LMFDB - Embedded newform 5225.2.a.y.1.11

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5225,2,Mod(1,5225)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5225.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5225 = 5^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5225.a (trivial)

Newform invariants

comment:select newform

sage:traces = [15,5,4,17,0,-1,21] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$41.7218350561$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 5 x^{14} - 11 x^{13} + 87 x^{12} - 4 x^{11} - 545 x^{10} + 431 x^{9} + 1480 x^{8} - 1763 x^{7} + \cdots + 15 $ x^15 - 5x^14 - 11x^13 + 87x^12 - 4x^11 - 545x^10 + 431x^9 + 1480x^8 - 1763x^7 - 1609x^6 + 2516x^5 + 391x^4 - 1081x^3 - 45x^2 + 153x + 15
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$1.83402$ of defining polynomial
Character	$\chi$	$=$	5225.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.83402 q^{2} -0.596609 q^{3} +1.36363 q^{4} -1.09419 q^{6} -2.23486 q^{7} -1.16712 q^{8} -2.64406 q^{9} +1.00000 q^{11} -0.813554 q^{12} +3.03166 q^{13} -4.09879 q^{14} -4.86777 q^{16} -2.88338 q^{17} -4.84926 q^{18} +1.00000 q^{19} +1.33334 q^{21} +1.83402 q^{22} +4.68074 q^{23} +0.696311 q^{24} +5.56013 q^{26} +3.36730 q^{27} -3.04753 q^{28} +5.80281 q^{29} -3.86669 q^{31} -6.59336 q^{32} -0.596609 q^{33} -5.28817 q^{34} -3.60552 q^{36} -2.50615 q^{37} +1.83402 q^{38} -1.80872 q^{39} +2.12459 q^{41} +2.44537 q^{42} -7.08607 q^{43} +1.36363 q^{44} +8.58457 q^{46} +8.99585 q^{47} +2.90416 q^{48} -2.00538 q^{49} +1.72025 q^{51} +4.13407 q^{52} +10.8443 q^{53} +6.17569 q^{54} +2.60834 q^{56} -0.596609 q^{57} +10.6425 q^{58} -1.87114 q^{59} +10.3158 q^{61} -7.09159 q^{62} +5.90911 q^{63} -2.35682 q^{64} -1.09419 q^{66} -2.18744 q^{67} -3.93186 q^{68} -2.79257 q^{69} +1.25183 q^{71} +3.08592 q^{72} +7.41454 q^{73} -4.59633 q^{74} +1.36363 q^{76} -2.23486 q^{77} -3.31723 q^{78} -3.19265 q^{79} +5.92321 q^{81} +3.89653 q^{82} +17.3292 q^{83} +1.81818 q^{84} -12.9960 q^{86} -3.46201 q^{87} -1.16712 q^{88} +10.8617 q^{89} -6.77536 q^{91} +6.38280 q^{92} +2.30690 q^{93} +16.4986 q^{94} +3.93366 q^{96} -16.9653 q^{97} -3.67791 q^{98} -2.64406 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 5 q^{2} + 4 q^{3} + 17 q^{4} - q^{6} + 21 q^{7} + 9 q^{8} + 15 q^{9} + 15 q^{11} + 11 q^{12} + 13 q^{13} + 9 q^{14} + 21 q^{16} + 17 q^{17} + 22 q^{18} + 15 q^{19} + 6 q^{21} + 5 q^{22} + 26 q^{23}+ \cdots + 15 q^{99}+O(q^{100}) $ 15 * q + 5 * q^2 + 4 * q^3 + 17 * q^4 - q^6 + 21 * q^7 + 9 * q^8 + 15 * q^9 + 15 * q^11 + 11 * q^12 + 13 * q^13 + 9 * q^14 + 21 * q^16 + 17 * q^17 + 22 * q^18 + 15 * q^19 + 6 * q^21 + 5 * q^22 + 26 * q^23 + q^24 + 3 * q^26 + q^27 + 46 * q^28 + 9 * q^29 + 14 * q^31 + 18 * q^32 + 4 * q^33 - 13 * q^34 + 12 * q^36 + 9 * q^37 + 5 * q^38 - 22 * q^39 + 4 * q^41 - 6 * q^42 + 28 * q^43 + 17 * q^44 + 27 * q^46 + 14 * q^47 - 4 * q^48 + 32 * q^49 - 40 * q^51 + 14 * q^52 + 3 * q^53 - 39 * q^54 + 34 * q^56 + 4 * q^57 + 26 * q^58 + q^59 + 2 * q^61 - 3 * q^62 + 45 * q^63 + 5 * q^64 - q^66 + 37 * q^67 + 26 * q^68 - 7 * q^69 - 7 * q^71 + 16 * q^72 + 42 * q^73 - 43 * q^74 + 17 * q^76 + 21 * q^77 - 64 * q^78 - 10 * q^79 + 31 * q^81 + 22 * q^82 + 14 * q^83 - 32 * q^84 + 37 * q^86 + 29 * q^87 + 9 * q^88 + 15 * q^89 - 22 * q^91 + 26 * q^92 - 18 * q^93 - 44 * q^94 + 71 * q^96 + 8 * q^97 - 10 * q^98 + 15 * 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 $

$2$	1.83402	1.29685	0.648424	−	0.761279i	$-0.275428\pi$
$2$	1.83402	1.29685	0.648424	+	0.761279i	$0.275428\pi$
$3$	−0.596609	−0.344452	−0.172226	−	0.985057i	$-0.555096\pi$
$3$	−0.596609	−0.344452	−0.172226	+	0.985057i	$0.555096\pi$
$4$	1.36363	0.681815
$5$	0	0
$5$	0	0
$6$	−1.09419	−0.446702
$7$	−2.23486	−0.844699	−0.422350	−	0.906433i	$-0.638795\pi$
$7$	−2.23486	−0.844699	−0.422350	+	0.906433i	$0.638795\pi$
$8$	−1.16712	−0.412637
$9$	−2.64406	−0.881353
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−0.813554	−0.234853
$13$	3.03166	0.840832	0.420416	−	0.907331i	$-0.361884\pi$
$13$	3.03166	0.840832	0.420416	+	0.907331i	$0.361884\pi$
$14$	−4.09879	−1.09545
$15$	0	0
$16$	−4.86777	−1.21694
$17$	−2.88338	−0.699321	−0.349661	−	0.936876i	$-0.613703\pi$
$17$	−2.88338	−0.699321	−0.349661	+	0.936876i	$0.613703\pi$
$18$	−4.84926	−1.14298
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	1.33334	0.290959
$22$	1.83402	0.391014
$23$	4.68074	0.976002	0.488001	−	0.872843i	$-0.337726\pi$
$23$	4.68074	0.976002	0.488001	+	0.872843i	$0.337726\pi$
$24$	0.696311	0.142134
$25$	0	0
$26$	5.56013	1.09043
$27$	3.36730	0.648036
$28$	−3.04753	−0.575929
$29$	5.80281	1.07756	0.538778	−	0.842448i	$-0.318886\pi$
$29$	5.80281	1.07756	0.538778	+	0.842448i	$0.318886\pi$
$30$	0	0
$31$	−3.86669	−0.694478	−0.347239	−	0.937777i	$-0.612881\pi$
$31$	−3.86669	−0.694478	−0.347239	+	0.937777i	$0.612881\pi$
$32$	−6.59336	−1.16555
$33$	−0.596609	−0.103856
$34$	−5.28817	−0.906913
$35$	0	0
$36$	−3.60552	−0.600919
$37$	−2.50615	−0.412008	−0.206004	−	0.978551i	$-0.566046\pi$
$37$	−2.50615	−0.412008	−0.206004	+	0.978551i	$0.566046\pi$
$38$	1.83402	0.297517
$39$	−1.80872	−0.289627
$40$	0	0
$41$	2.12459	0.331805	0.165902	−	0.986142i	$-0.446946\pi$
$41$	2.12459	0.331805	0.165902	+	0.986142i	$0.446946\pi$
$42$	2.44537	0.377329
$43$	−7.08607	−1.08062	−0.540308	−	0.841467i	$-0.681692\pi$
$43$	−7.08607	−1.08062	−0.540308	+	0.841467i	$0.681692\pi$
$44$	1.36363	0.205575
$45$	0	0
