LMFDB - Embedded newform 4925.2.a.r.1.24

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4925,2,Mod(1,4925)] 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("4925.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4925, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4925 = 5^{2} \cdot 197 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4925.a (trivial)

Newform invariants

comment:select newform

sage:traces = [49,-5,-22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$39.3263229955$
Analytic rank:	$1$
Dimension:	$49$
Twist minimal:	no (minimal twist has level 985)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.24
Character	$\chi$	$=$	4925.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.321757 q^{2} +2.86064 q^{3} -1.89647 q^{4} -0.920431 q^{6} -0.829142 q^{7} +1.25372 q^{8} +5.18323 q^{9} -1.61793 q^{11} -5.42511 q^{12} -3.76162 q^{13} +0.266783 q^{14} +3.38955 q^{16} -2.25602 q^{17} -1.66774 q^{18} +5.21309 q^{19} -2.37187 q^{21} +0.520580 q^{22} +0.268138 q^{23} +3.58643 q^{24} +1.21033 q^{26} +6.24543 q^{27} +1.57245 q^{28} -6.37430 q^{29} +0.201911 q^{31} -3.59805 q^{32} -4.62830 q^{33} +0.725890 q^{34} -9.82986 q^{36} -7.50915 q^{37} -1.67735 q^{38} -10.7606 q^{39} +2.01101 q^{41} +0.763168 q^{42} -8.49353 q^{43} +3.06836 q^{44} -0.0862754 q^{46} +1.06230 q^{47} +9.69627 q^{48} -6.31252 q^{49} -6.45364 q^{51} +7.13381 q^{52} +10.6764 q^{53} -2.00951 q^{54} -1.03951 q^{56} +14.9127 q^{57} +2.05098 q^{58} -12.4971 q^{59} +1.31673 q^{61} -0.0649663 q^{62} -4.29764 q^{63} -5.62140 q^{64} +1.48919 q^{66} -2.57912 q^{67} +4.27847 q^{68} +0.767045 q^{69} +6.89919 q^{71} +6.49832 q^{72} +4.02613 q^{73} +2.41612 q^{74} -9.88647 q^{76} +1.34149 q^{77} +3.46231 q^{78} +12.1773 q^{79} +2.31621 q^{81} -0.647057 q^{82} -7.56524 q^{83} +4.49819 q^{84} +2.73286 q^{86} -18.2345 q^{87} -2.02843 q^{88} -8.82490 q^{89} +3.11892 q^{91} -0.508517 q^{92} +0.577593 q^{93} -0.341802 q^{94} -10.2927 q^{96} +1.91883 q^{97} +2.03110 q^{98} -8.38610 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 49 q - 5 q^{2} - 22 q^{3} + 49 q^{4} + 2 q^{6} - 32 q^{7} - 15 q^{8} + 51 q^{9} - 2 q^{11} - 44 q^{12} - 32 q^{13} - 8 q^{14} + 49 q^{16} - 14 q^{17} - 25 q^{18} + 4 q^{19} + 10 q^{21} - 38 q^{22} - 24 q^{23}+ \cdots + 22 q^{99}+O(q^{100}) $ 49 * q - 5 * q^2 - 22 * q^3 + 49 * q^4 + 2 * q^6 - 32 * q^7 - 15 * q^8 + 51 * q^9 - 2 * q^11 - 44 * q^12 - 32 * q^13 - 8 * q^14 + 49 * q^16 - 14 * q^17 - 25 * q^18 + 4 * q^19 + 10 * q^21 - 38 * q^22 - 24 * q^23 - 2 * q^26 - 88 * q^27 - 80 * q^28 + 6 * q^29 - 2 * q^31 - 35 * q^32 - 48 * q^33 - 12 * q^34 + 43 * q^36 - 60 * q^37 - 34 * q^38 - 2 * q^41 - 36 * q^42 - 80 * q^43 - 8 * q^44 - 22 * q^46 - 48 * q^47 - 88 * q^48 + 65 * q^49 + 16 * q^51 - 96 * q^52 - 18 * q^53 + 42 * q^54 - 22 * q^56 - 40 * q^57 - 52 * q^58 + 8 * q^59 + 16 * q^61 - 38 * q^62 - 82 * q^63 + 55 * q^64 + 42 * q^66 - 146 * q^67 + 68 * q^68 - 6 * q^69 - 75 * q^72 - 78 * q^73 - 24 * q^74 - 4 * q^76 - 32 * q^77 + 38 * q^78 + 10 * q^79 + 41 * q^81 - 96 * q^82 - 64 * q^83 + 42 * q^84 - 28 * q^86 - 56 * q^87 - 72 * q^88 + 4 * q^89 - 60 * q^92 + 28 * q^93 - 8 * q^94 - 34 * q^96 - 124 * q^97 + 19 * q^98 + 22 * 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 $

$2$	−0.321757	−0.227517	−0.113758	−	0.993508i	$-0.536289\pi$
$2$	−0.321757	−0.227517	−0.113758	+	0.993508i	$0.536289\pi$
$3$	2.86064	1.65159	0.825794	−	0.563972i	$-0.190727\pi$
$3$	2.86064	1.65159	0.825794	+	0.563972i	$0.190727\pi$
$4$	−1.89647	−0.948236
$5$	0	0
$5$	0	0
$6$	−0.920431	−0.375764
$7$	−0.829142	−0.313386	−0.156693	−	0.987647i	$-0.550083\pi$
$7$	−0.829142	−0.313386	−0.156693	+	0.987647i	$0.550083\pi$
$8$	1.25372	0.443257
$9$	5.18323	1.72774
$10$	0	0
$11$	−1.61793	−0.487824	−0.243912	−	0.969797i	$-0.578431\pi$
$11$	−1.61793	−0.487824	−0.243912	+	0.969797i	$0.578431\pi$
$12$	−5.42511	−1.56610
$13$	−3.76162	−1.04329	−0.521643	−	0.853164i	$-0.674681\pi$
$13$	−3.76162	−1.04329	−0.521643	+	0.853164i	$0.674681\pi$
$14$	0.266783	0.0713007
$15$	0	0
$16$	3.38955	0.847388
$17$	−2.25602	−0.547165	−0.273582	−	0.961849i	$-0.588209\pi$
$17$	−2.25602	−0.547165	−0.273582	+	0.961849i	$0.588209\pi$
$18$	−1.66774	−0.393091
$19$	5.21309	1.19596	0.597982	−	0.801510i	$-0.295969\pi$
$19$	5.21309	1.19596	0.597982	+	0.801510i	$0.295969\pi$
$20$	0	0
$21$	−2.37187	−0.517585
$22$	0.520580	0.110988
$23$	0.268138	0.0559107	0.0279553	−	0.999609i	$-0.491100\pi$
$23$	0.268138	0.0559107	0.0279553	+	0.999609i	$0.491100\pi$
$24$	3.58643	0.732077
$25$	0	0
$26$	1.21033	0.237365
$27$	6.24543	1.20193
$28$	1.57245	0.297164
$29$	−6.37430	−1.18368	−0.591839	−	0.806056i	$-0.701598\pi$
$29$	−6.37430	−1.18368	−0.591839	+	0.806056i	$0.701598\pi$
$30$	0	0
$31$	0.201911	0.0362643	0.0181321	−	0.999836i	$-0.494228\pi$
$31$	0.201911	0.0362643	0.0181321	+	0.999836i	$0.494228\pi$
$32$	−3.59805	−0.636052
$33$	−4.62830	−0.805684
$34$	0.725890	0.124489
$35$	0	0
$36$	−9.82986	−1.63831
$37$	−7.50915	−1.23450	−0.617248	−	0.786769i	$-0.711752\pi$
$37$	−7.50915	−1.23450	−0.617248	+	0.786769i	$0.711752\pi$
$38$	−1.67735	−0.272102
$39$	−10.7606	−1.72308
$40$	0	0
$41$	2.01101	0.314067	0.157033	−	0.987593i	$-0.449807\pi$
$41$	2.01101	0.314067	0.157033	+	0.987593i	$0.449807\pi$
$42$	0.763168	0.117759
$43$	−8.49353	−1.29525	−0.647626	−	0.761959i	$-0.724238\pi$
$43$	−8.49353	−1.29525	−0.647626	+	0.761959i	$0.724238\pi$
$44$	3.06836	0.462572
$45$	0	0
$46$	−0.0862754	−0.0127206
