LMFDB - Embedded newform 525.4.a.l.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [525,4,Mod(1,525)] 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("525.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(525, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 525 = 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	525.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,2,6,2,0,6,14,18,18,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$30.9760027530$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	525.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+3.82843 q^{2} +3.00000 q^{3} +6.65685 q^{4} +11.4853 q^{6} +7.00000 q^{7} -5.14214 q^{8} +9.00000 q^{9} +48.5685 q^{11} +19.9706 q^{12} +43.6569 q^{13} +26.7990 q^{14} -72.9411 q^{16} +67.6569 q^{17} +34.4558 q^{18} -93.2548 q^{19} +21.0000 q^{21} +185.941 q^{22} +104.167 q^{23} -15.4264 q^{24} +167.137 q^{26} +27.0000 q^{27} +46.5980 q^{28} -58.7351 q^{29} -9.08831 q^{31} -238.113 q^{32} +145.706 q^{33} +259.019 q^{34} +59.9117 q^{36} +252.676 q^{37} -357.019 q^{38} +130.971 q^{39} +276.274 q^{41} +80.3970 q^{42} +92.6375 q^{43} +323.314 q^{44} +398.794 q^{46} +582.794 q^{47} -218.823 q^{48} +49.0000 q^{49} +202.971 q^{51} +290.617 q^{52} -623.019 q^{53} +103.368 q^{54} -35.9949 q^{56} -279.765 q^{57} -224.863 q^{58} -524.999 q^{59} -352.794 q^{61} -34.7939 q^{62} +63.0000 q^{63} -328.068 q^{64} +557.823 q^{66} +736.520 q^{67} +450.382 q^{68} +312.500 q^{69} -492.264 q^{71} -46.2792 q^{72} -1164.75 q^{73} +967.352 q^{74} -620.784 q^{76} +339.980 q^{77} +501.411 q^{78} -872.195 q^{79} +81.0000 q^{81} +1057.70 q^{82} +529.588 q^{83} +139.794 q^{84} +354.656 q^{86} -176.205 q^{87} -249.746 q^{88} -385.216 q^{89} +305.598 q^{91} +693.421 q^{92} -27.2649 q^{93} +2231.18 q^{94} -714.338 q^{96} +463.892 q^{97} +187.593 q^{98} +437.117 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 6 q^{3} + 2 q^{4} + 6 q^{6} + 14 q^{7} + 18 q^{8} + 18 q^{9} - 16 q^{11} + 6 q^{12} + 76 q^{13} + 14 q^{14} - 78 q^{16} + 124 q^{17} + 18 q^{18} - 96 q^{19} + 42 q^{21} + 304 q^{22} + 16 q^{23}+ \cdots - 144 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 6 * q^3 + 2 * q^4 + 6 * q^6 + 14 * q^7 + 18 * q^8 + 18 * q^9 - 16 * q^11 + 6 * q^12 + 76 * q^13 + 14 * q^14 - 78 * q^16 + 124 * q^17 + 18 * q^18 - 96 * q^19 + 42 * q^21 + 304 * q^22 + 16 * q^23 + 54 * q^24 + 108 * q^26 + 54 * q^27 + 14 * q^28 + 188 * q^29 - 120 * q^31 - 414 * q^32 - 48 * q^33 + 156 * q^34 + 18 * q^36 + 132 * q^37 - 352 * q^38 + 228 * q^39 + 100 * q^41 + 42 * q^42 + 536 * q^43 + 624 * q^44 + 560 * q^46 + 928 * q^47 - 234 * q^48 + 98 * q^49 + 372 * q^51 + 140 * q^52 - 884 * q^53 + 54 * q^54 + 126 * q^56 - 288 * q^57 - 676 * q^58 + 104 * q^59 - 468 * q^61 + 168 * q^62 + 126 * q^63 + 34 * q^64 + 912 * q^66 + 1688 * q^67 + 188 * q^68 + 48 * q^69 - 136 * q^71 + 162 * q^72 - 508 * q^73 + 1188 * q^74 - 608 * q^76 - 112 * q^77 + 324 * q^78 - 432 * q^79 + 162 * q^81 + 1380 * q^82 + 584 * q^83 + 42 * q^84 - 456 * q^86 + 564 * q^87 - 1744 * q^88 - 1404 * q^89 + 532 * q^91 + 1104 * q^92 - 360 * q^93 + 1600 * q^94 - 1242 * q^96 + 1188 * q^97 + 98 * q^98 - 144 * 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 $

$2$	3.82843	1.35355	0.676777	−	0.736188i	$-0.263376\pi$
$2$	3.82843	1.35355	0.676777	+	0.736188i	$0.263376\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	6.65685	0.832107
$5$	0	0
$5$	0	0
$6$	11.4853	0.781474
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−5.14214	−0.227252
$9$	9.00000	0.333333
$10$	0	0
$11$	48.5685	1.33127	0.665635	−	0.746278i	$-0.268161\pi$
$11$	48.5685	1.33127	0.665635	+	0.746278i	$0.268161\pi$
$12$	19.9706	0.480417
$13$	43.6569	0.931403	0.465701	−	0.884942i	$-0.345802\pi$
$13$	43.6569	0.931403	0.465701	+	0.884942i	$0.345802\pi$
$14$	26.7990	0.511595
$15$	0	0
$16$	−72.9411	−1.13971
$17$	67.6569	0.965247	0.482623	−	0.875828i	$-0.339684\pi$
$17$	67.6569	0.965247	0.482623	+	0.875828i	$0.339684\pi$
$18$	34.4558	0.451184
$19$	−93.2548	−1.12601	−0.563003	−	0.826455i	$-0.690354\pi$
$19$	−93.2548	−1.12601	−0.563003	+	0.826455i	$0.690354\pi$
$20$	0	0
$21$	21.0000	0.218218
$22$	185.941	1.80194
$23$	104.167	0.944357	0.472179	−	0.881503i	$-0.343468\pi$
$23$	104.167	0.944357	0.472179	+	0.881503i	$0.343468\pi$
$24$	−15.4264	−0.131204
$25$	0	0
$26$	167.137	1.26070
$27$	27.0000	0.192450
$28$	46.5980	0.314507
$29$	−58.7351	−0.376098	−0.188049	−	0.982160i	$-0.560216\pi$
$29$	−58.7351	−0.376098	−0.188049	+	0.982160i	$0.560216\pi$
$30$	0	0
$31$	−9.08831	−0.0526551	−0.0263276	−	0.999653i	$-0.508381\pi$
$31$	−9.08831	−0.0526551	−0.0263276	+	0.999653i	$0.508381\pi$
$32$	−238.113	−1.31540
$33$	145.706	0.768609
$34$	259.019	1.30651
$35$	0	0
$36$	59.9117	0.277369
$37$	252.676	1.12269	0.561347	−	0.827580i	$-0.310283\pi$
$37$	252.676	1.12269	0.561347	+	0.827580i	$0.310283\pi$
$38$	−357.019	−1.52411
$39$	130.971	0.537745
$40$	0	0
$41$	276.274	1.05236	0.526180	−	0.850373i	$-0.323624\pi$
$41$	276.274	1.05236	0.526180	+	0.850373i	$0.323624\pi$
$42$	80.3970	0.295370
$43$	92.6375	0.328537	0.164268	−	0.986416i	$-0.447474\pi$
$43$	92.6375	0.328537	0.164268	+	0.986416i	$0.447474\pi$
$44$	323.314	1.10776
$45$	0	0
$46$	398.794	1.27824
$47$	582.794	1.80871	0.904354	−	0.426784i	$-0.140354\pi$
$47$	582.794	1.80871	0.904354	+	0.426784i	$0.140354\pi$
$48$	−218.823	−0.658009
$49$	49.0000	0.142857
$50$	0	0
$51$	202.971	0.557286
$52$	290.617	0.775026
$53$	−623.019	−1.61468	−0.807342	−	0.590083i	$-0.799095\pi$