$46$	8.58457	1.26573
$47$	8.99585	1.31218	0.656090	−	0.754683i	$-0.272209\pi$
$47$	8.99585	1.31218	0.656090	+	0.754683i	$0.272209\pi$
$48$	2.90416	0.419179
$49$	−2.00538	−0.286483
$50$	0	0
$51$	1.72025	0.240883
$52$	4.13407	0.573292
$53$	10.8443	1.48957	0.744787	−	0.667302i	$-0.232551\pi$
$53$	10.8443	1.48957	0.744787	+	0.667302i	$0.232551\pi$
$54$	6.17569	0.840405
$55$	0	0
$56$	2.60834	0.348555
$57$	−0.596609	−0.0790228
$58$	10.6425	1.39743
$59$	−1.87114	−0.243602	−0.121801	−	0.992555i	$-0.538867\pi$
$59$	−1.87114	−0.243602	−0.121801	+	0.992555i	$0.538867\pi$
$60$	0	0
$61$	10.3158	1.32080	0.660401	−	0.750913i	$-0.270386\pi$
$61$	10.3158	1.32080	0.660401	+	0.750913i	$0.270386\pi$
$62$	−7.09159	−0.900633
$63$	5.90911	0.744478
$64$	−2.35682	−0.294602
$65$	0	0
$66$	−1.09419	−0.134686
$67$	−2.18744	−0.267239	−0.133619	−	0.991033i	$-0.542660\pi$
$67$	−2.18744	−0.267239	−0.133619	+	0.991033i	$0.542660\pi$
$68$	−3.93186	−0.476808
$69$	−2.79257	−0.336186
$70$	0	0
$71$	1.25183	0.148565	0.0742827	−	0.997237i	$-0.476333\pi$
$71$	1.25183	0.148565	0.0742827	+	0.997237i	$0.476333\pi$
$72$	3.08592	0.363679
$73$	7.41454	0.867806	0.433903	−	0.900960i	$-0.357136\pi$
$73$	7.41454	0.867806	0.433903	+	0.900960i	$0.357136\pi$
$74$	−4.59633	−0.534312
$75$	0	0
$76$	1.36363	0.156419
$77$	−2.23486	−0.254686
$78$	−3.31723	−0.375602
$79$	−3.19265	−0.359201	−0.179600	−	0.983740i	$-0.557480\pi$
$79$	−3.19265	−0.359201	−0.179600	+	0.983740i	$0.557480\pi$
$80$	0	0
$81$	5.92321	0.658135
$82$	3.89653	0.430300
$83$	17.3292	1.90213	0.951063	−	0.308998i	$-0.0999936\pi$
$83$	17.3292	1.90213	0.951063	+	0.308998i	$0.0999936\pi$
$84$	1.81818	0.198380
$85$	0	0
$86$	−12.9960	−1.40139
$87$	−3.46201	−0.371166
$88$	−1.16712	−0.124415
$89$	10.8617	1.15134	0.575668	−	0.817684i	$-0.304742\pi$
$89$	10.8617	1.15134	0.575668	+	0.817684i	$0.304742\pi$
$90$	0	0
$91$	−6.77536	−0.710250
$92$	6.38280	0.665453
$93$	2.30690	0.239215
$94$	16.4986	1.70170
$95$	0	0
$96$	3.93366	0.401478
$97$	−16.9653	−1.72257	−0.861283	−	0.508125i	$-0.830339\pi$
$97$	−16.9653	−1.72257	−0.861283	+	0.508125i	$0.830339\pi$
$98$	−3.67791	−0.371525
$99$	−2.64406	−0.265738
$100$	0	0
$101$	−10.6526	−1.05998	−0.529989	−	0.848005i	$-0.677804\pi$
$101$	−10.6526	−1.05998	−0.529989	+	0.848005i	$0.677804\pi$
$102$	3.15497	0.312389
$103$	15.6734	1.54435	0.772175	−	0.635410i	$-0.219169\pi$
$103$	15.6734	1.54435	0.772175	+	0.635410i	$0.219169\pi$
$104$	−3.53830	−0.346959
$105$	0	0
$106$	19.8886	1.93175
$107$	3.35117	0.323970	0.161985	−	0.986793i	$-0.448210\pi$
$107$	3.35117	0.323970	0.161985	+	0.986793i	$0.448210\pi$
$108$	4.59175	0.441841
$109$	−16.9337	−1.62196	−0.810978	−	0.585077i	$-0.801064\pi$
$109$	−16.9337	−1.62196	−0.810978	+	0.585077i	$0.801064\pi$
$110$	0	0
$111$	1.49519	0.141917
$112$	10.8788	1.02795
$113$	19.3034	1.81591	0.907954	−	0.419070i	$-0.137644\pi$
$113$	19.3034	1.81591	0.907954	+	0.419070i	$0.137644\pi$
$114$	−1.09419	−0.102481
$115$	0	0
$116$	7.91289	0.734693
$117$	−8.01589	−0.741070
$118$	−3.43172	−0.315915
$119$	6.44395	0.590716
$120$	0	0
$121$	1.00000	0.0909091
$122$	18.9194	1.71288
$123$	−1.26755	−0.114291
$124$	−5.27274	−0.473506
$125$	0	0
$126$	10.8374	0.965475
$127$	9.32764	0.827694	0.413847	−	0.910346i	$-0.364185\pi$
$127$	9.32764	0.827694	0.413847	+	0.910346i	$0.364185\pi$
$128$	8.86428	0.783499
$129$	4.22762	0.372221
$130$	0	0
$131$	20.8478	1.82148	0.910741	−	0.412977i	$-0.135511\pi$
$131$	20.8478	1.82148	0.910741	+	0.412977i	$0.135511\pi$
$132$	−0.813554	−0.0708108
$133$	−2.23486	−0.193787
$134$	−4.01182	−0.346568
$135$	0	0
$136$	3.36523	0.288566
$137$	−11.3437	−0.969158	−0.484579	−	0.874748i	$-0.661027\pi$
$137$	−11.3437	−0.969158	−0.484579	+	0.874748i	$0.661027\pi$
$138$	−5.12163	−0.435982
$139$	−1.37966	−0.117021	−0.0585105	−	0.998287i	$-0.518635\pi$
$139$	−1.37966	−0.117021	−0.0585105	+	0.998287i	$0.518635\pi$
$140$	0	0
$141$	−5.36701	−0.451983
$142$	2.29589	0.192667
$143$	3.03166	0.253520
$144$	12.8707	1.07256
$145$	0	0
$146$	13.5984	1.12541
$147$	1.19643	0.0986797
$148$	−3.41746	−0.280913
$149$	18.5982	1.52362	0.761811	−	0.647800i	$-0.224311\pi$
$149$	18.5982	1.52362	0.761811	+	0.647800i	$0.224311\pi$
$150$	0	0
$151$	7.69001	0.625805	0.312902	−	0.949785i	$-0.398699\pi$
$151$	7.69001	0.625805	0.312902	+	0.949785i	$0.398699\pi$
$152$	−1.16712	−0.0946655
$153$	7.62381	0.616349
$154$	−4.09879	−0.330290
$155$	0	0
$156$	−2.46642	−0.197472
$157$	9.60663	0.766693	0.383346	−	0.923605i	$-0.374772\pi$
$157$	9.60663	0.766693	0.383346	+	0.923605i	$0.374772\pi$
$158$	−5.85538	−0.465829
$159$	−6.46979	−0.513088
$160$	0	0
$161$	−10.4608	−0.824428
$162$	10.8633	0.853501
$163$	11.0474	0.865298	0.432649	−	0.901562i	$-0.357579\pi$
$163$	11.0474	0.865298	0.432649	+	0.901562i	$0.357579\pi$
$164$	2.89715	0.226229
$165$	0	0
$166$	31.7821	2.46677
$167$	−8.27878	−0.640631	−0.320316	−	0.947311i	$-0.603789\pi$
$167$	−8.27878	−0.640631	−0.320316	+	0.947311i	$0.603789\pi$
$168$	−1.55616	−0.120060
$169$	−3.80902	−0.293001
$170$	0	0
$171$	−2.64406	−0.202196
$172$	−9.66278	−0.736780
$173$	5.41017	0.411328	0.205664	−	0.978623i	$-0.434065\pi$
$173$	5.41017	0.411328	0.205664	+	0.978623i	$0.434065\pi$
$174$	−6.34940	−0.481346
$175$	0	0
$176$	−4.86777	−0.366922
$177$	1.11634	0.0839094
$178$	19.9205	1.49311
$179$	14.5286	1.08592	0.542959	−	0.839759i	$-0.317304\pi$
$179$	14.5286	1.08592	0.542959	+	0.839759i	$0.317304\pi$
$180$	0	0
$181$	−8.39894	−0.624288	−0.312144	−	0.950035i	$-0.601047\pi$
