LMFDB - Embedded newform 5225.2.a.k.1.3

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			1.3
Root			$0.759131$ 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)

$f(q)$	$=$	\(q-0.759131 q^{2} +2.25829 q^{3} -1.42372 q^{4} -1.71434 q^{6} -4.95189 q^{7} +2.59905 q^{8} +2.09989 q^{9} -1.00000 q^{11} -3.21518 q^{12} -0.499163 q^{13} +3.75913 q^{14} +0.874419 q^{16} +4.34828 q^{17} -1.59409 q^{18} -1.00000 q^{19} -11.1828 q^{21} +0.759131 q^{22} +2.78646 q^{23} +5.86942 q^{24} +0.378930 q^{26} -2.03271 q^{27} +7.05010 q^{28} +8.84908 q^{29} +1.67449 q^{31} -5.86190 q^{32} -2.25829 q^{33} -3.30091 q^{34} -2.98966 q^{36} +1.81215 q^{37} +0.759131 q^{38} -1.12726 q^{39} +10.5406 q^{41} +8.48922 q^{42} -2.65924 q^{43} +1.42372 q^{44} -2.11529 q^{46} -11.6115 q^{47} +1.97470 q^{48} +17.5212 q^{49} +9.81969 q^{51} +0.710668 q^{52} -10.6541 q^{53} +1.54310 q^{54} -12.8702 q^{56} -2.25829 q^{57} -6.71761 q^{58} +10.0174 q^{59} -12.5990 q^{61} -1.27116 q^{62} -10.3984 q^{63} +2.70111 q^{64} +1.71434 q^{66} -2.84997 q^{67} -6.19073 q^{68} +6.29265 q^{69} -13.3351 q^{71} +5.45772 q^{72} +13.2131 q^{73} -1.37566 q^{74} +1.42372 q^{76} +4.95189 q^{77} +0.855735 q^{78} -7.08995 q^{79} -10.8901 q^{81} -8.00173 q^{82} -15.1160 q^{83} +15.9212 q^{84} +2.01871 q^{86} +19.9838 q^{87} -2.59905 q^{88} -7.25826 q^{89} +2.47180 q^{91} -3.96714 q^{92} +3.78150 q^{93} +8.81467 q^{94} -13.2379 q^{96} -0.0194069 q^{97} -13.3009 q^{98} -2.09989 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.759131	−0.536787	−0.268393	−	0.963309i	$-0.586493\pi$
$2$	−0.759131	−0.536787	−0.268393	+	0.963309i	$0.586493\pi$
$3$	2.25829	1.30383	0.651913	−	0.758294i	$-0.273967\pi$
$3$	2.25829	1.30383	0.651913	+	0.758294i	$0.273967\pi$
$4$	−1.42372	−0.711860
$5$	0	0
$5$	0	0
$6$	−1.71434	−0.699877
$7$	−4.95189	−1.87164	−0.935819	−	0.352482i	$-0.885338\pi$
$7$	−4.95189	−1.87164	−0.935819	+	0.352482i	$0.885338\pi$
$8$	2.59905	0.918904
$9$	2.09989	0.699963
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−3.21518	−0.928142
$13$	−0.499163	−0.138443	−0.0692214	−	0.997601i	$-0.522051\pi$
$13$	−0.499163	−0.138443	−0.0692214	+	0.997601i	$0.522051\pi$
$14$	3.75913	1.00467
$15$	0	0
$16$	0.874419	0.218605
$17$	4.34828	1.05461	0.527306	−	0.849675i	$-0.323202\pi$
$17$	4.34828	1.05461	0.527306	+	0.849675i	$0.323202\pi$
$18$	−1.59409	−0.375731
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−11.1828	−2.44029
$22$	0.759131	0.161847
$23$	2.78646	0.581017	0.290509	−	0.956872i	$-0.406176\pi$
$23$	2.78646	0.581017	0.290509	+	0.956872i	$0.406176\pi$
$24$	5.86942	1.19809
$25$	0	0
$26$	0.378930	0.0743142
$27$	−2.03271	−0.391196
$28$	7.05010	1.33234
$29$	8.84908	1.64323	0.821616	−	0.570041i	$-0.193073\pi$
$29$	8.84908	1.64323	0.821616	+	0.570041i	$0.193073\pi$
$30$	0	0
$31$	1.67449	0.300748	0.150374	−	0.988629i	$-0.451952\pi$
$31$	1.67449	0.300748	0.150374	+	0.988629i	$0.451952\pi$
$32$	−5.86190	−1.03625
$33$	−2.25829	−0.393118
$34$	−3.30091	−0.566102
$35$	0	0
$36$	−2.98966	−0.498276
$37$	1.81215	0.297916	0.148958	−	0.988844i	$-0.452408\pi$
$37$	1.81215	0.297916	0.148958	+	0.988844i	$0.452408\pi$
$38$	0.759131	0.123147
$39$	−1.12726	−0.180505
$40$	0	0
$41$	10.5406	1.64617	0.823086	−	0.567916i	$-0.192250\pi$
$41$	10.5406	1.64617	0.823086	+	0.567916i	$0.192250\pi$
$42$	8.48922	1.30992
$43$	−2.65924	−0.405531	−0.202765	−	0.979227i	$-0.564993\pi$
$43$	−2.65924	−0.405531	−0.202765	+	0.979227i	$0.564993\pi$
$44$	1.42372	0.214634
$45$	0	0
$46$	−2.11529	−0.311882
$47$	−11.6115	−1.69372	−0.846858	−	0.531819i	$-0.821509\pi$
$47$	−11.6115	−1.69372	−0.846858	+	0.531819i	$0.821509\pi$
$48$	1.97470	0.285023
$49$	17.5212	2.50303
$50$	0	0
$51$	9.81969	1.37503
$52$	0.710668	0.0985519
$53$	−10.6541	−1.46345	−0.731726	−	0.681598i	$-0.761285\pi$
$53$	−10.6541	−1.46345	−0.731726	+	0.681598i	$0.761285\pi$
$54$	1.54310	0.209989
$55$	0	0
$56$	−12.8702	−1.71985
$57$	−2.25829	−0.299118
$58$	−6.71761	−0.882065
$59$	10.0174	1.30416	0.652079	−	0.758151i	$-0.273897\pi$
$59$	10.0174	1.30416	0.652079	+	0.758151i	$0.273897\pi$
$60$	0	0
$61$	−12.5990	−1.61314	−0.806569	−	0.591140i	$-0.798678\pi$
$61$	−12.5990	−1.61314	−0.806569	+	0.591140i	$0.798678\pi$
$62$	−1.27116	−0.161438
$63$	−10.3984	−1.31008
$64$	2.70111	0.337639
$65$	0	0
$66$	1.71434	0.211021
$67$	−2.84997	−0.348180	−0.174090	−	0.984730i	$-0.555698\pi$
$67$	−2.84997	−0.348180	−0.174090	+	0.984730i	$0.555698\pi$
$68$	−6.19073	−0.750736
$69$	6.29265	0.757546
$70$	0	0
$71$	−13.3351	−1.58258	−0.791291	−	0.611440i	$-0.790590\pi$
$71$	−13.3351	−1.58258	−0.791291	+	0.611440i	$0.790590\pi$
$72$	5.45772	0.643199
$73$	13.2131	1.54647	0.773236	−	0.634118i	$-0.218637\pi$
$73$	13.2131	1.54647	0.773236	+	0.634118i	$0.218637\pi$
$74$	−1.37566	−0.159917
$75$	0	0
$76$	1.42372	0.163312
$77$	4.95189	0.564320
$78$	0.855735	0.0968929
$79$	−7.08995	−0.797681	−0.398841	−	0.917020i	$-0.630587\pi$
$79$	−7.08995	−0.797681	−0.398841	+	0.917020i	$0.630587\pi$
$80$	0	0
$81$	−10.8901	−1.21001
$82$	−8.00173	−0.883643
$83$	−15.1160	−1.65920	−0.829600	−	0.558359i	$-0.811431\pi$
$83$	−15.1160	−1.65920	−0.829600	+	0.558359i	$0.811431\pi$
