LMFDB - Embedded newform 725.2.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [725,2,Mod(1,725)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(725, 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("725.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 725 = 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	725.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:	$5.78915414654$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 4 $ x^4 - 7*x^2 + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 145)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.52434$ of defining polynomial
Character	$\chi$	$=$	725.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.73205 q^{2} +2.52434 q^{3} +1.00000 q^{4} -4.37228 q^{6} +1.58457 q^{7} +1.73205 q^{8} +3.37228 q^{9} +6.37228 q^{11} +2.52434 q^{12} -0.939764 q^{13} -2.74456 q^{14} -5.00000 q^{16} -5.04868 q^{17} -5.84096 q^{18} +4.00000 q^{19} +4.00000 q^{21} -11.0371 q^{22} -3.46410 q^{23} +4.37228 q^{24} +1.62772 q^{26} +0.939764 q^{27} +1.58457 q^{28} -1.00000 q^{29} +2.37228 q^{31} +5.19615 q^{32} +16.0858 q^{33} +8.74456 q^{34} +3.37228 q^{36} +10.0974 q^{37} -6.92820 q^{38} -2.37228 q^{39} +6.74456 q^{41} -6.92820 q^{42} -5.69349 q^{43} +6.37228 q^{44} +6.00000 q^{46} -5.69349 q^{47} -12.6217 q^{48} -4.48913 q^{49} -12.7446 q^{51} -0.939764 q^{52} -0.939764 q^{53} -1.62772 q^{54} +2.74456 q^{56} +10.0974 q^{57} +1.73205 q^{58} +0.744563 q^{59} +6.00000 q^{61} -4.10891 q^{62} +5.34363 q^{63} +1.00000 q^{64} -27.8614 q^{66} +8.51278 q^{67} -5.04868 q^{68} -8.74456 q^{69} -4.74456 q^{71} +5.84096 q^{72} +6.92820 q^{73} -17.4891 q^{74} +4.00000 q^{76} +10.0974 q^{77} +4.10891 q^{78} +5.62772 q^{79} -7.74456 q^{81} -11.6819 q^{82} -16.7306 q^{83} +4.00000 q^{84} +9.86141 q^{86} -2.52434 q^{87} +11.0371 q^{88} +10.7446 q^{89} -1.48913 q^{91} -3.46410 q^{92} +5.98844 q^{93} +9.86141 q^{94} +13.1168 q^{96} +6.92820 q^{97} +7.77539 q^{98} +21.4891 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4} - 6 q^{6} + 2 q^{9} + 14 q^{11} + 12 q^{14} - 20 q^{16} + 16 q^{19} + 16 q^{21} + 6 q^{24} + 18 q^{26} - 4 q^{29} - 2 q^{31} + 12 q^{34} + 2 q^{36} + 2 q^{39} + 4 q^{41} + 14 q^{44} + 24 q^{46}+ \cdots + 40 q^{99}+O(q^{100}) $ 4 * q + 4 * q^4 - 6 * q^6 + 2 * q^9 + 14 * q^11 + 12 * q^14 - 20 * q^16 + 16 * q^19 + 16 * q^21 + 6 * q^24 + 18 * q^26 - 4 * q^29 - 2 * q^31 + 12 * q^34 + 2 * q^36 + 2 * q^39 + 4 * q^41 + 14 * q^44 + 24 * q^46 + 28 * q^49 - 28 * q^51 - 18 * q^54 - 12 * q^56 - 20 * q^59 + 24 * q^61 + 4 * q^64 - 54 * q^66 - 12 * q^69 + 4 * q^71 - 24 * q^74 + 16 * q^76 + 34 * q^79 - 8 * q^81 + 16 * q^84 - 18 * q^86 + 20 * q^89 + 40 * q^91 - 18 * q^94 + 18 * q^96 + 40 * q^99

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 $

$2$	−1.73205	−1.22474	−0.612372	−	0.790569i	$-0.709785\pi$
$2$	−1.73205	−1.22474	−0.612372	+	0.790569i	$0.709785\pi$
$3$	2.52434	1.45743	0.728714	−	0.684819i	$-0.240119\pi$
$3$	2.52434	1.45743	0.728714	+	0.684819i	$0.240119\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−4.37228	−1.78498
$7$	1.58457	0.598913	0.299456	−	0.954110i	$-0.403195\pi$
$7$	1.58457	0.598913	0.299456	+	0.954110i	$0.403195\pi$
$8$	1.73205	0.612372
$9$	3.37228	1.12409
$10$	0	0
$11$	6.37228	1.92132	0.960658	−	0.277736i	$-0.0895839\pi$
$11$	6.37228	1.92132	0.960658	+	0.277736i	$0.0895839\pi$
$12$	2.52434	0.728714
$13$	−0.939764	−0.260644	−0.130322	−	0.991472i	$-0.541601\pi$
$13$	−0.939764	−0.260644	−0.130322	+	0.991472i	$0.541601\pi$
$14$	−2.74456	−0.733515
$15$	0	0
$16$	−5.00000	−1.25000
$17$	−5.04868	−1.22448	−0.612242	−	0.790671i	$-0.709732\pi$
$17$	−5.04868	−1.22448	−0.612242	+	0.790671i	$0.709732\pi$
$18$	−5.84096	−1.37673
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	4.00000	0.872872
$22$	−11.0371	−2.35312
$23$	−3.46410	−0.722315	−0.361158	−	0.932505i	$-0.617618\pi$
$23$	−3.46410	−0.722315	−0.361158	+	0.932505i	$0.617618\pi$
$24$	4.37228	0.892488
$25$	0	0
$26$	1.62772	0.319222
$27$	0.939764	0.180858
$28$	1.58457	0.299456
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	2.37228	0.426074	0.213037	−	0.977044i	$-0.431664\pi$
$31$	2.37228	0.426074	0.213037	+	0.977044i	$0.431664\pi$
$32$	5.19615	0.918559
$33$	16.0858	2.80018
$34$	8.74456	1.49968
$35$	0	0
$36$	3.37228	0.562047
$37$	10.0974	1.65999	0.829997	−	0.557768i	$-0.188342\pi$
$37$	10.0974	1.65999	0.829997	+	0.557768i	$0.188342\pi$
$38$	−6.92820	−1.12390
$39$	−2.37228	−0.379869
$40$	0	0
$41$	6.74456	1.05332	0.526662	−	0.850075i	$-0.323443\pi$
$41$	6.74456	1.05332	0.526662	+	0.850075i	$0.323443\pi$
$42$	−6.92820	−1.06904
$43$	−5.69349	−0.868248	−0.434124	−	0.900853i	$-0.642942\pi$
$43$	−5.69349	−0.868248	−0.434124	+	0.900853i	$0.642942\pi$
$44$	6.37228	0.960658
$45$	0	0
$46$	6.00000	0.884652
$47$	−5.69349	−0.830480	−0.415240	−	0.909712i	$-0.636302\pi$
$47$	−5.69349	−0.830480	−0.415240	+	0.909712i	$0.636302\pi$
$48$	−12.6217	−1.82178
$49$	−4.48913	−0.641304
$50$	0	0
$51$	−12.7446	−1.78460
$52$	−0.939764	−0.130322
$53$	−0.939764	−0.129086	−0.0645432	−	0.997915i	$-0.520559\pi$
$53$	−0.939764	−0.129086	−0.0645432	+	0.997915i	$0.520559\pi$
$54$	−1.62772	−0.221504
$55$	0	0
$56$	2.74456	0.366758
$57$	10.0974	1.33743
$58$	1.73205	0.227429
$59$	0.744563	0.0969338	0.0484669	−	0.998825i	$-0.484566\pi$