$181$	−8.39894	−0.624288	−0.312144	+	0.950035i	$0.601047\pi$
$182$	−12.4261	−0.921087
$183$	−6.15450	−0.454954
$184$	−5.46296	−0.402735
$185$	0	0
$186$	4.23091	0.310225
$187$	−2.88338	−0.210853
$188$	12.2670	0.894664
$189$	−7.52545	−0.547396
$190$	0	0
$191$	0.488866	0.0353731	0.0176865	−	0.999844i	$-0.494370\pi$
$191$	0.488866	0.0353731	0.0176865	+	0.999844i	$0.494370\pi$
$192$	1.40610	0.101476
$193$	−10.9732	−0.789868	−0.394934	−	0.918710i	$-0.629232\pi$
$193$	−10.9732	−0.789868	−0.394934	+	0.918710i	$0.629232\pi$
$194$	−31.1147	−2.23391
$195$	0	0
$196$	−2.73460	−0.195328
$197$	6.81170	0.485314	0.242657	−	0.970112i	$-0.421981\pi$
$197$	6.81170	0.485314	0.242657	+	0.970112i	$0.421981\pi$
$198$	−4.84926	−0.344622
$199$	−7.26864	−0.515260	−0.257630	−	0.966244i	$-0.582942\pi$
$199$	−7.26864	−0.515260	−0.257630	+	0.966244i	$0.582942\pi$
$200$	0	0
$201$	1.30505	0.0920511
$202$	−19.5372	−1.37463
$203$	−12.9685	−0.910210
$204$	2.34578	0.164238
$205$	0	0
$206$	28.7454	2.00279
$207$	−12.3761	−0.860202
$208$	−14.7574	−1.02325
$209$	1.00000	0.0691714
$210$	0	0
$211$	−4.55702	−0.313718	−0.156859	−	0.987621i	$-0.550137\pi$
$211$	−4.55702	−0.313718	−0.156859	+	0.987621i	$0.550137\pi$
$212$	14.7876	1.01561
$213$	−0.746856	−0.0511737
$214$	6.14611	0.420139
$215$	0	0
$216$	−3.93002	−0.267404
$217$	8.64153	0.586625
$218$	−31.0568	−2.10343
$219$	−4.42358	−0.298918
$220$	0	0
$221$	−8.74142	−0.588012
$222$	2.74221	0.184045
$223$	−3.44256	−0.230531	−0.115265	−	0.993335i	$-0.536772\pi$
$223$	−3.44256	−0.230531	−0.115265	+	0.993335i	$0.536772\pi$
$224$	14.7353	0.984542
$225$	0	0
$226$	35.4028	2.35496
$227$	17.7892	1.18071	0.590356	−	0.807143i	$-0.298988\pi$
$227$	17.7892	1.18071	0.590356	+	0.807143i	$0.298988\pi$
$228$	−0.813554	−0.0538789
$229$	−19.4695	−1.28658	−0.643291	−	0.765622i	$-0.722431\pi$
$229$	−19.4695	−1.28658	−0.643291	+	0.765622i	$0.722431\pi$
$230$	0	0
$231$	1.33334	0.0877274
$232$	−6.77255	−0.444640
$233$	−7.70793	−0.504963	−0.252481	−	0.967602i	$-0.581247\pi$
$233$	−7.70793	−0.504963	−0.252481	+	0.967602i	$0.581247\pi$
$234$	−14.7013	−0.961055
$235$	0	0
$236$	−2.55155	−0.166092
$237$	1.90476	0.123728
$238$	11.8183	0.766069
$239$	−23.6770	−1.53154	−0.765768	−	0.643117i	$-0.777641\pi$
$239$	−23.6770	−1.53154	−0.765768	+	0.643117i	$0.777641\pi$
$240$	0	0
$241$	8.87417	0.571635	0.285818	−	0.958284i	$-0.407735\pi$
$241$	8.87417	0.571635	0.285818	+	0.958284i	$0.407735\pi$
$242$	1.83402	0.117895
$243$	−13.6357	−0.874733
$244$	14.0669	0.900543
$245$	0	0
$246$	−2.32471	−0.148218
$247$	3.03166	0.192900
$248$	4.51287	0.286568
$249$	−10.3387	−0.655192
$250$	0	0
$251$	−6.84752	−0.432212	−0.216106	−	0.976370i	$-0.569336\pi$
$251$	−6.84752	−0.432212	−0.216106	+	0.976370i	$0.569336\pi$
$252$	8.05784	0.507596
$253$	4.68074	0.294276
$254$	17.1071	1.07339
$255$	0	0
$256$	20.9709	1.31068
$257$	−6.94874	−0.433450	−0.216725	−	0.976233i	$-0.569538\pi$
$257$	−6.94874	−0.433450	−0.216725	+	0.976233i	$0.569538\pi$
$258$	7.75353	0.482714
$259$	5.60090	0.348023
$260$	0	0
$261$	−15.3430	−0.949706
$262$	38.2353	2.36219
$263$	−25.2403	−1.55638	−0.778191	−	0.628028i	$-0.783862\pi$
$263$	−25.2403	−1.55638	−0.778191	+	0.628028i	$0.783862\pi$
$264$	0.696311	0.0428550
$265$	0	0
$266$	−4.09879	−0.251313
$267$	−6.48018	−0.396580
$268$	−2.98287	−0.182207
$269$	10.3247	0.629506	0.314753	−	0.949174i	$-0.398078\pi$
$269$	10.3247	0.629506	0.314753	+	0.949174i	$0.398078\pi$
$270$	0	0
$271$	−29.1786	−1.77247	−0.886236	−	0.463235i	$-0.846689\pi$
$271$	−29.1786	−1.77247	−0.886236	+	0.463235i	$0.846689\pi$
$272$	14.0356	0.851034
$273$	4.04224	0.244647
$274$	−20.8046	−1.25685
$275$	0	0
$276$	−3.80804	−0.229217
$277$	11.8590	0.712540	0.356270	−	0.934383i	$-0.384048\pi$
$277$	11.8590	0.712540	0.356270	+	0.934383i	$0.384048\pi$
$278$	−2.53032	−0.151758
$279$	10.2238	0.612080
$280$	0	0
$281$	9.45893	0.564272	0.282136	−	0.959374i	$-0.408957\pi$
$281$	9.45893	0.564272	0.282136	+	0.959374i	$0.408957\pi$
$282$	−9.84320	−0.586154
$283$	18.6561	1.10899	0.554494	−	0.832188i	$-0.312912\pi$
$283$	18.6561	1.10899	0.554494	+	0.832188i	$0.312912\pi$
$284$	1.70704	0.101294
$285$	0	0
$286$	5.56013	0.328778
$287$	−4.74816	−0.280275
$288$	17.4332	1.02726
$289$	−8.68615	−0.510950
$290$	0	0
$291$	10.1217	0.593342
$292$	10.1107	0.591683
$293$	−28.1893	−1.64684	−0.823418	−	0.567435i	$-0.807936\pi$
$293$	−28.1893	−1.64684	−0.823418	+	0.567435i	$0.807936\pi$
$294$	2.19427	0.127973
$295$	0	0
$296$	2.92496	0.170010
$297$	3.36730	0.195390
$298$	34.1094	1.97591
$299$	14.1904	0.820654
$300$	0	0
$301$	15.8364	0.912796
$302$	14.1036	0.811573
$303$	6.35546	0.365112
$304$	−4.86777	−0.279186
$305$	0	0
$306$	13.9822	0.799310
$307$	8.93464	0.509927	0.254963	−	0.966951i	$-0.417937\pi$
$307$	8.93464	0.509927	0.254963	+	0.966951i	$0.417937\pi$
$308$	−3.04753	−0.173649
$309$	−9.35091	−0.531955
$310$	0	0
$311$	13.8365	0.784595	0.392297	−	0.919838i	$-0.371680\pi$
$311$	13.8365	0.784595	0.392297	+	0.919838i	$0.371680\pi$
$312$	2.11098	0.119511
$313$	12.0282	0.679873	0.339936	−	0.940448i	$-0.389594\pi$
$313$	12.0282	0.679873	0.339936	+	0.940448i	$0.389594\pi$
$314$	17.6188	0.994284
$315$	0	0
$316$	−4.35359	−0.244908
$317$	12.8073	0.719329	0.359665	−	0.933082i	$-0.382891\pi$
$317$	12.8073	0.719329	0.359665	+	0.933082i	$0.382891\pi$
$318$	−11.8657	−0.665397
$319$	5.80281	0.324895
$320$	0	0
$321$	−1.99934	−0.111592
$322$	−19.1854	−1.06916
$323$	−2.88338	−0.160435