$47$	1.06230	0.154952	0.0774759	−	0.996994i	$-0.475314\pi$
$47$	1.06230	0.154952	0.0774759	+	0.996994i	$0.475314\pi$
$48$	9.69627	1.39954
$49$	−6.31252	−0.901789
$50$	0	0
$51$	−6.45364	−0.903691
$52$	7.13381	0.989282
$53$	10.6764	1.46652	0.733260	−	0.679948i	$-0.237998\pi$
$53$	10.6764	1.46652	0.733260	+	0.679948i	$0.237998\pi$
$54$	−2.00951	−0.273460
$55$	0	0
$56$	−1.03951	−0.138911
$57$	14.9127	1.97524
$58$	2.05098	0.269307
$59$	−12.4971	−1.62699	−0.813494	−	0.581574i	$-0.802437\pi$
$59$	−12.4971	−1.62699	−0.813494	+	0.581574i	$0.802437\pi$
$60$	0	0
$61$	1.31673	0.168590	0.0842950	−	0.996441i	$-0.473136\pi$
$61$	1.31673	0.168590	0.0842950	+	0.996441i	$0.473136\pi$
$62$	−0.0649663	−0.00825073
$63$	−4.29764	−0.541451
$64$	−5.62140	−0.702675
$65$	0	0
$66$	1.48919	0.183307
$67$	−2.57912	−0.315090	−0.157545	−	0.987512i	$-0.550358\pi$
$67$	−2.57912	−0.315090	−0.157545	+	0.987512i	$0.550358\pi$
$68$	4.27847	0.518841
$69$	0.767045	0.0923414
$70$	0	0
$71$	6.89919	0.818783	0.409391	−	0.912359i	$-0.365741\pi$
$71$	6.89919	0.818783	0.409391	+	0.912359i	$0.365741\pi$
$72$	6.49832	0.765834
$73$	4.02613	0.471223	0.235611	−	0.971847i	$-0.424291\pi$
$73$	4.02613	0.471223	0.235611	+	0.971847i	$0.424291\pi$
$74$	2.41612	0.280869
$75$	0	0
$76$	−9.88647	−1.13406
$77$	1.34149	0.152877
$78$	3.46231	0.392030
$79$	12.1773	1.37006	0.685028	−	0.728517i	$-0.259790\pi$
$79$	12.1773	1.37006	0.685028	+	0.728517i	$0.259790\pi$
$80$	0	0
$81$	2.31621	0.257356
$82$	−0.647057	−0.0714555
$83$	−7.56524	−0.830393	−0.415196	−	0.909732i	$-0.636287\pi$
$83$	−7.56524	−0.830393	−0.415196	+	0.909732i	$0.636287\pi$
$84$	4.49819	0.490793
$85$	0	0
$86$	2.73286	0.294692
$87$	−18.2345	−1.95495
$88$	−2.02843	−0.216231
$89$	−8.82490	−0.935437	−0.467719	−	0.883877i	$-0.654924\pi$
$89$	−8.82490	−0.935437	−0.467719	+	0.883877i	$0.654924\pi$
$90$	0	0
$91$	3.11892	0.326952
$92$	−0.508517	−0.0530165
$93$	0.577593	0.0598936
$94$	−0.341802	−0.0352541
$95$	0	0
$96$	−10.2927	−1.05050
$97$	1.91883	0.194828	0.0974140	−	0.995244i	$-0.468943\pi$
$97$	1.91883	0.194828	0.0974140	+	0.995244i	$0.468943\pi$
$98$	2.03110	0.205172
$99$	−8.38610	−0.842835
$100$	0	0
$101$	−16.5253	−1.64433	−0.822166	−	0.569247i	$-0.807235\pi$
$101$	−16.5253	−1.64433	−0.822166	+	0.569247i	$0.807235\pi$
$102$	2.07651	0.205605
$103$	−14.1484	−1.39409	−0.697043	−	0.717029i	$-0.745501\pi$
$103$	−14.1484	−1.39409	−0.697043	+	0.717029i	$0.745501\pi$
$104$	−4.71602	−0.462444
$105$	0	0
$106$	−3.43522	−0.333658
$107$	−0.511628	−0.0494610	−0.0247305	−	0.999694i	$-0.507873\pi$
$107$	−0.511628	−0.0494610	−0.0247305	+	0.999694i	$0.507873\pi$
$108$	−11.8443	−1.13972
$109$	7.68077	0.735685	0.367842	−	0.929888i	$-0.380097\pi$
$109$	7.68077	0.735685	0.367842	+	0.929888i	$0.380097\pi$
$110$	0	0
$111$	−21.4809	−2.03888
$112$	−2.81042	−0.265560
$113$	10.3875	0.977174	0.488587	−	0.872515i	$-0.337512\pi$
$113$	10.3875	0.977174	0.488587	+	0.872515i	$0.337512\pi$
$114$	−4.79828	−0.449400
$115$	0	0
$116$	12.0887	1.12241
$117$	−19.4974	−1.80253
$118$	4.02104	0.370167
$119$	1.87056	0.171474
$120$	0	0
$121$	−8.38231	−0.762028
$122$	−0.423668	−0.0383571
$123$	5.75276	0.518709
$124$	−0.382918	−0.0343871
$125$	0	0
$126$	1.38280	0.123189
$127$	−3.46077	−0.307093	−0.153547	−	0.988141i	$-0.549070\pi$
$127$	−3.46077	−0.307093	−0.153547	+	0.988141i	$0.549070\pi$
$128$	9.00483	0.795922
$129$	−24.2969	−2.13922
$130$	0	0
$131$	3.36240	0.293774	0.146887	−	0.989153i	$-0.453075\pi$
$131$	3.36240	0.293774	0.146887	+	0.989153i	$0.453075\pi$
$132$	8.77745	0.763979
$133$	−4.32239	−0.374799
$134$	0.829852	0.0716883
$135$	0	0
$136$	−2.82841	−0.242534
$137$	−10.4401	−0.891959	−0.445979	−	0.895043i	$-0.647145\pi$
$137$	−10.4401	−0.891959	−0.445979	+	0.895043i	$0.647145\pi$
$138$	−0.246803	−0.0210092
$139$	−17.9388	−1.52155	−0.760776	−	0.649015i	$-0.775181\pi$
$139$	−17.9388	−1.52155	−0.760776	+	0.649015i	$0.775181\pi$
$140$	0	0
$141$	3.03884	0.255917
$142$	−2.21986	−0.186287
$143$	6.08603	0.508940
$144$	17.5688	1.46407
$145$	0	0
$146$	−1.29544	−0.107211
$147$	−18.0578	−1.48938
$148$	14.2409	1.17059
$149$	−2.23651	−0.183222	−0.0916110	−	0.995795i	$-0.529202\pi$
$149$	−2.23651	−0.183222	−0.0916110	+	0.995795i	$0.529202\pi$
$150$	0	0
$151$	−1.36330	−0.110944	−0.0554718	−	0.998460i	$-0.517666\pi$
$151$	−1.36330	−0.110944	−0.0554718	+	0.998460i	$0.517666\pi$
$152$	6.53574	0.530119
$153$	−11.6935	−0.945360
$154$	−0.431635	−0.0347822
$155$	0	0
$156$	20.4072	1.63389
$157$	−22.9949	−1.83520	−0.917598	−	0.397510i	$-0.869874\pi$
$157$	−22.9949	−1.83520	−0.917598	+	0.397510i	$0.869874\pi$
$158$	−3.91814	−0.311711
$159$	30.5414	2.42209
$160$	0	0
$161$	−0.222325	−0.0175216
$162$	−0.745256	−0.0585529
$163$	−8.59975	−0.673585	−0.336792	−	0.941579i	$-0.609342\pi$
$163$	−8.59975	−0.673585	−0.336792	+	0.941579i	$0.609342\pi$
$164$	−3.81382	−0.297810
$165$	0	0
$166$	2.43417	0.188928
$167$	9.06718	0.701640	0.350820	−	0.936443i	$-0.385903\pi$
$167$	9.06718	0.701640	0.350820	+	0.936443i	$0.385903\pi$
$168$	−2.97366	−0.229423
$169$	1.14980	0.0884463
$170$	0	0
$171$	27.0206	2.06632
$172$	16.1077	1.22820
$173$	−18.4600	−1.40349	−0.701745	−	0.712428i	$-0.747596\pi$
$173$	−18.4600	−1.40349	−0.701745	+	0.712428i	$0.747596\pi$
$174$	5.86710	0.444784
$175$	0	0
$176$	−5.48405	−0.413376
$177$	−35.7497	−2.68711
$178$	2.83948	0.212828
$179$	−6.33568	−0.473551	−0.236776	−	0.971564i	$-0.576091\pi$
$179$	−6.33568	−0.473551	−0.236776	+	0.971564i	$0.576091\pi$
$180$	0	0
$181$	21.2003	1.57580	0.787902	−	0.615801i	$-0.211167\pi$