$59$	0.744563	0.0969338	0.0484669	+	0.998825i	$0.484566\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	−4.10891	−0.521832
$63$	5.34363	0.673234
$64$	1.00000	0.125000
$65$	0	0
$66$	−27.8614	−3.42950
$67$	8.51278	1.04000	0.520001	−	0.854166i	$-0.325932\pi$
$67$	8.51278	1.04000	0.520001	+	0.854166i	$0.325932\pi$
$68$	−5.04868	−0.612242
$69$	−8.74456	−1.05272
$70$	0	0
$71$	−4.74456	−0.563076	−0.281538	−	0.959550i	$-0.590845\pi$
$71$	−4.74456	−0.563076	−0.281538	+	0.959550i	$0.590845\pi$
$72$	5.84096	0.688364
$73$	6.92820	0.810885	0.405442	−	0.914121i	$-0.367117\pi$
$73$	6.92820	0.810885	0.405442	+	0.914121i	$0.367117\pi$
$74$	−17.4891	−2.03307
$75$	0	0
$76$	4.00000	0.458831
$77$	10.0974	1.15070
$78$	4.10891	0.465243
$79$	5.62772	0.633168	0.316584	−	0.948565i	$-0.397464\pi$
$79$	5.62772	0.633168	0.316584	+	0.948565i	$0.397464\pi$
$80$	0	0
$81$	−7.74456	−0.860507
$82$	−11.6819	−1.29005
$83$	−16.7306	−1.83642	−0.918211	−	0.396092i	$-0.870366\pi$
$83$	−16.7306	−1.83642	−0.918211	+	0.396092i	$0.870366\pi$
$84$	4.00000	0.436436
$85$	0	0
$86$	9.86141	1.06338
$87$	−2.52434	−0.270637
$88$	11.0371	1.17656
$89$	10.7446	1.13892	0.569461	−	0.822019i	$-0.307152\pi$
$89$	10.7446	1.13892	0.569461	+	0.822019i	$0.307152\pi$
$90$	0	0
$91$	−1.48913	−0.156103
$92$	−3.46410	−0.361158
$93$	5.98844	0.620972
$94$	9.86141	1.01713
$95$	0	0
$96$	13.1168	1.33873
$97$	6.92820	0.703452	0.351726	−	0.936103i	$-0.385595\pi$
$97$	6.92820	0.703452	0.351726	+	0.936103i	$0.385595\pi$
$98$	7.77539	0.785433
$99$	21.4891	2.15974
$100$	0	0
$101$	−14.7446	−1.46714	−0.733569	−	0.679615i	$-0.762147\pi$
$101$	−14.7446	−1.46714	−0.733569	+	0.679615i	$0.762147\pi$
$102$	22.0742	2.18567
$103$	−3.46410	−0.341328	−0.170664	−	0.985329i	$-0.554591\pi$
$103$	−3.46410	−0.341328	−0.170664	+	0.985329i	$0.554591\pi$
$104$	−1.62772	−0.159611
$105$	0	0
$106$	1.62772	0.158098
$107$	−1.58457	−0.153187	−0.0765933	−	0.997062i	$-0.524404\pi$
$107$	−1.58457	−0.153187	−0.0765933	+	0.997062i	$0.524404\pi$
$108$	0.939764	0.0904288
$109$	1.11684	0.106974	0.0534871	−	0.998569i	$-0.482966\pi$
$109$	1.11684	0.106974	0.0534871	+	0.998569i	$0.482966\pi$
$110$	0	0
$111$	25.4891	2.41932
$112$	−7.92287	−0.748641
$113$	8.80773	0.828562	0.414281	−	0.910149i	$-0.364033\pi$
$113$	8.80773	0.828562	0.414281	+	0.910149i	$0.364033\pi$
$114$	−17.4891	−1.63801
$115$	0	0
$116$	−1.00000	−0.0928477
$117$	−3.16915	−0.292988
$118$	−1.28962	−0.118719
$119$	−8.00000	−0.733359
$120$	0	0
$121$	29.6060	2.69145
$122$	−10.3923	−0.940875
$123$	17.0256	1.53514
$124$	2.37228	0.213037
$125$	0	0
$126$	−9.25544	−0.824540
$127$	−3.46410	−0.307389	−0.153695	−	0.988118i	$-0.549117\pi$
$127$	−3.46410	−0.307389	−0.153695	+	0.988118i	$0.549117\pi$
$128$	−12.1244	−1.07165
$129$	−14.3723	−1.26541
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	16.0858	1.40009
$133$	6.33830	0.549600
$134$	−14.7446	−1.27374
$135$	0	0
$136$	−8.74456	−0.749840
$137$	−6.92820	−0.591916	−0.295958	−	0.955201i	$-0.595639\pi$
$137$	−6.92820	−0.591916	−0.295958	+	0.955201i	$0.595639\pi$
$138$	15.1460	1.28932
$139$	8.74456	0.741704	0.370852	−	0.928692i	$-0.379066\pi$
$139$	8.74456	0.741704	0.370852	+	0.928692i	$0.379066\pi$
$140$	0	0
$141$	−14.3723	−1.21036
$142$	8.21782	0.689624
$143$	−5.98844	−0.500778
$144$	−16.8614	−1.40512
$145$	0	0
$146$	−12.0000	−0.993127
$147$	−11.3321	−0.934653
$148$	10.0974	0.829997
$149$	−21.1168	−1.72996	−0.864980	−	0.501807i	$-0.832669\pi$
$149$	−21.1168	−1.72996	−0.864980	+	0.501807i	$0.832669\pi$
$150$	0	0
$151$	−9.48913	−0.772214	−0.386107	−	0.922454i	$-0.626180\pi$
$151$	−9.48913	−0.772214	−0.386107	+	0.922454i	$0.626180\pi$
$152$	6.92820	0.561951
$153$	−17.0256	−1.37643
$154$	−17.4891	−1.40931
$155$	0	0
$156$	−2.37228	−0.189935
$157$	−13.8564	−1.10586	−0.552931	−	0.833227i	$-0.686491\pi$
$157$	−13.8564	−1.10586	−0.552931	+	0.833227i	$0.686491\pi$
$158$	−9.74749	−0.775469
$159$	−2.37228	−0.188134
$160$	0	0
$161$	−5.48913	−0.432604
$162$	13.4140	1.05390
$163$	6.28339	0.492153	0.246077	−	0.969250i	$-0.420858\pi$
$163$	6.28339	0.492153	0.246077	+	0.969250i	$0.420858\pi$
$164$	6.74456	0.526662
$165$	0	0
$166$	28.9783	2.24915
$167$	9.80240	0.758532	0.379266	−	0.925288i	$-0.376176\pi$
$167$	9.80240	0.758532	0.379266	+	0.925288i	$0.376176\pi$
$168$	6.92820	0.534522
$169$	−12.1168	−0.932065
$170$	0	0
$171$	13.4891	1.03154
$172$	−5.69349	−0.434124
$173$	−10.6873	−0.812537	−0.406269	−	0.913754i	$-0.633170\pi$
$173$	−10.6873	−0.812537	−0.406269	+	0.913754i	$0.633170\pi$
$174$	4.37228	0.331462
$175$	0	0
$176$	−31.8614	−2.40164
$177$	1.87953	0.141274
$178$	−18.6101	−1.39489
$179$	16.7446	1.25155	0.625774	−	0.780005i	$-0.284783\pi$
$179$	16.7446	1.25155	0.625774	+	0.780005i	$0.284783\pi$
$180$	0	0
$181$	−13.8614	−1.03031	−0.515155	−	0.857097i	$-0.672266\pi$
$181$	−13.8614	−1.03031	−0.515155	+	0.857097i	$0.672266\pi$
$182$	2.57924	0.191186
$183$	15.1460	1.11963
$184$	−6.00000	−0.442326
$185$	0	0
$186$	−10.3723	−0.760533
$187$	−32.1716	−2.35262
$188$	−5.69349	−0.415240
$189$	1.48913	0.108318
$190$	0	0
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	2.52434	0.182178
$193$	−6.92820	−0.498703	−0.249351	−	0.968413i	$-0.580217\pi$
$193$	−6.92820	−0.498703	−0.249351	+	0.968413i	$0.580217\pi$
$194$	−12.0000	−0.861550
$195$	0	0
$196$	−4.48913	−0.320652