$324$	8.07707	0.448726
$325$	0	0
$326$	20.2611	1.12216
$327$	10.1028	0.558686
$328$	−2.47964	−0.136915
$329$	−20.1045	−1.10840
$330$	0	0
$331$	22.7336	1.24955	0.624776	−	0.780804i	$-0.285190\pi$
$331$	22.7336	1.24955	0.624776	+	0.780804i	$0.285190\pi$
$332$	23.6306	1.29690
$333$	6.62640	0.363124
$334$	−15.1834	−0.830801
$335$	0	0
$336$	−6.49040	−0.354080
$337$	20.2511	1.10315	0.551573	−	0.834127i	$-0.314028\pi$
$337$	20.2511	1.10315	0.551573	+	0.834127i	$0.314028\pi$
$338$	−6.98581	−0.379978
$339$	−11.5166	−0.625494
$340$	0	0
$341$	−3.86669	−0.209393
$342$	−4.84926	−0.262218
$343$	20.1258	1.08669
$344$	8.27026	0.445903
$345$	0	0
$346$	9.92237	0.533430
$347$	−34.5032	−1.85223	−0.926116	−	0.377240i	$-0.876873\pi$
$347$	−34.5032	−1.85223	−0.926116	+	0.377240i	$0.876873\pi$
$348$	−4.72090	−0.253067
$349$	−17.7850	−0.952010	−0.476005	−	0.879442i	$-0.657916\pi$
$349$	−17.7850	−0.952010	−0.476005	+	0.879442i	$0.657916\pi$
$350$	0	0
$351$	10.2085	0.544890
$352$	−6.59336	−0.351427
$353$	−4.43853	−0.236239	−0.118120	−	0.992999i	$-0.537687\pi$
$353$	−4.43853	−0.236239	−0.118120	+	0.992999i	$0.537687\pi$
$354$	2.04739	0.108818
$355$	0	0
$356$	14.8113	0.784998
$357$	−3.84452	−0.203474
$358$	26.6458	1.40827
$359$	−25.2069	−1.33037	−0.665184	−	0.746679i	$-0.731647\pi$
$359$	−25.2069	−1.33037	−0.665184	+	0.746679i	$0.731647\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−15.4038	−0.809607
$363$	−0.596609	−0.0313139
$364$	−9.23908	−0.484260
$365$	0	0
$366$	−11.2875	−0.590006
$367$	4.79279	0.250182	0.125091	−	0.992145i	$-0.460078\pi$
$367$	4.79279	0.250182	0.125091	+	0.992145i	$0.460078\pi$
$368$	−22.7848	−1.18774
$369$	−5.61753	−0.292437
$370$	0	0
$371$	−24.2355	−1.25824
$372$	3.14576	0.163100
$373$	32.2378	1.66921	0.834604	−	0.550851i	$-0.185697\pi$
$373$	32.2378	1.66921	0.834604	+	0.550851i	$0.185697\pi$
$374$	−5.28817	−0.273445
$375$	0	0
$376$	−10.4992	−0.541454
$377$	17.5922	0.906043
$378$	−13.8018	−0.709889
$379$	−29.2707	−1.50354	−0.751769	−	0.659427i	$-0.770799\pi$
$379$	−29.2707	−1.50354	−0.751769	+	0.659427i	$0.770799\pi$
$380$	0	0
$381$	−5.56496	−0.285101
$382$	0.896590	0.0458735
$383$	−22.2407	−1.13644	−0.568222	−	0.822875i	$-0.692369\pi$
$383$	−22.2407	−1.13644	−0.568222	+	0.822875i	$0.692369\pi$
$384$	−5.28851	−0.269878
$385$	0	0
$386$	−20.1251	−1.02434
$387$	18.7360	0.952404
$388$	−23.1344	−1.17447
$389$	25.1911	1.27724	0.638619	−	0.769523i	$-0.279506\pi$
$389$	25.1911	1.27724	0.638619	+	0.769523i	$0.279506\pi$
$390$	0	0
$391$	−13.4963	−0.682539
$392$	2.34051	0.118214
$393$	−12.4380	−0.627414
$394$	12.4928	0.629378
$395$	0	0
$396$	−3.60552	−0.181184
$397$	2.48439	0.124688	0.0623440	−	0.998055i	$-0.480142\pi$
$397$	2.48439	0.124688	0.0623440	+	0.998055i	$0.480142\pi$
$398$	−13.3308	−0.668214
$399$	1.33334	0.0667505
$400$	0	0
$401$	−5.61818	−0.280559	−0.140279	−	0.990112i	$-0.544800\pi$
$401$	−5.61818	−0.280559	−0.140279	+	0.990112i	$0.544800\pi$
$402$	2.39349	0.119376
$403$	−11.7225	−0.583940
$404$	−14.5263	−0.722709
$405$	0	0
$406$	−23.7845	−1.18040
$407$	−2.50615	−0.124225
$408$	−2.00773	−0.0993973
$409$	13.3011	0.657696	0.328848	−	0.944383i	$-0.393340\pi$
$409$	13.3011	0.657696	0.328848	+	0.944383i	$0.393340\pi$
$410$	0	0
$411$	6.76776	0.333829
$412$	21.3728	1.05296
$413$	4.18175	0.205771
$414$	−22.6981	−1.11555
$415$	0	0
$416$	−19.9889	−0.980035
$417$	0.823115	0.0403081
$418$	1.83402	0.0897049
$419$	−26.8456	−1.31150	−0.655748	−	0.754980i	$-0.727646\pi$
$419$	−26.8456	−1.31150	−0.655748	+	0.754980i	$0.727646\pi$
$420$	0	0
$421$	−37.7283	−1.83877	−0.919383	−	0.393364i	$-0.871311\pi$
$421$	−37.7283	−1.83877	−0.919383	+	0.393364i	$0.871311\pi$
$422$	−8.35767	−0.406845
$423$	−23.7855	−1.15649
$424$	−12.6565	−0.614654
$425$	0	0
$426$	−1.36975	−0.0663645
$427$	−23.0544	−1.11568
$428$	4.56975	0.220887
$429$	−1.80872	−0.0873257
$430$	0	0
$431$	−22.4835	−1.08299	−0.541495	−	0.840704i	$-0.682141\pi$
$431$	−22.4835	−1.08299	−0.541495	+	0.840704i	$0.682141\pi$
$432$	−16.3912	−0.788624
$433$	26.0421	1.25150	0.625751	−	0.780023i	$-0.284793\pi$
$433$	26.0421	1.25150	0.625751	+	0.780023i	$0.284793\pi$
$434$	15.8487	0.760764
$435$	0	0
$436$	−23.0913	−1.10587
$437$	4.68074	0.223910
$438$	−8.11294	−0.387651
$439$	31.6821	1.51210	0.756052	−	0.654512i	$-0.227126\pi$
$439$	31.6821	1.51210	0.756052	+	0.654512i	$0.227126\pi$
$440$	0	0
$441$	5.30234	0.252492
$442$	−16.0319	−0.762562
$443$	30.5809	1.45294	0.726471	−	0.687197i	$-0.241159\pi$
$443$	30.5809	1.45294	0.726471	+	0.687197i	$0.241159\pi$
$444$	2.03889	0.0967613
$445$	0	0
$446$	−6.31372	−0.298964
$447$	−11.0958	−0.524815
$448$	5.26717	0.248850
$449$	20.8054	0.981868	0.490934	−	0.871197i	$-0.336656\pi$
$449$	20.8054	0.981868	0.490934	+	0.871197i	$0.336656\pi$
$450$	0	0
$451$	2.12459	0.100043
$452$	26.3227	1.23811
$453$	−4.58793	−0.215560
$454$	32.6258	1.53120
$455$	0	0
$456$	0.696311	0.0326078
$457$	2.88348	0.134883	0.0674417	−	0.997723i	$-0.478516\pi$
$457$	2.88348	0.134883	0.0674417	+	0.997723i	$0.478516\pi$
$458$	−35.7075	−1.66850
$459$	−9.70918	−0.453186
$460$	0	0
$461$	−21.9150	−1.02069	−0.510343	−	0.859971i	$-0.670481\pi$
$461$	−21.9150	−1.02069	−0.510343	+	0.859971i	$0.670481\pi$
$462$	2.44537	0.113769
$463$	5.49470	0.255360	0.127680	−	0.991815i	$-0.459247\pi$
$463$	5.49470	0.255360	0.127680	+	0.991815i	$0.459247\pi$
$464$	−28.2468	−1.31132
$465$	0	0
$466$	−14.1365	−0.654860
$467$	18.2802	0.845905	0.422952	−	0.906152i	$-0.360994\pi$