$181$	21.2003	1.57580	0.787902	+	0.615801i	$0.211167\pi$
$182$	−1.00354	−0.0743870
$183$	3.76668	0.278441
$184$	0.336170	0.0247828
$185$	0	0
$186$	−0.185845	−0.0136268
$187$	3.65007	0.266920
$188$	−2.01461	−0.146931
$189$	−5.17835	−0.376670
$190$	0	0
$191$	16.9636	1.22744	0.613722	−	0.789522i	$-0.289672\pi$
$191$	16.9636	1.22744	0.613722	+	0.789522i	$0.289672\pi$
$192$	−16.0808	−1.16053
$193$	2.68943	0.193589	0.0967946	−	0.995304i	$-0.469141\pi$
$193$	2.68943	0.193589	0.0967946	+	0.995304i	$0.469141\pi$
$194$	−0.617399	−0.0443267
$195$	0	0
$196$	11.9715	0.855109
$197$	1.00000	0.0712470
$197$	1.00000	0.0712470
$198$	2.69829	0.191759
$199$	5.68325	0.402875	0.201438	−	0.979501i	$-0.435439\pi$
$199$	5.68325	0.402875	0.201438	+	0.979501i	$0.435439\pi$
$200$	0	0
$201$	−7.37793	−0.520399
$202$	5.31715	0.374113
$203$	5.28520	0.370948
$204$	12.2392	0.856912
$205$	0	0
$206$	4.55236	0.317178
$207$	1.38982	0.0965993
$208$	−12.7502	−0.884068
$209$	−8.43440	−0.583419
$210$	0	0
$211$	24.4504	1.68324	0.841618	−	0.540073i	$-0.181603\pi$
$211$	24.4504	1.68324	0.841618	+	0.540073i	$0.181603\pi$
$212$	−20.2475	−1.39061
$213$	19.7361	1.35229
$214$	0.164620	0.0112532
$215$	0	0
$216$	7.83002	0.532765
$217$	−0.167413	−0.0113647
$218$	−2.47135	−0.167381
$219$	11.5173	0.778266
$220$	0	0
$221$	8.48628	0.570849
$222$	6.91165	0.463879
$223$	−11.7779	−0.788710	−0.394355	−	0.918958i	$-0.629032\pi$
$223$	−11.7779	−0.788710	−0.394355	+	0.918958i	$0.629032\pi$
$224$	2.98330	0.199330
$225$	0	0
$226$	−3.34226	−0.222324
$227$	−21.2068	−1.40755	−0.703773	−	0.710425i	$-0.748503\pi$
$227$	−21.2068	−1.40755	−0.703773	+	0.710425i	$0.748503\pi$
$228$	−28.2816	−1.87299
$229$	2.42971	0.160560	0.0802799	−	0.996772i	$-0.474419\pi$
$229$	2.42971	0.160560	0.0802799	+	0.996772i	$0.474419\pi$
$230$	0	0
$231$	3.83752	0.252490
$232$	−7.99158	−0.524673
$233$	16.1706	1.05937	0.529685	−	0.848194i	$-0.322310\pi$
$233$	16.1706	1.05937	0.529685	+	0.848194i	$0.322310\pi$
$234$	6.27342	0.410106
$235$	0	0
$236$	23.7004	1.54277
$237$	34.8349	2.26277
$238$	−0.601866	−0.0390132
$239$	15.1153	0.977725	0.488862	−	0.872361i	$-0.337412\pi$
$239$	15.1153	0.977725	0.488862	+	0.872361i	$0.337412\pi$
$240$	0	0
$241$	−26.2341	−1.68989	−0.844943	−	0.534856i	$-0.820366\pi$
$241$	−26.2341	−1.68989	−0.844943	+	0.534856i	$0.820366\pi$
$242$	2.69707	0.173374
$243$	−12.1105	−0.776888
$244$	−2.49714	−0.159863
$245$	0	0
$246$	−1.85099	−0.118015
$247$	−19.6097	−1.24773
$248$	0.253139	0.0160744
$249$	−21.6414	−1.37147
$250$	0	0
$251$	−14.6337	−0.923669	−0.461835	−	0.886966i	$-0.652809\pi$
$251$	−14.6337	−0.923669	−0.461835	+	0.886966i	$0.652809\pi$
$252$	8.15035	0.513424
$253$	−0.433828	−0.0272745
$254$	1.11353	0.0698689
$255$	0	0
$256$	8.34543	0.521590
$257$	10.8280	0.675430	0.337715	−	0.941248i	$-0.390346\pi$
$257$	10.8280	0.675430	0.337715	+	0.941248i	$0.390346\pi$
$258$	7.81771	0.486709
$259$	6.22615	0.386874
$260$	0	0
$261$	−33.0395	−2.04509
$262$	−1.08188	−0.0668386
$263$	11.9857	0.739068	0.369534	−	0.929217i	$-0.379517\pi$
$263$	11.9857	0.739068	0.369534	+	0.929217i	$0.379517\pi$
$264$	−5.80259	−0.357125
$265$	0	0
$266$	1.39076	0.0852730
$267$	−25.2448	−1.54496
$268$	4.89124	0.298780
$269$	−27.6252	−1.68434	−0.842169	−	0.539213i	$-0.818722\pi$
$269$	−27.6252	−1.68434	−0.842169	+	0.539213i	$0.818722\pi$
$270$	0	0
$271$	9.65133	0.586277	0.293138	−	0.956070i	$-0.405300\pi$
$271$	9.65133	0.586277	0.293138	+	0.956070i	$0.405300\pi$
$272$	−7.64689	−0.463661
$273$	8.92209	0.539990
$274$	3.35918	0.202936
$275$	0	0
$276$	−1.45468	−0.0875615
$277$	9.47891	0.569532	0.284766	−	0.958597i	$-0.408084\pi$
$277$	9.47891	0.569532	0.284766	+	0.958597i	$0.408084\pi$
$278$	5.77195	0.346179
$279$	1.04655	0.0626554
$280$	0	0
$281$	1.28220	0.0764899	0.0382449	−	0.999268i	$-0.487823\pi$
$281$	1.28220	0.0764899	0.0382449	+	0.999268i	$0.487823\pi$
$282$	−0.977769	−0.0582253
$283$	−16.1549	−0.960310	−0.480155	−	0.877184i	$-0.659420\pi$
$283$	−16.1549	−0.960310	−0.480155	+	0.877184i	$0.659420\pi$
$284$	−13.0841	−0.776399
$285$	0	0
$286$	−1.95823	−0.115792
$287$	−1.66741	−0.0984243
$288$	−18.6495	−1.09893
$289$	−11.9104	−0.700611
$290$	0	0
$291$	5.48908	0.321776
$292$	−7.63544	−0.446830
$293$	−23.3783	−1.36578	−0.682889	−	0.730522i	$-0.739277\pi$
$293$	−23.3783	−1.36578	−0.682889	+	0.730522i	$0.739277\pi$
$294$	5.81024	0.338860
$295$	0	0
$296$	−9.41436	−0.547198
$297$	−10.1047	−0.586332
$298$	0.719613	0.0416861
$299$	−1.00863	−0.0583308
$300$	0	0
$301$	7.04235	0.405914
$302$	0.438651	0.0252415
$303$	−47.2730	−2.71576
$304$	17.6700	1.01345
$305$	0	0
$306$	3.76246	0.215085
$307$	22.3400	1.27501	0.637504	−	0.770447i	$-0.279967\pi$
$307$	22.3400	1.27501	0.637504	+	0.770447i	$0.279967\pi$
$308$	−2.54410	−0.144964
$309$	−40.4735	−2.30246
$310$	0	0
$311$	23.5861	1.33745	0.668723	−	0.743512i	$-0.266841\pi$
$311$	23.5861	1.33745	0.668723	+	0.743512i	$0.266841\pi$
$312$	−13.4908	−0.763766
$313$	−21.3334	−1.20583	−0.602917	−	0.797804i	$-0.705995\pi$
$313$	−21.3334	−1.20583	−0.602917	+	0.797804i	$0.705995\pi$
$314$	7.39879	0.417538
$315$	0	0
$316$	−23.0939	−1.29914
$317$	−31.4080	−1.76405	−0.882025	−	0.471202i	$-0.843820\pi$
$317$	−31.4080	−1.76405	−0.882025	+	0.471202i	$0.843820\pi$
$318$	−9.82691	−0.551066
$319$	10.3132	0.577426
$320$	0	0
$321$	−1.46358	−0.0816891
$322$	0.0715346	0.00398647
$323$	−11.7608	−0.654389
$324$	−4.39262	−0.244034
$325$	0	0
$326$	2.76704	0.153252
$327$	21.9719	1.21505
$328$	2.52124	0.139212
