LMFDB - Embedded newform 1881.2.a.k.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1881,2,Mod(1,1881)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1881, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1881.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1881 = 3^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1881.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:	$15.0198606202$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.246832.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 $ x^5 - 2x^4 - 5x^3 + 6x^2 + 7x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 209)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.71250$ of defining polynomial
Character	$\chi$	$=$	1881.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.779856 q^{2} -1.39182 q^{4} +3.49235 q^{5} +1.06736 q^{7} -2.64513 q^{8} +O(q^{10})$	\(q+0.779856 q^{2} -1.39182 q^{4} +3.49235 q^{5} +1.06736 q^{7} -2.64513 q^{8} +2.72353 q^{10} -1.00000 q^{11} -0.0563258 q^{13} +0.832387 q^{14} +0.720827 q^{16} +4.53628 q^{17} -1.00000 q^{19} -4.86074 q^{20} -0.779856 q^{22} +1.07949 q^{23} +7.19651 q^{25} -0.0439260 q^{26} -1.48558 q^{28} -0.299905 q^{29} +9.18548 q^{31} +5.85241 q^{32} +3.53764 q^{34} +3.72760 q^{35} +4.50448 q^{37} -0.779856 q^{38} -9.23774 q^{40} -12.0009 q^{41} +10.7260 q^{43} +1.39182 q^{44} +0.841844 q^{46} -2.89630 q^{47} -5.86074 q^{49} +5.61224 q^{50} +0.0783957 q^{52} +12.3213 q^{53} -3.49235 q^{55} -2.82331 q^{56} -0.233882 q^{58} +1.14582 q^{59} -8.09599 q^{61} +7.16335 q^{62} +3.12238 q^{64} -0.196709 q^{65} +11.2733 q^{67} -6.31370 q^{68} +2.90699 q^{70} -13.4948 q^{71} -11.1470 q^{73} +3.51284 q^{74} +1.39182 q^{76} -1.06736 q^{77} +11.4250 q^{79} +2.51738 q^{80} -9.35899 q^{82} +13.9802 q^{83} +15.8423 q^{85} +8.36475 q^{86} +2.64513 q^{88} +0.183185 q^{89} -0.0601200 q^{91} -1.50246 q^{92} -2.25870 q^{94} -3.49235 q^{95} +5.66263 q^{97} -4.57053 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{2} + 6 q^{4} + 5 q^{5} + 6 q^{7} - 6 q^{8}+O(q^{10}) $ 5 * q - 2 * q^2 + 6 * q^4 + 5 * q^5 + 6 * q^7 - 6 * q^8	$ 5 q - 2 q^{2} + 6 q^{4} + 5 q^{5} + 6 q^{7} - 6 q^{8} + 12 q^{10} - 5 q^{11} + 4 q^{13} + 14 q^{14} + 8 q^{16} + 4 q^{17} - 5 q^{19} + 8 q^{20} + 2 q^{22} - 3 q^{23} + 6 q^{25} + 6 q^{26} - 10 q^{28} - 10 q^{29} + 11 q^{31} - 14 q^{32} - 4 q^{34} + 8 q^{35} + q^{37} + 2 q^{38} - 16 q^{40} - 2 q^{41} + 20 q^{43} - 6 q^{44} - 4 q^{46} + 20 q^{47} + 3 q^{49} + 32 q^{50} + 6 q^{52} + 14 q^{53} - 5 q^{55} + 38 q^{56} - 6 q^{58} - 3 q^{59} - 10 q^{61} + 6 q^{62} + 9 q^{67} - 24 q^{68} + 50 q^{70} - 23 q^{71} - 8 q^{74} - 6 q^{76} - 6 q^{77} + 44 q^{79} + 18 q^{80} - 30 q^{82} + 14 q^{83} - 12 q^{85} - 52 q^{86} + 6 q^{88} + 27 q^{89} + 24 q^{91} - 58 q^{92} - 8 q^{94} - 5 q^{95} + 15 q^{97} + 10 q^{98}+O(q^{100}) $ 5 * q - 2 * q^2 + 6 * q^4 + 5 * q^5 + 6 * q^7 - 6 * q^8 + 12 * q^10 - 5 * q^11 + 4 * q^13 + 14 * q^14 + 8 * q^16 + 4 * q^17 - 5 * q^19 + 8 * q^20 + 2 * q^22 - 3 * q^23 + 6 * q^25 + 6 * q^26 - 10 * q^28 - 10 * q^29 + 11 * q^31 - 14 * q^32 - 4 * q^34 + 8 * q^35 + q^37 + 2 * q^38 - 16 * q^40 - 2 * q^41 + 20 * q^43 - 6 * q^44 - 4 * q^46 + 20 * q^47 + 3 * q^49 + 32 * q^50 + 6 * q^52 + 14 * q^53 - 5 * q^55 + 38 * q^56 - 6 * q^58 - 3 * q^59 - 10 * q^61 + 6 * q^62 + 9 * q^67 - 24 * q^68 + 50 * q^70 - 23 * q^71 - 8 * q^74 - 6 * q^76 - 6 * q^77 + 44 * q^79 + 18 * q^80 - 30 * q^82 + 14 * q^83 - 12 * q^85 - 52 * q^86 + 6 * q^88 + 27 * q^89 + 24 * q^91 - 58 * q^92 - 8 * q^94 - 5 * q^95 + 15 * q^97 + 10 * 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 $

$2$	0.779856	0.551441	0.275721	−	0.961238i	$-0.411084\pi$
$2$	0.779856	0.551441	0.275721	+	0.961238i	$0.411084\pi$
$3$	0	0
$3$	0	0
$4$	−1.39182	−0.695912
$5$	3.49235	1.56183	0.780913	−	0.624639i	$-0.214754\pi$
$5$	3.49235	1.56183	0.780913	+	0.624639i	$0.214754\pi$
$6$	0	0
$7$	1.06736	0.403424	0.201712	−	0.979445i	$-0.435349\pi$
$7$	1.06736	0.403424	0.201712	+	0.979445i	$0.435349\pi$
$8$	−2.64513	−0.935196
$9$	0	0
$10$	2.72353	0.861256
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−0.0563258	−0.0156220	−0.00781098	−	0.999969i	$-0.502486\pi$
$13$	−0.0563258	−0.0156220	−0.00781098	+	0.999969i	$0.502486\pi$
$14$	0.832387	0.222465
$15$	0	0
$16$	0.720827	0.180207
$17$	4.53628	1.10021	0.550104	−	0.835096i	$-0.314588\pi$
$17$	4.53628	1.10021	0.550104	+	0.835096i	$0.314588\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−4.86074	−1.08689
$21$	0	0
$22$	−0.779856	−0.166266
$23$	1.07949	0.225088	0.112544	−	0.993647i	$-0.464100\pi$
$23$	1.07949	0.225088	0.112544	+	0.993647i	$0.464100\pi$
$24$	0	0
$25$	7.19651	1.43930
$26$	−0.0439260	−0.00861460
$27$	0	0
$28$	−1.48558	−0.280748
$29$	−0.299905	−0.0556909	−0.0278455	−	0.999612i	$-0.508865\pi$
$29$	−0.299905	−0.0556909	−0.0278455	+	0.999612i	$0.508865\pi$
$30$	0	0
$31$	9.18548	1.64976	0.824880	−	0.565307i	$-0.191242\pi$
$31$	9.18548	1.64976	0.824880	+	0.565307i	$0.191242\pi$
$32$	5.85241	1.03457
$33$	0	0
$34$	3.53764	0.606701
$35$	3.72760	0.630079
$36$	0	0
$37$	4.50448	0.740531	0.370266	−	0.928926i	$-0.379267\pi$
$37$	4.50448	0.740531	0.370266	+	0.928926i	$0.379267\pi$
$38$	−0.779856	−0.126509
$39$	0	0
$40$	−9.23774	−1.46061
$41$	−12.0009	−1.87423	−0.937115	−	0.349020i	$-0.886514\pi$
$41$	−12.0009	−1.87423	−0.937115	+	0.349020i	$0.886514\pi$