$197$	−27.1229	−1.93243	−0.966214	−	0.257743i	$-0.917021\pi$
$197$	−27.1229	−1.93243	−0.966214	+	0.257743i	$0.917021\pi$
$198$	−37.2203	−2.64513
$199$	−9.48913	−0.672666	−0.336333	−	0.941743i	$-0.609187\pi$
$199$	−9.48913	−0.672666	−0.336333	+	0.941743i	$0.609187\pi$
$200$	0	0
$201$	21.4891	1.51573
$202$	25.5383	1.79687
$203$	−1.58457	−0.111215
$204$	−12.7446	−0.892298
$205$	0	0
$206$	6.00000	0.418040
$207$	−11.6819	−0.811950
$208$	4.69882	0.325804
$209$	25.4891	1.76312
$210$	0	0
$211$	−11.1168	−0.765315	−0.382658	−	0.923890i	$-0.624991\pi$
$211$	−11.1168	−0.765315	−0.382658	+	0.923890i	$0.624991\pi$
$212$	−0.939764	−0.0645432
$213$	−11.9769	−0.820642
$214$	2.74456	0.187614
$215$	0	0
$216$	1.62772	0.110752
$217$	3.75906	0.255181
$218$	−1.93443	−0.131016
$219$	17.4891	1.18181
$220$	0	0
$221$	4.74456	0.319154
$222$	−44.1485	−2.96305
$223$	10.3923	0.695920	0.347960	−	0.937509i	$-0.386874\pi$
$223$	10.3923	0.695920	0.347960	+	0.937509i	$0.386874\pi$
$224$	8.23369	0.550136
$225$	0	0
$226$	−15.2554	−1.01478
$227$	−13.5615	−0.900105	−0.450053	−	0.893002i	$-0.648595\pi$
$227$	−13.5615	−0.900105	−0.450053	+	0.893002i	$0.648595\pi$
$228$	10.0974	0.668713
$229$	−1.25544	−0.0829616	−0.0414808	−	0.999139i	$-0.513208\pi$
$229$	−1.25544	−0.0829616	−0.0414808	+	0.999139i	$0.513208\pi$
$230$	0	0
$231$	25.4891	1.67706
$232$	−1.73205	−0.113715
$233$	16.0858	1.05382	0.526908	−	0.849923i	$-0.323351\pi$
$233$	16.0858	1.05382	0.526908	+	0.849923i	$0.323351\pi$
$234$	5.48913	0.358835
$235$	0	0
$236$	0.744563	0.0484669
$237$	14.2063	0.922796
$238$	13.8564	0.898177
$239$	−6.23369	−0.403224	−0.201612	−	0.979465i	$-0.564618\pi$
$239$	−6.23369	−0.403224	−0.201612	+	0.979465i	$0.564618\pi$
$240$	0	0
$241$	12.3723	0.796969	0.398484	−	0.917175i	$-0.369536\pi$
$241$	12.3723	0.796969	0.398484	+	0.917175i	$0.369536\pi$
$242$	−51.2790	−3.29634
$243$	−22.3692	−1.43498
$244$	6.00000	0.384111
$245$	0	0
$246$	−29.4891	−1.88016
$247$	−3.75906	−0.239183
$248$	4.10891	0.260916
$249$	−42.2337	−2.67645
$250$	0	0
$251$	1.62772	0.102741	0.0513703	−	0.998680i	$-0.483641\pi$
$251$	1.62772	0.102741	0.0513703	+	0.998680i	$0.483641\pi$
$252$	5.34363	0.336617
$253$	−22.0742	−1.38779
$254$	6.00000	0.376473
$255$	0	0
$256$	19.0000	1.18750
$257$	12.3267	0.768921	0.384460	−	0.923142i	$-0.374388\pi$
$257$	12.3267	0.768921	0.384460	+	0.923142i	$0.374388\pi$
$258$	24.8935	1.54980
$259$	16.0000	0.994192
$260$	0	0
$261$	−3.37228	−0.208739
$262$	6.92820	0.428026
$263$	−26.4781	−1.63271	−0.816355	−	0.577551i	$-0.804008\pi$
$263$	−26.4781	−1.63271	−0.816355	+	0.577551i	$0.804008\pi$
$264$	27.8614	1.71475
$265$	0	0
$266$	−10.9783	−0.673120
$267$	27.1229	1.65989
$268$	8.51278	0.520001
$269$	0.510875	0.0311486	0.0155743	−	0.999879i	$-0.495042\pi$
$269$	0.510875	0.0311486	0.0155743	+	0.999879i	$0.495042\pi$
$270$	0	0
$271$	19.8614	1.20649	0.603247	−	0.797554i	$-0.293873\pi$
$271$	19.8614	1.20649	0.603247	+	0.797554i	$0.293873\pi$
$272$	25.2434	1.53060
$273$	−3.75906	−0.227508
$274$	12.0000	0.724947
$275$	0	0
$276$	−8.74456	−0.526361
$277$	6.92820	0.416275	0.208138	−	0.978100i	$-0.433260\pi$
$277$	6.92820	0.416275	0.208138	+	0.978100i	$0.433260\pi$
$278$	−15.1460	−0.908398
$279$	8.00000	0.478947
$280$	0	0
$281$	4.37228	0.260828	0.130414	−	0.991460i	$-0.458369\pi$
$281$	4.37228	0.260828	0.130414	+	0.991460i	$0.458369\pi$
$282$	24.8935	1.48239
$283$	−30.5870	−1.81821	−0.909104	−	0.416568i	$-0.863233\pi$
$283$	−30.5870	−1.81821	−0.909104	+	0.416568i	$0.863233\pi$
$284$	−4.74456	−0.281538
$285$	0	0
$286$	10.3723	0.613326
$287$	10.6873	0.630849
$288$	17.5229	1.03255
$289$	8.48913	0.499360
$290$	0	0
$291$	17.4891	1.02523
$292$	6.92820	0.405442
$293$	22.0742	1.28959	0.644795	−	0.764355i	$-0.276943\pi$
$293$	22.0742	1.28959	0.644795	+	0.764355i	$0.276943\pi$
$294$	19.6277	1.14471
$295$	0	0
$296$	17.4891	1.01653
$297$	5.98844	0.347484
$298$	36.5754	2.11876
$299$	3.25544	0.188267
$300$	0	0
$301$	−9.02175	−0.520005
$302$	16.4356	0.945765
$303$	−37.2203	−2.13825
$304$	−20.0000	−1.14708
$305$	0	0
$306$	29.4891	1.68578
$307$	15.7908	0.901231	0.450615	−	0.892718i	$-0.351205\pi$
$307$	15.7908	0.901231	0.450615	+	0.892718i	$0.351205\pi$
$308$	10.0974	0.575350
$309$	−8.74456	−0.497461
$310$	0	0
$311$	−18.9783	−1.07616	−0.538079	−	0.842894i	$-0.680850\pi$
$311$	−18.9783	−1.07616	−0.538079	+	0.842894i	$0.680850\pi$
$312$	−4.10891	−0.232621
$313$	−2.22938	−0.126012	−0.0630061	−	0.998013i	$-0.520069\pi$
$313$	−2.22938	−0.126012	−0.0630061	+	0.998013i	$0.520069\pi$
$314$	24.0000	1.35440
$315$	0	0
$316$	5.62772	0.316584
$317$	15.7359	0.883818	0.441909	−	0.897060i	$-0.354301\pi$
$317$	15.7359	0.883818	0.441909	+	0.897060i	$0.354301\pi$
$318$	4.10891	0.230416
$319$	−6.37228	−0.356779
$320$	0	0
$321$	−4.00000	−0.223258
$322$	9.50744	0.529829
$323$	−20.1947	−1.12366
$324$	−7.74456	−0.430253
$325$	0	0
$326$	−10.8832	−0.602762
$327$	2.81929	0.155907
$328$	11.6819	0.645026
$329$	−9.02175	−0.497385
$330$	0	0
$331$	0.138593	0.00761778	0.00380889	−	0.999993i	$-0.498788\pi$
$331$	0.138593	0.00761778	0.00380889	+	0.999993i	$0.498788\pi$
$332$	−16.7306	−0.918211
$333$	34.0511	1.86599
$334$	−16.9783	−0.929009
$335$	0	0
$336$	−20.0000	−1.09109
$337$	−25.2434	−1.37509	−0.687547	−	0.726140i	$-0.741313\pi$
$337$	−25.2434	−1.37509	−0.687547	+	0.726140i	$0.741313\pi$
$338$	20.9870	1.14154
