LMFDB - Embedded newform 5225.2.a.k.1.5

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 = [6,0,-3,8,0,2,-5] 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:	$1$
Dimension:	$6$
Coefficient field:	6.6.131947641.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 10x^{4} + 25x^{2} - 3x - 9 $ x^6 - 10x^4 + 25x^2 - 3*x - 9
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1045)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-2.04201$ 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)

$f(q)$	$=$	\(q+2.04201 q^{2} +1.09835 q^{3} +2.16980 q^{4} +2.24284 q^{6} +0.469142 q^{7} +0.346728 q^{8} -1.79363 q^{9} -1.00000 q^{11} +2.38320 q^{12} -2.14036 q^{13} +0.957992 q^{14} -3.63157 q^{16} -4.47995 q^{17} -3.66261 q^{18} -1.00000 q^{19} +0.515282 q^{21} -2.04201 q^{22} -0.200995 q^{23} +0.380828 q^{24} -4.37063 q^{26} -5.26508 q^{27} +1.01794 q^{28} -6.92379 q^{29} +4.65688 q^{31} -8.10916 q^{32} -1.09835 q^{33} -9.14810 q^{34} -3.89181 q^{36} +7.22421 q^{37} -2.04201 q^{38} -2.35086 q^{39} -3.34355 q^{41} +1.05221 q^{42} -3.75162 q^{43} -2.16980 q^{44} -0.410434 q^{46} +2.69748 q^{47} -3.98874 q^{48} -6.77991 q^{49} -4.92055 q^{51} -4.64414 q^{52} -10.4652 q^{53} -10.7513 q^{54} +0.162665 q^{56} -1.09835 q^{57} -14.1384 q^{58} +6.05634 q^{59} -5.04325 q^{61} +9.50939 q^{62} -0.841467 q^{63} -9.29582 q^{64} -2.24284 q^{66} -7.47221 q^{67} -9.72059 q^{68} -0.220763 q^{69} +15.9494 q^{71} -0.621901 q^{72} +15.5740 q^{73} +14.7519 q^{74} -2.16980 q^{76} -0.469142 q^{77} -4.80047 q^{78} +5.88178 q^{79} -0.402006 q^{81} -6.82756 q^{82} -12.9799 q^{83} +1.11806 q^{84} -7.66084 q^{86} -7.60473 q^{87} -0.346728 q^{88} -0.794873 q^{89} -1.00413 q^{91} -0.436119 q^{92} +5.11488 q^{93} +5.50828 q^{94} -8.90669 q^{96} -17.5493 q^{97} -13.8446 q^{98} +1.79363 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{3} + 8 q^{4} + 2 q^{6} - 5 q^{7} + 9 q^{9} - 6 q^{11} - 19 q^{12} + 9 q^{13} + 18 q^{14} + 4 q^{16} + 5 q^{17} - 2 q^{18} - 6 q^{19} + 3 q^{21} - 8 q^{23} - 7 q^{24} - 22 q^{26} - 30 q^{27} - 10 q^{28}+ \cdots - 9 q^{99}+O(q^{100}) $ 6 * q - 3 * q^3 + 8 * q^4 + 2 * q^6 - 5 * q^7 + 9 * q^9 - 6 * q^11 - 19 * q^12 + 9 * q^13 + 18 * q^14 + 4 * q^16 + 5 * q^17 - 2 * q^18 - 6 * q^19 + 3 * q^21 - 8 * q^23 - 7 * q^24 - 22 * q^26 - 30 * q^27 - 10 * q^28 - 5 * q^29 - q^31 - 15 * q^32 + 3 * q^33 - 22 * q^34 + 12 * q^36 - 9 * q^37 - 32 * q^39 + 25 * q^41 - 11 * q^42 - 15 * q^43 - 8 * q^44 - 16 * q^46 - 24 * q^47 + 4 * q^48 + 13 * q^49 + 27 * q^52 - 5 * q^53 - 11 * q^54 - 12 * q^56 + 3 * q^57 - 13 * q^58 + 39 * q^59 - 11 * q^61 + 42 * q^62 - 38 * q^63 - 14 * q^64 - 2 * q^66 - 24 * q^67 - 45 * q^68 + 14 * q^69 - 24 * q^71 + 61 * q^72 + 26 * q^73 + q^74 - 8 * q^76 + 5 * q^77 + 29 * q^78 + 11 * q^79 + 30 * q^81 - 8 * q^82 - 39 * q^83 + 25 * q^84 + 18 * q^86 + 16 * q^87 + 22 * q^89 - 26 * q^91 + 11 * q^92 + 6 * q^93 - 30 * q^94 - 15 * q^96 - 22 * q^97 - 33 * q^98 - 9 * 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 $

$2$	2.04201	1.44392	0.721959	−	0.691936i	$-0.243242\pi$
$2$	2.04201	1.44392	0.721959	+	0.691936i	$0.243242\pi$
$3$	1.09835	0.634132	0.317066	−	0.948403i	$-0.397302\pi$
$3$	1.09835	0.634132	0.317066	+	0.948403i	$0.397302\pi$
$4$	2.16980	1.08490
$5$	0	0
$5$	0	0
$6$	2.24284	0.915635
$7$	0.469142	0.177319	0.0886595	−	0.996062i	$-0.471742\pi$
$7$	0.469142	0.177319	0.0886595	+	0.996062i	$0.471742\pi$
$8$	0.346728	0.122587
$9$	−1.79363	−0.597876
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	2.38320	0.687969
$13$	−2.14036	−0.593628	−0.296814	−	0.954935i	$-0.595924\pi$
$13$	−2.14036	−0.593628	−0.296814	+	0.954935i	$0.595924\pi$
$14$	0.957992	0.256034
$15$	0	0
$16$	−3.63157	−0.907893
$17$	−4.47995	−1.08655	−0.543274	−	0.839555i	$-0.682816\pi$
$17$	−4.47995	−1.08655	−0.543274	+	0.839555i	$0.682816\pi$
$18$	−3.66261	−0.863284
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0.515282	0.112444
$22$	−2.04201	−0.435358
$23$	−0.200995	−0.0419105	−0.0209552	−	0.999780i	$-0.506671\pi$
$23$	−0.200995	−0.0419105	−0.0209552	+	0.999780i	$0.506671\pi$
$24$	0.380828	0.0777362
$25$	0	0
$26$	−4.37063	−0.857150
$27$	−5.26508	−1.01326
$28$	1.01794	0.192373
$29$	−6.92379	−1.28571	−0.642857	−	0.765986i	$-0.722251\pi$
$29$	−6.92379	−1.28571	−0.642857	+	0.765986i	$0.722251\pi$
$30$	0	0
$31$	4.65688	0.836401	0.418200	−	0.908355i	$-0.362661\pi$
$31$	4.65688	0.836401	0.418200	+	0.908355i	$0.362661\pi$
$32$	−8.10916	−1.43351
$33$	−1.09835	−0.191198
$34$	−9.14810	−1.56889
$35$	0	0
$36$	−3.89181	−0.648635
$37$	7.22421	1.18765	0.593826	−	0.804593i	$-0.297617\pi$
$37$	7.22421	1.18765	0.593826	+	0.804593i	$0.297617\pi$
$38$	−2.04201	−0.331257
$39$	−2.35086	−0.376439
$40$	0	0
$41$	−3.34355	−0.522175	−0.261088	−	0.965315i	$-0.584081\pi$
$41$	−3.34355	−0.522175	−0.261088	+	0.965315i	$0.584081\pi$
$42$	1.05221	0.162359
$43$	−3.75162	−0.572117	−0.286058	−	0.958212i	$-0.592345\pi$
$43$	−3.75162	−0.572117	−0.286058	+	0.958212i	$0.592345\pi$
$44$	−2.16980	−0.327109
$45$	0	0
$46$	−0.410434	−0.0605152
$47$	2.69748	0.393468	0.196734	−	0.980457i	$-0.436966\pi$
$47$	2.69748	0.393468	0.196734	+	0.980457i	$0.436966\pi$
$48$	−3.98874	−0.575724
$49$	−6.77991	−0.968558
$50$	0	0
$51$	−4.92055	−0.689015
$52$	−4.64414	−0.644027
$53$	−10.4652	−1.43751	−0.718754	−	0.695264i	$-0.755287\pi$
$53$	−10.4652	−1.43751	−0.718754	+	0.695264i	$0.755287\pi$
$54$	−10.7513	−1.46307
$55$	0	0
