LMFDB - Embedded newform 325.6.a.h.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,6,Mod(1,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("325.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	325.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:	$52.1247414392$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4 x^{8} - 181 x^{7} + 688 x^{6} + 10455 x^{5} - 37904 x^{4} - 197375 x^{3} + 702868 x^{2} + \cdots - 366960 $ x^9 - 4x^8 - 181x^7 + 688x^6 + 10455x^5 - 37904x^4 - 197375x^3 + 702868x^2 + 262812x - 366960
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 5 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$4.20685$ of defining polynomial
Character	$\chi$	$=$	325.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+3.20685 q^{2} -29.5845 q^{3} -21.7161 q^{4} -94.8732 q^{6} -11.5734 q^{7} -172.260 q^{8} +632.245 q^{9} -596.700 q^{11} +642.461 q^{12} +169.000 q^{13} -37.1142 q^{14} +142.504 q^{16} +2093.43 q^{17} +2027.52 q^{18} +35.4251 q^{19} +342.394 q^{21} -1913.53 q^{22} +2783.24 q^{23} +5096.22 q^{24} +541.958 q^{26} -11515.6 q^{27} +251.329 q^{28} -370.244 q^{29} +5055.06 q^{31} +5969.30 q^{32} +17653.1 q^{33} +6713.33 q^{34} -13729.9 q^{36} +4124.15 q^{37} +113.603 q^{38} -4999.79 q^{39} -18104.1 q^{41} +1098.01 q^{42} -7906.34 q^{43} +12958.0 q^{44} +8925.44 q^{46} -13185.2 q^{47} -4215.93 q^{48} -16673.1 q^{49} -61933.3 q^{51} -3670.02 q^{52} +38295.0 q^{53} -36929.0 q^{54} +1993.63 q^{56} -1048.03 q^{57} -1187.32 q^{58} +17886.6 q^{59} +7392.00 q^{61} +16210.8 q^{62} -7317.23 q^{63} +14582.5 q^{64} +56610.8 q^{66} -23357.0 q^{67} -45461.2 q^{68} -82340.9 q^{69} -33166.9 q^{71} -108910. q^{72} -10831.7 q^{73} +13225.5 q^{74} -769.295 q^{76} +6905.85 q^{77} -16033.6 q^{78} -17755.2 q^{79} +187049. q^{81} -58057.3 q^{82} -84243.8 q^{83} -7435.46 q^{84} -25354.5 q^{86} +10953.5 q^{87} +102787. q^{88} -46498.8 q^{89} -1955.91 q^{91} -60441.1 q^{92} -149552. q^{93} -42283.1 q^{94} -176599. q^{96} +154213. q^{97} -53468.0 q^{98} -377260. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 5 q^{2} - 11 q^{3} + 91 q^{4} - 83 q^{6} + 12 q^{7} - 639 q^{8} + 562 q^{9} - 1422 q^{11} + 1567 q^{12} + 1521 q^{13} - 342 q^{14} - 1061 q^{16} + 648 q^{17} + 418 q^{18} - 408 q^{19} - 3912 q^{21}+ \cdots - 757776 q^{99}+O(q^{100}) $ 9 * q - 5 * q^2 - 11 * q^3 + 91 * q^4 - 83 * q^6 + 12 * q^7 - 639 * q^8 + 562 * q^9 - 1422 * q^11 + 1567 * q^12 + 1521 * q^13 - 342 * q^14 - 1061 * q^16 + 648 * q^17 + 418 * q^18 - 408 * q^19 - 3912 * q^21 + 4345 * q^22 + 1839 * q^23 - 8469 * q^24 - 845 * q^26 - 7649 * q^27 - 2836 * q^28 - 8737 * q^29 + 748 * q^31 - 423 * q^32 - 356 * q^33 - 17789 * q^34 + 512 * q^36 - 15486 * q^37 - 3425 * q^38 - 1859 * q^39 - 28676 * q^41 + 6876 * q^42 + 28665 * q^43 - 30599 * q^44 - 12056 * q^46 - 29452 * q^47 + 64759 * q^48 - 40907 * q^49 - 31006 * q^51 + 15379 * q^52 + 75977 * q^53 - 102761 * q^54 - 23002 * q^56 - 38038 * q^57 + 142384 * q^58 - 88142 * q^59 + 28165 * q^61 - 137308 * q^62 + 41492 * q^63 - 100845 * q^64 + 42577 * q^66 - 94754 * q^67 + 89267 * q^68 - 181747 * q^69 - 70562 * q^71 - 263778 * q^72 + 60602 * q^73 - 135676 * q^74 + 46373 * q^76 - 140292 * q^77 - 14027 * q^78 - 164073 * q^79 - 69935 * q^81 - 72887 * q^82 - 22458 * q^83 - 345656 * q^84 - 294920 * q^86 - 87031 * q^87 + 430607 * q^88 - 252698 * q^89 + 2028 * q^91 - 237824 * q^92 + 56556 * q^93 - 501606 * q^94 - 319181 * q^96 + 137986 * q^97 + 378699 * q^98 - 757776 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	3.20685	0.566897	0.283448	−	0.958988i	$-0.408522\pi$
$2$	3.20685	0.566897	0.283448	+	0.958988i	$0.408522\pi$
$3$	−29.5845	−1.89785	−0.948925	−	0.315503i	$-0.897827\pi$
$3$	−29.5845	−1.89785	−0.948925	+	0.315503i	$0.897827\pi$
$4$	−21.7161	−0.678628
$5$	0	0
$5$	0	0
$6$	−94.8732	−1.07588
$7$	−11.5734	−0.0892722	−0.0446361	−	0.999003i	$-0.514213\pi$
$7$	−11.5734	−0.0892722	−0.0446361	+	0.999003i	$0.514213\pi$
$8$	−172.260	−0.951609
$9$	632.245	2.60183
$10$	0	0
$11$	−596.700	−1.48687	−0.743437	−	0.668806i	$-0.766806\pi$
$11$	−596.700	−1.48687	−0.743437	+	0.668806i	$0.766806\pi$
$12$	642.461	1.28793
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	−37.1142	−0.0506081