$84$	15.9212	1.73715
$85$	0	0
$86$	2.01871	0.217683
$87$	19.9838	2.14249
$88$	−2.59905	−0.277060
$89$	−7.25826	−0.769374	−0.384687	−	0.923047i	$-0.625691\pi$
$89$	−7.25826	−0.769374	−0.384687	+	0.923047i	$0.625691\pi$
$90$	0	0
$91$	2.47180	0.259115
$92$	−3.96714	−0.413603
$93$	3.78150	0.392123
$94$	8.81467	0.909164
$95$	0	0
$96$	−13.2379	−1.35109
$97$	−0.0194069	−0.00197047	−0.000985234	−	1.00000i	$-0.500314\pi$
$97$	−0.0194069	−0.00197047	−0.000985234		1.00000i	$0.500314\pi$
$98$	−13.3009	−1.34359
$99$	−2.09989	−0.211047
$100$	0	0
$101$	10.2786	1.02276	0.511379	−	0.859355i	$-0.329135\pi$
$101$	10.2786	1.02276	0.511379	+	0.859355i	$0.329135\pi$
$102$	−7.45443	−0.738098
$103$	5.23959	0.516272	0.258136	−	0.966109i	$-0.416892\pi$
$103$	5.23959	0.516272	0.258136	+	0.966109i	$0.416892\pi$
$104$	−1.29735	−0.127216
$105$	0	0
$106$	8.08786	0.785562
$107$	−16.5236	−1.59739	−0.798697	−	0.601733i	$-0.794477\pi$
$107$	−16.5236	−1.59739	−0.798697	+	0.601733i	$0.794477\pi$
$108$	2.89402	0.278477
$109$	−0.934463	−0.0895053	−0.0447527	−	0.998998i	$-0.514250\pi$
$109$	−0.934463	−0.0895053	−0.0447527	+	0.998998i	$0.514250\pi$
$110$	0	0
$111$	4.09237	0.388431
$112$	−4.33003	−0.409149
$113$	−17.0120	−1.60036	−0.800179	−	0.599761i	$-0.795262\pi$
$113$	−17.0120	−1.60036	−0.800179	+	0.599761i	$0.795262\pi$
$114$	1.71434	0.160563
$115$	0	0
$116$	−12.5986	−1.16975
$117$	−1.04819	−0.0969048
$118$	−7.60454	−0.700054
$119$	−21.5322	−1.97385
$120$	0	0
$121$	1.00000	0.0909091
$122$	9.56430	0.865911
$123$	23.8039	2.14632
$124$	−2.38401	−0.214091
$125$	0	0
$126$	7.89376	0.703232
$127$	−15.2360	−1.35197	−0.675987	−	0.736914i	$-0.736282\pi$
$127$	−15.2360	−1.35197	−0.675987	+	0.736914i	$0.736282\pi$
$128$	9.67331	0.855008
$129$	−6.00535	−0.528741
$130$	0	0
$131$	−5.16429	−0.451206	−0.225603	−	0.974219i	$-0.572435\pi$
$131$	−5.16429	−0.451206	−0.225603	+	0.974219i	$0.572435\pi$
$132$	3.21518	0.279845
$133$	4.95189	0.429383
$134$	2.16350	0.186898
$135$	0	0
$136$	11.3014	0.969087
$137$	14.1861	1.21200	0.606001	−	0.795464i	$-0.292773\pi$
$137$	14.1861	1.21200	0.606001	+	0.795464i	$0.292773\pi$
$138$	−4.77694	−0.406640
$139$	−1.20156	−0.101915	−0.0509573	−	0.998701i	$-0.516227\pi$
$139$	−1.20156	−0.101915	−0.0509573	+	0.998701i	$0.516227\pi$
$140$	0	0
$141$	−26.2222	−2.20831
$142$	10.1231	0.849509
$143$	0.499163	0.0417421
$144$	1.83618	0.153015
$145$	0	0
$146$	−10.0304	−0.830126
$147$	39.5680	3.26351
$148$	−2.58000	−0.212074
$149$	3.25736	0.266853	0.133427	−	0.991059i	$-0.457402\pi$
$149$	3.25736	0.266853	0.133427	+	0.991059i	$0.457402\pi$
$150$	0	0
$151$	14.9900	1.21987	0.609933	−	0.792453i	$-0.291196\pi$
$151$	14.9900	1.21987	0.609933	+	0.792453i	$0.291196\pi$
$152$	−2.59905	−0.210811
$153$	9.13090	0.738190
$154$	−3.75913	−0.302919
$155$	0	0
$156$	1.60490	0.128495
$157$	−2.08129	−0.166105	−0.0830524	−	0.996545i	$-0.526467\pi$
$157$	−2.08129	−0.166105	−0.0830524	+	0.996545i	$0.526467\pi$
$158$	5.38220	0.428185
$159$	−24.0601	−1.90809
$160$	0	0
$161$	−13.7982	−1.08745
$162$	8.26704	0.649520
$163$	−13.2889	−1.04087	−0.520434	−	0.853902i	$-0.674230\pi$
$163$	−13.2889	−1.04087	−0.520434	+	0.853902i	$0.674230\pi$
$164$	−15.0069	−1.17184
$165$	0	0
$166$	11.4750	0.890636
$167$	−23.3753	−1.80883	−0.904416	−	0.426652i	$-0.859693\pi$
$167$	−23.3753	−1.80883	−0.904416	+	0.426652i	$0.859693\pi$
$168$	−29.0647	−2.24239
$169$	−12.7508	−0.980834
$170$	0	0
$171$	−2.09989	−0.160583
$172$	3.78602	0.288681
$173$	−11.6401	−0.884983	−0.442491	−	0.896773i	$-0.645905\pi$
$173$	−11.6401	−0.884983	−0.442491	+	0.896773i	$0.645905\pi$
$174$	−15.1703	−1.15006
$175$	0	0
$176$	−0.874419	−0.0659118
$177$	22.6223	1.70040
$178$	5.50997	0.412990
$179$	−15.0945	−1.12822	−0.564110	−	0.825700i	$-0.690781\pi$
$179$	−15.0945	−1.12822	−0.564110	+	0.825700i	$0.690781\pi$
$180$	0	0
$181$	−15.0754	−1.12055	−0.560273	−	0.828308i	$-0.689304\pi$
$181$	−15.0754	−1.12055	−0.560273	+	0.828308i	$0.689304\pi$
$182$	−1.87642	−0.139089
$183$	−28.4523	−2.10325
$184$	7.24216	0.533899
$185$	0	0
$186$	−2.87065	−0.210487
$187$	−4.34828	−0.317978
$188$	16.5316	1.20569
$189$	10.0658	0.732177
$190$	0	0
$191$	3.60614	0.260931	0.130466	−	0.991453i	$-0.458353\pi$
$191$	3.60614	0.260931	0.130466	+	0.991453i	$0.458353\pi$
$192$	6.09991	0.440223
$193$	24.4479	1.75980	0.879900	−	0.475158i	$-0.157609\pi$
$193$	24.4479	1.75980	0.879900	+	0.475158i	$0.157609\pi$
$194$	0.0147323	0.00105772
$195$	0	0
$196$	−24.9453	−1.78180
$197$	15.4415	1.10016	0.550082	−	0.835111i	$-0.314597\pi$
$197$	15.4415	1.10016	0.550082	+	0.835111i	$0.314597\pi$
$198$	1.59409	0.113287
$199$	−19.3689	−1.37302	−0.686512	−	0.727119i	$-0.740859\pi$
$199$	−19.3689	−1.37302	−0.686512	+	0.727119i	$0.740859\pi$
$200$	0	0
$201$	−6.43607	−0.453966
$202$	−7.80280	−0.549003
$203$	−43.8196	−3.07554
$204$	−13.9805	−0.978830
$205$	0	0
$206$	−3.97753	−0.277128
$207$	5.85126	0.406691
$208$	−0.436477	−0.0302643
$209$	1.00000	0.0691714
$210$	0	0
$211$	12.4253	0.855393	0.427697	−	0.903922i	$-0.359325\pi$
$211$	12.4253	0.855393	0.427697	+	0.903922i	$0.359325\pi$
$212$	15.1685	1.04177
$213$	−30.1145	−2.06341
