−39.9874	−2.30102
$303$	0	0
$304$	14.6847	0.842227
$305$	3.25000	0.186095
$306$	0	0
$307$	−33.0818	−1.88808	−0.944040	−	0.329832i	$-0.893008\pi$
$307$	−33.0818	−1.88808	−0.944040	+	0.329832i	$0.893008\pi$
$308$	−28.4997	−1.62392
$309$	0	0
$310$	−5.27440	−0.299566
$311$	−18.0798	−1.02521	−0.512605	−	0.858624i	$-0.671320\pi$
$311$	−18.0798	−1.02521	−0.512605	+	0.858624i	$0.671320\pi$
$312$	0	0
$313$	10.3661	0.585927	0.292963	−	0.956124i	$-0.405359\pi$
$313$	10.3661	0.585927	0.292963	+	0.956124i	$0.405359\pi$
$314$	45.6815	2.57796
$315$	0	0
$316$	13.2020	0.742672
$317$	−3.76302	−0.211352	−0.105676	−	0.994401i	$-0.533701\pi$
$317$	−3.76302	−0.211352	−0.105676	+	0.994401i	$0.533701\pi$
$318$	0	0
$319$	17.2740	0.967158
$320$	5.30455	0.296534
$321$	0	0
$322$	22.2839	1.24183
$323$	16.3158	0.907833
$324$	0	0
$325$	11.3712	0.630762
$326$	−9.66297	−0.535183
$327$	0	0
$328$	−0.868172	−0.0479368
$329$	−2.88013	−0.158787
$330$	0	0
$331$	−13.0196	−0.715623	−0.357811	−	0.933794i	$-0.616477\pi$
$331$	−13.0196	−0.715623	−0.357811	+	0.933794i	$0.616477\pi$
$332$	−25.8510	−1.41876
$333$	0	0
$334$	−9.27828	−0.507685
$335$	−9.68543	−0.529172
$336$	0	0
$337$	27.1502	1.47897	0.739483	−	0.673175i	$-0.235070\pi$
$337$	27.1502	1.47897	0.739483	+	0.673175i	$0.235070\pi$
$338$	−13.0738	−0.711121
$339$	0	0
$340$	6.31761	0.342620
$341$	−12.5285	−0.678458
$342$	0	0
$343$	23.4339	1.26531
$344$	0.323140	0.0174225
$345$	0	0
$346$	16.8585	0.906321
$347$	−36.7795	−1.97443	−0.987213	−	0.159405i	$-0.949042\pi$
$347$	−36.7795	−1.97443	−0.987213	+	0.159405i	$0.949042\pi$
$348$	0	0
$349$	28.0449	1.50121	0.750606	−	0.660750i	$-0.229762\pi$
$349$	28.0449	1.50121	0.750606	+	0.660750i	$0.229762\pi$
$350$	−39.1658	−2.09350
$351$	0	0
$352$	26.5826	1.41686
$353$	14.9368	0.795005	0.397502	−	0.917601i	$-0.369877\pi$
$353$	14.9368	0.795005	0.397502	+	0.917601i	$0.369877\pi$
$354$	0	0
$355$	6.72786	0.357078
$356$	29.7664	1.57761
$357$	0	0
$358$	−17.4519	−0.922360
$359$	19.5526	1.03195	0.515973	−	0.856605i	$-0.327430\pi$
$359$	19.5526	1.03195	0.515973	+	0.856605i	$0.327430\pi$
$360$	0	0
$361$	−6.36711	−0.335111
$362$	11.2028	0.588807
$363$	0	0
$364$	−21.4990	−1.12685
$365$	6.36976	0.333408
$366$	0	0
$367$	9.76632	0.509798	0.254899	−	0.966968i	$-0.417958\pi$
$367$	9.76632	0.509798	0.254899	+	0.966968i	$0.417958\pi$
$368$	−10.5606	−0.550509
$369$	0	0
$370$	14.7689	0.767799
$371$	8.37127	0.434615
$372$	0	0
$373$	−14.7414	−0.763280	−0.381640	−	0.924311i	$-0.624641\pi$
$373$	−14.7414	−0.763280	−0.381640	+	0.924311i	$0.624641\pi$
$374$	30.5420	1.57929
$375$	0	0
$376$	0.0884615	0.00456205
$377$	13.0308	0.671120
$378$	0	0
$379$	11.0837	0.569329	0.284665	−	0.958627i	$-0.408118\pi$
$379$	11.0837	0.569329	0.284665	+	0.958627i	$0.408118\pi$
$380$	4.89157	0.250932
$381$	0	0
$382$	−4.07295	−0.208390
$383$	−31.2011	−1.59430	−0.797151	−	0.603780i	$-0.793660\pi$
$383$	−31.2011	−1.59430	−0.797151	+	0.603780i	$0.793660\pi$
$384$	0	0
$385$	10.5092	0.535597
$386$	−36.3276	−1.84903
$387$	0	0
$388$	12.7316	0.646350
$389$	−18.7044	−0.948350	−0.474175	−	0.880431i	$-0.657254\pi$
$389$	−18.7044	−0.948350	−0.474175	+	0.880431i	$0.657254\pi$
$390$	0	0
$391$	−11.7336	−0.593391
$392$	−1.66503	−0.0840967
$393$	0	0
$394$	−44.0931	−2.22138
$395$	−4.86821	−0.244946
$396$	0	0
$397$	9.10573	0.457003	0.228502	−	0.973544i	$-0.426617\pi$