$467$	18.2802	0.845905	0.422952	+	0.906152i	$0.360994\pi$
$468$	−10.9307	−0.505272
$469$	4.88864	0.225736
$470$	0	0
$471$	−5.73140	−0.264089
$472$	2.18384	0.100519
$473$	−7.08607	−0.325818
$474$	3.49337	0.160456
$475$	0	0
$476$	8.78717	0.402759
$477$	−28.6729	−1.31284
$478$	−43.4240	−1.98617
$479$	11.9035	0.543884	0.271942	−	0.962314i	$-0.412334\pi$
$479$	11.9035	0.543884	0.271942	+	0.962314i	$0.412334\pi$
$480$	0	0
$481$	−7.59780	−0.346430
$482$	16.2754	0.741324
$483$	6.24102	0.283976
$484$	1.36363	0.0619832
$485$	0	0
$486$	−25.0082	−1.13440
$487$	11.3635	0.514931	0.257465	−	0.966288i	$-0.417113\pi$
$487$	11.3635	0.514931	0.257465	+	0.966288i	$0.417113\pi$
$488$	−12.0397	−0.545013
$489$	−6.59097	−0.298054
$490$	0	0
$491$	13.4818	0.608423	0.304212	−	0.952604i	$-0.401607\pi$
$491$	13.4818	0.608423	0.304212	+	0.952604i	$0.401607\pi$
$492$	−1.72847	−0.0779253
$493$	−16.7317	−0.753557
$494$	5.56013	0.250162
$495$	0	0
$496$	18.8222	0.845141
$497$	−2.79768	−0.125493
$498$	−18.9615	−0.849684
$499$	−3.27906	−0.146791	−0.0733955	−	0.997303i	$-0.523384\pi$
$499$	−3.27906	−0.146791	−0.0733955	+	0.997303i	$0.523384\pi$
$500$	0	0
$501$	4.93919	0.220667
$502$	−12.5585	−0.560513
$503$	−14.6071	−0.651300	−0.325650	−	0.945490i	$-0.605583\pi$
$503$	−14.6071	−0.651300	−0.325650	+	0.945490i	$0.605583\pi$
$504$	−6.89661	−0.307200
$505$	0	0
$506$	8.58457	0.381631
$507$	2.27249	0.100925
$508$	12.7195	0.564335
$509$	20.6334	0.914562	0.457281	−	0.889322i	$-0.348823\pi$
$509$	20.6334	0.914562	0.457281	+	0.889322i	$0.348823\pi$
$510$	0	0
$511$	−16.5705	−0.733035
$512$	20.7325	0.916256
$513$	3.36730	0.148670
$514$	−12.7441	−0.562119
$515$	0	0
$516$	5.76490	0.253786
$517$	8.99585	0.395637
$518$	10.2722	0.451333
$519$	−3.22776	−0.141683
$520$	0	0
$521$	−4.29362	−0.188107	−0.0940533	−	0.995567i	$-0.529982\pi$
$521$	−4.29362	−0.188107	−0.0940533	+	0.995567i	$0.529982\pi$
$522$	−28.1393	−1.23162
$523$	15.4694	0.676431	0.338216	−	0.941069i	$-0.390177\pi$
$523$	15.4694	0.676431	0.338216	+	0.941069i	$0.390177\pi$
$524$	28.4287	1.24191
$525$	0	0
$526$	−46.2912	−2.01839
$527$	11.1491	0.485663
$528$	2.90416	0.126387
$529$	−1.09067	−0.0474206
$530$	0	0
$531$	4.94741	0.214699
$532$	−3.04753	−0.132127
$533$	6.44103	0.278992
$534$	−11.8848	−0.514305
$535$	0	0
$536$	2.55300	0.110273
$537$	−8.66790	−0.374047
$538$	18.9356	0.816373
$539$	−2.00538	−0.0863778
$540$	0	0
$541$	−41.9419	−1.80322	−0.901611	−	0.432548i	$-0.857615\pi$
$541$	−41.9419	−1.80322	−0.901611	+	0.432548i	$0.857615\pi$
$542$	−53.5141	−2.29863
$543$	5.01088	0.215037
$544$	19.0111	0.815096
$545$	0	0
$546$	7.41355	0.317271
$547$	43.0246	1.83960	0.919800	−	0.392388i	$-0.128351\pi$
$547$	43.0246	1.83960	0.919800	+	0.392388i	$0.128351\pi$
$548$	−15.4686	−0.660787
$549$	−27.2756	−1.16409
$550$	0	0
$551$	5.80281	0.247208
$552$	3.25925	0.138723
$553$	7.13513	0.303417
$554$	21.7497	0.924056
$555$	0	0
$556$	−1.88134	−0.0797866
$557$	31.8516	1.34960	0.674799	−	0.738002i	$-0.264230\pi$
$557$	31.8516	1.34960	0.674799	+	0.738002i	$0.264230\pi$
$558$	18.7506	0.793775
$559$	−21.4826	−0.908617
$560$	0	0
$561$	1.72025	0.0726289
$562$	17.3479	0.731776
$563$	8.58917	0.361991	0.180995	−	0.983484i	$-0.442068\pi$
$563$	8.58917	0.361991	0.180995	+	0.983484i	$0.442068\pi$
$564$	−7.31861	−0.308169
$565$	0	0
$566$	34.2156	1.43819
$567$	−13.2376	−0.555926
$568$	−1.46103	−0.0613037
$569$	−7.45323	−0.312456	−0.156228	−	0.987721i	$-0.549933\pi$
$569$	−7.45323	−0.312456	−0.156228	+	0.987721i	$0.549933\pi$
$570$	0	0
$571$	28.2372	1.18169	0.590846	−	0.806785i	$-0.298794\pi$
$571$	28.2372	1.18169	0.590846	+	0.806785i	$0.298794\pi$
$572$	4.13407	0.172854
$573$	−0.291662	−0.0121843
$574$	−8.70823	−0.363474
$575$	0	0
$576$	6.23156	0.259648
$577$	−12.5237	−0.521367	−0.260684	−	0.965424i	$-0.583948\pi$
$577$	−12.5237	−0.521367	−0.260684	+	0.965424i	$0.583948\pi$
$578$	−15.9306	−0.662624
$579$	6.54670	0.272072
$580$	0	0
$581$	−38.7284	−1.60672
$582$	18.5633	0.769475
$583$	10.8443	0.449124
$584$	−8.65362	−0.358089
$585$	0	0
$586$	−51.6997	−2.13570
$587$	−12.9723	−0.535422	−0.267711	−	0.963499i	$-0.586267\pi$
$587$	−12.9723	−0.535422	−0.267711	+	0.963499i	$0.586267\pi$
$588$	1.63149	0.0672813
$589$	−3.86669	−0.159324
$590$	0	0
$591$	−4.06392	−0.167167
$592$	12.1994	0.501390
$593$	23.9722	0.984421	0.492210	−	0.870476i	$-0.336189\pi$
$593$	23.9722	0.984421	0.492210	+	0.870476i	$0.336189\pi$
$594$	6.17569	0.253392
$595$	0	0
$596$	25.3610	1.03883
$597$	4.33653	0.177483
$598$	26.0255	1.06426
$599$	5.94996	0.243109	0.121554	−	0.992585i	$-0.461212\pi$
$599$	5.94996	0.243109	0.121554	+	0.992585i	$0.461212\pi$
$600$	0	0
$601$	−7.87983	−0.321425	−0.160712	−	0.987001i	$-0.551379\pi$
$601$	−7.87983	−0.321425	−0.160712	+	0.987001i	$0.551379\pi$
$602$	29.0443	1.18376
$603$	5.78373	0.235532
$604$	10.4863	0.426683
$605$	0	0
$606$	11.6560	0.473495
$607$	−41.1891	−1.67181	−0.835906	−	0.548872i	$-0.815057\pi$
$607$	−41.1891	−1.67181	−0.835906	+	0.548872i	$0.815057\pi$
$608$	−6.59336	−0.267396
$609$	7.73712	0.313524
$610$	0	0
$611$	27.2724	1.10332
$612$	10.3961	0.420236
$613$	−12.7666	−0.515639	−0.257819	−	0.966193i	$-0.583004\pi$
$613$	−12.7666	−0.515639	−0.257819	+	0.966193i	$0.583004\pi$
$614$	16.3863	0.661298
$615$	0	0
$616$	2.60834	0.105093
$617$	−16.0423	−0.645840	−0.322920	−	0.946426i	$-0.604665\pi$
$617$	−16.0423	−0.645840	−0.322920	+	0.946426i	$0.604665\pi$
$618$	−17.1498	−0.689865