$214$	12.5436	0.857460
$215$	0	0
$216$	−5.28313	−0.359471
$217$	−8.29191	−0.562891
$218$	0.709380	0.0480453
$219$	29.8390	2.01633
$220$	0	0
$221$	−2.17050	−0.146003
$222$	−3.10664	−0.208504
$223$	9.46847	0.634055	0.317028	−	0.948416i	$-0.397315\pi$
$223$	9.46847	0.634055	0.317028	+	0.948416i	$0.397315\pi$
$224$	29.0275	1.93948
$225$	0	0
$226$	12.9144	0.859051
$227$	−1.20914	−0.0802532	−0.0401266	−	0.999195i	$-0.512776\pi$
$227$	−1.20914	−0.0802532	−0.0401266	+	0.999195i	$0.512776\pi$
$228$	3.21518	0.212930
$229$	−14.5551	−0.961831	−0.480916	−	0.876767i	$-0.659696\pi$
$229$	−14.5551	−0.961831	−0.480916	+	0.876767i	$0.659696\pi$
$230$	0	0
$231$	11.1828	0.735775
$232$	22.9992	1.50997
$233$	14.0283	0.919022	0.459511	−	0.888172i	$-0.348025\pi$
$233$	14.0283	0.919022	0.459511	+	0.888172i	$0.348025\pi$
$234$	0.795711	0.0520172
$235$	0	0
$236$	−14.2620	−0.928378
$237$	−16.0112	−1.04004
$238$	16.3457	1.05954
$239$	−3.88964	−0.251600	−0.125800	−	0.992056i	$-0.540150\pi$
$239$	−3.88964	−0.251600	−0.125800	+	0.992056i	$0.540150\pi$
$240$	0	0
$241$	−10.6210	−0.684160	−0.342080	−	0.939671i	$-0.611131\pi$
$241$	−10.6210	−0.684160	−0.342080	+	0.939671i	$0.611131\pi$
$242$	−0.759131	−0.0487988
$243$	−18.4950	−1.18645
$244$	17.9375	1.14833
$245$	0	0
$246$	−18.0703	−1.15212
$247$	0.499163	0.0317610
$248$	4.35210	0.276358
$249$	−34.1364	−2.16331
$250$	0	0
$251$	−2.97829	−0.187988	−0.0939940	−	0.995573i	$-0.529963\pi$
$251$	−2.97829	−0.187988	−0.0939940	+	0.995573i	$0.529963\pi$
$252$	14.8044	0.932592
$253$	−2.78646	−0.175183
$254$	11.5661	0.725721
$255$	0	0
$256$	−12.7455	−0.796596
$257$	−6.16276	−0.384423	−0.192211	−	0.981354i	$-0.561566\pi$
$257$	−6.16276	−0.384423	−0.192211	+	0.981354i	$0.561566\pi$
$258$	4.55885	0.283821
$259$	−8.97357	−0.557591
$260$	0	0
$261$	18.5821	1.15020
$262$	3.92037	0.242202
$263$	18.8622	1.16310	0.581548	−	0.813512i	$-0.302447\pi$
$263$	18.8622	1.16310	0.581548	+	0.813512i	$0.302447\pi$
$264$	−5.86942	−0.361238
$265$	0	0
$266$	−3.75913	−0.230487
$267$	−16.3913	−1.00313
$268$	4.05756	0.247855
$269$	−0.262397	−0.0159986	−0.00799932	−	0.999968i	$-0.502546\pi$
$269$	−0.262397	−0.0159986	−0.00799932	+	0.999968i	$0.502546\pi$
$270$	0	0
$271$	27.2164	1.65328	0.826638	−	0.562734i	$-0.190250\pi$
$271$	27.2164	1.65328	0.826638	+	0.562734i	$0.190250\pi$
$272$	3.80222	0.230543
$273$	5.58204	0.337841
$274$	−10.7691	−0.650587
$275$	0	0
$276$	−8.95897	−0.539266
$277$	−22.1763	−1.33244	−0.666222	−	0.745754i	$-0.732090\pi$
$277$	−22.1763	−1.33244	−0.666222	+	0.745754i	$0.732090\pi$
$278$	0.912138	0.0547064
$279$	3.51625	0.210513
$280$	0	0
$281$	−15.9540	−0.951737	−0.475868	−	0.879517i	$-0.657866\pi$
$281$	−15.9540	−0.951737	−0.475868	+	0.879517i	$0.657866\pi$
$282$	19.9061	1.18539
$283$	2.99709	0.178159	0.0890793	−	0.996025i	$-0.471608\pi$
$283$	2.99709	0.178159	0.0890793	+	0.996025i	$0.471608\pi$
$284$	18.9854	1.12658
$285$	0	0
$286$	−0.378930	−0.0224066
$287$	−52.1961	−3.08104
$288$	−12.3093	−0.725335
$289$	1.90752	0.112207
$290$	0	0
$291$	−0.0438264	−0.00256915
$292$	−18.8117	−1.10087
$293$	−2.71694	−0.158725	−0.0793625	−	0.996846i	$-0.525288\pi$
$293$	−2.71694	−0.158725	−0.0793625	+	0.996846i	$0.525288\pi$
$294$	−30.0373	−1.75181
$295$	0	0
$296$	4.70988	0.273756
$297$	2.03271	0.117950
$298$	−2.47276	−0.143243
$299$	−1.39090	−0.0804376
$300$	0	0
$301$	13.1683	0.759006
$302$	−11.3793	−0.654808
$303$	23.2121	1.33350
$304$	−0.874419	−0.0501514
$305$	0	0
$306$	−6.93155	−0.396250
$307$	4.88778	0.278961	0.139480	−	0.990225i	$-0.455457\pi$
$307$	4.88778	0.278961	0.139480	+	0.990225i	$0.455457\pi$
$308$	−7.05010	−0.401717
$309$	11.8325	0.673129
$310$	0	0
$311$	−16.6843	−0.946082	−0.473041	−	0.881040i	$-0.656844\pi$
$311$	−16.6843	−0.946082	−0.473041	+	0.881040i	$0.656844\pi$
$312$	−2.92980	−0.165867
$313$	−10.5924	−0.598716	−0.299358	−	0.954141i	$-0.596773\pi$
$313$	−10.5924	−0.598716	−0.299358	+	0.954141i	$0.596773\pi$
$314$	1.57997	0.0891628
$315$	0	0
$316$	10.0941	0.567838
$317$	−1.42787	−0.0801974	−0.0400987	−	0.999196i	$-0.512767\pi$
$317$	−1.42787	−0.0801974	−0.0400987	+	0.999196i	$0.512767\pi$
$318$	18.2648	1.02424
$319$	−8.84908	−0.495453
$320$	0	0
$321$	−37.3151	−2.08272
$322$	10.4747	0.583731
$323$	−4.34828	−0.241945
$324$	15.5045	0.861361
$325$	0	0
$326$	10.0880	0.558724
$327$	−2.11029	−0.116699
$328$	27.3957	1.51267
$329$	57.4990	3.17002
$330$	0	0
$331$	−3.79639	−0.208669	−0.104334	−	0.994542i	$-0.533271\pi$
$331$	−3.79639	−0.208669	−0.104334	+	0.994542i	$0.533271\pi$
$332$	21.5210	1.18112
$333$	3.80532	0.208530
$334$	17.7449	0.970957
$335$	0	0
$336$	−9.77847	−0.533459
$337$	22.1330	1.20566	0.602830	−	0.797869i	$-0.294040\pi$
$337$	22.1330	1.20566	0.602830	+	0.797869i	$0.294040\pi$
$338$	9.67956	0.526498
$339$	−38.4182	−2.08659
$340$	0	0
$341$	−1.67449	−0.0906790
$342$	1.59409	0.0861986
$343$	−52.0997	−2.81312
$344$	−6.91151	−0.372643
$345$	0	0
$346$	8.83639	0.475047
$347$	−1.70802	−0.0916913	−0.0458457	−	0.998949i	$-0.514598\pi$
$347$	−1.70802	−0.0916913	−0.0458457	+	0.998949i	$0.514598\pi$
$348$	−28.4514	−1.52515
$349$	−3.58382	−0.191837	−0.0959186	−	0.995389i	$-0.530579\pi$