$56$	0.162665	0.0217370
$57$	−1.09835	−0.145480
$58$	−14.1384	−1.85647
$59$	6.05634	0.788468	0.394234	−	0.919010i	$-0.371010\pi$
$59$	6.05634	0.788468	0.394234	+	0.919010i	$0.371010\pi$
$60$	0	0
$61$	−5.04325	−0.645722	−0.322861	−	0.946446i	$-0.604645\pi$
$61$	−5.04325	−0.645722	−0.322861	+	0.946446i	$0.604645\pi$
$62$	9.50939	1.20769
$63$	−0.841467	−0.106015
$64$	−9.29582	−1.16198
$65$	0	0
$66$	−2.24284	−0.276074
$67$	−7.47221	−0.912876	−0.456438	−	0.889755i	$-0.650875\pi$
$67$	−7.47221	−0.912876	−0.456438	+	0.889755i	$0.650875\pi$
$68$	−9.72059	−1.17879
$69$	−0.220763	−0.0265768
$70$	0	0
$71$	15.9494	1.89284	0.946421	−	0.322936i	$-0.104670\pi$
$71$	15.9494	1.89284	0.946421	+	0.322936i	$0.104670\pi$
$72$	−0.621901	−0.0732918
$73$	15.5740	1.82280	0.911400	−	0.411522i	$-0.135003\pi$
$73$	15.5740	1.82280	0.911400	+	0.411522i	$0.135003\pi$
$74$	14.7519	1.71487
$75$	0	0
$76$	−2.16980	−0.248893
$77$	−0.469142	−0.0534637
$78$	−4.80047	−0.543547
$79$	5.88178	0.661752	0.330876	−	0.943674i	$-0.392656\pi$
$79$	5.88178	0.661752	0.330876	+	0.943674i	$0.392656\pi$
$80$	0	0
$81$	−0.402006	−0.0446673
$82$	−6.82756	−0.753978
$83$	−12.9799	−1.42473	−0.712364	−	0.701810i	$-0.752376\pi$
$83$	−12.9799	−1.42473	−0.712364	+	0.701810i	$0.752376\pi$
$84$	1.11806	0.121990
$85$	0	0
$86$	−7.66084	−0.826090
$87$	−7.60473	−0.815313
$88$	−0.346728	−0.0369613
$89$	−0.794873	−0.0842564	−0.0421282	−	0.999112i	$-0.513414\pi$
$89$	−0.794873	−0.0842564	−0.0421282	+	0.999112i	$0.513414\pi$
$90$	0	0
$91$	−1.00413	−0.105262
$92$	−0.436119	−0.0454686
$93$	5.11488	0.530389
$94$	5.50828	0.568136
$95$	0	0
$96$	−8.90669	−0.909035
$97$	−17.5493	−1.78186	−0.890931	−	0.454139i	$-0.849947\pi$
$97$	−17.5493	−1.78186	−0.890931	+	0.454139i	$0.849947\pi$
$98$	−13.8446	−1.39852
$99$	1.79363	0.180267
$100$	0	0
$101$	14.0995	1.40295	0.701476	−	0.712693i	$-0.252525\pi$
$101$	14.0995	1.40295	0.701476	+	0.712693i	$0.252525\pi$
$102$	−10.0478	−0.994881
$103$	−14.7905	−1.45735	−0.728673	−	0.684861i	$-0.759863\pi$
$103$	−14.7905	−1.45735	−0.728673	+	0.684861i	$0.759863\pi$
$104$	−0.742122	−0.0727710
$105$	0	0
$106$	−21.3701	−2.07564
$107$	−3.73310	−0.360892	−0.180446	−	0.983585i	$-0.557754\pi$
$107$	−3.73310	−0.360892	−0.180446	+	0.983585i	$0.557754\pi$
$108$	−11.4242	−1.09929
$109$	0.525483	0.0503321	0.0251661	−	0.999683i	$-0.491989\pi$
$109$	0.525483	0.0503321	0.0251661	+	0.999683i	$0.491989\pi$
$110$	0	0
$111$	7.93470	0.753129
$112$	−1.70372	−0.160987
$113$	−9.63228	−0.906128	−0.453064	−	0.891478i	$-0.649669\pi$
$113$	−9.63228	−0.906128	−0.453064	+	0.891478i	$0.649669\pi$
$114$	−2.24284	−0.210061
$115$	0	0
$116$	−15.0232	−1.39487
$117$	3.83901	0.354916
$118$	12.3671	1.13848
$119$	−2.10173	−0.192666
$120$	0	0
$121$	1.00000	0.0909091
$122$	−10.2984	−0.932370
$123$	−3.67239	−0.331128
$124$	10.1045	0.907410
$125$	0	0
$126$	−1.71828	−0.153077
$127$	17.6992	1.57055	0.785276	−	0.619146i	$-0.212521\pi$
$127$	17.6992	1.57055	0.785276	+	0.619146i	$0.212521\pi$
$128$	−2.76383	−0.244290
$129$	−4.12059	−0.362798
$130$	0	0
$131$	19.8018	1.73009	0.865046	−	0.501693i	$-0.167289\pi$
$131$	19.8018	1.73009	0.865046	+	0.501693i	$0.167289\pi$
$132$	−2.38320	−0.207430
$133$	−0.469142	−0.0406798
$134$	−15.2583	−1.31812
$135$	0	0
$136$	−1.55332	−0.133196
$137$	−9.38019	−0.801403	−0.400702	−	0.916209i	$-0.631234\pi$
$137$	−9.38019	−0.801403	−0.400702	+	0.916209i	$0.631234\pi$
$138$	−0.450800	−0.0383747
$139$	−13.6770	−1.16007	−0.580034	−	0.814592i	$-0.696961\pi$
$139$	−13.6770	−1.16007	−0.580034	+	0.814592i	$0.696961\pi$
$140$	0	0
$141$	2.96278	0.249511
$142$	32.5687	2.73311
$143$	2.14036	0.178986
$144$	6.51370	0.542808
$145$	0	0
$146$	31.8022	2.63197
$147$	−7.44670	−0.614194
$148$	15.6751	1.28848
$149$	9.34304	0.765412	0.382706	−	0.923870i	$-0.374992\pi$
$149$	9.34304	0.765412	0.382706	+	0.923870i	$0.374992\pi$
$150$	0	0
$151$	7.72133	0.628353	0.314176	−	0.949365i	$-0.398272\pi$
$151$	7.72133	0.628353	0.314176	+	0.949365i	$0.398272\pi$
$152$	−0.346728	−0.0281233
$153$	8.03537	0.649621
$154$	−0.957992	−0.0771972
$155$	0	0
$156$	−5.10089	−0.408398
$157$	−2.86254	−0.228455	−0.114228	−	0.993455i	$-0.536439\pi$
$157$	−2.86254	−0.228455	−0.114228	+	0.993455i	$0.536439\pi$
$158$	12.0106	0.955515
$159$	−11.4945	−0.911570
$160$	0	0
$161$	−0.0942954	−0.00743152
$162$	−0.820899	−0.0644959
$163$	−10.9042	−0.854083	−0.427041	−	0.904232i	$-0.640444\pi$
$163$	−10.9042	−0.854083	−0.427041	+	0.904232i	$0.640444\pi$
$164$	−7.25483	−0.566507
$165$	0	0
$166$	−26.5050	−2.05719
$167$	15.7161	1.21615	0.608074	−	0.793880i	$-0.291942\pi$
$167$	15.7161	1.21615	0.608074	+	0.793880i	$0.291942\pi$
$168$	0.178663	0.0137841
$169$	−8.41887	−0.647605
$170$	0	0
$171$	1.79363	0.137162
$172$	−8.14026	−0.620689
$173$	−3.84278	−0.292161	−0.146080	−	0.989273i	$-0.546666\pi$
$173$	−3.84278	−0.292161	−0.146080	+	0.989273i	$0.546666\pi$
$174$	−15.5289	−1.17724
$175$	0	0
$176$	3.63157	0.273740
$177$	6.65198	0.499993
$178$	−1.62314	−0.121659
$179$	19.8584	1.48429	0.742143	−	0.670242i	$-0.233810\pi$
$179$	19.8584	1.48429	0.742143	+	0.670242i	$0.233810\pi$
$180$	0	0
$181$	−15.0067	−1.11544	−0.557719	−	0.830030i	$-0.688323\pi$
$181$	−15.0067	−1.11544	−0.557719	+	0.830030i	$0.688323\pi$
$182$	−2.05044	−0.151989
$183$	−5.53925	−0.409473
$184$	−0.0696907	−0.00513767
$185$	0	0
$186$	10.4446	0.765838
$187$	4.47995	0.327607
$188$	5.85299	0.426873
$189$	−2.47007	−0.179671
$190$	0	0