$619$	12.9063	0.518750	0.259375	−	0.965777i	$-0.416483\pi$
$619$	12.9063	0.518750	0.259375	+	0.965777i	$0.416483\pi$
$620$	0	0
$621$	15.7614	0.632485
$622$	25.3764	1.01750
$623$	−24.2744	−0.972533
$624$	8.80443	0.352459
$625$	0	0
$626$	22.0599	0.881692
$627$	−0.596609	−0.0238263
$628$	13.0999	0.522743
$629$	7.22616	0.288126
$630$	0	0
$631$	−2.75194	−0.109553	−0.0547766	−	0.998499i	$-0.517445\pi$
$631$	−2.75194	−0.109553	−0.0547766	+	0.998499i	$0.517445\pi$
$632$	3.72618	0.148220
$633$	2.71876	0.108061
$634$	23.4888	0.932861
$635$	0	0
$636$	−8.82240	−0.349831
$637$	−6.07964	−0.240884
$638$	10.6425	0.421340
$639$	−3.30992	−0.130939
$640$	0	0
$641$	−25.7864	−1.01850	−0.509252	−	0.860618i	$-0.670078\pi$
$641$	−25.7864	−1.01850	−0.509252	+	0.860618i	$0.670078\pi$
$642$	−3.66682	−0.144718
$643$	32.1532	1.26800	0.634000	−	0.773333i	$-0.281412\pi$
$643$	32.1532	1.26800	0.634000	+	0.773333i	$0.281412\pi$
$644$	−14.2647	−0.562108
$645$	0	0
$646$	−5.28817	−0.208060
$647$	22.2482	0.874666	0.437333	−	0.899300i	$-0.355923\pi$
$647$	22.2482	0.874666	0.437333	+	0.899300i	$0.355923\pi$
$648$	−6.91307	−0.271571
$649$	−1.87114	−0.0734488
$650$	0	0
$651$	−5.15562	−0.202065
$652$	15.0646	0.589974
$653$	0.228402	0.00893805	0.00446902	−	0.999990i	$-0.498577\pi$
$653$	0.228402	0.00893805	0.00446902	+	0.999990i	$0.498577\pi$
$654$	18.5288	0.724531
$655$	0	0
$656$	−10.3420	−0.403787
$657$	−19.6045	−0.764843
$658$	−36.8721	−1.43742
$659$	10.3016	0.401294	0.200647	−	0.979664i	$-0.435696\pi$
$659$	10.3016	0.401294	0.200647	+	0.979664i	$0.435696\pi$
$660$	0	0
$661$	4.41139	0.171583	0.0857916	−	0.996313i	$-0.472658\pi$
$661$	4.41139	0.171583	0.0857916	+	0.996313i	$0.472658\pi$
$662$	41.6939	1.62048
$663$	5.21521	0.202542
$664$	−20.2252	−0.784888
$665$	0	0
$666$	12.1529	0.470917
$667$	27.1615	1.05170
$668$	−11.2892	−0.436792
$669$	2.05386	0.0794069
$670$	0	0
$671$	10.3158	0.398237
$672$	−8.79120	−0.339128
$673$	−29.0701	−1.12057	−0.560285	−	0.828300i	$-0.689308\pi$
$673$	−29.0701	−1.12057	−0.560285	+	0.828300i	$0.689308\pi$
$674$	37.1408	1.43061
$675$	0	0
$676$	−5.19409	−0.199773
$677$	28.5867	1.09868	0.549338	−	0.835600i	$-0.314880\pi$
$677$	28.5867	1.09868	0.549338	+	0.835600i	$0.314880\pi$
$678$	−21.1216	−0.811171
$679$	37.9152	1.45505
$680$	0	0
$681$	−10.6132	−0.406699
$682$	−7.09159	−0.271551
$683$	6.47289	0.247678	0.123839	−	0.992302i	$-0.460479\pi$
$683$	6.47289	0.247678	0.123839	+	0.992302i	$0.460479\pi$
$684$	−3.60552	−0.137860
$685$	0	0
$686$	36.9111	1.40927
$687$	11.6157	0.443166
$688$	34.4934	1.31505
$689$	32.8762	1.25248
$690$	0	0
$691$	14.8056	0.563231	0.281616	−	0.959527i	$-0.409130\pi$
$691$	14.8056	0.563231	0.281616	+	0.959527i	$0.409130\pi$
$692$	7.37747	0.280450
$693$	5.90911	0.224469
$694$	−63.2796	−2.40206
$695$	0	0
$696$	4.04056	0.153157
$697$	−6.12598	−0.232038
$698$	−32.6181	−1.23461
$699$	4.59862	0.173936
$700$	0	0
$701$	10.0602	0.379968	0.189984	−	0.981787i	$-0.439156\pi$
$701$	10.0602	0.379968	0.189984	+	0.981787i	$0.439156\pi$
$702$	18.7226	0.706639
$703$	−2.50615	−0.0945211
$704$	−2.35682	−0.0888259
$705$	0	0
$706$	−8.14036	−0.306367
$707$	23.8072	0.895363
$708$	1.52228	0.0572107
$709$	−42.9072	−1.61141	−0.805707	−	0.592314i	$-0.798214\pi$
$709$	−42.9072	−1.61141	−0.805707	+	0.592314i	$0.798214\pi$
$710$	0	0
$711$	8.44154	0.316582
$712$	−12.6768	−0.475084
$713$	−18.0990	−0.677812
$714$	−7.05093	−0.263874
$715$	0	0
$716$	19.8116	0.740396
$717$	14.1259	0.527541
$718$	−46.2299	−1.72529
$719$	16.6275	0.620099	0.310050	−	0.950720i	$-0.399654\pi$
$719$	16.6275	0.620099	0.310050	+	0.950720i	$0.399654\pi$
$720$	0	0
$721$	−35.0280	−1.30451
$722$	1.83402	0.0682552
$723$	−5.29441	−0.196901
$724$	−11.4530	−0.425649
$725$	0	0
$726$	−1.09419	−0.0406093
$727$	−10.0688	−0.373431	−0.186715	−	0.982414i	$-0.559784\pi$
$727$	−10.0688	−0.373431	−0.186715	+	0.982414i	$0.559784\pi$
$728$	7.90762	0.293076
$729$	−9.63444	−0.356831
$730$	0	0
$731$	20.4318	0.755698
$732$	−8.39246	−0.310194
$733$	23.6368	0.873044	0.436522	−	0.899694i	$-0.356210\pi$
$733$	23.6368	0.873044	0.436522	+	0.899694i	$0.356210\pi$
$734$	8.79008	0.324448
$735$	0	0
$736$	−30.8618	−1.13758
$737$	−2.18744	−0.0805755
$738$	−10.3027	−0.379246
$739$	23.5085	0.864776	0.432388	−	0.901688i	$-0.357671\pi$
$739$	23.5085	0.864776	0.432388	+	0.901688i	$0.357671\pi$
$740$	0	0
$741$	−1.80872	−0.0664449
$742$	−44.4483	−1.63175
$743$	7.37487	0.270558	0.135279	−	0.990808i	$-0.456807\pi$
$743$	7.37487	0.270558	0.135279	+	0.990808i	$0.456807\pi$
$744$	−2.69242	−0.0987090
$745$	0	0
$746$	59.1247	2.16471
$747$	−45.8194	−1.67644
$748$	−3.93186	−0.143763
$749$	−7.48941	−0.273657
$750$	0	0
$751$	−16.5579	−0.604208	−0.302104	−	0.953275i	$-0.597689\pi$
$751$	−16.5579	−0.604208	−0.302104	+	0.953275i	$0.597689\pi$
$752$	−43.7898	−1.59685
$753$	4.08529	0.148876
$754$	32.2644	1.17500
$755$	0	0
$756$	−10.2619	−0.373223
$757$	45.2848	1.64590	0.822952	−	0.568111i	$-0.192325\pi$
$757$	45.2848	1.64590	0.822952	+	0.568111i	$0.192325\pi$
$758$	−53.6831	−1.94986
$759$	−2.79257	−0.101364
$760$	0	0
$761$	−32.4848	−1.17757	−0.588787	−	0.808288i	$-0.700394\pi$
$761$	−32.4848	−1.17757	−0.588787	+	0.808288i	$0.700394\pi$
$762$	−10.2062	−0.369733
$763$	37.8446	1.37006
$764$	0.666632	0.0241179
$765$	0	0
$766$	−40.7898	−1.47380
$767$	−5.67268	−0.204829
$768$	−12.5114	−0.451467
$769$	−34.7020	−1.25139	−0.625693	−	0.780070i	$-0.715184\pi$
$769$	−34.7020	−1.25139	−0.625693	+	0.780070i	$0.715184\pi$
