LMFDB - Embedded newform 1525.2.a.m.1.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1525,2,Mod(1,1525)] 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("1525.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1525, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1525 = 5^{2} \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1525.a (trivial)

Newform invariants

comment:select newform

sage:traces = [10,5,7,11,0,4,6] 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:	$12.1771863082$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 5x^{9} - 3x^{8} + 39x^{7} - 9x^{6} - 106x^{5} + 37x^{4} + 118x^{3} - 39x^{2} - 45x + 13 $ x^10 - 5x^9 - 3x^8 + 39x^7 - 9x^6 - 106x^5 + 37x^4 + 118x^3 - 39x^2 - 45*x + 13
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			1.9
Root			$-1.27001$ of defining polynomial
Character	$\chi$	$=$	1525.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.27001 q^{2} -2.54073 q^{3} +3.15294 q^{4} -5.76747 q^{6} +1.79213 q^{7} +2.61719 q^{8} +3.45529 q^{9} -1.88895 q^{11} -8.01076 q^{12} +1.55451 q^{13} +4.06815 q^{14} -0.364847 q^{16} +3.70745 q^{17} +7.84355 q^{18} +2.30546 q^{19} -4.55332 q^{21} -4.28792 q^{22} +5.82712 q^{23} -6.64955 q^{24} +3.52874 q^{26} -1.15678 q^{27} +5.65048 q^{28} +2.39615 q^{29} +1.26955 q^{31} -6.06258 q^{32} +4.79930 q^{33} +8.41594 q^{34} +10.8943 q^{36} +1.78665 q^{37} +5.23341 q^{38} -3.94958 q^{39} -3.18643 q^{41} -10.3361 q^{42} +3.67580 q^{43} -5.95573 q^{44} +13.2276 q^{46} +7.50956 q^{47} +0.926976 q^{48} -3.78826 q^{49} -9.41961 q^{51} +4.90127 q^{52} +8.06339 q^{53} -2.62590 q^{54} +4.69034 q^{56} -5.85753 q^{57} +5.43929 q^{58} +2.46134 q^{59} +1.00000 q^{61} +2.88189 q^{62} +6.19234 q^{63} -13.0324 q^{64} +10.8944 q^{66} +15.5651 q^{67} +11.6894 q^{68} -14.8051 q^{69} +3.85186 q^{71} +9.04315 q^{72} -10.3093 q^{73} +4.05570 q^{74} +7.26896 q^{76} -3.38524 q^{77} -8.96557 q^{78} +14.0572 q^{79} -7.42683 q^{81} -7.23322 q^{82} +4.14791 q^{83} -14.3563 q^{84} +8.34409 q^{86} -6.08797 q^{87} -4.94372 q^{88} -18.1462 q^{89} +2.78588 q^{91} +18.3726 q^{92} -3.22558 q^{93} +17.0468 q^{94} +15.4034 q^{96} -9.12471 q^{97} -8.59939 q^{98} -6.52686 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 5 q^{2} + 7 q^{3} + 11 q^{4} + 4 q^{6} + 6 q^{7} + 15 q^{8} + 11 q^{9} + 19 q^{12} + 3 q^{13} - 5 q^{14} + 17 q^{16} + 11 q^{17} + 5 q^{18} - 2 q^{19} + 9 q^{21} - 6 q^{22} + 33 q^{23} - 7 q^{24}+ \cdots - 12 q^{99}+O(q^{100}) $ 10 * q + 5 * q^2 + 7 * q^3 + 11 * q^4 + 4 * q^6 + 6 * q^7 + 15 * q^8 + 11 * q^9 + 19 * q^12 + 3 * q^13 - 5 * q^14 + 17 * q^16 + 11 * q^17 + 5 * q^18 - 2 * q^19 + 9 * q^21 - 6 * q^22 + 33 * q^23 - 7 * q^24 + 12 * q^26 + 19 * q^27 + 21 * q^28 - 3 * q^29 + 20 * q^32 + 7 * q^33 - 3 * q^34 + 45 * q^36 - q^37 + 11 * q^38 - 22 * q^39 + 4 * q^41 - 16 * q^42 + 30 * q^43 - 22 * q^44 + 23 * q^46 + 5 * q^47 + 20 * q^48 - 10 * q^49 - q^51 - 35 * q^52 + 35 * q^53 - 52 * q^54 + 13 * q^56 + 4 * q^57 + 26 * q^58 - 4 * q^59 + 10 * q^61 + 22 * q^62 + 47 * q^63 + q^64 + 25 * q^66 + 28 * q^68 + 26 * q^69 - 11 * q^71 - 27 * q^72 - 9 * q^73 - 41 * q^74 + 44 * q^76 - 4 * q^77 + 51 * q^78 + 16 * q^79 + 14 * q^81 - 22 * q^82 + 18 * q^83 + 2 * q^84 + 7 * q^86 - q^87 - 32 * q^88 - 7 * q^89 - 12 * q^91 + 80 * q^92 - 3 * q^93 - 73 * q^94 + 53 * q^96 + 8 * q^97 - 8 * q^98 - 12 * 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 $

$2$	2.27001	1.60514	0.802569	−	0.596559i	$-0.203466\pi$
$2$	2.27001	1.60514	0.802569	+	0.596559i	$0.203466\pi$
$3$	−2.54073	−1.46689	−0.733445	−	0.679749i	$-0.762089\pi$
$3$	−2.54073	−1.46689	−0.733445	+	0.679749i	$0.762089\pi$
$4$	3.15294	1.57647
$5$	0	0
$5$	0	0
$6$	−5.76747	−2.35456
$7$	1.79213	0.677362	0.338681	−	0.940901i	$-0.390019\pi$
$7$	1.79213	0.677362	0.338681	+	0.940901i	$0.390019\pi$
$8$	2.61719	0.925315
$9$	3.45529	1.15176
$10$	0	0
$11$	−1.88895	−0.569539	−0.284769	−	0.958596i	$-0.591917\pi$
$11$	−1.88895	−0.569539	−0.284769	+	0.958596i	$0.591917\pi$
$12$	−8.01076	−2.31251
$13$	1.55451	0.431142	0.215571	−	0.976488i	$-0.430839\pi$
$13$	1.55451	0.431142	0.215571	+	0.976488i	$0.430839\pi$
$14$	4.06815	1.08726
$15$	0	0
$16$	−0.364847	−0.0912117
$17$	3.70745	0.899188	0.449594	−	0.893233i	$-0.351569\pi$
$17$	3.70745	0.899188	0.449594	+	0.893233i	$0.351569\pi$
$18$	7.84355	1.84874
$19$	2.30546	0.528908	0.264454	−	0.964398i	$-0.414808\pi$
$19$	2.30546	0.528908	0.264454	+	0.964398i	$0.414808\pi$
$20$	0	0
$21$	−4.55332	−0.993615
$22$	−4.28792	−0.914188
$23$	5.82712	1.21504	0.607519	−	0.794305i	$-0.292165\pi$
$23$	5.82712	1.21504	0.607519	+	0.794305i	$0.292165\pi$
$24$	−6.64955	−1.35733
$25$	0	0
$26$	3.52874	0.692043
$27$	−1.15678	−0.222622
$28$	5.65048	1.06784
$29$	2.39615	0.444954	0.222477	−	0.974938i	$-0.428586\pi$
$29$	2.39615	0.444954	0.222477	+	0.974938i	$0.428586\pi$
$30$	0	0
$31$	1.26955	0.228018	0.114009	−	0.993480i	$-0.463631\pi$
$31$	1.26955	0.228018	0.114009	+	0.993480i	$0.463631\pi$
$32$	−6.06258	−1.07172
$33$	4.79930	0.835450
$34$	8.41594	1.44332
$35$	0	0
$36$	10.8943	1.81572
$37$	1.78665	0.293723	0.146861	−	0.989157i	$-0.453083\pi$
$37$	1.78665	0.293723	0.146861	+	0.989157i	$0.453083\pi$
$38$	5.23341	0.848970
$39$	−3.94958	−0.632438
$40$	0	0
$41$	−3.18643	−0.497636	−0.248818	−	0.968550i	$-0.580042\pi$
$41$	−3.18643	−0.497636	−0.248818	+	0.968550i	$0.580042\pi$
$42$	−10.3361	−1.59489
$43$	3.67580	0.560554	0.280277	−	0.959919i	$-0.409574\pi$
$43$	3.67580	0.560554	0.280277	+	0.959919i	$0.409574\pi$