$191$	14.4281	1.04398	0.521990	−	0.852951i	$-0.325190\pi$
$191$	14.4281	1.04398	0.521990	+	0.852951i	$0.325190\pi$
$192$	−10.2101	−0.736847
$193$	−23.3554	−1.68116	−0.840580	−	0.541688i	$-0.817785\pi$
$193$	−23.3554	−1.68116	−0.840580	+	0.541688i	$0.817785\pi$
$194$	−35.8358	−2.57286
$195$	0	0
$196$	−14.7110	−1.05079
$197$	10.3230	0.735484	0.367742	−	0.929928i	$-0.380131\pi$
$197$	10.3230	0.735484	0.367742	+	0.929928i	$0.380131\pi$
$198$	3.66261	0.260290
$199$	−10.5459	−0.747579	−0.373789	−	0.927514i	$-0.621942\pi$
$199$	−10.5459	−0.747579	−0.373789	+	0.927514i	$0.621942\pi$
$200$	0	0
$201$	−8.20710	−0.578884
$202$	28.7913	2.02575
$203$	−3.24824	−0.227982
$204$	−10.6766	−0.747512
$205$	0	0
$206$	−30.2022	−2.10429
$207$	0.360511	0.0250573
$208$	7.77287	0.538951
$209$	1.00000	0.0691714
$210$	0	0
$211$	7.35896	0.506612	0.253306	−	0.967386i	$-0.418482\pi$
$211$	7.35896	0.506612	0.253306	+	0.967386i	$0.418482\pi$
$212$	−22.7074	−1.55955
$213$	17.5180	1.20031
$214$	−7.62302	−0.521099
$215$	0	0
$216$	−1.82555	−0.124213
$217$	2.18474	0.148310
$218$	1.07304	0.0726755
$219$	17.1057	1.15590
$220$	0	0
$221$	9.58870	0.645006
$222$	16.2027	1.08746
$223$	−14.4057	−0.964674	−0.482337	−	0.875986i	$-0.660212\pi$
$223$	−14.4057	−0.964674	−0.482337	+	0.875986i	$0.660212\pi$
$224$	−3.80435	−0.254189
$225$	0	0
$226$	−19.6692	−1.30837
$227$	29.7139	1.97218	0.986091	−	0.166204i	$-0.0531510\pi$
$227$	29.7139	1.97218	0.986091	+	0.166204i	$0.0531510\pi$
$228$	−2.38320	−0.157831
$229$	−11.0142	−0.727836	−0.363918	−	0.931431i	$-0.618561\pi$
$229$	−11.0142	−0.727836	−0.363918	+	0.931431i	$0.618561\pi$
$230$	0	0
$231$	−0.515282	−0.0339030
$232$	−2.40067	−0.157612
$233$	−9.53690	−0.624783	−0.312391	−	0.949953i	$-0.601130\pi$
$233$	−9.53690	−0.624783	−0.312391	+	0.949953i	$0.601130\pi$
$234$	7.83929	0.512470
$235$	0	0
$236$	13.1410	0.855408
$237$	6.46024	0.419638
$238$	−4.29176	−0.278193
$239$	4.02979	0.260665	0.130333	−	0.991470i	$-0.458395\pi$
$239$	4.02979	0.260665	0.130333	+	0.991470i	$0.458395\pi$
$240$	0	0
$241$	6.06607	0.390750	0.195375	−	0.980729i	$-0.437408\pi$
$241$	6.06607	0.390750	0.195375	+	0.980729i	$0.437408\pi$
$242$	2.04201	0.131265
$243$	15.3537	0.984940
$244$	−10.9428	−0.700543
$245$	0	0
$246$	−7.49905	−0.478122
$247$	2.14036	0.136188
$248$	1.61467	0.102532
$249$	−14.2565	−0.903466
$250$	0	0
$251$	−20.7498	−1.30971	−0.654857	−	0.755753i	$-0.727271\pi$
$251$	−20.7498	−1.30971	−0.654857	+	0.755753i	$0.727271\pi$
$252$	−1.82581	−0.115015
$253$	0.200995	0.0126365
$254$	36.1420	2.26775
$255$	0	0
$256$	12.9479	0.809243
$257$	−21.3702	−1.33304	−0.666519	−	0.745488i	$-0.732216\pi$
$257$	−21.3702	−1.33304	−0.666519	+	0.745488i	$0.732216\pi$
$258$	−8.41428	−0.523850
$259$	3.38918	0.210593
$260$	0	0
$261$	12.4187	0.768699
$262$	40.4355	2.49811
$263$	2.10893	0.130042	0.0650212	−	0.997884i	$-0.479288\pi$
$263$	2.10893	0.130042	0.0650212	+	0.997884i	$0.479288\pi$
$264$	−0.380828	−0.0234384
$265$	0	0
$266$	−0.957992	−0.0587382
$267$	−0.873048	−0.0534297
$268$	−16.2132	−0.990378
$269$	20.3100	1.23833	0.619163	−	0.785263i	$-0.287472\pi$
$269$	20.3100	1.23833	0.619163	+	0.785263i	$0.287472\pi$
$270$	0	0
$271$	−2.81864	−0.171220	−0.0856100	−	0.996329i	$-0.527284\pi$
$271$	−2.81864	−0.171220	−0.0856100	+	0.996329i	$0.527284\pi$
$272$	16.2693	0.986470
$273$	−1.10289	−0.0667497
$274$	−19.1544	−1.15716
$275$	0	0
$276$	−0.479011	−0.0288331
$277$	16.6723	1.00174	0.500872	−	0.865522i	$-0.333013\pi$
$277$	16.6723	1.00174	0.500872	+	0.865522i	$0.333013\pi$
$278$	−27.9285	−1.67504
$279$	−8.35272	−0.500064
$280$	0	0
$281$	−27.9281	−1.66605	−0.833026	−	0.553234i	$-0.813393\pi$
$281$	−27.9281	−1.66605	−0.833026	+	0.553234i	$0.813393\pi$
$282$	6.05001	0.360273
$283$	4.88494	0.290380	0.145190	−	0.989404i	$-0.453621\pi$
$283$	4.88494	0.290380	0.145190	+	0.989404i	$0.453621\pi$
$284$	34.6069	2.05354
$285$	0	0
$286$	4.37063	0.258441
$287$	−1.56860	−0.0925916
$288$	14.5448	0.857062
$289$	3.06997	0.180587
$290$	0	0
$291$	−19.2753	−1.12994
$292$	33.7924	1.97755
$293$	−0.648185	−0.0378674	−0.0189337	−	0.999821i	$-0.506027\pi$
$293$	−0.648185	−0.0378674	−0.0189337	+	0.999821i	$0.506027\pi$
$294$	−15.2062	−0.886845
$295$	0	0
$296$	2.50483	0.145591
$297$	5.26508	0.305511
$298$	19.0786	1.10519
$299$	0.430202	0.0248792
$300$	0	0
$301$	−1.76004	−0.101447
$302$	15.7670	0.907290
$303$	15.4862	0.889657
$304$	3.63157	0.208285
$305$	0	0
$306$	16.4083	0.938000
$307$	−11.8075	−0.673891	−0.336945	−	0.941524i	$-0.609394\pi$
$307$	−11.8075	−0.673891	−0.336945	+	0.941524i	$0.609394\pi$
$308$	−1.01794	−0.0580027
$309$	−16.2451	−0.924150
$310$	0	0
$311$	−14.4213	−0.817756	−0.408878	−	0.912589i	$-0.634080\pi$
$311$	−14.4213	−0.817756	−0.408878	+	0.912589i	$0.634080\pi$
$312$	−0.815108	−0.0461464
$313$	5.49337	0.310504	0.155252	−	0.987875i	$-0.450381\pi$
$313$	5.49337	0.310504	0.155252	+	0.987875i	$0.450381\pi$
$314$	−5.84533	−0.329871
$315$	0	0
$316$	12.7623	0.717934
$317$	−13.4742	−0.756789	−0.378395	−	0.925644i	$-0.623524\pi$
$317$	−13.4742	−0.756789	−0.378395	+	0.925644i	$0.623524\pi$
$318$	−23.4718	−1.31623
$319$	6.92379	0.387658
$320$	0	0
$321$	−4.10024	−0.228853
$322$	−0.192552	−0.0107305
$323$	4.47995	0.249271
$324$	−0.872271	−0.0484595
$325$	0	0
$326$	−22.2665	−1.23323
$327$	0.577164	0.0319172
$328$	−1.15930	−0.0640118
$329$	1.26550	0.0697694
$330$	0	0
$331$	10.6406	0.584859	0.292430	−	0.956287i	$-0.405536\pi$