$339$	22.2337	1.20757
$340$	0	0
$341$	15.1168	0.818623
$342$	−23.3639	−1.26337
$343$	−18.2054	−0.982998
$344$	−9.86141	−0.531691
$345$	0	0
$346$	18.5109	0.995151
$347$	−19.8997	−1.06827	−0.534137	−	0.845398i	$-0.679363\pi$
$347$	−19.8997	−1.06827	−0.534137	+	0.845398i	$0.679363\pi$
$348$	−2.52434	−0.135319
$349$	−1.86141	−0.0996388	−0.0498194	−	0.998758i	$-0.515865\pi$
$349$	−1.86141	−0.0996388	−0.0498194	+	0.998758i	$0.515865\pi$
$350$	0	0
$351$	−0.883156	−0.0471394
$352$	33.1113	1.76484
$353$	23.9538	1.27493	0.637465	−	0.770479i	$-0.279983\pi$
$353$	23.9538	1.27493	0.637465	+	0.770479i	$0.279983\pi$
$354$	−3.25544	−0.173025
$355$	0	0
$356$	10.7446	0.569461
$357$	−20.1947	−1.06882
$358$	−29.0024	−1.53283
$359$	−31.1168	−1.64228	−0.821142	−	0.570724i	$-0.806663\pi$
$359$	−31.1168	−1.64228	−0.821142	+	0.570724i	$0.806663\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	24.0087	1.26187
$363$	74.7355	3.92259
$364$	−1.48913	−0.0780514
$365$	0	0
$366$	−26.2337	−1.37126
$367$	10.3923	0.542474	0.271237	−	0.962513i	$-0.412567\pi$
$367$	10.3923	0.542474	0.271237	+	0.962513i	$0.412567\pi$
$368$	17.3205	0.902894
$369$	22.7446	1.18403
$370$	0	0
$371$	−1.48913	−0.0773115
$372$	5.98844	0.310486
$373$	5.39853	0.279525	0.139763	−	0.990185i	$-0.455366\pi$
$373$	5.39853	0.279525	0.139763	+	0.990185i	$0.455366\pi$
$374$	55.7228	2.88136
$375$	0	0
$376$	−9.86141	−0.508563
$377$	0.939764	0.0484003
$378$	−2.57924	−0.132662
$379$	−21.4891	−1.10382	−0.551911	−	0.833903i	$-0.686101\pi$
$379$	−21.4891	−1.10382	−0.551911	+	0.833903i	$0.686101\pi$
$380$	0	0
$381$	−8.74456	−0.447998
$382$	27.7128	1.41791
$383$	18.6101	0.950933	0.475467	−	0.879734i	$-0.342279\pi$
$383$	18.6101	0.950933	0.475467	+	0.879734i	$0.342279\pi$
$384$	−30.6060	−1.56185
$385$	0	0
$386$	12.0000	0.610784
$387$	−19.2000	−0.975993
$388$	6.92820	0.351726
$389$	−2.74456	−0.139155	−0.0695774	−	0.997577i	$-0.522165\pi$
$389$	−2.74456	−0.139155	−0.0695774	+	0.997577i	$0.522165\pi$
$390$	0	0
$391$	17.4891	0.884463
$392$	−7.77539	−0.392717
$393$	−10.0974	−0.509344
$394$	46.9783	2.36673
$395$	0	0
$396$	21.4891	1.07987
$397$	−7.27806	−0.365275	−0.182638	−	0.983180i	$-0.558464\pi$
$397$	−7.27806	−0.365275	−0.182638	+	0.983180i	$0.558464\pi$
$398$	16.4356	0.823845
$399$	16.0000	0.801002
$400$	0	0
$401$	−13.1168	−0.655024	−0.327512	−	0.944847i	$-0.606210\pi$
$401$	−13.1168	−0.655024	−0.327512	+	0.944847i	$0.606210\pi$
$402$	−37.2203	−1.85638
$403$	−2.22938	−0.111054
$404$	−14.7446	−0.733569
$405$	0	0
$406$	2.74456	0.136210
$407$	64.3432	3.18937
$408$	−22.0742	−1.09284
$409$	−30.4674	−1.50651	−0.753257	−	0.657726i	$-0.771519\pi$
$409$	−30.4674	−1.50651	−0.753257	+	0.657726i	$0.771519\pi$
$410$	0	0
$411$	−17.4891	−0.862675
$412$	−3.46410	−0.170664
$413$	1.17981	0.0580549
$414$	20.2337	0.994432
$415$	0	0
$416$	−4.88316	−0.239416
$417$	22.0742	1.08098
$418$	−44.1485	−2.15937
$419$	24.7446	1.20885	0.604425	−	0.796662i	$-0.293403\pi$
$419$	24.7446	1.20885	0.604425	+	0.796662i	$0.293403\pi$
$420$	0	0
$421$	−20.9783	−1.02242	−0.511209	−	0.859457i	$-0.670802\pi$
$421$	−20.9783	−1.02242	−0.511209	+	0.859457i	$0.670802\pi$
$422$	19.2549	0.937316
$423$	−19.2000	−0.933538
$424$	−1.62772	−0.0790490
$425$	0	0
$426$	20.7446	1.00508
$427$	9.50744	0.460097
$428$	−1.58457	−0.0765933
$429$	−15.1168	−0.729848
$430$	0	0
$431$	−11.2554	−0.542155	−0.271078	−	0.962557i	$-0.587380\pi$
$431$	−11.2554	−0.542155	−0.271078	+	0.962557i	$0.587380\pi$
$432$	−4.69882	−0.226072
$433$	14.4463	0.694246	0.347123	−	0.937820i	$-0.387159\pi$
$433$	14.4463	0.694246	0.347123	+	0.937820i	$0.387159\pi$
$434$	−6.51087	−0.312532
$435$	0	0
$436$	1.11684	0.0534871
$437$	−13.8564	−0.662842
$438$	−30.2921	−1.44741
$439$	30.2337	1.44298	0.721488	−	0.692427i	$-0.243459\pi$
$439$	30.2337	1.44298	0.721488	+	0.692427i	$0.243459\pi$
$440$	0	0
$441$	−15.1386	−0.720885
$442$	−8.21782	−0.390882
$443$	−7.22316	−0.343183	−0.171591	−	0.985168i	$-0.554891\pi$
$443$	−7.22316	−0.343183	−0.171591	+	0.985168i	$0.554891\pi$
$444$	25.4891	1.20966
$445$	0	0
$446$	−18.0000	−0.852325
$447$	−53.3060	−2.52129
$448$	1.58457	0.0748641
$449$	34.4674	1.62662	0.813308	−	0.581833i	$-0.197664\pi$
$449$	34.4674	1.62662	0.813308	+	0.581833i	$0.197664\pi$
$450$	0	0
$451$	42.9783	2.02377
$452$	8.80773	0.414281
$453$	−23.9538	−1.12545
$454$	23.4891	1.10240
$455$	0	0
$456$	17.4891	0.819003
$457$	2.57924	0.120652	0.0603259	−	0.998179i	$-0.480786\pi$
$457$	2.57924	0.120652	0.0603259	+	0.998179i	$0.480786\pi$
$458$	2.17448	0.101607
$459$	−4.74456	−0.221457
$460$	0	0
$461$	4.51087	0.210092	0.105046	−	0.994467i	$-0.466501\pi$
$461$	4.51087	0.210092	0.105046	+	0.994467i	$0.466501\pi$
$462$	−44.1485	−2.05397
$463$	20.4897	0.952235	0.476118	−	0.879382i	$-0.342044\pi$
$463$	20.4897	0.952235	0.476118	+	0.879382i	$0.342044\pi$
$464$	5.00000	0.232119
$465$	0	0
$466$	−27.8614	−1.29065
$467$	39.7446	1.83916	0.919580	−	0.392902i	$-0.128529\pi$
$467$	39.7446	1.83916	0.919580	+	0.392902i	$0.128529\pi$
$468$	−3.16915	−0.146494
$469$	13.4891	0.622870
$470$	0	0
$471$	−34.9783	−1.61171
$472$	1.28962	0.0593596
$473$	−36.2805	−1.66818
$474$	−24.6060	−1.13019
$475$	0	0
$476$	−8.00000	−0.366679
$477$	−3.16915	−0.145105
$478$	10.7971	0.493846
$479$	−10.3723	−0.473922	−0.236961	−	0.971519i	$-0.576151\pi$
