LMFDB - Embedded newform 245.4.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [245,4,Mod(1,245)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(245, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("245.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 245 = 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	245.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$14.4554679514$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	245.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.41421 q^{2} +3.24264 q^{3} +11.4853 q^{4} +5.00000 q^{5} -14.3137 q^{6} -15.3848 q^{8} -16.4853 q^{9} -22.0711 q^{10} -0.142136 q^{11} +37.2426 q^{12} -32.1421 q^{13} +16.2132 q^{15} -23.9706 q^{16} -114.368 q^{17} +72.7696 q^{18} +43.2304 q^{19} +57.4264 q^{20} +0.627417 q^{22} +154.463 q^{23} -49.8873 q^{24} +25.0000 q^{25} +141.882 q^{26} -141.007 q^{27} -40.1472 q^{29} -71.5685 q^{30} +75.4214 q^{31} +228.889 q^{32} -0.460895 q^{33} +504.843 q^{34} -189.338 q^{36} -400.333 q^{37} -190.828 q^{38} -104.225 q^{39} -76.9239 q^{40} -95.4264 q^{41} -340.071 q^{43} -1.63247 q^{44} -82.4264 q^{45} -681.833 q^{46} -7.49033 q^{47} -77.7279 q^{48} -110.355 q^{50} -370.853 q^{51} -369.161 q^{52} -676.818 q^{53} +622.436 q^{54} -0.710678 q^{55} +140.181 q^{57} +177.218 q^{58} +796.181 q^{59} +186.213 q^{60} -757.102 q^{61} -332.926 q^{62} -818.602 q^{64} -160.711 q^{65} +2.03449 q^{66} -740.581 q^{67} -1313.54 q^{68} +500.868 q^{69} +37.0253 q^{71} +253.622 q^{72} +80.8772 q^{73} +1767.16 q^{74} +81.0660 q^{75} +496.514 q^{76} +460.073 q^{78} -317.358 q^{79} -119.853 q^{80} -12.1329 q^{81} +421.233 q^{82} -945.929 q^{83} -571.838 q^{85} +1501.15 q^{86} -130.183 q^{87} +2.18672 q^{88} +783.205 q^{89} +363.848 q^{90} +1774.05 q^{92} +244.564 q^{93} +33.0639 q^{94} +216.152 q^{95} +742.206 q^{96} +393.107 q^{97} +2.34315 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{2} - 2 q^{3} + 6 q^{4} + 10 q^{5} - 6 q^{6} + 6 q^{8} - 16 q^{9} - 30 q^{10} + 28 q^{11} + 66 q^{12} - 36 q^{13} - 10 q^{15} - 14 q^{16} - 76 q^{17} + 72 q^{18} + 160 q^{19} + 30 q^{20} - 44 q^{22}+ \cdots + 16 q^{99}+O(q^{100}) $ 2 * q - 6 * q^2 - 2 * q^3 + 6 * q^4 + 10 * q^5 - 6 * q^6 + 6 * q^8 - 16 * q^9 - 30 * q^10 + 28 * q^11 + 66 * q^12 - 36 * q^13 - 10 * q^15 - 14 * q^16 - 76 * q^17 + 72 * q^18 + 160 * q^19 + 30 * q^20 - 44 * q^22 - 22 * q^23 - 162 * q^24 + 50 * q^25 + 148 * q^26 - 2 * q^27 - 250 * q^29 - 30 * q^30 - 132 * q^31 + 42 * q^32 - 148 * q^33 + 444 * q^34 - 192 * q^36 - 416 * q^37 - 376 * q^38 - 84 * q^39 + 30 * q^40 - 106 * q^41 - 666 * q^43 - 156 * q^44 - 80 * q^45 - 402 * q^46 - 196 * q^47 - 130 * q^48 - 150 * q^50 - 572 * q^51 - 348 * q^52 - 952 * q^53 + 402 * q^54 + 140 * q^55 - 472 * q^57 + 510 * q^58 + 840 * q^59 + 330 * q^60 + 98 * q^61 - 4 * q^62 - 602 * q^64 - 180 * q^65 + 236 * q^66 - 1286 * q^67 - 1524 * q^68 + 1426 * q^69 + 1064 * q^71 + 264 * q^72 - 172 * q^73 + 1792 * q^74 - 50 * q^75 - 144 * q^76 + 428 * q^78 - 1240 * q^79 - 70 * q^80 - 754 * q^81 + 438 * q^82 - 1906 * q^83 - 380 * q^85 + 2018 * q^86 + 970 * q^87 + 604 * q^88 + 650 * q^89 + 360 * q^90 + 2742 * q^92 + 1332 * q^93 + 332 * q^94 + 800 * q^95 + 1722 * q^96 - 628 * q^97 + 16 * 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 $

$2$	−4.41421	−1.56066	−0.780330	−	0.625368i	$-0.784949\pi$
$2$	−4.41421	−1.56066	−0.780330	+	0.625368i	$0.784949\pi$
$3$	3.24264	0.624046	0.312023	−	0.950074i	$-0.398993\pi$
$3$	3.24264	0.624046	0.312023	+	0.950074i	$0.398993\pi$
$4$	11.4853	1.43566
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	−14.3137	−0.973925
$7$	0	0
$7$	0	0
$8$	−15.3848	−0.679917
$9$	−16.4853	−0.610566
$10$	−22.0711	−0.697948
$11$	−0.142136	−0.00389595	−0.00194798	−	0.999998i	$-0.500620\pi$
$11$	−0.142136	−0.00389595	−0.00194798	+	0.999998i	$0.500620\pi$
$12$	37.2426	0.895919
$13$	−32.1421	−0.685740	−0.342870	−	0.939383i	$-0.611399\pi$
$13$	−32.1421	−0.685740	−0.342870	+	0.939383i	$0.611399\pi$
$14$	0	0
$15$	16.2132	0.279082
$16$	−23.9706	−0.374540
$17$	−114.368	−1.63166	−0.815829	−	0.578293i	$-0.803719\pi$
$17$	−114.368	−1.63166	−0.815829	+	0.578293i	$0.803719\pi$
$18$	72.7696	0.952886
$19$	43.2304	0.521987	0.260993	−	0.965341i	$-0.415950\pi$
$19$	43.2304	0.521987	0.260993	+	0.965341i	$0.415950\pi$
$20$	57.4264	0.642047
$21$	0	0
$22$	0.627417	0.00608026
$23$	154.463	1.40034	0.700169	−	0.713977i	$-0.253108\pi$
$23$	154.463	1.40034	0.700169	+	0.713977i	$0.253108\pi$
$24$	−49.8873	−0.424300
$25$	25.0000	0.200000
$26$	141.882	1.07021
$27$	−141.007	−1.00507
$28$	0	0
$29$	−40.1472	−0.257074	−0.128537	−	0.991705i	$-0.541028\pi$
$29$	−40.1472	−0.257074	−0.128537	+	0.991705i	$0.541028\pi$
$30$	−71.5685	−0.435552
$31$	75.4214	0.436970	0.218485	−	0.975840i	$-0.429888\pi$
$31$	75.4214	0.436970	0.218485	+	0.975840i	$0.429888\pi$
$32$	228.889	1.26445
$33$	−0.460895	−0.00243126
$34$	504.843	2.54647
$35$	0	0
$36$	−189.338	−0.876565
$37$	−400.333	−1.77877	−0.889383	−	0.457163i	$-0.848866\pi$
$37$	−400.333	−1.77877	−0.889383	+	0.457163i	$0.848866\pi$
$38$	−190.828	−0.814644
$39$	−104.225	−0.427934
$40$	−76.9239	−0.304068
$41$	−95.4264	−0.363490	−0.181745	−	0.983346i	$-0.558175\pi$
$41$	−95.4264	−0.363490	−0.181745	+	0.983346i	$0.558175\pi$
$42$	0	0
$43$	−340.071	−1.20605	−0.603027	−	0.797721i	$-0.706039\pi$
$43$	−340.071	−1.20605	−0.603027	+	0.797721i	$0.706039\pi$
$44$	−1.63247	−0.00559327
$45$	−82.4264	−0.273053
$46$	−681.833	−2.18545
$47$	−7.49033	−0.0232463	−0.0116232	−	0.999932i	$-0.503700\pi$
$47$	−7.49033	−0.0232463	−0.0116232	+	0.999932i	$0.503700\pi$
$48$	−77.7279	−0.233730
$49$	0	0
$50$	−110.355	−0.312132
$51$	−370.853	−1.01823
$52$	−369.161	−0.984490
$53$	−676.818	−1.75412	−0.877058	−	0.480385i	$-0.840497\pi$