$53$	−623.019	−1.61468	−0.807342	+	0.590083i	$0.799095\pi$
$54$	103.368	0.260491
$55$	0	0
$56$	−35.9949	−0.0858933
$57$	−279.765	−0.650100
$58$	−224.863	−0.509068
$59$	−524.999	−1.15846	−0.579229	−	0.815165i	$-0.696646\pi$
$59$	−524.999	−1.15846	−0.579229	+	0.815165i	$0.696646\pi$
$60$	0	0
$61$	−352.794	−0.740502	−0.370251	−	0.928932i	$-0.620728\pi$
$61$	−352.794	−0.740502	−0.370251	+	0.928932i	$0.620728\pi$
$62$	−34.7939	−0.0712715
$63$	63.0000	0.125988
$64$	−328.068	−0.640758
$65$	0	0
$66$	557.823	1.04035
$67$	736.520	1.34299	0.671494	−	0.741010i	$-0.265653\pi$
$67$	736.520	1.34299	0.671494	+	0.741010i	$0.265653\pi$
$68$	450.382	0.803188
$69$	312.500	0.545225
$70$	0	0
$71$	−492.264	−0.822831	−0.411415	−	0.911448i	$-0.634965\pi$
$71$	−492.264	−0.822831	−0.411415	+	0.911448i	$0.634965\pi$
$72$	−46.2792	−0.0757508
$73$	−1164.75	−1.86745	−0.933727	−	0.357987i	$-0.883463\pi$
$73$	−1164.75	−1.86745	−0.933727	+	0.357987i	$0.883463\pi$
$74$	967.352	1.51963
$75$	0	0
$76$	−620.784	−0.936958
$77$	339.980	0.503173
$78$	501.411	0.727867
$79$	−872.195	−1.24215	−0.621074	−	0.783752i	$-0.713303\pi$
$79$	−872.195	−1.24215	−0.621074	+	0.783752i	$0.713303\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	1057.70	1.42443
$83$	529.588	0.700359	0.350180	−	0.936683i	$-0.386121\pi$
$83$	529.588	0.700359	0.350180	+	0.936683i	$0.386121\pi$
$84$	139.794	0.181581
$85$	0	0
$86$	354.656	0.444692
$87$	−176.205	−0.217140
$88$	−249.746	−0.302534
$89$	−385.216	−0.458796	−0.229398	−	0.973333i	$-0.573676\pi$
$89$	−385.216	−0.458796	−0.229398	+	0.973333i	$0.573676\pi$
$90$	0	0
$91$	305.598	0.352037
$92$	693.421	0.785806
$93$	−27.2649	−0.0304005
$94$	2231.18	2.44818
$95$	0	0
$96$	−714.338	−0.759446
$97$	463.892	0.485579	0.242789	−	0.970079i	$-0.421938\pi$
$97$	463.892	0.485579	0.242789	+	0.970079i	$0.421938\pi$
$98$	187.593	0.193365
$99$	437.117	0.443757
$100$	0	0
$101$	−432.725	−0.426314	−0.213157	−	0.977018i	$-0.568375\pi$
$101$	−432.725	−0.426314	−0.213157	+	0.977018i	$0.568375\pi$
$102$	777.058	0.754316
$103$	−512.626	−0.490393	−0.245197	−	0.969473i	$-0.578853\pi$
$103$	−512.626	−0.490393	−0.245197	+	0.969473i	$0.578853\pi$
$104$	−224.489	−0.211663
$105$	0	0
$106$	−2385.18	−2.18556
$107$	−1963.09	−1.77363	−0.886817	−	0.462122i	$-0.847088\pi$
$107$	−1963.09	−1.77363	−0.886817	+	0.462122i	$0.847088\pi$
$108$	179.735	0.160139
$109$	545.176	0.479068	0.239534	−	0.970888i	$-0.423005\pi$
$109$	545.176	0.479068	0.239534	+	0.970888i	$0.423005\pi$
$110$	0	0
$111$	758.029	0.648188
$112$	−510.588	−0.430768
$113$	231.823	0.192992	0.0964961	−	0.995333i	$-0.469236\pi$
$113$	231.823	0.192992	0.0964961	+	0.995333i	$0.469236\pi$
$114$	−1071.06	−0.879945
$115$	0	0
$116$	−390.991	−0.312953
$117$	392.912	0.310468
$118$	−2009.92	−1.56804
$119$	473.598	0.364829
$120$	0	0
$121$	1027.90	0.772279
$122$	−1350.65	−1.00231
$123$	828.823	0.607581
$124$	−60.4996	−0.0438147
$125$	0	0
$126$	241.191	0.170532
$127$	−2372.90	−1.65796	−0.828979	−	0.559280i	$-0.811078\pi$
$127$	−2372.90	−1.65796	−0.828979	+	0.559280i	$0.811078\pi$
$128$	648.917	0.448099
$129$	277.913	0.189681
$130$	0	0
$131$	1200.04	0.800364	0.400182	−	0.916436i	$-0.368947\pi$
$131$	1200.04	0.800364	0.400182	+	0.916436i	$0.368947\pi$
$132$	969.941	0.639565
$133$	−652.784	−0.425591
$134$	2819.71	1.81781
$135$	0	0
$136$	−347.901	−0.219355
$137$	−2781.25	−1.73444	−0.867221	−	0.497924i	$-0.834096\pi$
$137$	−2781.25	−1.73444	−0.867221	+	0.497924i	$0.834096\pi$
$138$	1196.38	0.737991
$139$	−1245.60	−0.760078	−0.380039	−	0.924971i	$-0.624089\pi$
$139$	−1245.60	−0.760078	−0.380039	+	0.924971i	$0.624089\pi$
$140$	0	0
$141$	1748.38	1.04426
$142$	−1884.60	−1.11375
$143$	2120.35	1.23995
$144$	−656.470	−0.379902
$145$	0	0
$146$	−4459.17	−2.52770
$147$	147.000	0.0824786
$148$	1682.03	0.934202
$149$	19.4046	0.0106690	0.00533452	−	0.999986i	$-0.498302\pi$
$149$	19.4046	0.0106690	0.00533452	+	0.999986i	$0.498302\pi$
$150$	0	0
$151$	−2349.80	−1.26638	−0.633192	−	0.773995i	$-0.718256\pi$
$151$	−2349.80	−1.26638	−0.633192	+	0.773995i	$0.718256\pi$
$152$	479.529	0.255888
$153$	608.912	0.321749
$154$	1301.59	0.681071
$155$	0	0
$156$	871.852	0.447462
$157$	3898.46	1.98172	0.990862	−	0.134880i	$-0.0430650\pi$
$157$	3898.46	1.98172	0.990862	+	0.134880i	$0.0430650\pi$
$158$	−3339.14	−1.68131
$159$	−1869.06	−0.932239
$160$	0	0
$161$	729.166	0.356934
$162$	310.103	0.150395
$163$	−1527.54	−0.734024	−0.367012	−	0.930216i	$-0.619619\pi$
$163$	−1527.54	−0.734024	−0.367012	+	0.930216i	$0.619619\pi$
$164$	1839.12	0.875676
$165$	0	0
$166$	2027.49	0.947974
$167$	998.518	0.462681	0.231340	−	0.972873i	$-0.425689\pi$
$167$	998.518	0.462681	0.231340	+	0.972873i	$0.425689\pi$
$168$	−107.985	−0.0495905
$169$	−291.079	−0.132489
$170$	0	0
$171$	−839.294	−0.375336
$172$	616.674	0.273378
$173$	−685.253	−0.301149	−0.150575	−	0.988599i	$-0.548112\pi$
$173$	−685.253	−0.301149	−0.150575	+	0.988599i	$0.548112\pi$
$174$	−674.589	−0.293911
$175$	0	0
$176$	−3542.64	−1.51725
$177$	−1575.00	−0.668836
$178$	−1474.77	−0.621005
$179$	1025.58	0.428245	0.214122	−	0.976807i	$-0.431311\pi$
$179$	1025.58	0.428245	0.214122	+	0.976807i	$0.431311\pi$
$180$	0	0
$181$	2899.40	1.19067	0.595333	−	0.803479i	$-0.297020\pi$
$181$	2899.40	1.19067	0.595333	+	0.803479i	$0.297020\pi$
$182$	1169.96	0.476501
$183$	−1058.38	−0.427529
$184$	−535.638	−0.214608
$185$	0	0
$186$	−104.382	−0.0411486
$187$	3285.99	1.28500
$188$	3879.57	1.50504
$189$	189.000	0.0727393
$190$	0	0