$329$	−0.880794	−0.0485598
$330$	0	0
$331$	−13.1941	−0.725213	−0.362607	−	0.931942i	$-0.618113\pi$
$331$	−13.1941	−0.725213	−0.362607	+	0.931942i	$0.618113\pi$
$332$	14.3473	0.787408
$333$	−38.9216	−2.13289
$334$	−2.91743	−0.159635
$335$	0	0
$336$	−8.03959	−0.438595
$337$	−16.8702	−0.918977	−0.459488	−	0.888184i	$-0.651967\pi$
$337$	−16.8702	−0.918977	−0.459488	+	0.888184i	$0.651967\pi$
$338$	−0.369957	−0.0201230
$339$	29.7149	1.61389
$340$	0	0
$341$	−0.326677	−0.0176906
$342$	−8.69409	−0.470123
$343$	11.0380	0.595995
$344$	−10.6485	−0.574129
$345$	0	0
$346$	5.93965	0.319318
$347$	12.0168	0.645098	0.322549	−	0.946553i	$-0.395460\pi$
$347$	12.0168	0.645098	0.322549	+	0.946553i	$0.395460\pi$
$348$	34.5813	1.85375
$349$	9.96763	0.533555	0.266778	−	0.963758i	$-0.414041\pi$
$349$	9.96763	0.533555	0.266778	+	0.963758i	$0.414041\pi$
$350$	0	0
$351$	−23.4930	−1.25396
$352$	5.82139	0.310281
$353$	−12.5743	−0.669262	−0.334631	−	0.942349i	$-0.608612\pi$
$353$	−12.5743	−0.669262	−0.334631	+	0.942349i	$0.608612\pi$
$354$	11.5027	0.611364
$355$	0	0
$356$	16.7362	0.887015
$357$	5.35099	0.283204
$358$	2.03855	0.107741
$359$	35.1287	1.85402	0.927011	−	0.375034i	$-0.122369\pi$
$359$	35.1287	1.85402	0.927011	+	0.375034i	$0.122369\pi$
$360$	0	0
$361$	8.17626	0.430330
$362$	−6.82135	−0.358522
$363$	−23.9787	−1.25856
$364$	−5.91494	−0.310027
$365$	0	0
$366$	−1.21196	−0.0633501
$367$	6.50600	0.339610	0.169805	−	0.985478i	$-0.445686\pi$
$367$	6.50600	0.339610	0.169805	+	0.985478i	$0.445686\pi$
$368$	0.908868	0.0473780
$369$	10.4235	0.542627
$370$	0	0
$371$	−8.85228	−0.459587
$372$	−1.09539	−0.0567933
$373$	11.7116	0.606402	0.303201	−	0.952927i	$-0.401945\pi$
$373$	11.7116	0.606402	0.303201	+	0.952927i	$0.401945\pi$
$374$	−1.17444	−0.0607288
$375$	0	0
$376$	1.33182	0.0686834
$377$	23.9777	1.23491
$378$	1.66617	0.0856987
$379$	26.0985	1.34059	0.670294	−	0.742095i	$-0.266168\pi$
$379$	26.0985	1.34059	0.670294	+	0.742095i	$0.266168\pi$
$380$	0	0
$381$	−9.89999	−0.507192
$382$	−5.45817	−0.279264
$383$	21.6030	1.10386	0.551931	−	0.833890i	$-0.313891\pi$
$383$	21.6030	1.10386	0.551931	+	0.833890i	$0.313891\pi$
$384$	25.7595	1.31454
$385$	0	0
$386$	−0.865343	−0.0440448
$387$	−44.0240	−2.23786
$388$	−3.63902	−0.184743
$389$	−7.79774	−0.395361	−0.197681	−	0.980266i	$-0.563341\pi$
$389$	−7.79774	−0.395361	−0.197681	+	0.980266i	$0.563341\pi$
$390$	0	0
$391$	−0.604924	−0.0305923
$392$	−7.91413	−0.399724
$393$	9.61861	0.485194
$394$	−0.321757	−0.0162099
$395$	0	0
$396$	15.9040	0.799206
$397$	31.1670	1.56423	0.782113	−	0.623137i	$-0.214142\pi$
$397$	31.1670	1.56423	0.782113	+	0.623137i	$0.214142\pi$
$398$	−1.82863	−0.0916609
$399$	−12.3648	−0.619013
$400$	0	0
$401$	11.3173	0.565158	0.282579	−	0.959244i	$-0.408810\pi$
$401$	11.3173	0.565158	0.282579	+	0.959244i	$0.408810\pi$
$402$	2.37390	0.118400
$403$	−0.759512	−0.0378340
$404$	31.3398	1.55922
$405$	0	0
$406$	−1.70055	−0.0843970
$407$	12.1493	0.602216
$408$	−8.09105	−0.400567
$409$	−3.75436	−0.185641	−0.0928205	−	0.995683i	$-0.529588\pi$
$409$	−3.75436	−0.185641	−0.0928205	+	0.995683i	$0.529588\pi$
$410$	0	0
$411$	−29.8653	−1.47315
$412$	26.8321	1.32192
$413$	10.3619	0.509875
$414$	−0.447186	−0.0219780
$415$	0	0
$416$	13.5345	0.663584
$417$	−51.3164	−2.51298
$418$	2.71383	0.132738
$419$	−3.49970	−0.170972	−0.0854859	−	0.996339i	$-0.527244\pi$
$419$	−3.49970	−0.170972	−0.0854859	+	0.996339i	$0.527244\pi$
$420$	0	0
$421$	24.1260	1.17583	0.587914	−	0.808923i	$-0.299949\pi$
$421$	24.1260	1.17583	0.587914	+	0.808923i	$0.299949\pi$
$422$	−7.86710	−0.382965
$423$	5.50613	0.267717
$424$	13.3852	0.650045
$425$	0	0
$426$	−6.35022	−0.307669
$427$	−1.09176	−0.0528338
$428$	0.970288	0.0469007
$429$	17.4099	0.840559
$430$	0	0
$431$	−18.2590	−0.879505	−0.439753	−	0.898119i	$-0.644934\pi$
$431$	−18.2590	−0.879505	−0.439753	+	0.898119i	$0.644934\pi$
$432$	21.1692	1.01850
$433$	33.8382	1.62616	0.813080	−	0.582152i	$-0.197789\pi$
$433$	33.8382	1.62616	0.813080	+	0.582152i	$0.197789\pi$
$434$	0.0538663	0.00258567
$435$	0	0
$436$	−14.5664	−0.697603
$437$	1.39783	0.0668671
$438$	−3.70577	−0.177069
$439$	10.2163	0.487596	0.243798	−	0.969826i	$-0.421607\pi$
$439$	10.2163	0.487596	0.243798	+	0.969826i	$0.421607\pi$
$440$	0	0
$441$	−32.7193	−1.55806
$442$	−2.73053	−0.129878
$443$	−22.6314	−1.07525	−0.537624	−	0.843184i	$-0.680678\pi$
$443$	−22.6314	−1.07525	−0.537624	+	0.843184i	$0.680678\pi$
$444$	40.7380	1.93334
$445$	0	0
$446$	3.78964	0.179445
$447$	−6.39784	−0.302607
$448$	4.66094	0.220209
$449$	1.25782	0.0593604	0.0296802	−	0.999559i	$-0.490551\pi$
$449$	1.25782	0.0593604	0.0296802	+	0.999559i	$0.490551\pi$
$450$	0	0
$451$	−3.25367	−0.153209
$452$	−19.6996	−0.926592
$453$	−3.89989	−0.183233
$454$	6.82345	0.320240
$455$	0	0
$456$	18.6964	0.875538
$457$	−12.2454	−0.572815	−0.286407	−	0.958108i	$-0.592461\pi$
$457$	−12.2454	−0.572815	−0.286407	+	0.958108i	$0.592461\pi$
$458$	−0.781778	−0.0365301
$459$	−14.0898	−0.657656
$460$	0	0
$461$	20.0515	0.933892	0.466946	−	0.884286i	$-0.345354\pi$
$461$	20.0515	0.933892	0.466946	+	0.884286i	$0.345354\pi$
$462$	−1.23475	−0.0574458
$463$	−4.09512	−0.190316	−0.0951582	−	0.995462i	$-0.530336\pi$
$463$	−4.09512	−0.190316	−0.0951582	+	0.995462i	$0.530336\pi$
$464$	−21.6060	−1.00303
$465$	0	0
$466$	−5.20300	−0.241025
$467$	−17.2068	−0.796234	−0.398117	−	0.917335i	$-0.630336\pi$
$467$	−17.2068	−0.796234	−0.398117	+	0.917335i	$0.630336\pi$
$468$	36.9762	1.70923
$469$	2.13846	0.0987449
$470$	0	0