$53$	−676.818	−1.75412	−0.877058	+	0.480385i	$0.840497\pi$
$54$	622.436	1.56857
$55$	−0.710678	−0.00174232
$56$	0	0
$57$	140.181	0.325744
$58$	177.218	0.401205
$59$	796.181	1.75685	0.878423	−	0.477884i	$-0.158596\pi$
$59$	796.181	1.75685	0.878423	+	0.477884i	$0.158596\pi$
$60$	186.213	0.400667
$61$	−757.102	−1.58913	−0.794565	−	0.607179i	$-0.792301\pi$
$61$	−757.102	−1.58913	−0.794565	+	0.607179i	$0.792301\pi$
$62$	−332.926	−0.681962
$63$	0	0
$64$	−818.602	−1.59883
$65$	−160.711	−0.306672
$66$	2.03449	0.00379437
$67$	−740.581	−1.35039	−0.675197	−	0.737638i	$-0.735941\pi$
$67$	−740.581	−1.35039	−0.675197	+	0.737638i	$0.735941\pi$
$68$	−1313.54	−2.34251
$69$	500.868	0.873876
$70$	0	0
$71$	37.0253	0.0618886	0.0309443	−	0.999521i	$-0.490149\pi$
$71$	37.0253	0.0618886	0.0309443	+	0.999521i	$0.490149\pi$
$72$	253.622	0.415134
$73$	80.8772	0.129671	0.0648353	−	0.997896i	$-0.479348\pi$
$73$	80.8772	0.129671	0.0648353	+	0.997896i	$0.479348\pi$
$74$	1767.16	2.77605
$75$	81.0660	0.124809
$76$	496.514	0.749395
$77$	0	0
$78$	460.073	0.667859
$79$	−317.358	−0.451970	−0.225985	−	0.974131i	$-0.572560\pi$
$79$	−317.358	−0.451970	−0.225985	+	0.974131i	$0.572560\pi$
$80$	−119.853	−0.167499
$81$	−12.1329	−0.0166432
$82$	421.233	0.567285
$83$	−945.929	−1.25095	−0.625477	−	0.780243i	$-0.715096\pi$
$83$	−945.929	−1.25095	−0.625477	+	0.780243i	$0.715096\pi$
$84$	0	0
$85$	−571.838	−0.729700
$86$	1501.15	1.88224
$87$	−130.183	−0.160426
$88$	2.18672	0.00264893
$89$	783.205	0.932804	0.466402	−	0.884573i	$-0.345550\pi$
$89$	783.205	0.932804	0.466402	+	0.884573i	$0.345550\pi$
$90$	363.848	0.426144
$91$	0	0
$92$	1774.05	2.01041
$93$	244.564	0.272690
$94$	33.0639	0.0362796
$95$	216.152	0.233439
$96$	742.206	0.789074
$97$	393.107	0.411484	0.205742	−	0.978606i	$-0.434039\pi$
$97$	393.107	0.411484	0.205742	+	0.978606i	$0.434039\pi$
$98$	0	0
$99$	2.34315	0.00237874
$100$	287.132	0.287132
$101$	454.442	0.447709	0.223855	−	0.974623i	$-0.428136\pi$
$101$	454.442	0.447709	0.223855	+	0.974623i	$0.428136\pi$
$102$	1637.02	1.58911
$103$	−703.699	−0.673180	−0.336590	−	0.941651i	$-0.609274\pi$
$103$	−703.699	−0.673180	−0.336590	+	0.941651i	$0.609274\pi$
$104$	494.500	0.466247
$105$	0	0
$106$	2987.62	2.73758
$107$	954.997	0.862832	0.431416	−	0.902153i	$-0.358014\pi$
$107$	954.997	0.862832	0.431416	+	0.902153i	$0.358014\pi$
$108$	−1619.51	−1.44294
$109$	288.927	0.253891	0.126946	−	0.991910i	$-0.459483\pi$
$109$	288.927	0.253891	0.126946	+	0.991910i	$0.459483\pi$
$110$	3.13708	0.00271918
$111$	−1298.14	−1.11003
$112$	0	0
$113$	1251.24	1.04165	0.520826	−	0.853663i	$-0.325624\pi$
$113$	1251.24	1.04165	0.520826	+	0.853663i	$0.325624\pi$
$114$	−618.788	−0.508376
$115$	772.315	0.626250
$116$	−461.102	−0.369071
$117$	529.872	0.418690
$118$	−3514.51	−2.74184
$119$	0	0
$120$	−249.437	−0.189753
$121$	−1330.98	−0.999985
$122$	3342.01	2.48009
$123$	−309.434	−0.226835
$124$	866.235	0.627341
$125$	125.000	0.0894427
$126$	0	0
$127$	−1947.62	−1.36082	−0.680408	−	0.732833i	$-0.738198\pi$
$127$	−1947.62	−1.36082	−0.680408	+	0.732833i	$0.738198\pi$
$128$	1782.37	1.23079
$129$	−1102.73	−0.752634
$130$	709.411	0.478611
$131$	1061.65	0.708064	0.354032	−	0.935233i	$-0.384810\pi$
$131$	1061.65	0.708064	0.354032	+	0.935233i	$0.384810\pi$
$132$	−5.29351	−0.00349046
$133$	0	0
$134$	3269.08	2.10750
$135$	−705.036	−0.449480
$136$	1759.52	1.10939
$137$	−1439.34	−0.897601	−0.448800	−	0.893632i	$-0.648149\pi$
$137$	−1439.34	−0.897601	−0.448800	+	0.893632i	$0.648149\pi$
$138$	−2210.94	−1.36382
$139$	662.132	0.404038	0.202019	−	0.979382i	$-0.435250\pi$
$139$	662.132	0.404038	0.202019	+	0.979382i	$0.435250\pi$
$140$	0	0
$141$	−24.2885	−0.0145068
$142$	−163.437	−0.0965870
$143$	4.56854	0.00267161
$144$	395.161	0.228681
$145$	−200.736	−0.114967
$146$	−357.009	−0.202372
$147$	0	0
$148$	−4597.94	−2.55370
$149$	−787.110	−0.432769	−0.216384	−	0.976308i	$-0.569426\pi$
$149$	−787.110	−0.432769	−0.216384	+	0.976308i	$0.569426\pi$
$150$	−357.843	−0.194785
$151$	−848.355	−0.457206	−0.228603	−	0.973520i	$-0.573416\pi$
$151$	−848.355	−0.457206	−0.228603	+	0.973520i	$0.573416\pi$
$152$	−665.091	−0.354908
$153$	1885.38	0.996235
$154$	0	0
$155$	377.107	0.195419
$156$	−1197.06	−0.614368
$157$	954.922	0.485421	0.242710	−	0.970099i	$-0.421964\pi$
$157$	954.922	0.485421	0.242710	+	0.970099i	$0.421964\pi$
$158$	1400.89	0.705371
$159$	−2194.68	−1.09465
$160$	1144.45	0.565478
$161$	0	0
$162$	53.5572	0.0259744
$163$	−7.36421	−0.00353871	−0.00176936	−	0.999998i	$-0.500563\pi$
$163$	−7.36421	−0.00353871	−0.00176936	+	0.999998i	$0.500563\pi$
$164$	−1096.00	−0.521848
$165$	−2.30447	−0.00108729
$166$	4175.53	1.95231
$167$	2921.40	1.35368	0.676841	−	0.736129i	$-0.263349\pi$
$167$	2921.40	1.35368	0.676841	+	0.736129i	$0.263349\pi$
$168$	0	0
$169$	−1163.88	−0.529760
$170$	2524.21	1.13881
$171$	−712.666	−0.318707
$172$	−3905.81	−1.73148
$173$	−3969.57	−1.74451	−0.872256	−	0.489049i	$-0.837344\pi$
$173$	−3969.57	−1.74451	−0.872256	+	0.489049i	$0.837344\pi$
$174$	574.655	0.250371
$175$	0	0
$176$	3.40707	0.00145919
$177$	2581.73	1.09635
$178$	−3457.23	−1.45579
$179$	−1105.95	−0.461803	−0.230901	−	0.972977i	$-0.574167\pi$
$179$	−1105.95	−0.461803	−0.230901	+	0.972977i	$0.574167\pi$
$180$	−946.690	−0.392012
$181$	−117.214	−0.0481349	−0.0240674	−	0.999710i	$-0.507662\pi$
$181$	−117.214	−0.0481349	−0.0240674	+	0.999710i	$0.507662\pi$
$182$	0	0
$183$	−2455.01	−0.991691
$184$	−2376.38	−0.952114
$185$	−2001.67	−0.795488
$186$	−1079.56	−0.425576
$187$	16.2557	0.00635687
$188$	−86.0286	−0.0333738
$189$	0	0
$190$	−954.142	−0.364320
$191$	1002.60	0.379820	0.189910	−	0.981802i	$-0.439180\pi$
$191$	1002.60	0.379820	0.189910	+	0.981802i	$0.439180\pi$
$192$	−2654.43	−0.997746
$193$	3257.76	1.21502	0.607510	−	0.794312i	$-0.292169\pi$