$397$	9.10573	0.457003	0.228502	+	0.973544i	$0.426617\pi$
$398$	7.33481	0.367661
$399$	0	0
$400$	18.5611	0.928056
$401$	26.3060	1.31366	0.656831	−	0.754038i	$-0.271897\pi$
$401$	26.3060	1.31366	0.656831	+	0.754038i	$0.271897\pi$
$402$	0	0
$403$	−9.45100	−0.470788
$404$	15.8129	0.786723
$405$	0	0
$406$	−44.8817	−2.22744
$407$	35.0812	1.73891
$408$	0	0
$409$	−30.1487	−1.49076	−0.745378	−	0.666642i	$-0.767731\pi$
$409$	−30.1487	−1.49076	−0.745378	+	0.666642i	$0.767731\pi$
$410$	−9.08157	−0.448507
$411$	0	0
$412$	−1.90488	−0.0938468
$413$	−50.8178	−2.50058
$414$	0	0
$415$	9.53248	0.467931
$416$	20.0528	0.983170
$417$	0	0
$418$	23.6479	1.15666
$419$	19.2797	0.941875	0.470937	−	0.882167i	$-0.343916\pi$
$419$	19.2797	0.941875	0.470937	+	0.882167i	$0.343916\pi$
$420$	0	0
$421$	−22.0329	−1.07382	−0.536910	−	0.843639i	$-0.680409\pi$
$421$	−22.0329	−1.07382	−0.536910	+	0.843639i	$0.680409\pi$
$422$	35.3470	1.72067
$423$	0	0
$424$	−0.257119	−0.0124868
$425$	20.6227	1.00035
$426$	0	0
$427$	−20.0580	−0.970674
$428$	26.1304	1.26306
$429$	0	0
$430$	3.38023	0.163009
$431$	14.5238	0.699585	0.349793	−	0.936827i	$-0.386252\pi$
$431$	14.5238	0.699585	0.349793	+	0.936827i	$0.386252\pi$
$432$	0	0
$433$	−14.0023	−0.672908	−0.336454	−	0.941700i	$-0.609228\pi$
$433$	−14.0023	−0.672908	−0.336454	+	0.941700i	$0.609228\pi$
$434$	32.5519	1.56254
$435$	0	0
$436$	24.0360	1.15112
$437$	−9.08500	−0.434595
$438$	0	0
$439$	36.3096	1.73296	0.866482	−	0.499208i	$-0.166376\pi$
$439$	36.3096	1.73296	0.866482	+	0.499208i	$0.166376\pi$
$440$	−0.322783	−0.0153881
$441$	0	0
$442$	23.0396	1.09588
$443$	14.0802	0.668972	0.334486	−	0.942401i	$-0.391437\pi$
$443$	14.0802	0.668972	0.334486	+	0.942401i	$0.391437\pi$
$444$	0	0
$445$	−10.9763	−0.520324
$446$	−36.7641	−1.74083
$447$	0	0
$448$	−32.7380	−1.54673
$449$	−0.723191	−0.0341295	−0.0170647	−	0.999854i	$-0.505432\pi$
$449$	−0.723191	−0.0341295	−0.0170647	+	0.999854i	$0.505432\pi$
$450$	0	0
$451$	−21.5719	−1.01578
$452$	13.2051	0.621114
$453$	0	0
$454$	3.48721	0.163663
$455$	7.92769	0.371656
$456$	0	0
$457$	25.8037	1.20705	0.603523	−	0.797346i	$-0.293763\pi$
$457$	25.8037	1.20705	0.603523	+	0.797346i	$0.293763\pi$
$458$	−15.3287	−0.716261
$459$	0	0
$460$	−3.51779	−0.164018
$461$	−14.5026	−0.675453	−0.337726	−	0.941244i	$-0.609658\pi$
$461$	−14.5026	−0.675453	−0.337726	+	0.941244i	$0.609658\pi$
$462$	0	0
$463$	36.0906	1.67727	0.838636	−	0.544693i	$-0.183354\pi$
$463$	36.0906	1.67727	0.838636	+	0.544693i	$0.183354\pi$
$464$	21.2700	0.987434
$465$	0	0
$466$	−45.2140	−2.09450
$467$	−35.9415	−1.66318	−0.831588	−	0.555393i	$-0.812568\pi$
$467$	−35.9415	−1.66318	−0.831588	+	0.555393i	$0.812568\pi$
$468$	0	0
$469$	59.7754	2.76017
$470$	0.925357	0.0426835
$471$	0	0
$472$	1.56084	0.0718434
$473$	8.02920	0.369183
$474$	0	0
$475$	15.9676	0.732646
$476$	−38.9903	−1.78712
$477$	0	0
$478$	27.9925	1.28035
$479$	31.8677	1.45607	0.728037	−	0.685538i	$-0.240433\pi$
$479$	31.8677	1.45607	0.728037	+	0.685538i	$0.240433\pi$
$480$	0	0
$481$	26.4638	1.20665
$482$	−9.85435	−0.448853
$483$	0	0
$484$	0.499386	0.0226994
$485$	−4.69475	−0.213177
$486$	0	0
$487$	12.6821	0.574681	0.287341	−	0.957828i	$-0.407229\pi$
$487$	12.6821	0.574681	0.287341	+	0.957828i	$0.407229\pi$
$488$	0.616069	0.0278882
$489$	0	0
$490$	−17.4172	−0.786827
$491$	−33.7881	−1.52483	−0.762417	−	0.647086i	$-0.775987\pi$