$191$	1074.18	0.406939	0.203469	−	0.979081i	$-0.434778\pi$
$191$	1074.18	0.406939	0.203469	+	0.979081i	$0.434778\pi$
$192$	−984.204	−0.369942
$193$	−898.999	−0.335292	−0.167646	−	0.985847i	$-0.553617\pi$
$193$	−898.999	−0.335292	−0.167646	+	0.985847i	$0.553617\pi$
$194$	1775.98	0.657257
$195$	0	0
$196$	326.186	0.118872
$197$	−3063.63	−1.10799	−0.553996	−	0.832519i	$-0.686898\pi$
$197$	−3063.63	−1.10799	−0.553996	+	0.832519i	$0.686898\pi$
$198$	1673.47	0.600648
$199$	−949.522	−0.338240	−0.169120	−	0.985595i	$-0.554093\pi$
$199$	−949.522	−0.338240	−0.169120	+	0.985595i	$0.554093\pi$
$200$	0	0
$201$	2209.56	0.775375
$202$	−1656.66	−0.577039
$203$	−411.145	−0.142151
$204$	1351.15	0.463721
$205$	0	0
$206$	−1962.55	−0.663774
$207$	937.499	0.314786
$208$	−3184.38	−1.06152
$209$	−4529.25	−1.49902
$210$	0	0
$211$	2306.64	0.752587	0.376294	−	0.926500i	$-0.377198\pi$
$211$	2306.64	0.752587	0.376294	+	0.926500i	$0.377198\pi$
$212$	−4147.35	−1.34359
$213$	−1476.79	−0.475062
$214$	−7515.53	−2.40071
$215$	0	0
$216$	−138.838	−0.0437348
$217$	−63.6182	−0.0199018
$218$	2087.17	0.648444
$219$	−3494.26	−1.07817
$220$	0	0
$221$	2953.69	0.899033
$222$	2902.06	0.877357
$223$	3227.61	0.969222	0.484611	−	0.874730i	$-0.338961\pi$
$223$	3227.61	0.969222	0.484611	+	0.874730i	$0.338961\pi$
$224$	−1666.79	−0.497174
$225$	0	0
$226$	887.519	0.261225
$227$	637.820	0.186492	0.0932458	−	0.995643i	$-0.470276\pi$
$227$	637.820	0.186492	0.0932458	+	0.995643i	$0.470276\pi$
$228$	−1862.35	−0.540953
$229$	544.774	0.157204	0.0786019	−	0.996906i	$-0.474954\pi$
$229$	544.774	0.157204	0.0786019	+	0.996906i	$0.474954\pi$
$230$	0	0
$231$	1019.94	0.290507
$232$	302.024	0.0854691
$233$	5748.54	1.61631	0.808154	−	0.588972i	$-0.200467\pi$
$233$	5748.54	1.61631	0.808154	+	0.588972i	$0.200467\pi$
$234$	1504.23	0.420234
$235$	0	0
$236$	−3494.84	−0.963961
$237$	−2616.59	−0.717154
$238$	1813.14	0.493816
$239$	−2678.10	−0.724820	−0.362410	−	0.932019i	$-0.618046\pi$
$239$	−2678.10	−0.724820	−0.362410	+	0.932019i	$0.618046\pi$
$240$	0	0
$241$	−2202.16	−0.588604	−0.294302	−	0.955713i	$-0.595087\pi$
$241$	−2202.16	−0.588604	−0.294302	+	0.955713i	$0.595087\pi$
$242$	3935.25	1.04532
$243$	243.000	0.0641500
$244$	−2348.50	−0.616177
$245$	0	0
$246$	3173.09	0.822393
$247$	−4071.21	−1.04877
$248$	46.7333	0.0119660
$249$	1588.76	0.404353
$250$	0	0
$251$	−5716.90	−1.43764	−0.718820	−	0.695196i	$-0.755317\pi$
$251$	−5716.90	−1.43764	−0.718820	+	0.695196i	$0.755317\pi$
$252$	419.382	0.104836
$253$	5059.22	1.25719
$254$	−9084.47	−2.24413
$255$	0	0
$256$	5108.88	1.24728
$257$	−4724.29	−1.14666	−0.573332	−	0.819323i	$-0.694350\pi$
$257$	−4724.29	−1.14666	−0.573332	+	0.819323i	$0.694350\pi$
$258$	1063.97	0.256743
$259$	1768.73	0.424339
$260$	0	0
$261$	−528.616	−0.125366
$262$	4594.25	1.08334
$263$	−5975.36	−1.40097	−0.700487	−	0.713665i	$-0.747034\pi$
$263$	−5975.36	−1.40097	−0.700487	+	0.713665i	$0.747034\pi$
$264$	−749.238	−0.174668
$265$	0	0
$266$	−2499.14	−0.576059
$267$	−1155.65	−0.264886
$268$	4902.90	1.11751
$269$	4486.11	1.01681	0.508407	−	0.861117i	$-0.330234\pi$
$269$	4486.11	1.01681	0.508407	+	0.861117i	$0.330234\pi$
$270$	0	0
$271$	3827.68	0.857989	0.428994	−	0.903307i	$-0.358868\pi$
$271$	3827.68	0.857989	0.428994	+	0.903307i	$0.358868\pi$
$272$	−4934.97	−1.10010
$273$	916.794	0.203249
$274$	−10647.8	−2.34766
$275$	0	0
$276$	2080.26	0.453685
$277$	3420.54	0.741950	0.370975	−	0.928643i	$-0.379024\pi$
$277$	3420.54	0.741950	0.370975	+	0.928643i	$0.379024\pi$
$278$	−4768.71	−1.02881
$279$	−81.7948	−0.0175517
$280$	0	0
$281$	5235.92	1.11156	0.555781	−	0.831329i	$-0.312419\pi$
$281$	5235.92	1.11156	0.555781	+	0.831329i	$0.312419\pi$
$282$	6693.55	1.41346
$283$	6985.88	1.46738	0.733688	−	0.679486i	$-0.237797\pi$
$283$	6985.88	1.46738	0.733688	+	0.679486i	$0.237797\pi$
$284$	−3276.93	−0.684683
$285$	0	0
$286$	8117.60	1.67834
$287$	1933.92	0.397755
$288$	−2143.01	−0.438466
$289$	−335.550	−0.0682984
$290$	0	0
$291$	1391.68	0.280349
$292$	−7753.59	−1.55392
$293$	−7399.70	−1.47541	−0.737705	−	0.675123i	$-0.764090\pi$
$293$	−7399.70	−1.47541	−0.737705	+	0.675123i	$0.764090\pi$
$294$	562.779	0.111639
$295$	0	0
$296$	−1299.30	−0.255135
$297$	1311.35	0.256203
$298$	74.2892	0.0144411
$299$	4547.58	0.879577
$300$	0	0
$301$	648.463	0.124175
$302$	−8996.04	−1.71412
$303$	−1298.17	−0.246133
$304$	6802.11	1.28332
$305$	0	0
$306$	2331.17	0.435504
$307$	−2668.64	−0.496116	−0.248058	−	0.968745i	$-0.579792\pi$
$307$	−2668.64	−0.496116	−0.248058	+	0.968745i	$0.579792\pi$
$308$	2263.20	0.418693
$309$	−1537.88	−0.283129
$310$	0	0
$311$	6189.25	1.12849	0.564244	−	0.825608i	$-0.309168\pi$
$311$	6189.25	1.12849	0.564244	+	0.825608i	$0.309168\pi$
$312$	−673.468	−0.122204
$313$	2921.59	0.527598	0.263799	−	0.964578i	$-0.415024\pi$
$313$	2921.59	0.527598	0.263799	+	0.964578i	$0.415024\pi$
$314$	14925.0	2.68237
$315$	0	0
$316$	−5806.08	−1.03360
$317$	−9825.56	−1.74088	−0.870439	−	0.492276i	$-0.836165\pi$
$317$	−9825.56	−1.74088	−0.870439	+	0.492276i	$0.836165\pi$
$318$	−7155.55	−1.26183
$319$	−2852.68	−0.500687
$320$	0	0
$321$	−5889.26	−1.02401
$322$	2791.56	0.483129
$323$	−6309.33	−1.08687
$324$	539.205	0.0924563
$325$	0	0
$326$	−5848.07	−0.993541
$327$	1635.53	0.276590
$328$	−1420.64	−0.239151
$329$	4079.56	0.683627
$330$	0	0
$331$	−9258.17	−1.53739	−0.768693	−	0.639618i	$-0.779093\pi$
$331$	−9258.17	−1.53739	−0.768693	+	0.639618i	$0.779093\pi$
$332$	3525.39	0.582774
$333$	2274.09	0.374232
$334$	3822.75	0.626263