$193$	3257.76	1.21502	0.607510	+	0.794312i	$0.292169\pi$
$194$	−1735.26	−0.642187
$195$	−521.127	−0.191378
$196$	0	0
$197$	−41.6955	−0.0150796	−0.00753981	−	0.999972i	$-0.502400\pi$
$197$	−41.6955	−0.0150796	−0.00753981	+	0.999972i	$0.502400\pi$
$198$	−10.3431	−0.00371240
$199$	2405.61	0.856932	0.428466	−	0.903558i	$-0.359054\pi$
$199$	2405.61	0.856932	0.428466	+	0.903558i	$0.359054\pi$
$200$	−384.619	−0.135983
$201$	−2401.44	−0.842708
$202$	−2006.00	−0.698722
$203$	0	0
$204$	−4259.35	−1.46183
$205$	−477.132	−0.162558
$206$	3106.28	1.05061
$207$	−2546.37	−0.854998
$208$	770.465	0.256837
$209$	−6.14459	−0.00203364
$210$	0	0
$211$	1211.24	0.395190	0.197595	−	0.980284i	$-0.436687\pi$
$211$	1211.24	0.395190	0.197595	+	0.980284i	$0.436687\pi$
$212$	−7773.45	−2.51831
$213$	120.060	0.0386214
$214$	−4215.56	−1.34659
$215$	−1700.36	−0.539364
$216$	2169.36	0.683363
$217$	0	0
$218$	−1275.38	−0.396238
$219$	262.256	0.0809205
$220$	−8.16234	−0.00250139
$221$	3676.02	1.11889
$222$	5730.25	1.73238
$223$	4164.99	1.25071	0.625354	−	0.780341i	$-0.284954\pi$
$223$	4164.99	1.25071	0.625354	+	0.780341i	$0.284954\pi$
$224$	0	0
$225$	−412.132	−0.122113
$226$	−5523.24	−1.62566
$227$	2356.24	0.688940	0.344470	−	0.938797i	$-0.388059\pi$
$227$	2356.24	0.688940	0.344470	+	0.938797i	$0.388059\pi$
$228$	1610.02	0.467658
$229$	4345.16	1.25387	0.626935	−	0.779071i	$-0.284309\pi$
$229$	4345.16	1.25387	0.626935	+	0.779071i	$0.284309\pi$
$230$	−3409.16	−0.977363
$231$	0	0
$232$	617.655	0.174789
$233$	2458.25	0.691182	0.345591	−	0.938385i	$-0.387679\pi$
$233$	2458.25	0.691182	0.345591	+	0.938385i	$0.387679\pi$
$234$	−2338.97	−0.653432
$235$	−37.4517	−0.0103961
$236$	9144.36	2.52223
$237$	−1029.08	−0.282050
$238$	0	0
$239$	322.304	0.0872307	0.0436154	−	0.999048i	$-0.486112\pi$
$239$	322.304	0.0872307	0.0436154	+	0.999048i	$0.486112\pi$
$240$	−388.640	−0.104527
$241$	−5011.24	−1.33943	−0.669714	−	0.742619i	$-0.733583\pi$
$241$	−5011.24	−1.33943	−0.669714	+	0.742619i	$0.733583\pi$
$242$	5875.23	1.56064
$243$	3767.85	0.994682
$244$	−8695.53	−2.28145
$245$	0	0
$246$	1365.91	0.354012
$247$	−1389.52	−0.357947
$248$	−1160.34	−0.297104
$249$	−3067.31	−0.780654
$250$	−551.777	−0.139590
$251$	3936.95	0.990031	0.495015	−	0.868884i	$-0.335162\pi$
$251$	3936.95	0.990031	0.495015	+	0.868884i	$0.335162\pi$
$252$	0	0
$253$	−21.9547	−0.00545565
$254$	8597.23	2.12377
$255$	−1854.26	−0.455367
$256$	−1318.94	−0.322008
$257$	−3626.16	−0.880131	−0.440066	−	0.897966i	$-0.645045\pi$
$257$	−3626.16	−0.880131	−0.440066	+	0.897966i	$0.645045\pi$
$258$	4867.68	1.17461
$259$	0	0
$260$	−1845.81	−0.440277
$261$	661.838	0.156961
$262$	−4686.33	−1.10505
$263$	−786.555	−0.184415	−0.0922074	−	0.995740i	$-0.529392\pi$
$263$	−786.555	−0.184415	−0.0922074	+	0.995740i	$0.529392\pi$
$264$	7.09076	0.00165305
$265$	−3384.09	−0.784465
$266$	0	0
$267$	2539.65	0.582113
$268$	−8505.78	−1.93871
$269$	4057.94	0.919766	0.459883	−	0.887979i	$-0.347891\pi$
$269$	4057.94	0.919766	0.459883	+	0.887979i	$0.347891\pi$
$270$	3112.18	0.701486
$271$	2899.42	0.649916	0.324958	−	0.945728i	$-0.394650\pi$
$271$	2899.42	0.649916	0.324958	+	0.945728i	$0.394650\pi$
$272$	2741.45	0.611122
$273$	0	0
$274$	6353.56	1.40085
$275$	−3.55339	−0.000779191 0
$276$	5752.61	1.25459
$277$	5293.96	1.14831	0.574157	−	0.818745i	$-0.305330\pi$
$277$	5293.96	1.14831	0.574157	+	0.818745i	$0.305330\pi$
$278$	−2922.79	−0.630566
$279$	−1243.34	−0.266799
$280$	0	0
$281$	2359.40	0.500890	0.250445	−	0.968131i	$-0.419423\pi$
$281$	2359.40	0.500890	0.250445	+	0.968131i	$0.419423\pi$
$282$	107.214	0.0226402
$283$	−2881.35	−0.605224	−0.302612	−	0.953114i	$-0.597859\pi$
$283$	−2881.35	−0.605224	−0.302612	+	0.953114i	$0.597859\pi$
$284$	425.245	0.0888510
$285$	700.904	0.145677
$286$	−20.1665	−0.00416948
$287$	0	0
$288$	−3773.31	−0.772028
$289$	8166.93	1.66231
$290$	886.091	0.179424
$291$	1274.70	0.256785
$292$	928.897	0.186163
$293$	760.730	0.151680	0.0758402	−	0.997120i	$-0.475836\pi$
$293$	760.730	0.151680	0.0758402	+	0.997120i	$0.475836\pi$
$294$	0	0
$295$	3980.90	0.785685
$296$	6159.03	1.20941
$297$	20.0421	0.00391570
$298$	3474.47	0.675405
$299$	−4964.77	−0.960268
$300$	931.066	0.179184
$301$	0	0
$302$	3744.82	0.713544
$303$	1473.59	0.279391
$304$	−1036.26	−0.195505
$305$	−3785.51	−0.710681
$306$	−8322.47	−1.55478
$307$	−4968.57	−0.923685	−0.461842	−	0.886962i	$-0.652811\pi$
$307$	−4968.57	−0.923685	−0.461842	+	0.886962i	$0.652811\pi$
$308$	0	0
$309$	−2281.84	−0.420096
$310$	−1664.63	−0.304983
$311$	−5033.96	−0.917845	−0.458922	−	0.888476i	$-0.651764\pi$
$311$	−5033.96	−0.917845	−0.458922	+	0.888476i	$0.651764\pi$
$312$	1603.48	0.290960
$313$	3599.20	0.649965	0.324983	−	0.945720i	$-0.394642\pi$
$313$	3599.20	0.649965	0.324983	+	0.945720i	$0.394642\pi$
$314$	−4215.23	−0.757577
$315$	0	0
$316$	−3644.95	−0.648875
$317$	3541.56	0.627488	0.313744	−	0.949508i	$-0.398417\pi$
$317$	3541.56	0.627488	0.313744	+	0.949508i	$0.398417\pi$
$318$	9687.78	1.70838
$319$	5.70635	0.00100155
$320$	−4093.01	−0.715020
$321$	3096.71	0.538447
$322$	0	0
$323$	−4944.16	−0.851704
$324$	−139.350	−0.0238940
$325$	−803.553	−0.137148
$326$	32.5072	0.00552272
$327$	936.886	0.158440
$328$	1468.11	0.247143
$329$	0	0
$330$	10.1724	0.00169689
$331$	−7357.96	−1.22184	−0.610922	−	0.791691i	$-0.709201\pi$
$331$	−7357.96	−1.22184	−0.610922	+	0.791691i	$0.709201\pi$
$332$	−10864.3	−1.79594
$333$	6599.60	1.08605
$334$	−12895.7	−2.11264
$335$	−3702.90	−0.603914
$336$	0	0
$337$	−2323.24	−0.375534	−0.187767	−	0.982214i	$-0.560125\pi$
$337$	−2323.24	−0.375534	−0.187767	+	0.982214i	$0.560125\pi$
$338$	5137.63	0.826776
$339$	4057.32	0.650039
$340$	−6567.72	−1.04760
$341$	−10.7201	−0.00170242
$342$	3145.86	0.497394
$343$	0	0