$44$	−5.95573	−0.897861
$45$	0	0
$46$	13.2276	1.95030
$47$	7.50956	1.09538	0.547691	−	0.836681i	$-0.315507\pi$
$47$	7.50956	1.09538	0.547691	+	0.836681i	$0.315507\pi$
$48$	0.926976	0.133798
$49$	−3.78826	−0.541181
$50$	0	0
$51$	−9.41961	−1.31901
$52$	4.90127	0.679683
$53$	8.06339	1.10759	0.553796	−	0.832652i	$-0.313179\pi$
$53$	8.06339	1.10759	0.553796	+	0.832652i	$0.313179\pi$
$54$	−2.62590	−0.357339
$55$	0	0
$56$	4.69034	0.626773
$57$	−5.85753	−0.775849
$58$	5.43929	0.714214
$59$	2.46134	0.320439	0.160220	−	0.987081i	$-0.448780\pi$
$59$	2.46134	0.320439	0.160220	+	0.987081i	$0.448780\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	2.88189	0.366001
$63$	6.19234	0.780162
$64$	−13.0324	−1.62905
$65$	0	0
$66$	10.8944	1.34101
$67$	15.5651	1.90158	0.950789	−	0.309840i	$-0.100275\pi$
$67$	15.5651	1.90158	0.950789	+	0.309840i	$0.100275\pi$
$68$	11.6894	1.41754
$69$	−14.8051	−1.78233
$70$	0	0
$71$	3.85186	0.457132	0.228566	−	0.973528i	$-0.426596\pi$
$71$	3.85186	0.457132	0.228566	+	0.973528i	$0.426596\pi$
$72$	9.04315	1.06574
$73$	−10.3093	−1.20661	−0.603307	−	0.797509i	$-0.706151\pi$
$73$	−10.3093	−1.20661	−0.603307	+	0.797509i	$0.706151\pi$
$74$	4.05570	0.471466
$75$	0	0
$76$	7.26896	0.833807
$77$	−3.38524	−0.385784
$78$	−8.96557	−1.01515
$79$	14.0572	1.58156	0.790779	−	0.612102i	$-0.209676\pi$
$79$	14.0572	1.58156	0.790779	+	0.612102i	$0.209676\pi$
$80$	0	0
$81$	−7.42683	−0.825203
$82$	−7.23322	−0.798775
$83$	4.14791	0.455292	0.227646	−	0.973744i	$-0.426897\pi$
$83$	4.14791	0.455292	0.227646	+	0.973744i	$0.426897\pi$
$84$	−14.3563	−1.56640
$85$	0	0
$86$	8.34409	0.899766
$87$	−6.08797	−0.652699
$88$	−4.94372	−0.527002
$89$	−18.1462	−1.92349	−0.961747	−	0.273940i	$-0.911673\pi$
$89$	−18.1462	−1.92349	−0.961747	+	0.273940i	$0.911673\pi$
$90$	0	0
$91$	2.78588	0.292040
$92$	18.3726	1.91547
$93$	−3.22558	−0.334477
$94$	17.0468	1.75824
$95$	0	0
$96$	15.4034	1.57210
$97$	−9.12471	−0.926474	−0.463237	−	0.886234i	$-0.653312\pi$
$97$	−9.12471	−0.926474	−0.463237	+	0.886234i	$0.653312\pi$
$98$	−8.59939	−0.868670
$99$	−6.52686	−0.655974
$100$	0	0
$101$	−5.50449	−0.547717	−0.273858	−	0.961770i	$-0.588300\pi$
$101$	−5.50449	−0.547717	−0.273858	+	0.961770i	$0.588300\pi$
$102$	−21.3826	−2.11719
$103$	−12.2900	−1.21097	−0.605487	−	0.795855i	$-0.707022\pi$
$103$	−12.2900	−1.21097	−0.605487	+	0.795855i	$0.707022\pi$
$104$	4.06843	0.398943
$105$	0	0
$106$	18.3040	1.77784
$107$	6.21969	0.601281	0.300640	−	0.953738i	$-0.402800\pi$
$107$	6.21969	0.601281	0.300640	+	0.953738i	$0.402800\pi$
$108$	−3.64725	−0.350957
$109$	19.3394	1.85238	0.926190	−	0.377058i	$-0.123064\pi$
$109$	19.3394	1.85238	0.926190	+	0.377058i	$0.123064\pi$
$110$	0	0
$111$	−4.53938	−0.430859
$112$	−0.653854	−0.0617834
$113$	−4.20114	−0.395210	−0.197605	−	0.980282i	$-0.563316\pi$
$113$	−4.20114	−0.395210	−0.197605	+	0.980282i	$0.563316\pi$
$114$	−13.2967	−1.24535
$115$	0	0
$116$	7.55493	0.701458
$117$	5.37128	0.496575
$118$	5.58726	0.514349
$119$	6.64424	0.609076
$120$	0	0
$121$	−7.43188	−0.675626
$122$	2.27001	0.205517
$123$	8.09584	0.729977
$124$	4.00282	0.359464
$125$	0	0
$126$	14.0567	1.25227
$127$	−3.34841	−0.297124	−0.148562	−	0.988903i	$-0.547464\pi$
$127$	−3.34841	−0.297124	−0.148562	+	0.988903i	$0.547464\pi$
$128$	−17.4585	−1.54313
$129$	−9.33920	−0.822270
$130$	0	0
$131$	2.67400	0.233629	0.116814	−	0.993154i	$-0.462732\pi$
$131$	2.67400	0.233629	0.116814	+	0.993154i	$0.462732\pi$
$132$	15.1319	1.31706
$133$	4.13168	0.358262
$134$	35.3329	3.05230
$135$	0	0
$136$	9.70308	0.832032
$137$	3.28253	0.280445	0.140223	−	0.990120i	$-0.455218\pi$
$137$	3.28253	0.280445	0.140223	+	0.990120i	$0.455218\pi$
$138$	−33.6077	−2.86088
$139$	−10.3692	−0.879502	−0.439751	−	0.898120i	$-0.644933\pi$
$139$	−10.3692	−0.879502	−0.439751	+	0.898120i	$0.644933\pi$
$140$	0	0
$141$	−19.0797	−1.60680
$142$	8.74375	0.733760
$143$	−2.93638	−0.245552
$144$	−1.26065	−0.105054
$145$	0	0
$146$	−23.4022	−1.93678
$147$	9.62495	0.793852
$148$	5.63319	0.463045
$149$	18.7734	1.53797	0.768987	−	0.639264i	$-0.220761\pi$
$149$	18.7734	1.53797	0.768987	+	0.639264i	$0.220761\pi$
$150$	0	0
$151$	−17.5482	−1.42805	−0.714027	−	0.700118i	$-0.753131\pi$
$151$	−17.5482	−1.42805	−0.714027	+	0.700118i	$0.753131\pi$
$152$	6.03381	0.489406
$153$	12.8103	1.03565
$154$	−7.68452	−0.619237
$155$	0	0
$156$	−12.4528	−0.997020
$157$	−0.824434	−0.0657970	−0.0328985	−	0.999459i	$-0.510474\pi$
$157$	−0.824434	−0.0657970	−0.0328985	+	0.999459i	$0.510474\pi$
$158$	31.9100	2.53862
$159$	−20.4869	−1.62471
$160$	0	0
$161$	10.4430	0.823021
$162$	−16.8590	−1.32456
$163$	−13.7176	−1.07445	−0.537224	−	0.843439i	$-0.680527\pi$
$163$	−13.7176	−1.07445	−0.537224	+	0.843439i	$0.680527\pi$
$164$	−10.0466	−0.784509
$165$	0	0
$166$	9.41579	0.730807
$167$	1.43039	0.110687	0.0553433	−	0.998467i	$-0.482375\pi$
$167$	1.43039	0.110687	0.0553433	+	0.998467i	$0.482375\pi$
$168$	−11.9169	−0.919407
$169$	−10.5835	−0.814116
$170$	0	0
$171$	7.96603	0.609177
$172$	11.5896	0.883696
$173$	−5.68777	−0.432433	−0.216216	−	0.976345i	$-0.569372\pi$
$173$	−5.68777	−0.432433	−0.216216	+	0.976345i	$0.569372\pi$
$174$	−13.8197	−1.04767
$175$	0	0
$176$	0.689176	0.0519486
$177$	−6.25359	−0.470049
$178$	−41.1920	−3.08747
$179$	−24.9959	−1.86828	−0.934141	−	0.356905i	$-0.883832\pi$
$179$	−24.9959	−1.86828	−0.934141	+	0.356905i	$0.883832\pi$
$180$	0	0
$181$	−15.2043	−1.13013	−0.565065	−	0.825046i	$-0.691149\pi$