$42$	0	0
$43$	10.7260	1.63570	0.817851	−	0.575430i	$-0.195165\pi$
$43$	10.7260	1.63570	0.817851	+	0.575430i	$0.195165\pi$
$44$	1.39182	0.209826
$45$	0	0
$46$	0.841844	0.124123
$47$	−2.89630	−0.422469	−0.211235	−	0.977435i	$-0.567748\pi$
$47$	−2.89630	−0.422469	−0.211235	+	0.977435i	$0.567748\pi$
$48$	0	0
$49$	−5.86074	−0.837249
$50$	5.61224	0.793691
$51$	0	0
$52$	0.0783957	0.0108715
$53$	12.3213	1.69246	0.846230	−	0.532818i	$-0.178867\pi$
$53$	12.3213	1.69246	0.846230	+	0.532818i	$0.178867\pi$
$54$	0	0
$55$	−3.49235	−0.470908
$56$	−2.82331	−0.377281
$57$	0	0
$58$	−0.233882	−0.0307103
$59$	1.14582	0.149173	0.0745863	−	0.997215i	$-0.476236\pi$
$59$	1.14582	0.149173	0.0745863	+	0.997215i	$0.476236\pi$
$60$	0	0
$61$	−8.09599	−1.03659	−0.518293	−	0.855203i	$-0.673432\pi$
$61$	−8.09599	−1.03659	−0.518293	+	0.855203i	$0.673432\pi$
$62$	7.16335	0.909746
$63$	0	0
$64$	3.12238	0.390298
$65$	−0.196709	−0.0243988
$66$	0	0
$67$	11.2733	1.37726	0.688628	−	0.725115i	$-0.258213\pi$
$67$	11.2733	1.37726	0.688628	+	0.725115i	$0.258213\pi$
$68$	−6.31370	−0.765649
$69$	0	0
$70$	2.90699	0.347452
$71$	−13.4948	−1.60154	−0.800771	−	0.598970i	$-0.795577\pi$
$71$	−13.4948	−1.60154	−0.800771	+	0.598970i	$0.795577\pi$
$72$	0	0
$73$	−11.1470	−1.30466	−0.652330	−	0.757935i	$-0.726208\pi$
$73$	−11.1470	−1.30466	−0.652330	+	0.757935i	$0.726208\pi$
$74$	3.51284	0.408360
$75$	0	0
$76$	1.39182	0.159653
$77$	−1.06736	−0.121637
$78$	0	0
$79$	11.4250	1.28541	0.642706	−	0.766113i	$-0.277812\pi$
$79$	11.4250	1.28541	0.642706	+	0.766113i	$0.277812\pi$
$80$	2.51738	0.281452
$81$	0	0
$82$	−9.35899	−1.03353
$83$	13.9802	1.53452	0.767261	−	0.641335i	$-0.221619\pi$
$83$	13.9802	1.53452	0.767261	+	0.641335i	$0.221619\pi$
$84$	0	0
$85$	15.8423	1.71834
$86$	8.36475	0.901994
$87$	0	0
$88$	2.64513	0.281972
$89$	0.183185	0.0194176	0.00970878	−	0.999953i	$-0.496910\pi$
$89$	0.183185	0.0194176	0.00970878	+	0.999953i	$0.496910\pi$
$90$	0	0
$91$	−0.0601200	−0.00630228
$92$	−1.50246	−0.156642
$93$	0	0
$94$	−2.25870	−0.232967
$95$	−3.49235	−0.358308
$96$	0	0
$97$	5.66263	0.574953	0.287477	−	0.957788i	$-0.407184\pi$
$97$	5.66263	0.574953	0.287477	+	0.957788i	$0.407184\pi$
$98$	−4.57053	−0.461694
$99$	0	0
$100$	−10.0163	−1.00163
$101$	−8.00759	−0.796785	−0.398392	−	0.917215i	$-0.630432\pi$
$101$	−8.00759	−0.796785	−0.398392	+	0.917215i	$0.630432\pi$
$102$	0	0
$103$	6.18725	0.609648	0.304824	−	0.952409i	$-0.401402\pi$
$103$	6.18725	0.609648	0.304824	+	0.952409i	$0.401402\pi$
$104$	0.148989	0.0146096
$105$	0	0
$106$	9.60883	0.933292
$107$	7.29027	0.704777	0.352388	−	0.935854i	$-0.385370\pi$
$107$	7.29027	0.704777	0.352388	+	0.935854i	$0.385370\pi$
$108$	0	0
$109$	7.79895	0.747004	0.373502	−	0.927629i	$-0.378157\pi$
$109$	7.79895	0.747004	0.373502	+	0.927629i	$0.378157\pi$
$110$	−2.72353	−0.259678
$111$	0	0
$112$	0.769382	0.0726998
$113$	0.430558	0.0405035	0.0202517	−	0.999795i	$-0.493553\pi$
$113$	0.430558	0.0405035	0.0202517	+	0.999795i	$0.493553\pi$
$114$	0	0
$115$	3.76995	0.351549
$116$	0.417415	0.0387560
$117$	0	0
$118$	0.893572	0.0822600
$119$	4.84184	0.443851
$120$	0	0
$121$	1.00000	0.0909091
$122$	−6.31370	−0.571616
$123$	0	0
$124$	−12.7846	−1.14809
$125$	7.67100	0.686115
$126$	0	0
$127$	−3.13888	−0.278531	−0.139265	−	0.990255i	$-0.544474\pi$
$127$	−3.13888	−0.278531	−0.139265	+	0.990255i	$0.544474\pi$
$128$	−9.26981	−0.819343
$129$	0	0
$130$	−0.153405	−0.0134545
$131$	−12.4315	−1.08615	−0.543075	−	0.839684i	$-0.682740\pi$
$131$	−12.4315	−1.08615	−0.543075	+	0.839684i	$0.682740\pi$
$132$	0	0
$133$	−1.06736	−0.0925519
$134$	8.79157	0.759476
$135$	0	0
$136$	−11.9991	−1.02891
$137$	−13.8301	−1.18159	−0.590794	−	0.806822i	$-0.701186\pi$
$137$	−13.8301	−1.18159	−0.590794	+	0.806822i	$0.701186\pi$
$138$	0	0
$139$	15.9968	1.35683	0.678417	−	0.734677i	$-0.262666\pi$
$139$	15.9968	1.35683	0.678417	+	0.734677i	$0.262666\pi$
$140$	−5.18816	−0.438480
$141$	0	0
$142$	−10.5240	−0.883157
$143$	0.0563258	0.00471020
$144$	0	0
$145$	−1.04737	−0.0869796
$146$	−8.69307	−0.719443
$147$	0	0
$148$	−6.26944	−0.515345
$149$	−11.3620	−0.930812	−0.465406	−	0.885097i	$-0.654092\pi$
$149$	−11.3620	−0.930812	−0.465406	+	0.885097i	$0.654092\pi$
$150$	0	0
$151$	4.66341	0.379503	0.189751	−	0.981832i	$-0.439232\pi$
$151$	4.66341	0.379503	0.189751	+	0.981832i	$0.439232\pi$
$152$	2.64513	0.214549
$153$	0	0
$154$	−0.832387	−0.0670757
$155$	32.0789	2.57664
$156$	0	0
$157$	−9.05206	−0.722433	−0.361217	−	0.932482i	$-0.617639\pi$
$157$	−9.05206	−0.722433	−0.361217	+	0.932482i	$0.617639\pi$
$158$	8.90984	0.708829
$159$	0	0
$160$	20.4387	1.61582
$161$	1.15220	0.0908062
$162$	0	0
$163$	2.36761	0.185446	0.0927229	−	0.995692i	$-0.470443\pi$
$163$	2.36761	0.185446	0.0927229	+	0.995692i	$0.470443\pi$
$164$	16.7032	1.30430
$165$	0	0
$166$	10.9025	0.846199
$167$	9.27361	0.717613	0.358807	−	0.933412i	$-0.383184\pi$
$167$	9.27361	0.717613	0.358807	+	0.933412i	$0.383184\pi$
$168$	0	0
$169$	−12.9968	−0.999756
$170$	12.3547	0.947561
$171$	0	0
$172$	−14.9287	−1.13831
$173$	−10.7172	−0.814812	−0.407406	−	0.913247i	$-0.633567\pi$
$173$	−10.7172	−0.814812	−0.407406	+	0.913247i	$0.633567\pi$
$174$	0	0
$175$	7.68128	0.580650
$176$	−0.720827	−0.0543344
$177$	0	0
$178$	0.142858	0.0107076
$179$	22.9070	1.71215	0.856073	−	0.516854i	$-0.172897\pi$
$179$	22.9070	1.71215	0.856073	+	0.516854i	$0.172897\pi$