$479$	−10.3723	−0.473922	−0.236961	+	0.971519i	$0.576151\pi$
$480$	0	0
$481$	−9.48913	−0.432667
$482$	−21.4294	−0.976083
$483$	−13.8564	−0.630488
$484$	29.6060	1.34573
$485$	0	0
$486$	38.7446	1.75749
$487$	9.10268	0.412482	0.206241	−	0.978501i	$-0.433877\pi$
$487$	9.10268	0.412482	0.206241	+	0.978501i	$0.433877\pi$
$488$	10.3923	0.470438
$489$	15.8614	0.717278
$490$	0	0
$491$	−17.6277	−0.795528	−0.397764	−	0.917488i	$-0.630214\pi$
$491$	−17.6277	−0.795528	−0.397764	+	0.917488i	$0.630214\pi$
$492$	17.0256	0.767571
$493$	5.04868	0.227381
$494$	6.51087	0.292938
$495$	0	0
$496$	−11.8614	−0.532593
$497$	−7.51811	−0.337233
$498$	73.1509	3.27797
$499$	16.7446	0.749590	0.374795	−	0.927108i	$-0.377713\pi$
$499$	16.7446	0.749590	0.374795	+	0.927108i	$0.377713\pi$
$500$	0	0
$501$	24.7446	1.10551
$502$	−2.81929	−0.125831
$503$	39.7446	1.77212	0.886062	−	0.463567i	$-0.153431\pi$
$503$	39.7446	1.77212	0.886062	+	0.463567i	$0.153431\pi$
$504$	9.25544	0.412270
$505$	0	0
$506$	38.2337	1.69969
$507$	−30.5870	−1.35842
$508$	−3.46410	−0.153695
$509$	5.86141	0.259802	0.129901	−	0.991527i	$-0.458534\pi$
$509$	5.86141	0.259802	0.129901	+	0.991527i	$0.458534\pi$
$510$	0	0
$511$	10.9783	0.485649
$512$	−8.66025	−0.382733
$513$	3.75906	0.165966
$514$	−21.3505	−0.941732
$515$	0	0
$516$	−14.3723	−0.632704
$517$	−36.2805	−1.59561
$518$	−27.7128	−1.21763
$519$	−26.9783	−1.18421
$520$	0	0
$521$	−25.8614	−1.13301	−0.566504	−	0.824059i	$-0.691705\pi$
$521$	−25.8614	−1.13301	−0.566504	+	0.824059i	$0.691705\pi$
$522$	5.84096	0.255652
$523$	−4.75372	−0.207866	−0.103933	−	0.994584i	$-0.533143\pi$
$523$	−4.75372	−0.207866	−0.103933	+	0.994584i	$0.533143\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	45.8614	1.99965
$527$	−11.9769	−0.521721
$528$	−80.4290	−3.50022
$529$	−11.0000	−0.478261
$530$	0	0
$531$	2.51087	0.108963
$532$	6.33830	0.274800
$533$	−6.33830	−0.274542
$534$	−46.9783	−2.03295
$535$	0	0
$536$	14.7446	0.636868
$537$	42.2689	1.82404
$538$	−0.884861	−0.0381491
$539$	−28.6060	−1.23215
$540$	0	0
$541$	34.7446	1.49379	0.746893	−	0.664945i	$-0.231545\pi$
$541$	34.7446	1.49379	0.746893	+	0.664945i	$0.231545\pi$
$542$	−34.4010	−1.47765
$543$	−34.9909	−1.50160
$544$	−26.2337	−1.12476
$545$	0	0
$546$	6.51087	0.278640
$547$	4.75372	0.203254	0.101627	−	0.994823i	$-0.467595\pi$
$547$	4.75372	0.203254	0.101627	+	0.994823i	$0.467595\pi$
$548$	−6.92820	−0.295958
$549$	20.2337	0.863553
$550$	0	0
$551$	−4.00000	−0.170406
$552$	−15.1460	−0.644658
$553$	8.91754	0.379212
$554$	−12.0000	−0.509831
$555$	0	0
$556$	8.74456	0.370852
$557$	−30.8820	−1.30851	−0.654255	−	0.756274i	$-0.727018\pi$
$557$	−30.8820	−1.30851	−0.654255	+	0.756274i	$0.727018\pi$
$558$	−13.8564	−0.586588
$559$	5.35053	0.226303
$560$	0	0
$561$	−81.2119	−3.42877
$562$	−7.57301	−0.319448
$563$	−1.23472	−0.0520371	−0.0260186	−	0.999661i	$-0.508283\pi$
$563$	−1.23472	−0.0520371	−0.0260186	+	0.999661i	$0.508283\pi$
$564$	−14.3723	−0.605182
$565$	0	0
$566$	52.9783	2.22684
$567$	−12.2718	−0.515369
$568$	−8.21782	−0.344812
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	0	0
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	−5.98844	−0.250389
$573$	−40.3894	−1.68729
$574$	−18.5109	−0.772629
$575$	0	0
$576$	3.37228	0.140512
$577$	8.80773	0.366671	0.183335	−	0.983050i	$-0.441311\pi$
$577$	8.80773	0.366671	0.183335	+	0.983050i	$0.441311\pi$
$578$	−14.7036	−0.611589
$579$	−17.4891	−0.726823
$580$	0	0
$581$	−26.5109	−1.09986
$582$	−30.2921	−1.25565
$583$	−5.98844	−0.248016
$584$	12.0000	0.496564
$585$	0	0
$586$	−38.2337	−1.57942
$587$	36.9253	1.52407	0.762035	−	0.647536i	$-0.224200\pi$
$587$	36.9253	1.52407	0.762035	+	0.647536i	$0.224200\pi$
$588$	−11.3321	−0.467327
$589$	9.48913	0.390993
$590$	0	0
$591$	−68.4674	−2.81637
$592$	−50.4868	−2.07499
$593$	−14.2063	−0.583381	−0.291691	−	0.956513i	$-0.594218\pi$
$593$	−14.2063	−0.583381	−0.291691	+	0.956513i	$0.594218\pi$
$594$	−10.3723	−0.425580
$595$	0	0
$596$	−21.1168	−0.864980
$597$	−23.9538	−0.980362
$598$	−5.63858	−0.230579
$599$	−21.3505	−0.872359	−0.436180	−	0.899860i	$-0.643669\pi$
$599$	−21.3505	−0.872359	−0.436180	+	0.899860i	$0.643669\pi$
$600$	0	0
$601$	−18.7446	−0.764607	−0.382303	−	0.924037i	$-0.624869\pi$
$601$	−18.7446	−0.764607	−0.382303	+	0.924037i	$0.624869\pi$
$602$	15.6261	0.636873
$603$	28.7075	1.16906
$604$	−9.48913	−0.386107
$605$	0	0
$606$	64.4674	2.61881
$607$	−25.8882	−1.05077	−0.525385	−	0.850865i	$-0.676079\pi$
$607$	−25.8882	−1.05077	−0.525385	+	0.850865i	$0.676079\pi$
$608$	20.7846	0.842927
$609$	−4.00000	−0.162088
$610$	0	0
$611$	5.35053	0.216459
$612$	−17.0256	−0.688217
$613$	39.4496	1.59336	0.796678	−	0.604404i	$-0.206589\pi$
$613$	39.4496	1.59336	0.796678	+	0.604404i	$0.206589\pi$
$614$	−27.3505	−1.10378
$615$	0	0
$616$	17.4891	0.704657
$617$	24.5437	0.988091	0.494045	−	0.869436i	$-0.335518\pi$
$617$	24.5437	0.988091	0.494045	+	0.869436i	$0.335518\pi$
$618$	15.1460	0.609263
$619$	25.6277	1.03006	0.515032	−	0.857171i	$-0.327780\pi$
$619$	25.6277	1.03006	0.515032	+	0.857171i	$0.327780\pi$
$620$	0	0
$621$	−3.25544	−0.130636
$622$	32.8713	1.31802
$623$	17.0256	0.682114
$624$	11.8614	0.474836
$625$	0	0
$626$	3.86141	0.154333
$627$	64.3432	2.56962
$628$	−13.8564	−0.552931
$629$	−50.9783	−2.03264
$630$	0	0
$631$	6.23369	0.248159	0.124080	−	0.992272i	$-0.460402\pi$