$491$	−33.7881	−1.52483	−0.762417	+	0.647086i	$0.775987\pi$
$492$	0	0
$493$	23.6324	1.06435
$494$	17.8390	0.802615
$495$	0	0
$496$	−15.4267	−0.692681
$497$	−41.5222	−1.86253
$498$	0	0
$499$	−15.2141	−0.681077	−0.340538	−	0.940231i	$-0.610609\pi$
$499$	−15.2141	−0.681077	−0.340538	+	0.940231i	$0.610609\pi$
$500$	13.0641	0.584242
$501$	0	0
$502$	−12.5303	−0.559256
$503$	−18.6115	−0.829845	−0.414923	−	0.909857i	$-0.636191\pi$
$503$	−18.6115	−0.829845	−0.414923	+	0.909857i	$0.636191\pi$
$504$	0	0
$505$	−5.83097	−0.259475
$506$	−17.0065	−0.756031
$507$	0	0
$508$	−15.5067	−0.688000
$509$	4.75744	0.210870	0.105435	−	0.994426i	$-0.466377\pi$
$509$	4.75744	0.210870	0.105435	+	0.994426i	$0.466377\pi$
$510$	0	0
$511$	−39.3121	−1.73907
$512$	31.6161	1.39725
$513$	0	0
$514$	−32.0890	−1.41539
$515$	0.702419	0.0309523
$516$	0	0
$517$	2.19804	0.0966697
$518$	−91.1490	−4.00486
$519$	0	0
$520$	−0.243495	−0.0106779
$521$	37.2463	1.63179	0.815894	−	0.578202i	$-0.196245\pi$
$521$	37.2463	1.63179	0.815894	+	0.578202i	$0.196245\pi$
$522$	0	0
$523$	−31.6588	−1.38434	−0.692171	−	0.721734i	$-0.743345\pi$
$523$	−31.6588	−1.38434	−0.692171	+	0.721734i	$0.743345\pi$
$524$	14.4080	0.629417
$525$	0	0
$526$	33.7893	1.47328
$527$	−17.1402	−0.746638
$528$	0	0
$529$	−16.4665	−0.715934
$530$	−2.68961	−0.116829
$531$	0	0
$532$	−30.1892	−1.30887
$533$	−16.2729	−0.704858
$534$	0	0
$535$	−9.63549	−0.416579
$536$	−1.83597	−0.0793017
$537$	0	0
$538$	−29.5292	−1.27309
$539$	−41.3717	−1.78201
$540$	0	0
$541$	36.6462	1.57554	0.787771	−	0.615969i	$-0.211235\pi$
$541$	36.6462	1.57554	0.787771	+	0.615969i	$0.211235\pi$
$542$	−1.42193	−0.0610772
$543$	0	0
$544$	36.3675	1.55924
$545$	−8.86320	−0.379658
$546$	0	0
$547$	1.00000	0.0427569
$547$	1.00000	0.0427569
$548$	−10.1720	−0.434527
$549$	0	0
$550$	29.8903	1.27453
$551$	18.2980	0.779522
$552$	0	0
$553$	30.0450	1.27764
$554$	−18.1001	−0.769000
$555$	0	0
$556$	−32.2544	−1.36789
$557$	20.3359	0.861660	0.430830	−	0.902433i	$-0.358221\pi$
$557$	20.3359	0.861660	0.430830	+	0.902433i	$0.358221\pi$
$558$	0	0
$559$	6.05690	0.256179
$560$	12.9403	0.546826
$561$	0	0
$562$	52.3534	2.20840
$563$	4.03557	0.170079	0.0850394	−	0.996378i	$-0.472898\pi$
$563$	4.03557	0.170079	0.0850394	+	0.996378i	$0.472898\pi$
$564$	0	0
$565$	−4.86933	−0.204854
$566$	7.92815	0.333245
$567$	0	0
$568$	1.27533	0.0535117
$569$	−1.98791	−0.0833376	−0.0416688	−	0.999131i	$-0.513267\pi$
$569$	−1.98791	−0.0833376	−0.0416688	+	0.999131i	$0.513267\pi$
$570$	0	0
$571$	−30.1856	−1.26323	−0.631614	−	0.775283i	$-0.717607\pi$
$571$	−30.1856	−1.26323	−0.631614	+	0.775283i	$0.717607\pi$
$572$	16.4075	0.686032
$573$	0	0
$574$	56.0486	2.33942
$575$	−11.4832	−0.478883
$576$	0	0
$577$	31.1795	1.29802	0.649010	−	0.760780i	$-0.275183\pi$
$577$	31.1795	1.29802	0.649010	+	0.760780i	$0.275183\pi$
$578$	8.07494	0.335873
$579$	0	0
$580$	7.08516	0.294195
$581$	−58.8314	−2.44074
$582$	0	0
$583$	−6.38875	−0.264595
$584$	1.20745	0.0499646
$585$	0	0
$586$	−19.0963	−0.788861
$587$	29.3436	1.21114	0.605571	−	0.795792i	$-0.292945\pi$
$587$	29.3436	1.21114	0.605571	+	0.795792i	$0.292945\pi$
$588$	0	0
$589$	−13.2712	−0.546832
$590$	16.3273	0.672182
$591$	0	0
$592$	43.1966	1.77537
$593$	5.23773	0.215088	0.107544	−	0.994200i	$-0.465701\pi$
$593$	5.23773	0.215088	0.107544	+	0.994200i	$0.465701\pi$
$594$	0	0
$595$	14.3775	0.589421
$596$	14.9503	0.612389
$597$	0	0