$180$	0	0
$181$	5.90522	0.438931	0.219466	−	0.975620i	$-0.429569\pi$
$181$	5.90522	0.438931	0.219466	+	0.975620i	$0.429569\pi$
$182$	−0.0468849	−0.00347534
$183$	0	0
$184$	−2.85539	−0.210502
$185$	15.7312	1.15658
$186$	0	0
$187$	−4.53628	−0.331725
$188$	4.03114	0.294001
$189$	0	0
$190$	−2.72353	−0.197586
$191$	−6.44628	−0.466437	−0.233218	−	0.972424i	$-0.574926\pi$
$191$	−6.44628	−0.466437	−0.233218	+	0.972424i	$0.574926\pi$
$192$	0	0
$193$	1.43606	0.103370	0.0516849	−	0.998663i	$-0.483541\pi$
$193$	1.43606	0.103370	0.0516849	+	0.998663i	$0.483541\pi$
$194$	4.41604	0.317053
$195$	0	0
$196$	8.15713	0.582652
$197$	4.75399	0.338708	0.169354	−	0.985555i	$-0.445832\pi$
$197$	4.75399	0.338708	0.169354	+	0.985555i	$0.445832\pi$
$198$	0	0
$199$	−2.36002	−0.167298	−0.0836489	−	0.996495i	$-0.526657\pi$
$199$	−2.36002	−0.167298	−0.0836489	+	0.996495i	$0.526657\pi$
$200$	−19.0357	−1.34603
$201$	0	0
$202$	−6.24476	−0.439380
$203$	−0.320107	−0.0224671
$204$	0	0
$205$	−41.9115	−2.92722
$206$	4.82517	0.336185
$207$	0	0
$208$	−0.0406011	−0.00281518
$209$	1.00000	0.0691714
$210$	0	0
$211$	−24.5133	−1.68756	−0.843781	−	0.536687i	$-0.819676\pi$
$211$	−24.5133	−1.68756	−0.843781	+	0.536687i	$0.819676\pi$
$212$	−17.1491	−1.17780
$213$	0	0
$214$	5.68536	0.388643
$215$	37.4590	2.55468
$216$	0	0
$217$	9.80422	0.665554
$218$	6.08205	0.411929
$219$	0	0
$220$	4.86074	0.327711
$221$	−0.255509	−0.0171874
$222$	0	0
$223$	8.47427	0.567479	0.283740	−	0.958901i	$-0.408425\pi$
$223$	8.47427	0.567479	0.283740	+	0.958901i	$0.408425\pi$
$224$	6.24663	0.417371
$225$	0	0
$226$	0.335773	0.0223353
$227$	−13.7491	−0.912558	−0.456279	−	0.889837i	$-0.650818\pi$
$227$	−13.7491	−0.912558	−0.456279	+	0.889837i	$0.650818\pi$
$228$	0	0
$229$	−1.99640	−0.131926	−0.0659629	−	0.997822i	$-0.521012\pi$
$229$	−1.99640	−0.131926	−0.0659629	+	0.997822i	$0.521012\pi$
$230$	2.94001	0.193859
$231$	0	0
$232$	0.793288	0.0520819
$233$	−11.3481	−0.743437	−0.371718	−	0.928346i	$-0.621231\pi$
$233$	−11.3481	−0.743437	−0.371718	+	0.928346i	$0.621231\pi$
$234$	0	0
$235$	−10.1149	−0.659823
$236$	−1.59478	−0.103811
$237$	0	0
$238$	3.77594	0.244758
$239$	−5.21045	−0.337036	−0.168518	−	0.985699i	$-0.553898\pi$
$239$	−5.21045	−0.337036	−0.168518	+	0.985699i	$0.553898\pi$
$240$	0	0
$241$	6.60827	0.425676	0.212838	−	0.977087i	$-0.431729\pi$
$241$	6.60827	0.425676	0.212838	+	0.977087i	$0.431729\pi$
$242$	0.779856	0.0501310
$243$	0	0
$244$	11.2682	0.721373
$245$	−20.4678	−1.30764
$246$	0	0
$247$	0.0563258	0.00358393
$248$	−24.2968	−1.54285
$249$	0	0
$250$	5.98227	0.378352
$251$	−24.0024	−1.51502	−0.757510	−	0.652823i	$-0.773585\pi$
$251$	−24.0024	−1.51502	−0.757510	+	0.652823i	$0.773585\pi$
$252$	0	0
$253$	−1.07949	−0.0678667
$254$	−2.44788	−0.153593
$255$	0	0
$256$	−13.4739	−0.842117
$257$	5.68903	0.354872	0.177436	−	0.984132i	$-0.443220\pi$
$257$	5.68903	0.354872	0.177436	+	0.984132i	$0.443220\pi$
$258$	0	0
$259$	4.80790	0.298748
$260$	0.273785	0.0169794
$261$	0	0
$262$	−9.69481	−0.598948
$263$	−13.9857	−0.862395	−0.431197	−	0.902258i	$-0.641909\pi$
$263$	−13.9857	−0.862395	−0.431197	+	0.902258i	$0.641909\pi$
$264$	0	0
$265$	43.0303	2.64333
$266$	−0.832387	−0.0510369
$267$	0	0
$268$	−15.6905	−0.958450
$269$	−15.4020	−0.939075	−0.469538	−	0.882912i	$-0.655579\pi$
$269$	−15.4020	−0.939075	−0.469538	+	0.882912i	$0.655579\pi$
$270$	0	0
$271$	−25.3911	−1.54240	−0.771199	−	0.636595i	$-0.780342\pi$
$271$	−25.3911	−1.54240	−0.771199	+	0.636595i	$0.780342\pi$
$272$	3.26987	0.198265
$273$	0	0
$274$	−10.7855	−0.651577
$275$	−7.19651	−0.433966
$276$	0	0
$277$	19.9798	1.20047	0.600235	−	0.799824i	$-0.295074\pi$
$277$	19.9798	1.20047	0.600235	+	0.799824i	$0.295074\pi$
$278$	12.4752	0.748214
$279$	0	0
$280$	−9.86000	−0.589248
$281$	−6.18130	−0.368745	−0.184373	−	0.982856i	$-0.559025\pi$
$281$	−6.18130	−0.368745	−0.184373	+	0.982856i	$0.559025\pi$
$282$	0	0
$283$	−7.12127	−0.423316	−0.211658	−	0.977344i	$-0.567886\pi$
$283$	−7.12127	−0.423316	−0.211658	+	0.977344i	$0.567886\pi$
$284$	18.7825	1.11453
$285$	0	0
$286$	0.0439260	0.00259740
$287$	−12.8093	−0.756110
$288$	0	0
$289$	3.57781	0.210459
$290$	−0.816800	−0.0479641
$291$	0	0
$292$	15.5147	0.907929
$293$	−19.1342	−1.11783	−0.558916	−	0.829224i	$-0.688783\pi$
$293$	−19.1342	−1.11783	−0.558916	+	0.829224i	$0.688783\pi$
$294$	0	0
$295$	4.00159	0.232982
$296$	−11.9149	−0.692542
$297$	0	0
$298$	−8.86073	−0.513288
$299$	−0.0608030	−0.00351633
$300$	0	0
$301$	11.4485	0.659882
$302$	3.63679	0.209274
$303$	0	0
$304$	−0.720827	−0.0413422
$305$	−28.2740	−1.61897
$306$	0	0
$307$	6.82573	0.389565	0.194782	−	0.980846i	$-0.437600\pi$
$307$	6.82573	0.389565	0.194782	+	0.980846i	$0.437600\pi$
$308$	1.48558	0.0846487
$309$	0	0
$310$	25.0169	1.42087
$311$	12.8609	0.729276	0.364638	−	0.931149i	$-0.381193\pi$
$311$	12.8609	0.729276	0.364638	+	0.931149i	$0.381193\pi$
$312$	0	0
$313$	−3.01707	−0.170535	−0.0852673	−	0.996358i	$-0.527174\pi$
$313$	−3.01707	−0.170535	−0.0852673	+	0.996358i	$0.527174\pi$
$314$	−7.05930	−0.398380
$315$	0	0
$316$	−15.9016	−0.894534
$317$	−17.7857	−0.998943	−0.499471	−	0.866330i	$-0.666472\pi$
$317$	−17.7857	−0.998943	−0.499471	+	0.866330i	$0.666472\pi$
$318$	0	0
$319$	0.299905	0.0167914
$320$	10.9045	0.609577
$321$	0	0
$322$	0.898551	0.0500743
$323$	−4.53628	−0.252405
$324$	0	0
$325$	−0.405349	−0.0224847