$344$	5231.92	0.820018
$345$	2504.34	0.390809
$346$	17522.5	2.72259
$347$	−5447.79	−0.842803	−0.421401	−	0.906874i	$-0.638462\pi$
$347$	−5447.79	−0.842803	−0.421401	+	0.906874i	$0.638462\pi$
$348$	−1495.19	−0.230317
$349$	−1227.84	−0.188324	−0.0941618	−	0.995557i	$-0.530017\pi$
$349$	−1227.84	−0.188324	−0.0941618	+	0.995557i	$0.530017\pi$
$350$	0	0
$351$	4532.27	0.689216
$352$	−32.5333	−0.00492623
$353$	1981.86	0.298821	0.149411	−	0.988775i	$-0.452262\pi$
$353$	1981.86	0.298821	0.149411	+	0.988775i	$0.452262\pi$
$354$	−11396.3	−1.71104
$355$	185.126	0.0276774
$356$	8995.33	1.33919
$357$	0	0
$358$	4881.90	0.720717
$359$	−12218.1	−1.79622	−0.898112	−	0.439768i	$-0.855061\pi$
$359$	−12218.1	−1.79622	−0.898112	+	0.439768i	$0.855061\pi$
$360$	1268.11	0.185654
$361$	−4990.13	−0.727530
$362$	517.406	0.0751222
$363$	−4315.89	−0.624037
$364$	0	0
$365$	404.386	0.0579905
$366$	10836.9	1.54769
$367$	13647.7	1.94115	0.970577	−	0.240789i	$-0.0774062\pi$
$367$	13647.7	1.94115	0.970577	+	0.240789i	$0.0774062\pi$
$368$	−3702.56	−0.524482
$369$	1573.13	0.221935
$370$	8835.78	1.24149
$371$	0	0
$372$	2808.89	0.391490
$373$	−5544.03	−0.769594	−0.384797	−	0.923001i	$-0.625729\pi$
$373$	−5544.03	−0.769594	−0.384797	+	0.923001i	$0.625729\pi$
$374$	−71.7561	−0.00992091
$375$	405.330	0.0558164
$376$	115.237	0.0158056
$377$	1290.42	0.176286
$378$	0	0
$379$	−634.243	−0.0859601	−0.0429801	−	0.999076i	$-0.513685\pi$
$379$	−634.243	−0.0859601	−0.0429801	+	0.999076i	$0.513685\pi$
$380$	2482.57	0.335140
$381$	−6315.45	−0.849213
$382$	−4425.69	−0.592769
$383$	5790.93	0.772592	0.386296	−	0.922375i	$-0.373754\pi$
$383$	5790.93	0.772592	0.386296	+	0.922375i	$0.373754\pi$
$384$	5779.58	0.768068
$385$	0	0
$386$	−14380.5	−1.89623
$387$	5606.17	0.736376
$388$	4514.94	0.590751
$389$	1516.99	0.197724	0.0988619	−	0.995101i	$-0.468480\pi$
$389$	1516.99	0.197724	0.0988619	+	0.995101i	$0.468480\pi$
$390$	2300.37	0.298676
$391$	−17665.6	−2.28487
$392$	0	0
$393$	3442.53	0.441865
$394$	184.053	0.0235341
$395$	−1586.79	−0.202127
$396$	26.9117	0.00341506
$397$	−7874.94	−0.995547	−0.497773	−	0.867307i	$-0.665849\pi$
$397$	−7874.94	−0.995547	−0.497773	+	0.867307i	$0.665849\pi$
$398$	−10618.9	−1.33738
$399$	0	0
$400$	−599.264	−0.0749080
$401$	−11111.3	−1.38372	−0.691859	−	0.722033i	$-0.743208\pi$
$401$	−11111.3	−1.38372	−0.691859	+	0.722033i	$0.743208\pi$
$402$	10600.5	1.31518
$403$	−2424.20	−0.299648
$404$	5219.39	0.642758
$405$	−60.6645	−0.00744307
$406$	0	0
$407$	56.9016	0.00692999
$408$	5705.49	0.692313
$409$	−10710.1	−1.29482	−0.647411	−	0.762141i	$-0.724148\pi$
$409$	−10710.1	−1.29482	−0.647411	+	0.762141i	$0.724148\pi$
$410$	2106.16	0.253697
$411$	−4667.27	−0.560145
$412$	−8082.19	−0.966458
$413$	0	0
$414$	11240.2	1.33436
$415$	−4729.64	−0.559444
$416$	−7356.99	−0.867082
$417$	2147.06	0.252139
$418$	27.1235	0.00317381
$419$	8615.98	1.00458	0.502289	−	0.864700i	$-0.332491\pi$
$419$	8615.98	1.00458	0.502289	+	0.864700i	$0.332491\pi$
$420$	0	0
$421$	6689.72	0.774435	0.387217	−	0.921988i	$-0.373436\pi$
$421$	6689.72	0.774435	0.387217	+	0.921988i	$0.373436\pi$
$422$	−5346.67	−0.616758
$423$	123.480	0.0141934
$424$	10412.7	1.19265
$425$	−2859.19	−0.326332
$426$	−529.969	−0.0602748
$427$	0	0
$428$	10968.4	1.23873
$429$	14.8141	0.00166721
$430$	7505.73	0.841764
$431$	6170.64	0.689627	0.344814	−	0.938671i	$-0.387942\pi$
$431$	6170.64	0.689627	0.344814	+	0.938671i	$0.387942\pi$
$432$	3380.02	0.376438
$433$	14001.1	1.55392	0.776961	−	0.629548i	$-0.216760\pi$
$433$	14001.1	1.55392	0.776961	+	0.629548i	$0.216760\pi$
$434$	0	0
$435$	−650.914	−0.0717447
$436$	3318.41	0.364502
$437$	6677.50	0.730957
$438$	−1157.65	−0.126289
$439$	12207.0	1.32712	0.663562	−	0.748121i	$-0.269044\pi$
$439$	12207.0	1.32712	0.663562	+	0.748121i	$0.269044\pi$
$440$	10.9336	0.00118464
$441$	0	0
$442$	−16226.7	−1.74621
$443$	1227.76	0.131677	0.0658384	−	0.997830i	$-0.479028\pi$
$443$	1227.76	0.131677	0.0658384	+	0.997830i	$0.479028\pi$
$444$	−14909.5	−1.59363
$445$	3916.03	0.417163
$446$	−18385.1	−1.95193
$447$	−2552.32	−0.270068
$448$	0	0
$449$	−1151.70	−0.121051	−0.0605257	−	0.998167i	$-0.519278\pi$
$449$	−1151.70	−0.121051	−0.0605257	+	0.998167i	$0.519278\pi$
$450$	1819.24	0.190577
$451$	13.5635	0.00141614
$452$	14370.8	1.49546
$453$	−2750.91	−0.285318
$454$	−10401.0	−1.07520
$455$	0	0
$456$	−2156.65	−0.221479
$457$	18713.8	1.91553	0.957763	−	0.287558i	$-0.0928433\pi$
$457$	18713.8	1.91553	0.957763	+	0.287558i	$0.0928433\pi$
$458$	−19180.5	−1.95687
$459$	16126.6	1.63993
$460$	8870.25	0.899082
$461$	−3154.57	−0.318705	−0.159352	−	0.987222i	$-0.550941\pi$
$461$	−3154.57	−0.318705	−0.159352	+	0.987222i	$0.550941\pi$
$462$	0	0
$463$	4051.11	0.406633	0.203316	−	0.979113i	$-0.434828\pi$
$463$	4051.11	0.406633	0.203316	+	0.979113i	$0.434828\pi$
$464$	962.351	0.0962845
$465$	1222.82	0.121951
$466$	−10851.2	−1.07870
$467$	16341.6	1.61927	0.809635	−	0.586934i	$-0.199665\pi$
$467$	16341.6	1.61927	0.809635	+	0.586934i	$0.199665\pi$
$468$	6085.73	0.601096
$469$	0	0
$470$	165.320	0.0162247
$471$	3096.47	0.302925
$472$	−12249.1	−1.19451
$473$	48.3362	0.00469873
$474$	4542.57	0.440184
$475$	1080.76	0.104397
$476$	0	0
$477$	11157.5	1.07100
$478$	−1422.72	−0.136138
$479$	2531.45	0.241472	0.120736	−	0.992685i	$-0.461475\pi$
$479$	2531.45	0.241472	0.120736	+	0.992685i	$0.461475\pi$
$480$	3711.03	0.352885
$481$	12867.6	1.21977
$482$	22120.7	2.09039
$483$	0	0
$484$	−15286.7	−1.43564
$485$	1965.53	0.184021
$486$	−16632.1	−1.55236
$487$	−8442.18	−0.785527	−0.392763	−	0.919640i	$-0.628481\pi$
$487$	−8442.18	−0.785527	−0.392763	+	0.919640i	$0.628481\pi$
$488$	11647.8	1.08048
$489$	−23.8795	−0.00220832
$490$	0	0
$491$	−15223.9	−1.39928	−0.699640	−	0.714496i	$-0.746656\pi$