$598$	−12.8290	−0.524617
$599$	13.9395	0.569554	0.284777	−	0.958594i	$-0.408080\pi$
$599$	13.9395	0.569554	0.284777	+	0.958594i	$0.408080\pi$
$600$	0	0
$601$	−17.3985	−0.709701	−0.354851	−	0.934923i	$-0.615468\pi$
$601$	−17.3985	−0.709701	−0.354851	+	0.934923i	$0.615468\pi$
$602$	−20.8617	−0.850258
$603$	0	0
$604$	−38.9588	−1.58521
$605$	−0.184147	−0.00748664
$606$	0	0
$607$	5.13339	0.208358	0.104179	−	0.994559i	$-0.466779\pi$
$607$	5.13339	0.208358	0.104179	+	0.994559i	$0.466779\pi$
$608$	28.1584	1.14198
$609$	0	0
$610$	6.44444	0.260928
$611$	1.65811	0.0670800
$612$	0	0
$613$	8.90668	0.359737	0.179869	−	0.983691i	$-0.442433\pi$
$613$	8.90668	0.359737	0.179869	+	0.983691i	$0.442433\pi$
$614$	−65.5980	−2.64732
$615$	0	0
$616$	1.99212	0.0802647
$617$	−22.7075	−0.914171	−0.457085	−	0.889423i	$-0.651107\pi$
$617$	−22.7075	−0.914171	−0.457085	+	0.889423i	$0.651107\pi$
$618$	0	0
$619$	−18.2517	−0.733600	−0.366800	−	0.930300i	$-0.619547\pi$
$619$	−18.2517	−0.733600	−0.366800	+	0.930300i	$0.619547\pi$
$620$	−5.13874	−0.206377
$621$	0	0
$622$	−35.8504	−1.43747
$623$	67.7419	2.71402
$624$	0	0
$625$	17.6452	0.705809
$626$	20.5550	0.821541
$627$	0	0
$628$	44.5065	1.77600
$629$	47.9944	1.91366
$630$	0	0
$631$	23.0859	0.919037	0.459518	−	0.888168i	$-0.348022\pi$
$631$	23.0859	0.919037	0.459518	+	0.888168i	$0.348022\pi$
$632$	−0.922816	−0.0367076
$633$	0	0
$634$	−7.46170	−0.296342
$635$	5.71806	0.226914
$636$	0	0
$637$	−31.2091	−1.23655
$638$	34.2526	1.35607
$639$	0	0
$640$	−0.769147	−0.0304032
$641$	5.14884	0.203367	0.101683	−	0.994817i	$-0.467577\pi$
$641$	5.14884	0.203367	0.101683	+	0.994817i	$0.467577\pi$
$642$	0	0
$643$	25.7206	1.01432	0.507160	−	0.861852i	$-0.330695\pi$
$643$	25.7206	1.01432	0.507160	+	0.861852i	$0.330695\pi$
$644$	21.7107	0.855522
$645$	0	0
$646$	32.3525	1.27289
$647$	−5.81471	−0.228600	−0.114300	−	0.993446i	$-0.536463\pi$
$647$	−5.81471	−0.228600	−0.114300	+	0.993446i	$0.536463\pi$
$648$	0	0
$649$	38.7828	1.52236
$650$	22.5480	0.884406
$651$	0	0
$652$	−9.41443	−0.368697
$653$	−15.9057	−0.622436	−0.311218	−	0.950339i	$-0.600737\pi$
$653$	−15.9057	−0.622436	−0.311218	+	0.950339i	$0.600737\pi$
$654$	0	0
$655$	−5.31291	−0.207592
$656$	−26.5621	−1.03707
$657$	0	0
$658$	−5.71101	−0.222638
$659$	30.7546	1.19803	0.599014	−	0.800739i	$-0.295560\pi$
$659$	30.7546	1.19803	0.599014	+	0.800739i	$0.295560\pi$
$660$	0	0
$661$	−31.9625	−1.24320	−0.621598	−	0.783336i	$-0.713516\pi$
$661$	−31.9625	−1.24320	−0.621598	+	0.783336i	$0.713516\pi$
$662$	−25.8166	−1.00339
$663$	0	0
$664$	1.80697	0.0701241
$665$	11.1322	0.431687
$666$	0	0
$667$	−13.1591	−0.509522
$668$	−9.03963	−0.349754
$669$	0	0
$670$	−19.2053	−0.741964
$671$	15.3078	0.590949
$672$	0	0
$673$	5.89825	0.227361	0.113680	−	0.993517i	$-0.463736\pi$
$673$	5.89825	0.227361	0.113680	+	0.993517i	$0.463736\pi$
$674$	53.8362	2.07369
$675$	0	0
$676$	−12.7375	−0.489905
$677$	32.3269	1.24243	0.621213	−	0.783642i	$-0.286640\pi$
$677$	32.3269	1.24243	0.621213	+	0.783642i	$0.286640\pi$
$678$	0	0
$679$	28.9745	1.11194
$680$	−0.441598	−0.0169345
$681$	0	0
$682$	−24.8428	−0.951281
$683$	45.4717	1.73993	0.869963	−	0.493117i	$-0.164143\pi$
$683$	45.4717	1.73993	0.869963	+	0.493117i	$0.164143\pi$
$684$	0	0
$685$	3.75090	0.143314
$686$	46.4671	1.77412
$687$	0	0
$688$	9.88659	0.376923
$689$	−4.81941	−0.183605
$690$	0	0