$331$	10.6406	0.584859	0.292430	+	0.956287i	$0.405536\pi$
$332$	−28.1637	−1.54569
$333$	−12.9576	−0.710069
$334$	32.0924	1.75602
$335$	0	0
$336$	−1.87128	−0.102087
$337$	2.99281	0.163029	0.0815145	−	0.996672i	$-0.474024\pi$
$337$	2.99281	0.163029	0.0815145	+	0.996672i	$0.474024\pi$
$338$	−17.1914	−0.935089
$339$	−10.5796	−0.574605
$340$	0	0
$341$	−4.65688	−0.252184
$342$	3.66261	0.198051
$343$	−6.46473	−0.349063
$344$	−1.30079	−0.0701340
$345$	0	0
$346$	−7.84698	−0.421856
$347$	−1.57779	−0.0847001	−0.0423500	−	0.999103i	$-0.513484\pi$
$347$	−1.57779	−0.0847001	−0.0423500	+	0.999103i	$0.513484\pi$
$348$	−16.5007	−0.884532
$349$	12.9750	0.694534	0.347267	−	0.937766i	$-0.387110\pi$
$349$	12.9750	0.694534	0.347267	+	0.937766i	$0.387110\pi$
$350$	0	0
$351$	11.2691	0.601503
$352$	8.10916	0.432220
$353$	13.5044	0.718768	0.359384	−	0.933190i	$-0.382987\pi$
$353$	13.5044	0.718768	0.359384	+	0.933190i	$0.382987\pi$
$354$	13.5834	0.721949
$355$	0	0
$356$	−1.72471	−0.0914096
$357$	−2.30844	−0.122175
$358$	40.5510	2.14319
$359$	−22.9543	−1.21148	−0.605741	−	0.795662i	$-0.707123\pi$
$359$	−22.9543	−1.21148	−0.605741	+	0.795662i	$0.707123\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−30.6438	−1.61060
$363$	1.09835	0.0576484
$364$	−2.17876	−0.114198
$365$	0	0
$366$	−11.3112	−0.591246
$367$	−5.77896	−0.301660	−0.150830	−	0.988560i	$-0.548195\pi$
$367$	−5.77896	−0.301660	−0.150830	+	0.988560i	$0.548195\pi$
$368$	0.729930	0.0380502
$369$	5.99710	0.312196
$370$	0	0
$371$	−4.90967	−0.254898
$372$	11.0983	0.575418
$373$	−32.2936	−1.67210	−0.836051	−	0.548652i	$-0.815141\pi$
$373$	−32.2936	−1.67210	−0.836051	+	0.548652i	$0.815141\pi$
$374$	9.14810	0.473037
$375$	0	0
$376$	0.935292	0.0482340
$377$	14.8194	0.763237
$378$	−5.04390	−0.259430
$379$	−12.2031	−0.626831	−0.313415	−	0.949616i	$-0.601473\pi$
$379$	−12.2031	−0.626831	−0.313415	+	0.949616i	$0.601473\pi$
$380$	0	0
$381$	19.4399	0.995938
$382$	29.4623	1.50742
$383$	−11.5424	−0.589787	−0.294894	−	0.955530i	$-0.595284\pi$
$383$	−11.5424	−0.589787	−0.294894	+	0.955530i	$0.595284\pi$
$384$	−3.03565	−0.154912
$385$	0	0
$386$	−47.6919	−2.42746
$387$	6.72902	0.342055
$388$	−38.0784	−1.93314
$389$	−4.65210	−0.235871	−0.117935	−	0.993021i	$-0.537628\pi$
$389$	−4.65210	−0.235871	−0.117935	+	0.993021i	$0.537628\pi$
$390$	0	0
$391$	0.900450	0.0455377
$392$	−2.35078	−0.118732
$393$	21.7493	1.09711
$394$	21.0797	1.06198
$395$	0	0
$396$	3.89181	0.195571
$397$	23.8285	1.19592	0.597960	−	0.801526i	$-0.295978\pi$
$397$	23.8285	1.19592	0.597960	+	0.801526i	$0.295978\pi$
$398$	−21.5348	−1.07944
$399$	−0.515282	−0.0257963
$400$	0	0
$401$	−5.11101	−0.255232	−0.127616	−	0.991824i	$-0.540732\pi$
$401$	−5.11101	−0.255232	−0.127616	+	0.991824i	$0.540732\pi$
$402$	−16.7590	−0.835861
$403$	−9.96739	−0.496511
$404$	30.5930	1.52206
$405$	0	0
$406$	−6.63293	−0.329187
$407$	−7.22421	−0.358091
$408$	−1.70609	−0.0844642
$409$	−20.0164	−0.989748	−0.494874	−	0.868965i	$-0.664786\pi$
$409$	−20.0164	−0.989748	−0.494874	+	0.868965i	$0.664786\pi$
$410$	0	0
$411$	−10.3027	−0.508196
$412$	−32.0923	−1.58107
$413$	2.84128	0.139810
$414$	0.736167	0.0361806
$415$	0	0
$416$	17.3565	0.850972
$417$	−15.0221	−0.735636
$418$	2.04201	0.0998779
$419$	34.3463	1.67793	0.838964	−	0.544188i	$-0.183162\pi$
$419$	34.3463	1.67793	0.838964	+	0.544188i	$0.183162\pi$
$420$	0	0
$421$	31.9075	1.55508	0.777539	−	0.628835i	$-0.216468\pi$
$421$	31.9075	1.55508	0.777539	+	0.628835i	$0.216468\pi$
$422$	15.0271	0.731506
$423$	−4.83828	−0.235245
$424$	−3.62858	−0.176220
$425$	0	0
$426$	35.7718	1.73315
$427$	−2.36600	−0.114499
$428$	−8.10006	−0.391531
$429$	2.35086	0.113501
$430$	0	0
$431$	38.2858	1.84416	0.922080	−	0.386998i	$-0.126488\pi$
$431$	38.2858	1.84416	0.922080	+	0.386998i	$0.126488\pi$
$432$	19.1205	0.919936
$433$	20.3681	0.978827	0.489413	−	0.872052i	$-0.337211\pi$
$433$	20.3681	0.978827	0.489413	+	0.872052i	$0.337211\pi$
$434$	4.46126	0.214147
$435$	0	0
$436$	1.14019	0.0546053
$437$	0.200995	0.00961492
$438$	34.9300	1.66902
$439$	−24.9107	−1.18892	−0.594462	−	0.804123i	$-0.702635\pi$
$439$	−24.9107	−1.18892	−0.594462	+	0.804123i	$0.702635\pi$
$440$	0	0
$441$	12.1606	0.579078
$442$	19.5802	0.931335
$443$	22.7243	1.07966	0.539832	−	0.841773i	$-0.318488\pi$
$443$	22.7243	1.07966	0.539832	+	0.841773i	$0.318488\pi$
$444$	17.2167	0.817068
$445$	0	0
$446$	−29.4165	−1.39291
$447$	10.2619	0.485372
$448$	−4.36106	−0.206041
$449$	−41.1163	−1.94040	−0.970198	−	0.242313i	$-0.922094\pi$
$449$	−41.1163	−1.94040	−0.970198	+	0.242313i	$0.922094\pi$
$450$	0	0
$451$	3.34355	0.157442
$452$	−20.9001	−0.983057
$453$	8.48071	0.398459
$454$	60.6761	2.84767
$455$	0	0
$456$	−0.380828	−0.0178339
$457$	−6.07340	−0.284101	−0.142051	−	0.989859i	$-0.545370\pi$
$457$	−6.07340	−0.284101	−0.142051	+	0.989859i	$0.545370\pi$
$458$	−22.4910	−1.05094
$459$	23.5873	1.10096
$460$	0	0
$461$	−6.74719	−0.314248	−0.157124	−	0.987579i	$-0.550222\pi$
$461$	−6.74719	−0.314248	−0.157124	+	0.987579i	$0.550222\pi$
$462$	−1.05221	−0.0489532
$463$	−37.3772	−1.73707	−0.868534	−	0.495630i	$-0.834937\pi$
$463$	−37.3772	−1.73707	−0.868534	+	0.495630i	$0.834937\pi$
$464$	25.1442	1.16729
$465$	0	0
$466$	−19.4744	−0.902135
$467$	−28.4403	−1.31606	−0.658030	−	0.752992i	$-0.728610\pi$
$467$	−28.4403	−1.31606	−0.658030	+	0.752992i	$0.728610\pi$
$468$	8.32987	0.385048
$469$	−3.50553	−0.161870
$470$	0	0
$471$	−3.14407	−0.144871
$472$	2.09990	0.0966558