$770$	0	0
$771$	4.14568	0.149303
$772$	−14.9634	−0.538544
$773$	36.4485	1.31096	0.655481	−	0.755212i	$-0.272466\pi$
$773$	36.4485	1.31096	0.655481	+	0.755212i	$0.272466\pi$
$774$	34.3622	1.23512
$775$	0	0
$776$	19.8005	0.710796
$777$	−3.34155	−0.119877
$778$	46.2010	1.65638
$779$	2.12459	0.0761212
$780$	0	0
$781$	1.25183	0.0447942
$782$	−24.7525	−0.885149
$783$	19.5398	0.698295
$784$	9.76173	0.348633
$785$	0	0
$786$	−22.8115	−0.813661
$787$	13.7303	0.489433	0.244717	−	0.969595i	$-0.421305\pi$
$787$	13.7303	0.489433	0.244717	+	0.969595i	$0.421305\pi$
$788$	9.28864	0.330894
$789$	15.0586	0.536100
$790$	0	0
$791$	−43.1404	−1.53390
$792$	3.08592	0.109653
$793$	31.2740	1.11057
$794$	4.55642	0.161701
$795$	0	0
$796$	−9.91173	−0.351312
$797$	18.7861	0.665438	0.332719	−	0.943026i	$-0.392034\pi$
$797$	18.7861	0.665438	0.332719	+	0.943026i	$0.392034\pi$
$798$	2.44537	0.0865653
$799$	−25.9384	−0.917635
$800$	0	0
$801$	−28.7189	−1.01473
$802$	−10.3039	−0.363842
$803$	7.41454	0.261653
$804$	1.77960	0.0627618
$805$	0	0
$806$	−21.4993	−0.757281
$807$	−6.15978	−0.216835
$808$	12.4329	0.437386
$809$	−19.5394	−0.686968	−0.343484	−	0.939159i	$-0.611607\pi$
$809$	−19.5394	−0.686968	−0.343484	+	0.939159i	$0.611607\pi$
$810$	0	0
$811$	30.9500	1.08680	0.543400	−	0.839474i	$-0.317137\pi$
$811$	30.9500	1.08680	0.543400	+	0.839474i	$0.317137\pi$
$812$	−17.6842	−0.620595
$813$	17.4082	0.610532
$814$	−4.59633	−0.161101
$815$	0	0
$816$	−8.37378	−0.293141
$817$	−7.08607	−0.247910
$818$	24.3944	0.852932
$819$	17.9144	0.625981
$820$	0	0
$821$	−41.5384	−1.44970	−0.724850	−	0.688907i	$-0.758091\pi$
$821$	−41.5384	−1.44970	−0.724850	+	0.688907i	$0.758091\pi$
$822$	12.4122	0.432925
$823$	1.03434	0.0360549	0.0180274	−	0.999837i	$-0.494261\pi$
$823$	1.03434	0.0360549	0.0180274	+	0.999837i	$0.494261\pi$
$824$	−18.2927	−0.637256
$825$	0	0
$826$	7.66942	0.266853
$827$	28.5352	0.992266	0.496133	−	0.868246i	$-0.334753\pi$
$827$	28.5352	0.992266	0.496133	+	0.868246i	$0.334753\pi$
$828$	−16.8765	−0.586498
$829$	45.3435	1.57485	0.787423	−	0.616413i	$-0.211415\pi$
$829$	45.3435	1.57485	0.787423	+	0.616413i	$0.211415\pi$
$830$	0	0
$831$	−7.07520	−0.245436
$832$	−7.14508	−0.247711
$833$	5.78226	0.200344
$834$	1.50961	0.0522735
$835$	0	0
$836$	1.36363	0.0471621
$837$	−13.0203	−0.450047
$838$	−49.2354	−1.70081
$839$	−54.9859	−1.89832	−0.949162	−	0.314787i	$-0.898067\pi$
$839$	−54.9859	−1.89832	−0.949162	+	0.314787i	$0.898067\pi$
$840$	0	0
$841$	4.67262	0.161125
$842$	−69.1945	−2.38460
$843$	−5.64328	−0.194365
$844$	−6.21409	−0.213898
$845$	0	0
$846$	−43.6232	−1.49980
$847$	−2.23486	−0.0767909
$848$	−52.7874	−1.81273
$849$	−11.1304	−0.381993
$850$	0	0
$851$	−11.7306	−0.402121
$852$	−1.01844	−0.0348910
$853$	−31.0326	−1.06254	−0.531268	−	0.847204i	$-0.678284\pi$
$853$	−31.0326	−1.06254	−0.531268	+	0.847204i	$0.678284\pi$
$854$	−42.2823	−1.44687
$855$	0	0
$856$	−3.91120	−0.133682
$857$	−34.7215	−1.18606	−0.593032	−	0.805179i	$-0.702069\pi$
$857$	−34.7215	−1.18606	−0.593032	+	0.805179i	$0.702069\pi$
$858$	−3.31723	−0.113248
$859$	−25.1407	−0.857790	−0.428895	−	0.903354i	$-0.641097\pi$
$859$	−25.1407	−0.857790	−0.428895	+	0.903354i	$0.641097\pi$
$860$	0	0
$861$	2.83280	0.0965414
$862$	−41.2351	−1.40447
$863$	−14.7102	−0.500743	−0.250371	−	0.968150i	$-0.580553\pi$
$863$	−14.7102	−0.500743	−0.250371	+	0.968150i	$0.580553\pi$
$864$	−22.2018	−0.755321
$865$	0	0
$866$	47.7617	1.62301
$867$	5.18223	0.175998
$868$	11.7839	0.399970
$869$	−3.19265	−0.108303
$870$	0	0
$871$	−6.63159	−0.224703
$872$	19.7636	0.669280
$873$	44.8573	1.51819
$874$	8.58457	0.290377
$875$	0	0
$876$	−6.03213	−0.203807
$877$	41.6310	1.40578	0.702890	−	0.711298i	$-0.251893\pi$
$877$	41.6310	1.40578	0.702890	+	0.711298i	$0.251893\pi$
$878$	58.1056	1.96097
$879$	16.8180	0.567257
$880$	0	0
$881$	40.2961	1.35761	0.678805	−	0.734319i	$-0.262498\pi$
$881$	40.2961	1.35761	0.678805	+	0.734319i	$0.262498\pi$
$882$	9.72460	0.327444
$883$	−33.4683	−1.12630	−0.563149	−	0.826355i	$-0.690410\pi$
$883$	−33.4683	−1.12630	−0.563149	+	0.826355i	$0.690410\pi$
$884$	−11.9201	−0.400915
$885$	0	0
$886$	56.0860	1.88425
$887$	26.0465	0.874557	0.437278	−	0.899326i	$-0.355942\pi$
$887$	26.0465	0.874557	0.437278	+	0.899326i	$0.355942\pi$
$888$	−1.74506	−0.0585603
$889$	−20.8460	−0.699153
$890$	0	0
$891$	5.92321	0.198435
$892$	−4.69438	−0.157179
$893$	8.99585	0.301035
$894$	−20.3500	−0.680606
$895$	0	0
$896$	−19.8105	−0.661821
$897$	−8.46614	−0.282676
$898$	38.1575	1.27333
$899$	−22.4377	−0.748339
$900$	0	0
$901$	−31.2681	−1.04169
$902$	3.89653	0.129740
$903$	−9.44815	−0.314415
$904$	−22.5292	−0.749312
$905$	0	0
$906$	−8.41436	−0.279548
$907$	−30.2463	−1.00431	−0.502157	−	0.864777i	$-0.667460\pi$
$907$	−30.2463	−1.00431	−0.502157	+	0.864777i	$0.667460\pi$
$908$	24.2579	0.805027
$909$	28.1662	0.934214
$910$	0	0
$911$	−13.9292	−0.461494	−0.230747	−	0.973014i	$-0.574117\pi$
$911$	−13.9292	−0.461494	−0.230747	+	0.973014i	$0.574117\pi$
$912$	2.90416	0.0961663
$913$	17.3292	0.573512
$914$	5.28836	0.174923
$915$	0	0
$916$	−26.5492	−0.877211
$917$	−46.5921	−1.53861
$918$	−17.8068	−0.587713
$919$	31.6978	1.04562	0.522808	−	0.852451i	$-0.324885\pi$
$919$	31.6978	1.04562	0.522808	+	0.852451i	$0.324885\pi$
$920$	0	0
$921$	−5.33049	−0.175646
$922$	−40.1926	−1.32367
$923$	3.79514	0.124919
$924$	1.81818	0.0598138
$925$	0	0
$926$	10.0774	0.331164
$927$	−41.4415	−1.36112
$928$	−38.2600	−1.25595