$691$	−17.8493	−0.679021	−0.339510	−	0.940602i	$-0.610261\pi$
$691$	−17.8493	−0.679021	−0.339510	+	0.940602i	$0.610261\pi$
$692$	16.4249	0.624382
$693$	0	0
$694$	−72.9301	−2.76839
$695$	11.8937	0.451154
$696$	0	0
$697$	−29.5123	−1.11786
$698$	55.6104	2.10488
$699$	0	0
$700$	−38.1584	−1.44225
$701$	−22.0209	−0.831717	−0.415858	−	0.909429i	$-0.636519\pi$
$701$	−22.0209	−0.831717	−0.415858	+	0.909429i	$0.636519\pi$
$702$	0	0
$703$	37.1609	1.40155
$704$	24.9848	0.941651
$705$	0	0
$706$	29.6182	1.11469
$707$	35.9869	1.35343
$708$	0	0
$709$	47.4912	1.78357	0.891785	−	0.452459i	$-0.149453\pi$
$709$	47.4912	1.78357	0.891785	+	0.452459i	$0.149453\pi$
$710$	13.3407	0.500667
$711$	0	0
$712$	−2.08066	−0.0779758
$713$	9.54407	0.357428
$714$	0	0
$715$	−6.05022	−0.226265
$716$	−17.0030	−0.635432
$717$	0	0
$718$	38.7709	1.44692
$719$	−17.4092	−0.649253	−0.324626	−	0.945842i	$-0.605239\pi$
$719$	−17.4092	−0.649253	−0.324626	+	0.945842i	$0.605239\pi$
$720$	0	0
$721$	−4.33511	−0.161448
$722$	−12.6253	−0.469867
$723$	0	0
$724$	10.9147	0.405640
$725$	23.1282	0.858960
$726$	0	0
$727$	13.0030	0.482254	0.241127	−	0.970494i	$-0.422483\pi$
$727$	13.0030	0.482254	0.241127	+	0.970494i	$0.422483\pi$
$728$	1.50277	0.0556964
$729$	0	0
$730$	12.6306	0.467480
$731$	10.9847	0.406283
$732$	0	0
$733$	−6.25826	−0.231154	−0.115577	−	0.993299i	$-0.536872\pi$
$733$	−6.25826	−0.231154	−0.115577	+	0.993299i	$0.536872\pi$
$734$	19.3656	0.714799
$735$	0	0
$736$	−20.2503	−0.746435
$737$	−45.6191	−1.68040
$738$	0	0
$739$	44.0177	1.61922	0.809608	−	0.586971i	$-0.199680\pi$
$739$	44.0177	1.61922	0.809608	+	0.586971i	$0.199680\pi$
$740$	14.3890	0.528951
$741$	0	0
$742$	16.5994	0.609383
$743$	38.5408	1.41392	0.706962	−	0.707251i	$-0.250065\pi$
$743$	38.5408	1.41392	0.706962	+	0.707251i	$0.250065\pi$
$744$	0	0
$745$	−5.51288	−0.201976
$746$	−29.2307	−1.07021
$747$	0	0
$748$	29.7564	1.08800
$749$	59.4672	2.17288
$750$	0	0
$751$	−32.4423	−1.18383	−0.591917	−	0.805999i	$-0.701629\pi$
$751$	−32.4423	−1.18383	−0.591917	+	0.805999i	$0.701629\pi$
$752$	2.70651	0.0986964
$753$	0	0
$754$	25.8388	0.940993
$755$	14.3660	0.522831
$756$	0	0
$757$	12.2641	0.445745	0.222873	−	0.974848i	$-0.428457\pi$
$757$	12.2641	0.445745	0.222873	+	0.974848i	$0.428457\pi$
$758$	21.9778	0.798270
$759$	0	0
$760$	−0.341918	−0.0124027
$761$	−21.1292	−0.765932	−0.382966	−	0.923762i	$-0.625097\pi$
$761$	−21.1292	−0.765932	−0.382966	+	0.923762i	$0.625097\pi$
$762$	0	0
$763$	54.7009	1.98030
$764$	−3.96819	−0.143564
$765$	0	0
$766$	−61.8687	−2.23541
$767$	29.2562	1.05638
$768$	0	0
$769$	12.5981	0.454300	0.227150	−	0.973860i	$-0.427059\pi$
$769$	12.5981	0.454300	0.227150	+	0.973860i	$0.427059\pi$
$770$	20.8387	0.750973
$771$	0	0
$772$	−35.3932	−1.27383
$773$	−17.2688	−0.621116	−0.310558	−	0.950554i	$-0.600516\pi$
$773$	−17.2688	−0.621116	−0.310558	+	0.950554i	$0.600516\pi$
$774$	0	0
$775$	−16.7745	−0.602557
$776$	−0.889935	−0.0319468
$777$	0	0
$778$	−37.0890	−1.32970
$779$	−22.8507	−0.818710
$780$	0	0
$781$	31.6887	1.13391
$782$	−23.2665	−0.832007
$783$	0	0
$784$	−50.9422	−1.81937
$785$	−16.4116	−0.585756
$786$	0	0
$787$	−44.8720	−1.59951	−0.799756	−	0.600325i	$-0.795038\pi$
$787$	−44.8720	−1.59951	−0.799756	+	0.600325i	$0.795038\pi$
$788$	−42.9590	−1.53035
$789$	0	0
$790$	−9.65317	−0.343444
$791$	30.0519	1.06852
$792$	0	0
$793$	11.5475	0.410065
$794$	18.0558	0.640775
$795$	0	0