$473$	3.75162	0.172500
$474$	13.1919	0.605923
$475$	0	0
$476$	−4.56034	−0.209023
$477$	18.7707	0.859453
$478$	8.22886	0.376379
$479$	35.7235	1.63225	0.816124	−	0.577877i	$-0.196119\pi$
$479$	35.7235	1.63225	0.816124	+	0.577877i	$0.196119\pi$
$480$	0	0
$481$	−15.4624	−0.705024
$482$	12.3870	0.564210
$483$	−0.103569	−0.00471257
$484$	2.16980	0.0986272
$485$	0	0
$486$	31.3524	1.42217
$487$	9.68607	0.438918	0.219459	−	0.975622i	$-0.429571\pi$
$487$	9.68607	0.438918	0.219459	+	0.975622i	$0.429571\pi$
$488$	−1.74864	−0.0791570
$489$	−11.9766	−0.541601
$490$	0	0
$491$	−8.75793	−0.395240	−0.197620	−	0.980279i	$-0.563321\pi$
$491$	−8.75793	−0.395240	−0.197620	+	0.980279i	$0.563321\pi$
$492$	−7.96834	−0.359241
$493$	31.0182	1.39699
$494$	4.37063	0.196644
$495$	0	0
$496$	−16.9118	−0.759363
$497$	7.48252	0.335637
$498$	−29.1118	−1.30453
$499$	15.3711	0.688104	0.344052	−	0.938951i	$-0.388200\pi$
$499$	15.3711	0.688104	0.344052	+	0.938951i	$0.388200\pi$
$500$	0	0
$501$	17.2618	0.771198
$502$	−42.3712	−1.89112
$503$	−22.5117	−1.00375	−0.501874	−	0.864941i	$-0.667356\pi$
$503$	−22.5117	−1.00375	−0.501874	+	0.864941i	$0.667356\pi$
$504$	−0.291760	−0.0129960
$505$	0	0
$506$	0.410434	0.0182460
$507$	−9.24686	−0.410667
$508$	38.4037	1.70389
$509$	−30.6457	−1.35835	−0.679173	−	0.733978i	$-0.737661\pi$
$509$	−30.6457	−1.35835	−0.679173	+	0.733978i	$0.737661\pi$
$510$	0	0
$511$	7.30642	0.323217
$512$	31.9673	1.41277
$513$	5.26508	0.232459
$514$	−43.6381	−1.92480
$515$	0	0
$516$	−8.94084	−0.393599
$517$	−2.69748	−0.118635
$518$	6.92073	0.304079
$519$	−4.22071	−0.185269
$520$	0	0
$521$	6.68929	0.293063	0.146532	−	0.989206i	$-0.453189\pi$
$521$	6.68929	0.293063	0.146532	+	0.989206i	$0.453189\pi$
$522$	25.3591	1.10994
$523$	−5.06543	−0.221496	−0.110748	−	0.993849i	$-0.535325\pi$
$523$	−5.06543	−0.221496	−0.110748	+	0.993849i	$0.535325\pi$
$524$	42.9659	1.87697
$525$	0	0
$526$	4.30646	0.187771
$527$	−20.8626	−0.908790
$528$	3.98874	0.173587
$529$	−22.9596	−0.998244
$530$	0	0
$531$	−10.8628	−0.471407
$532$	−1.01794	−0.0441334
$533$	7.15640	0.309978
$534$	−1.78277	−0.0771481
$535$	0	0
$536$	−2.59082	−0.111907
$537$	21.8114	0.941233
$538$	41.4733	1.78804
$539$	6.77991	0.292031
$540$	0	0
$541$	−45.1702	−1.94202	−0.971010	−	0.239038i	$-0.923168\pi$
$541$	−45.1702	−1.94202	−0.971010	+	0.239038i	$0.923168\pi$
$542$	−5.75568	−0.247227
$543$	−16.4826	−0.707335
$544$	36.3286	1.55758
$545$	0	0
$546$	−2.25210	−0.0963812
$547$	34.5604	1.47769	0.738847	−	0.673873i	$-0.235371\pi$
$547$	34.5604	1.47769	0.738847	+	0.673873i	$0.235371\pi$
$548$	−20.3531	−0.869442
$549$	9.04572	0.386062
$550$	0	0
$551$	6.92379	0.294963
$552$	−0.0765447	−0.00325796
$553$	2.75939	0.117341
$554$	34.0450	1.44643
$555$	0	0
$556$	−29.6763	−1.25856
$557$	−26.7387	−1.13296	−0.566478	−	0.824077i	$-0.691695\pi$
$557$	−26.7387	−1.13296	−0.566478	+	0.824077i	$0.691695\pi$
$558$	−17.0563	−0.722052
$559$	8.02981	0.339625
$560$	0	0
$561$	4.92055	0.207746
$562$	−57.0294	−2.40564
$563$	3.43066	0.144585	0.0722925	−	0.997383i	$-0.476968\pi$
$563$	3.43066	0.144585	0.0722925	+	0.997383i	$0.476968\pi$
$564$	6.42862	0.270694
$565$	0	0
$566$	9.97510	0.419285
$567$	−0.188598	−0.00792036
$568$	5.53009	0.232037
$569$	6.20817	0.260260	0.130130	−	0.991497i	$-0.458461\pi$
$569$	6.20817	0.260260	0.130130	+	0.991497i	$0.458461\pi$
$570$	0	0
$571$	−4.29662	−0.179808	−0.0899040	−	0.995950i	$-0.528656\pi$
$571$	−4.29662	−0.179808	−0.0899040	+	0.995950i	$0.528656\pi$
$572$	4.64414	0.194181
$573$	15.8471	0.662022
$574$	−3.20310	−0.133695
$575$	0	0
$576$	16.6733	0.694719
$577$	−24.6963	−1.02812	−0.514060	−	0.857754i	$-0.671859\pi$
$577$	−24.6963	−1.02812	−0.514060	+	0.857754i	$0.671859\pi$
$578$	6.26891	0.260752
$579$	−25.6524	−1.06608
$580$	0	0
$581$	−6.08941	−0.252631
$582$	−39.3602	−1.63153
$583$	10.4652	0.433425
$584$	5.39994	0.223451
$585$	0	0
$586$	−1.32360	−0.0546774
$587$	−9.31384	−0.384423	−0.192212	−	0.981353i	$-0.561566\pi$
$587$	−9.31384	−0.384423	−0.192212	+	0.981353i	$0.561566\pi$
$588$	−16.1578	−0.666338
$589$	−4.65688	−0.191884
$590$	0	0
$591$	11.3383	0.466394
$592$	−26.2352	−1.07826
$593$	25.8934	1.06331	0.531656	−	0.846960i	$-0.321570\pi$
$593$	25.8934	1.06331	0.531656	+	0.846960i	$0.321570\pi$
$594$	10.7513	0.441133
$595$	0	0
$596$	20.2725	0.830394
$597$	−11.5831	−0.474064
$598$	0.878476	0.0359236
$599$	24.8277	1.01443	0.507215	−	0.861819i	$-0.330675\pi$
$599$	24.8277	1.01443	0.507215	+	0.861819i	$0.330675\pi$
$600$	0	0
$601$	9.59884	0.391545	0.195772	−	0.980649i	$-0.437279\pi$
$601$	9.59884	0.391545	0.195772	+	0.980649i	$0.437279\pi$
$602$	−3.59402	−0.146481
$603$	13.4024	0.545787
$604$	16.7537	0.681699
$605$	0	0
$606$	31.6229	1.28459
$607$	11.9356	0.484452	0.242226	−	0.970220i	$-0.422123\pi$
$607$	11.9356	0.484452	0.242226	+	0.970220i	$0.422123\pi$
$608$	8.10916	0.328870
$609$	−3.56770	−0.144570
$610$	0	0
$611$	−5.77357	−0.233574
$612$	17.4351	0.704774
$613$	−9.27813	−0.374740	−0.187370	−	0.982289i	$-0.559996\pi$
$613$	−9.27813	−0.374740	−0.187370	+	0.982289i	$0.559996\pi$
$614$	−24.1111	−0.973043
$615$	0	0
$616$	−0.162665	−0.00655394
$617$	−4.13222	−0.166357	−0.0831784	−	0.996535i	$-0.526507\pi$
$617$	−4.13222	−0.166357	−0.0831784	+	0.996535i	$0.526507\pi$
$618$	−33.1726	−1.33440
$619$	−27.0790	−1.08840	−0.544198	−	0.838957i	$-0.683166\pi$
$619$	−27.0790	−1.08840	−0.544198	+	0.838957i	$0.683166\pi$
$620$	0	0
$621$	1.05826	0.0424664