$471$	−65.7801	−3.03099
$472$	−15.6679	−0.721173
$473$	13.7419	0.631854
$474$	−11.2084	−0.514818
$475$	0	0
$476$	−3.54746	−0.162598
$477$	55.3384	2.53377
$478$	−4.86345	−0.222449
$479$	−2.67160	−0.122068	−0.0610342	−	0.998136i	$-0.519440\pi$
$479$	−2.67160	−0.122068	−0.0610342	+	0.998136i	$0.519440\pi$
$480$	0	0
$481$	28.2466	1.28793
$482$	8.44101	0.384478
$483$	−0.635990	−0.0289385
$484$	15.8968	0.722582
$485$	0	0
$486$	3.89664	0.176755
$487$	26.5967	1.20521	0.602605	−	0.798040i	$-0.294130\pi$
$487$	26.5967	1.20521	0.602605	+	0.798040i	$0.294130\pi$
$488$	1.65081	0.0747286
$489$	−24.6008	−1.11248
$490$	0	0
$491$	−18.3164	−0.826608	−0.413304	−	0.910593i	$-0.635625\pi$
$491$	−18.3164	−0.826608	−0.413304	+	0.910593i	$0.635625\pi$
$492$	−10.9100	−0.491859
$493$	14.3805	0.647666
$494$	6.30955	0.283880
$495$	0	0
$496$	0.684387	0.0307299
$497$	−5.72041	−0.256595
$498$	6.96328	0.312032
$499$	9.49828	0.425201	0.212601	−	0.977139i	$-0.431807\pi$
$499$	9.49828	0.425201	0.212601	+	0.977139i	$0.431807\pi$
$500$	0	0
$501$	25.9379	1.15882
$502$	4.70849	0.210150
$503$	−10.0197	−0.446756	−0.223378	−	0.974732i	$-0.571708\pi$
$503$	−10.0197	−0.446756	−0.223378	+	0.974732i	$0.571708\pi$
$504$	−5.38803	−0.240002
$505$	0	0
$506$	0.139587	0.00620542
$507$	3.28916	0.146077
$508$	6.56325	0.291197
$509$	8.65419	0.383590	0.191795	−	0.981435i	$-0.438569\pi$
$509$	8.65419	0.383590	0.191795	+	0.981435i	$0.438569\pi$
$510$	0	0
$511$	−3.33823	−0.147675
$512$	−20.6949	−0.914592
$513$	32.5580	1.43747
$514$	−3.48398	−0.153672
$515$	0	0
$516$	46.0784	2.02849
$517$	−1.71872	−0.0755891
$518$	−2.00331	−0.0880204
$519$	−52.8074	−2.31799
$520$	0	0
$521$	9.60415	0.420765	0.210383	−	0.977619i	$-0.432529\pi$
$521$	9.60415	0.420765	0.210383	+	0.977619i	$0.432529\pi$
$522$	10.6307	0.465293
$523$	4.62627	0.202293	0.101146	−	0.994872i	$-0.467749\pi$
$523$	4.62627	0.202293	0.101146	+	0.994872i	$0.467749\pi$
$524$	−6.37670	−0.278568
$525$	0	0
$526$	−3.85648	−0.168150
$527$	−0.455514	−0.0198425
$528$	−15.6879	−0.682727
$529$	−22.9281	−0.996874
$530$	0	0
$531$	−64.7755	−2.81102
$532$	8.19729	0.355398
$533$	−7.56466	−0.327662
$534$	8.12271	0.351504
$535$	0	0
$536$	−3.23350	−0.139666
$537$	−18.1241	−0.782112
$538$	8.88862	0.383216
$539$	10.2132	0.439914
$540$	0	0
$541$	21.8706	0.940290	0.470145	−	0.882589i	$-0.344202\pi$
$541$	21.8706	0.940290	0.470145	+	0.882589i	$0.344202\pi$
$542$	−3.10539	−0.133388
$543$	60.6463	2.60258
$544$	8.11727	0.348025
$545$	0	0
$546$	−2.87075	−0.122857
$547$	24.0932	1.03015	0.515076	−	0.857145i	$-0.327764\pi$
$547$	24.0932	1.03015	0.515076	+	0.857145i	$0.327764\pi$
$548$	19.7994	0.845787
$549$	6.82492	0.291280
$550$	0	0
$551$	−33.2298	−1.41564
$552$	0.961659	0.0409309
$553$	−10.0967	−0.429357
$554$	−3.04991	−0.129578
$555$	0	0
$556$	34.0205	1.44279
$557$	−15.3101	−0.648710	−0.324355	−	0.945935i	$-0.605147\pi$
$557$	−15.3101	−0.648710	−0.324355	+	0.945935i	$0.605147\pi$
$558$	−0.336735	−0.0142551
$559$	31.9495	1.35132
$560$	0	0
$561$	10.4415	0.440842
$562$	−0.412559	−0.0174027
$563$	−34.1032	−1.43728	−0.718638	−	0.695384i	$-0.755234\pi$
$563$	−34.1032	−1.43728	−0.718638	+	0.695384i	$0.755234\pi$
$564$	−5.76308	−0.242669
$565$	0	0
$566$	5.19797	0.218487
$567$	−1.92046	−0.0806519
$568$	8.64964	0.362931
$569$	30.2451	1.26794	0.633970	−	0.773358i	$-0.281424\pi$
$569$	30.2451	1.26794	0.633970	+	0.773358i	$0.281424\pi$
$570$	0	0
$571$	17.6296	0.737776	0.368888	−	0.929474i	$-0.379739\pi$
$571$	17.6296	0.737776	0.368888	+	0.929474i	$0.379739\pi$
$572$	−11.5420	−0.482595
$573$	48.5267	2.02723
$574$	0.536502	0.0223932
$575$	0	0
$576$	−29.1370	−1.21404
$577$	−28.4571	−1.18468	−0.592342	−	0.805687i	$-0.701797\pi$
$577$	−28.4571	−1.18468	−0.592342	+	0.805687i	$0.701797\pi$
$578$	3.83226	0.159401
$579$	7.69347	0.319730
$580$	0	0
$581$	6.27266	0.260234
$582$	−1.76615	−0.0732094
$583$	−17.2737	−0.715403
$584$	5.04763	0.208873
$585$	0	0
$586$	7.52216	0.310737
$587$	−12.1815	−0.502785	−0.251392	−	0.967885i	$-0.580888\pi$
$587$	−12.1815	−0.502785	−0.251392	+	0.967885i	$0.580888\pi$
$588$	34.2462	1.41229
$589$	1.05258	0.0433707
$590$	0	0
$591$	2.86064	0.117671
$592$	−25.4526	−1.04610
$593$	15.5316	0.637807	0.318904	−	0.947787i	$-0.396685\pi$
$593$	15.5316	0.637807	0.318904	+	0.947787i	$0.396685\pi$
$594$	3.25125	0.133400
$595$	0	0
$596$	4.24148	0.173738
$597$	16.2577	0.665384
$598$	0.324536	0.0132712
$599$	17.5452	0.716877	0.358438	−	0.933553i	$-0.383309\pi$
$599$	17.5452	0.716877	0.358438	+	0.933553i	$0.383309\pi$
$600$	0	0
$601$	0.948894	0.0387062	0.0193531	−	0.999813i	$-0.493839\pi$
$601$	0.948894	0.0387062	0.0193531	+	0.999813i	$0.493839\pi$
$602$	−2.26593	−0.0923523
$603$	−13.3682	−0.544395
$604$	2.58545	0.105201
$605$	0	0
$606$	15.2104	0.617881
$607$	42.6889	1.73269	0.866345	−	0.499446i	$-0.166463\pi$
$607$	42.6889	1.73269	0.866345	+	0.499446i	$0.166463\pi$
$608$	−18.7569	−0.760695
$609$	15.1190	0.612654
$610$	0	0
$611$	−3.99596	−0.161659
$612$	22.1763	0.896425
$613$	−41.1196	−1.66080	−0.830402	−	0.557164i	$-0.811889\pi$
$613$	−41.1196	−1.66080	−0.830402	+	0.557164i	$0.811889\pi$
$614$	−7.18805	−0.290086
$615$	0	0
$616$	1.68185	0.0677638
$617$	37.4229	1.50659	0.753294	−	0.657684i	$-0.228464\pi$
$617$	37.4229	1.50659	0.753294	+	0.657684i	$0.228464\pi$
$618$	13.0226	0.523848
$619$	9.19699	0.369658	0.184829	−	0.982771i	$-0.440827\pi$
$619$	9.19699	0.369658	0.184829	+	0.982771i	$0.440827\pi$
$620$	0	0
$621$	1.67464	0.0672009
$622$	−7.58901	−0.304291
$623$	7.31709	0.293153