$491$	−15223.9	−1.39928	−0.699640	+	0.714496i	$0.746656\pi$
$492$	−3553.93	−0.325658
$493$	4591.53	0.419457
$494$	6133.63	0.558634
$495$	11.7157	0.00106380
$496$	−1807.89	−0.163663
$497$	0	0
$498$	13539.8	1.21833
$499$	−16622.6	−1.49125	−0.745623	−	0.666368i	$-0.767848\pi$
$499$	−16622.6	−1.49125	−0.745623	+	0.666368i	$0.767848\pi$
$500$	1435.66	0.128409
$501$	9473.06	0.844760
$502$	−17378.5	−1.54510
$503$	−17506.6	−1.55185	−0.775923	−	0.630828i	$-0.782715\pi$
$503$	−17506.6	−1.55185	−0.775923	+	0.630828i	$0.782715\pi$
$504$	0	0
$505$	2272.21	0.200222
$506$	96.9127	0.00851442
$507$	−3774.05	−0.330595
$508$	−22369.0	−1.95367
$509$	12383.2	1.07834	0.539169	−	0.842198i	$-0.318739\pi$
$509$	12383.2	1.07834	0.539169	+	0.842198i	$0.318739\pi$
$510$	8185.12	0.710673
$511$	0	0
$512$	−8436.86	−0.728242
$513$	−6095.80	−0.524632
$514$	16006.7	1.37359
$515$	−3518.50	−0.301055
$516$	−12665.1	−1.08053
$517$	1.06464	9.05666e−5 0
$518$	0	0
$519$	−12871.9	−1.08866
$520$	2472.50	0.208512
$521$	−14979.0	−1.25958	−0.629790	−	0.776765i	$-0.716859\pi$
$521$	−14979.0	−1.25958	−0.629790	+	0.776765i	$0.716859\pi$
$522$	−2921.49	−0.244962
$523$	3719.11	0.310947	0.155474	−	0.987840i	$-0.450310\pi$
$523$	3719.11	0.310947	0.155474	+	0.987840i	$0.450310\pi$
$524$	12193.3	1.01654
$525$	0	0
$526$	3472.02	0.287809
$527$	−8625.75	−0.712986
$528$	11.0479	0.000910603 0
$529$	11691.8	0.960945
$530$	14938.1	1.22428
$531$	−13125.3	−1.07267
$532$	0	0
$533$	3067.21	0.249260
$534$	−11210.6	−0.908481
$535$	4774.99	0.385870
$536$	11393.7	0.918156
$537$	−3586.20	−0.288186
$538$	−17912.6	−1.43544
$539$	0	0
$540$	−8097.53	−0.645301
$541$	−244.726	−0.0194484	−0.00972420	−	0.999953i	$-0.503095\pi$
$541$	−244.726	−0.0194484	−0.00972420	+	0.999953i	$0.503095\pi$
$542$	−12798.7	−1.01430
$543$	−380.081	−0.0300384
$544$	−26177.5	−2.06315
$545$	1444.63	0.113544
$546$	0	0
$547$	−12685.4	−0.991568	−0.495784	−	0.868446i	$-0.665119\pi$
$547$	−12685.4	−0.991568	−0.495784	+	0.868446i	$0.665119\pi$
$548$	−16531.3	−1.28865
$549$	12481.0	0.970269
$550$	15.6854	0.00121605
$551$	−1735.58	−0.134189
$552$	−7705.74	−0.594163
$553$	0	0
$554$	−23368.7	−1.79213
$555$	−6490.68	−0.496422
$556$	7604.77	0.580062
$557$	−22062.7	−1.67833	−0.839164	−	0.543879i	$-0.816955\pi$
$557$	−22062.7	−1.67833	−0.839164	+	0.543879i	$0.816955\pi$
$558$	5488.38	0.416383
$559$	10930.6	0.827040
$560$	0	0
$561$	52.7114	0.00396698
$562$	−10414.9	−0.781719
$563$	7862.20	0.588547	0.294274	−	0.955721i	$-0.404922\pi$
$563$	7862.20	0.588547	0.294274	+	0.955721i	$0.404922\pi$
$564$	−278.960	−0.0208268
$565$	6256.19	0.465841
$566$	12718.9	0.944549
$567$	0	0
$568$	−569.625	−0.0420791
$569$	20941.5	1.54291	0.771453	−	0.636286i	$-0.219530\pi$
$569$	20941.5	1.54291	0.771453	+	0.636286i	$0.219530\pi$
$570$	−3093.94	−0.227352
$571$	−7131.66	−0.522681	−0.261341	−	0.965247i	$-0.584165\pi$
$571$	−7131.66	−0.522681	−0.261341	+	0.965247i	$0.584165\pi$
$572$	52.4710	0.00383553
$573$	3251.07	0.237025
$574$	0	0
$575$	3861.57	0.280067
$576$	13494.9	0.976193
$577$	−15633.0	−1.12792	−0.563961	−	0.825801i	$-0.690723\pi$
$577$	−15633.0	−1.12792	−0.563961	+	0.825801i	$0.690723\pi$
$578$	−36050.6	−2.59430
$579$	10563.7	0.758229
$580$	−2305.51	−0.165054
$581$	0	0
$582$	−5626.82	−0.400754
$583$	96.2000	0.00683396
$584$	−1244.28	−0.0881654
$585$	2649.36	0.187244
$586$	−3358.03	−0.236721
$587$	−19406.2	−1.36453	−0.682266	−	0.731104i	$-0.739005\pi$
$587$	−19406.2	−1.36453	−0.682266	+	0.731104i	$0.739005\pi$
$588$	0	0
$589$	3260.50	0.228093
$590$	−17572.6	−1.22619
$591$	−135.204	−0.00941038
$592$	9596.21	0.666219
$593$	−14859.6	−1.02903	−0.514513	−	0.857483i	$-0.672027\pi$
$593$	−14859.6	−1.02903	−0.514513	+	0.857483i	$0.672027\pi$
$594$	−88.4703	−0.00611108
$595$	0	0
$596$	−9040.18	−0.621309
$597$	7800.54	0.534765
$598$	21915.6	1.49865
$599$	−10529.9	−0.718261	−0.359131	−	0.933287i	$-0.616927\pi$
$599$	−10529.9	−0.718261	−0.359131	+	0.933287i	$0.616927\pi$
$600$	−1247.18	−0.0848600
$601$	11595.2	0.786984	0.393492	−	0.919328i	$-0.371267\pi$
$601$	11595.2	0.786984	0.393492	+	0.919328i	$0.371267\pi$
$602$	0	0
$603$	12208.7	0.824504
$604$	−9743.60	−0.656393
$605$	−6654.90	−0.447207
$606$	−6504.74	−0.436035
$607$	−23282.1	−1.55682	−0.778411	−	0.627755i	$-0.783974\pi$
$607$	−23282.1	−1.55682	−0.778411	+	0.627755i	$0.783974\pi$
$608$	9894.99	0.660024
$609$	0	0
$610$	16710.0	1.10913
$611$	240.755	0.0159409
$612$	21654.1	1.43026
$613$	−7387.98	−0.486782	−0.243391	−	0.969928i	$-0.578260\pi$
$613$	−7387.98	−0.486782	−0.243391	+	0.969928i	$0.578260\pi$
$614$	21932.3	1.44156
$615$	−1547.17	−0.101444
$616$	0	0
$617$	667.085	0.0435265	0.0217632	−	0.999763i	$-0.493072\pi$
$617$	667.085	0.0435265	0.0217632	+	0.999763i	$0.493072\pi$
$618$	10072.5	0.655627
$619$	−9425.78	−0.612042	−0.306021	−	0.952025i	$-0.598998\pi$
$619$	−9425.78	−0.612042	−0.306021	+	0.952025i	$0.598998\pi$
$620$	4331.18	0.280555
$621$	−21780.4	−1.40743
$622$	22221.0	1.43244
$623$	0	0
$624$	2498.34	0.160278
$625$	625.000	0.0400000
$626$	−15887.7	−1.01437
$627$	−19.9247	−0.00126908
$628$	10967.5	0.696899
$629$	45785.1	2.90234
$630$	0	0
$631$	28047.3	1.76949	0.884744	−	0.466078i	$-0.154333\pi$
$631$	28047.3	1.76949	0.884744	+	0.466078i	$0.154333\pi$
$632$	4882.49	0.307302
$633$	3927.61	0.246617
$634$	−15633.2	−0.979295
$635$	−9738.12	−0.608576
$636$	−25206.5	−1.57155
$637$	0	0
$638$	−25.1890	−0.00156308
$639$	−610.372	−0.0377871
$640$	8911.85	0.550425
$641$	23208.3	1.43007	0.715034	−	0.699090i	$-0.246411\pi$
$641$	23208.3	1.43007	0.715034	+	0.699090i	$0.246411\pi$
$642$	−13669.5	−0.840333
$643$	−4294.22	−0.263371	−0.131686	−	0.991292i	$-0.542039\pi$
$643$	−4294.22	−0.263371	−0.131686	+	0.991292i	$0.542039\pi$
$644$	0	0
$645$	−5513.64	−0.336588
$646$	21824.6	1.32922