$15$	0	0
$16$	142.504	0.139164
$17$	2093.43	1.75686	0.878429	−	0.477872i	$-0.158592\pi$
$17$	2093.43	1.75686	0.878429	+	0.477872i	$0.158592\pi$
$18$	2027.52	1.47497
$19$	35.4251	0.0225127	0.0112563	−	0.999937i	$-0.496417\pi$
$19$	35.4251	0.0225127	0.0112563	+	0.999937i	$0.496417\pi$
$20$	0	0
$21$	342.394	0.169425
$22$	−1913.53	−0.842904
$23$	2783.24	1.09706	0.548531	−	0.836130i	$-0.315187\pi$
$23$	2783.24	1.09706	0.548531	+	0.836130i	$0.315187\pi$
$24$	5096.22	1.80601
$25$	0	0
$26$	541.958	0.157229
$27$	−11515.6	−3.04004
$28$	251.329	0.0605826
$29$	−370.244	−0.0817511	−0.0408755	−	0.999164i	$-0.513015\pi$
$29$	−370.244	−0.0817511	−0.0408755	+	0.999164i	$0.513015\pi$
$30$	0	0
$31$	5055.06	0.944761	0.472381	−	0.881395i	$-0.343395\pi$
$31$	5055.06	0.944761	0.472381	+	0.881395i	$0.343395\pi$
$32$	5969.30	1.03050
$33$	17653.1	2.82186
$34$	6713.33	0.995957
$35$	0	0
$36$	−13729.9	−1.76568
$37$	4124.15	0.495256	0.247628	−	0.968855i	$-0.420349\pi$
$37$	4124.15	0.495256	0.247628	+	0.968855i	$0.420349\pi$
$38$	113.603	0.0127624
$39$	−4999.79	−0.526369
$40$	0	0
$41$	−18104.1	−1.68197	−0.840985	−	0.541058i	$-0.818024\pi$
$41$	−18104.1	−1.68197	−0.840985	+	0.541058i	$0.818024\pi$
$42$	1098.01	0.0960466
$43$	−7906.34	−0.652085	−0.326042	−	0.945355i	$-0.605715\pi$
$43$	−7906.34	−0.652085	−0.326042	+	0.945355i	$0.605715\pi$
$44$	12958.0	1.00903
$45$	0	0
$46$	8925.44	0.621921
$47$	−13185.2	−0.870650	−0.435325	−	0.900273i	$-0.643366\pi$
$47$	−13185.2	−0.870650	−0.435325	+	0.900273i	$0.643366\pi$
$48$	−4215.93	−0.264113
$49$	−16673.1	−0.992030
$50$	0	0
$51$	−61933.3	−3.33425
$52$	−3670.02	−0.188218
$53$	38295.0	1.87263	0.936316	−	0.351160i	$-0.114213\pi$
$53$	38295.0	1.87263	0.936316	+	0.351160i	$0.114213\pi$
$54$	−36929.0	−1.72339
$55$	0	0
$56$	1993.63	0.0849522
$57$	−1048.03	−0.0427256
$58$	−1187.32	−0.0463444
$59$	17886.6	0.668955	0.334477	−	0.942404i	$-0.391440\pi$
$59$	17886.6	0.668955	0.334477	+	0.942404i	$0.391440\pi$
$60$	0	0
$61$	7392.00	0.254353	0.127177	−	0.991880i	$-0.459408\pi$
$61$	7392.00	0.254353	0.127177	+	0.991880i	$0.459408\pi$
$62$	16210.8	0.535582
$63$	−7317.23	−0.232271
$64$	14582.5	0.445023
$65$	0	0
$66$	56610.8	1.59970
$67$	−23357.0	−0.635667	−0.317834	−	0.948147i	$-0.602955\pi$
$67$	−23357.0	−0.635667	−0.317834	+	0.948147i	$0.602955\pi$
$68$	−45461.2	−1.19225
$69$	−82340.9	−2.08206
$70$	0	0
$71$	−33166.9	−0.780835	−0.390417	−	0.920638i	$-0.627669\pi$
$71$	−33166.9	−0.780835	−0.390417	+	0.920638i	$0.627669\pi$
$72$	−108910.	−2.47593
$73$	−10831.7	−0.237898	−0.118949	−	0.992900i	$-0.537953\pi$
$73$	−10831.7	−0.237898	−0.118949	+	0.992900i	$0.537953\pi$
$74$	13225.5	0.280759
$75$	0	0
$76$	−769.295	−0.0152777
$77$	6905.85	0.132736
$78$	−16033.6	−0.298397
$79$	−17755.2	−0.320079	−0.160039	−	0.987111i	$-0.551162\pi$
$79$	−17755.2	−0.320079	−0.160039	+	0.987111i	$0.551162\pi$
$80$	0	0
$81$	187049.	3.16770
$82$	−58057.3	−0.953503
$83$	−84243.8	−1.34228	−0.671139	−	0.741331i	$-0.734195\pi$
$83$	−84243.8	−1.34228	−0.671139	+	0.741331i	$0.734195\pi$
$84$	−7435.46	−0.114977
$85$	0	0
$86$	−25354.5	−0.369665
$87$	10953.5	0.155151
$88$	102787.	1.41492
$89$	−46498.8	−0.622253	−0.311126	−	0.950369i	$-0.600706\pi$
$89$	−46498.8	−0.622253	−0.311126	+	0.950369i	$0.600706\pi$
$90$	0	0
$91$	−1955.91	−0.0247597
$92$	−60441.1	−0.744497
$93$	−149552.	−1.79301
$94$	−42283.1	−0.493568
$95$	0	0
$96$	−176599.	−1.95573
$97$	154213.	1.66414	0.832072	−	0.554667i	$-0.187155\pi$
$97$	154213.	1.66414	0.832072	+	0.554667i	$0.187155\pi$
$98$	−53468.0	−0.562379
$99$	−377260.	−3.86860
$100$	0	0
$101$	−110873.	−1.08149	−0.540745	−	0.841187i	$-0.681858\pi$
$101$	−110873.	−1.08149	−0.540745	+	0.841187i	$0.681858\pi$
$102$	−198611.	−1.89018
$103$	104407.	0.969699	0.484849	−	0.874598i	$-0.338874\pi$
$103$	104407.	0.969699	0.484849	+	0.874598i	$0.338874\pi$
$104$	−29111.9	−0.263929
$105$	0	0
$106$	122806.	1.06159
$107$	−2229.04	−0.0188217	−0.00941084	−	0.999956i	$-0.502996\pi$
$107$	−2229.04	−0.0188217	−0.00941084	+	0.999956i	$0.502996\pi$
$108$	250075.	2.06305
$109$	−47011.7	−0.379000	−0.189500	−	0.981881i	$-0.560687\pi$
$109$	−47011.7	−0.379000	−0.189500	+	0.981881i	$0.560687\pi$