$181$	−15.2043	−1.13013	−0.565065	+	0.825046i	$0.691149\pi$
$182$	6.32397	0.468764
$183$	−2.54073	−0.187816
$184$	15.2506	1.12429
$185$	0	0
$186$	−7.32210	−0.536883
$187$	−7.00317	−0.512122
$188$	23.6772	1.72684
$189$	−2.07310	−0.150796
$190$	0	0
$191$	−10.4139	−0.753525	−0.376762	−	0.926310i	$-0.622963\pi$
$191$	−10.4139	−0.753525	−0.376762	+	0.926310i	$0.622963\pi$
$192$	33.1118	2.38964
$193$	−5.74404	−0.413465	−0.206733	−	0.978397i	$-0.566283\pi$
$193$	−5.74404	−0.413465	−0.206733	+	0.978397i	$0.566283\pi$
$194$	−20.7132	−1.48712
$195$	0	0
$196$	−11.9442	−0.853155
$197$	2.74698	0.195714	0.0978570	−	0.995200i	$-0.468801\pi$
$197$	2.74698	0.195714	0.0978570	+	0.995200i	$0.468801\pi$
$198$	−14.8160	−1.05293
$199$	−18.2071	−1.29066	−0.645332	−	0.763902i	$-0.723281\pi$
$199$	−18.2071	−1.29066	−0.645332	+	0.763902i	$0.723281\pi$
$200$	0	0
$201$	−39.5466	−2.78940
$202$	−12.4952	−0.879161
$203$	4.29422	0.301395
$204$	−29.6995	−2.07938
$205$	0	0
$206$	−27.8985	−1.94378
$207$	20.1344	1.39944
$208$	−0.567157	−0.0393252
$209$	−4.35488	−0.301233
$210$	0	0
$211$	16.0535	1.10517	0.552584	−	0.833457i	$-0.313642\pi$
$211$	16.0535	1.10517	0.552584	+	0.833457i	$0.313642\pi$
$212$	25.4234	1.74609
$213$	−9.78652	−0.670561
$214$	14.1188	0.965139
$215$	0	0
$216$	−3.02750	−0.205995
$217$	2.27520	0.154451
$218$	43.9006	2.97333
$219$	26.1931	1.76997
$220$	0	0
$221$	5.76325	0.387678
$222$	−10.3044	−0.691588
$223$	−15.8606	−1.06210	−0.531051	−	0.847340i	$-0.678203\pi$
$223$	−15.8606	−1.06210	−0.531051	+	0.847340i	$0.678203\pi$
$224$	−10.8649	−0.725944
$225$	0	0
$226$	−9.53663	−0.634367
$227$	13.4225	0.890886	0.445443	−	0.895310i	$-0.353046\pi$
$227$	13.4225	0.890886	0.445443	+	0.895310i	$0.353046\pi$
$228$	−18.4685	−1.22310
$229$	23.7938	1.57234	0.786169	−	0.618012i	$-0.212062\pi$
$229$	23.7938	1.57234	0.786169	+	0.618012i	$0.212062\pi$
$230$	0	0
$231$	8.60097	0.565902
$232$	6.27118	0.411723
$233$	21.8103	1.42884	0.714421	−	0.699716i	$-0.246690\pi$
$233$	21.8103	1.42884	0.714421	+	0.699716i	$0.246690\pi$
$234$	12.1928	0.797071
$235$	0	0
$236$	7.76046	0.505163
$237$	−35.7155	−2.31997
$238$	15.0825	0.977652
$239$	−3.55949	−0.230244	−0.115122	−	0.993351i	$-0.536726\pi$
$239$	−3.55949	−0.230244	−0.115122	+	0.993351i	$0.536726\pi$
$240$	0	0
$241$	−9.64534	−0.621311	−0.310655	−	0.950523i	$-0.600549\pi$
$241$	−9.64534	−0.621311	−0.310655	+	0.950523i	$0.600549\pi$
$242$	−16.8704	−1.08447
$243$	22.3399	1.43310
$244$	3.15294	0.201846
$245$	0	0
$246$	18.3776	1.17172
$247$	3.58385	0.228035
$248$	3.32265	0.210989
$249$	−10.5387	−0.667863
$250$	0	0
$251$	−14.9873	−0.945988	−0.472994	−	0.881066i	$-0.656827\pi$
$251$	−14.9873	−0.945988	−0.472994	+	0.881066i	$0.656827\pi$
$252$	19.5241	1.22990
$253$	−11.0071	−0.692011
$254$	−7.60093	−0.476925
$255$	0	0
$256$	−13.5662	−0.847888
$257$	−20.6149	−1.28592	−0.642961	−	0.765899i	$-0.722294\pi$
$257$	−20.6149	−1.28592	−0.642961	+	0.765899i	$0.722294\pi$
$258$	−21.2001	−1.31986
$259$	3.20191	0.198957
$260$	0	0
$261$	8.27941	0.512483
$262$	6.07001	0.375006
$263$	17.0831	1.05339	0.526696	−	0.850054i	$-0.323431\pi$
$263$	17.0831	1.05339	0.526696	+	0.850054i	$0.323431\pi$
$264$	12.5606	0.773054
$265$	0	0
$266$	9.37895	0.575060
$267$	46.1045	2.82155
$268$	49.0758	2.99778
$269$	10.2560	0.625319	0.312659	−	0.949865i	$-0.398780\pi$
$269$	10.2560	0.625319	0.312659	+	0.949865i	$0.398780\pi$
$270$	0	0
$271$	23.9865	1.45707	0.728537	−	0.685006i	$-0.240201\pi$
$271$	23.9865	1.45707	0.728537	+	0.685006i	$0.240201\pi$
$272$	−1.35265	−0.0820165
$273$	−7.07816	−0.428390
$274$	7.45136	0.450153
$275$	0	0
$276$	−46.6797	−2.80978
$277$	−4.91767	−0.295474	−0.147737	−	0.989027i	$-0.547199\pi$
$277$	−4.91767	−0.295474	−0.147737	+	0.989027i	$0.547199\pi$
$278$	−23.5381	−1.41172
$279$	4.38667	0.262623
$280$	0	0
$281$	−6.37605	−0.380363	−0.190182	−	0.981749i	$-0.560908\pi$
$281$	−6.37605	−0.380363	−0.190182	+	0.981749i	$0.560908\pi$
$282$	−43.3112	−2.57914
$283$	−7.52098	−0.447076	−0.223538	−	0.974695i	$-0.571761\pi$
$283$	−7.52098	−0.447076	−0.223538	+	0.974695i	$0.571761\pi$
$284$	12.1447	0.720654
$285$	0	0
$286$	−6.66560	−0.394145
$287$	−5.71050	−0.337080
$288$	−20.9480	−1.23437
$289$	−3.25483	−0.191460
$290$	0	0
$291$	23.1834	1.35904
$292$	−32.5046	−1.90219
$293$	23.2634	1.35906	0.679532	−	0.733646i	$-0.262183\pi$
$293$	23.2634	1.35906	0.679532	+	0.733646i	$0.262183\pi$
$294$	21.8487	1.27424
$295$	0	0
$296$	4.67599	0.271786
$297$	2.18509	0.126792
$298$	42.6157	2.46866
$299$	9.05829	0.523855
$300$	0	0
$301$	6.58751	0.379698
$302$	−39.8346	−2.29223
$303$	13.9854	0.803440
$304$	−0.841138	−0.0482426
$305$	0	0
$306$	29.0796	1.66237
$307$	−1.66925	−0.0952690	−0.0476345	−	0.998865i	$-0.515168\pi$
$307$	−1.66925	−0.0952690	−0.0476345	+	0.998865i	$0.515168\pi$
$308$	−10.6735	−0.608177
$309$	31.2257	1.77637
$310$	0	0
$311$	−9.05812	−0.513639	−0.256819	−	0.966459i	$-0.582675\pi$
$311$	−9.05812	−0.513639	−0.256819	+	0.966459i	$0.582675\pi$
$312$	−10.3368	−0.585205
$313$	3.29555	0.186275	0.0931377	−	0.995653i	$-0.470310\pi$
$313$	3.29555	0.186275	0.0931377	+	0.995653i	$0.470310\pi$
$314$	−1.87147	−0.105613
$315$	0	0
$316$	44.3215	2.49328
$317$	−19.4725	−1.09369	−0.546843	−	0.837235i	$-0.684171\pi$
$317$	−19.4725	−1.09369	−0.546843	+	0.837235i	$0.684171\pi$
$318$	−46.5054	−2.60789
$319$	−4.52620	−0.253419
$320$	0	0
$321$	−15.8025	−0.882012
$322$	23.7056	1.32106
$323$	8.54736	0.475588
$324$	−23.4163	−1.30091
$325$	0	0