$631$	6.23369	0.248159	0.124080	+	0.992272i	$0.460402\pi$
$632$	9.74749	0.387735
$633$	−28.0627	−1.11539
$634$	−27.2554	−1.08245
$635$	0	0
$636$	−2.37228	−0.0940671
$637$	4.21872	0.167152
$638$	11.0371	0.436964
$639$	−16.0000	−0.632950
$640$	0	0
$641$	3.48913	0.137812	0.0689061	−	0.997623i	$-0.478049\pi$
$641$	3.48913	0.137812	0.0689061	+	0.997623i	$0.478049\pi$
$642$	6.92820	0.273434
$643$	−4.75372	−0.187468	−0.0937342	−	0.995597i	$-0.529880\pi$
$643$	−4.75372	−0.187468	−0.0937342	+	0.995597i	$0.529880\pi$
$644$	−5.48913	−0.216302
$645$	0	0
$646$	34.9783	1.37620
$647$	27.4179	1.07791	0.538954	−	0.842335i	$-0.318820\pi$
$647$	27.4179	1.07791	0.538954	+	0.842335i	$0.318820\pi$
$648$	−13.4140	−0.526951
$649$	4.74456	0.186240
$650$	0	0
$651$	9.48913	0.371908
$652$	6.28339	0.246077
$653$	17.6155	0.689346	0.344673	−	0.938723i	$-0.387990\pi$
$653$	17.6155	0.689346	0.344673	+	0.938723i	$0.387990\pi$
$654$	−4.88316	−0.190947
$655$	0	0
$656$	−33.7228	−1.31665
$657$	23.3639	0.911511
$658$	15.6261	0.609170
$659$	−41.3505	−1.61079	−0.805394	−	0.592740i	$-0.798046\pi$
$659$	−41.3505	−1.61079	−0.805394	+	0.592740i	$0.798046\pi$
$660$	0	0
$661$	−16.5109	−0.642199	−0.321099	−	0.947046i	$-0.604052\pi$
$661$	−16.5109	−0.642199	−0.321099	+	0.947046i	$0.604052\pi$
$662$	−0.240051	−0.00932984
$663$	11.9769	0.465143
$664$	−28.9783	−1.12457
$665$	0	0
$666$	−58.9783	−2.28536
$667$	3.46410	0.134131
$668$	9.80240	0.379266
$669$	26.2337	1.01425
$670$	0	0
$671$	38.2337	1.47600
$672$	20.7846	0.801784
$673$	3.40920	0.131415	0.0657075	−	0.997839i	$-0.479070\pi$
$673$	3.40920	0.131415	0.0657075	+	0.997839i	$0.479070\pi$
$674$	43.7228	1.68414
$675$	0	0
$676$	−12.1168	−0.466032
$677$	−0.699713	−0.0268922	−0.0134461	−	0.999910i	$-0.504280\pi$
$677$	−0.699713	−0.0268922	−0.0134461	+	0.999910i	$0.504280\pi$
$678$	−38.5099	−1.47896
$679$	10.9783	0.421307
$680$	0	0
$681$	−34.2337	−1.31184
$682$	−26.1831	−1.00260
$683$	31.8766	1.21973	0.609863	−	0.792507i	$-0.291225\pi$
$683$	31.8766	1.21973	0.609863	+	0.792507i	$0.291225\pi$
$684$	13.4891	0.515770
$685$	0	0
$686$	31.5326	1.20392
$687$	−3.16915	−0.120911
$688$	28.4674	1.08531
$689$	0.883156	0.0336456
$690$	0	0
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	−10.6873	−0.406269
$693$	34.0511	1.29349
$694$	34.4674	1.30836
$695$	0	0
$696$	−4.37228	−0.165731
$697$	−34.0511	−1.28978
$698$	3.22405	0.122032
$699$	40.6060	1.53586
$700$	0	0
$701$	−17.1168	−0.646494	−0.323247	−	0.946315i	$-0.604775\pi$
$701$	−17.1168	−0.646494	−0.323247	+	0.946315i	$0.604775\pi$
$702$	1.52967	0.0577337
$703$	40.3894	1.52332
$704$	6.37228	0.240164
$705$	0	0
$706$	−41.4891	−1.56146
$707$	−23.3639	−0.878688
$708$	1.87953	0.0706370
$709$	28.0951	1.05513	0.527567	−	0.849514i	$-0.323104\pi$
$709$	28.0951	1.05513	0.527567	+	0.849514i	$0.323104\pi$
$710$	0	0
$711$	18.9783	0.711740
$712$	18.6101	0.697444
$713$	−8.21782	−0.307760
$714$	34.9783	1.30903
$715$	0	0
$716$	16.7446	0.625774
$717$	−15.7359	−0.587669
$718$	53.8960	2.01138
$719$	3.25544	0.121407	0.0607037	−	0.998156i	$-0.480666\pi$
$719$	3.25544	0.121407	0.0607037	+	0.998156i	$0.480666\pi$
$720$	0	0
$721$	−5.48913	−0.204426
$722$	5.19615	0.193381
$723$	31.2318	1.16152
$724$	−13.8614	−0.515155
$725$	0	0
$726$	−129.446	−4.80418
$727$	0.294954	0.0109392	0.00546961	−	0.999985i	$-0.498259\pi$
$727$	0.294954	0.0109392	0.00546961	+	0.999985i	$0.498259\pi$
$728$	−2.57924	−0.0955930
$729$	−33.2337	−1.23088
$730$	0	0
$731$	28.7446	1.06316
$732$	15.1460	0.559813
$733$	−27.7128	−1.02360	−0.511798	−	0.859106i	$-0.671020\pi$
$733$	−27.7128	−1.02360	−0.511798	+	0.859106i	$0.671020\pi$
$734$	−18.0000	−0.664392
$735$	0	0
$736$	−18.0000	−0.663489
$737$	54.2458	1.99817
$738$	−39.3947	−1.45014
$739$	9.62772	0.354161	0.177081	−	0.984196i	$-0.443335\pi$
$739$	9.62772	0.354161	0.177081	+	0.984196i	$0.443335\pi$
$740$	0	0
$741$	−9.48913	−0.348592
$742$	2.57924	0.0946869
$743$	17.3205	0.635428	0.317714	−	0.948187i	$-0.397085\pi$
$743$	17.3205	0.635428	0.317714	+	0.948187i	$0.397085\pi$
$744$	10.3723	0.380266
$745$	0	0
$746$	−9.35053	−0.342347
$747$	−56.4203	−2.06431
$748$	−32.1716	−1.17631
$749$	−2.51087	−0.0917454
$750$	0	0
$751$	−20.4674	−0.746865	−0.373433	−	0.927657i	$-0.621819\pi$
$751$	−20.4674	−0.746865	−0.373433	+	0.927657i	$0.621819\pi$
$752$	28.4674	1.03810
$753$	4.10891	0.149737
$754$	−1.62772	−0.0592780
$755$	0	0
$756$	1.48913	0.0541590
$757$	29.5923	1.07555	0.537776	−	0.843088i	$-0.319265\pi$
$757$	29.5923	1.07555	0.537776	+	0.843088i	$0.319265\pi$
$758$	37.2203	1.35190
$759$	−55.7228	−2.02261
$760$	0	0
$761$	38.4674	1.39444	0.697221	−	0.716857i	$-0.254420\pi$
$761$	38.4674	1.39444	0.697221	+	0.716857i	$0.254420\pi$
$762$	15.1460	0.548683
$763$	1.76972	0.0640682
$764$	−16.0000	−0.578860
$765$	0	0
$766$	−32.2337	−1.16465
$767$	−0.699713	−0.0252652
$768$	47.9624	1.73069
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	0	0
$771$	31.1168	1.12065
$772$	−6.92820	−0.249351
$773$	−51.6666	−1.85832	−0.929159	−	0.369681i	$-0.879467\pi$
$773$	−51.6666	−1.85832	−0.929159	+	0.369681i	$0.879467\pi$
$774$	33.2554	1.19534
$775$	0	0
$776$	12.0000	0.430775
$777$	40.3894	1.44896
$778$	4.75372	0.170429
$779$	26.9783	0.966596
$780$	0	0
$781$	−30.2337	−1.08185
$782$	−30.2921	−1.08324
$783$	−0.939764	−0.0335844
$784$	22.4456	0.801630
$785$	0	0