$335$	0	0
$336$	−1531.76	−0.248704
$337$	3693.98	0.597103	0.298552	−	0.954394i	$-0.403497\pi$
$337$	3693.98	0.597103	0.298552	+	0.954394i	$0.403497\pi$
$338$	−1114.38	−0.179331
$339$	695.470	0.111424
$340$	0	0
$341$	−441.406	−0.0700982
$342$	−3213.17	−0.508037
$343$	343.000	0.0539949
$344$	−476.355	−0.0746608
$345$	0	0
$346$	−2623.44	−0.407622
$347$	−3832.83	−0.592960	−0.296480	−	0.955039i	$-0.595813\pi$
$347$	−3832.83	−0.592960	−0.296480	+	0.955039i	$0.595813\pi$
$348$	−1172.97	−0.180684
$349$	8325.22	1.27690	0.638451	−	0.769662i	$-0.279575\pi$
$349$	8325.22	1.27690	0.638451	+	0.769662i	$0.279575\pi$
$350$	0	0
$351$	1178.74	0.179248
$352$	−11564.8	−1.75115
$353$	8991.52	1.35572	0.677862	−	0.735189i	$-0.262907\pi$
$353$	8991.52	1.35572	0.677862	+	0.735189i	$0.262907\pi$
$354$	−6029.76	−0.905306
$355$	0	0
$356$	−2564.33	−0.381767
$357$	1420.79	0.210634
$358$	3926.38	0.579652
$359$	−12893.8	−1.89557	−0.947783	−	0.318917i	$-0.896681\pi$
$359$	−12893.8	−1.89557	−0.947783	+	0.318917i	$0.896681\pi$
$360$	0	0
$361$	1837.46	0.267891
$362$	11100.1	1.61163
$363$	3083.71	0.445875
$364$	2034.32	0.292932
$365$	0	0
$366$	−4051.94	−0.578684
$367$	7480.17	1.06393	0.531964	−	0.846767i	$-0.321454\pi$
$367$	7480.17	1.06393	0.531964	+	0.846767i	$0.321454\pi$
$368$	−7598.02	−1.07629
$369$	2486.47	0.350787
$370$	0	0
$371$	−4361.14	−0.610293
$372$	−181.499	−0.0252964
$373$	3523.32	0.489090	0.244545	−	0.969638i	$-0.421361\pi$
$373$	3523.32	0.489090	0.244545	+	0.969638i	$0.421361\pi$
$374$	12580.2	1.73932
$375$	0	0
$376$	−2996.81	−0.411033
$377$	−2564.19	−0.350298
$378$	723.573	0.0984565
$379$	13515.4	1.83177	0.915886	−	0.401438i	$-0.131490\pi$
$379$	13515.4	1.83177	0.915886	+	0.401438i	$0.131490\pi$
$380$	0	0
$381$	−7118.69	−0.957222
$382$	4112.44	0.550813
$383$	657.182	0.0876774	0.0438387	−	0.999039i	$-0.486041\pi$
$383$	657.182	0.0876774	0.0438387	+	0.999039i	$0.486041\pi$
$384$	1946.75	0.258710
$385$	0	0
$386$	−3441.75	−0.453836
$387$	833.738	0.109512
$388$	3088.06	0.404053
$389$	−9741.87	−1.26975	−0.634875	−	0.772615i	$-0.718948\pi$
$389$	−9741.87	−1.26975	−0.634875	+	0.772615i	$0.718948\pi$
$390$	0	0
$391$	7047.58	0.911538
$392$	−251.965	−0.0324646
$393$	3600.11	0.462091
$394$	−11728.9	−1.49973
$395$	0	0
$396$	2909.82	0.369253
$397$	−4407.42	−0.557184	−0.278592	−	0.960410i	$-0.589868\pi$
$397$	−4407.42	−0.557184	−0.278592	+	0.960410i	$0.589868\pi$
$398$	−3635.18	−0.457827
$399$	−1958.35	−0.245715
$400$	0	0
$401$	−11569.5	−1.44078	−0.720391	−	0.693568i	$-0.756037\pi$
$401$	−11569.5	−1.44078	−0.720391	+	0.693568i	$0.756037\pi$
$402$	8459.14	1.04951
$403$	−396.767	−0.0490431
$404$	−2880.59	−0.354739
$405$	0	0
$406$	−1574.04	−0.192410
$407$	12272.1	1.49461
$408$	−1043.70	−0.126645
$409$	−3083.03	−0.372729	−0.186364	−	0.982481i	$-0.559670\pi$
$409$	−3083.03	−0.372729	−0.186364	+	0.982481i	$0.559670\pi$
$410$	0	0
$411$	−8343.76	−1.00138
$412$	−3412.47	−0.408060
$413$	−3674.99	−0.437856
$414$	3589.15	0.426079
$415$	0	0
$416$	−10395.3	−1.22517
$417$	−3736.81	−0.438831
$418$	−17339.9	−2.02900
$419$	−5415.21	−0.631385	−0.315692	−	0.948862i	$-0.602237\pi$
$419$	−5415.21	−0.631385	−0.315692	+	0.948862i	$0.602237\pi$
$420$	0	0
$421$	4188.34	0.484863	0.242432	−	0.970168i	$-0.422055\pi$
$421$	4188.34	0.484863	0.242432	+	0.970168i	$0.422055\pi$
$422$	8830.82	1.01867
$423$	5245.15	0.602902
$424$	3203.65	0.366941
$425$	0	0
$426$	−5653.79	−0.643021
$427$	−2469.56	−0.279884
$428$	−13068.0	−1.47585
$429$	6361.05	0.715884
$430$	0	0
$431$	−9108.41	−1.01795	−0.508975	−	0.860781i	$-0.669976\pi$
$431$	−9108.41	−1.01795	−0.508975	+	0.860781i	$0.669976\pi$
$432$	−1969.41	−0.219336
$433$	16847.0	1.86979	0.934893	−	0.354930i	$-0.115495\pi$
$433$	16847.0	1.86979	0.934893	+	0.354930i	$0.115495\pi$
$434$	−243.558	−0.0269381
$435$	0	0
$436$	3629.16	0.398635
$437$	−9714.03	−1.06335
$438$	−13377.5	−1.45937
$439$	8434.14	0.916946	0.458473	−	0.888708i	$-0.348397\pi$
$439$	8434.14	0.916946	0.458473	+	0.888708i	$0.348397\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	11308.0	1.21689
$443$	4298.49	0.461010	0.230505	−	0.973071i	$-0.425962\pi$
$443$	4298.49	0.461010	0.230505	+	0.973071i	$0.425962\pi$
$444$	5046.09	0.539362
$445$	0	0
$446$	12356.7	1.31189
$447$	58.2139	0.00615978
$448$	−2296.48	−0.242184
$449$	10545.0	1.10835	0.554173	−	0.832402i	$-0.313035\pi$
$449$	10545.0	1.10835	0.554173	+	0.832402i	$0.313035\pi$
$450$	0	0
$451$	13418.2	1.40098
$452$	1543.21	0.160590
$453$	−7049.40	−0.731147
$454$	2441.85	0.252426
$455$	0	0
$456$	1438.59	0.147737
$457$	−11952.4	−1.22344	−0.611719	−	0.791075i	$-0.709522\pi$
$457$	−11952.4	−1.22344	−0.611719	+	0.791075i	$0.709522\pi$
$458$	2085.63	0.212784
$459$	1826.74	0.185762
$460$	0	0
$461$	−17200.9	−1.73780	−0.868900	−	0.494988i	$-0.835173\pi$
$461$	−17200.9	−1.73780	−0.868900	+	0.494988i	$0.835173\pi$
$462$	3904.76	0.393217
$463$	10368.7	1.04076	0.520381	−	0.853934i	$-0.325790\pi$
$463$	10368.7	1.04076	0.520381	+	0.853934i	$0.325790\pi$
$464$	4284.20	0.428640
$465$	0	0
$466$	22007.9	2.18776
$467$	16879.5	1.67257	0.836284	−	0.548296i	$-0.184723\pi$
$467$	16879.5	1.67257	0.836284	+	0.548296i	$0.184723\pi$
$468$	2615.56	0.258342
$469$	5155.64	0.507602
$470$	0	0
$471$	11695.4	1.14415
$472$	2699.62	0.263263
$473$	4499.27	0.437371
$474$	−10017.4	−0.970706
$475$	0	0
$476$	3152.67	0.303577
$477$	−5607.17	−0.538228
$478$	−10252.9	−0.981082
$479$	7329.12	0.699115	0.349558	−	0.936915i	$-0.386332\pi$