$349$	−3.58382	−0.191837	−0.0959186	+	0.995389i	$0.530579\pi$
$350$	0	0
$351$	1.01465	0.0541583
$352$	5.86190	0.312440
$353$	2.41512	0.128544	0.0642719	−	0.997932i	$-0.479528\pi$
$353$	2.41512	0.128544	0.0642719	+	0.997932i	$0.479528\pi$
$354$	−17.1733	−0.912749
$355$	0	0
$356$	10.3337	0.547686
$357$	−48.6260	−2.57356
$358$	11.4587	0.605613
$359$	−18.6345	−0.983489	−0.491745	−	0.870739i	$-0.663641\pi$
$359$	−18.6345	−0.983489	−0.491745	+	0.870739i	$0.663641\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	11.4442	0.601494
$363$	2.25829	0.118530
$364$	−3.51915	−0.184453
$365$	0	0
$366$	21.5990	1.12900
$367$	1.21394	0.0633673	0.0316837	−	0.999498i	$-0.489913\pi$
$367$	1.21394	0.0633673	0.0316837	+	0.999498i	$0.489913\pi$
$368$	2.43654	0.127013
$369$	22.1342	1.15226
$370$	0	0
$371$	52.7579	2.73905
$372$	−5.38380	−0.279137
$373$	30.3171	1.56976	0.784881	−	0.619646i	$-0.212724\pi$
$373$	30.3171	1.56976	0.784881	+	0.619646i	$0.212724\pi$
$374$	3.30091	0.170686
$375$	0	0
$376$	−30.1790	−1.55636
$377$	−4.41713	−0.227494
$378$	−7.64124	−0.393023
$379$	15.3776	0.789897	0.394948	−	0.918703i	$-0.370762\pi$
$379$	15.3776	0.789897	0.394948	+	0.918703i	$0.370762\pi$
$380$	0	0
$381$	−34.4073	−1.76274
$382$	−2.73753	−0.140064
$383$	−1.62199	−0.0828798	−0.0414399	−	0.999141i	$-0.513195\pi$
$383$	−1.62199	−0.0828798	−0.0414399	+	0.999141i	$0.513195\pi$
$384$	21.8452	1.11478
$385$	0	0
$386$	−18.5592	−0.944638
$387$	−5.58411	−0.283856
$388$	0.0276299	0.00140270
$389$	−33.6559	−1.70642	−0.853210	−	0.521567i	$-0.825348\pi$
$389$	−33.6559	−1.70642	−0.853210	+	0.521567i	$0.825348\pi$
$390$	0	0
$391$	12.1163	0.612748
$392$	45.5385	2.30004
$393$	−11.6625	−0.588295
$394$	−11.7221	−0.590553
$395$	0	0
$396$	2.98966	0.150236
$397$	6.60727	0.331609	0.165805	−	0.986159i	$-0.446978\pi$
$397$	6.60727	0.331609	0.165805	+	0.986159i	$0.446978\pi$
$398$	14.7035	0.737021
$399$	11.1828	0.559841
$400$	0	0
$401$	0.424693	0.0212082	0.0106041	−	0.999944i	$-0.496625\pi$
$401$	0.424693	0.0212082	0.0106041	+	0.999944i	$0.496625\pi$
$402$	4.88582	0.243683
$403$	−0.835845	−0.0416364
$404$	−14.6339	−0.728061
$405$	0	0
$406$	33.2648	1.65091
$407$	−1.81215	−0.0898250
$408$	25.5219	1.26352
$409$	2.86729	0.141779	0.0708893	−	0.997484i	$-0.477416\pi$
$409$	2.86729	0.141779	0.0708893	+	0.997484i	$0.477416\pi$
$410$	0	0
$411$	32.0364	1.58024
$412$	−7.45971	−0.367513
$413$	−49.6052	−2.44091
$414$	−4.44187	−0.218306
$415$	0	0
$416$	2.92604	0.143461
$417$	−2.71347	−0.132879
$418$	−0.759131	−0.0371303
$419$	26.1407	1.27706	0.638529	−	0.769597i	$-0.279543\pi$
$419$	26.1407	1.27706	0.638529	+	0.769597i	$0.279543\pi$
$420$	0	0
$421$	15.0032	0.731211	0.365606	−	0.930770i	$-0.380862\pi$
$421$	15.0032	0.731211	0.365606	+	0.930770i	$0.380862\pi$
$422$	−9.43243	−0.459164
$423$	−24.3829	−1.18554
$424$	−27.6906	−1.34477
$425$	0	0
$426$	22.8608	1.10761
$427$	62.3889	3.01921
$428$	23.5249	1.13712
$429$	1.12726	0.0544244
$430$	0	0
$431$	−4.12884	−0.198879	−0.0994397	−	0.995044i	$-0.531705\pi$
$431$	−4.12884	−0.198879	−0.0994397	+	0.995044i	$0.531705\pi$
$432$	−1.77744	−0.0855173
$433$	−2.75298	−0.132300	−0.0661499	−	0.997810i	$-0.521072\pi$
$433$	−2.75298	−0.132300	−0.0661499	+	0.997810i	$0.521072\pi$
$434$	6.29464	0.302153
$435$	0	0
$436$	1.33041	0.0637153
$437$	−2.78646	−0.133294
$438$	−22.6517	−1.08234
$439$	20.8996	0.997485	0.498743	−	0.866750i	$-0.333795\pi$
$439$	20.8996	0.997485	0.498743	+	0.866750i	$0.333795\pi$
$440$	0	0
$441$	36.7926	1.75203
$442$	1.64769	0.0783727
$443$	7.56488	0.359418	0.179709	−	0.983720i	$-0.442484\pi$
$443$	7.56488	0.359418	0.179709	+	0.983720i	$0.442484\pi$
$444$	−5.82639	−0.276508
$445$	0	0
$446$	−7.18781	−0.340352
$447$	7.35607	0.347930
$448$	−13.3756	−0.631938
$449$	−0.886578	−0.0418402	−0.0209201	−	0.999781i	$-0.506660\pi$
$449$	−0.886578	−0.0418402	−0.0209201	+	0.999781i	$0.506660\pi$
$450$	0	0
$451$	−10.5406	−0.496340
$452$	24.2204	1.13923
$453$	33.8517	1.59049
$454$	0.917893	0.0430788
$455$	0	0
$456$	−5.86942	−0.274861
$457$	−11.1530	−0.521716	−0.260858	−	0.965377i	$-0.584005\pi$
$457$	−11.1530	−0.521716	−0.260858	+	0.965377i	$0.584005\pi$
$458$	11.0493	0.516298
$459$	−8.83880	−0.412560
$460$	0	0
$461$	42.0232	1.95721	0.978607	−	0.205737i	$-0.0659591\pi$
$461$	42.0232	1.95721	0.978607	+	0.205737i	$0.0659591\pi$
$462$	−8.48922	−0.394954
$463$	−16.2714	−0.756197	−0.378098	−	0.925765i	$-0.623422\pi$
$463$	−16.2714	−0.756197	−0.378098	+	0.925765i	$0.623422\pi$
$464$	7.73781	0.359219
$465$	0	0
$466$	−10.6493	−0.493319
$467$	31.3740	1.45182	0.725909	−	0.687791i	$-0.241419\pi$
$467$	31.3740	1.45182	0.725909	+	0.687791i	$0.241419\pi$
$468$	1.49232	0.0689827
$469$	14.1127	0.651666
$470$	0	0
$471$	−4.70016	−0.216572
$472$	26.0358	1.19840
$473$	2.65924	0.122272
$474$	12.1546	0.558279
$475$	0	0
$476$	30.6558	1.40511
$477$	−22.3724	−1.02436
$478$	2.95274	0.135055
$479$	−8.96367	−0.409561	−0.204780	−	0.978808i	$-0.565648\pi$
$479$	−8.96367	−0.409561	−0.204780	+	0.978808i	$0.565648\pi$
$480$	0	0
$481$	−0.904558	−0.0412443
$482$	8.06275	0.367248
$483$	−31.1605	−1.41785
$484$	−1.42372	−0.0647146
$485$	0	0