$326$	−31.1392	−1.72464
$327$	−49.1362	−2.71724
$328$	−8.33947	−0.460470
$329$	13.4581	0.741970
$330$	0	0
$331$	1.91682	0.105358	0.0526789	−	0.998612i	$-0.483224\pi$
$331$	1.91682	0.105358	0.0526789	+	0.998612i	$0.483224\pi$
$332$	13.0781	0.717755
$333$	6.17339	0.338300
$334$	3.24699	0.177667
$335$	0	0
$336$	1.66126	0.0906294
$337$	−22.9030	−1.24760	−0.623802	−	0.781583i	$-0.714413\pi$
$337$	−22.9030	−1.24760	−0.623802	+	0.781583i	$0.714413\pi$
$338$	−24.0247	−1.30677
$339$	10.6740	0.579730
$340$	0	0
$341$	−2.39811	−0.129865
$342$	18.0830	0.977814
$343$	−19.3340	−1.04394
$344$	9.62024	0.518689
$345$	0	0
$346$	−12.9113	−0.694115
$347$	−8.99834	−0.483056	−0.241528	−	0.970394i	$-0.577649\pi$
$347$	−8.99834	−0.483056	−0.241528	+	0.970394i	$0.577649\pi$
$348$	−19.1950	−1.02896
$349$	−23.7367	−1.27060	−0.635298	−	0.772267i	$-0.719123\pi$
$349$	−23.7367	−1.27060	−0.635298	+	0.772267i	$0.719123\pi$
$350$	0	0
$351$	−1.79822	−0.0959818
$352$	11.4519	0.610387
$353$	24.4479	1.30123	0.650615	−	0.759408i	$-0.274511\pi$
$353$	24.4479	1.30123	0.650615	+	0.759408i	$0.274511\pi$
$354$	−14.1957	−0.754494
$355$	0	0
$356$	−57.2139	−3.03233
$357$	−16.8812	−0.893447
$358$	−56.7409	−2.99885
$359$	22.4903	1.18699	0.593496	−	0.804837i	$-0.297747\pi$
$359$	22.4903	1.18699	0.593496	+	0.804837i	$0.297747\pi$
$360$	0	0
$361$	−13.6849	−0.720257
$362$	−34.5140	−1.81402
$363$	18.8824	0.991068
$364$	8.78371	0.460392
$365$	0	0
$366$	−5.76747	−0.301471
$367$	33.3961	1.74326	0.871632	−	0.490161i	$-0.163062\pi$
$367$	33.3961	1.74326	0.871632	+	0.490161i	$0.163062\pi$
$368$	−2.12601	−0.110826
$369$	−11.0100	−0.573160
$370$	0	0
$371$	14.4507	0.750241
$372$	−10.1701	−0.527294
$373$	−20.2965	−1.05091	−0.525456	−	0.850821i	$-0.676105\pi$
$373$	−20.2965	−1.05091	−0.525456	+	0.850821i	$0.676105\pi$
$374$	−15.8973	−0.822028
$375$	0	0
$376$	19.6539	1.01357
$377$	3.72484	0.191839
$378$	−4.70595	−0.242048
$379$	27.4669	1.41088	0.705439	−	0.708771i	$-0.250750\pi$
$379$	27.4669	1.41088	0.705439	+	0.708771i	$0.250750\pi$
$380$	0	0
$381$	8.50740	0.435847
$382$	−23.6397	−1.20951
$383$	−24.4475	−1.24921	−0.624605	−	0.780941i	$-0.714740\pi$
$383$	−24.4475	−1.24921	−0.624605	+	0.780941i	$0.714740\pi$
$384$	44.3574	2.26360
$385$	0	0
$386$	−13.0390	−0.663669
$387$	12.7010	0.645626
$388$	−28.7697	−1.46056
$389$	−8.38339	−0.425055	−0.212527	−	0.977155i	$-0.568169\pi$
$389$	−8.38339	−0.425055	−0.212527	+	0.977155i	$0.568169\pi$
$390$	0	0
$391$	21.6037	1.09255
$392$	−9.91459	−0.500762
$393$	−6.79391	−0.342707
$394$	6.23566	0.314148
$395$	0	0
$396$	−20.5788	−1.03412
$397$	5.36898	0.269462	0.134731	−	0.990882i	$-0.456983\pi$
$397$	5.36898	0.269462	0.134731	+	0.990882i	$0.456983\pi$
$398$	−41.3302	−2.07169
$399$	−10.4975	−0.525531
$400$	0	0
$401$	19.6244	0.979997	0.489998	−	0.871723i	$-0.336997\pi$
$401$	19.6244	0.979997	0.489998	+	0.871723i	$0.336997\pi$
$402$	−89.7712	−4.47738
$403$	1.97353	0.0983083
$404$	−17.3553	−0.863459
$405$	0	0
$406$	9.74792	0.483781
$407$	−3.37488	−0.167287
$408$	−24.6529	−1.22050
$409$	−10.5028	−0.519328	−0.259664	−	0.965699i	$-0.583612\pi$
$409$	−10.5028	−0.519328	−0.259664	+	0.965699i	$0.583612\pi$
$410$	0	0
$411$	−8.34000	−0.411382
$412$	−38.7498	−1.90906
$413$	4.41105	0.217053
$414$	45.7053	2.24629
$415$	0	0
$416$	−9.42431	−0.462065
$417$	26.3452	1.29013
$418$	−9.88562	−0.483521
$419$	24.9607	1.21941	0.609704	−	0.792629i	$-0.291288\pi$
$419$	24.9607	1.21941	0.609704	+	0.792629i	$0.291288\pi$
$420$	0	0
$421$	35.6151	1.73577	0.867886	−	0.496763i	$-0.165478\pi$
$421$	35.6151	1.73577	0.867886	+	0.496763i	$0.165478\pi$
$422$	36.4416	1.77395
$423$	25.9477	1.26162
$424$	21.1034	1.02487
$425$	0	0
$426$	−22.2155	−1.07634
$427$	1.79213	0.0867273
$428$	19.6103	0.947901
$429$	7.46054	0.360198
$430$	0	0
$431$	38.5146	1.85518	0.927592	−	0.373596i	$-0.121875\pi$
$431$	38.5146	1.85518	0.927592	+	0.373596i	$0.121875\pi$
$432$	0.422047	0.0203057
$433$	−33.3139	−1.60096	−0.800482	−	0.599356i	$-0.795423\pi$
$433$	−33.3139	−1.60096	−0.800482	+	0.599356i	$0.795423\pi$
$434$	5.16473	0.247915
$435$	0	0
$436$	60.9760	2.92022
$437$	13.4342	0.642643
$438$	59.4587	2.84105
$439$	−6.08439	−0.290392	−0.145196	−	0.989403i	$-0.546381\pi$
$439$	−6.08439	−0.290392	−0.145196	+	0.989403i	$0.546381\pi$
$440$	0	0
$441$	−13.0896	−0.623313
$442$	13.0826	0.622277
$443$	−2.20245	−0.104642	−0.0523209	−	0.998630i	$-0.516662\pi$
$443$	−2.20245	−0.104642	−0.0523209	+	0.998630i	$0.516662\pi$
$444$	−14.3124	−0.679236
$445$	0	0
$446$	−36.0037	−1.70482
$447$	−47.6980	−2.25604
$448$	−23.3558	−1.10346
$449$	40.5612	1.91420	0.957101	−	0.289756i	$-0.0935741\pi$
$449$	40.5612	1.91420	0.957101	+	0.289756i	$0.0935741\pi$
$450$	0	0
$451$	6.01899	0.283423
$452$	−13.2460	−0.623037
$453$	44.5853	2.09480
$454$	30.4693	1.42999
$455$	0	0
$456$	−15.3303	−0.717905
$457$	6.08807	0.284788	0.142394	−	0.989810i	$-0.454520\pi$
$457$	6.08807	0.284788	0.142394	+	0.989810i	$0.454520\pi$
$458$	54.0121	2.52382
$459$	−4.28869	−0.200179
$460$	0	0
$461$	−30.7208	−1.43081	−0.715406	−	0.698709i	$-0.753758\pi$
$461$	−30.7208	−1.43081	−0.715406	+	0.698709i	$0.753758\pi$
$462$	19.5243	0.908352
$463$	7.44698	0.346091	0.173045	−	0.984914i	$-0.444639\pi$
$463$	7.44698	0.346091	0.173045	+	0.984914i	$0.444639\pi$
$464$	−0.874229	−0.0405851
$465$	0	0
$466$	49.5097	2.29349
$467$	2.52820	0.116991	0.0584955	−	0.998288i	$-0.481370\pi$
$467$	2.52820	0.116991	0.0584955	+	0.998288i	$0.481370\pi$
$468$	16.9353	0.782835