$796$	7.14615	0.253289
$797$	−30.5278	−1.08135	−0.540675	−	0.841232i	$-0.681831\pi$
$797$	−30.5278	−1.08135	−0.540675	+	0.841232i	$0.681831\pi$
$798$	0	0
$799$	3.00713	0.106384
$800$	35.5915	1.25835
$801$	0	0
$802$	52.1623	1.84191
$803$	30.0020	1.05875
$804$	0	0
$805$	−8.00575	−0.282166
$806$	−18.7404	−0.660103
$807$	0	0
$808$	−1.10532	−0.0388849
$809$	33.9180	1.19249	0.596247	−	0.802801i	$-0.296658\pi$
$809$	33.9180	1.19249	0.596247	+	0.802801i	$0.296658\pi$
$810$	0	0
$811$	−43.9741	−1.54414	−0.772069	−	0.635539i	$-0.780778\pi$
$811$	−43.9741	−1.54414	−0.772069	+	0.635539i	$0.780778\pi$
$812$	−43.7273	−1.53453
$813$	0	0
$814$	69.5626	2.43817
$815$	3.47154	0.121603
$816$	0	0
$817$	8.50518	0.297559
$818$	−59.7818	−2.09022
$819$	0	0
$820$	−8.84798	−0.308985
$821$	−50.2569	−1.75398	−0.876989	−	0.480510i	$-0.840452\pi$
$821$	−50.2569	−1.75398	−0.876989	+	0.480510i	$0.840452\pi$
$822$	0	0
$823$	45.2402	1.57697	0.788487	−	0.615051i	$-0.210865\pi$
$823$	45.2402	1.57697	0.788487	+	0.615051i	$0.210865\pi$
$824$	0.133150	0.00463851
$825$	0	0
$826$	−100.767	−3.50612
$827$	1.56540	0.0544343	0.0272172	−	0.999630i	$-0.491335\pi$
$827$	1.56540	0.0544343	0.0272172	+	0.999630i	$0.491335\pi$
$828$	0	0
$829$	2.28756	0.0794504	0.0397252	−	0.999211i	$-0.487352\pi$
$829$	2.28756	0.0794504	0.0397252	+	0.999211i	$0.487352\pi$
$830$	18.9020	0.656097
$831$	0	0
$832$	18.8475	0.653421
$833$	−56.6004	−1.96109
$834$	0	0
$835$	3.33333	0.115355
$836$	23.0397	0.796843
$837$	0	0
$838$	38.2297	1.32062
$839$	−19.8628	−0.685740	−0.342870	−	0.939383i	$-0.611399\pi$
$839$	−19.8628	−0.685740	−0.342870	+	0.939383i	$0.611399\pi$
$840$	0	0
$841$	−2.49638	−0.0860822
$842$	−43.6892	−1.50563
$843$	0	0
$844$	34.4378	1.18540
$845$	4.69692	0.161579
$846$	0	0
$847$	1.13650	0.0390505
$848$	−7.86665	−0.270142
$849$	0	0
$850$	40.8927	1.40261
$851$	−26.7244	−0.916101
$852$	0	0
$853$	−9.92703	−0.339895	−0.169948	−	0.985453i	$-0.554360\pi$
$853$	−9.92703	−0.339895	−0.169948	+	0.985453i	$0.554360\pi$
$854$	−39.7730	−1.36100
$855$	0	0
$856$	−1.82650	−0.0624285
$857$	−17.2232	−0.588334	−0.294167	−	0.955754i	$-0.595042\pi$
$857$	−17.2232	−0.588334	−0.294167	+	0.955754i	$0.595042\pi$
$858$	0	0
$859$	37.5632	1.28164	0.640820	−	0.767692i	$-0.278595\pi$
$859$	37.5632	1.28164	0.640820	+	0.767692i	$0.278595\pi$
$860$	3.29328	0.112300
$861$	0	0
$862$	28.7992	0.980904
$863$	−10.5951	−0.360662	−0.180331	−	0.983606i	$-0.557717\pi$
$863$	−10.5951	−0.360662	−0.180331	+	0.983606i	$0.557717\pi$
$864$	0	0
$865$	−6.05664	−0.205932
$866$	−27.7652	−0.943500
$867$	0	0
$868$	31.7147	1.07647
$869$	−22.9296	−0.777834
$870$	0	0
$871$	−34.4132	−1.16605
$872$	−1.68011	−0.0568956
$873$	0	0
$874$	−18.0147	−0.609355
$875$	29.7310	1.00509
$876$	0	0
$877$	15.5414	0.524797	0.262398	−	0.964960i	$-0.415487\pi$
$877$	15.5414	0.524797	0.262398	+	0.964960i	$0.415487\pi$
$878$	71.9984	2.42983
$879$	0	0
$880$	−9.87569	−0.332909
$881$	−17.9521	−0.604823	−0.302411	−	0.953177i	$-0.597792\pi$
$881$	−17.9521	−0.604823	−0.302411	+	0.953177i	$0.597792\pi$
$882$	0	0
$883$	−45.2199	−1.52177	−0.760885	−	0.648887i	$-0.775235\pi$
$883$	−45.2199	−1.52177	−0.760885	+	0.648887i	$0.775235\pi$
$884$	22.4470	0.754974
$885$	0	0
$886$	27.9197	0.937981
$887$	−16.0659	−0.539439	−0.269720	−	0.962939i	$-0.586931\pi$
$887$	−16.0659	−0.539439	−0.269720	+	0.962939i	$0.586931\pi$
$888$	0	0