$479$	7329.12	0.699115	0.349558	+	0.936915i	$0.386332\pi$
$480$	0	0
$481$	11031.0	1.04568
$482$	−8430.80	−0.796706
$483$	2187.50	0.206076
$484$	6842.60	0.642619
$485$	0	0
$486$	930.308	0.0868305
$487$	17209.5	1.60131	0.800655	−	0.599125i	$-0.204485\pi$
$487$	17209.5	1.60131	0.800655	+	0.599125i	$0.204485\pi$
$488$	1814.11	0.168281
$489$	−4582.61	−0.423789
$490$	0	0
$491$	11392.5	1.04712	0.523560	−	0.851989i	$-0.324604\pi$
$491$	11392.5	1.04712	0.523560	+	0.851989i	$0.324604\pi$
$492$	5517.35	0.505572
$493$	−3973.83	−0.363027
$494$	−15586.3	−1.41956
$495$	0	0
$496$	662.912	0.0600113
$497$	−3445.85	−0.311001
$498$	6082.47	0.547313
$499$	19079.4	1.71164	0.855822	−	0.517271i	$-0.173052\pi$
$499$	19079.4	1.71164	0.855822	+	0.517271i	$0.173052\pi$
$500$	0	0
$501$	2995.55	0.267129
$502$	−21886.7	−1.94592
$503$	13499.4	1.19663	0.598317	−	0.801259i	$-0.295836\pi$
$503$	13499.4	1.19663	0.598317	+	0.801259i	$0.295836\pi$
$504$	−323.955	−0.0286311
$505$	0	0
$506$	19368.8	1.70168
$507$	−873.237	−0.0764928
$508$	−15796.0	−1.37960
$509$	−4328.85	−0.376960	−0.188480	−	0.982077i	$-0.560356\pi$
$509$	−4328.85	−0.376960	−0.188480	+	0.982077i	$0.560356\pi$
$510$	0	0
$511$	−8153.27	−0.705831
$512$	14367.6	1.24017
$513$	−2517.88	−0.216700
$514$	−18086.6	−1.55207
$515$	0	0
$516$	1850.02	0.157835
$517$	28305.5	2.40788
$518$	6771.47	0.574365
$519$	−2055.76	−0.173869
$520$	0	0
$521$	19395.7	1.63098	0.815490	−	0.578771i	$-0.196468\pi$
$521$	19395.7	1.63098	0.815490	+	0.578771i	$0.196468\pi$
$522$	−2023.77	−0.169689
$523$	−20413.8	−1.70675	−0.853377	−	0.521294i	$-0.825450\pi$
$523$	−20413.8	−1.70675	−0.853377	+	0.521294i	$0.825450\pi$
$524$	7988.47	0.665989
$525$	0	0
$526$	−22876.2	−1.89629
$527$	−614.887	−0.0508252
$528$	−10627.9	−0.875987
$529$	−1316.34	−0.108189
$530$	0	0
$531$	−4724.99	−0.386153
$532$	−4345.49	−0.354137
$533$	12061.3	0.980171
$534$	−4424.32	−0.358537
$535$	0	0
$536$	−3787.28	−0.305197
$537$	3076.75	0.247247
$538$	17174.8	1.37631
$539$	2379.86	0.190181
$540$	0	0
$541$	−4919.18	−0.390928	−0.195464	−	0.980711i	$-0.562621\pi$
$541$	−4919.18	−0.390928	−0.195464	+	0.980711i	$0.562621\pi$
$542$	14654.0	1.16133
$543$	8698.19	0.687431
$544$	−16110.0	−1.26969
$545$	0	0
$546$	3509.88	0.275108
$547$	−15334.2	−1.19862	−0.599308	−	0.800518i	$-0.704558\pi$
$547$	−15334.2	−1.19862	−0.599308	+	0.800518i	$0.704558\pi$
$548$	−18514.4	−1.44324
$549$	−3175.15	−0.246834
$550$	0	0
$551$	5477.33	0.423488
$552$	−1606.92	−0.123904
$553$	−6105.37	−0.469487
$554$	13095.3	1.00427
$555$	0	0
$556$	−8291.81	−0.632466
$557$	8613.78	0.655256	0.327628	−	0.944807i	$-0.393751\pi$
$557$	8613.78	0.655256	0.327628	+	0.944807i	$0.393751\pi$
$558$	−313.145	−0.0237572
$559$	4044.26	0.306000
$560$	0	0
$561$	9857.98	0.741897
$562$	20045.3	1.50456
$563$	2320.81	0.173731	0.0868654	−	0.996220i	$-0.472315\pi$
$563$	2320.81	0.173731	0.0868654	+	0.996220i	$0.472315\pi$
$564$	11638.7	0.868934
$565$	0	0
$566$	26744.9	1.98617
$567$	567.000	0.0419961
$568$	2531.29	0.186990
$569$	−1736.04	−0.127906	−0.0639529	−	0.997953i	$-0.520371\pi$
$569$	−1736.04	−0.127906	−0.0639529	+	0.997953i	$0.520371\pi$
$570$	0	0
$571$	23897.8	1.75148	0.875738	−	0.482786i	$-0.160375\pi$
$571$	23897.8	1.75148	0.875738	+	0.482786i	$0.160375\pi$
$572$	14114.9	1.03177
$573$	3222.55	0.234946
$574$	7403.87	0.538382
$575$	0	0
$576$	−2952.61	−0.213586
$577$	−8029.26	−0.579311	−0.289655	−	0.957131i	$-0.593541\pi$
$577$	−8029.26	−0.579311	−0.289655	+	0.957131i	$0.593541\pi$
$578$	−1284.63	−0.0924455
$579$	−2697.00	−0.193581
$580$	0	0
$581$	3707.12	0.264711
$582$	5327.93	0.379467
$583$	−30259.1	−2.14958
$584$	5989.32	0.424383
$585$	0	0
$586$	−28329.2	−1.99705
$587$	8015.14	0.563578	0.281789	−	0.959476i	$-0.409072\pi$
$587$	8015.14	0.563578	0.281789	+	0.959476i	$0.409072\pi$
$588$	978.558	0.0686310
$589$	847.529	0.0592900
$590$	0	0
$591$	−9190.88	−0.639700
$592$	−18430.5	−1.27954
$593$	−12820.3	−0.887801	−0.443901	−	0.896076i	$-0.646406\pi$
$593$	−12820.3	−0.887801	−0.443901	+	0.896076i	$0.646406\pi$
$594$	5020.41	0.346784
$595$	0	0
$596$	129.174	0.00887779
$597$	−2848.57	−0.195283
$598$	17410.1	1.19055
$599$	−15330.5	−1.04572	−0.522860	−	0.852418i	$-0.675135\pi$
$599$	−15330.5	−1.04572	−0.522860	+	0.852418i	$0.675135\pi$
$600$	0	0
$601$	107.658	0.00730694	0.00365347	−	0.999993i	$-0.498837\pi$
$601$	107.658	0.00730694	0.00365347	+	0.999993i	$0.498837\pi$
$602$	2482.59	0.168078
$603$	6628.68	0.447663
$604$	−15642.3	−1.05377
$605$	0	0
$606$	−4969.97	−0.333154
$607$	8213.06	0.549189	0.274595	−	0.961560i	$-0.411456\pi$
$607$	8213.06	0.549189	0.274595	+	0.961560i	$0.411456\pi$
$608$	22205.2	1.48115
$609$	−1233.44	−0.0820712
$610$	0	0
$611$	25443.0	1.68463
$612$	4053.44	0.267729
$613$	−1242.60	−0.0818732	−0.0409366	−	0.999162i	$-0.513034\pi$
$613$	−1242.60	−0.0818732	−0.0409366	+	0.999162i	$0.513034\pi$
$614$	−10216.7	−0.671519
$615$	0	0
$616$	−1748.22	−0.114347
$617$	13170.6	0.859367	0.429683	−	0.902980i	$-0.358625\pi$
$617$	13170.6	0.859367	0.429683	+	0.902980i	$0.358625\pi$
$618$	−5887.65	−0.383230
$619$	19774.1	1.28399	0.641993	−	0.766711i	$-0.278108\pi$
$619$	19774.1	1.28399	0.641993	+	0.766711i	$0.278108\pi$
$620$	0	0
$621$	2812.50	0.181742
$622$	23695.1	1.52747
$623$	−2696.51	−0.173409
$624$	−9553.14	−0.612871
$625$	0	0
$626$	11185.1	0.714132
$627$	−13587.8	−0.865459
$628$	25951.5	1.64901
$629$	17095.3	1.08368
$630$	0	0
$631$	−14308.8	−0.902735	−0.451367	−	0.892338i	$-0.649064\pi$
$631$	−14308.8	−0.902735	−0.451367	+	0.892338i	$0.649064\pi$