$624$	−36.4737	−1.46012
$625$	0	0
$626$	6.86417	0.274347
$627$	−24.1277	−0.963569
$628$	43.6093	1.74020
$629$	16.9408	0.675472
$630$	0	0
$631$	−19.3908	−0.771937	−0.385968	−	0.922512i	$-0.626133\pi$
$631$	−19.3908	−0.771937	−0.385968	+	0.922512i	$0.626133\pi$
$632$	15.2669	0.607286
$633$	69.9437	2.78001
$634$	10.1058	0.401351
$635$	0	0
$636$	−57.9208	−2.29671
$637$	23.7453	0.940824
$638$	−3.31834	−0.131374
$639$	35.7601	1.41465
$640$	0	0
$641$	13.3926	0.528977	0.264488	−	0.964389i	$-0.414797\pi$
$641$	13.3926	0.528977	0.264488	+	0.964389i	$0.414797\pi$
$642$	0.470918	0.0185857
$643$	−25.7558	−1.01571	−0.507855	−	0.861442i	$-0.669562\pi$
$643$	−25.7558	−1.01571	−0.507855	+	0.861442i	$0.669562\pi$
$644$	0.421633	0.0166146
$645$	0	0
$646$	3.78413	0.148885
$647$	−18.6648	−0.733790	−0.366895	−	0.930262i	$-0.619579\pi$
$647$	−18.6648	−0.733790	−0.366895	+	0.930262i	$0.619579\pi$
$648$	2.90387	0.114075
$649$	20.2194	0.793683
$650$	0	0
$651$	−0.478907	−0.0187698
$652$	16.3092	0.638717
$653$	−23.0659	−0.902638	−0.451319	−	0.892363i	$-0.649046\pi$
$653$	−23.0659	−0.902638	−0.451319	+	0.892363i	$0.649046\pi$
$654$	−7.06962	−0.276444
$655$	0	0
$656$	6.81642	0.266136
$657$	20.8684	0.814153
$658$	0.283402	0.0110482
$659$	14.3959	0.560784	0.280392	−	0.959886i	$-0.409536\pi$
$659$	14.3959	0.560784	0.280392	+	0.959886i	$0.409536\pi$
$660$	0	0
$661$	−48.3648	−1.88117	−0.940586	−	0.339555i	$-0.889724\pi$
$661$	−48.3648	−1.88117	−0.940586	+	0.339555i	$0.889724\pi$
$662$	4.24530	0.164998
$663$	24.2762	0.942808
$664$	−9.48468	−0.368077
$665$	0	0
$666$	12.5233	0.485269
$667$	−1.70919	−0.0661802
$668$	−17.1957	−0.665320
$669$	−33.6924	−1.30262
$670$	0	0
$671$	−2.13037	−0.0822422
$672$	8.53412	0.329211
$673$	6.79907	0.262085	0.131042	−	0.991377i	$-0.458168\pi$
$673$	6.79907	0.262085	0.131042	+	0.991377i	$0.458168\pi$
$674$	5.42810	0.209083
$675$	0	0
$676$	−2.18057	−0.0838680
$677$	49.8767	1.91692	0.958459	−	0.285230i	$-0.0920699\pi$
$677$	49.8767	1.91692	0.958459	+	0.285230i	$0.0920699\pi$
$678$	−9.56098	−0.367187
$679$	−1.59099	−0.0610565
$680$	0	0
$681$	−60.6649	−2.32469
$682$	0.105111	0.00402490
$683$	2.51134	0.0960938	0.0480469	−	0.998845i	$-0.484700\pi$
$683$	2.51134	0.0960938	0.0480469	+	0.998845i	$0.484700\pi$
$684$	−51.2439	−1.95936
$685$	0	0
$686$	−3.55155	−0.135599
$687$	6.95052	0.265179
$688$	−28.7893	−1.09758
$689$	−40.1607	−1.53000
$690$	0	0
$691$	−23.6282	−0.898861	−0.449430	−	0.893315i	$-0.648373\pi$
$691$	−23.6282	−0.898861	−0.449430	+	0.893315i	$0.648373\pi$
$692$	35.0089	1.33084
$693$	6.95327	0.264133
$694$	−3.86651	−0.146771
$695$	0	0
$696$	−22.8610	−0.866544
$697$	−4.53687	−0.171846
$698$	−3.20716	−0.121393
$699$	46.2581	1.74964
$700$	0	0
$701$	26.3338	0.994614	0.497307	−	0.867575i	$-0.334322\pi$
$701$	26.3338	0.994614	0.497307	+	0.867575i	$0.334322\pi$
$702$	7.55903	0.285297
$703$	−39.1458	−1.47641
$704$	9.09502	0.342782
$705$	0	0
$706$	4.04587	0.152268
$707$	13.7019	0.515311
$708$	67.7983	2.54802
$709$	−31.8692	−1.19687	−0.598436	−	0.801170i	$-0.704211\pi$
$709$	−31.8692	−1.19687	−0.598436	+	0.801170i	$0.704211\pi$
$710$	0	0
$711$	63.1179	2.36710
$712$	−11.0639	−0.414639
$713$	0.0541400	0.00202756
$714$	−1.72172	−0.0644337
$715$	0	0
$716$	12.0154	0.449038
$717$	43.2392	1.61480
$718$	−11.3029	−0.421821
$719$	−20.2244	−0.754242	−0.377121	−	0.926164i	$-0.623086\pi$
$719$	−20.2244	−0.754242	−0.377121	+	0.926164i	$0.623086\pi$
$720$	0	0
$721$	11.7311	0.436887
$722$	−2.63077	−0.0979072
$723$	−75.0461	−2.79100
$724$	−40.2057	−1.49423
$725$	0	0
$726$	7.71533	0.286343
$727$	9.29216	0.344627	0.172314	−	0.985042i	$-0.444876\pi$
$727$	9.29216	0.344627	0.172314	+	0.985042i	$0.444876\pi$
$728$	3.91025	0.144923
$729$	−41.5923	−1.54045
$730$	0	0
$731$	19.1616	0.708716
$732$	−7.14341	−0.264028
$733$	−22.2835	−0.823061	−0.411530	−	0.911396i	$-0.635006\pi$
$733$	−22.2835	−0.823061	−0.411530	+	0.911396i	$0.635006\pi$
$734$	−2.09335	−0.0772671
$735$	0	0
$736$	−0.964775	−0.0355621
$737$	4.17284	0.153708
$738$	−3.35385	−0.123457
$739$	−8.92753	−0.328405	−0.164202	−	0.986427i	$-0.552505\pi$
$739$	−8.92753	−0.328405	−0.164202	+	0.986427i	$0.552505\pi$
$740$	0	0
$741$	−56.0961	−2.06074
$742$	2.84829	0.104564
$743$	21.2507	0.779613	0.389807	−	0.920897i	$-0.372542\pi$
$743$	21.2507	0.779613	0.389807	+	0.920897i	$0.372542\pi$
$744$	0.724139	0.0265482
$745$	0	0
$746$	−3.76828	−0.137967
$747$	−39.2124	−1.43471
$748$	−6.92226	−0.253103
$749$	0.424212	0.0155004
$750$	0	0
$751$	35.7550	1.30472	0.652360	−	0.757910i	$-0.273779\pi$
$751$	35.7550	1.30472	0.652360	+	0.757910i	$0.273779\pi$
$752$	3.60071	0.131304
$753$	−41.8616	−1.52552
$754$	−7.71501	−0.280964
$755$	0	0
$756$	9.82060	0.357172
$757$	3.60221	0.130925	0.0654623	−	0.997855i	$-0.479148\pi$
$757$	3.60221	0.130925	0.0654623	+	0.997855i	$0.479148\pi$
$758$	−8.39738	−0.305007
$759$	−1.24102	−0.0450463
$760$	0	0
$761$	38.7623	1.40513	0.702567	−	0.711618i	$-0.252037\pi$
$761$	38.7623	1.40513	0.702567	+	0.711618i	$0.252037\pi$
$762$	3.18539	0.115395
$763$	−6.36845	−0.230554
$764$	−32.1710	−1.16391
$765$	0	0
$766$	−6.95093	−0.251147
$767$	47.0095	1.69741
$768$	23.8732	0.861451
$769$	−23.0743	−0.832080	−0.416040	−	0.909346i	$-0.636582\pi$
$769$	−23.0743	−0.832080	−0.416040	+	0.909346i	$0.636582\pi$
$770$	0	0
$771$	30.9749	1.11553
$772$	−5.10042	−0.183568
$773$	−35.9068	−1.29148	−0.645738	−	0.763559i	$-0.723450\pi$
$773$	−35.9068	−1.29148	−0.645738	+	0.763559i	$0.723450\pi$
$774$	14.1650	0.509152