$110$	0	0
$111$	−122011.	−0.939921
$112$	−1649.26	−0.0124235
$113$	198706.	1.46391	0.731956	−	0.681352i	$-0.238608\pi$
$113$	198706.	1.46391	0.731956	+	0.681352i	$0.238608\pi$
$114$	−3360.89	−0.0242210
$115$	0	0
$116$	8040.27	0.0554786
$117$	106849.	0.721618
$118$	57359.5	0.379228
$119$	−24228.2	−0.156839
$120$	0	0
$121$	194999.	1.21079
$122$	23705.0	0.144192
$123$	535603.	3.19213
$124$	−109776.	−0.641142
$125$	0	0
$126$	−23465.3	−0.131674
$127$	87410.5	0.480899	0.240450	−	0.970662i	$-0.422705\pi$
$127$	87410.5	0.480899	0.240450	+	0.970662i	$0.422705\pi$
$128$	−144254.	−0.778219
$129$	233905.	1.23756
$130$	0	0
$131$	−19351.5	−0.0985226	−0.0492613	−	0.998786i	$-0.515687\pi$
$131$	−19351.5	−0.0985226	−0.0492613	+	0.998786i	$0.515687\pi$
$132$	−383356.	−1.91500
$133$	−409.989	−0.00200975
$134$	−74902.4	−0.360358
$135$	0	0
$136$	−360614.	−1.67184
$137$	−79960.9	−0.363979	−0.181989	−	0.983300i	$-0.558254\pi$
$137$	−79960.9	−0.363979	−0.181989	+	0.983300i	$0.558254\pi$
$138$	−264055.	−1.18031
$139$	285962.	1.25537	0.627684	−	0.778468i	$-0.284003\pi$
$139$	285962.	1.25537	0.627684	+	0.778468i	$0.284003\pi$
$140$	0	0
$141$	390079.	1.65236
$142$	−106361.	−0.442652
$143$	−100842.	−0.412385
$144$	90097.7	0.362083
$145$	0	0
$146$	−34735.8	−0.134864
$147$	493265.	1.88272
$148$	−89560.4	−0.336095
$149$	506571.	1.86928	0.934641	−	0.355593i	$-0.115721\pi$
$149$	506571.	1.86928	0.934641	+	0.355593i	$0.115721\pi$
$150$	0	0
$151$	−353322.	−1.26104	−0.630519	−	0.776174i	$-0.717158\pi$
$151$	−353322.	−1.26104	−0.630519	+	0.776174i	$0.717158\pi$
$152$	−6102.31	−0.0214232
$153$	1.32356e6	4.57105
$154$	22146.0	0.0752479
$155$	0	0
$156$	108576.	0.357209
$157$	222777.	0.721308	0.360654	−	0.932700i	$-0.382554\pi$
$157$	222777.	0.721308	0.360654	+	0.932700i	$0.382554\pi$
$158$	−56938.2	−0.181452
$159$	−1.13294e6	−3.55397
$160$	0	0
$161$	−32211.6	−0.0979371
$162$	599840.	1.79576
$163$	−485830.	−1.43224	−0.716120	−	0.697978i	$-0.754083\pi$
$163$	−485830.	−1.43224	−0.716120	+	0.697978i	$0.754083\pi$
$164$	393151.	1.14143
$165$	0	0
$166$	−270157.	−0.760933
$167$	492737.	1.36717	0.683587	−	0.729869i	$-0.260419\pi$
$167$	492737.	1.36717	0.683587	+	0.729869i	$0.260419\pi$
$168$	−58980.7	−0.161226
$169$	28561.0	0.0769231
$170$	0	0
$171$	22397.3	0.0585742
$172$	171695.	0.442523
$173$	−608498.	−1.54577	−0.772883	−	0.634549i	$-0.781186\pi$
$173$	−608498.	−1.54577	−0.772883	+	0.634549i	$0.781186\pi$
$174$	35126.3	0.0879547
$175$	0	0
$176$	−85032.3	−0.206920
$177$	−529166.	−1.26958
$178$	−149115.	−0.352753
$179$	−270050.	−0.629958	−0.314979	−	0.949099i	$-0.601997\pi$
$179$	−270050.	−0.629958	−0.314979	+	0.949099i	$0.601997\pi$
$180$	0	0
$181$	−602323.	−1.36657	−0.683286	−	0.730151i	$-0.739450\pi$
$181$	−602323.	−1.36657	−0.683286	+	0.730151i	$0.739450\pi$
$182$	−6272.30	−0.0140362
$183$	−218689.	−0.482724
$184$	−479440.	−1.04397
$185$	0	0
$186$	−479590.	−1.01645
$187$	−1.24915e6	−2.61223
$188$	286332.	0.590847
$189$	133275.	0.271391
$190$	0	0
$191$	−138491.	−0.274688	−0.137344	−	0.990523i	$-0.543857\pi$
$191$	−138491.	−0.274688	−0.137344	+	0.990523i	$0.543857\pi$
$192$	−431417.	−0.844586
$193$	−58857.2	−0.113738	−0.0568691	−	0.998382i	$-0.518112\pi$
$193$	−58857.2	−0.113738	−0.0568691	+	0.998382i	$0.518112\pi$
$194$	494537.	0.943398
$195$	0	0
$196$	362074.	0.673220
$197$	−271860.	−0.499091	−0.249545	−	0.968363i	$-0.580281\pi$
$197$	−271860.	−0.499091	−0.249545	+	0.968363i	$0.580281\pi$
$198$	−1.20982e6	−2.19309
$199$	−71334.8	−0.127694	−0.0638468	−	0.997960i	$-0.520337\pi$
$199$	−71334.8	−0.127694	−0.0638468	+	0.997960i	$0.520337\pi$
$200$	0	0
$201$	691006.	1.20640
$202$	−355553.	−0.613093
$203$	4284.99	0.00729810
$204$	1.34495e6	2.26272
$205$	0	0
$206$	334818.	0.549719
$207$	1.75969e6	2.85437
$208$	24083.2	0.0385973
$209$	−21138.1	−0.0334735
$210$	0	0
$211$	480921.	0.743648	0.371824	−	0.928303i	$-0.378732\pi$
$211$	480921.	0.743648	0.371824	+	0.928303i	$0.378732\pi$
$212$	−831618.	−1.27082
$213$	981228.	1.48191
$214$	−7148.20	−0.0106699
$215$	0	0
$216$	1.98368e6	2.89292
$217$	−58504.3	−0.0843409
$218$	−150759.	−0.214854
$219$	320452.	0.451495
$220$	0	0
$221$	353790.	0.487265
$222$	−391271.	−0.532838