$326$	1.84640	0.102262
$327$	0	0
$328$	31.7441	1.75277
$329$	−3.09140	−0.170434
$330$	0	0
$331$	−26.3860	−1.45030	−0.725152	−	0.688589i	$-0.758230\pi$
$331$	−26.3860	−1.45030	−0.725152	+	0.688589i	$0.758230\pi$
$332$	−19.4579	−1.06789
$333$	0	0
$334$	7.23207	0.395722
$335$	39.3704	2.15104
$336$	0	0
$337$	−13.2024	−0.719181	−0.359590	−	0.933110i	$-0.617084\pi$
$337$	−13.2024	−0.719181	−0.359590	+	0.933110i	$0.617084\pi$
$338$	−10.1357	−0.551307
$339$	0	0
$340$	−22.0497	−1.19581
$341$	−9.18548	−0.497422
$342$	0	0
$343$	−13.7271	−0.741191
$344$	−28.3718	−1.52970
$345$	0	0
$346$	−8.35786	−0.449321
$347$	−16.4809	−0.884740	−0.442370	−	0.896833i	$-0.645862\pi$
$347$	−16.4809	−0.884740	−0.442370	+	0.896833i	$0.645862\pi$
$348$	0	0
$349$	−19.0165	−1.01793	−0.508966	−	0.860787i	$-0.669972\pi$
$349$	−19.0165	−1.01793	−0.508966	+	0.860787i	$0.669972\pi$
$350$	5.99029	0.320194
$351$	0	0
$352$	−5.85241	−0.311934
$353$	−13.1955	−0.702325	−0.351162	−	0.936315i	$-0.614213\pi$
$353$	−13.1955	−0.702325	−0.351162	+	0.936315i	$0.614213\pi$
$354$	0	0
$355$	−47.1287	−2.50133
$356$	−0.254961	−0.0135129
$357$	0	0
$358$	17.8641	0.944148
$359$	2.88756	0.152400	0.0761998	−	0.997093i	$-0.475721\pi$
$359$	2.88756	0.152400	0.0761998	+	0.997093i	$0.475721\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	4.60522	0.242045
$363$	0	0
$364$	0.0836765	0.00438584
$365$	−38.9293	−2.03765
$366$	0	0
$367$	−2.10833	−0.110054	−0.0550269	−	0.998485i	$-0.517524\pi$
$367$	−2.10833	−0.110054	−0.0550269	+	0.998485i	$0.517524\pi$
$368$	0.778123	0.0405624
$369$	0	0
$370$	12.2681	0.637787
$371$	13.1513	0.682780
$372$	0	0
$373$	8.07305	0.418007	0.209003	−	0.977915i	$-0.432978\pi$
$373$	8.07305	0.418007	0.209003	+	0.977915i	$0.432978\pi$
$374$	−3.53764	−0.182927
$375$	0	0
$376$	7.66111	0.395091
$377$	0.0168924	0.000870002 0
$378$	0	0
$379$	14.3355	0.736365	0.368183	−	0.929754i	$-0.379980\pi$
$379$	14.3355	0.736365	0.368183	+	0.929754i	$0.379980\pi$
$380$	4.86074	0.249351
$381$	0	0
$382$	−5.02717	−0.257212
$383$	9.79867	0.500689	0.250344	−	0.968157i	$-0.419456\pi$
$383$	9.79867	0.500689	0.250344	+	0.968157i	$0.419456\pi$
$384$	0	0
$385$	−3.72760	−0.189976
$386$	1.11992	0.0570024
$387$	0	0
$388$	−7.88139	−0.400117
$389$	19.9236	1.01016	0.505082	−	0.863071i	$-0.331462\pi$
$389$	19.9236	1.01016	0.505082	+	0.863071i	$0.331462\pi$
$390$	0	0
$391$	4.89685	0.247644
$392$	15.5024	0.782992
$393$	0	0
$394$	3.70743	0.186778
$395$	39.9001	2.00759
$396$	0	0
$397$	13.9941	0.702342	0.351171	−	0.936311i	$-0.385784\pi$
$397$	13.9941	0.702342	0.351171	+	0.936311i	$0.385784\pi$
$398$	−1.84048	−0.0922549
$399$	0	0
$400$	5.18744	0.259372
$401$	18.6293	0.930301	0.465151	−	0.885232i	$-0.346000\pi$
$401$	18.6293	0.930301	0.465151	+	0.885232i	$0.346000\pi$
$402$	0	0
$403$	−0.517380	−0.0257725
$404$	11.1452	0.554492
$405$	0	0
$406$	−0.249637	−0.0123893
$407$	−4.50448	−0.223279
$408$	0	0
$409$	−30.8428	−1.52508	−0.762540	−	0.646941i	$-0.776048\pi$
$409$	−30.8428	−1.52508	−0.762540	+	0.646941i	$0.776048\pi$
$410$	−32.6849	−1.61419
$411$	0	0
$412$	−8.61157	−0.424262
$413$	1.22300	0.0601799
$414$	0	0
$415$	48.8236	2.39666
$416$	−0.329642	−0.0161620
$417$	0	0
$418$	0.779856	0.0381440
$419$	−37.9347	−1.85323	−0.926616	−	0.376009i	$-0.877296\pi$
$419$	−37.9347	−1.85323	−0.926616	+	0.376009i	$0.877296\pi$
$420$	0	0
$421$	−18.4642	−0.899891	−0.449945	−	0.893056i	$-0.648557\pi$
$421$	−18.4642	−0.899891	−0.449945	+	0.893056i	$0.648557\pi$
$422$	−19.1168	−0.930592
$423$	0	0
$424$	−32.5915	−1.58278
$425$	32.6454	1.58353
$426$	0	0
$427$	−8.64134	−0.418184
$428$	−10.1468	−0.490463
$429$	0	0
$430$	29.2126	1.40876
$431$	−33.6742	−1.62203	−0.811013	−	0.585027i	$-0.801084\pi$
$431$	−33.6742	−1.62203	−0.811013	+	0.585027i	$0.801084\pi$
$432$	0	0
$433$	11.4041	0.548047	0.274024	−	0.961723i	$-0.411645\pi$
$433$	11.4041	0.548047	0.274024	+	0.961723i	$0.411645\pi$
$434$	7.64588	0.367014
$435$	0	0
$436$	−10.8548	−0.519849
$437$	−1.07949	−0.0516388
$438$	0	0
$439$	−3.76890	−0.179880	−0.0899399	−	0.995947i	$-0.528667\pi$
$439$	−3.76890	−0.179880	−0.0899399	+	0.995947i	$0.528667\pi$
$440$	9.23774	0.440392
$441$	0	0
$442$	−0.199261	−0.00947786
$443$	−4.51841	−0.214676	−0.107338	−	0.994223i	$-0.534233\pi$
$443$	−4.51841	−0.214676	−0.107338	+	0.994223i	$0.534233\pi$
$444$	0	0
$445$	0.639746	0.0303269
$446$	6.60871	0.312931
$447$	0	0
$448$	3.33271	0.157456
$449$	37.3611	1.76318	0.881590	−	0.472016i	$-0.156474\pi$
$449$	37.3611	1.76318	0.881590	+	0.472016i	$0.156474\pi$
$450$	0	0
$451$	12.0009	0.565102
$452$	−0.599261	−0.0281869
$453$	0	0
$454$	−10.7223	−0.503222
$455$	−0.209960	−0.00984308
$456$	0	0
$457$	9.37773	0.438672	0.219336	−	0.975649i	$-0.429611\pi$
$457$	9.37773	0.438672	0.219336	+	0.975649i	$0.429611\pi$
$458$	−1.55690	−0.0727494
$459$	0	0
$460$	−5.24710	−0.244648
$461$	−31.6785	−1.47542	−0.737708	−	0.675120i	$-0.764092\pi$
$461$	−31.6785	−1.47542	−0.737708	+	0.675120i	$0.764092\pi$
$462$	0	0
$463$	−37.1284	−1.72550	−0.862752	−	0.505628i	$-0.831261\pi$
$463$	−37.1284	−1.72550	−0.862752	+	0.505628i	$0.831261\pi$
$464$	−0.216179	−0.0100359
$465$	0	0
$466$	−8.84986	−0.409962
$467$	15.9678	0.738902	0.369451	−	0.929250i	$-0.379546\pi$
$467$	15.9678	0.738902	0.369451	+	0.929250i	$0.379546\pi$
$468$	0	0
$469$	12.0327	0.555619
$470$	−7.88816	−0.363854
$471$	0	0