$632$	4484.95	0.282281
$633$	6919.93	0.434507
$634$	−37616.4	−2.35637
$635$	0	0
$636$	−12442.0	−0.775722
$637$	2139.19	0.133058
$638$	−10921.3	−0.677707
$639$	−4430.38	−0.274277
$640$	0	0
$641$	11537.5	0.710925	0.355463	−	0.934691i	$-0.384323\pi$
$641$	11537.5	0.710925	0.355463	+	0.934691i	$0.384323\pi$
$642$	−22546.6	−1.38605
$643$	−19603.0	−1.20228	−0.601139	−	0.799144i	$-0.705286\pi$
$643$	−19603.0	−1.20228	−0.601139	+	0.799144i	$0.705286\pi$
$644$	4853.95	0.297007
$645$	0	0
$646$	−24154.8	−1.47114
$647$	−21650.0	−1.31553	−0.657765	−	0.753223i	$-0.728498\pi$
$647$	−21650.0	−1.31553	−0.657765	+	0.753223i	$0.728498\pi$
$648$	−416.513	−0.0252503
$649$	−25498.4	−1.54222
$650$	0	0
$651$	−190.855	−0.0114903
$652$	−10168.6	−0.610787
$653$	2927.33	0.175429	0.0877145	−	0.996146i	$-0.472044\pi$
$653$	2927.33	0.175429	0.0877145	+	0.996146i	$0.472044\pi$
$654$	6261.50	0.374379
$655$	0	0
$656$	−20151.7	−1.19938
$657$	−10482.8	−0.622484
$658$	15618.3	0.925326
$659$	−4778.76	−0.282480	−0.141240	−	0.989975i	$-0.545109\pi$
$659$	−4778.76	−0.282480	−0.141240	+	0.989975i	$0.545109\pi$
$660$	0	0
$661$	−31510.3	−1.85417	−0.927086	−	0.374849i	$-0.877695\pi$
$661$	−31510.3	−1.85417	−0.927086	+	0.374849i	$0.877695\pi$
$662$	−35444.2	−2.08093
$663$	8861.06	0.519057
$664$	−2723.21	−0.159158
$665$	0	0
$666$	8706.17	0.506542
$667$	−6118.23	−0.355170
$668$	6646.99	0.385000
$669$	9682.82	0.559581
$670$	0	0
$671$	−17134.7	−0.985808
$672$	−5000.37	−0.287044
$673$	8992.15	0.515040	0.257520	−	0.966273i	$-0.417095\pi$
$673$	8992.15	0.515040	0.257520	+	0.966273i	$0.417095\pi$
$674$	14142.1	0.808211
$675$	0	0
$676$	−1937.67	−0.110245
$677$	−19340.8	−1.09797	−0.548985	−	0.835832i	$-0.684986\pi$
$677$	−19340.8	−1.09797	−0.548985	+	0.835832i	$0.684986\pi$
$678$	2662.56	0.150818
$679$	3247.25	0.183531
$680$	0	0
$681$	1913.46	0.107671
$682$	−1689.89	−0.0948816
$683$	4255.14	0.238387	0.119194	−	0.992871i	$-0.461969\pi$
$683$	4255.14	0.238387	0.119194	+	0.992871i	$0.461969\pi$
$684$	−5587.05	−0.312319
$685$	0	0
$686$	1313.15	0.0730850
$687$	1634.32	0.0907616
$688$	−6757.08	−0.374435
$689$	−27199.1	−1.50392
$690$	0	0
$691$	−17505.5	−0.963733	−0.481867	−	0.876245i	$-0.660041\pi$
$691$	−17505.5	−0.963733	−0.481867	+	0.876245i	$0.660041\pi$
$692$	−4561.63	−0.250588
$693$	3059.82	0.167724
$694$	−14673.7	−0.802603
$695$	0	0
$696$	906.071	0.0493456
$697$	18691.8	1.01579
$698$	31872.5	1.72836
$699$	17245.6	0.933175
$700$	0	0
$701$	−3240.77	−0.174611	−0.0873054	−	0.996182i	$-0.527826\pi$
$701$	−3240.77	−0.174611	−0.0873054	+	0.996182i	$0.527826\pi$
$702$	4512.70	0.242622
$703$	−23563.3	−1.26416
$704$	−15933.8	−0.853022
$705$	0	0
$706$	34423.4	1.83504
$707$	−3029.07	−0.161132
$708$	−10484.5	−0.556543
$709$	19949.3	1.05672	0.528358	−	0.849022i	$-0.322808\pi$
$709$	19949.3	1.05672	0.528358	+	0.849022i	$0.322808\pi$
$710$	0	0
$711$	−7849.76	−0.414049
$712$	1980.83	0.104262
$713$	−946.698	−0.0497253
$714$	5439.41	0.285105
$715$	0	0
$716$	6827.17	0.356345
$717$	−8034.30	−0.418475
$718$	−49362.9	−2.56575
$719$	−11259.4	−0.584011	−0.292006	−	0.956417i	$-0.594323\pi$
$719$	−11259.4	−0.584011	−0.292006	+	0.956417i	$0.594323\pi$
$720$	0	0
$721$	−3588.38	−0.185351
$722$	7034.60	0.362605
$723$	−6606.47	−0.339830
$724$	19300.9	0.990761
$725$	0	0
$726$	11805.8	0.603516
$727$	12228.5	0.623840	0.311920	−	0.950108i	$-0.399028\pi$
$727$	12228.5	0.623840	0.311920	+	0.950108i	$0.399028\pi$
$728$	−1571.43	−0.0800013
$729$	729.000	0.0370370
$730$	0	0
$731$	6267.56	0.317119
$732$	−7045.49	−0.355750
$733$	26635.1	1.34214	0.671072	−	0.741392i	$-0.265834\pi$
$733$	26635.1	1.34214	0.671072	+	0.741392i	$0.265834\pi$
$734$	28637.3	1.44008
$735$	0	0
$736$	−24803.4	−1.24221
$737$	35771.7	1.78788
$738$	9519.26	0.474809
$739$	−6074.00	−0.302349	−0.151174	−	0.988507i	$-0.548306\pi$
$739$	−6074.00	−0.302349	−0.151174	+	0.988507i	$0.548306\pi$
$740$	0	0
$741$	−12213.6	−0.605505
$742$	−16696.3	−0.826065
$743$	4016.87	0.198337	0.0991686	−	0.995071i	$-0.468382\pi$
$743$	4016.87	0.198337	0.0991686	+	0.995071i	$0.468382\pi$
$744$	140.200	0.00690858
$745$	0	0
$746$	13488.8	0.662010
$747$	4766.29	0.233453
$748$	21874.4	1.06926
$749$	−13741.6	−0.670370
$750$	0	0
$751$	−23913.2	−1.16192	−0.580962	−	0.813931i	$-0.697324\pi$
$751$	−23913.2	−1.16192	−0.580962	+	0.813931i	$0.697324\pi$
$752$	−42509.6	−2.06139
$753$	−17150.7	−0.830022
$754$	−9816.81	−0.474147
$755$	0	0
$756$	1258.15	0.0605269
$757$	−31044.9	−1.49055	−0.745275	−	0.666758i	$-0.767682\pi$
$757$	−31044.9	−1.49055	−0.745275	+	0.666758i	$0.767682\pi$
$758$	51742.9	2.47940
$759$	15177.6	0.725842
$760$	0	0
$761$	14011.8	0.667446	0.333723	−	0.942671i	$-0.391695\pi$
$761$	14011.8	0.667446	0.333723	+	0.942671i	$0.391695\pi$
$762$	−27253.4	−1.29565
$763$	3816.23	0.181071
$764$	7150.69	0.338616
$765$	0	0
$766$	2515.97	0.118676
$767$	−22919.8	−1.07899
$768$	15326.6	0.720120
$769$	3342.49	0.156740	0.0783701	−	0.996924i	$-0.475028\pi$
$769$	3342.49	0.156740	0.0783701	+	0.996924i	$0.475028\pi$
$770$	0	0
$771$	−14172.9	−0.662027
$772$	−5984.51	−0.278999
$773$	21074.6	0.980594	0.490297	−	0.871555i	$-0.336888\pi$
$773$	21074.6	0.980594	0.490297	+	0.871555i	$0.336888\pi$
$774$	3191.90	0.148231
$775$	0	0
$776$	−2385.40	−0.110349
$777$	5306.20	0.244992
$778$	−37296.0	−1.71867
$779$	−25763.9	−1.18496
$780$	0	0
$781$	−23908.5	−1.09541
$782$	26981.1	1.23382
$783$	−1585.85	−0.0723800
$784$	−3574.12	−0.162815
$785$	0	0
$786$	13782.8	0.625464