$223$	−1.34684e6	−1.81365	−0.906823	−	0.421511i	$-0.861500\pi$
$223$	−1.34684e6	−1.81365	−0.906823	+	0.421511i	$0.861500\pi$
$224$	−69085.1	−0.0919950
$225$	0	0
$226$	637221.	0.829887
$227$	−719772.	−0.927108	−0.463554	−	0.886069i	$-0.653426\pi$
$227$	−719772.	−0.927108	−0.463554	+	0.886069i	$0.653426\pi$
$228$	22759.2	0.0289948
$229$	−568410.	−0.716264	−0.358132	−	0.933671i	$-0.616586\pi$
$229$	−568410.	−0.716264	−0.358132	+	0.933671i	$0.616586\pi$
$230$	0	0
$231$	−204306.	−0.251914
$232$	63778.2	0.0777951
$233$	282144.	0.340472	0.170236	−	0.985403i	$-0.445547\pi$
$233$	282144.	0.340472	0.170236	+	0.985403i	$0.445547\pi$
$234$	342650.	0.409083
$235$	0	0
$236$	−388426.	−0.453972
$237$	525278.	0.607461
$238$	−77696.1	−0.0889113
$239$	−143244.	−0.162211	−0.0811056	−	0.996706i	$-0.525845\pi$
$239$	−143244.	−0.162211	−0.0811056	+	0.996706i	$0.525845\pi$
$240$	0	0
$241$	−369914.	−0.410259	−0.205130	−	0.978735i	$-0.565762\pi$
$241$	−369914.	−0.410259	−0.205130	+	0.978735i	$0.565762\pi$
$242$	625334.	0.686395
$243$	−2.73547e6	−2.97178
$244$	−160525.	−0.172611
$245$	0	0
$246$	1.71760e6	1.80961
$247$	5986.84	0.00624389
$248$	−870783.	−0.899043
$249$	2.49231e6	2.54744
$250$	0	0
$251$	−731868.	−0.733244	−0.366622	−	0.930370i	$-0.619486\pi$
$251$	−731868.	−0.733244	−0.366622	+	0.930370i	$0.619486\pi$
$252$	158902.	0.157626
$253$	−1.66076e6	−1.63119
$254$	280312.	0.272620
$255$	0	0
$256$	−929240.	−0.886192
$257$	1.05879e6	0.999951	0.499975	−	0.866040i	$-0.333342\pi$
$257$	1.05879e6	0.999951	0.499975	+	0.866040i	$0.333342\pi$
$258$	750100.	0.701568
$259$	−47730.4	−0.0442126
$260$	0	0
$261$	−234085.	−0.212703
$262$	−62057.3	−0.0558521
$263$	318172.	0.283644	0.141822	−	0.989892i	$-0.454704\pi$
$263$	318172.	0.283644	0.141822	+	0.989892i	$0.454704\pi$
$264$	−3.04091e6	−2.68531
$265$	0	0
$266$	−1314.77	−0.00113932
$267$	1.37565e6	1.18094
$268$	507223.	0.431382
$269$	1.13453e6	0.955954	0.477977	−	0.878372i	$-0.341370\pi$
$269$	1.13453e6	0.955954	0.477977	+	0.878372i	$0.341370\pi$
$270$	0	0
$271$	−1.05400e6	−0.871805	−0.435902	−	0.899994i	$-0.643571\pi$
$271$	−1.05400e6	−0.871805	−0.435902	+	0.899994i	$0.643571\pi$
$272$	298324.	0.244492
$273$	57864.6	0.0469901
$274$	−256423.	−0.206338
$275$	0	0
$276$	1.78812e6	1.41294
$277$	−2.18170e6	−1.70843	−0.854214	−	0.519922i	$-0.825961\pi$
$277$	−2.18170e6	−1.70843	−0.854214	+	0.519922i	$0.825961\pi$
$278$	917037.	0.711664
$279$	3.19604e6	2.45811
$280$	0	0
$281$	−2.11562e6	−1.59835	−0.799176	−	0.601097i	$-0.794730\pi$
$281$	−2.11562e6	−1.59835	−0.799176	+	0.601097i	$0.794730\pi$
$282$	1.25093e6	0.936718
$283$	−390968.	−0.290185	−0.145092	−	0.989418i	$-0.546348\pi$
$283$	−390968.	−0.290185	−0.145092	+	0.989418i	$0.546348\pi$
$284$	720256.	0.529896
$285$	0	0
$286$	−323386.	−0.233779
$287$	209527.	0.150153
$288$	3.77406e6	2.68119
$289$	2.96261e6	2.08655
$290$	0	0
$291$	−4.56231e6	−3.15830
$292$	235223.	0.161444
$293$	−2.44684e6	−1.66508	−0.832542	−	0.553961i	$-0.813116\pi$
$293$	−2.44684e6	−1.66508	−0.832542	+	0.553961i	$0.813116\pi$
$294$	1.58183e6	1.06731
$295$	0	0
$296$	−710424.	−0.471290
$297$	6.87138e6	4.52015
$298$	1.62450e6	1.05969
$299$	470368.	0.304270
$300$	0	0
$301$	91503.3	0.0582131
$302$	−1.13305e6	−0.714878
$303$	3.28013e6	2.05251
$304$	5048.23	0.00313296
$305$	0	0
$306$	4.24447e6	2.59131
$307$	−1.58927e6	−0.962390	−0.481195	−	0.876614i	$-0.659797\pi$
$307$	−1.58927e6	−0.962390	−0.481195	+	0.876614i	$0.659797\pi$
$308$	−149968.	−0.0900787
$309$	−3.08884e6	−1.84034
$310$	0	0
$311$	3.15211e6	1.84799	0.923996	−	0.382403i	$-0.124903\pi$
$311$	3.15211e6	1.84799	0.923996	+	0.382403i	$0.124903\pi$
$312$	861261.	0.500897
$313$	1.58939e6	0.917001	0.458500	−	0.888694i	$-0.348387\pi$
$313$	1.58939e6	0.917001	0.458500	+	0.888694i	$0.348387\pi$
$314$	714412.	0.408907
$315$	0	0
$316$	385573.	0.217215
$317$	−1.45529e6	−0.813396	−0.406698	−	0.913563i	$-0.633320\pi$
$317$	−1.45529e6	−0.813396	−0.406698	+	0.913563i	$0.633320\pi$
$318$	−3.63317e6	−2.01473
$319$	220925.	0.121554
$320$	0	0
$321$	65945.1	0.0357207
$322$	−103298.	−0.0555202
$323$	74160.0	0.0395516
$324$	−4.06198e6	−2.14969
$325$	0	0
$326$	−1.55799e6	−0.811932
$327$	1.39082e6	0.719285