$622$	−29.4484	−1.18077
$623$	−0.372908	−0.0149403
$624$	8.53732	0.341766
$625$	0	0
$626$	11.2175	0.448342
$627$	1.09835	0.0438638
$628$	−6.21113	−0.247851
$629$	−32.3641	−1.29044
$630$	0	0
$631$	−2.93731	−0.116933	−0.0584663	−	0.998289i	$-0.518621\pi$
$631$	−2.93731	−0.116933	−0.0584663	+	0.998289i	$0.518621\pi$
$632$	2.03938	0.0811220
$633$	8.08271	0.321259
$634$	−27.5145	−1.09274
$635$	0	0
$636$	−24.9407	−0.988962
$637$	14.5114	0.574963
$638$	14.1384	0.559746
$639$	−28.6072	−1.13169
$640$	0	0
$641$	−4.32999	−0.171024	−0.0855122	−	0.996337i	$-0.527253\pi$
$641$	−4.32999	−0.171024	−0.0855122	+	0.996337i	$0.527253\pi$
$642$	−8.37273	−0.330445
$643$	5.02416	0.198134	0.0990668	−	0.995081i	$-0.468414\pi$
$643$	5.02416	0.198134	0.0990668	+	0.995081i	$0.468414\pi$
$644$	−0.204602	−0.00806245
$645$	0	0
$646$	9.14810	0.359927
$647$	−10.7629	−0.423133	−0.211566	−	0.977364i	$-0.567856\pi$
$647$	−10.7629	−0.423133	−0.211566	+	0.977364i	$0.567856\pi$
$648$	−0.139387	−0.00547562
$649$	−6.05634	−0.237732
$650$	0	0
$651$	2.39961	0.0940480
$652$	−23.6599	−0.926593
$653$	35.3311	1.38261	0.691307	−	0.722561i	$-0.257035\pi$
$653$	35.3311	1.38261	0.691307	+	0.722561i	$0.257035\pi$
$654$	1.17857	0.0460858
$655$	0	0
$656$	12.1424	0.474080
$657$	−27.9340	−1.08981
$658$	2.58416	0.100741
$659$	−18.0616	−0.703580	−0.351790	−	0.936079i	$-0.614427\pi$
$659$	−18.0616	−0.703580	−0.351790	+	0.936079i	$0.614427\pi$
$660$	0	0
$661$	10.6516	0.414300	0.207150	−	0.978309i	$-0.433581\pi$
$661$	10.6516	0.414300	0.207150	+	0.978309i	$0.433581\pi$
$662$	21.7282	0.844489
$663$	10.5317	0.409019
$664$	−4.50049	−0.174653
$665$	0	0
$666$	−26.4594	−1.02528
$667$	1.39165	0.0538849
$668$	34.1007	1.31940
$669$	−15.8224	−0.611731
$670$	0	0
$671$	5.04325	0.194693
$672$	−4.17850	−0.161189
$673$	−7.05682	−0.272020	−0.136010	−	0.990707i	$-0.543428\pi$
$673$	−7.05682	−0.272020	−0.136010	+	0.990707i	$0.543428\pi$
$674$	6.11135	0.235401
$675$	0	0
$676$	−18.2672	−0.702586
$677$	17.0051	0.653560	0.326780	−	0.945100i	$-0.394036\pi$
$677$	17.0051	0.653560	0.326780	+	0.945100i	$0.394036\pi$
$678$	−21.6036	−0.829682
$679$	−8.23312	−0.315958
$680$	0	0
$681$	32.6363	1.25062
$682$	−9.50939	−0.364133
$683$	32.7782	1.25422	0.627112	−	0.778929i	$-0.284237\pi$
$683$	32.7782	1.25422	0.627112	+	0.778929i	$0.284237\pi$
$684$	3.89181	0.148807
$685$	0	0
$686$	−13.2010	−0.504018
$687$	−12.0974	−0.461544
$688$	13.6243	0.519421
$689$	22.3993	0.853346
$690$	0	0
$691$	23.4551	0.892274	0.446137	−	0.894965i	$-0.352799\pi$
$691$	23.4551	0.892274	0.446137	+	0.894965i	$0.352799\pi$
$692$	−8.33805	−0.316965
$693$	0.841467	0.0319647
$694$	−3.22185	−0.122300
$695$	0	0
$696$	−2.63677	−0.0999466
$697$	14.9790	0.567369
$698$	26.4950	1.00285
$699$	−10.4748	−0.396195
$700$	0	0
$701$	6.40155	0.241783	0.120892	−	0.992666i	$-0.461425\pi$
$701$	6.40155	0.241783	0.120892	+	0.992666i	$0.461425\pi$
$702$	23.0117	0.868520
$703$	−7.22421	−0.272466
$704$	9.29582	0.350349
$705$	0	0
$706$	27.5761	1.03784
$707$	6.61466	0.248770
$708$	14.4334	0.542442
$709$	16.2437	0.610044	0.305022	−	0.952345i	$-0.401336\pi$
$709$	16.2437	0.610044	0.305022	+	0.952345i	$0.401336\pi$
$710$	0	0
$711$	−10.5497	−0.395646
$712$	−0.275605	−0.0103287
$713$	−0.936012	−0.0350539
$714$	−4.71385	−0.176411
$715$	0	0
$716$	43.0887	1.61030
$717$	4.42611	0.165296
$718$	−46.8729	−1.74928
$719$	−2.10857	−0.0786363	−0.0393181	−	0.999227i	$-0.512519\pi$
$719$	−2.10857	−0.0786363	−0.0393181	+	0.999227i	$0.512519\pi$
$720$	0	0
$721$	−6.93882	−0.258415
$722$	2.04201	0.0759957
$723$	6.66266	0.247787
$724$	−32.5615	−1.21014
$725$	0	0
$726$	2.24284	0.0832395
$727$	−1.59364	−0.0591048	−0.0295524	−	0.999563i	$-0.509408\pi$
$727$	−1.59364	−0.0591048	−0.0295524	+	0.999563i	$0.509408\pi$
$728$	−0.348160	−0.0129037
$729$	18.0697	0.669249
$730$	0	0
$731$	16.8071	0.621632
$732$	−12.0191	−0.444237
$733$	−10.6588	−0.393692	−0.196846	−	0.980434i	$-0.563070\pi$
$733$	−10.6588	−0.393692	−0.196846	+	0.980434i	$0.563070\pi$
$734$	−11.8007	−0.435572
$735$	0	0
$736$	1.62990	0.0600791
$737$	7.47221	0.275242
$738$	12.2461	0.450786
$739$	3.32501	0.122313	0.0611563	−	0.998128i	$-0.480521\pi$
$739$	3.32501	0.122313	0.0611563	+	0.998128i	$0.480521\pi$
$740$	0	0
$741$	2.35086	0.0863610
$742$	−10.0256	−0.368051
$743$	51.3600	1.88421	0.942107	−	0.335311i	$-0.108841\pi$
$743$	51.3600	1.88421	0.942107	+	0.335311i	$0.108841\pi$
$744$	1.77347	0.0650187
$745$	0	0
$746$	−65.9439	−2.41438
$747$	23.2811	0.851812
$748$	9.72059	0.355420
$749$	−1.75135	−0.0639930
$750$	0	0
$751$	−22.1565	−0.808502	−0.404251	−	0.914648i	$-0.632468\pi$
$751$	−22.1565	−0.808502	−0.404251	+	0.914648i	$0.632468\pi$
$752$	−9.79610	−0.357227
$753$	−22.7905	−0.830532
$754$	30.2613	1.10205
$755$	0	0
$756$	−5.35955	−0.194925
$757$	−3.94366	−0.143335	−0.0716674	−	0.997429i	$-0.522832\pi$
$757$	−3.94366	−0.143335	−0.0716674	+	0.997429i	$0.522832\pi$
$758$	−24.9188	−0.905092
$759$	0.220763	0.00801320
$760$	0	0
$761$	−48.5719	−1.76073	−0.880366	−	0.474296i	$-0.842703\pi$
$761$	−48.5719	−1.76073	−0.880366	+	0.474296i	$0.842703\pi$
$762$	39.6965	1.43805
$763$	0.246526	0.00892484
$764$	31.3061	1.13261
$765$	0	0
$766$	−23.5696	−0.851604
$767$	−12.9627	−0.468057
$768$	14.2213	0.513167
$769$	−34.5865	−1.24722	−0.623611	−	0.781735i	$-0.714335\pi$
$769$	−34.5865	−1.24722	−0.623611	+	0.781735i	$0.714335\pi$
$770$	0	0
$771$	−23.4720	−0.845322
$772$	−50.6765	−1.82389