$469$	27.8947	1.28806
$470$	0	0
$471$	2.09466	0.0965169
$472$	6.44178	0.296507
$473$	−6.94338	−0.319257
$474$	−81.0745	−3.72388
$475$	0	0
$476$	20.9489	0.960190
$477$	27.8614	1.27568
$478$	−8.08007	−0.369574
$479$	26.5409	1.21268	0.606342	−	0.795204i	$-0.292636\pi$
$479$	26.5409	1.21268	0.606342	+	0.795204i	$0.292636\pi$
$480$	0	0
$481$	2.77735	0.126636
$482$	−21.8950	−0.997290
$483$	−26.5327	−1.20728
$484$	−23.4323	−1.06510
$485$	0	0
$486$	50.7117	2.30033
$487$	37.3298	1.69157	0.845787	−	0.533520i	$-0.179131\pi$
$487$	37.3298	1.69157	0.845787	+	0.533520i	$0.179131\pi$
$488$	2.61719	0.118474
$489$	34.8528	1.57610
$490$	0	0
$491$	−8.37678	−0.378039	−0.189019	−	0.981973i	$-0.560531\pi$
$491$	−8.37678	−0.378039	−0.189019	+	0.981973i	$0.560531\pi$
$492$	25.5257	1.15079
$493$	8.88361	0.400098
$494$	8.13536	0.366027
$495$	0	0
$496$	−0.463192	−0.0207979
$497$	6.90304	0.309644
$498$	−23.9230	−1.07201
$499$	−35.4688	−1.58780	−0.793902	−	0.608046i	$-0.791954\pi$
$499$	−35.4688	−1.58780	−0.793902	+	0.608046i	$0.791954\pi$
$500$	0	0
$501$	−3.63422	−0.162365
$502$	−34.0213	−1.51844
$503$	−11.7634	−0.524505	−0.262252	−	0.964999i	$-0.584465\pi$
$503$	−11.7634	−0.524505	−0.262252	+	0.964999i	$0.584465\pi$
$504$	16.2065	0.721895
$505$	0	0
$506$	−24.9862	−1.11077
$507$	26.8898	1.19422
$508$	−10.5573	−0.468406
$509$	3.50030	0.155148	0.0775740	−	0.996987i	$-0.475283\pi$
$509$	3.50030	0.155148	0.0775740	+	0.996987i	$0.475283\pi$
$510$	0	0
$511$	−18.4756	−0.817314
$512$	4.12166	0.182153
$513$	−2.66690	−0.117747
$514$	−46.7960	−2.06408
$515$	0	0
$516$	−29.4459	−1.29628
$517$	−14.1852	−0.623862
$518$	7.26835	0.319353
$519$	14.4511	0.634331
$520$	0	0
$521$	−10.2188	−0.447693	−0.223847	−	0.974624i	$-0.571861\pi$
$521$	−10.2188	−0.447693	−0.223847	+	0.974624i	$0.571861\pi$
$522$	18.7943	0.822606
$523$	5.57480	0.243769	0.121884	−	0.992544i	$-0.461106\pi$
$523$	5.57480	0.243769	0.121884	+	0.992544i	$0.461106\pi$
$524$	8.43097	0.368308
$525$	0	0
$526$	38.7789	1.69084
$527$	4.70680	0.205031
$528$	−1.75101	−0.0762028
$529$	10.9553	0.476317
$530$	0	0
$531$	8.50465	0.369071
$532$	13.0269	0.564790
$533$	−4.95332	−0.214552
$534$	104.658	4.52898
$535$	0	0
$536$	40.7367	1.75956
$537$	63.5078	2.74056
$538$	23.2812	1.00372
$539$	7.15583	0.308223
$540$	0	0
$541$	−26.1711	−1.12518	−0.562592	−	0.826735i	$-0.690196\pi$
$541$	−26.1711	−1.12518	−0.562592	+	0.826735i	$0.690196\pi$
$542$	54.4495	2.33881
$543$	38.6301	1.65778
$544$	−22.4767	−0.963680
$545$	0	0
$546$	−16.0675	−0.687625
$547$	16.3234	0.697940	0.348970	−	0.937134i	$-0.386531\pi$
$547$	16.3234	0.697940	0.348970	+	0.937134i	$0.386531\pi$
$548$	10.3496	0.442113
$549$	3.45529	0.147468
$550$	0	0
$551$	5.52422	0.235340
$552$	−38.7477	−1.64921
$553$	25.1923	1.07129
$554$	−11.1632	−0.474277
$555$	0	0
$556$	−32.6934	−1.38651
$557$	35.2354	1.49297	0.746485	−	0.665402i	$-0.231740\pi$
$557$	35.2354	1.49297	0.746485	+	0.665402i	$0.231740\pi$
$558$	9.95779	0.421547
$559$	5.71405	0.241679
$560$	0	0
$561$	17.7931	0.751227
$562$	−14.4737	−0.610536
$563$	−0.551922	−0.0232608	−0.0116304	−	0.999932i	$-0.503702\pi$
$563$	−0.551922	−0.0232608	−0.0116304	+	0.999932i	$0.503702\pi$
$564$	−60.1573	−2.53308
$565$	0	0
$566$	−17.0727	−0.717619
$567$	−13.3098	−0.558961
$568$	10.0810	0.422991
$569$	−20.2087	−0.847193	−0.423596	−	0.905851i	$-0.639233\pi$
$569$	−20.2087	−0.847193	−0.423596	+	0.905851i	$0.639233\pi$
$570$	0	0
$571$	32.3598	1.35422	0.677108	−	0.735883i	$-0.263233\pi$
$571$	32.3598	1.35422	0.677108	+	0.735883i	$0.263233\pi$
$572$	−9.25823	−0.387106
$573$	26.4589	1.10534
$574$	−12.9629	−0.541060
$575$	0	0
$576$	−45.0308	−1.87628
$577$	26.9025	1.11997	0.559984	−	0.828504i	$-0.310807\pi$
$577$	26.9025	1.11997	0.559984	+	0.828504i	$0.310807\pi$
$578$	−7.38849	−0.307320
$579$	14.5940	0.606508
$580$	0	0
$581$	7.43360	0.308398
$582$	52.6265	2.18144
$583$	−15.2313	−0.630816
$584$	−26.9814	−1.11650
$585$	0	0
$586$	52.8082	2.18149
$587$	−14.2747	−0.589182	−0.294591	−	0.955623i	$-0.595183\pi$
$587$	−14.2747	−0.589182	−0.294591	+	0.955623i	$0.595183\pi$
$588$	30.3469	1.25148
$589$	2.92689	0.120601
$590$	0	0
$591$	−6.97932	−0.287091
$592$	−0.651852	−0.0267910
$593$	−42.2892	−1.73661	−0.868305	−	0.496031i	$-0.834790\pi$
$593$	−42.2892	−1.73661	−0.868305	+	0.496031i	$0.834790\pi$
$594$	4.96018	0.203518
$595$	0	0
$596$	59.1913	2.42457
$597$	46.2592	1.89326
$598$	20.5624	0.840859
$599$	8.34712	0.341054	0.170527	−	0.985353i	$-0.445453\pi$
$599$	8.34712	0.341054	0.170527	+	0.985353i	$0.445453\pi$
$600$	0	0
$601$	−9.59151	−0.391246	−0.195623	−	0.980679i	$-0.562673\pi$
$601$	−9.59151	−0.391246	−0.195623	+	0.980679i	$0.562673\pi$
$602$	14.9537	0.609468
$603$	53.7819	2.19017
$604$	−55.3285	−2.25129
$605$	0	0
$606$	31.7470	1.28963
$607$	18.5438	0.752669	0.376334	−	0.926484i	$-0.377184\pi$
$607$	18.5438	0.752669	0.376334	+	0.926484i	$0.377184\pi$
$608$	−13.9770	−0.566842
$609$	−10.9104	−0.442114
$610$	0	0
$611$	11.6737	0.472266
$612$	40.3902	1.63268
$613$	2.92204	0.118020	0.0590101	−	0.998257i	$-0.481206\pi$
$613$	2.92204	0.118020	0.0590101	+	0.998257i	$0.481206\pi$
$614$	−3.78921	−0.152920
$615$	0	0
$616$	−8.85980	−0.356971
$617$	−10.0848	−0.405997	−0.202999	−	0.979179i	$-0.565069\pi$
$617$	−10.0848	−0.405997	−0.202999	+	0.979179i	$0.565069\pi$
$618$	70.8825	2.85131
$619$	−4.52404	−0.181836	−0.0909182	−	0.995858i	$-0.528980\pi$
$619$	−4.52404	−0.181836	−0.0909182	+	0.995858i	$0.528980\pi$
$620$	0	0
$621$	−6.74068	−0.270494