$486$	14.0401	0.636872
$487$	−7.53795	−0.341577	−0.170788	−	0.985308i	$-0.554631\pi$
$487$	−7.53795	−0.341577	−0.170788	+	0.985308i	$0.554631\pi$
$488$	−32.7455	−1.48232
$489$	−30.0103	−1.35711
$490$	0	0
$491$	−0.0747970	−0.00337554	−0.00168777	−	0.999999i	$-0.500537\pi$
$491$	−0.0747970	−0.00337554	−0.00168777	+	0.999999i	$0.500537\pi$
$492$	−33.8901	−1.52788
$493$	38.4783	1.73297
$494$	−0.378930	−0.0170489
$495$	0	0
$496$	1.46421	0.0657450
$497$	66.0337	2.96202
$498$	25.9140	1.16123
$499$	14.9395	0.668784	0.334392	−	0.942434i	$-0.391469\pi$
$499$	14.9395	0.668784	0.334392	+	0.942434i	$0.391469\pi$
$500$	0	0
$501$	−52.7882	−2.35840
$502$	2.26091	0.100909
$503$	−26.9195	−1.20028	−0.600140	−	0.799895i	$-0.704888\pi$
$503$	−26.9195	−1.20028	−0.600140	+	0.799895i	$0.704888\pi$
$504$	−27.0260	−1.20383
$505$	0	0
$506$	2.11529	0.0940360
$507$	−28.7951	−1.27884
$508$	21.6918	0.962416
$509$	−4.53071	−0.200820	−0.100410	−	0.994946i	$-0.532015\pi$
$509$	−4.53071	−0.200820	−0.100410	+	0.994946i	$0.532015\pi$
$510$	0	0
$511$	−65.4296	−2.89444
$512$	−9.67109	−0.427406
$513$	2.03271	0.0897465
$514$	4.67834	0.206353
$515$	0	0
$516$	8.54993	0.376390
$517$	11.6115	0.510675
$518$	6.81212	0.299307
$519$	−26.2868	−1.15386
$520$	0	0
$521$	−3.01018	−0.131879	−0.0659393	−	0.997824i	$-0.521004\pi$
$521$	−3.01018	−0.131879	−0.0659393	+	0.997824i	$0.521004\pi$
$522$	−14.1062	−0.617413
$523$	20.3779	0.891063	0.445531	−	0.895266i	$-0.353015\pi$
$523$	20.3779	0.891063	0.445531	+	0.895266i	$0.353015\pi$
$524$	7.35251	0.321196
$525$	0	0
$526$	−14.3189	−0.624334
$527$	7.28117	0.317173
$528$	−1.97470	−0.0859376
$529$	−15.2356	−0.662419
$530$	0	0
$531$	21.0355	0.912862
$532$	−7.05010	−0.305661
$533$	−5.26150	−0.227901
$534$	12.4431	0.538467
$535$	0	0
$536$	−7.40723	−0.319943
$537$	−34.0879	−1.47100
$538$	0.199194	0.00858786
$539$	−17.5212	−0.754691
$540$	0	0
$541$	−19.9886	−0.859377	−0.429689	−	0.902977i	$-0.641377\pi$
$541$	−19.9886	−0.859377	−0.429689	+	0.902977i	$0.641377\pi$
$542$	−20.6608	−0.887457
$543$	−34.0447	−1.46100
$544$	−25.4892	−1.09284
$545$	0	0
$546$	−4.23750	−0.181348
$547$	−4.16030	−0.177881	−0.0889407	−	0.996037i	$-0.528348\pi$
$547$	−4.16030	−0.177881	−0.0889407	+	0.996037i	$0.528348\pi$
$548$	−20.1971	−0.862776
$549$	−26.4565	−1.12914
$550$	0	0
$551$	−8.84908	−0.376983
$552$	16.3549	0.696111
$553$	35.1086	1.49297
$554$	16.8347	0.715238
$555$	0	0
$556$	1.71068	0.0725490
$557$	−2.83353	−0.120061	−0.0600303	−	0.998197i	$-0.519120\pi$
$557$	−2.83353	−0.120061	−0.0600303	+	0.998197i	$0.519120\pi$
$558$	−2.66930	−0.113000
$559$	1.32739	0.0561428
$560$	0	0
$561$	−9.81969	−0.414588
$562$	12.1112	0.510880
$563$	−20.3978	−0.859663	−0.429831	−	0.902909i	$-0.641427\pi$
$563$	−20.3978	−0.859663	−0.429831	+	0.902909i	$0.641427\pi$
$564$	37.3331	1.57201
$565$	0	0
$566$	−2.27518	−0.0956331
$567$	53.9267	2.26471
$568$	−34.6585	−1.45424
$569$	−2.20481	−0.0924303	−0.0462152	−	0.998932i	$-0.514716\pi$
$569$	−2.20481	−0.0924303	−0.0462152	+	0.998932i	$0.514716\pi$
$570$	0	0
$571$	−33.3008	−1.39360	−0.696798	−	0.717268i	$-0.745392\pi$
$571$	−33.3008	−1.39360	−0.696798	+	0.717268i	$0.745392\pi$
$572$	−0.710668	−0.0297145
$573$	8.14372	0.340209
$574$	39.6237	1.65386
$575$	0	0
$576$	5.67204	0.236335
$577$	17.1388	0.713496	0.356748	−	0.934201i	$-0.383885\pi$
$577$	17.1388	0.713496	0.356748	+	0.934201i	$0.383885\pi$
$578$	−1.44806	−0.0602313
$579$	55.2106	2.29447
$580$	0	0
$581$	74.8529	3.10542
$582$	0.0332700	0.00137908
$583$	10.6541	0.441248
$584$	34.3414	1.42106
$585$	0	0
$586$	2.06251	0.0852015
$587$	−12.4171	−0.512510	−0.256255	−	0.966609i	$-0.582489\pi$
$587$	−12.4171	−0.512510	−0.256255	+	0.966609i	$0.582489\pi$
$588$	−56.3337	−2.32316
$589$	−1.67449	−0.0689963
$590$	0	0
$591$	34.8715	1.43442
$592$	1.58458	0.0651259
$593$	−11.0150	−0.452332	−0.226166	−	0.974089i	$-0.572619\pi$
$593$	−11.0150	−0.452332	−0.226166	+	0.974089i	$0.572619\pi$
$594$	−1.54310	−0.0633140
$595$	0	0
$596$	−4.63756	−0.189962
$597$	−43.7406	−1.79018
$598$	1.05587	0.0431778
$599$	−0.143692	−0.00587109	−0.00293555	−	0.999996i	$-0.500934\pi$
$599$	−0.143692	−0.00587109	−0.00293555	+	0.999996i	$0.500934\pi$
$600$	0	0
$601$	−22.1248	−0.902490	−0.451245	−	0.892400i	$-0.649020\pi$
$601$	−22.1248	−0.902490	−0.451245	+	0.892400i	$0.649020\pi$
$602$	−9.99644	−0.407424
$603$	−5.98463	−0.243713
$604$	−21.3415	−0.868374
$605$	0	0
$606$	−17.6210	−0.715805
$607$	−44.0261	−1.78696	−0.893481	−	0.449101i	$-0.851745\pi$
$607$	−44.0261	−1.78696	−0.893481	+	0.449101i	$0.851745\pi$
$608$	5.86190	0.237732
$609$	−98.9576	−4.00996
$610$	0	0
$611$	5.79604	0.234483
$612$	−12.9999	−0.525488
$613$	7.70425	0.311172	0.155586	−	0.987822i	$-0.450273\pi$
$613$	7.70425	0.311172	0.155586	+	0.987822i	$0.450273\pi$
$614$	−3.71047	−0.149742
$615$	0	0
$616$	12.8702	0.518556
$617$	−22.4763	−0.904864	−0.452432	−	0.891799i	$-0.649443\pi$
$617$	−22.4763	−0.904864	−0.452432	+	0.891799i	$0.649443\pi$
$618$	−8.98244	−0.361327
$619$	−19.9890	−0.803425	−0.401712	−	0.915766i	$-0.631585\pi$
$619$	−19.9890	−0.803425	−0.401712	+	0.915766i	$0.631585\pi$
$620$	0	0
$621$	−5.66408	−0.227292