$773$	−9.98101	−0.358992	−0.179496	−	0.983759i	$-0.557447\pi$
$773$	−9.98101	−0.358992	−0.179496	+	0.983759i	$0.557447\pi$
$774$	13.7407	0.493900
$775$	0	0
$776$	−6.08483	−0.218433
$777$	3.72250	0.133544
$778$	−9.49962	−0.340578
$779$	3.34355	0.119795
$780$	0	0
$781$	−15.9494	−0.570713
$782$	1.83873	0.0657527
$783$	36.4543	1.30277
$784$	24.6217	0.879347
$785$	0	0
$786$	44.4122	1.58413
$787$	−8.47966	−0.302267	−0.151134	−	0.988513i	$-0.548292\pi$
$787$	−8.47966	−0.302267	−0.151134	+	0.988513i	$0.548292\pi$
$788$	22.3988	0.797926
$789$	2.31635	0.0824641
$790$	0	0
$791$	−4.51890	−0.160674
$792$	0.621901	0.0220983
$793$	10.7944	0.383319
$794$	48.6581	1.72681
$795$	0	0
$796$	−22.8825	−0.811047
$797$	−50.8181	−1.80007	−0.900034	−	0.435820i	$-0.856459\pi$
$797$	−50.8181	−1.80007	−0.900034	+	0.435820i	$0.856459\pi$
$798$	−1.05221	−0.0372478
$799$	−12.0846	−0.427522
$800$	0	0
$801$	1.42571	0.0503749
$802$	−10.4367	−0.368534
$803$	−15.5740	−0.549595
$804$	−17.8077	−0.628030
$805$	0	0
$806$	−20.3535	−0.716921
$807$	22.3075	0.785262
$808$	4.88869	0.171983
$809$	26.2792	0.923926	0.461963	−	0.886899i	$-0.347145\pi$
$809$	26.2792	0.923926	0.461963	+	0.886899i	$0.347145\pi$
$810$	0	0
$811$	44.2582	1.55412	0.777058	−	0.629429i	$-0.216711\pi$
$811$	44.2582	1.55412	0.777058	+	0.629429i	$0.216711\pi$
$812$	−7.04802	−0.247337
$813$	−3.09585	−0.108576
$814$	−14.7519	−0.517054
$815$	0	0
$816$	17.8693	0.625552
$817$	3.75162	0.131253
$818$	−40.8737	−1.42912
$819$	1.80104	0.0629334
$820$	0	0
$821$	−45.4098	−1.58481	−0.792406	−	0.609994i	$-0.791172\pi$
$821$	−45.4098	−1.58481	−0.792406	+	0.609994i	$0.791172\pi$
$822$	−21.0382	−0.733793
$823$	−20.0801	−0.699950	−0.349975	−	0.936759i	$-0.613810\pi$
$823$	−20.0801	−0.699950	−0.349975	+	0.936759i	$0.613810\pi$
$824$	−5.12826	−0.178652
$825$	0	0
$826$	5.80192	0.201875
$827$	−2.70217	−0.0939637	−0.0469818	−	0.998896i	$-0.514960\pi$
$827$	−2.70217	−0.0939637	−0.0469818	+	0.998896i	$0.514960\pi$
$828$	0.782237	0.0271846
$829$	43.8322	1.52236	0.761178	−	0.648543i	$-0.224621\pi$
$829$	43.8322	1.52236	0.761178	+	0.648543i	$0.224621\pi$
$830$	0	0
$831$	18.3120	0.635237
$832$	19.8964	0.689783
$833$	30.3737	1.05238
$834$	−30.6753	−1.06220
$835$	0	0
$836$	2.16980	0.0750440
$837$	−24.5189	−0.847495
$838$	70.1354	2.42279
$839$	12.7989	0.441866	0.220933	−	0.975289i	$-0.429090\pi$
$839$	12.7989	0.441866	0.220933	+	0.975289i	$0.429090\pi$
$840$	0	0
$841$	18.9388	0.653062
$842$	65.1554	2.24540
$843$	−30.6748	−1.05650
$844$	15.9675	0.549623
$845$	0	0
$846$	−9.87981	−0.339675
$847$	0.469142	0.0161199
$848$	38.0052	1.30510
$849$	5.36537	0.184139
$850$	0	0
$851$	−1.45203	−0.0497750
$852$	38.0104	1.30222
$853$	30.8728	1.05707	0.528533	−	0.848913i	$-0.322742\pi$
$853$	30.8728	1.05707	0.528533	+	0.848913i	$0.322742\pi$
$854$	−4.83139	−0.165327
$855$	0	0
$856$	−1.29437	−0.0442406
$857$	18.5582	0.633935	0.316967	−	0.948436i	$-0.397335\pi$
$857$	18.5582	0.633935	0.316967	+	0.948436i	$0.397335\pi$
$858$	4.80047	0.163885
$859$	−0.760628	−0.0259523	−0.0129762	−	0.999916i	$-0.504131\pi$
$859$	−0.760628	−0.0259523	−0.0129762	+	0.999916i	$0.504131\pi$
$860$	0	0
$861$	−1.72287	−0.0587153
$862$	78.1799	2.66282
$863$	46.7268	1.59060	0.795300	−	0.606216i	$-0.207313\pi$
$863$	46.7268	1.59060	0.795300	+	0.606216i	$0.207313\pi$
$864$	42.6954	1.45253
$865$	0	0
$866$	41.5918	1.41335
$867$	3.37190	0.114516
$868$	4.74044	0.160901
$869$	−5.88178	−0.199526
$870$	0	0
$871$	15.9932	0.541909
$872$	0.182200	0.00617006
$873$	31.4769	1.06533
$874$	0.410434	0.0138831
$875$	0	0
$876$	37.1159	1.25403
$877$	−38.0871	−1.28611	−0.643056	−	0.765819i	$-0.722334\pi$
$877$	−38.0871	−1.28611	−0.643056	+	0.765819i	$0.722334\pi$
$878$	−50.8679	−1.71671
$879$	−0.711934	−0.0240129
$880$	0	0
$881$	−14.4600	−0.487170	−0.243585	−	0.969880i	$-0.578323\pi$
$881$	−14.4600	−0.487170	−0.243585	+	0.969880i	$0.578323\pi$
$882$	24.8321	0.836141
$883$	−0.159279	−0.00536017	−0.00268008	−	0.999996i	$-0.500853\pi$
$883$	−0.159279	−0.00536017	−0.00268008	+	0.999996i	$0.500853\pi$
$884$	20.8055	0.699766
$885$	0	0
$886$	46.4032	1.55895
$887$	9.95048	0.334104	0.167052	−	0.985948i	$-0.446575\pi$
$887$	9.95048	0.334104	0.167052	+	0.985948i	$0.446575\pi$
$888$	2.75118	0.0923236
$889$	8.30345	0.278489
$890$	0	0
$891$	0.402006	0.0134677
$892$	−31.2574	−1.04657
$893$	−2.69748	−0.0902678
$894$	20.9549	0.700837
$895$	0	0
$896$	−1.29663	−0.0433173
$897$	0.472512	0.0157767
$898$	−83.9597	−2.80177
$899$	−32.2433	−1.07537
$900$	0	0
$901$	46.8837	1.56192
$902$	6.82756	0.227333
$903$	−1.93314	−0.0643309
$904$	−3.33978	−0.111079
$905$	0	0
$906$	17.3177	0.575342
$907$	−51.7590	−1.71863	−0.859315	−	0.511448i	$-0.829109\pi$
$907$	−51.7590	−1.71863	−0.859315	+	0.511448i	$0.829109\pi$
$908$	64.4732	2.13962
$909$	−25.2893	−0.838792
$910$	0	0
$911$	−29.1062	−0.964332	−0.482166	−	0.876080i	$-0.660150\pi$
$911$	−29.1062	−0.964332	−0.482166	+	0.876080i	$0.660150\pi$
$912$	3.98874	0.132080
$913$	12.9799	0.429572
$914$	−12.4019	−0.410219
$915$	0	0
$916$	−23.8985	−0.789629
$917$	9.28986	0.306778
$918$	48.1655	1.58970
$919$	−12.7499	−0.420581	−0.210290	−	0.977639i	$-0.567441\pi$
$919$	−12.7499	−0.420581	−0.210290	+	0.977639i	$0.567441\pi$
$920$	0	0
$921$	−12.9688	−0.427336
$922$	−13.7778	−0.453748
$923$	−34.1373	−1.12364
$924$	−1.11806	−0.0367814
$925$	0	0
$926$	−76.3246	−2.50818
$927$	26.5286	0.871313
$928$	56.1461	1.84309