$622$	−20.5620	−0.824461
$623$	−32.5204	−1.30290
$624$	1.44099	0.0576858
$625$	0	0
$626$	7.48093	0.298998
$627$	11.0646	0.441876
$628$	−2.59939	−0.103727
$629$	6.62390	0.264112
$630$	0	0
$631$	−10.0048	−0.398286	−0.199143	−	0.979970i	$-0.563816\pi$
$631$	−10.0048	−0.398286	−0.199143	+	0.979970i	$0.563816\pi$
$632$	36.7903	1.46344
$633$	−40.7876	−1.62116
$634$	−44.2028	−1.75552
$635$	0	0
$636$	−64.5939	−2.56131
$637$	−5.88888	−0.233326
$638$	−10.2745	−0.406772
$639$	13.3093	0.526508
$640$	0	0
$641$	1.42539	0.0562995	0.0281498	−	0.999604i	$-0.491038\pi$
$641$	1.42539	0.0562995	0.0281498	+	0.999604i	$0.491038\pi$
$642$	−35.8719	−1.41575
$643$	−32.3254	−1.27479	−0.637395	−	0.770538i	$-0.719988\pi$
$643$	−32.3254	−1.27479	−0.637395	+	0.770538i	$0.719988\pi$
$644$	32.9260	1.29747
$645$	0	0
$646$	19.4026	0.763384
$647$	12.3726	0.486419	0.243209	−	0.969974i	$-0.421800\pi$
$647$	12.3726	0.486419	0.243209	+	0.969974i	$0.421800\pi$
$648$	−19.4374	−0.763572
$649$	−4.64934	−0.182502
$650$	0	0
$651$	−5.78067	−0.226562
$652$	−43.2509	−1.69384
$653$	−23.1110	−0.904403	−0.452201	−	0.891916i	$-0.649361\pi$
$653$	−23.1110	−0.904403	−0.452201	+	0.891916i	$0.649361\pi$
$654$	−111.540	−4.36154
$655$	0	0
$656$	1.16256	0.0453903
$657$	−35.6217	−1.38973
$658$	30.5501	1.19096
$659$	−17.5368	−0.683138	−0.341569	−	0.939857i	$-0.610958\pi$
$659$	−17.5368	−0.683138	−0.341569	+	0.939857i	$0.610958\pi$
$660$	0	0
$661$	22.5115	0.875594	0.437797	−	0.899074i	$-0.355759\pi$
$661$	22.5115	0.875594	0.437797	+	0.899074i	$0.355759\pi$
$662$	4.35119	0.169114
$663$	−14.6429	−0.568681
$664$	10.8558	0.421289
$665$	0	0
$666$	14.0136	0.543018
$667$	13.9627	0.540637
$668$	4.50992	0.174494
$669$	40.2974	1.55799
$670$	0	0
$671$	−1.88895	−0.0729219
$672$	27.6048	1.06488
$673$	−39.6265	−1.52749	−0.763745	−	0.645518i	$-0.776642\pi$
$673$	−39.6265	−1.52749	−0.763745	+	0.645518i	$0.776642\pi$
$674$	−51.9899	−2.00258
$675$	0	0
$676$	−33.3692	−1.28343
$677$	−23.4938	−0.902938	−0.451469	−	0.892287i	$-0.649100\pi$
$677$	−23.4938	−0.902938	−0.451469	+	0.892287i	$0.649100\pi$
$678$	24.2300	0.930546
$679$	−16.3527	−0.627559
$680$	0	0
$681$	−34.1030	−1.30683
$682$	−5.44374	−0.208452
$683$	−25.2179	−0.964938	−0.482469	−	0.875913i	$-0.660260\pi$
$683$	−25.2179	−0.964938	−0.482469	+	0.875913i	$0.660260\pi$
$684$	25.1164	0.960350
$685$	0	0
$686$	−43.8883	−1.67566
$687$	−60.4535	−2.30645
$688$	−1.34110	−0.0511291
$689$	12.5346	0.477530
$690$	0	0
$691$	−2.09722	−0.0797819	−0.0398910	−	0.999204i	$-0.512701\pi$
$691$	−2.09722	−0.0797819	−0.0398910	+	0.999204i	$0.512701\pi$
$692$	−17.9332	−0.681718
$693$	−11.6970	−0.444332
$694$	−20.4263	−0.775372
$695$	0	0
$696$	−15.9333	−0.603952
$697$	−11.8135	−0.447469
$698$	−53.8825	−2.03948
$699$	−55.4141	−2.09595
$700$	0	0
$701$	−31.9346	−1.20615	−0.603076	−	0.797684i	$-0.706058\pi$
$701$	−31.9346	−1.20615	−0.603076	+	0.797684i	$0.706058\pi$
$702$	−4.08197	−0.154064
$703$	4.11903	0.155352
$704$	24.6175	0.927808
$705$	0	0
$706$	55.4969	2.08865
$707$	−9.86476	−0.371003
$708$	−19.7172	−0.741018
$709$	7.35176	0.276101	0.138051	−	0.990425i	$-0.455916\pi$
$709$	7.35176	0.276101	0.138051	+	0.990425i	$0.455916\pi$
$710$	0	0
$711$	48.5717	1.82158
$712$	−47.4920	−1.77984
$713$	7.39783	0.277051
$714$	−38.3204	−1.43411
$715$	0	0
$716$	−78.8106	−2.94529
$717$	9.04369	0.337743
$718$	51.0532	1.90529
$719$	9.98991	0.372561	0.186280	−	0.982497i	$-0.440357\pi$
$719$	9.98991	0.372561	0.186280	+	0.982497i	$0.440357\pi$
$720$	0	0
$721$	−22.0254	−0.820268
$722$	−31.0648	−1.15611
$723$	24.5062	0.911394
$724$	−47.9384	−1.78162
$725$	0	0
$726$	42.8632	1.59080
$727$	42.0264	1.55867	0.779337	−	0.626606i	$-0.215556\pi$
$727$	42.0264	1.55867	0.779337	+	0.626606i	$0.215556\pi$
$728$	7.29116	0.270229
$729$	−34.4790	−1.27700
$730$	0	0
$731$	13.6278	0.504043
$732$	−8.01076	−0.296086
$733$	13.2083	0.487858	0.243929	−	0.969793i	$-0.421564\pi$
$733$	13.2083	0.487858	0.243929	+	0.969793i	$0.421564\pi$
$734$	75.8095	2.79818
$735$	0	0
$736$	−35.3273	−1.30218
$737$	−29.4016	−1.08302
$738$	−24.9929	−0.920001
$739$	−9.40521	−0.345976	−0.172988	−	0.984924i	$-0.555342\pi$
$739$	−9.40521	−0.345976	−0.172988	+	0.984924i	$0.555342\pi$
$740$	0	0
$741$	−9.10557	−0.334502
$742$	32.8031	1.20424
$743$	28.5157	1.04614	0.523069	−	0.852290i	$-0.324787\pi$
$743$	28.5157	1.04614	0.523069	+	0.852290i	$0.324787\pi$
$744$	−8.44195	−0.309497
$745$	0	0
$746$	−46.0732	−1.68686
$747$	14.3322	0.524390
$748$	−22.0806	−0.807346
$749$	11.1465	0.407285
$750$	0	0
$751$	−12.5609	−0.458353	−0.229177	−	0.973385i	$-0.573603\pi$
$751$	−12.5609	−0.458353	−0.229177	+	0.973385i	$0.573603\pi$
$752$	−2.73984	−0.0999117
$753$	38.0786	1.38766
$754$	8.45541	0.307928
$755$	0	0
$756$	−6.53636	−0.237725
$757$	−35.6264	−1.29486	−0.647432	−	0.762124i	$-0.724157\pi$
$757$	−35.6264	−1.29486	−0.647432	+	0.762124i	$0.724157\pi$
$758$	62.3500	2.26465
$759$	27.9661	1.01510
$760$	0	0
$761$	46.8725	1.69913	0.849563	−	0.527487i	$-0.176866\pi$
$761$	46.8725	1.69913	0.849563	+	0.527487i	$0.176866\pi$
$762$	19.3119	0.699596
$763$	34.6588	1.25473
$764$	−32.8345	−1.18791
$765$	0	0
$766$	−55.4961	−2.00516
$767$	3.82617	0.138155
$768$	34.4680	1.24376
$769$	−13.6421	−0.491946	−0.245973	−	0.969277i	$-0.579107\pi$
$769$	−13.6421	−0.491946	−0.245973	+	0.969277i	$0.579107\pi$
$770$	0	0
$771$	52.3768	1.88630
$772$	−18.1106	−0.651816
$773$	−48.4545	−1.74279	−0.871393	−	0.490585i	$-0.836783\pi$