$622$	12.6656	0.507844
$623$	35.9421	1.43999
$624$	−0.985694	−0.0394593
$625$	0	0
$626$	8.04100	0.321383
$627$	2.25829	0.0901876
$628$	2.96317	0.118243
$629$	7.87974	0.314186
$630$	0	0
$631$	4.75533	0.189307	0.0946534	−	0.995510i	$-0.469826\pi$
$631$	4.75533	0.189307	0.0946534	+	0.995510i	$0.469826\pi$
$632$	−18.4271	−0.732992
$633$	28.0600	1.11528
$634$	1.08394	0.0430489
$635$	0	0
$636$	34.2548	1.35829
$637$	−8.74592	−0.346526
$638$	6.71761	0.265953
$639$	−28.0022	−1.10775
$640$	0	0
$641$	0.415256	0.0164016	0.00820081	−	0.999966i	$-0.497390\pi$
$641$	0.415256	0.0164016	0.00820081	+	0.999966i	$0.497390\pi$
$642$	28.3270	1.11798
$643$	−22.0655	−0.870180	−0.435090	−	0.900387i	$-0.643283\pi$
$643$	−22.0655	−0.870180	−0.435090	+	0.900387i	$0.643283\pi$
$644$	19.6448	0.774115
$645$	0	0
$646$	3.30091	0.129873
$647$	−39.4539	−1.55109	−0.775547	−	0.631290i	$-0.782526\pi$
$647$	−39.4539	−1.55109	−0.775547	+	0.631290i	$0.782526\pi$
$648$	−28.3040	−1.11189
$649$	−10.0174	−0.393218
$650$	0	0
$651$	−18.7256	−0.733913
$652$	18.9197	0.740952
$653$	−34.3144	−1.34283	−0.671413	−	0.741084i	$-0.734312\pi$
$653$	−34.3144	−1.34283	−0.671413	+	0.741084i	$0.734312\pi$
$654$	1.60199	0.0626427
$655$	0	0
$656$	9.21695	0.359861
$657$	27.7460	1.08247
$658$	−43.6493	−1.70163
$659$	12.6713	0.493604	0.246802	−	0.969066i	$-0.420620\pi$
$659$	12.6713	0.493604	0.246802	+	0.969066i	$0.420620\pi$
$660$	0	0
$661$	27.5364	1.07104	0.535520	−	0.844522i	$-0.320115\pi$
$661$	27.5364	1.07104	0.535520	+	0.844522i	$0.320115\pi$
$662$	2.88196	0.112011
$663$	−4.90162	−0.190363
$664$	−39.2873	−1.52464
$665$	0	0
$666$	−2.88873	−0.111936
$667$	24.6576	0.954746
$668$	33.2798	1.28764
$669$	21.3826	0.826698
$670$	0	0
$671$	12.5990	0.486380
$672$	65.5526	2.52875
$673$	20.7136	0.798449	0.399225	−	0.916853i	$-0.369279\pi$
$673$	20.7136	0.798449	0.399225	+	0.916853i	$0.369279\pi$
$674$	−16.8018	−0.647183
$675$	0	0
$676$	18.1536	0.698216
$677$	11.5968	0.445701	0.222850	−	0.974853i	$-0.428464\pi$
$677$	11.5968	0.445701	0.222850	+	0.974853i	$0.428464\pi$
$678$	29.1644	1.12005
$679$	0.0961006	0.00368800
$680$	0	0
$681$	−2.73058	−0.104636
$682$	1.27116	0.0486753
$683$	21.3445	0.816727	0.408363	−	0.912820i	$-0.366100\pi$
$683$	21.3445	0.816727	0.408363	+	0.912820i	$0.366100\pi$
$684$	2.98966	0.114312
$685$	0	0
$686$	39.5505	1.51005
$687$	−32.8698	−1.25406
$688$	−2.32529	−0.0886509
$689$	5.31813	0.202604
$690$	0	0
$691$	36.1885	1.37668	0.688338	−	0.725390i	$-0.258340\pi$
$691$	36.1885	1.37668	0.688338	+	0.725390i	$0.258340\pi$
$692$	16.5723	0.629984
$693$	10.3984	0.395003
$694$	1.29661	0.0492187
$695$	0	0
$696$	51.9390	1.96874
$697$	45.8337	1.73607
$698$	2.72059	0.102976
$699$	31.6799	1.19824
$700$	0	0
$701$	−16.3009	−0.615677	−0.307839	−	0.951439i	$-0.599606\pi$
$701$	−16.3009	−0.615677	−0.307839	+	0.951439i	$0.599606\pi$
$702$	−0.770256	−0.0290714
$703$	−1.81215	−0.0683466
$704$	−2.70111	−0.101802
$705$	0	0
$706$	−1.83339	−0.0690006
$707$	−50.8985	−1.91423
$708$	−32.2078	−1.21044
$709$	43.5634	1.63606	0.818030	−	0.575176i	$-0.195066\pi$
$709$	43.5634	1.63606	0.818030	+	0.575176i	$0.195066\pi$
$710$	0	0
$711$	−14.8881	−0.558348
$712$	−18.8646	−0.706980
$713$	4.66591	0.174740
$714$	36.9135	1.38145
$715$	0	0
$716$	21.4904	0.803134
$717$	−8.78394	−0.328042
$718$	14.1460	0.527924
$719$	−11.2024	−0.417779	−0.208889	−	0.977939i	$-0.566985\pi$
$719$	−11.2024	−0.417779	−0.208889	+	0.977939i	$0.566985\pi$
$720$	0	0
$721$	−25.9459	−0.966274
$722$	−0.759131	−0.0282519
$723$	−23.9854	−0.892026
$724$	21.4632	0.797672
$725$	0	0
$726$	−1.71434	−0.0636251
$727$	−40.2384	−1.49236	−0.746180	−	0.665744i	$-0.768114\pi$
$727$	−40.2384	−1.49236	−0.746180	+	0.665744i	$0.768114\pi$
$728$	6.42433	0.238101
$729$	−9.09668	−0.336914
$730$	0	0
$731$	−11.5631	−0.427677
$732$	40.5081	1.49722
$733$	12.7812	0.472084	0.236042	−	0.971743i	$-0.424150\pi$
$733$	12.7812	0.472084	0.236042	+	0.971743i	$0.424150\pi$
$734$	−0.921542	−0.0340147
$735$	0	0
$736$	−16.3340	−0.602078
$737$	2.84997	0.104980
$738$	−16.8028	−0.618518
$739$	−2.61487	−0.0961895	−0.0480948	−	0.998843i	$-0.515315\pi$
$739$	−2.61487	−0.0961895	−0.0480948	+	0.998843i	$0.515315\pi$
$740$	0	0
$741$	1.12726	0.0414108
$742$	−40.0502	−1.47029
$743$	26.0025	0.953937	0.476969	−	0.878920i	$-0.341736\pi$
$743$	26.0025	0.953937	0.476969	+	0.878920i	$0.341736\pi$
$744$	9.82831	0.360323
$745$	0	0
$746$	−23.0147	−0.842628
$747$	−31.7420	−1.16138
$748$	6.19073	0.226356
$749$	81.8229	2.98974
$750$	0	0
$751$	−50.2644	−1.83417	−0.917086	−	0.398689i	$-0.869465\pi$
$751$	−50.2644	−1.83417	−0.917086	+	0.398689i	$0.869465\pi$
$752$	−10.1533	−0.370255
$753$	−6.72585	−0.245104
$754$	3.35318	0.122116
$755$	0	0
$756$	−14.3308	−0.521208
$757$	0.0174245	0.000633303 0	0.000316651	−	1.00000i	$-0.499899\pi$
$757$	0.0174245	0.000633303 0	0.000316651		1.00000i	$0.499899\pi$
$758$	−11.6736	−0.424006
$759$	−6.29265	−0.228409
$760$	0	0
$761$	22.6593	0.821398	0.410699	−	0.911771i	$-0.365285\pi$
$761$	22.6593	0.821398	0.410699	+	0.911771i	$0.365285\pi$
$762$	26.1196	0.946215
$763$	4.62735	0.167521
$764$	−5.13413	−0.185746
$765$	0	0