$328$	3.11861e6	1.60058
$329$	152598.	0.0777248
$330$	0	0
$331$	3.21928e6	1.61506	0.807530	−	0.589826i	$-0.200804\pi$
$331$	3.21928e6	1.61506	0.807530	+	0.589826i	$0.200804\pi$
$332$	1.82945e6	0.910908
$333$	2.60747e6	1.28857
$334$	1.58013e6	0.775046
$335$	0	0
$336$	48792.7	0.0235780
$337$	−3.07320e6	−1.47406	−0.737031	−	0.675859i	$-0.763773\pi$
$337$	−3.07320e6	−1.47406	−0.737031	+	0.675859i	$0.763773\pi$
$338$	91590.9	0.0436074
$339$	−5.87863e6	−2.77829
$340$	0	0
$341$	−3.01635e6	−1.40474
$342$	71824.9	0.0332055
$343$	387478.	0.177833
$344$	1.36194e6	0.620530
$345$	0	0
$346$	−1.95136e6	−0.876289
$347$	−679230.	−0.302826	−0.151413	−	0.988471i	$-0.548382\pi$
$347$	−679230.	−0.302826	−0.151413	+	0.988471i	$0.548382\pi$
$348$	−237868.	−0.105290
$349$	3.00182e6	1.31923	0.659616	−	0.751603i	$-0.270719\pi$
$349$	3.00182e6	1.31923	0.659616	+	0.751603i	$0.270719\pi$
$350$	0	0
$351$	−1.94614e6	−0.843154
$352$	−3.56188e6	−1.53222
$353$	2.13910e6	0.913682	0.456841	−	0.889548i	$-0.348981\pi$
$353$	2.13910e6	0.913682	0.456841	+	0.889548i	$0.348981\pi$
$354$	−1.69696e6	−0.719718
$355$	0	0
$356$	1.00977e6	0.422278
$357$	716779.	0.297656
$358$	−866010.	−0.357121
$359$	4.04695e6	1.65726	0.828631	−	0.559795i	$-0.189120\pi$
$359$	4.04695e6	1.65726	0.828631	+	0.559795i	$0.189120\pi$
$360$	0	0
$361$	−2.47484e6	−0.999493
$362$	−1.93156e6	−0.774705
$363$	−5.76897e6	−2.29790
$364$	42474.7	0.0168026
$365$	0	0
$366$	−701303.	−0.273655
$367$	1.59021e6	0.616295	0.308148	−	0.951339i	$-0.400291\pi$
$367$	1.59021e6	0.616295	0.308148	+	0.951339i	$0.400291\pi$
$368$	396624.	0.152672
$369$	−1.14463e7	−4.37620
$370$	0	0
$371$	−443204.	−0.167174
$372$	3.24768e6	1.21679
$373$	−3.36023e6	−1.25054	−0.625270	−	0.780409i	$-0.715011\pi$
$373$	−3.36023e6	−1.25054	−0.625270	+	0.780409i	$0.715011\pi$
$374$	−4.00584e6	−1.48086
$375$	0	0
$376$	2.27128e6	0.828518
$377$	−62571.3	−0.0226737
$378$	427394.	0.153850
$379$	849174.	0.303668	0.151834	−	0.988406i	$-0.451482\pi$
$379$	849174.	0.303668	0.151834	+	0.988406i	$0.451482\pi$
$380$	0	0
$381$	−2.58600e6	−0.912674
$382$	−444121.	−0.155720
$383$	−4.04395e6	−1.40867	−0.704335	−	0.709868i	$-0.748755\pi$
$383$	−4.04395e6	−1.40867	−0.704335	+	0.709868i	$0.748755\pi$
$384$	4.26768e6	1.47694
$385$	0	0
$386$	−188746.	−0.0644777
$387$	−4.99874e6	−1.69662
$388$	−3.34890e6	−1.12934
$389$	−2.94042e6	−0.985226	−0.492613	−	0.870249i	$-0.663958\pi$
$389$	−2.94042e6	−0.985226	−0.492613	+	0.870249i	$0.663958\pi$
$390$	0	0
$391$	5.82653e6	1.92738
$392$	2.87209e6	0.944025
$393$	572504.	0.186981
$394$	−871814.	−0.282933
$395$	0	0
$396$	8.19263e6	2.62534
$397$	−1.24060e6	−0.395053	−0.197526	−	0.980298i	$-0.563291\pi$
$397$	−1.24060e6	−0.395053	−0.197526	+	0.980298i	$0.563291\pi$
$398$	−228760.	−0.0723890
$399$	12129.3	0.00381421
$400$	0	0
$401$	−972004.	−0.301861	−0.150931	−	0.988544i	$-0.548227\pi$
$401$	−972004.	−0.301861	−0.150931	+	0.988544i	$0.548227\pi$
$402$	2.21595e6	0.683904
$403$	854305.	0.262030
$404$	2.40773e6	0.733930
$405$	0	0
$406$	13741.3	0.00413727
$407$	−2.46088e6	−0.736383
$408$	1.06686e7	3.17290
$409$	571971.	0.169070	0.0845348	−	0.996421i	$-0.473060\pi$
$409$	571971.	0.169070	0.0845348	+	0.996421i	$0.473060\pi$
$410$	0	0
$411$	2.36561e6	0.690777
$412$	−2.26731e6	−0.658065
$413$	−207009.	−0.0597191
$414$	5.64306e6	1.61813
$415$	0	0
$416$	1.00881e6	0.285809
$417$	−8.46005e6	−2.38250
$418$	−67786.9	−0.0189760
$419$	−4.06416e6	−1.13093	−0.565465	−	0.824772i	$-0.691303\pi$
$419$	−4.06416e6	−1.13093	−0.565465	+	0.824772i	$0.691303\pi$
$420$	0	0
$421$	−3.73588e6	−1.02728	−0.513639	−	0.858007i	$-0.671703\pi$
$421$	−3.73588e6	−1.02728	−0.513639	+	0.858007i	$0.671703\pi$
$422$	1.54224e6	0.421572
$423$	−8.33631e6	−2.26528
$424$	−6.59668e6	−1.78201
$425$	0	0
$426$	3.14665e6	0.840088
$427$	−85550.6	−0.0227067
$428$	48406.0	0.0127729
$429$	2.98337e6	0.782644
$430$	0	0
$431$	−1.53689e6	−0.398520	−0.199260	−	0.979947i	$-0.563854\pi$
$431$	−1.53689e6	−0.398520	−0.199260	+	0.979947i	$0.563854\pi$
$432$	−1.64103e6	−0.423065
$433$	112160.	0.0287487	0.0143744	−	0.999897i	$-0.495424\pi$
$433$	112160.	0.0287487	0.0143744	+	0.999897i	$0.495424\pi$
$434$	−187615.	−0.0478126