$775$	0	0
$776$	2.40568	0.0863588
$777$	17.8107	0.638957
$778$	2.50898	0.0899514
$779$	10.4836	0.375613
$780$	0	0
$781$	−11.1624	−0.399422
$782$	0.194639	0.00696027
$783$	−39.8103	−1.42270
$784$	−21.3966	−0.764165
$785$	0	0
$786$	−3.09486	−0.110390
$787$	47.6891	1.69993	0.849967	−	0.526836i	$-0.176622\pi$
$787$	47.6891	1.69993	0.849967	+	0.526836i	$0.176622\pi$
$788$	−1.89647	−0.0675590
$789$	34.2866	1.22064
$790$	0	0
$791$	−8.61272	−0.306233
$792$	−10.5138	−0.373592
$793$	−4.95304	−0.175888
$794$	−10.0282	−0.355888
$795$	0	0
$796$	−10.7781	−0.382021
$797$	37.3731	1.32382	0.661911	−	0.749582i	$-0.269745\pi$
$797$	37.3731	1.32382	0.661911	+	0.749582i	$0.269745\pi$
$798$	3.97846	0.140836
$799$	−2.39656	−0.0847841
$800$	0	0
$801$	−45.7415	−1.61620
$802$	−3.64142	−0.128583
$803$	−6.51399	−0.229874
$804$	13.9920	0.493461
$805$	0	0
$806$	0.244379	0.00860787
$807$	−79.0257	−2.78183
$808$	−20.7181	−0.728861
$809$	11.5270	0.405269	0.202635	−	0.979254i	$-0.435050\pi$
$809$	11.5270	0.405269	0.202635	+	0.979254i	$0.435050\pi$
$810$	0	0
$811$	−54.6969	−1.92067	−0.960334	−	0.278852i	$-0.910046\pi$
$811$	−54.6969	−1.92067	−0.960334	+	0.278852i	$0.910046\pi$
$812$	−10.0232	−0.351747
$813$	27.6089	0.968288
$814$	−3.90911	−0.137014
$815$	0	0
$816$	−21.8749	−0.765776
$817$	−44.2775	−1.54907
$818$	1.20799	0.0422365
$819$	16.1661	0.564889
$820$	0	0
$821$	37.6766	1.31492	0.657462	−	0.753488i	$-0.271630\pi$
$821$	37.6766	1.31492	0.657462	+	0.753488i	$0.271630\pi$
$822$	9.60939	0.335166
$823$	−42.6684	−1.48733	−0.743664	−	0.668554i	$-0.766914\pi$
$823$	−42.6684	−1.48733	−0.743664	+	0.668554i	$0.766914\pi$
$824$	−17.7381	−0.617938
$825$	0	0
$826$	−3.33402	−0.116005
$827$	−9.65451	−0.335720	−0.167860	−	0.985811i	$-0.553686\pi$
$827$	−9.65451	−0.335720	−0.167860	+	0.985811i	$0.553686\pi$
$828$	−2.63576	−0.0915990
$829$	56.0163	1.94553	0.972763	−	0.231803i	$-0.0744624\pi$
$829$	56.0163	1.94553	0.972763	+	0.231803i	$0.0744624\pi$
$830$	0	0
$831$	27.1157	0.940633
$832$	21.1456	0.733091
$833$	14.2412	0.493427
$834$	16.5114	0.571745
$835$	0	0
$836$	15.9956	0.553219
$837$	1.26102	0.0435872
$838$	1.12606	0.0388990
$839$	−36.1692	−1.24870	−0.624350	−	0.781144i	$-0.714636\pi$
$839$	−36.1692	−1.24870	−0.624350	+	0.781144i	$0.714636\pi$
$840$	0	0
$841$	11.6317	0.401093
$842$	−7.76271	−0.267521
$843$	3.66792	0.126330
$844$	−46.3695	−1.59611
$845$	0	0
$846$	−1.77164	−0.0609101
$847$	6.95013	0.238809
$848$	36.1883	1.24271
$849$	−46.2133	−1.58604
$850$	0	0
$851$	−2.01349	−0.0690215
$852$	−37.4289	−1.28229
$853$	18.1371	0.621002	0.310501	−	0.950573i	$-0.399503\pi$
$853$	18.1371	0.621002	0.310501	+	0.950573i	$0.399503\pi$
$854$	0.351281	0.0120206
$855$	0	0
$856$	−0.641438	−0.0219239
$857$	−13.9610	−0.476898	−0.238449	−	0.971155i	$-0.576639\pi$
$857$	−13.9610	−0.476898	−0.238449	+	0.971155i	$0.576639\pi$
$858$	−5.60177	−0.191241
$859$	−5.11540	−0.174535	−0.0872676	−	0.996185i	$-0.527814\pi$
$859$	−5.11540	−0.174535	−0.0872676	+	0.996185i	$0.527814\pi$
$860$	0	0
$861$	−4.76986	−0.162556
$862$	5.87497	0.200102
$863$	25.1731	0.856901	0.428451	−	0.903565i	$-0.359060\pi$
$863$	25.1731	0.856901	0.428451	+	0.903565i	$0.359060\pi$
$864$	−22.4714	−0.764492
$865$	0	0
$866$	−10.8877	−0.369979
$867$	−34.0713	−1.15712
$868$	0.317494	0.0107764
$869$	−19.7020	−0.668345
$870$	0	0
$871$	9.70169	0.328729
$872$	9.62953	0.326097
$873$	9.94576	0.336613
$874$	−0.449761	−0.0152134
$875$	0	0
$876$	−21.8422	−0.737980
$877$	−11.0092	−0.371755	−0.185878	−	0.982573i	$-0.559513\pi$
$877$	−11.0092	−0.371755	−0.185878	+	0.982573i	$0.559513\pi$
$878$	−3.28716	−0.110936
$879$	−66.8769	−2.25570
$880$	0	0
$881$	4.71494	0.158850	0.0794252	−	0.996841i	$-0.474692\pi$
$881$	4.71494	0.158850	0.0794252	+	0.996841i	$0.474692\pi$
$882$	10.5277	0.354485
$883$	16.4519	0.553651	0.276826	−	0.960920i	$-0.410718\pi$
$883$	16.4519	0.553651	0.276826	+	0.960920i	$0.410718\pi$
$884$	−16.0940	−0.541300
$885$	0	0
$886$	7.28181	0.244637
$887$	−47.2680	−1.58710	−0.793552	−	0.608502i	$-0.791771\pi$
$887$	−47.2680	−1.58710	−0.793552	+	0.608502i	$0.791771\pi$
$888$	−26.9310	−0.903747
$889$	2.86947	0.0962388
$890$	0	0
$891$	−3.74745	−0.125544
$892$	22.3365	0.747883
$893$	5.53784	0.185317
$894$	2.05855	0.0688483
$895$	0	0
$896$	−7.46628	−0.249431
$897$	−2.88534	−0.0963385
$898$	−0.404714	−0.0135055
$899$	−1.28704	−0.0429252
$900$	0	0
$901$	−24.0862	−0.802428
$902$	1.04689	0.0348577
$903$	20.1456	0.670403
$904$	13.0230	0.433139
$905$	0	0
$906$	1.25482	0.0416886
$907$	25.1296	0.834416	0.417208	−	0.908811i	$-0.363009\pi$
$907$	25.1296	0.834416	0.417208	+	0.908811i	$0.363009\pi$
$908$	40.2181	1.33469
$909$	−85.6547	−2.84099
$910$	0	0
$911$	11.7520	0.389361	0.194681	−	0.980867i	$-0.437633\pi$
$911$	11.7520	0.389361	0.194681	+	0.980867i	$0.437633\pi$
$912$	50.5475	1.67379
$913$	12.2400	0.405085
$914$	3.94004	0.130325
$915$	0	0
$916$	−4.60788	−0.152249
$917$	−2.78791	−0.0920649
$918$	4.53350	0.149628
$919$	−40.4198	−1.33333	−0.666663	−	0.745359i	$-0.732278\pi$
$919$	−40.4198	−1.33333	−0.666663	+	0.745359i	$0.732278\pi$
$920$	0	0
$921$	63.9065	2.10579
$922$	−6.45172	−0.212476
$923$	−25.9521	−0.854225
$924$	−7.27775	−0.239420
$925$	0	0
$926$	1.31764	0.0433002
$927$	−73.3346	−2.40862
$928$	22.9351	0.752880
$929$	25.2709	0.829112	0.414556	−	0.910024i	$-0.363937\pi$
$929$	25.2709	0.829112	0.414556	+	0.910024i	$0.363937\pi$
$930$	0	0
$931$	−32.9077	−1.07851
$932$	−30.6671	−1.00453
$933$	67.4713	2.20891
$934$	5.53641	0.181157
$935$	0	0