$889$	−35.2900	−1.18359
$890$	−21.7648	−0.729559
$891$	0	0
$892$	−35.8185	−1.19929
$893$	2.32834	0.0779151
$894$	0	0
$895$	6.26980	0.209576
$896$	4.74693	0.158584
$897$	0	0
$898$	−1.43402	−0.0478537
$899$	−19.2226	−0.641110
$900$	0	0
$901$	−8.74040	−0.291185
$902$	−42.7749	−1.42425
$903$	0	0
$904$	−0.923028	−0.0306995
$905$	−4.02475	−0.133787
$906$	0	0
$907$	−8.12301	−0.269720	−0.134860	−	0.990865i	$-0.543059\pi$
$907$	−8.12301	−0.269720	−0.134860	+	0.990865i	$0.543059\pi$
$908$	3.39751	0.112750
$909$	0	0
$910$	15.7198	0.521107
$911$	−10.5301	−0.348876	−0.174438	−	0.984668i	$-0.555811\pi$
$911$	−10.5301	−0.348876	−0.174438	+	0.984668i	$0.555811\pi$
$912$	0	0
$913$	44.8987	1.48593
$914$	51.1662	1.69243
$915$	0	0
$916$	−14.9344	−0.493446
$917$	32.7896	1.08281
$918$	0	0
$919$	−31.0906	−1.02559	−0.512793	−	0.858513i	$-0.671389\pi$
$919$	−31.0906	−1.02559	−0.512793	+	0.858513i	$0.671389\pi$
$920$	0.245892	0.00810683
$921$	0	0
$922$	−28.7572	−0.947068
$923$	23.9047	0.786831
$924$	0	0
$925$	46.9704	1.54438
$926$	71.5641	2.35174
$927$	0	0
$928$	40.7859	1.33886
$929$	29.4503	0.966232	0.483116	−	0.875556i	$-0.339505\pi$
$929$	29.4503	0.966232	0.483116	+	0.875556i	$0.339505\pi$
$930$	0	0
$931$	−43.8243	−1.43628
$932$	−44.0510	−1.44294
$933$	0	0
$934$	−71.2685	−2.33198
$935$	−10.9726	−0.358842
$936$	0	0
$937$	31.6441	1.03377	0.516884	−	0.856055i	$-0.327092\pi$
$937$	31.6441	1.03377	0.516884	+	0.856055i	$0.327092\pi$
$938$	118.529	3.87010
$939$	0	0
$940$	0.901556	0.0294055
$941$	30.6375	0.998753	0.499377	−	0.866385i	$-0.333562\pi$
$941$	30.6375	0.998753	0.499377	+	0.866385i	$0.333562\pi$
$942$	0	0
$943$	16.4332	0.535137
$944$	47.7545	1.55428
$945$	0	0
$946$	15.9211	0.517640
$947$	−41.1947	−1.33865	−0.669323	−	0.742971i	$-0.733416\pi$
$947$	−41.1947	−1.33865	−0.669323	+	0.742971i	$0.733416\pi$
$948$	0	0
$949$	22.6323	0.734675
$950$	31.6623	1.02726
$951$	0	0
$952$	2.72540	0.0883307
$953$	−47.8644	−1.55048	−0.775240	−	0.631666i	$-0.782371\pi$
$953$	−47.8644	−1.55048	−0.775240	+	0.631666i	$0.782371\pi$
$954$	0	0
$955$	1.46326	0.0473499
$956$	27.2725	0.882056
$957$	0	0
$958$	63.1906	2.04159
$959$	−23.1493	−0.747531
$960$	0	0
$961$	−17.0582	−0.550264
$962$	52.4752	1.69187
$963$	0	0
$964$	−9.60088	−0.309224
$965$	13.0511	0.420131
$966$	0	0
$967$	15.1929	0.488570	0.244285	−	0.969703i	$-0.421447\pi$
$967$	15.1929	0.488570	0.244285	+	0.969703i	$0.421447\pi$
$968$	−0.0349068	−0.00112195
$969$	0	0
$970$	−9.30922	−0.298901
$971$	−29.2489	−0.938641	−0.469321	−	0.883028i	$-0.655501\pi$
$971$	−29.2489	−0.938641	−0.469321	+	0.883028i	$0.655501\pi$
$972$	0	0
$973$	−73.4042	−2.35323
$974$	25.1474	0.805774
$975$	0	0
$976$	18.8489	0.603338
$977$	−14.1107	−0.451441	−0.225720	−	0.974192i	$-0.572474\pi$
$977$	−14.1107	−0.451441	−0.225720	+	0.974192i	$0.572474\pi$
$978$	0	0
$979$	−51.6990	−1.65231
$980$	−16.9692	−0.542060
$981$	0	0
$982$	−66.9984	−2.13800
$983$	−6.41828	−0.204711	−0.102356	−	0.994748i	$-0.532638\pi$
$983$	−6.41828	−0.204711	−0.102356	+	0.994748i	$0.532638\pi$
$984$	0	0
$985$	15.8410	0.504736
$986$	46.8608	1.49235
$987$	0	0
$988$	17.3802	0.552937
$989$	−6.11654	−0.194495
$990$	0	0
$991$	34.3307	1.09055	0.545275	−	0.838257i	$-0.316425\pi$
$991$	34.3307	1.09055	0.545275	+	0.838257i	$0.316425\pi$
$992$	−29.5813	−0.939207
$993$	0	0
$994$	−82.3344	−2.61149
$995$	−2.63512	−0.0835389
$996$	0	0