$929$	−29.7520	−0.976132	−0.488066	−	0.872807i	$-0.662298\pi$
$929$	−29.7520	−0.976132	−0.488066	+	0.872807i	$0.662298\pi$
$930$	0	0
$931$	−2.00538	−0.0657237
$932$	−10.5108	−0.344291
$933$	−8.25497	−0.270256
$934$	33.5262	1.09701
$935$	0	0
$936$	9.35547	0.305793
$937$	14.5055	0.473873	0.236937	−	0.971525i	$-0.423857\pi$
$937$	14.5055	0.473873	0.236937	+	0.971525i	$0.423857\pi$
$938$	8.96587	0.292746
$939$	−7.17612	−0.234184
$940$	0	0
$941$	26.4270	0.861497	0.430749	−	0.902472i	$-0.358250\pi$
$941$	26.4270	0.861497	0.430749	+	0.902472i	$0.358250\pi$
$942$	−10.5115	−0.342484
$943$	9.94463	0.323842
$944$	9.10831	0.296450
$945$	0	0
$946$	−12.9960	−0.422536
$947$	47.6292	1.54774	0.773870	−	0.633344i	$-0.218318\pi$
$947$	47.6292	1.54774	0.773870	+	0.633344i	$0.218318\pi$
$948$	2.59739	0.0843593
$949$	22.4784	0.729679
$950$	0	0
$951$	−7.64095	−0.247775
$952$	−7.52083	−0.243752
$953$	26.1535	0.847195	0.423598	−	0.905850i	$-0.360767\pi$
$953$	26.1535	0.847195	0.423598	+	0.905850i	$0.360767\pi$
$954$	−52.5866	−1.70255
$955$	0	0
$956$	−32.2866	−1.04422
$957$	−3.46201	−0.111911
$958$	21.8312	0.705335
$959$	25.3516	0.818647
$960$	0	0
$961$	−16.0487	−0.517700
$962$	−13.9345	−0.449267
$963$	−8.86068	−0.285531
$964$	12.1011	0.389750
$965$	0	0
$966$	11.4462	0.368274
$967$	21.0604	0.677256	0.338628	−	0.940920i	$-0.390037\pi$
$967$	21.0604	0.677256	0.338628	+	0.940920i	$0.390037\pi$
$968$	−1.16712	−0.0375125
$969$	1.72025	0.0552623
$970$	0	0
$971$	−23.9140	−0.767436	−0.383718	−	0.923450i	$-0.625356\pi$
$971$	−23.9140	−0.767436	−0.383718	+	0.923450i	$0.625356\pi$
$972$	−18.5941	−0.596406
$973$	3.08335	0.0988475
$974$	20.8409	0.667787
$975$	0	0
$976$	−50.2150	−1.60734
$977$	−51.3119	−1.64161	−0.820806	−	0.571207i	$-0.806475\pi$
$977$	−51.3119	−1.64161	−0.820806	+	0.571207i	$0.806475\pi$
$978$	−12.0880	−0.386531
$979$	10.8617	0.347141
$980$	0	0
$981$	44.7737	1.42951
$982$	24.7258	0.789033
$983$	28.6268	0.913053	0.456527	−	0.889710i	$-0.349093\pi$
$983$	28.6268	0.913053	0.456527	+	0.889710i	$0.349093\pi$
$984$	1.47937	0.0471607
$985$	0	0
$986$	−30.6862	−0.977249
$987$	11.9945	0.381790
$988$	4.13407	0.131522
$989$	−33.1681	−1.05468
$990$	0	0
$991$	34.0379	1.08125	0.540625	−	0.841263i	$-0.318188\pi$
$991$	34.0379	1.08125	0.540625	+	0.841263i	$0.318188\pi$
$992$	25.4945	0.809451
$993$	−13.5631	−0.430411
$994$	−5.13100	−0.162746
$995$	0	0
$996$	−14.0982	−0.446720
$997$	−37.9979	−1.20340	−0.601702	−	0.798721i	$-0.705511\pi$
$997$	−37.9979	−1.20340	−0.601702	+	0.798721i	$0.705511\pi$
$998$	−6.01387	−0.190366
$999$	−8.43894	−0.266996

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5225.2.a.y.1.11	yes	15
5.4	even	2		5225.2.a.r.1.5	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5225.2.a.r.1.5	✓	15	5.4	even	2
5225.2.a.y.1.11	yes	15	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5225.a (trivial)

\(f(q)\)	\(=\)	\(q+1.83402 q^{2} -0.596609 q^{3} +1.36363 q^{4} -1.09419 q^{6} -2.23486 q^{7} -1.16712 q^{8} -2.64406 q^{9} +1.00000 q^{11} -0.813554 q^{12} +3.03166 q^{13} -4.09879 q^{14} -4.86777 q^{16} -2.88338 q^{17} -4.84926 q^{18} +1.00000 q^{19} +1.33334 q^{21} +1.83402 q^{22} +4.68074 q^{23} +0.696311 q^{24} +5.56013 q^{26} +3.36730 q^{27} -3.04753 q^{28} +5.80281 q^{29} -3.86669 q^{31} -6.59336 q^{32} -0.596609 q^{33} -5.28817 q^{34} -3.60552 q^{36} -2.50615 q^{37} +1.83402 q^{38} -1.80872 q^{39} +2.12459 q^{41} +2.44537 q^{42} -7.08607 q^{43} +1.36363 q^{44} +8.58457 q^{46} +8.99585 q^{47} +2.90416 q^{48} -2.00538 q^{49} +1.72025 q^{51} +4.13407 q^{52} +10.8443 q^{53} +6.17569 q^{54} +2.60834 q^{56} -0.596609 q^{57} +10.6425 q^{58} -1.87114 q^{59} +10.3158 q^{61} -7.09159 q^{62} +5.90911 q^{63} -2.35682 q^{64} -1.09419 q^{66} -2.18744 q^{67} -3.93186 q^{68} -2.79257 q^{69} +1.25183 q^{71} +3.08592 q^{72} +7.41454 q^{73} -4.59633 q^{74} +1.36363 q^{76} -2.23486 q^{77} -3.31723 q^{78} -3.19265 q^{79} +5.92321 q^{81} +3.89653 q^{82} +17.3292 q^{83} +1.81818 q^{84} -12.9960 q^{86} -3.46201 q^{87} -1.16712 q^{88} +10.8617 q^{89} -6.77536 q^{91} +6.38280 q^{92} +2.30690 q^{93} +16.4986 q^{94} +3.93366 q^{96} -16.9653 q^{97} -3.67791 q^{98} -2.64406 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 5 q^{2} + 4 q^{3} + 17 q^{4} - q^{6} + 21 q^{7} + 9 q^{8} + 15 q^{9} + 15 q^{11} + 11 q^{12} + 13 q^{13} + 9 q^{14} + 21 q^{16} + 17 q^{17} + 22 q^{18} + 15 q^{19} + 6 q^{21} + 5 q^{22} + 26 q^{23}+ \cdots + 15 q^{99}+O(q^{100}) \) 15 * q + 5 * q^2 + 4 * q^3 + 17 * q^4 - q^6 + 21 * q^7 + 9 * q^8 + 15 * q^9 + 15 * q^11 + 11 * q^12 + 13 * q^13 + 9 * q^14 + 21 * q^16 + 17 * q^17 + 22 * q^18 + 15 * q^19 + 6 * q^21 + 5 * q^22 + 26 * q^23 + q^24 + 3 * q^26 + q^27 + 46 * q^28 + 9 * q^29 + 14 * q^31 + 18 * q^32 + 4 * q^33 - 13 * q^34 + 12 * q^36 + 9 * q^37 + 5 * q^38 - 22 * q^39 + 4 * q^41 - 6 * q^42 + 28 * q^43 + 17 * q^44 + 27 * q^46 + 14 * q^47 - 4 * q^48 + 32 * q^49 - 40 * q^51 + 14 * q^52 + 3 * q^53 - 39 * q^54 + 34 * q^56 + 4 * q^57 + 26 * q^58 + q^59 + 2 * q^61 - 3 * q^62 + 45 * q^63 + 5 * q^64 - q^66 + 37 * q^67 + 26 * q^68 - 7 * q^69 - 7 * q^71 + 16 * q^72 + 42 * q^73 - 43 * q^74 + 17 * q^76 + 21 * q^77 - 64 * q^78 - 10 * q^79 + 31 * q^81 + 22 * q^82 + 14 * q^83 - 32 * q^84 + 37 * q^86 + 29 * q^87 + 9 * q^88 + 15 * q^89 - 22 * q^91 + 26 * q^92 - 18 * q^93 - 44 * q^94 + 71 * q^96 + 8 * q^97 - 10 * q^98 + 15 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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