$472$	−3.03084	−0.139506
$473$	−10.7260	−0.493183
$474$	0	0
$475$	−7.19651	−0.330199
$476$	−6.73900	−0.308882
$477$	0	0
$478$	−4.06340	−0.185855
$479$	−28.1729	−1.28725	−0.643627	−	0.765340i	$-0.722571\pi$
$479$	−28.1729	−1.28725	−0.643627	+	0.765340i	$0.722571\pi$
$480$	0	0
$481$	−0.253718	−0.0115686
$482$	5.15350	0.234735
$483$	0	0
$484$	−1.39182	−0.0632648
$485$	19.7759	0.897977
$486$	0	0
$487$	24.0613	1.09032	0.545161	−	0.838331i	$-0.316468\pi$
$487$	24.0613	1.09032	0.545161	+	0.838331i	$0.316468\pi$
$488$	21.4150	0.969410
$489$	0	0
$490$	−15.9619	−0.721085
$491$	29.0197	1.30964	0.654821	−	0.755784i	$-0.272744\pi$
$491$	29.0197	1.30964	0.654821	+	0.755784i	$0.272744\pi$
$492$	0	0
$493$	−1.36045	−0.0612716
$494$	0.0439260	0.00197632
$495$	0	0
$496$	6.62114	0.297298
$497$	−14.4039	−0.646102
$498$	0	0
$499$	−15.0257	−0.672642	−0.336321	−	0.941747i	$-0.609183\pi$
$499$	−15.0257	−0.672642	−0.336321	+	0.941747i	$0.609183\pi$
$500$	−10.6767	−0.477476
$501$	0	0
$502$	−18.7184	−0.835445
$503$	28.7785	1.28317	0.641585	−	0.767052i	$-0.278277\pi$
$503$	28.7785	1.28317	0.641585	+	0.767052i	$0.278277\pi$
$504$	0	0
$505$	−27.9653	−1.24444
$506$	−0.841844	−0.0374245
$507$	0	0
$508$	4.36878	0.193833
$509$	10.4772	0.464395	0.232197	−	0.972669i	$-0.425408\pi$
$509$	10.4772	0.464395	0.232197	+	0.972669i	$0.425408\pi$
$510$	0	0
$511$	−11.8979	−0.526332
$512$	8.03194	0.354965
$513$	0	0
$514$	4.43662	0.195691
$515$	21.6081	0.952165
$516$	0	0
$517$	2.89630	0.127379
$518$	3.74947	0.164742
$519$	0	0
$520$	0.520323	0.0228177
$521$	24.7590	1.08471	0.542357	−	0.840148i	$-0.317532\pi$
$521$	24.7590	1.08471	0.542357	+	0.840148i	$0.317532\pi$
$522$	0	0
$523$	14.8566	0.649635	0.324818	−	0.945777i	$-0.394697\pi$
$523$	14.8566	0.649635	0.324818	+	0.945777i	$0.394697\pi$
$524$	17.3025	0.755865
$525$	0	0
$526$	−10.9068	−0.475560
$527$	41.6679	1.81508
$528$	0	0
$529$	−21.8347	−0.949335
$530$	33.5574	1.45764
$531$	0	0
$532$	1.48558	0.0644080
$533$	0.675962	0.0292792
$534$	0	0
$535$	25.4602	1.10074
$536$	−29.8195	−1.28801
$537$	0	0
$538$	−12.0113	−0.517845
$539$	5.86074	0.252440
$540$	0	0
$541$	3.88960	0.167227	0.0836134	−	0.996498i	$-0.473354\pi$
$541$	3.88960	0.167227	0.0836134	+	0.996498i	$0.473354\pi$
$542$	−19.8014	−0.850541
$543$	0	0
$544$	26.5481	1.13824
$545$	27.2367	1.16669
$546$	0	0
$547$	17.5180	0.749015	0.374507	−	0.927224i	$-0.377812\pi$
$547$	17.5180	0.749015	0.374507	+	0.927224i	$0.377812\pi$
$548$	19.2491	0.822283
$549$	0	0
$550$	−5.61224	−0.239307
$551$	0.299905	0.0127764
$552$	0	0
$553$	12.1946	0.518567
$554$	15.5814	0.661988
$555$	0	0
$556$	−22.2648	−0.944237
$557$	−28.9860	−1.22818	−0.614088	−	0.789237i	$-0.710476\pi$
$557$	−28.9860	−1.22818	−0.614088	+	0.789237i	$0.710476\pi$
$558$	0	0
$559$	−0.604152	−0.0255529
$560$	2.68695	0.113544
$561$	0	0
$562$	−4.82052	−0.203341
$563$	−29.6112	−1.24796	−0.623982	−	0.781438i	$-0.714486\pi$
$563$	−29.6112	−1.24796	−0.623982	+	0.781438i	$0.714486\pi$
$564$	0	0
$565$	1.50366	0.0632594
$566$	−5.55356	−0.233434
$567$	0	0
$568$	35.6957	1.49776
$569$	4.05818	0.170128	0.0850640	−	0.996375i	$-0.472891\pi$
$569$	4.05818	0.170128	0.0850640	+	0.996375i	$0.472891\pi$
$570$	0	0
$571$	20.3380	0.851117	0.425559	−	0.904931i	$-0.360078\pi$
$571$	20.3380	0.851117	0.425559	+	0.904931i	$0.360078\pi$
$572$	−0.0783957	−0.00327789
$573$	0	0
$574$	−9.98942	−0.416950
$575$	7.76854	0.323971
$576$	0	0
$577$	24.4768	1.01898	0.509492	−	0.860476i	$-0.329833\pi$
$577$	24.4768	1.01898	0.509492	+	0.860476i	$0.329833\pi$
$578$	2.79017	0.116056
$579$	0	0
$580$	1.45776	0.0605302
$581$	14.9219	0.619064
$582$	0	0
$583$	−12.3213	−0.510296
$584$	29.4854	1.22011
$585$	0	0
$586$	−14.9219	−0.616419
$587$	3.34628	0.138116	0.0690579	−	0.997613i	$-0.478001\pi$
$587$	3.34628	0.138116	0.0690579	+	0.997613i	$0.478001\pi$
$588$	0	0
$589$	−9.18548	−0.378481
$590$	3.12067	0.128476
$591$	0	0
$592$	3.24695	0.133449
$593$	39.2063	1.61001	0.805006	−	0.593267i	$-0.202162\pi$
$593$	39.2063	1.61001	0.805006	+	0.593267i	$0.202162\pi$
$594$	0	0
$595$	16.9094	0.693219
$596$	15.8139	0.647764
$597$	0	0
$598$	−0.0474175	−0.00193905
$599$	−5.72987	−0.234116	−0.117058	−	0.993125i	$-0.537346\pi$
$599$	−5.72987	−0.234116	−0.117058	+	0.993125i	$0.537346\pi$
$600$	0	0
$601$	13.9163	0.567656	0.283828	−	0.958875i	$-0.408395\pi$
$601$	13.9163	0.567656	0.283828	+	0.958875i	$0.408395\pi$
$602$	8.92820	0.363886
$603$	0	0
$604$	−6.49065	−0.264101
$605$	3.49235	0.141984
$606$	0	0
$607$	0.156175	0.00633897	0.00316948	−	0.999995i	$-0.498991\pi$
$607$	0.156175	0.00633897	0.00316948	+	0.999995i	$0.498991\pi$
$608$	−5.85241	−0.237347
$609$	0	0
$610$	−22.0497	−0.892765
$611$	0.163137	0.00659980
$612$	0	0
$613$	40.1902	1.62327	0.811634	−	0.584166i	$-0.198578\pi$
$613$	40.1902	1.62327	0.811634	+	0.584166i	$0.198578\pi$
$614$	5.32308	0.214822
$615$	0	0
$616$	2.82331	0.113755
$617$	−12.8606	−0.517747	−0.258874	−	0.965911i	$-0.583351\pi$
$617$	−12.8606	−0.517747	−0.258874	+	0.965911i	$0.583351\pi$
$618$	0	0
$619$	−2.03398	−0.0817526	−0.0408763	−	0.999164i	$-0.513015\pi$
$619$	−2.03398	−0.0817526	−0.0408763	+	0.999164i	$0.513015\pi$
$620$	−44.6482	−1.79312
$621$	0	0
$622$	10.0297	0.402153
$623$	0.195524	0.00783352
$624$	0	0
$625$	−9.19275	−0.367710
$626$	−2.35288	−0.0940398
$627$	0	0
$628$	12.5989	0.502750
$629$	20.4336	0.814739
$630$	0	0