$773$	−48.4545	−1.74279	−0.871393	+	0.490585i	$0.836783\pi$
$774$	28.8313	1.03632
$775$	0	0
$776$	−23.8811	−0.857280
$777$	−8.13517	−0.291848
$778$	−19.0304	−0.682272
$779$	−7.34617	−0.263204
$780$	0	0
$781$	−7.27595	−0.260354
$782$	49.0407	1.75369
$783$	−2.77182	−0.0990567
$784$	1.38214	0.0493620
$785$	0	0
$786$	−15.4222	−0.550093
$787$	−25.7677	−0.918518	−0.459259	−	0.888302i	$-0.651885\pi$
$787$	−25.7677	−0.918518	−0.459259	+	0.888302i	$0.651885\pi$
$788$	8.66106	0.308537
$789$	−43.4036	−1.54521
$790$	0	0
$791$	−7.52900	−0.267700
$792$	−17.0820	−0.606983
$793$	1.55451	0.0552021
$794$	12.1876	0.432523
$795$	0	0
$796$	−57.4058	−2.03469
$797$	−18.8374	−0.667255	−0.333628	−	0.942705i	$-0.608273\pi$
$797$	−18.8374	−0.667255	−0.333628	+	0.942705i	$0.608273\pi$
$798$	−23.8294	−0.843550
$799$	27.8413	0.984955
$800$	0	0
$801$	−62.7005	−2.21541
$802$	44.5476	1.57303
$803$	19.4737	0.687213
$804$	−124.688	−4.39741
$805$	0	0
$806$	4.47992	0.157798
$807$	−26.0577	−0.917274
$808$	−14.4063	−0.506810
$809$	−29.5515	−1.03898	−0.519488	−	0.854478i	$-0.673877\pi$
$809$	−29.5515	−1.03898	−0.519488	+	0.854478i	$0.673877\pi$
$810$	0	0
$811$	7.53611	0.264629	0.132314	−	0.991208i	$-0.457759\pi$
$811$	7.53611	0.264629	0.132314	+	0.991208i	$0.457759\pi$
$812$	13.5394	0.475141
$813$	−60.9431	−2.13737
$814$	−7.66100	−0.268518
$815$	0	0
$816$	3.43672	0.120309
$817$	8.47439	0.296481
$818$	−23.8413	−0.833593
$819$	9.62603	0.336361
$820$	0	0
$821$	2.74112	0.0956659	0.0478329	−	0.998855i	$-0.484768\pi$
$821$	2.74112	0.0956659	0.0478329	+	0.998855i	$0.484768\pi$
$822$	−18.9319	−0.660325
$823$	−48.8628	−1.70325	−0.851626	−	0.524150i	$-0.824383\pi$
$823$	−48.8628	−1.70325	−0.851626	+	0.524150i	$0.824383\pi$
$824$	−32.1653	−1.12053
$825$	0	0
$826$	10.0131	0.348401
$827$	−24.8813	−0.865208	−0.432604	−	0.901584i	$-0.642405\pi$
$827$	−24.8813	−0.865208	−0.432604	+	0.901584i	$0.642405\pi$
$828$	63.4826	2.20617
$829$	31.5594	1.09610	0.548052	−	0.836445i	$-0.315370\pi$
$829$	31.5594	1.09610	0.548052	+	0.836445i	$0.315370\pi$
$830$	0	0
$831$	12.4945	0.433428
$832$	−20.2590	−0.702353
$833$	−14.0448	−0.486623
$834$	59.8039	2.07084
$835$	0	0
$836$	−13.7307	−0.474886
$837$	−1.46859	−0.0507619
$838$	56.6609	1.95732
$839$	−6.86045	−0.236849	−0.118425	−	0.992963i	$-0.537784\pi$
$839$	−6.86045	−0.236849	−0.118425	+	0.992963i	$0.537784\pi$
$840$	0	0
$841$	−23.2585	−0.802016
$842$	80.8465	2.78616
$843$	16.1998	0.557951
$844$	50.6157	1.74227
$845$	0	0
$846$	58.9016	2.02508
$847$	−13.3189	−0.457643
$848$	−2.94190	−0.101025
$849$	19.1088	0.655811
$850$	0	0
$851$	10.4110	0.356884
$852$	−30.8563	−1.05712
$853$	23.9434	0.819806	0.409903	−	0.912129i	$-0.365563\pi$
$853$	23.9434	0.819806	0.409903	+	0.912129i	$0.365563\pi$
$854$	4.06815	0.139209
$855$	0	0
$856$	16.2781	0.556374
$857$	25.9248	0.885575	0.442787	−	0.896627i	$-0.353990\pi$
$857$	25.9248	0.885575	0.442787	+	0.896627i	$0.353990\pi$
$858$	16.9355	0.578168
$859$	−48.8250	−1.66589	−0.832943	−	0.553358i	$-0.813346\pi$
$859$	−48.8250	−1.66589	−0.832943	+	0.553358i	$0.813346\pi$
$860$	0	0
$861$	14.5088	0.494459
$862$	87.4285	2.97783
$863$	−19.5973	−0.667100	−0.333550	−	0.942732i	$-0.608247\pi$
$863$	−19.5973	−0.667100	−0.333550	+	0.942732i	$0.608247\pi$
$864$	7.01305	0.238589
$865$	0	0
$866$	−75.6229	−2.56977
$867$	8.26963	0.280851
$868$	7.17358	0.243487
$869$	−26.5533	−0.900758
$870$	0	0
$871$	24.1960	0.819851
$872$	50.6148	1.71403
$873$	−31.5286	−1.06708
$874$	30.4957	1.03153
$875$	0	0
$876$	82.5854	2.79030
$877$	−5.66776	−0.191387	−0.0956934	−	0.995411i	$-0.530507\pi$
$877$	−5.66776	−0.191387	−0.0956934	+	0.995411i	$0.530507\pi$
$878$	−13.8116	−0.466119
$879$	−59.1060	−1.99360
$880$	0	0
$881$	34.6346	1.16687	0.583435	−	0.812160i	$-0.301708\pi$
$881$	34.6346	1.16687	0.583435	+	0.812160i	$0.301708\pi$
$882$	−29.7134	−1.00050
$883$	−16.3424	−0.549966	−0.274983	−	0.961449i	$-0.588672\pi$
$883$	−16.3424	−0.549966	−0.274983	+	0.961449i	$0.588672\pi$
$884$	18.1712	0.611163
$885$	0	0
$886$	−4.99959	−0.167964
$887$	−35.6633	−1.19746	−0.598729	−	0.800952i	$-0.704327\pi$
$887$	−35.6633	−1.19746	−0.598729	+	0.800952i	$0.704327\pi$
$888$	−11.8804	−0.398680
$889$	−6.00080	−0.201260
$890$	0	0
$891$	14.0289	0.469985
$892$	−50.0075	−1.67437
$893$	17.3130	0.579356
$894$	−108.275	−3.62126
$895$	0	0
$896$	−31.2880	−1.04526
$897$	−23.0146	−0.768437
$898$	92.0743	3.07256
$899$	3.04204	0.101458
$900$	0	0
$901$	29.8946	0.995933
$902$	13.6632	0.454933
$903$	−16.7371	−0.556975
$904$	−10.9952	−0.365694
$905$	0	0
$906$	101.209	3.36244
$907$	22.7809	0.756426	0.378213	−	0.925719i	$-0.376539\pi$
$907$	22.7809	0.756426	0.378213	+	0.925719i	$0.376539\pi$
$908$	42.3205	1.40445
$909$	−19.0196	−0.630841
$910$	0	0
$911$	13.7781	0.456490	0.228245	−	0.973604i	$-0.426701\pi$
$911$	13.7781	0.456490	0.228245	+	0.973604i	$0.426701\pi$
$912$	2.13710	0.0707665
$913$	−7.83518	−0.259307
$914$	13.8200	0.457124
$915$	0	0
$916$	75.0204	2.47874
$917$	4.79216	0.158251
$918$	−9.73537	−0.321315
$919$	−33.1608	−1.09388	−0.546938	−	0.837173i	$-0.684207\pi$
$919$	−33.1608	−1.09388	−0.546938	+	0.837173i	$0.684207\pi$
$920$	0	0
$921$	4.24110	0.139749
$922$	−69.7366	−2.29665
$923$	5.98774	0.197089
$924$	27.1183	0.892128
$925$	0	0
$926$	16.9047	0.555524
$927$	−42.4657	−1.39476
$928$	−14.5269	−0.476868
$929$	−57.9781	−1.90220	−0.951100	−	0.308885i	$-0.900044\pi$
$929$	−57.9781	−1.90220	−0.951100	+	0.308885i	$0.900044\pi$
$930$	0	0
$931$	−8.73368	−0.286235
$932$	68.7667	2.25253