$435$	0	0
$436$	1.02091e6	0.257200
$437$	98596.5	0.0246978
$438$	1.02764e6	0.255951
$439$	−164279.	−0.0406837	−0.0203418	−	0.999793i	$-0.506475\pi$
$439$	−164279.	−0.0406837	−0.0203418	+	0.999793i	$0.506475\pi$
$440$	0	0
$441$	−1.05415e7	−2.58110
$442$	1.13455e6	0.276229
$443$	4.62345e6	1.11933	0.559664	−	0.828720i	$-0.310930\pi$
$443$	4.62345e6	1.11933	0.559664	+	0.828720i	$0.310930\pi$
$444$	2.64960e6	0.637857
$445$	0	0
$446$	−4.31910e6	−1.02815
$447$	−1.49867e7	−3.54761
$448$	−168769.	−0.0397282
$449$	421554.	0.0986820	0.0493410	−	0.998782i	$-0.484288\pi$
$449$	421554.	0.0986820	0.0493410	+	0.998782i	$0.484288\pi$
$450$	0	0
$451$	1.08027e7	2.50088
$452$	−4.31512e6	−0.993452
$453$	1.04529e7	2.39326
$454$	−2.30820e6	−0.525574
$455$	0	0
$456$	180534.	0.0406581
$457$	5.34433e6	1.19702	0.598512	−	0.801114i	$-0.295759\pi$
$457$	5.34433e6	1.19702	0.598512	+	0.801114i	$0.295759\pi$
$458$	−1.82281e6	−0.406047
$459$	−2.41072e7	−5.34091
$460$	0	0
$461$	−4.95164e6	−1.08517	−0.542584	−	0.840002i	$-0.682554\pi$
$461$	−4.95164e6	−1.08517	−0.542584	+	0.840002i	$0.682554\pi$
$462$	−655180.	−0.142809
$463$	−3.87726e6	−0.840567	−0.420283	−	0.907393i	$-0.638069\pi$
$463$	−3.87726e6	−0.840567	−0.420283	+	0.907393i	$0.638069\pi$
$464$	−52761.5	−0.0113768
$465$	0	0
$466$	904795.	0.193013
$467$	−1.64031e6	−0.348043	−0.174022	−	0.984742i	$-0.555676\pi$
$467$	−1.64031e6	−0.348043	−0.174022	+	0.984742i	$0.555676\pi$
$468$	−2.32035e6	−0.489711
$469$	270320.	0.0567474
$470$	0	0
$471$	−6.59075e6	−1.36893
$472$	−3.08113e6	−0.636583
$473$	4.71771e6	0.969568
$474$	1.68449e6	0.344368
$475$	0	0
$476$	526141.	0.106435
$477$	2.42118e7	4.87227
$478$	−459361.	−0.0919570
$479$	2.63694e6	0.525123	0.262561	−	0.964915i	$-0.415433\pi$
$479$	2.63694e6	0.525123	0.262561	+	0.964915i	$0.415433\pi$
$480$	0	0
$481$	696981.	0.137359
$482$	−1.18626e6	−0.232575
$483$	952965.	0.185870
$484$	−4.23463e6	−0.821678
$485$	0	0
$486$	−8.77225e6	−1.68469
$487$	−3.47150e6	−0.663277	−0.331639	−	0.943407i	$-0.607601\pi$
$487$	−3.47150e6	−0.663277	−0.331639	+	0.943407i	$0.607601\pi$
$488$	−1.27334e6	−0.242045
$489$	1.43731e7	2.71817
$490$	0	0
$491$	−2.74272e6	−0.513425	−0.256713	−	0.966488i	$-0.582639\pi$
$491$	−2.74272e6	−0.513425	−0.256713	+	0.966488i	$0.582639\pi$
$492$	−1.16312e7	−2.16627
$493$	−775082.	−0.143625
$494$	19198.9	0.00353964
$495$	0	0
$496$	720368.	0.131477
$497$	383854.	0.0697068
$498$	7.99248e6	1.44414
$499$	−1.00565e7	−1.80799	−0.903996	−	0.427540i	$-0.859380\pi$
$499$	−1.00565e7	−1.80799	−0.903996	+	0.427540i	$0.859380\pi$
$500$	0	0
$501$	−1.45774e7	−2.59469
$502$	−2.34699e6	−0.415674
$503$	−5.52516e6	−0.973699	−0.486850	−	0.873486i	$-0.661854\pi$
$503$	−5.52516e6	−0.973699	−0.486850	+	0.873486i	$0.661854\pi$
$504$	1.26046e6	0.221031
$505$	0	0
$506$	−5.32580e6	−0.924717
$507$	−844964.	−0.145988
$508$	−1.89821e6	−0.326352
$509$	−6.59944e6	−1.12905	−0.564524	−	0.825416i	$-0.690940\pi$
$509$	−6.59944e6	−1.12905	−0.564524	+	0.825416i	$0.690940\pi$
$510$	0	0
$511$	125360.	0.0212377
$512$	1.63618e6	0.275839
$513$	−407943.	−0.0684393
$514$	3.39540e6	0.566869
$515$	0	0
$516$	−5.07951e6	−0.839842
$517$	7.86763e6	1.29455
$518$	−153064.	−0.0250640
$519$	1.80021e7	2.93363
$520$	0	0
$521$	−653159.	−0.105420	−0.0527102	−	0.998610i	$-0.516786\pi$
$521$	−653159.	−0.105420	−0.0527102	+	0.998610i	$0.516786\pi$
$522$	−750677.	−0.120580
$523$	−3.60676e6	−0.576584	−0.288292	−	0.957543i	$-0.593087\pi$
$523$	−3.60676e6	−0.576584	−0.288292	+	0.957543i	$0.593087\pi$
$524$	420239.	0.0668602
$525$	0	0
$526$	1.02033e6	0.160797
$527$	1.05824e7	1.65981
$528$	2.51564e6	0.392703
$529$	1.31008e6	0.203544
$530$	0	0
$531$	1.13087e7	1.74051
$532$	8903.36	0.00136388
$533$	−3.05960e6	−0.466495
$534$	4.41149e6	0.669472
$535$	0	0
$536$	4.02346e6	0.604906
$537$	7.98931e6	1.19557
$538$	3.63828e6	0.541927
$539$	9.94881e6	1.47502
$540$	0	0
$541$	−1.92001e6	−0.282040	−0.141020	−	0.990007i	$-0.545038\pi$
$541$	−1.92001e6	−0.282040	−0.141020	+	0.990007i	$0.545038\pi$
$542$	−3.38004e6	−0.494223
$543$	1.78194e7	2.59355
$544$	1.24963e7	1.81044
$545$	0	0
$546$	185563.	0.0266385
$547$	7.41583e6	1.05972	0.529860	−	0.848085i	$-0.322244\pi$