$787$	−21394.8	−0.969048	−0.484524	−	0.874778i	$-0.661007\pi$
$787$	−21394.8	−0.969048	−0.484524	+	0.874778i	$0.661007\pi$
$788$	−20394.1	−0.921968
$789$	−17926.1	−0.808853
$790$	0	0
$791$	1622.76	0.0729442
$792$	−2247.71	−0.100845
$793$	−15401.9	−0.689706
$794$	−16873.5	−0.754179
$795$	0	0
$796$	−6320.83	−0.281452
$797$	−20645.0	−0.917547	−0.458773	−	0.888553i	$-0.651711\pi$
$797$	−20645.0	−0.917547	−0.458773	+	0.888553i	$0.651711\pi$
$798$	−7497.41	−0.332588
$799$	39430.0	1.74585
$800$	0	0
$801$	−3466.95	−0.152932
$802$	−44293.0	−1.95017
$803$	−56570.4	−2.48608
$804$	14708.7	0.645194
$805$	0	0
$806$	−1518.99	−0.0663825
$807$	13458.3	0.587058
$808$	2225.13	0.0968810
$809$	−15939.0	−0.692688	−0.346344	−	0.938108i	$-0.612577\pi$
$809$	−15939.0	−0.692688	−0.346344	+	0.938108i	$0.612577\pi$
$810$	0	0
$811$	22829.2	0.988460	0.494230	−	0.869331i	$-0.335450\pi$
$811$	22829.2	0.988460	0.494230	+	0.869331i	$0.335450\pi$
$812$	−2736.94	−0.118285
$813$	11483.0	0.495360
$814$	46982.9	2.02303
$815$	0	0
$816$	−14804.9	−0.635141
$817$	−8638.90	−0.369935
$818$	−11803.2	−0.504508
$819$	2750.38	0.117346
$820$	0	0
$821$	−5700.22	−0.242313	−0.121157	−	0.992633i	$-0.538660\pi$
$821$	−5700.22	−0.242313	−0.121157	+	0.992633i	$0.538660\pi$
$822$	−31943.5	−1.35542
$823$	32438.0	1.37390	0.686948	−	0.726707i	$-0.258950\pi$
$823$	32438.0	1.37390	0.686948	+	0.726707i	$0.258950\pi$
$824$	2635.99	0.111443
$825$	0	0
$826$	−14069.4	−0.592662
$827$	12762.6	0.536638	0.268319	−	0.963330i	$-0.413532\pi$
$827$	12762.6	0.536638	0.268319	+	0.963330i	$0.413532\pi$
$828$	6240.79	0.261935
$829$	−30766.7	−1.28899	−0.644494	−	0.764609i	$-0.722932\pi$
$829$	−30766.7	−1.28899	−0.644494	+	0.764609i	$0.722932\pi$
$830$	0	0
$831$	10261.6	0.428365
$832$	−14322.4	−0.596804
$833$	3315.19	0.137892
$834$	−14306.1	−0.593981
$835$	0	0
$836$	−30150.6	−1.24734
$837$	−245.384	−0.0101335
$838$	−20731.7	−0.854613
$839$	−9779.71	−0.402423	−0.201212	−	0.979548i	$-0.564488\pi$
$839$	−9779.71	−0.402423	−0.201212	+	0.979548i	$0.564488\pi$
$840$	0	0
$841$	−20939.2	−0.858551
$842$	16034.8	0.656288
$843$	15707.8	0.641760
$844$	15355.0	0.626233
$845$	0	0
$846$	20080.7	0.816061
$847$	7195.32	0.291894
$848$	45443.7	1.84026
$849$	20957.6	0.847190
$850$	0	0
$851$	26320.4	1.06023
$852$	−9830.79	−0.395302
$853$	−24201.5	−0.971445	−0.485723	−	0.874113i	$-0.661444\pi$
$853$	−24201.5	−0.971445	−0.485723	+	0.874113i	$0.661444\pi$
$854$	−9454.52	−0.378837
$855$	0	0
$856$	10094.5	0.403062
$857$	−21036.7	−0.838507	−0.419254	−	0.907869i	$-0.637708\pi$
$857$	−21036.7	−0.838507	−0.419254	+	0.907869i	$0.637708\pi$
$858$	24352.8	0.968988
$859$	−6179.19	−0.245438	−0.122719	−	0.992441i	$-0.539161\pi$
$859$	−6179.19	−0.245438	−0.122719	+	0.992441i	$0.539161\pi$
$860$	0	0
$861$	5801.76	0.229644
$862$	−34870.9	−1.37785
$863$	−50256.2	−1.98232	−0.991160	−	0.132671i	$-0.957644\pi$
$863$	−50256.2	−1.98232	−0.991160	+	0.132671i	$0.957644\pi$
$864$	−6429.04	−0.253149
$865$	0	0
$866$	64497.7	2.53085
$867$	−1006.65	−0.0394321
$868$	−423.497	−0.0165604
$869$	−42361.2	−1.65363
$870$	0	0
$871$	32154.1	1.25086
$872$	−2803.37	−0.108869
$873$	4175.03	0.161860
$874$	−37189.5	−1.43930
$875$	0	0
$876$	−23260.8	−0.897156
$877$	9175.95	0.353306	0.176653	−	0.984273i	$-0.443473\pi$
$877$	9175.95	0.353306	0.176653	+	0.984273i	$0.443473\pi$
$878$	32289.5	1.24114
$879$	−22199.1	−0.851828
$880$	0	0
$881$	−26172.7	−1.00089	−0.500444	−	0.865769i	$-0.666830\pi$
$881$	−26172.7	−1.00089	−0.500444	+	0.865769i	$0.666830\pi$
$882$	1688.34	0.0644549
$883$	−18615.5	−0.709471	−0.354736	−	0.934967i	$-0.615429\pi$
$883$	−18615.5	−0.709471	−0.354736	+	0.934967i	$0.615429\pi$
$884$	19662.3	0.748092
$885$	0	0
$886$	16456.4	0.624001
$887$	12837.8	0.485964	0.242982	−	0.970031i	$-0.421874\pi$
$887$	12837.8	0.485964	0.242982	+	0.970031i	$0.421874\pi$
$888$	−3897.89	−0.147302
$889$	−16610.3	−0.626649
$890$	0	0
$891$	3934.05	0.147919
$892$	21485.7	0.806496
$893$	−54348.4	−2.03662
$894$	222.868	0.00833759
$895$	0	0
$896$	4542.42	0.169366
$897$	13642.7	0.507824
$898$	40370.6	1.50020
$899$	533.803	0.0198035
$900$	0	0
$901$	−42151.5	−1.55857
$902$	51370.7	1.89630
$903$	1945.39	0.0716926
$904$	−1192.07	−0.0438579
$905$	0	0
$906$	−26988.1	−0.989647
$907$	26766.1	0.979885	0.489942	−	0.871755i	$-0.337018\pi$
$907$	26766.1	0.979885	0.489942	+	0.871755i	$0.337018\pi$
$908$	4245.87	0.155181
$909$	−3894.52	−0.142105
$910$	0	0
$911$	5022.67	0.182666	0.0913328	−	0.995820i	$-0.470887\pi$
$911$	5022.67	0.182666	0.0913328	+	0.995820i	$0.470887\pi$
$912$	20406.3	0.740923
$913$	25721.3	0.932367
$914$	−45759.0	−1.65599
$915$	0	0
$916$	3626.48	0.130810
$917$	8400.26	0.302509
$918$	6993.52	0.251439
$919$	9541.79	0.342497	0.171248	−	0.985228i	$-0.445220\pi$
$919$	9541.79	0.342497	0.171248	+	0.985228i	$0.445220\pi$
$920$	0	0
$921$	−8005.93	−0.286432
$922$	−65852.4	−2.35220
$923$	−21490.7	−0.766387
$924$	6789.59	0.241733
$925$	0	0
$926$	39695.7	1.40873
$927$	−4613.63	−0.163464
$928$	13985.6	0.494718
$929$	−25479.6	−0.899846	−0.449923	−	0.893067i	$-0.648549\pi$
$929$	−25479.6	−0.899846	−0.449923	+	0.893067i	$0.648549\pi$
$930$	0	0
$931$	−4569.49	−0.160858
$932$	38267.2	1.34494
$933$	18567.7	0.651533
$934$	64621.9	2.26391
$935$	0	0
$936$	−2020.41	−0.0705545
$937$	33608.3	1.17176	0.585878	−	0.810399i	$-0.300750\pi$
$937$	33608.3	1.17176	0.585878	+	0.810399i	$0.300750\pi$
$938$	19738.0	0.687066
$939$	8764.77	0.304609
$940$	0	0
$941$	−19173.6	−0.664232	−0.332116	−	0.943239i	$-0.607762\pi$