$933$	23.0142	0.753451
$934$	5.73903	0.187787
$935$	0	0
$936$	14.0576	0.459488
$937$	45.7039	1.49308	0.746541	−	0.665340i	$-0.231713\pi$
$937$	45.7039	1.49308	0.746541	+	0.665340i	$0.231713\pi$
$938$	63.3212	2.06751
$939$	−8.37309	−0.273245
$940$	0	0
$941$	34.5669	1.12685	0.563424	−	0.826168i	$-0.309484\pi$
$941$	34.5669	1.12685	0.563424	+	0.826168i	$0.309484\pi$
$942$	4.75490	0.154923
$943$	−18.5677	−0.604647
$944$	−0.898012	−0.0292278
$945$	0	0
$946$	−15.7615	−0.512452
$947$	32.0073	1.04010	0.520048	−	0.854137i	$-0.325914\pi$
$947$	32.0073	1.04010	0.520048	+	0.854137i	$0.325914\pi$
$948$	−112.609	−3.65737
$949$	−16.0259	−0.520222
$950$	0	0
$951$	49.4744	1.60432
$952$	17.3892	0.563587
$953$	−46.7613	−1.51475	−0.757373	−	0.652982i	$-0.773518\pi$
$953$	−46.7613	−1.51475	−0.757373	+	0.652982i	$0.773518\pi$
$954$	63.2456	2.04765
$955$	0	0
$956$	−11.2229	−0.362973
$957$	11.4998	0.371737
$958$	60.2481	1.94653
$959$	5.88272	0.189963
$960$	0	0
$961$	−29.3882	−0.948008
$962$	6.30462	0.203269
$963$	21.4909	0.692534
$964$	−30.4112	−0.979478
$965$	0	0
$966$	−60.2295	−1.93785
$967$	−40.0623	−1.28832	−0.644159	−	0.764892i	$-0.722792\pi$
$967$	−40.0623	−1.28832	−0.644159	+	0.764892i	$0.722792\pi$
$968$	−19.4506	−0.625167
$969$	−21.7165	−0.697635
$970$	0	0
$971$	−39.8182	−1.27783	−0.638913	−	0.769279i	$-0.720615\pi$
$971$	−39.8182	−1.27783	−0.638913	+	0.769279i	$0.720615\pi$
$972$	70.4363	2.25924
$973$	−18.5829	−0.595741
$974$	84.7390	2.71521
$975$	0	0
$976$	−0.364847	−0.0116785
$977$	52.3259	1.67405	0.837026	−	0.547162i	$-0.184292\pi$
$977$	52.3259	1.67405	0.837026	+	0.547162i	$0.184292\pi$
$978$	79.1161	2.52985
$979$	34.2772	1.09550
$980$	0	0
$981$	66.8234	2.13351
$982$	−19.0154	−0.606805
$983$	31.4856	1.00423	0.502117	−	0.864800i	$-0.332555\pi$
$983$	31.4856	1.00423	0.502117	+	0.864800i	$0.332555\pi$
$984$	21.1883	0.675459
$985$	0	0
$986$	20.1659	0.642213
$987$	−34.1934	−1.08839
$988$	11.2997	0.359490
$989$	21.4193	0.681094
$990$	0	0
$991$	16.6395	0.528570	0.264285	−	0.964445i	$-0.414864\pi$
$991$	16.6395	0.528570	0.264285	+	0.964445i	$0.414864\pi$
$992$	−7.69675	−0.244372
$993$	−4.87011	−0.154548
$994$	15.6700	0.497021
$995$	0	0
$996$	−33.2279	−1.05287
$997$	−13.4063	−0.424582	−0.212291	−	0.977206i	$-0.568093\pi$
$997$	−13.4063	−0.424582	−0.212291	+	0.977206i	$0.568093\pi$
$998$	−80.5146	−2.54865
$999$	−2.06675	−0.0653892

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1525.2.a.m.1.9	yes	10
5.2	odd	4		1525.2.b.h.1099.18		20
5.3	odd	4		1525.2.b.h.1099.3		20
5.4	even	2		1525.2.a.l.1.2	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1525.2.a.l.1.2	✓	10	5.4	even	2
1525.2.a.m.1.9	yes	10	1.1	even	1	trivial
1525.2.b.h.1099.3		20	5.3	odd	4
1525.2.b.h.1099.18		20	5.2	odd	4

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:	\( N \)	\(=\)	\( 1525 = 5^{2} \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1525.a (trivial)

\(f(q)\)	\(=\)	\(q+2.27001 q^{2} -2.54073 q^{3} +3.15294 q^{4} -5.76747 q^{6} +1.79213 q^{7} +2.61719 q^{8} +3.45529 q^{9} -1.88895 q^{11} -8.01076 q^{12} +1.55451 q^{13} +4.06815 q^{14} -0.364847 q^{16} +3.70745 q^{17} +7.84355 q^{18} +2.30546 q^{19} -4.55332 q^{21} -4.28792 q^{22} +5.82712 q^{23} -6.64955 q^{24} +3.52874 q^{26} -1.15678 q^{27} +5.65048 q^{28} +2.39615 q^{29} +1.26955 q^{31} -6.06258 q^{32} +4.79930 q^{33} +8.41594 q^{34} +10.8943 q^{36} +1.78665 q^{37} +5.23341 q^{38} -3.94958 q^{39} -3.18643 q^{41} -10.3361 q^{42} +3.67580 q^{43} -5.95573 q^{44} +13.2276 q^{46} +7.50956 q^{47} +0.926976 q^{48} -3.78826 q^{49} -9.41961 q^{51} +4.90127 q^{52} +8.06339 q^{53} -2.62590 q^{54} +4.69034 q^{56} -5.85753 q^{57} +5.43929 q^{58} +2.46134 q^{59} +1.00000 q^{61} +2.88189 q^{62} +6.19234 q^{63} -13.0324 q^{64} +10.8944 q^{66} +15.5651 q^{67} +11.6894 q^{68} -14.8051 q^{69} +3.85186 q^{71} +9.04315 q^{72} -10.3093 q^{73} +4.05570 q^{74} +7.26896 q^{76} -3.38524 q^{77} -8.96557 q^{78} +14.0572 q^{79} -7.42683 q^{81} -7.23322 q^{82} +4.14791 q^{83} -14.3563 q^{84} +8.34409 q^{86} -6.08797 q^{87} -4.94372 q^{88} -18.1462 q^{89} +2.78588 q^{91} +18.3726 q^{92} -3.22558 q^{93} +17.0468 q^{94} +15.4034 q^{96} -9.12471 q^{97} -8.59939 q^{98} -6.52686 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 5 q^{2} + 7 q^{3} + 11 q^{4} + 4 q^{6} + 6 q^{7} + 15 q^{8} + 11 q^{9} + 19 q^{12} + 3 q^{13} - 5 q^{14} + 17 q^{16} + 11 q^{17} + 5 q^{18} - 2 q^{19} + 9 q^{21} - 6 q^{22} + 33 q^{23} - 7 q^{24}+ \cdots - 12 q^{99}+O(q^{100}) \) 10 * q + 5 * q^2 + 7 * q^3 + 11 * q^4 + 4 * q^6 + 6 * q^7 + 15 * q^8 + 11 * q^9 + 19 * q^12 + 3 * q^13 - 5 * q^14 + 17 * q^16 + 11 * q^17 + 5 * q^18 - 2 * q^19 + 9 * q^21 - 6 * q^22 + 33 * q^23 - 7 * q^24 + 12 * q^26 + 19 * q^27 + 21 * q^28 - 3 * q^29 + 20 * q^32 + 7 * q^33 - 3 * q^34 + 45 * q^36 - q^37 + 11 * q^38 - 22 * q^39 + 4 * q^41 - 16 * q^42 + 30 * q^43 - 22 * q^44 + 23 * q^46 + 5 * q^47 + 20 * q^48 - 10 * q^49 - q^51 - 35 * q^52 + 35 * q^53 - 52 * q^54 + 13 * q^56 + 4 * q^57 + 26 * q^58 - 4 * q^59 + 10 * q^61 + 22 * q^62 + 47 * q^63 + q^64 + 25 * q^66 + 28 * q^68 + 26 * q^69 - 11 * q^71 - 27 * q^72 - 9 * q^73 - 41 * q^74 + 44 * q^76 - 4 * q^77 + 51 * q^78 + 16 * q^79 + 14 * q^81 - 22 * q^82 + 18 * q^83 + 2 * q^84 + 7 * q^86 - q^87 - 32 * q^88 - 7 * q^89 - 12 * q^91 + 80 * q^92 - 3 * q^93 - 73 * q^94 + 53 * q^96 + 8 * q^97 - 8 * q^98 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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