$936$	−24.4442	−0.798984
$937$	−47.3151	−1.54572	−0.772858	−	0.634579i	$-0.781174\pi$
$937$	−47.3151	−1.54572	−0.772858	+	0.634579i	$0.781174\pi$
$938$	−0.688066	−0.0224661
$939$	−61.0270	−1.99154
$940$	0	0
$941$	25.0662	0.817136	0.408568	−	0.912728i	$-0.366028\pi$
$941$	25.0662	0.817136	0.408568	+	0.912728i	$0.366028\pi$
$942$	21.1652	0.689601
$943$	0.539228	0.0175597
$944$	−42.3596	−1.37869
$945$	0	0
$946$	−4.42157	−0.143758
$947$	−41.4584	−1.34722	−0.673609	−	0.739088i	$-0.735257\pi$
$947$	−41.4584	−1.34722	−0.673609	+	0.739088i	$0.735257\pi$
$948$	−66.0633	−2.14564
$949$	−15.1448	−0.491620
$950$	0	0
$951$	−89.8469	−2.91348
$952$	2.34516	0.0760069
$953$	−10.9171	−0.353640	−0.176820	−	0.984243i	$-0.556581\pi$
$953$	−10.9171	−0.353640	−0.176820	+	0.984243i	$0.556581\pi$
$954$	−17.8055	−0.576476
$955$	0	0
$956$	−28.6657	−0.927114
$957$	29.5022	0.953670
$958$	0.859606	0.0277726
$959$	8.65633	0.279528
$960$	0	0
$961$	−30.9592	−0.998685
$962$	−9.08854	−0.293026
$963$	−2.65189	−0.0854559
$964$	49.7522	1.60241
$965$	0	0
$966$	0.204634	0.00658400
$967$	15.3317	0.493035	0.246517	−	0.969138i	$-0.420714\pi$
$967$	15.3317	0.493035	0.246517	+	0.969138i	$0.420714\pi$
$968$	−10.5091	−0.337774
$969$	−33.6434	−1.08078
$970$	0	0
$971$	59.2555	1.90160	0.950800	−	0.309804i	$-0.100264\pi$
$971$	59.2555	1.90160	0.950800	+	0.309804i	$0.100264\pi$
$972$	22.9672	0.736673
$973$	14.8738	0.476833
$974$	−8.55767	−0.274205
$975$	0	0
$976$	4.46312	0.142861
$977$	40.8266	1.30616	0.653080	−	0.757289i	$-0.273477\pi$
$977$	40.8266	1.30616	0.653080	+	0.757289i	$0.273477\pi$
$978$	7.91548	0.253109
$979$	14.2780	0.456328
$980$	0	0
$981$	39.8112	1.27108
$982$	5.89344	0.188067
$983$	41.8978	1.33633	0.668167	−	0.744011i	$-0.267079\pi$
$983$	41.8978	1.33633	0.668167	+	0.744011i	$0.267079\pi$
$984$	7.21235	0.229921
$985$	0	0
$986$	−4.62704	−0.147355
$987$	−2.51963	−0.0802007
$988$	37.1892	1.18315
$989$	−2.27744	−0.0724184
$990$	0	0
$991$	15.5921	0.495301	0.247650	−	0.968849i	$-0.420342\pi$
$991$	15.5921	0.495301	0.247650	+	0.968849i	$0.420342\pi$
$992$	−0.726485	−0.0230659
$993$	−37.7435	−1.19775
$994$	1.84058	0.0583798
$995$	0	0
$996$	41.0423	1.30047
$997$	1.65744	0.0524916	0.0262458	−	0.999656i	$-0.491645\pi$
$997$	1.65744	0.0524916	0.0262458	+	0.999656i	$0.491645\pi$
$998$	−3.05614	−0.0967405
$999$	−46.8979	−1.48378

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4925.2.a.r.1.24		49
5.2	odd	4		985.2.b.a.789.45	✓	98
5.3	odd	4		985.2.b.a.789.54	yes	98
5.4	even	2		4925.2.a.s.1.26		49

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
985.2.b.a.789.45	✓	98	5.2	odd	4
985.2.b.a.789.54	yes	98	5.3	odd	4
4925.2.a.r.1.24		49	1.1	even	1	trivial
4925.2.a.s.1.26		49	5.4	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:	\( N \)	\(=\)	\( 4925 = 5^{2} \cdot 197 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4925.a (trivial)

\(f(q)\)	\(=\)	\(q-0.321757 q^{2} +2.86064 q^{3} -1.89647 q^{4} -0.920431 q^{6} -0.829142 q^{7} +1.25372 q^{8} +5.18323 q^{9} -1.61793 q^{11} -5.42511 q^{12} -3.76162 q^{13} +0.266783 q^{14} +3.38955 q^{16} -2.25602 q^{17} -1.66774 q^{18} +5.21309 q^{19} -2.37187 q^{21} +0.520580 q^{22} +0.268138 q^{23} +3.58643 q^{24} +1.21033 q^{26} +6.24543 q^{27} +1.57245 q^{28} -6.37430 q^{29} +0.201911 q^{31} -3.59805 q^{32} -4.62830 q^{33} +0.725890 q^{34} -9.82986 q^{36} -7.50915 q^{37} -1.67735 q^{38} -10.7606 q^{39} +2.01101 q^{41} +0.763168 q^{42} -8.49353 q^{43} +3.06836 q^{44} -0.0862754 q^{46} +1.06230 q^{47} +9.69627 q^{48} -6.31252 q^{49} -6.45364 q^{51} +7.13381 q^{52} +10.6764 q^{53} -2.00951 q^{54} -1.03951 q^{56} +14.9127 q^{57} +2.05098 q^{58} -12.4971 q^{59} +1.31673 q^{61} -0.0649663 q^{62} -4.29764 q^{63} -5.62140 q^{64} +1.48919 q^{66} -2.57912 q^{67} +4.27847 q^{68} +0.767045 q^{69} +6.89919 q^{71} +6.49832 q^{72} +4.02613 q^{73} +2.41612 q^{74} -9.88647 q^{76} +1.34149 q^{77} +3.46231 q^{78} +12.1773 q^{79} +2.31621 q^{81} -0.647057 q^{82} -7.56524 q^{83} +4.49819 q^{84} +2.73286 q^{86} -18.2345 q^{87} -2.02843 q^{88} -8.82490 q^{89} +3.11892 q^{91} -0.508517 q^{92} +0.577593 q^{93} -0.341802 q^{94} -10.2927 q^{96} +1.91883 q^{97} +2.03110 q^{98} -8.38610 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 49 q - 5 q^{2} - 22 q^{3} + 49 q^{4} + 2 q^{6} - 32 q^{7} - 15 q^{8} + 51 q^{9} - 2 q^{11} - 44 q^{12} - 32 q^{13} - 8 q^{14} + 49 q^{16} - 14 q^{17} - 25 q^{18} + 4 q^{19} + 10 q^{21} - 38 q^{22} - 24 q^{23}+ \cdots + 22 q^{99}+O(q^{100}) \) 49 * q - 5 * q^2 - 22 * q^3 + 49 * q^4 + 2 * q^6 - 32 * q^7 - 15 * q^8 + 51 * q^9 - 2 * q^11 - 44 * q^12 - 32 * q^13 - 8 * q^14 + 49 * q^16 - 14 * q^17 - 25 * q^18 + 4 * q^19 + 10 * q^21 - 38 * q^22 - 24 * q^23 - 2 * q^26 - 88 * q^27 - 80 * q^28 + 6 * q^29 - 2 * q^31 - 35 * q^32 - 48 * q^33 - 12 * q^34 + 43 * q^36 - 60 * q^37 - 34 * q^38 - 2 * q^41 - 36 * q^42 - 80 * q^43 - 8 * q^44 - 22 * q^46 - 48 * q^47 - 88 * q^48 + 65 * q^49 + 16 * q^51 - 96 * q^52 - 18 * q^53 + 42 * q^54 - 22 * q^56 - 40 * q^57 - 52 * q^58 + 8 * q^59 + 16 * q^61 - 38 * q^62 - 82 * q^63 + 55 * q^64 + 42 * q^66 - 146 * q^67 + 68 * q^68 - 6 * q^69 - 75 * q^72 - 78 * q^73 - 24 * q^74 - 4 * q^76 - 32 * q^77 + 38 * q^78 + 10 * q^79 + 41 * q^81 - 96 * q^82 - 64 * q^83 + 42 * q^84 - 28 * q^86 - 56 * q^87 - 72 * q^88 + 4 * q^89 - 60 * q^92 + 28 * q^93 - 8 * q^94 - 34 * q^96 - 124 * q^97 + 19 * q^98 + 22 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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