$631$	37.6984	1.50075	0.750375	−	0.661012i	$-0.229873\pi$
$631$	37.6984	1.50075	0.750375	+	0.661012i	$0.229873\pi$
$632$	−30.2206	−1.20211
$633$	0	0
$634$	−13.8703	−0.550858
$635$	−10.9621	−0.435017
$636$	0	0
$637$	0.330111	0.0130795
$638$	0.233882	0.00925950
$639$	0	0
$640$	−32.3734	−1.27967
$641$	−2.84360	−0.112315	−0.0561576	−	0.998422i	$-0.517885\pi$
$641$	−2.84360	−0.112315	−0.0561576	+	0.998422i	$0.517885\pi$
$642$	0	0
$643$	30.6389	1.20828	0.604140	−	0.796878i	$-0.293517\pi$
$643$	30.6389	1.20828	0.604140	+	0.796878i	$0.293517\pi$
$644$	−1.60366	−0.0631932
$645$	0	0
$646$	−3.53764	−0.139187
$647$	6.76617	0.266006	0.133003	−	0.991116i	$-0.457538\pi$
$647$	6.76617	0.266006	0.133003	+	0.991116i	$0.457538\pi$
$648$	0	0
$649$	−1.14582	−0.0449772
$650$	−0.316114	−0.0123990
$651$	0	0
$652$	−3.29530	−0.129054
$653$	−11.9015	−0.465743	−0.232872	−	0.972508i	$-0.574812\pi$
$653$	−11.9015	−0.465743	−0.232872	+	0.972508i	$0.574812\pi$
$654$	0	0
$655$	−43.4153	−1.69638
$656$	−8.65059	−0.337749
$657$	0	0
$658$	−2.41085	−0.0939845
$659$	11.2353	0.437664	0.218832	−	0.975763i	$-0.429775\pi$
$659$	11.2353	0.437664	0.218832	+	0.975763i	$0.429775\pi$
$660$	0	0
$661$	−22.9273	−0.891768	−0.445884	−	0.895091i	$-0.647111\pi$
$661$	−22.9273	−0.891768	−0.445884	+	0.895091i	$0.647111\pi$
$662$	−20.5772	−0.799757
$663$	0	0
$664$	−36.9794	−1.43508
$665$	−3.72760	−0.144550
$666$	0	0
$667$	−0.323743	−0.0125354
$668$	−12.9072	−0.499396
$669$	0	0
$670$	30.7033	1.18617
$671$	8.09599	0.312542
$672$	0	0
$673$	−3.09828	−0.119430	−0.0597150	−	0.998215i	$-0.519019\pi$
$673$	−3.09828	−0.119430	−0.0597150	+	0.998215i	$0.519019\pi$
$674$	−10.2960	−0.396586
$675$	0	0
$676$	18.0893	0.695743
$677$	−33.3985	−1.28361	−0.641804	−	0.766868i	$-0.721814\pi$
$677$	−33.3985	−1.28361	−0.641804	+	0.766868i	$0.721814\pi$
$678$	0	0
$679$	6.04407	0.231950
$680$	−41.9049	−1.60698
$681$	0	0
$682$	−7.16335	−0.274299
$683$	−6.01634	−0.230209	−0.115104	−	0.993353i	$-0.536720\pi$
$683$	−6.01634	−0.230209	−0.115104	+	0.993353i	$0.536720\pi$
$684$	0	0
$685$	−48.2997	−1.84544
$686$	−10.7051	−0.408723
$687$	0	0
$688$	7.73160	0.294765
$689$	−0.694007	−0.0264396
$690$	0	0
$691$	−15.6730	−0.596227	−0.298114	−	0.954530i	$-0.596357\pi$
$691$	−15.6730	−0.596227	−0.298114	+	0.954530i	$0.596357\pi$
$692$	14.9164	0.567038
$693$	0	0
$694$	−12.8527	−0.487882
$695$	55.8665	2.11914
$696$	0	0
$697$	−54.4395	−2.06204
$698$	−14.8302	−0.561330
$699$	0	0
$700$	−10.6910	−0.404082
$701$	−18.0567	−0.681992	−0.340996	−	0.940065i	$-0.610764\pi$
$701$	−18.0567	−0.681992	−0.340996	+	0.940065i	$0.610764\pi$
$702$	0	0
$703$	−4.50448	−0.169890
$704$	−3.12238	−0.117679
$705$	0	0
$706$	−10.2906	−0.387291
$707$	−8.54699	−0.321442
$708$	0	0
$709$	16.2048	0.608586	0.304293	−	0.952579i	$-0.401580\pi$
$709$	16.2048	0.608586	0.304293	+	0.952579i	$0.401580\pi$
$710$	−36.7536	−1.37934
$711$	0	0
$712$	−0.484549	−0.0181592
$713$	9.91560	0.371342
$714$	0	0
$715$	0.196709	0.00735652
$716$	−31.8825	−1.19150
$717$	0	0
$718$	2.25188	0.0840395
$719$	27.9403	1.04200	0.520998	−	0.853558i	$-0.325560\pi$
$719$	27.9403	1.04200	0.520998	+	0.853558i	$0.325560\pi$
$720$	0	0
$721$	6.60403	0.245947
$722$	0.779856	0.0290232
$723$	0	0
$724$	−8.21903	−0.305458
$725$	−2.15827	−0.0801561
$726$	0	0
$727$	−30.7020	−1.13867	−0.569337	−	0.822104i	$-0.692800\pi$
$727$	−30.7020	−1.13867	−0.569337	+	0.822104i	$0.692800\pi$
$728$	0.159025	0.00589387
$729$	0	0
$730$	−30.3592	−1.12365
$731$	48.6562	1.79961
$732$	0	0
$733$	44.9333	1.65965	0.829826	−	0.558023i	$-0.188440\pi$
$733$	44.9333	1.65965	0.829826	+	0.558023i	$0.188440\pi$
$734$	−1.64419	−0.0606882
$735$	0	0
$736$	6.31760	0.232870
$737$	−11.2733	−0.415258
$738$	0	0
$739$	−3.19172	−0.117409	−0.0587047	−	0.998275i	$-0.518697\pi$
$739$	−3.19172	−0.117409	−0.0587047	+	0.998275i	$0.518697\pi$
$740$	−21.8951	−0.804880
$741$	0	0
$742$	10.2561	0.376513
$743$	−17.4348	−0.639619	−0.319810	−	0.947482i	$-0.603619\pi$
$743$	−17.4348	−0.639619	−0.319810	+	0.947482i	$0.603619\pi$
$744$	0	0
$745$	−39.6801	−1.45377
$746$	6.29581	0.230506
$747$	0	0
$748$	6.31370	0.230852
$749$	7.78135	0.284324
$750$	0	0
$751$	9.55633	0.348715	0.174358	−	0.984682i	$-0.444215\pi$
$751$	9.55633	0.348715	0.174358	+	0.984682i	$0.444215\pi$
$752$	−2.08773	−0.0761317
$753$	0	0
$754$	0.0131736	0.000479755 0
$755$	16.2863	0.592718
$756$	0	0
$757$	−48.5259	−1.76370	−0.881852	−	0.471527i	$-0.843703\pi$
$757$	−48.5259	−1.76370	−0.881852	+	0.471527i	$0.843703\pi$
$758$	11.1796	0.406062
$759$	0	0
$760$	9.23774	0.335088
$761$	−38.6035	−1.39937	−0.699687	−	0.714449i	$-0.746677\pi$
$761$	−38.6035	−1.39937	−0.699687	+	0.714449i	$0.746677\pi$
$762$	0	0
$763$	8.32429	0.301360
$764$	8.97210	0.324599
$765$	0	0
$766$	7.64155	0.276101
$767$	−0.0645391	−0.00233037
$768$	0	0
$769$	53.2658	1.92081	0.960406	−	0.278603i	$-0.0898712\pi$
$769$	53.2658	1.92081	0.960406	+	0.278603i	$0.0898712\pi$
$770$	−2.90699	−0.104761
$771$	0	0
$772$	−1.99874	−0.0719363
$773$	−9.31748	−0.335126	−0.167563	−	0.985861i	$-0.553590\pi$
$773$	−9.31748	−0.335126	−0.167563	+	0.985861i	$0.553590\pi$
$774$	0	0
$775$	66.1034	2.37451
$776$	−14.9784	−0.537694
$777$	0	0
$778$	15.5375	0.557047
$779$	12.0009	0.429978
$780$	0	0
$781$	13.4948	0.482883
$782$	3.81884	0.136561
$783$	0	0
$784$	−4.22458	−0.150878
$785$	−31.6130	−1.12832
$786$	0	0