$941$	−19173.6	−0.664232	−0.332116	+	0.943239i	$0.607762\pi$
$942$	44774.9	1.54867
$943$	28778.5	0.993804
$944$	38294.0	1.32030
$945$	0	0
$946$	17225.1	0.592005
$947$	979.315	0.0336045	0.0168023	−	0.999859i	$-0.494651\pi$
$947$	979.315	0.0336045	0.0168023	+	0.999859i	$0.494651\pi$
$948$	−17418.2	−0.596749
$949$	−50849.5	−1.73935
$950$	0	0
$951$	−29476.7	−1.00510
$952$	−2435.31	−0.0829083
$953$	−3048.61	−0.103624	−0.0518122	−	0.998657i	$-0.516500\pi$
$953$	−3048.61	−0.103624	−0.0518122	+	0.998657i	$0.516500\pi$
$954$	−21466.7	−0.728521
$955$	0	0
$956$	−17827.7	−0.603127
$957$	−8558.03	−0.289072
$958$	28059.0	0.946290
$959$	−19468.8	−0.655557
$960$	0	0
$961$	−29708.4	−0.997227
$962$	42231.6	1.41538
$963$	−17667.8	−0.591211
$964$	−14659.4	−0.489781
$965$	0	0
$966$	8374.67	0.278934
$967$	14467.9	0.481133	0.240567	−	0.970633i	$-0.422667\pi$
$967$	14467.9	0.481133	0.240567	+	0.970633i	$0.422667\pi$
$968$	−5285.62	−0.175502
$969$	−18928.0	−0.627507
$970$	0	0
$971$	12952.8	0.428090	0.214045	−	0.976824i	$-0.431336\pi$
$971$	12952.8	0.428090	0.214045	+	0.976824i	$0.431336\pi$
$972$	1617.62	0.0533797
$973$	−8719.23	−0.287282
$974$	65885.4	2.16746
$975$	0	0
$976$	25733.2	0.843954
$977$	−47244.2	−1.54706	−0.773529	−	0.633760i	$-0.781511\pi$
$977$	−47244.2	−1.54706	−0.773529	+	0.633760i	$0.781511\pi$
$978$	−17544.2	−0.573621
$979$	−18709.4	−0.610781
$980$	0	0
$981$	4906.58	0.159689
$982$	43615.3	1.41733
$983$	1536.84	0.0498651	0.0249326	−	0.999689i	$-0.492063\pi$
$983$	1536.84	0.0498651	0.0249326	+	0.999689i	$0.492063\pi$
$984$	−4261.92	−0.138074
$985$	0	0
$986$	−15213.5	−0.491376
$987$	12238.7	0.394692
$988$	−27101.5	−0.872685
$989$	9649.73	0.310256
$990$	0	0
$991$	3785.22	0.121334	0.0606668	−	0.998158i	$-0.480677\pi$
$991$	3785.22	0.121334	0.0606668	+	0.998158i	$0.480677\pi$
$992$	2164.04	0.0692625
$993$	−27774.5	−0.887610
$994$	−13192.2	−0.420956
$995$	0	0
$996$	10576.2	0.336465
$997$	25894.9	0.822566	0.411283	−	0.911508i	$-0.365081\pi$
$997$	25894.9	0.822566	0.411283	+	0.911508i	$0.365081\pi$
$998$	73044.0	2.31680
$999$	6822.26	0.216063

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.4.a.l.1.2		2
3.2	odd	2		1575.4.a.q.1.1		2
5.2	odd	4		525.4.d.l.274.4		4
5.3	odd	4		525.4.d.l.274.1		4
5.4	even	2		105.4.a.e.1.1	✓	2
15.14	odd	2		315.4.a.k.1.2		2
20.19	odd	2		1680.4.a.bo.1.1		2
35.34	odd	2		735.4.a.o.1.1		2
105.104	even	2		2205.4.a.bb.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.a.e.1.1	✓	2	5.4	even	2
315.4.a.k.1.2		2	15.14	odd	2
525.4.a.l.1.2		2	1.1	even	1	trivial
525.4.d.l.274.1		4	5.3	odd	4
525.4.d.l.274.4		4	5.2	odd	4
735.4.a.o.1.1		2	35.34	odd	2
1575.4.a.q.1.1		2	3.2	odd	2
1680.4.a.bo.1.1		2	20.19	odd	2
2205.4.a.bb.1.2		2	105.104	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	525.a (trivial)

\(f(q)\)	\(=\)	\(q+3.82843 q^{2} +3.00000 q^{3} +6.65685 q^{4} +11.4853 q^{6} +7.00000 q^{7} -5.14214 q^{8} +9.00000 q^{9} +48.5685 q^{11} +19.9706 q^{12} +43.6569 q^{13} +26.7990 q^{14} -72.9411 q^{16} +67.6569 q^{17} +34.4558 q^{18} -93.2548 q^{19} +21.0000 q^{21} +185.941 q^{22} +104.167 q^{23} -15.4264 q^{24} +167.137 q^{26} +27.0000 q^{27} +46.5980 q^{28} -58.7351 q^{29} -9.08831 q^{31} -238.113 q^{32} +145.706 q^{33} +259.019 q^{34} +59.9117 q^{36} +252.676 q^{37} -357.019 q^{38} +130.971 q^{39} +276.274 q^{41} +80.3970 q^{42} +92.6375 q^{43} +323.314 q^{44} +398.794 q^{46} +582.794 q^{47} -218.823 q^{48} +49.0000 q^{49} +202.971 q^{51} +290.617 q^{52} -623.019 q^{53} +103.368 q^{54} -35.9949 q^{56} -279.765 q^{57} -224.863 q^{58} -524.999 q^{59} -352.794 q^{61} -34.7939 q^{62} +63.0000 q^{63} -328.068 q^{64} +557.823 q^{66} +736.520 q^{67} +450.382 q^{68} +312.500 q^{69} -492.264 q^{71} -46.2792 q^{72} -1164.75 q^{73} +967.352 q^{74} -620.784 q^{76} +339.980 q^{77} +501.411 q^{78} -872.195 q^{79} +81.0000 q^{81} +1057.70 q^{82} +529.588 q^{83} +139.794 q^{84} +354.656 q^{86} -176.205 q^{87} -249.746 q^{88} -385.216 q^{89} +305.598 q^{91} +693.421 q^{92} -27.2649 q^{93} +2231.18 q^{94} -714.338 q^{96} +463.892 q^{97} +187.593 q^{98} +437.117 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 6 q^{3} + 2 q^{4} + 6 q^{6} + 14 q^{7} + 18 q^{8} + 18 q^{9} - 16 q^{11} + 6 q^{12} + 76 q^{13} + 14 q^{14} - 78 q^{16} + 124 q^{17} + 18 q^{18} - 96 q^{19} + 42 q^{21} + 304 q^{22} + 16 q^{23}+ \cdots - 144 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 6 * q^3 + 2 * q^4 + 6 * q^6 + 14 * q^7 + 18 * q^8 + 18 * q^9 - 16 * q^11 + 6 * q^12 + 76 * q^13 + 14 * q^14 - 78 * q^16 + 124 * q^17 + 18 * q^18 - 96 * q^19 + 42 * q^21 + 304 * q^22 + 16 * q^23 + 54 * q^24 + 108 * q^26 + 54 * q^27 + 14 * q^28 + 188 * q^29 - 120 * q^31 - 414 * q^32 - 48 * q^33 + 156 * q^34 + 18 * q^36 + 132 * q^37 - 352 * q^38 + 228 * q^39 + 100 * q^41 + 42 * q^42 + 536 * q^43 + 624 * q^44 + 560 * q^46 + 928 * q^47 - 234 * q^48 + 98 * q^49 + 372 * q^51 + 140 * q^52 - 884 * q^53 + 54 * q^54 + 126 * q^56 - 288 * q^57 - 676 * q^58 + 104 * q^59 - 468 * q^61 + 168 * q^62 + 126 * q^63 + 34 * q^64 + 912 * q^66 + 1688 * q^67 + 188 * q^68 + 48 * q^69 - 136 * q^71 + 162 * q^72 - 508 * q^73 + 1188 * q^74 - 608 * q^76 - 112 * q^77 + 324 * q^78 - 432 * q^79 + 162 * q^81 + 1380 * q^82 + 584 * q^83 + 42 * q^84 - 456 * q^86 + 564 * q^87 - 1744 * q^88 - 1404 * q^89 + 532 * q^91 + 1104 * q^92 - 360 * q^93 + 1600 * q^94 - 1242 * q^96 + 1188 * q^97 + 98 * q^98 - 144 * q^99

Introduction

L-functions

Modular forms

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