$997$	−23.9065	−0.757127	−0.378564	−	0.925575i	$-0.623582\pi$
$997$	−23.9065	−0.757127	−0.378564	+	0.925575i	$0.623582\pi$
$998$	−30.1681	−0.954953
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4923.2.a.n.1.21		25
3.2	odd	2		547.2.a.c.1.5	✓	25
12.11	even	2		8752.2.a.v.1.6		25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.a.c.1.5	✓	25	3.2	odd	2
4923.2.a.n.1.21		25	1.1	even	1	trivial
8752.2.a.v.1.6		25	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)

\(f(q)\)	\(=\)	\(q+1.98290 q^{2} +1.93190 q^{4} -0.712381 q^{5} +4.39659 q^{7} -0.135039 q^{8} +O(q^{10})\)	\(q+1.98290 q^{2} +1.93190 q^{4} -0.712381 q^{5} +4.39659 q^{7} -0.135039 q^{8} -1.41258 q^{10} -3.35537 q^{11} -2.53115 q^{13} +8.71801 q^{14} -4.13157 q^{16} -4.59046 q^{17} -3.55428 q^{19} -1.37625 q^{20} -6.65336 q^{22} +2.55608 q^{23} -4.49251 q^{25} -5.01902 q^{26} +8.49377 q^{28} -5.14817 q^{29} +3.73387 q^{31} -7.92241 q^{32} -9.10242 q^{34} -3.13205 q^{35} -10.4553 q^{37} -7.04778 q^{38} +0.0961991 q^{40} +6.42906 q^{41} -2.39294 q^{43} -6.48223 q^{44} +5.06845 q^{46} -0.655082 q^{47} +12.3300 q^{49} -8.90821 q^{50} -4.88993 q^{52} +1.90404 q^{53} +2.39030 q^{55} -0.593710 q^{56} -10.2083 q^{58} -11.5584 q^{59} -4.56217 q^{61} +7.40391 q^{62} -7.44623 q^{64} +1.80315 q^{65} +13.5959 q^{67} -8.86829 q^{68} -6.21055 q^{70} -9.44418 q^{71} -8.94150 q^{73} -20.7317 q^{74} -6.86650 q^{76} -14.7522 q^{77} +6.83371 q^{79} +2.94325 q^{80} +12.7482 q^{82} -13.3811 q^{83} +3.27016 q^{85} -4.74497 q^{86} +0.453105 q^{88} +15.4078 q^{89} -11.1284 q^{91} +4.93808 q^{92} -1.29896 q^{94} +2.53200 q^{95} +6.59022 q^{97} +24.4492 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 4 q^{2} + 26 q^{4} - 29 q^{5} + 5 q^{7} - 6 q^{8}+O(q^{10}) \) 25 * q - 4 * q^2 + 26 * q^4 - 29 * q^5 + 5 * q^7 - 6 * q^8	\( 25 q - 4 q^{2} + 26 q^{4} - 29 q^{5} + 5 q^{7} - 6 q^{8} - q^{10} - 10 q^{11} + 19 q^{13} - 9 q^{14} + 16 q^{16} - 40 q^{17} - 33 q^{20} - 10 q^{22} - 26 q^{23} + 36 q^{25} + 8 q^{26} - 8 q^{28} - 30 q^{29} - 5 q^{31} - 6 q^{32} - 7 q^{34} - 11 q^{35} + 26 q^{37} - 25 q^{38} - 25 q^{40} - 9 q^{41} - 10 q^{43} - 34 q^{46} - 28 q^{47} + 20 q^{49} + 9 q^{50} - 2 q^{52} - 80 q^{53} - q^{55} - 7 q^{56} - 24 q^{58} + 2 q^{59} + 22 q^{61} - 36 q^{62} - 28 q^{64} - 30 q^{65} - 16 q^{67} - 59 q^{68} - 61 q^{70} + q^{71} + 2 q^{73} + 8 q^{74} - 46 q^{76} - 67 q^{77} - 34 q^{79} - 30 q^{80} - 4 q^{82} - 15 q^{83} + 15 q^{85} + 44 q^{86} - 55 q^{88} - 38 q^{89} - 41 q^{91} - 40 q^{92} - 46 q^{94} + 46 q^{95} - 2 q^{97} + 14 q^{98}+O(q^{100}) \) 25 * q - 4 * q^2 + 26 * q^4 - 29 * q^5 + 5 * q^7 - 6 * q^8 - q^10 - 10 * q^11 + 19 * q^13 - 9 * q^14 + 16 * q^16 - 40 * q^17 - 33 * q^20 - 10 * q^22 - 26 * q^23 + 36 * q^25 + 8 * q^26 - 8 * q^28 - 30 * q^29 - 5 * q^31 - 6 * q^32 - 7 * q^34 - 11 * q^35 + 26 * q^37 - 25 * q^38 - 25 * q^40 - 9 * q^41 - 10 * q^43 - 34 * q^46 - 28 * q^47 + 20 * q^49 + 9 * q^50 - 2 * q^52 - 80 * q^53 - q^55 - 7 * q^56 - 24 * q^58 + 2 * q^59 + 22 * q^61 - 36 * q^62 - 28 * q^64 - 30 * q^65 - 16 * q^67 - 59 * q^68 - 61 * q^70 + q^71 + 2 * q^73 + 8 * q^74 - 46 * q^76 - 67 * q^77 - 34 * q^79 - 30 * q^80 - 4 * q^82 - 15 * q^83 + 15 * q^85 + 44 * q^86 - 55 * q^88 - 38 * q^89 - 41 * q^91 - 40 * q^92 - 46 * q^94 + 46 * q^95 - 2 * q^97 + 14 * q^98

Introduction

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