$787$	−15.0616	−0.536889	−0.268444	−	0.963295i	$-0.586510\pi$
$787$	−15.0616	−0.536889	−0.268444	+	0.963295i	$0.586510\pi$
$788$	−6.61672	−0.235711
$789$	0	0
$790$	31.1163	1.10707
$791$	0.459561	0.0163401
$792$	0	0
$793$	0.456013	0.0161935
$794$	10.9133	0.387300
$795$	0	0
$796$	3.28474	0.116425
$797$	19.0593	0.675114	0.337557	−	0.941305i	$-0.390399\pi$
$797$	19.0593	0.675114	0.337557	+	0.941305i	$0.390399\pi$
$798$	0	0
$799$	−13.1384	−0.464804
$800$	42.1169	1.48906
$801$	0	0
$802$	14.5281	0.513006
$803$	11.1470	0.393370
$804$	0	0
$805$	4.02389	0.141824
$806$	−0.403481	−0.0142120
$807$	0	0
$808$	21.1811	0.745150
$809$	15.2273	0.535363	0.267681	−	0.963507i	$-0.413743\pi$
$809$	15.2273	0.535363	0.267681	+	0.963507i	$0.413743\pi$
$810$	0	0
$811$	−43.3354	−1.52171	−0.760857	−	0.648920i	$-0.775221\pi$
$811$	−43.3354	−1.52171	−0.760857	+	0.648920i	$0.775221\pi$
$812$	0.445532	0.0156351
$813$	0	0
$814$	−3.51284	−0.123125
$815$	8.26854	0.289634
$816$	0	0
$817$	−10.7260	−0.375256
$818$	−24.0530	−0.840992
$819$	0	0
$820$	58.3334	2.03709
$821$	−19.4279	−0.678040	−0.339020	−	0.940779i	$-0.610095\pi$
$821$	−19.4279	−0.678040	−0.339020	+	0.940779i	$0.610095\pi$
$822$	0	0
$823$	2.35683	0.0821539	0.0410769	−	0.999156i	$-0.486921\pi$
$823$	2.35683	0.0821539	0.0410769	+	0.999156i	$0.486921\pi$
$824$	−16.3661	−0.570141
$825$	0	0
$826$	0.953763	0.0331857
$827$	29.8127	1.03669	0.518344	−	0.855172i	$-0.326549\pi$
$827$	29.8127	1.03669	0.518344	+	0.855172i	$0.326549\pi$
$828$	0	0
$829$	52.6510	1.82864	0.914322	−	0.404988i	$-0.132724\pi$
$829$	52.6510	1.82864	0.914322	+	0.404988i	$0.132724\pi$
$830$	38.0754	1.32162
$831$	0	0
$832$	−0.175871	−0.00609722
$833$	−26.5859	−0.921148
$834$	0	0
$835$	32.3867	1.12079
$836$	−1.39182	−0.0481373
$837$	0	0
$838$	−29.5836	−1.02195
$839$	−29.7892	−1.02844	−0.514218	−	0.857660i	$-0.671918\pi$
$839$	−29.7892	−1.02844	−0.514218	+	0.857660i	$0.671918\pi$
$840$	0	0
$841$	−28.9101	−0.996899
$842$	−14.3994	−0.496237
$843$	0	0
$844$	34.1182	1.17440
$845$	−45.3895	−1.56145
$846$	0	0
$847$	1.06736	0.0366749
$848$	8.88152	0.304992
$849$	0	0
$850$	25.4587	0.873226
$851$	4.86252	0.166685
$852$	0	0
$853$	−35.2393	−1.20657	−0.603285	−	0.797526i	$-0.706142\pi$
$853$	−35.2393	−1.20657	−0.603285	+	0.797526i	$0.706142\pi$
$854$	−6.73900	−0.230604
$855$	0	0
$856$	−19.2837	−0.659105
$857$	21.5877	0.737421	0.368711	−	0.929544i	$-0.379799\pi$
$857$	21.5877	0.737421	0.368711	+	0.929544i	$0.379799\pi$
$858$	0	0
$859$	27.7669	0.947396	0.473698	−	0.880687i	$-0.342919\pi$
$859$	27.7669	0.947396	0.473698	+	0.880687i	$0.342919\pi$
$860$	−52.1364	−1.77784
$861$	0	0
$862$	−26.2610	−0.894453
$863$	−7.11774	−0.242291	−0.121145	−	0.992635i	$-0.538657\pi$
$863$	−7.11774	−0.242291	−0.121145	+	0.992635i	$0.538657\pi$
$864$	0	0
$865$	−37.4282	−1.27260
$866$	8.89357	0.302216
$867$	0	0
$868$	−13.6458	−0.463167
$869$	−11.4250	−0.387566
$870$	0	0
$871$	−0.634980	−0.0215155
$872$	−20.6293	−0.698595
$873$	0	0
$874$	−0.841844	−0.0284758
$875$	8.18772	0.276796
$876$	0	0
$877$	−44.1031	−1.48925	−0.744627	−	0.667481i	$-0.767373\pi$
$877$	−44.1031	−1.48925	−0.744627	+	0.667481i	$0.767373\pi$
$878$	−2.93920	−0.0991931
$879$	0	0
$880$	−2.51738	−0.0848608
$881$	37.9204	1.27757	0.638785	−	0.769385i	$-0.279437\pi$
$881$	37.9204	1.27757	0.638785	+	0.769385i	$0.279437\pi$
$882$	0	0
$883$	4.85252	0.163300	0.0816502	−	0.996661i	$-0.473981\pi$
$883$	4.85252	0.163300	0.0816502	+	0.996661i	$0.473981\pi$
$884$	0.355624	0.0119609
$885$	0	0
$886$	−3.52371	−0.118381
$887$	−11.2754	−0.378591	−0.189295	−	0.981920i	$-0.560620\pi$
$887$	−11.2754	−0.378591	−0.189295	+	0.981920i	$0.560620\pi$
$888$	0	0
$889$	−3.35032	−0.112366
$890$	0.498910	0.0167235
$891$	0	0
$892$	−11.7947	−0.394916
$893$	2.89630	0.0969210
$894$	0	0
$895$	79.9991	2.67408
$896$	−9.89423	−0.330543
$897$	0	0
$898$	29.1363	0.972290
$899$	−2.75477	−0.0918767
$900$	0	0
$901$	55.8928	1.86206
$902$	9.35899	0.311620
$903$	0	0
$904$	−1.13888	−0.0378787
$905$	20.6231	0.685534
$906$	0	0
$907$	24.0191	0.797540	0.398770	−	0.917051i	$-0.369437\pi$
$907$	24.0191	0.797540	0.398770	+	0.917051i	$0.369437\pi$
$908$	19.1363	0.635061
$909$	0	0
$910$	−0.163739	−0.00542788
$911$	−7.48540	−0.248002	−0.124001	−	0.992282i	$-0.539573\pi$
$911$	−7.48540	−0.248002	−0.124001	+	0.992282i	$0.539573\pi$
$912$	0	0
$913$	−13.9802	−0.462676
$914$	7.31328	0.241902
$915$	0	0
$916$	2.77864	0.0918089
$917$	−13.2689	−0.438179
$918$	0	0
$919$	40.0187	1.32009	0.660047	−	0.751224i	$-0.270536\pi$
$919$	40.0187	1.32009	0.660047	+	0.751224i	$0.270536\pi$
$920$	−9.97201	−0.328768
$921$	0	0
$922$	−24.7047	−0.813605
$923$	0.760108	0.0250193
$924$	0	0
$925$	32.4165	1.06585
$926$	−28.9548	−0.951514
$927$	0	0
$928$	−1.75517	−0.0576161
$929$	32.3099	1.06005	0.530027	−	0.847981i	$-0.322182\pi$
$929$	32.3099	1.06005	0.530027	+	0.847981i	$0.322182\pi$
$930$	0	0
$931$	5.86074	0.192078
$932$	15.7945	0.517367
$933$	0	0
$934$	12.4526	0.407461
$935$	−15.8423	−0.518098
$936$	0	0
$937$	−54.1000	−1.76737	−0.883684	−	0.468083i	$-0.844945\pi$
$937$	−54.1000	−1.76737	−0.883684	+	0.468083i	$0.844945\pi$
$938$	9.38378	0.306391
$939$	0	0
$940$	14.0782	0.459179
$941$	−26.6955	−0.870248	−0.435124	−	0.900371i	$-0.643295\pi$
$941$	−26.6955	−0.870248	−0.435124	+	0.900371i	$0.643295\pi$
$942$	0	0
$943$	−12.9548	−0.421868
$944$	0.825935	0.0268819
$945$	0	0