$647$	−17393.4	−1.05689	−0.528443	−	0.848969i	$-0.677224\pi$
$647$	−17393.4	−1.05689	−0.528443	+	0.848969i	$0.677224\pi$
$648$	186.662	0.0113160
$649$	−113.166	−0.00684459
$650$	3547.06	0.214042
$651$	0	0
$652$	−84.5801	−0.00508039
$653$	14475.4	0.867486	0.433743	−	0.901037i	$-0.357193\pi$
$653$	14475.4	0.867486	0.433743	+	0.901037i	$0.357193\pi$
$654$	−4135.61	−0.247271
$655$	5308.23	0.316656
$656$	2287.42	0.136142
$657$	−1333.28	−0.0791725
$658$	0	0
$659$	620.113	0.0366558	0.0183279	−	0.999832i	$-0.494166\pi$
$659$	620.113	0.0366558	0.0183279	+	0.999832i	$0.494166\pi$
$660$	−26.4675	−0.00156098
$661$	6105.50	0.359268	0.179634	−	0.983733i	$-0.442509\pi$
$661$	6105.50	0.359268	0.179634	+	0.983733i	$0.442509\pi$
$662$	32479.6	1.90688
$663$	11920.0	0.698242
$664$	14552.9	0.850546
$665$	0	0
$666$	−29132.1	−1.69496
$667$	−6201.25	−0.359990
$668$	33553.1	1.94343
$669$	13505.6	0.780500
$670$	16345.4	0.942505
$671$	107.611	0.00619118
$672$	0	0
$673$	−16302.5	−0.933754	−0.466877	−	0.884322i	$-0.654621\pi$
$673$	−16302.5	−0.933754	−0.466877	+	0.884322i	$0.654621\pi$
$674$	10255.3	0.586081
$675$	−3525.18	−0.201014
$676$	−13367.5	−0.760556
$677$	−4819.09	−0.273579	−0.136789	−	0.990600i	$-0.543678\pi$
$677$	−4819.09	−0.273579	−0.136789	+	0.990600i	$0.543678\pi$
$678$	−17909.9	−1.01449
$679$	0	0
$680$	8797.59	0.496136
$681$	7640.45	0.429930
$682$	47.3206	0.00265689
$683$	−8190.32	−0.458849	−0.229425	−	0.973326i	$-0.573684\pi$
$683$	−8190.32	−0.458849	−0.229425	+	0.973326i	$0.573684\pi$
$684$	−8185.17	−0.457555
$685$	−7196.71	−0.401419
$686$	0	0
$687$	14089.8	0.782473
$688$	8151.69	0.451716
$689$	21754.4	1.20287
$690$	−11054.7	−0.609920
$691$	476.973	0.0262589	0.0131295	−	0.999914i	$-0.495821\pi$
$691$	476.973	0.0262589	0.0131295	+	0.999914i	$0.495821\pi$
$692$	−45591.6	−2.50453
$693$	0	0
$694$	24047.7	1.31533
$695$	3310.66	0.180691
$696$	2002.83	0.109077
$697$	10913.7	0.593092
$698$	5419.96	0.293909
$699$	7971.22	0.431329
$700$	0	0
$701$	−20366.7	−1.09735	−0.548673	−	0.836037i	$-0.684867\pi$
$701$	−20366.7	−1.09735	−0.548673	+	0.836037i	$0.684867\pi$
$702$	−20006.4	−1.07563
$703$	−17306.6	−0.928492
$704$	116.353	0.00622898
$705$	−121.442	−0.00648763
$706$	−8748.36	−0.466358
$707$	0	0
$708$	29651.9	1.57399
$709$	22415.1	1.18733	0.593664	−	0.804713i	$-0.297681\pi$
$709$	22415.1	1.18733	0.593664	+	0.804713i	$0.297681\pi$
$710$	−817.187	−0.0431950
$711$	5231.74	0.275957
$712$	−12049.4	−0.634230
$713$	11649.8	0.611906
$714$	0	0
$715$	22.8427	0.00119478
$716$	−12702.2	−0.662992
$717$	1045.12	0.0544360
$718$	53933.1	2.80329
$719$	−26156.0	−1.35668	−0.678340	−	0.734748i	$-0.737300\pi$
$719$	−26156.0	−1.35668	−0.678340	+	0.734748i	$0.737300\pi$
$720$	1975.81	0.102269
$721$	0	0
$722$	22027.5	1.13543
$723$	−16249.6	−0.835866
$724$	−1346.23	−0.0691053
$725$	−1003.68	−0.0514148
$726$	19051.3	0.973910
$727$	−13666.2	−0.697184	−0.348592	−	0.937275i	$-0.613340\pi$
$727$	−13666.2	−0.697184	−0.348592	+	0.937275i	$0.613340\pi$
$728$	0	0
$729$	12545.4	0.637371
$730$	−1785.05	−0.0905035
$731$	38893.1	1.96787
$732$	−28196.5	−1.42373
$733$	25038.2	1.26167	0.630836	−	0.775916i	$-0.282712\pi$
$733$	25038.2	1.26167	0.630836	+	0.775916i	$0.282712\pi$
$734$	−60243.8	−3.02948
$735$	0	0
$736$	35354.9	1.77065
$737$	105.263	0.00526107
$738$	−6944.14	−0.346365
$739$	3255.20	0.162036	0.0810178	−	0.996713i	$-0.474183\pi$
$739$	3255.20	0.162036	0.0810178	+	0.996713i	$0.474183\pi$
$740$	−22989.7	−1.14205
$741$	−4505.71	−0.223376
$742$	0	0
$743$	−8366.48	−0.413104	−0.206552	−	0.978436i	$-0.566224\pi$
$743$	−8366.48	−0.413104	−0.206552	+	0.978436i	$0.566224\pi$
$744$	−3762.57	−0.185406
$745$	−3935.55	−0.193540
$746$	24472.5	1.20108
$747$	15593.9	0.763790
$748$	186.701	0.00912630
$749$	0	0
$750$	−1789.21	−0.0871105
$751$	−10164.5	−0.493884	−0.246942	−	0.969030i	$-0.579426\pi$
$751$	−10164.5	−0.493884	−0.246942	+	0.969030i	$0.579426\pi$
$752$	179.547	0.00870668
$753$	12766.1	0.617825
$754$	−5696.17	−0.275123
$755$	−4241.77	−0.204469
$756$	0	0
$757$	9031.23	0.433614	0.216807	−	0.976214i	$-0.430436\pi$
$757$	9031.23	0.433614	0.216807	+	0.976214i	$0.430436\pi$
$758$	2799.68	0.134155
$759$	−71.1912	−0.00340458
$760$	−3325.45	−0.158720
$761$	−32434.7	−1.54502	−0.772508	−	0.635005i	$-0.780998\pi$
$761$	−32434.7	−1.54502	−0.772508	+	0.635005i	$0.780998\pi$
$762$	27877.7	1.32533
$763$	0	0
$764$	11515.1	0.545292
$765$	9426.90	0.445530
$766$	−25562.4	−1.20575
$767$	−25591.0	−1.20474
$768$	−4276.86	−0.200948
$769$	−22629.4	−1.06117	−0.530583	−	0.847633i	$-0.678027\pi$
$769$	−22629.4	−1.06117	−0.530583	+	0.847633i	$0.678027\pi$
$770$	0	0
$771$	−11758.3	−0.549243
$772$	37416.3	1.74436
$773$	2707.01	0.125956	0.0629782	−	0.998015i	$-0.479940\pi$
$773$	2707.01	0.125956	0.0629782	+	0.998015i	$0.479940\pi$
$774$	−24746.8	−1.14923
$775$	1885.53	0.0873940
$776$	−6047.86	−0.279775
$777$	0	0
$778$	−6696.33	−0.308580
$779$	−4125.33	−0.189737
$780$	−5985.29	−0.274754
$781$	−5.26261	−0.000241115 0
$782$	77979.5	3.56591
$783$	5661.04	0.258377
$784$	0	0
$785$	4774.61	0.217087
$786$	−15196.1	−0.689601
$787$	−25084.6	−1.13618	−0.568088	−	0.822968i	$-0.692317\pi$
$787$	−25084.6	−1.13618	−0.568088	+	0.822968i	$0.692317\pi$
$788$	−478.885	−0.0216492
$789$	−2550.52	−0.115083
$790$	7004.44	0.315451
$791$	0	0
$792$	−36.0488	−0.00161735
$793$	24334.9	1.08973
$794$	34761.7	1.55371
$795$	−10973.4	−0.489542
$796$	27629.1	1.23026
$797$	7374.99	0.327774	0.163887	−	0.986479i	$-0.447597\pi$
$797$	7374.99	0.327774	0.163887	+	0.986479i	$0.447597\pi$
$798$	0	0
$799$	856.651	0.0379301
$800$	5722.23	0.252889
$801$	−12911.4	−0.569539
$802$	49047.5	2.15951
$803$	−11.4955	−0.000505191 0
$804$	−27581.2	−1.20984
$805$	0	0
$806$	10701.0	0.467649
$807$	13158.5	0.573977