$547$	7.41583e6	1.05972	0.529860	+	0.848085i	$0.322244\pi$
$548$	1.73644e6	0.247006
$549$	4.67356e6	0.661784
$550$	0	0
$551$	−13115.9	−0.00184043
$552$	1.41840e7	1.98130
$553$	205488.	0.0285741
$554$	−6.99640e6	−0.968502
$555$	0	0
$556$	−6.20998e6	−0.851928
$557$	−3.52870e6	−0.481922	−0.240961	−	0.970535i	$-0.577463\pi$
$557$	−3.52870e6	−0.481922	−0.240961	+	0.970535i	$0.577463\pi$
$558$	1.02492e7	1.39349
$559$	−1.33617e6	−0.180856
$560$	0	0
$561$	3.69556e7	4.95761
$562$	−6.78449e6	−0.906100
$563$	4.31809e6	0.574144	0.287072	−	0.957909i	$-0.407318\pi$
$563$	4.31809e6	0.574144	0.287072	+	0.957909i	$0.407318\pi$
$564$	−8.47100e6	−1.12134
$565$	0	0
$566$	−1.25378e6	−0.164505
$567$	−2.16480e6	−0.282787
$568$	5.71332e6	0.743049
$569$	−3.99812e6	−0.517696	−0.258848	−	0.965918i	$-0.583343\pi$
$569$	−3.99812e6	−0.517696	−0.258848	+	0.965918i	$0.583343\pi$
$570$	0	0
$571$	298969.	0.0383739	0.0191869	−	0.999816i	$-0.493892\pi$
$571$	298969.	0.0383739	0.0191869	+	0.999816i	$0.493892\pi$
$572$	2.18990e6	0.279856
$573$	4.09720e6	0.521316
$574$	671921.	0.0851213
$575$	0	0
$576$	9.21972e6	1.15787
$577$	−2.22363e6	−0.278050	−0.139025	−	0.990289i	$-0.544397\pi$
$577$	−2.22363e6	−0.278050	−0.139025	+	0.990289i	$0.544397\pi$
$578$	9.50064e6	1.18286
$579$	1.74126e6	0.215858
$580$	0	0
$581$	974988.	0.119828
$582$	−1.46307e7	−1.79043
$583$	−2.28506e7	−2.78437
$584$	1.86587e6	0.226386
$585$	0	0
$586$	−7.84665e6	−0.943931
$587$	−3.25215e6	−0.389560	−0.194780	−	0.980847i	$-0.562399\pi$
$587$	−3.25215e6	−0.389560	−0.194780	+	0.980847i	$0.562399\pi$
$588$	−1.07118e7	−1.27767
$589$	179076.	0.0212691
$590$	0	0
$591$	8.04285e6	0.947199
$592$	587709.	0.0689220
$593$	−1.23502e7	−1.44224	−0.721121	−	0.692810i	$-0.756373\pi$
$593$	−1.23502e7	−1.44224	−0.721121	+	0.692810i	$0.756373\pi$
$594$	2.20355e7	2.56246
$595$	0	0
$596$	−1.10007e7	−1.26855
$597$	2.11041e6	0.242343
$598$	1.50840e6	0.172490
$599$	−1.26026e7	−1.43513	−0.717567	−	0.696490i	$-0.754744\pi$
$599$	−1.26026e7	−1.43513	−0.717567	+	0.696490i	$0.754744\pi$
$600$	0	0
$601$	1.10475e7	1.24760	0.623802	−	0.781582i	$-0.285587\pi$
$601$	1.10475e7	1.24760	0.623802	+	0.781582i	$0.285587\pi$
$602$	293437.	0.0330008
$603$	−1.47673e7	−1.65390
$604$	7.67277e6	0.855776
$605$	0	0
$606$	1.05189e7	1.16356
$607$	1.47315e7	1.62284	0.811421	−	0.584462i	$-0.198694\pi$
$607$	1.47315e7	1.62284	0.811421	+	0.584462i	$0.198694\pi$
$608$	211463.	0.0231993
$609$	−126769.	−0.0138507
$610$	0	0
$611$	−2.22831e6	−0.241475
$612$	−2.87426e7	−3.10204
$613$	6.38520e6	0.686315	0.343157	−	0.939278i	$-0.388504\pi$
$613$	6.38520e6	0.686315	0.343157	+	0.939278i	$0.388504\pi$
$614$	−5.09655e6	−0.545576
$615$	0	0
$616$	−1.18960e6	−0.126313
$617$	1.69836e7	1.79604	0.898020	−	0.439954i	$-0.145005\pi$
$617$	1.69836e7	1.79604	0.898020	+	0.439954i	$0.145005\pi$
$618$	−9.90544e6	−1.04328
$619$	−1.06421e7	−1.11635	−0.558175	−	0.829723i	$-0.688498\pi$
$619$	−1.06421e7	−1.11635	−0.558175	+	0.829723i	$0.688498\pi$
$620$	0	0
$621$	−3.20508e7	−3.33511
$622$	1.01083e7	1.04762
$623$	538150.	0.0555499
$624$	−712492.	−0.0732518
$625$	0	0
$626$	5.09694e6	0.519845
$627$	625362.	0.0635276
$628$	−4.83784e6	−0.489500
$629$	8.63362e6	0.870095
$630$	0	0
$631$	8.93725e6	0.893574	0.446787	−	0.894640i	$-0.352568\pi$
$631$	8.93725e6	0.893574	0.446787	+	0.894640i	$0.352568\pi$
$632$	3.05850e6	0.304590
$633$	−1.42278e7	−1.41133
$634$	−4.66691e6	−0.461111
$635$	0	0
$636$	2.46030e7	2.41183
$637$	−2.81775e6	−0.275140
$638$	708473.	0.0689083
$639$	−2.09696e7	−2.03160
$640$	0	0
$641$	−1.12700e7	−1.08337	−0.541686	−	0.840581i	$-0.682214\pi$
$641$	−1.12700e7	−1.08337	−0.541686	+	0.840581i	$0.682214\pi$
$642$	211476.	0.0202499
$643$	1.16584e7	1.11202	0.556008	−	0.831177i	$-0.312332\pi$
$643$	1.16584e7	1.11202	0.556008	+	0.831177i	$0.312332\pi$
$644$	699510.	0.0664629
$645$	0	0
$646$	237820.	0.0224217
$647$	8.09860e6	0.760587	0.380294	−	0.924866i	$-0.375823\pi$
$647$	8.09860e6	0.760587	0.380294	+	0.924866i	$0.375823\pi$
$648$	−3.22211e7	−3.01441
$649$	−1.06729e7	−0.994651
$650$	0	0
$651$	1.73082e6	0.160066
$652$	1.05503e7	0.971958
$653$	−2.50550e6	−0.229938	−0.114969	−	0.993369i	$-0.536677\pi$
$653$	−2.50550e6	−0.229938	−0.114969	+	0.993369i	$0.536677\pi$