$766$	1.23130	0.0444888
$767$	−5.00032	−0.180551
$768$	−28.7832	−1.03862
$769$	−43.4932	−1.56841	−0.784203	−	0.620504i	$-0.786928\pi$
$769$	−43.4932	−1.56841	−0.784203	+	0.620504i	$0.786928\pi$
$770$	0	0
$771$	−13.9173	−0.501220
$772$	−34.8070	−1.25273
$773$	−51.7597	−1.86167	−0.930833	−	0.365444i	$-0.880917\pi$
$773$	−51.7597	−1.86167	−0.930833	+	0.365444i	$0.880917\pi$
$774$	4.23907	0.152370
$775$	0	0
$776$	−0.0504394	−0.00181067
$777$	−20.2650	−0.727001
$778$	25.5492	0.915984
$779$	−10.5406	−0.377658
$780$	0	0
$781$	13.3351	0.477166
$782$	−9.19786	−0.328915
$783$	−17.9876	−0.642826
$784$	15.3209	0.547174
$785$	0	0
$786$	8.85336	0.315789
$787$	−30.7685	−1.09678	−0.548389	−	0.836223i	$-0.684759\pi$
$787$	−30.7685	−1.09678	−0.548389	+	0.836223i	$0.684759\pi$
$788$	−21.9844	−0.783163
$789$	42.5965	1.51647
$790$	0	0
$791$	84.2417	2.99529
$792$	−5.45772	−0.193932
$793$	6.28896	0.223327
$794$	−5.01578	−0.178003
$795$	0	0
$796$	27.5759	0.977401
$797$	−7.43505	−0.263363	−0.131682	−	0.991292i	$-0.542038\pi$
$797$	−7.43505	−0.263363	−0.131682	+	0.991292i	$0.542038\pi$
$798$	−8.48922	−0.300515
$799$	−50.4902	−1.78621
$800$	0	0
$801$	−15.2415	−0.538533
$802$	−0.322398	−0.0113843
$803$	−13.2131	−0.466279
$804$	9.16317	0.323160
$805$	0	0
$806$	0.634516	0.0223499
$807$	−0.592570	−0.0208595
$808$	26.7146	0.939817
$809$	−48.6194	−1.70937	−0.854683	−	0.519150i	$-0.826248\pi$
$809$	−48.6194	−1.70937	−0.854683	+	0.519150i	$0.826248\pi$
$810$	0	0
$811$	4.14175	0.145436	0.0727182	−	0.997353i	$-0.476833\pi$
$811$	4.14175	0.145436	0.0727182	+	0.997353i	$0.476833\pi$
$812$	62.3869	2.18935
$813$	61.4625	2.15559
$814$	1.37566	0.0482169
$815$	0	0
$816$	8.58653	0.300589
$817$	2.65924	0.0930351
$818$	−2.17665	−0.0761049
$819$	5.19050	0.181371
$820$	0	0
$821$	34.4147	1.20108	0.600541	−	0.799594i	$-0.294952\pi$
$821$	34.4147	1.20108	0.600541	+	0.799594i	$0.294952\pi$
$822$	−24.3199	−0.848252
$823$	−10.2755	−0.358183	−0.179091	−	0.983832i	$-0.557316\pi$
$823$	−10.2755	−0.358183	−0.179091	+	0.983832i	$0.557316\pi$
$824$	13.6180	0.474404
$825$	0	0
$826$	37.6568	1.31025
$827$	37.7462	1.31256	0.656282	−	0.754516i	$-0.272128\pi$
$827$	37.7462	1.31256	0.656282	+	0.754516i	$0.272128\pi$
$828$	−8.33056	−0.289507
$829$	−7.42020	−0.257714	−0.128857	−	0.991663i	$-0.541131\pi$
$829$	−7.42020	−0.257714	−0.128857	+	0.991663i	$0.541131\pi$
$830$	0	0
$831$	−50.0805	−1.73727
$832$	−1.34829	−0.0467437
$833$	76.1870	2.63972
$834$	2.05988	0.0713277
$835$	0	0
$836$	−1.42372	−0.0492404
$837$	−3.40377	−0.117651
$838$	−19.8442	−0.685508
$839$	11.4778	0.396259	0.198129	−	0.980176i	$-0.436513\pi$
$839$	11.4778	0.396259	0.198129	+	0.980176i	$0.436513\pi$
$840$	0	0
$841$	49.3062	1.70021
$842$	−11.3894	−0.392505
$843$	−36.0288	−1.24090
$844$	−17.6902	−0.608920
$845$	0	0
$846$	18.5098	0.636381
$847$	−4.95189	−0.170149
$848$	−9.31615	−0.319918
$849$	6.76831	0.232288
$850$	0	0
$851$	5.04949	0.173094
$852$	42.8746	1.46886
$853$	34.4829	1.18067	0.590336	−	0.807158i	$-0.298995\pi$
$853$	34.4829	1.18067	0.590336	+	0.807158i	$0.298995\pi$
$854$	−47.3613	−1.62067
$855$	0	0
$856$	−42.9456	−1.46785
$857$	−14.4282	−0.492859	−0.246429	−	0.969161i	$-0.579257\pi$
$857$	−14.4282	−0.492859	−0.246429	+	0.969161i	$0.579257\pi$
$858$	−0.855735	−0.0292143
$859$	−27.3349	−0.932653	−0.466327	−	0.884613i	$-0.654423\pi$
$859$	−27.3349	−0.932653	−0.466327	+	0.884613i	$0.654423\pi$
$860$	0	0
$861$	−117.874	−4.01714
$862$	3.13433	0.106756
$863$	7.77083	0.264522	0.132261	−	0.991215i	$-0.457776\pi$
$863$	7.77083	0.264522	0.132261	+	0.991215i	$0.457776\pi$
$864$	11.9156	0.405376
$865$	0	0
$866$	2.08987	0.0710167
$867$	4.30774	0.146299
$868$	11.8054	0.400700
$869$	7.08995	0.240510
$870$	0	0
$871$	1.42260	0.0482029
$872$	−2.42872	−0.0822468
$873$	−0.0407523	−0.00137925
$874$	2.11529	0.0715507
$875$	0	0
$876$	−42.4824	−1.43535
$877$	−33.4134	−1.12829	−0.564145	−	0.825676i	$-0.690794\pi$
$877$	−33.4134	−1.12829	−0.564145	+	0.825676i	$0.690794\pi$
$878$	−15.8656	−0.535437
$879$	−6.13564	−0.206950
$880$	0	0
$881$	−38.4814	−1.29647	−0.648236	−	0.761439i	$-0.724493\pi$
$881$	−38.4814	−1.29647	−0.648236	+	0.761439i	$0.724493\pi$
$882$	−27.9304	−0.940464
$883$	−6.45348	−0.217177	−0.108589	−	0.994087i	$-0.534633\pi$
$883$	−6.45348	−0.217177	−0.108589	+	0.994087i	$0.534633\pi$
$884$	3.09018	0.103934
$885$	0	0
$886$	−5.74273	−0.192931
$887$	51.6865	1.73546	0.867732	−	0.497032i	$-0.165577\pi$
$887$	51.6865	1.73546	0.867732	+	0.497032i	$0.165577\pi$
$888$	10.6363	0.356930
$889$	75.4468	2.53040
$890$	0	0
$891$	10.8901	0.364833
$892$	−13.4804	−0.451359
$893$	11.6115	0.388565
$894$	−5.58422	−0.186764
$895$	0	0
$896$	−47.9011	−1.60026
$897$	−3.14105	−0.104877
$898$	0.673029	0.0224593
$899$	14.8177	0.494199
$900$	0	0
$901$	−46.3270	−1.54338
$902$	8.00173	0.266429
$903$	29.7378	0.989612
$904$	−44.2152	−1.47057
$905$	0	0
$906$	−25.6979	−0.853756
$907$	51.3590	1.70535	0.852673	−	0.522445i	$-0.174980\pi$
$907$	51.3590	1.70535	0.852673	+	0.522445i	$0.174980\pi$
$908$	1.72147	0.0571290
$909$	21.5839	0.715894
$910$	0	0
$911$	−30.5365	−1.01172	−0.505859	−	0.862616i	$-0.668825\pi$