$786$	17.4891	0.623816
$787$	−13.5615	−0.483414	−0.241707	−	0.970349i	$-0.577707\pi$
$787$	−13.5615	−0.483414	−0.241707	+	0.970349i	$0.577707\pi$
$788$	−27.1229	−0.966214
$789$	−66.8397	−2.37955
$790$	0	0
$791$	13.9565	0.496236
$792$	37.2203	1.32256
$793$	−5.63858	−0.200232
$794$	12.6060	0.447369
$795$	0	0
$796$	−9.48913	−0.336333
$797$	−4.45877	−0.157938	−0.0789688	−	0.996877i	$-0.525163\pi$
$797$	−4.45877	−0.157938	−0.0789688	+	0.996877i	$0.525163\pi$
$798$	−27.7128	−0.981023
$799$	28.7446	1.01691
$800$	0	0
$801$	36.2337	1.28025
$802$	22.7190	0.802237
$803$	44.1485	1.55797
$804$	21.4891	0.757863
$805$	0	0
$806$	3.86141	0.136012
$807$	1.28962	0.0453968
$808$	−25.5383	−0.898435
$809$	1.25544	0.0441388	0.0220694	−	0.999756i	$-0.492975\pi$
$809$	1.25544	0.0441388	0.0220694	+	0.999756i	$0.492975\pi$
$810$	0	0
$811$	−20.0000	−0.702295	−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000	−0.702295	−0.351147	+	0.936320i	$0.614208\pi$
$812$	−1.58457	−0.0556076
$813$	50.1369	1.75838
$814$	−111.446	−3.90617
$815$	0	0
$816$	63.7228	2.23074
$817$	−22.7739	−0.796759
$818$	52.7710	1.84510
$819$	−5.02175	−0.175474
$820$	0	0
$821$	11.6277	0.405810	0.202905	−	0.979198i	$-0.434962\pi$
$821$	11.6277	0.405810	0.202905	+	0.979198i	$0.434962\pi$
$822$	30.2921	1.05656
$823$	−33.1662	−1.15610	−0.578051	−	0.816000i	$-0.696187\pi$
$823$	−33.1662	−1.15610	−0.578051	+	0.816000i	$0.696187\pi$
$824$	−6.00000	−0.209020
$825$	0	0
$826$	−2.04350	−0.0711024
$827$	33.4063	1.16165	0.580825	−	0.814028i	$-0.302730\pi$
$827$	33.4063	1.16165	0.580825	+	0.814028i	$0.302730\pi$
$828$	−11.6819	−0.405975
$829$	26.0000	0.903017	0.451509	−	0.892267i	$-0.350886\pi$
$829$	26.0000	0.903017	0.451509	+	0.892267i	$0.350886\pi$
$830$	0	0
$831$	17.4891	0.606691
$832$	−0.939764	−0.0325804
$833$	22.6641	0.785266
$834$	−38.2337	−1.32392
$835$	0	0
$836$	25.4891	0.881560
$837$	2.22938	0.0770588
$838$	−42.8588	−1.48053
$839$	−13.6277	−0.470481	−0.235241	−	0.971937i	$-0.575588\pi$
$839$	−13.6277	−0.470481	−0.235241	+	0.971937i	$0.575588\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	36.3354	1.25220
$843$	11.0371	0.380138
$844$	−11.1168	−0.382658
$845$	0	0
$846$	33.2554	1.14335
$847$	46.9128	1.61194
$848$	4.69882	0.161358
$849$	−77.2119	−2.64991
$850$	0	0
$851$	−34.9783	−1.19904
$852$	−11.9769	−0.410321
$853$	−5.63858	−0.193061	−0.0965307	−	0.995330i	$-0.530775\pi$
$853$	−5.63858	−0.193061	−0.0965307	+	0.995330i	$0.530775\pi$
$854$	−16.4674	−0.563502
$855$	0	0
$856$	−2.74456	−0.0938072
$857$	−29.9422	−1.02281	−0.511403	−	0.859341i	$-0.670874\pi$
$857$	−29.9422	−1.02281	−0.511403	+	0.859341i	$0.670874\pi$
$858$	26.1831	0.893878
$859$	11.3940	0.388759	0.194380	−	0.980926i	$-0.437731\pi$
$859$	11.3940	0.388759	0.194380	+	0.980926i	$0.437731\pi$
$860$	0	0
$861$	26.9783	0.919416
$862$	19.4950	0.664002
$863$	6.63325	0.225798	0.112899	−	0.993606i	$-0.463986\pi$
$863$	6.63325	0.225798	0.112899	+	0.993606i	$0.463986\pi$
$864$	4.88316	0.166128
$865$	0	0
$866$	−25.0217	−0.850274
$867$	21.4294	0.727781
$868$	3.75906	0.127591
$869$	35.8614	1.21651
$870$	0	0
$871$	−8.00000	−0.271070
$872$	1.93443	0.0655081
$873$	23.3639	0.790747
$874$	24.0000	0.811812
$875$	0	0
$876$	17.4891	0.590903
$877$	−41.3292	−1.39559	−0.697793	−	0.716299i	$-0.745835\pi$
$877$	−41.3292	−1.39559	−0.697793	+	0.716299i	$0.745835\pi$
$878$	−52.3663	−1.76728
$879$	55.7228	1.87948
$880$	0	0
$881$	−8.97825	−0.302485	−0.151242	−	0.988497i	$-0.548327\pi$
$881$	−8.97825	−0.302485	−0.151242	+	0.988497i	$0.548327\pi$
$882$	26.2208	0.882901
$883$	−18.6101	−0.626281	−0.313140	−	0.949707i	$-0.601381\pi$
$883$	−18.6101	−0.626281	−0.313140	+	0.949707i	$0.601381\pi$
$884$	4.74456	0.159577
$885$	0	0
$886$	12.5109	0.420311
$887$	−32.2265	−1.08206	−0.541030	−	0.841003i	$-0.681965\pi$
$887$	−32.2265	−1.08206	−0.541030	+	0.841003i	$0.681965\pi$
$888$	44.1485	1.48153
$889$	−5.48913	−0.184099
$890$	0	0
$891$	−49.3505	−1.65331
$892$	10.3923	0.347960
$893$	−22.7739	−0.762101
$894$	92.3288	3.08794
$895$	0	0
$896$	−19.2119	−0.641826
$897$	8.21782	0.274385
$898$	−59.6992	−1.99219
$899$	−2.37228	−0.0791200
$900$	0	0
$901$	4.74456	0.158064
$902$	−74.4405	−2.47860
$903$	−22.7739	−0.757869
$904$	15.2554	0.507388
$905$	0	0
$906$	41.4891	1.37838
$907$	−33.1662	−1.10127	−0.550634	−	0.834747i	$-0.685614\pi$
$907$	−33.1662	−1.10127	−0.550634	+	0.834747i	$0.685614\pi$
$908$	−13.5615	−0.450053
$909$	−49.7228	−1.64920
$910$	0	0
$911$	−8.88316	−0.294312	−0.147156	−	0.989113i	$-0.547012\pi$
$911$	−8.88316	−0.294312	−0.147156	+	0.989113i	$0.547012\pi$
$912$	−50.4868	−1.67178
$913$	−106.612	−3.52835
$914$	−4.46738	−0.147768
$915$	0	0
$916$	−1.25544	−0.0414808
$917$	−6.33830	−0.209309
$918$	8.21782	0.271229
$919$	17.7663	0.586057	0.293028	−	0.956104i	$-0.405337\pi$
$919$	17.7663	0.586057	0.293028	+	0.956104i	$0.405337\pi$
$920$	0	0
$921$	39.8614	1.31348
$922$	−7.81306	−0.257310
$923$	4.45877	0.146762
$924$	25.4891	0.838531
$925$	0	0
$926$	−35.4891	−1.16625
$927$	−11.6819	−0.383685
$928$	−5.19615	−0.170572
$929$	−24.5109	−0.804176	−0.402088	−	0.915601i	$-0.631715\pi$
$929$	−24.5109	−0.804176	−0.402088	+	0.915601i	$0.631715\pi$
$930$	0	0
$931$	−17.9565	−0.588501
$932$	16.0858	0.526908
$933$	−47.9075	−1.56842
$934$	−68.8397	−2.25250
$935$	0	0
$936$	−5.48913	−0.179418