$808$	−6991.48	−0.304405
$809$	2314.37	0.100580	0.0502899	−	0.998735i	$-0.483985\pi$
$809$	2314.37	0.100580	0.0502899	+	0.998735i	$0.483985\pi$
$810$	267.786	0.0116161
$811$	38300.1	1.65832	0.829160	−	0.559011i	$-0.188819\pi$
$811$	38300.1	1.65832	0.829160	+	0.559011i	$0.188819\pi$
$812$	0	0
$813$	9401.78	0.405578
$814$	−251.176	−0.0108154
$815$	−36.8211	−0.00158256
$816$	8889.55	0.381368
$817$	−14701.4	−0.629544
$818$	47276.8	2.02078
$819$	0	0
$820$	−5480.00	−0.233378
$821$	3942.93	0.167612	0.0838059	−	0.996482i	$-0.473292\pi$
$821$	3942.93	0.167612	0.0838059	+	0.996482i	$0.473292\pi$
$822$	20602.3	0.874195
$823$	18502.8	0.783679	0.391839	−	0.920034i	$-0.371839\pi$
$823$	18502.8	0.783679	0.391839	+	0.920034i	$0.371839\pi$
$824$	10826.3	0.457707
$825$	−11.5224	−0.000486251 0
$826$	0	0
$827$	−11965.2	−0.503110	−0.251555	−	0.967843i	$-0.580942\pi$
$827$	−11965.2	−0.503110	−0.251555	+	0.967843i	$0.580942\pi$
$828$	−29245.7	−1.22749
$829$	−35962.9	−1.50669	−0.753344	−	0.657626i	$-0.771561\pi$
$829$	−35962.9	−1.50669	−0.753344	+	0.657626i	$0.771561\pi$
$830$	20877.7	0.873101
$831$	17166.4	0.716602
$832$	26311.6	1.09638
$833$	0	0
$834$	−9477.56	−0.393503
$835$	14607.0	0.605385
$836$	−70.5723	−0.00291961
$837$	−10634.9	−0.439185
$838$	−38032.8	−1.56780
$839$	−20174.0	−0.830137	−0.415069	−	0.909790i	$-0.636242\pi$
$839$	−20174.0	−0.830137	−0.415069	+	0.909790i	$0.636242\pi$
$840$	0	0
$841$	−22777.2	−0.933913
$842$	−29529.9	−1.20863
$843$	7650.69	0.312579
$844$	13911.4	0.567359
$845$	−5819.42	−0.236916
$846$	−545.068	−0.0221511
$847$	0	0
$848$	16223.7	0.656987
$849$	−9343.17	−0.377688
$850$	12621.1	0.509293
$851$	−61836.6	−2.49087
$852$	1378.92	0.0554471
$853$	25297.5	1.01544	0.507719	−	0.861523i	$-0.330489\pi$
$853$	25297.5	1.01544	0.507719	+	0.861523i	$0.330489\pi$
$854$	0	0
$855$	−3563.33	−0.142530
$856$	−14692.4	−0.586655
$857$	39269.1	1.56523	0.782617	−	0.622504i	$-0.213884\pi$
$857$	39269.1	1.56523	0.782617	+	0.622504i	$0.213884\pi$
$858$	−65.3928	−0.00260195
$859$	−14499.1	−0.575905	−0.287953	−	0.957645i	$-0.592975\pi$
$859$	−14499.1	−0.575905	−0.287953	+	0.957645i	$0.592975\pi$
$860$	−19529.1	−0.774343
$861$	0	0
$862$	−27238.5	−1.07627
$863$	−24411.3	−0.962884	−0.481442	−	0.876478i	$-0.659887\pi$
$863$	−24411.3	−0.962884	−0.481442	+	0.876478i	$0.659887\pi$
$864$	−32275.0	−1.27086
$865$	−19847.8	−0.780170
$866$	−61803.7	−2.42515
$867$	26482.4	1.03736
$868$	0	0
$869$	45.1079	0.00176085
$870$	2873.28	0.111969
$871$	23803.8	0.926019
$872$	−4445.07	−0.172625
$873$	−6480.48	−0.251238
$874$	−29475.9	−1.14078
$875$	0	0
$876$	3012.08	0.116174
$877$	−5056.65	−0.194699	−0.0973494	−	0.995250i	$-0.531036\pi$
$877$	−5056.65	−0.194699	−0.0973494	+	0.995250i	$0.531036\pi$
$878$	−53884.2	−2.07119
$879$	2466.77	0.0946556
$880$	17.0354	0.000652570 0
$881$	13233.9	0.506086	0.253043	−	0.967455i	$-0.418569\pi$
$881$	13233.9	0.506086	0.253043	+	0.967455i	$0.418569\pi$
$882$	0	0
$883$	−13824.2	−0.526866	−0.263433	−	0.964678i	$-0.584855\pi$
$883$	−13824.2	−0.526866	−0.263433	+	0.964678i	$0.584855\pi$
$884$	42220.1	1.60635
$885$	12908.6	0.490304
$886$	−5419.61	−0.205503
$887$	−37819.3	−1.43162	−0.715811	−	0.698294i	$-0.753943\pi$
$887$	−37819.3	−1.43162	−0.715811	+	0.698294i	$0.753943\pi$
$888$	19971.5	0.754731
$889$	0	0
$890$	−17286.2	−0.651049
$891$	1.72452	6.48412e−5 0
$892$	47836.0	1.79559
$893$	−323.810	−0.0121343
$894$	11266.5	0.421484
$895$	−5529.76	−0.206524
$896$	0	0
$897$	−16099.0	−0.599252
$898$	5083.85	0.188920
$899$	−3027.96	−0.112334
$900$	−4733.45	−0.175313
$901$	77406.0	2.86212
$902$	−59.8721	−0.00221012
$903$	0	0
$904$	−19250.0	−0.708237
$905$	−586.068	−0.0215266
$906$	12143.1	0.445284
$907$	1052.75	0.0385404	0.0192702	−	0.999814i	$-0.493866\pi$
$907$	1052.75	0.0385404	0.0192702	+	0.999814i	$0.493866\pi$
$908$	27062.1	0.989083
$909$	−7491.60	−0.273356
$910$	0	0
$911$	7854.40	0.285651	0.142825	−	0.989748i	$-0.454381\pi$
$911$	7854.40	0.285651	0.142825	+	0.989748i	$0.454381\pi$
$912$	−3360.21	−0.122004
$913$	134.450	0.00487366
$914$	−82606.8	−2.98949
$915$	−12275.0	−0.443498
$916$	49905.4	1.80013
$917$	0	0
$918$	−71186.4	−2.55937
$919$	4935.85	0.177169	0.0885846	−	0.996069i	$-0.471766\pi$
$919$	4935.85	0.177169	0.0885846	+	0.996069i	$0.471766\pi$
$920$	−11881.9	−0.425798
$921$	−16111.3	−0.576422
$922$	13924.9	0.497390
$923$	−1190.07	−0.0424395
$924$	0	0
$925$	−10008.3	−0.355753
$926$	−17882.4	−0.634615
$927$	11600.7	0.411021
$928$	−9189.27	−0.325056
$929$	−15705.0	−0.554645	−0.277322	−	0.960777i	$-0.589447\pi$
$929$	−15705.0	−0.554645	−0.277322	+	0.960777i	$0.589447\pi$
$930$	−5397.80	−0.190323
$931$	0	0
$932$	28233.7	0.992302
$933$	−16323.3	−0.572778
$934$	−72135.3	−2.52713
$935$	81.2785	0.00284288
$936$	−8151.96	−0.284674
$937$	−43806.5	−1.52732	−0.763658	−	0.645621i	$-0.776599\pi$
$937$	−43806.5	−1.52732	−0.763658	+	0.645621i	$0.776599\pi$
$938$	0	0
$939$	11670.9	0.405608
$940$	−430.143	−0.0149252
$941$	4062.42	0.140734	0.0703672	−	0.997521i	$-0.477583\pi$
$941$	4062.42	0.140734	0.0703672	+	0.997521i	$0.477583\pi$
$942$	−13668.5	−0.472763
$943$	−14739.8	−0.509009
$944$	−19084.9	−0.658009
$945$	0	0
$946$	−213.366	−0.00733313
$947$	−4860.13	−0.166772	−0.0833859	−	0.996517i	$-0.526573\pi$
$947$	−4860.13	−0.166772	−0.0833859	+	0.996517i	$0.526573\pi$
$948$	−11819.3	−0.404928
$949$	−2599.57	−0.0889204
$950$	−4770.71	−0.162929
$951$	11484.0	0.391581
$952$	0	0
$953$	−18999.6	−0.645811	−0.322906	−	0.946431i	$-0.604660\pi$
$953$	−18999.6	−0.645811	−0.322906	+	0.946431i	$0.604660\pi$
$954$	−49251.8	−1.67147
$955$	5013.00	0.169861
$956$	3701.76	0.125234
$957$	18.5036	0.000625013 0
$958$	−11174.4	−0.376855
$959$	0	0
$960$	−13272.2	−0.446205
$961$	−24102.6	−0.809057
$962$	−56800.2	−1.90365
$963$	−15743.4	−0.526816