$911$	−30.5365	−1.01172	−0.505859	+	0.862616i	$0.668825\pi$
$912$	−1.97470	−0.0653887
$913$	15.1160	0.500267
$914$	8.46660	0.280050
$915$	0	0
$916$	20.7225	0.684689
$917$	25.5730	0.844495
$918$	6.70981	0.221457
$919$	−44.4425	−1.46602	−0.733011	−	0.680217i	$-0.761886\pi$
$919$	−44.4425	−1.46602	−0.733011	+	0.680217i	$0.761886\pi$
$920$	0	0
$921$	11.0380	0.363716
$922$	−31.9011	−1.05061
$923$	6.65637	0.219097
$924$	−15.9212	−0.523769
$925$	0	0
$926$	12.3521	0.405916
$927$	11.0026	0.361371
$928$	−51.8724	−1.70280
$929$	−4.42046	−0.145030	−0.0725152	−	0.997367i	$-0.523103\pi$
$929$	−4.42046	−0.145030	−0.0725152	+	0.997367i	$0.523103\pi$
$930$	0	0
$931$	−17.5212	−0.574234
$932$	−19.9723	−0.654215
$933$	−37.6781	−1.23353
$934$	−23.8170	−0.779316
$935$	0	0
$936$	−2.72429	−0.0890462
$937$	−57.9019	−1.89157	−0.945786	−	0.324789i	$-0.894707\pi$
$937$	−57.9019	−1.89157	−0.945786	+	0.324789i	$0.894707\pi$
$938$	−10.7134	−0.349806
$939$	−23.9207	−0.780622
$940$	0	0
$941$	−46.1004	−1.50283	−0.751415	−	0.659830i	$-0.770628\pi$
$941$	−46.1004	−1.50283	−0.751415	+	0.659830i	$0.770628\pi$
$942$	3.56803	0.116253
$943$	29.3711	0.956455
$944$	8.75943	0.285095
$945$	0	0
$946$	−2.01871	−0.0656340
$947$	−29.8328	−0.969434	−0.484717	−	0.874671i	$-0.661077\pi$
$947$	−29.8328	−0.969434	−0.484717	+	0.874671i	$0.661077\pi$
$948$	22.7954	0.740362
$949$	−6.59547	−0.214098
$950$	0	0
$951$	−3.22456	−0.104563
$952$	−55.9633	−1.81378
$953$	9.41874	0.305103	0.152551	−	0.988296i	$-0.451251\pi$
$953$	9.41874	0.305103	0.152551	+	0.988296i	$0.451251\pi$
$954$	16.9836	0.549864
$955$	0	0
$956$	5.53775	0.179104
$957$	−19.9838	−0.645985
$958$	6.80460	0.219847
$959$	−70.2481	−2.26843
$960$	0	0
$961$	−28.1961	−0.909551
$962$	0.686678	0.0221394
$963$	−34.6977	−1.11812
$964$	15.1214	0.487026
$965$	0	0
$966$	23.6549	0.761083
$967$	−51.5972	−1.65925	−0.829627	−	0.558318i	$-0.811447\pi$
$967$	−51.5972	−1.65925	−0.829627	+	0.558318i	$0.811447\pi$
$968$	2.59905	0.0835367
$969$	−9.81969	−0.315454
$970$	0	0
$971$	−4.76616	−0.152953	−0.0764767	−	0.997071i	$-0.524367\pi$
$971$	−4.76616	−0.152953	−0.0764767	+	0.997071i	$0.524367\pi$
$972$	26.3317	0.844589
$973$	5.94997	0.190747
$974$	5.72229	0.183354
$975$	0	0
$976$	−11.0168	−0.352640
$977$	1.46971	0.0470203	0.0235101	−	0.999724i	$-0.492516\pi$
$977$	1.46971	0.0470203	0.0235101	+	0.999724i	$0.492516\pi$
$978$	22.7817	0.728479
$979$	7.25826	0.231975
$980$	0	0
$981$	−1.96227	−0.0626504
$982$	0.0567807	0.00181195
$983$	−8.77114	−0.279756	−0.139878	−	0.990169i	$-0.544671\pi$
$983$	−8.77114	−0.279756	−0.139878	+	0.990169i	$0.544671\pi$
$984$	61.8675	1.97226
$985$	0	0
$986$	−29.2100	−0.930237
$987$	129.850	4.13316
$988$	−0.710668	−0.0226094
$989$	−7.40987	−0.235620
$990$	0	0
$991$	29.7744	0.945816	0.472908	−	0.881112i	$-0.343204\pi$
$991$	29.7744	0.945816	0.472908	+	0.881112i	$0.343204\pi$
$992$	−9.81572	−0.311650
$993$	−8.57337	−0.272068
$994$	−50.1283	−1.58997
$995$	0	0
$996$	48.6007	1.53997
$997$	−13.8558	−0.438818	−0.219409	−	0.975633i	$-0.570413\pi$
$997$	−13.8558	−0.438818	−0.219409	+	0.975633i	$0.570413\pi$
$998$	−11.3410	−0.358994
$999$	−3.68359	−0.116543

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5225.2.a.k.1.3		6
5.4	even	2		1045.2.a.g.1.4	✓	6
15.14	odd	2		9405.2.a.w.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.g.1.4	✓	6	5.4	even	2
5225.2.a.k.1.3		6	1.1	even	1	trivial
9405.2.a.w.1.3		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)

\(f(q)\)	\(=\)	\(q-0.759131 q^{2} +2.25829 q^{3} -1.42372 q^{4} -1.71434 q^{6} -4.95189 q^{7} +2.59905 q^{8} +2.09989 q^{9} -1.00000 q^{11} -3.21518 q^{12} -0.499163 q^{13} +3.75913 q^{14} +0.874419 q^{16} +4.34828 q^{17} -1.59409 q^{18} -1.00000 q^{19} -11.1828 q^{21} +0.759131 q^{22} +2.78646 q^{23} +5.86942 q^{24} +0.378930 q^{26} -2.03271 q^{27} +7.05010 q^{28} +8.84908 q^{29} +1.67449 q^{31} -5.86190 q^{32} -2.25829 q^{33} -3.30091 q^{34} -2.98966 q^{36} +1.81215 q^{37} +0.759131 q^{38} -1.12726 q^{39} +10.5406 q^{41} +8.48922 q^{42} -2.65924 q^{43} +1.42372 q^{44} -2.11529 q^{46} -11.6115 q^{47} +1.97470 q^{48} +17.5212 q^{49} +9.81969 q^{51} +0.710668 q^{52} -10.6541 q^{53} +1.54310 q^{54} -12.8702 q^{56} -2.25829 q^{57} -6.71761 q^{58} +10.0174 q^{59} -12.5990 q^{61} -1.27116 q^{62} -10.3984 q^{63} +2.70111 q^{64} +1.71434 q^{66} -2.84997 q^{67} -6.19073 q^{68} +6.29265 q^{69} -13.3351 q^{71} +5.45772 q^{72} +13.2131 q^{73} -1.37566 q^{74} +1.42372 q^{76} +4.95189 q^{77} +0.855735 q^{78} -7.08995 q^{79} -10.8901 q^{81} -8.00173 q^{82} -15.1160 q^{83} +15.9212 q^{84} +2.01871 q^{86} +19.9838 q^{87} -2.59905 q^{88} -7.25826 q^{89} +2.47180 q^{91} -3.96714 q^{92} +3.78150 q^{93} +8.81467 q^{94} -13.2379 q^{96} -0.0194069 q^{97} -13.3009 q^{98} -2.09989 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{3} + 8 q^{4} + 2 q^{6} - 5 q^{7} + 9 q^{9} - 6 q^{11} - 19 q^{12} + 9 q^{13} + 18 q^{14} + 4 q^{16} + 5 q^{17} - 2 q^{18} - 6 q^{19} + 3 q^{21} - 8 q^{23} - 7 q^{24} - 22 q^{26} - 30 q^{27} - 10 q^{28}+ \cdots - 9 q^{99}+O(q^{100}) \) 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

Related objects

Downloads

Learn more