$929$	2.50821	0.0822916	0.0411458	−	0.999153i	$-0.486899\pi$
$929$	2.50821	0.0822916	0.0411458	+	0.999153i	$0.486899\pi$
$930$	0	0
$931$	6.77991	0.222202
$932$	−20.6931	−0.677826
$933$	−15.8396	−0.518566
$934$	−58.0753	−1.90028
$935$	0	0
$936$	1.33109	0.0435081
$937$	−47.9802	−1.56744	−0.783722	−	0.621112i	$-0.786681\pi$
$937$	−47.9802	−1.56744	−0.783722	+	0.621112i	$0.786681\pi$
$938$	−7.15832	−0.233727
$939$	6.03364	0.196900
$940$	0	0
$941$	−44.4858	−1.45020	−0.725098	−	0.688645i	$-0.758206\pi$
$941$	−44.4858	−1.45020	−0.725098	+	0.688645i	$0.758206\pi$
$942$	−6.42021	−0.209182
$943$	0.672039	0.0218846
$944$	−21.9940	−0.715845
$945$	0	0
$946$	7.66084	0.249075
$947$	−6.70041	−0.217734	−0.108867	−	0.994056i	$-0.534722\pi$
$947$	−6.70041	−0.217734	−0.108867	+	0.994056i	$0.534722\pi$
$948$	14.0174	0.455265
$949$	−33.3339	−1.08207
$950$	0	0
$951$	−14.7994	−0.479904
$952$	−0.728730	−0.0236183
$953$	57.1810	1.85227	0.926136	−	0.377189i	$-0.123109\pi$
$953$	57.1810	1.85227	0.926136	+	0.377189i	$0.123109\pi$
$954$	38.3300	1.24098
$955$	0	0
$956$	8.74382	0.282796
$957$	7.60473	0.245826
$958$	72.9477	2.35683
$959$	−4.40064	−0.142104
$960$	0	0
$961$	−9.31344	−0.300434
$962$	−31.5743	−1.01800
$963$	6.69579	0.215769
$964$	13.1621	0.423924
$965$	0	0
$966$	−0.211489	−0.00680456
$967$	5.24741	0.168745	0.0843727	−	0.996434i	$-0.473111\pi$
$967$	5.24741	0.168745	0.0843727	+	0.996434i	$0.473111\pi$
$968$	0.346728	0.0111443
$969$	4.92055	0.158071
$970$	0	0
$971$	−27.4776	−0.881799	−0.440899	−	0.897557i	$-0.645340\pi$
$971$	−27.4776	−0.881799	−0.440899	+	0.897557i	$0.645340\pi$
$972$	33.3144	1.06856
$973$	−6.41645	−0.205702
$974$	19.7790	0.633761
$975$	0	0
$976$	18.3149	0.586247
$977$	−42.6475	−1.36442	−0.682208	−	0.731158i	$-0.738980\pi$
$977$	−42.6475	−1.36442	−0.682208	+	0.731158i	$0.738980\pi$
$978$	−24.4563	−0.782028
$979$	0.794873	0.0254043
$980$	0	0
$981$	−0.942521	−0.0300924
$982$	−17.8838	−0.570694
$983$	48.8786	1.55899	0.779493	−	0.626411i	$-0.215477\pi$
$983$	48.8786	1.55899	0.779493	+	0.626411i	$0.215477\pi$
$984$	−1.27332	−0.0405919
$985$	0	0
$986$	63.3395	2.01714
$987$	1.38996	0.0442430
$988$	4.64414	0.147750
$989$	0.754059	0.0239777
$990$	0	0
$991$	−60.2922	−1.91524	−0.957622	−	0.288027i	$-0.907001\pi$
$991$	−60.2922	−1.91524	−0.957622	+	0.288027i	$0.907001\pi$
$992$	−37.7634	−1.19899
$993$	11.6871	0.370878
$994$	15.2794	0.484632
$995$	0	0
$996$	−30.9336	−0.980169
$997$	6.50314	0.205957	0.102978	−	0.994684i	$-0.467163\pi$
$997$	6.50314	0.205957	0.102978	+	0.994684i	$0.467163\pi$
$998$	31.3879	0.993565
$999$	−38.0360	−1.20341

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5225.2.a.k.1.5		6
5.4	even	2		1045.2.a.g.1.2	✓	6
15.14	odd	2		9405.2.a.w.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.g.1.2	✓	6	5.4	even	2
5225.2.a.k.1.5		6	1.1	even	1	trivial
9405.2.a.w.1.5		6	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.04201 q^{2} +1.09835 q^{3} +2.16980 q^{4} +2.24284 q^{6} +0.469142 q^{7} +0.346728 q^{8} -1.79363 q^{9} -1.00000 q^{11} +2.38320 q^{12} -2.14036 q^{13} +0.957992 q^{14} -3.63157 q^{16} -4.47995 q^{17} -3.66261 q^{18} -1.00000 q^{19} +0.515282 q^{21} -2.04201 q^{22} -0.200995 q^{23} +0.380828 q^{24} -4.37063 q^{26} -5.26508 q^{27} +1.01794 q^{28} -6.92379 q^{29} +4.65688 q^{31} -8.10916 q^{32} -1.09835 q^{33} -9.14810 q^{34} -3.89181 q^{36} +7.22421 q^{37} -2.04201 q^{38} -2.35086 q^{39} -3.34355 q^{41} +1.05221 q^{42} -3.75162 q^{43} -2.16980 q^{44} -0.410434 q^{46} +2.69748 q^{47} -3.98874 q^{48} -6.77991 q^{49} -4.92055 q^{51} -4.64414 q^{52} -10.4652 q^{53} -10.7513 q^{54} +0.162665 q^{56} -1.09835 q^{57} -14.1384 q^{58} +6.05634 q^{59} -5.04325 q^{61} +9.50939 q^{62} -0.841467 q^{63} -9.29582 q^{64} -2.24284 q^{66} -7.47221 q^{67} -9.72059 q^{68} -0.220763 q^{69} +15.9494 q^{71} -0.621901 q^{72} +15.5740 q^{73} +14.7519 q^{74} -2.16980 q^{76} -0.469142 q^{77} -4.80047 q^{78} +5.88178 q^{79} -0.402006 q^{81} -6.82756 q^{82} -12.9799 q^{83} +1.11806 q^{84} -7.66084 q^{86} -7.60473 q^{87} -0.346728 q^{88} -0.794873 q^{89} -1.00413 q^{91} -0.436119 q^{92} +5.11488 q^{93} +5.50828 q^{94} -8.90669 q^{96} -17.5493 q^{97} -13.8446 q^{98} +1.79363 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{3} + 8 q^{4} + 2 q^{6} - 5 q^{7} + 9 q^{9} - 6 q^{11} - 19 q^{12} + 9 q^{13} + 18 q^{14} + 4 q^{16} + 5 q^{17} - 2 q^{18} - 6 q^{19} + 3 q^{21} - 8 q^{23} - 7 q^{24} - 22 q^{26} - 30 q^{27} - 10 q^{28}+ \cdots - 9 q^{99}+O(q^{100}) \) 6 * q - 3 * q^3 + 8 * q^4 + 2 * q^6 - 5 * q^7 + 9 * q^9 - 6 * q^11 - 19 * q^12 + 9 * q^13 + 18 * q^14 + 4 * q^16 + 5 * q^17 - 2 * q^18 - 6 * q^19 + 3 * q^21 - 8 * q^23 - 7 * q^24 - 22 * q^26 - 30 * q^27 - 10 * q^28 - 5 * q^29 - q^31 - 15 * q^32 + 3 * q^33 - 22 * q^34 + 12 * q^36 - 9 * q^37 - 32 * q^39 + 25 * q^41 - 11 * q^42 - 15 * q^43 - 8 * q^44 - 16 * q^46 - 24 * q^47 + 4 * q^48 + 13 * q^49 + 27 * q^52 - 5 * q^53 - 11 * q^54 - 12 * q^56 + 3 * q^57 - 13 * q^58 + 39 * q^59 - 11 * q^61 + 42 * q^62 - 38 * q^63 - 14 * q^64 - 2 * q^66 - 24 * q^67 - 45 * q^68 + 14 * q^69 - 24 * q^71 + 61 * q^72 + 26 * q^73 + q^74 - 8 * q^76 + 5 * q^77 + 29 * q^78 + 11 * q^79 + 30 * q^81 - 8 * q^82 - 39 * q^83 + 25 * q^84 + 18 * q^86 + 16 * q^87 + 22 * q^89 - 26 * q^91 + 11 * q^92 + 6 * q^93 - 30 * q^94 - 15 * q^96 - 22 * q^97 - 33 * q^98 - 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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