$964$	−57555.5	−1.92296
$965$	16288.8	0.543373
$966$	0	0
$967$	−32695.1	−1.08728	−0.543641	−	0.839318i	$-0.682955\pi$
$967$	−32695.1	−1.08728	−0.543641	+	0.839318i	$0.682955\pi$
$968$	20476.8	0.679907
$969$	−16032.1	−0.531503
$970$	−8676.29	−0.287195
$971$	−6189.69	−0.204569	−0.102285	−	0.994755i	$-0.532615\pi$
$971$	−6189.69	−0.204569	−0.102285	+	0.994755i	$0.532615\pi$
$972$	43274.8	1.42803
$973$	0	0
$974$	37265.6	1.22594
$975$	−2605.63	−0.0855868
$976$	18148.2	0.595193
$977$	2142.44	0.0701563	0.0350782	−	0.999385i	$-0.488832\pi$
$977$	2142.44	0.0701563	0.0350782	+	0.999385i	$0.488832\pi$
$978$	105.409	0.00344644
$979$	−111.321	−0.00363416
$980$	0	0
$981$	−4763.04	−0.155018
$982$	67201.7	2.18380
$983$	16808.2	0.545371	0.272685	−	0.962103i	$-0.412088\pi$
$983$	16808.2	0.545371	0.272685	+	0.962103i	$0.412088\pi$
$984$	4760.57	0.154229
$985$	−208.478	−0.00674381
$986$	−20268.0	−0.654630
$987$	0	0
$988$	−15959.0	−0.513891
$989$	−52528.4	−1.68888
$990$	−51.7157	−0.00166024
$991$	21250.8	0.681185	0.340593	−	0.940211i	$-0.389372\pi$
$991$	21250.8	0.681185	0.340593	+	0.940211i	$0.389372\pi$
$992$	17263.1	0.552526
$993$	−23859.2	−0.762487
$994$	0	0
$995$	12028.1	0.383231
$996$	−35228.9	−1.12075
$997$	−37639.9	−1.19566	−0.597828	−	0.801624i	$-0.703969\pi$
$997$	−37639.9	−1.19566	−0.597828	+	0.801624i	$0.703969\pi$
$998$	73375.9	2.32733
$999$	56449.8	1.78778

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.4.a.g.1.1		2
3.2	odd	2		2205.4.a.bf.1.2		2
5.4	even	2		1225.4.a.x.1.2		2
7.2	even	3		35.4.e.b.11.2	✓	4
7.3	odd	6		245.4.e.l.226.2		4
7.4	even	3		35.4.e.b.16.2	yes	4
7.5	odd	6		245.4.e.l.116.2		4
7.6	odd	2		245.4.a.h.1.1		2
21.2	odd	6		315.4.j.c.46.1		4
21.11	odd	6		315.4.j.c.226.1		4
21.20	even	2		2205.4.a.bg.1.2		2
28.11	odd	6		560.4.q.i.401.2		4
28.23	odd	6		560.4.q.i.81.2		4
35.2	odd	12		175.4.k.c.74.1		8
35.4	even	6		175.4.e.c.51.1		4
35.9	even	6		175.4.e.c.151.1		4
35.18	odd	12		175.4.k.c.149.1		8
35.23	odd	12		175.4.k.c.74.4		8
35.32	odd	12		175.4.k.c.149.4		8
35.34	odd	2		1225.4.a.v.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.4.e.b.11.2	✓	4	7.2	even	3
35.4.e.b.16.2	yes	4	7.4	even	3
175.4.e.c.51.1		4	35.4	even	6
175.4.e.c.151.1		4	35.9	even	6
175.4.k.c.74.1		8	35.2	odd	12
175.4.k.c.74.4		8	35.23	odd	12
175.4.k.c.149.1		8	35.18	odd	12
175.4.k.c.149.4		8	35.32	odd	12
245.4.a.g.1.1		2	1.1	even	1	trivial
245.4.a.h.1.1		2	7.6	odd	2
245.4.e.l.116.2		4	7.5	odd	6
245.4.e.l.226.2		4	7.3	odd	6
315.4.j.c.46.1		4	21.2	odd	6
315.4.j.c.226.1		4	21.11	odd	6
560.4.q.i.81.2		4	28.23	odd	6
560.4.q.i.401.2		4	28.11	odd	6
1225.4.a.v.1.2		2	35.34	odd	2
1225.4.a.x.1.2		2	5.4	even	2
2205.4.a.bf.1.2		2	3.2	odd	2
2205.4.a.bg.1.2		2	21.20	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 245 = 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	245.a (trivial)

\(f(q)\)	\(=\)	\(q-4.41421 q^{2} +3.24264 q^{3} +11.4853 q^{4} +5.00000 q^{5} -14.3137 q^{6} -15.3848 q^{8} -16.4853 q^{9} -22.0711 q^{10} -0.142136 q^{11} +37.2426 q^{12} -32.1421 q^{13} +16.2132 q^{15} -23.9706 q^{16} -114.368 q^{17} +72.7696 q^{18} +43.2304 q^{19} +57.4264 q^{20} +0.627417 q^{22} +154.463 q^{23} -49.8873 q^{24} +25.0000 q^{25} +141.882 q^{26} -141.007 q^{27} -40.1472 q^{29} -71.5685 q^{30} +75.4214 q^{31} +228.889 q^{32} -0.460895 q^{33} +504.843 q^{34} -189.338 q^{36} -400.333 q^{37} -190.828 q^{38} -104.225 q^{39} -76.9239 q^{40} -95.4264 q^{41} -340.071 q^{43} -1.63247 q^{44} -82.4264 q^{45} -681.833 q^{46} -7.49033 q^{47} -77.7279 q^{48} -110.355 q^{50} -370.853 q^{51} -369.161 q^{52} -676.818 q^{53} +622.436 q^{54} -0.710678 q^{55} +140.181 q^{57} +177.218 q^{58} +796.181 q^{59} +186.213 q^{60} -757.102 q^{61} -332.926 q^{62} -818.602 q^{64} -160.711 q^{65} +2.03449 q^{66} -740.581 q^{67} -1313.54 q^{68} +500.868 q^{69} +37.0253 q^{71} +253.622 q^{72} +80.8772 q^{73} +1767.16 q^{74} +81.0660 q^{75} +496.514 q^{76} +460.073 q^{78} -317.358 q^{79} -119.853 q^{80} -12.1329 q^{81} +421.233 q^{82} -945.929 q^{83} -571.838 q^{85} +1501.15 q^{86} -130.183 q^{87} +2.18672 q^{88} +783.205 q^{89} +363.848 q^{90} +1774.05 q^{92} +244.564 q^{93} +33.0639 q^{94} +216.152 q^{95} +742.206 q^{96} +393.107 q^{97} +2.34315 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{2} - 2 q^{3} + 6 q^{4} + 10 q^{5} - 6 q^{6} + 6 q^{8} - 16 q^{9} - 30 q^{10} + 28 q^{11} + 66 q^{12} - 36 q^{13} - 10 q^{15} - 14 q^{16} - 76 q^{17} + 72 q^{18} + 160 q^{19} + 30 q^{20} - 44 q^{22}+ \cdots + 16 q^{99}+O(q^{100}) \) 2 * q - 6 * q^2 - 2 * q^3 + 6 * q^4 + 10 * q^5 - 6 * q^6 + 6 * q^8 - 16 * q^9 - 30 * q^10 + 28 * q^11 + 66 * q^12 - 36 * q^13 - 10 * q^15 - 14 * q^16 - 76 * q^17 + 72 * q^18 + 160 * q^19 + 30 * q^20 - 44 * q^22 - 22 * q^23 - 162 * q^24 + 50 * q^25 + 148 * q^26 - 2 * q^27 - 250 * q^29 - 30 * q^30 - 132 * q^31 + 42 * q^32 - 148 * q^33 + 444 * q^34 - 192 * q^36 - 416 * q^37 - 376 * q^38 - 84 * q^39 + 30 * q^40 - 106 * q^41 - 666 * q^43 - 156 * q^44 - 80 * q^45 - 402 * q^46 - 196 * q^47 - 130 * q^48 - 150 * q^50 - 572 * q^51 - 348 * q^52 - 952 * q^53 + 402 * q^54 + 140 * q^55 - 472 * q^57 + 510 * q^58 + 840 * q^59 + 330 * q^60 + 98 * q^61 - 4 * q^62 - 602 * q^64 - 180 * q^65 + 236 * q^66 - 1286 * q^67 - 1524 * q^68 + 1426 * q^69 + 1064 * q^71 + 264 * q^72 - 172 * q^73 + 1792 * q^74 - 50 * q^75 - 144 * q^76 + 428 * q^78 - 1240 * q^79 - 70 * q^80 - 754 * q^81 + 438 * q^82 - 1906 * q^83 - 380 * q^85 + 2018 * q^86 + 970 * q^87 + 604 * q^88 + 650 * q^89 + 360 * q^90 + 2742 * q^92 + 1332 * q^93 + 332 * q^94 + 800 * q^95 + 1722 * q^96 - 628 * q^97 + 16 * q^99

Introduction

L-functions

Modular forms

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Database

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