$946$	−8.36475	−0.271961
$947$	39.4463	1.28183	0.640916	−	0.767611i	$-0.278555\pi$
$947$	39.4463	1.28183	0.640916	+	0.767611i	$0.278555\pi$
$948$	0	0
$949$	0.627865	0.0203814
$950$	−5.61224	−0.182085
$951$	0	0
$952$	−12.8073	−0.415088
$953$	48.7523	1.57924	0.789621	−	0.613595i	$-0.210277\pi$
$953$	48.7523	1.57924	0.789621	+	0.613595i	$0.210277\pi$
$954$	0	0
$955$	−22.5127	−0.728493
$956$	7.25203	0.234547
$957$	0	0
$958$	−21.9708	−0.709845
$959$	−14.7618	−0.476682
$960$	0	0
$961$	53.3730	1.72171
$962$	−0.197864	−0.00637938
$963$	0	0
$964$	−9.19755	−0.296233
$965$	5.01522	0.161446
$966$	0	0
$967$	−1.39478	−0.0448532	−0.0224266	−	0.999748i	$-0.507139\pi$
$967$	−1.39478	−0.0448532	−0.0224266	+	0.999748i	$0.507139\pi$
$968$	−2.64513	−0.0850178
$969$	0	0
$970$	15.4224	0.495182
$971$	3.11733	0.100040	0.0500200	−	0.998748i	$-0.484071\pi$
$971$	3.11733	0.100040	0.0500200	+	0.998748i	$0.484071\pi$
$972$	0	0
$973$	17.0744	0.547380
$974$	18.7644	0.601249
$975$	0	0
$976$	−5.83580	−0.186800
$977$	22.3690	0.715647	0.357823	−	0.933789i	$-0.383519\pi$
$977$	22.3690	0.715647	0.357823	+	0.933789i	$0.383519\pi$
$978$	0	0
$979$	−0.183185	−0.00585462
$980$	28.4875	0.910001
$981$	0	0
$982$	22.6312	0.722191
$983$	−42.2421	−1.34731	−0.673657	−	0.739044i	$-0.735277\pi$
$983$	−42.2421	−1.34731	−0.673657	+	0.739044i	$0.735277\pi$
$984$	0	0
$985$	16.6026	0.529003
$986$	−1.06096	−0.0337877
$987$	0	0
$988$	−0.0783957	−0.00249410
$989$	11.5786	0.368178
$990$	0	0
$991$	−1.69828	−0.0539477	−0.0269738	−	0.999636i	$-0.508587\pi$
$991$	−1.69828	−0.0539477	−0.0269738	+	0.999636i	$0.508587\pi$
$992$	53.7572	1.70679
$993$	0	0
$994$	−11.2329	−0.356287
$995$	−8.24203	−0.261290
$996$	0	0
$997$	18.9376	0.599759	0.299879	−	0.953977i	$-0.403054\pi$
$997$	18.9376	0.599759	0.299879	+	0.953977i	$0.403054\pi$
$998$	−11.7179	−0.370923
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1881.2.a.k.1.4		5
3.2	odd	2		209.2.a.c.1.2	✓	5
12.11	even	2		3344.2.a.t.1.5		5
15.14	odd	2		5225.2.a.h.1.4		5
33.32	even	2		2299.2.a.n.1.4		5
57.56	even	2		3971.2.a.h.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
209.2.a.c.1.2	✓	5	3.2	odd	2
1881.2.a.k.1.4		5	1.1	even	1	trivial
2299.2.a.n.1.4		5	33.32	even	2
3344.2.a.t.1.5		5	12.11	even	2
3971.2.a.h.1.4		5	57.56	even	2
5225.2.a.h.1.4		5	15.14	odd	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:	\( N \)	\(=\)	\( 1881 = 3^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1881.a (trivial)

\(f(q)\)	\(=\)	\(q+0.779856 q^{2} -1.39182 q^{4} +3.49235 q^{5} +1.06736 q^{7} -2.64513 q^{8} +O(q^{10})\)	\(q+0.779856 q^{2} -1.39182 q^{4} +3.49235 q^{5} +1.06736 q^{7} -2.64513 q^{8} +2.72353 q^{10} -1.00000 q^{11} -0.0563258 q^{13} +0.832387 q^{14} +0.720827 q^{16} +4.53628 q^{17} -1.00000 q^{19} -4.86074 q^{20} -0.779856 q^{22} +1.07949 q^{23} +7.19651 q^{25} -0.0439260 q^{26} -1.48558 q^{28} -0.299905 q^{29} +9.18548 q^{31} +5.85241 q^{32} +3.53764 q^{34} +3.72760 q^{35} +4.50448 q^{37} -0.779856 q^{38} -9.23774 q^{40} -12.0009 q^{41} +10.7260 q^{43} +1.39182 q^{44} +0.841844 q^{46} -2.89630 q^{47} -5.86074 q^{49} +5.61224 q^{50} +0.0783957 q^{52} +12.3213 q^{53} -3.49235 q^{55} -2.82331 q^{56} -0.233882 q^{58} +1.14582 q^{59} -8.09599 q^{61} +7.16335 q^{62} +3.12238 q^{64} -0.196709 q^{65} +11.2733 q^{67} -6.31370 q^{68} +2.90699 q^{70} -13.4948 q^{71} -11.1470 q^{73} +3.51284 q^{74} +1.39182 q^{76} -1.06736 q^{77} +11.4250 q^{79} +2.51738 q^{80} -9.35899 q^{82} +13.9802 q^{83} +15.8423 q^{85} +8.36475 q^{86} +2.64513 q^{88} +0.183185 q^{89} -0.0601200 q^{91} -1.50246 q^{92} -2.25870 q^{94} -3.49235 q^{95} +5.66263 q^{97} -4.57053 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} + 6 q^{4} + 5 q^{5} + 6 q^{7} - 6 q^{8}+O(q^{10}) \) 5 * q - 2 * q^2 + 6 * q^4 + 5 * q^5 + 6 * q^7 - 6 * q^8	\( 5 q - 2 q^{2} + 6 q^{4} + 5 q^{5} + 6 q^{7} - 6 q^{8} + 12 q^{10} - 5 q^{11} + 4 q^{13} + 14 q^{14} + 8 q^{16} + 4 q^{17} - 5 q^{19} + 8 q^{20} + 2 q^{22} - 3 q^{23} + 6 q^{25} + 6 q^{26} - 10 q^{28} - 10 q^{29} + 11 q^{31} - 14 q^{32} - 4 q^{34} + 8 q^{35} + q^{37} + 2 q^{38} - 16 q^{40} - 2 q^{41} + 20 q^{43} - 6 q^{44} - 4 q^{46} + 20 q^{47} + 3 q^{49} + 32 q^{50} + 6 q^{52} + 14 q^{53} - 5 q^{55} + 38 q^{56} - 6 q^{58} - 3 q^{59} - 10 q^{61} + 6 q^{62} + 9 q^{67} - 24 q^{68} + 50 q^{70} - 23 q^{71} - 8 q^{74} - 6 q^{76} - 6 q^{77} + 44 q^{79} + 18 q^{80} - 30 q^{82} + 14 q^{83} - 12 q^{85} - 52 q^{86} + 6 q^{88} + 27 q^{89} + 24 q^{91} - 58 q^{92} - 8 q^{94} - 5 q^{95} + 15 q^{97} + 10 q^{98}+O(q^{100}) \) 5 * q - 2 * q^2 + 6 * q^4 + 5 * q^5 + 6 * q^7 - 6 * q^8 + 12 * q^10 - 5 * q^11 + 4 * q^13 + 14 * q^14 + 8 * q^16 + 4 * q^17 - 5 * q^19 + 8 * q^20 + 2 * q^22 - 3 * q^23 + 6 * q^25 + 6 * q^26 - 10 * q^28 - 10 * q^29 + 11 * q^31 - 14 * q^32 - 4 * q^34 + 8 * q^35 + q^37 + 2 * q^38 - 16 * q^40 - 2 * q^41 + 20 * q^43 - 6 * q^44 - 4 * q^46 + 20 * q^47 + 3 * q^49 + 32 * q^50 + 6 * q^52 + 14 * q^53 - 5 * q^55 + 38 * q^56 - 6 * q^58 - 3 * q^59 - 10 * q^61 + 6 * q^62 + 9 * q^67 - 24 * q^68 + 50 * q^70 - 23 * q^71 - 8 * q^74 - 6 * q^76 - 6 * q^77 + 44 * q^79 + 18 * q^80 - 30 * q^82 + 14 * q^83 - 12 * q^85 - 52 * q^86 + 6 * q^88 + 27 * q^89 + 24 * q^91 - 58 * q^92 - 8 * q^94 - 5 * q^95 + 15 * q^97 + 10 * q^98

Introduction

L-functions

Modular forms

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