$937$	21.3745	0.698275	0.349138	−	0.937071i	$-0.386475\pi$
$937$	21.3745	0.698275	0.349138	+	0.937071i	$0.386475\pi$
$938$	−23.3639	−0.762857
$939$	−5.62772	−0.183654
$940$	0	0
$941$	19.6277	0.639845	0.319923	−	0.947444i	$-0.396343\pi$
$941$	19.6277	0.639845	0.319923	+	0.947444i	$0.396343\pi$
$942$	60.5841	1.97394
$943$	−23.3639	−0.760832
$944$	−3.72281	−0.121167
$945$	0	0
$946$	62.8397	2.04309
$947$	−41.0342	−1.33343	−0.666716	−	0.745311i	$-0.732301\pi$
$947$	−41.0342	−1.33343	−0.666716	+	0.745311i	$0.732301\pi$
$948$	14.2063	0.461398
$949$	−6.51087	−0.211352
$950$	0	0
$951$	39.7228	1.28810
$952$	−13.8564	−0.449089
$953$	40.0395	1.29701	0.648504	−	0.761211i	$-0.275395\pi$
$953$	40.0395	1.29701	0.648504	+	0.761211i	$0.275395\pi$
$954$	5.48913	0.177717
$955$	0	0
$956$	−6.23369	−0.201612
$957$	−16.0858	−0.519980
$958$	17.9653	0.580433
$959$	−10.9783	−0.354506
$960$	0	0
$961$	−25.3723	−0.818461
$962$	16.4356	0.529907
$963$	−5.34363	−0.172196
$964$	12.3723	0.398484
$965$	0	0
$966$	24.0000	0.772187
$967$	−4.99377	−0.160589	−0.0802945	−	0.996771i	$-0.525586\pi$
$967$	−4.99377	−0.160589	−0.0802945	+	0.996771i	$0.525586\pi$
$968$	51.2790	1.64817
$969$	−50.9783	−1.63766
$970$	0	0
$971$	38.9783	1.25087	0.625436	−	0.780276i	$-0.284921\pi$
$971$	38.9783	1.25087	0.625436	+	0.780276i	$0.284921\pi$
$972$	−22.3692	−0.717492
$973$	13.8564	0.444216
$974$	−15.7663	−0.505185
$975$	0	0
$976$	−30.0000	−0.960277
$977$	8.56768	0.274104	0.137052	−	0.990564i	$-0.456237\pi$
$977$	8.56768	0.274104	0.137052	+	0.990564i	$0.456237\pi$
$978$	−27.4728	−0.878482
$979$	68.4674	2.18823
$980$	0	0
$981$	3.76631	0.120249
$982$	30.5321	0.974319
$983$	56.1802	1.79187	0.895936	−	0.444184i	$-0.146506\pi$
$983$	56.1802	1.79187	0.895936	+	0.444184i	$0.146506\pi$
$984$	29.4891	0.940079
$985$	0	0
$986$	−8.74456	−0.278484
$987$	−22.7739	−0.724903
$988$	−3.75906	−0.119591
$989$	19.7228	0.627149
$990$	0	0
$991$	28.7446	0.913101	0.456551	−	0.889697i	$-0.349085\pi$
$991$	28.7446	0.913101	0.456551	+	0.889697i	$0.349085\pi$
$992$	12.3267	0.391374
$993$	0.349857	0.0111024
$994$	13.0217	0.413025
$995$	0	0
$996$	−42.2337	−1.33823
$997$	2.57924	0.0816854	0.0408427	−	0.999166i	$-0.486996\pi$
$997$	2.57924	0.0816854	0.0408427	+	0.999166i	$0.486996\pi$
$998$	−29.0024	−0.918056
$999$	9.48913	0.300223

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	725.2.a.g.1.2		4
3.2	odd	2		6525.2.a.bk.1.4		4
5.2	odd	4		145.2.b.a.59.1	✓	4
5.3	odd	4		145.2.b.a.59.4	yes	4
5.4	even	2	inner	725.2.a.g.1.3		4
15.2	even	4		1305.2.c.e.784.4		4
15.8	even	4		1305.2.c.e.784.2		4
15.14	odd	2		6525.2.a.bk.1.1		4
20.3	even	4		2320.2.d.c.929.1		4
20.7	even	4		2320.2.d.c.929.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.b.a.59.1	✓	4	5.2	odd	4
145.2.b.a.59.4	yes	4	5.3	odd	4
725.2.a.g.1.2		4	1.1	even	1	trivial
725.2.a.g.1.3		4	5.4	even	2	inner
1305.2.c.e.784.2		4	15.8	even	4
1305.2.c.e.784.4		4	15.2	even	4
2320.2.d.c.929.1		4	20.3	even	4
2320.2.d.c.929.4		4	20.7	even	4
6525.2.a.bk.1.1		4	15.14	odd	2
6525.2.a.bk.1.4		4	3.2	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:	\( N \)	\(=\)	\( 725 = 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	725.a (trivial)

\(f(q)\)	\(=\)	\(q-1.73205 q^{2} +2.52434 q^{3} +1.00000 q^{4} -4.37228 q^{6} +1.58457 q^{7} +1.73205 q^{8} +3.37228 q^{9} +6.37228 q^{11} +2.52434 q^{12} -0.939764 q^{13} -2.74456 q^{14} -5.00000 q^{16} -5.04868 q^{17} -5.84096 q^{18} +4.00000 q^{19} +4.00000 q^{21} -11.0371 q^{22} -3.46410 q^{23} +4.37228 q^{24} +1.62772 q^{26} +0.939764 q^{27} +1.58457 q^{28} -1.00000 q^{29} +2.37228 q^{31} +5.19615 q^{32} +16.0858 q^{33} +8.74456 q^{34} +3.37228 q^{36} +10.0974 q^{37} -6.92820 q^{38} -2.37228 q^{39} +6.74456 q^{41} -6.92820 q^{42} -5.69349 q^{43} +6.37228 q^{44} +6.00000 q^{46} -5.69349 q^{47} -12.6217 q^{48} -4.48913 q^{49} -12.7446 q^{51} -0.939764 q^{52} -0.939764 q^{53} -1.62772 q^{54} +2.74456 q^{56} +10.0974 q^{57} +1.73205 q^{58} +0.744563 q^{59} +6.00000 q^{61} -4.10891 q^{62} +5.34363 q^{63} +1.00000 q^{64} -27.8614 q^{66} +8.51278 q^{67} -5.04868 q^{68} -8.74456 q^{69} -4.74456 q^{71} +5.84096 q^{72} +6.92820 q^{73} -17.4891 q^{74} +4.00000 q^{76} +10.0974 q^{77} +4.10891 q^{78} +5.62772 q^{79} -7.74456 q^{81} -11.6819 q^{82} -16.7306 q^{83} +4.00000 q^{84} +9.86141 q^{86} -2.52434 q^{87} +11.0371 q^{88} +10.7446 q^{89} -1.48913 q^{91} -3.46410 q^{92} +5.98844 q^{93} +9.86141 q^{94} +13.1168 q^{96} +6.92820 q^{97} +7.77539 q^{98} +21.4891 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} - 6 q^{6} + 2 q^{9} + 14 q^{11} + 12 q^{14} - 20 q^{16} + 16 q^{19} + 16 q^{21} + 6 q^{24} + 18 q^{26} - 4 q^{29} - 2 q^{31} + 12 q^{34} + 2 q^{36} + 2 q^{39} + 4 q^{41} + 14 q^{44} + 24 q^{46}+ \cdots + 40 q^{99}+O(q^{100}) \) 4 * q + 4 * q^4 - 6 * q^6 + 2 * q^9 + 14 * q^11 + 12 * q^14 - 20 * q^16 + 16 * q^19 + 16 * q^21 + 6 * q^24 + 18 * q^26 - 4 * q^29 - 2 * q^31 + 12 * q^34 + 2 * q^36 + 2 * q^39 + 4 * q^41 + 14 * q^44 + 24 * q^46 + 28 * q^49 - 28 * q^51 - 18 * q^54 - 12 * q^56 - 20 * q^59 + 24 * q^61 + 4 * q^64 - 54 * q^66 - 12 * q^69 + 4 * q^71 - 24 * q^74 + 16 * q^76 + 34 * q^79 - 8 * q^81 + 16 * q^84 - 18 * q^86 + 20 * q^89 + 40 * q^91 - 18 * q^94 + 18 * q^96 + 40 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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