$654$	4.46015e6	0.407760
$655$	0	0
$656$	−2.57992e6	−0.234070
$657$	−6.84831e6	−0.618971
$658$	489360.	0.0440619
$659$	−2.14090e7	−1.92036	−0.960182	−	0.279376i	$-0.909873\pi$
$659$	−2.14090e7	−1.92036	−0.960182	+	0.279376i	$0.909873\pi$
$660$	0	0
$661$	1.59525e7	1.42012	0.710060	−	0.704141i	$-0.248668\pi$
$661$	1.59525e7	1.42012	0.710060	+	0.704141i	$0.248668\pi$
$662$	1.03238e7	0.915572
$663$	−1.04667e7	−0.924755
$664$	1.45118e7	1.27732
$665$	0	0
$666$	8.36177e6	0.730488
$667$	−1.03048e6	−0.0896860
$668$	−1.07003e7	−0.927803
$669$	3.98455e7	3.44203
$670$	0	0
$671$	−4.41080e6	−0.378191
$672$	2.04385e6	0.174593
$673$	9.71969e6	0.827208	0.413604	−	0.910457i	$-0.364270\pi$
$673$	9.71969e6	0.827208	0.413604	+	0.910457i	$0.364270\pi$
$674$	−9.85529e6	−0.835641
$675$	0	0
$676$	−620234.	−0.0522022
$677$	1.01440e7	0.850628	0.425314	−	0.905046i	$-0.360164\pi$
$677$	1.01440e7	0.850628	0.425314	+	0.905046i	$0.360164\pi$
$678$	−1.88519e7	−1.57500
$679$	−1.78477e6	−0.148562
$680$	0	0
$681$	2.12941e7	1.75951
$682$	−9.67300e6	−0.796343
$683$	−6.41414e6	−0.526122	−0.263061	−	0.964779i	$-0.584732\pi$
$683$	−6.41414e6	−0.526122	−0.263061	+	0.964779i	$0.584732\pi$
$684$	−486383.	−0.0397501
$685$	0	0
$686$	1.24259e6	0.100813
$687$	1.68161e7	1.35936
$688$	−1.12669e6	−0.0907471
$689$	6.47185e6	0.519374
$690$	0	0
$691$	−8.38877e6	−0.668349	−0.334175	−	0.942511i	$-0.608458\pi$
$691$	−8.38877e6	−0.668349	−0.334175	+	0.942511i	$0.608458\pi$
$692$	1.32142e7	1.04900
$693$	4.36619e6	0.345358
$694$	−2.17819e6	−0.171671
$695$	0	0
$696$	−1.88685e6	−0.147643
$697$	−3.78998e7	−2.95498
$698$	9.62639e6	0.747868
$699$	−8.34712e6	−0.646165
$700$	0	0
$701$	−5.05150e6	−0.388263	−0.194131	−	0.980976i	$-0.562189\pi$
$701$	−5.05150e6	−0.388263	−0.194131	+	0.980976i	$0.562189\pi$
$702$	−6.24099e6	−0.477981
$703$	146098.	0.0111495
$704$	−8.70138e6	−0.661693
$705$	0	0
$706$	6.85979e6	0.517963
$707$	1.28318e6	0.0965470
$708$	1.14914e7	0.861570
$709$	1.37386e7	1.02642	0.513212	−	0.858262i	$-0.328455\pi$
$709$	1.37386e7	1.02642	0.513212	+	0.858262i	$0.328455\pi$
$710$	0	0
$711$	−1.12256e7	−0.832791
$712$	8.00986e6	0.592141
$713$	1.40694e7	1.03646
$714$	2.29860e6	0.168740
$715$	0	0
$716$	5.86443e6	0.427507
$717$	4.23780e6	0.307852
$718$	1.29780e7	0.939496
$719$	−8.31655e6	−0.599958	−0.299979	−	0.953946i	$-0.596980\pi$
$719$	−8.31655e6	−0.599958	−0.299979	+	0.953946i	$0.596980\pi$
$720$	0	0
$721$	−1.20835e6	−0.0865671
$722$	−7.93646e6	−0.566609
$723$	1.09437e7	0.778610
$724$	1.30801e7	0.927395
$725$	0	0
$726$	−1.85002e7	−1.30267
$727$	4.72548e6	0.331596	0.165798	−	0.986160i	$-0.446980\pi$
$727$	4.72548e6	0.331596	0.165798	+	0.986160i	$0.446980\pi$
$728$	336924.	0.0235615
$729$	3.54746e7	2.47229
$730$	0	0
$731$	−1.65514e7	−1.14562
$732$	4.74907e6	0.327590
$733$	1.07477e6	0.0738847	0.0369423	−	0.999317i	$-0.488238\pi$
$733$	1.07477e6	0.0738847	0.0369423	+	0.999317i	$0.488238\pi$
$734$	5.09956e6	0.349376
$735$	0	0
$736$	1.66140e7	1.13052
$737$	1.39371e7	0.945157
$738$	−3.67064e7	−2.48086
$739$	−2.04180e7	−1.37531	−0.687657	−	0.726036i	$-0.741361\pi$
$739$	−2.04180e7	−1.37531	−0.687657	+	0.726036i	$0.741361\pi$
$740$	0	0
$741$	−177118.	−0.0118500
$742$	−1.42129e6	−0.0947703
$743$	−2.55138e7	−1.69552	−0.847762	−	0.530377i	$-0.822050\pi$
$743$	−2.55138e7	−1.69552	−0.847762	+	0.530377i	$0.822050\pi$
$744$	2.57617e7	1.70625
$745$	0	0
$746$	−1.07758e7	−0.708926
$747$	−5.32627e7	−3.49238
$748$	2.71267e7	1.77273
$749$	25797.6	0.00168025
$750$	0	0
$751$	2.09344e7	1.35444	0.677220	−	0.735780i	$-0.263184\pi$
$751$	2.09344e7	1.35444	0.677220	+	0.735780i	$0.263184\pi$
$752$	−1.87896e6	−0.121163
$753$	2.16520e7	1.39159
$754$	−200657.	−0.0128536
$755$	0	0
$756$	−2.89422e6	−0.184173
$757$	−6.00641e6	−0.380956	−0.190478	−	0.981691i	$-0.561004\pi$
$757$	−6.00641e6	−0.380956	−0.190478	+	0.981691i	$0.561004\pi$
$758$	2.72317e6	0.172148
$759$	4.91328e7	3.09576
$760$	0	0
$761$	−4.48972e6	−0.281033	−0.140517	−	0.990078i	$-0.544876\pi$
$761$	−4.48972e6	−0.281033	−0.140517	+	0.990078i	$0.544876\pi$
$762$	−8.29291e6	−0.517392
$763$	544085.	0.0338342
$764$	3.00749e6	0.186411
$765$	0	0
$766$	−1.29684e7	−0.798570
$767$	3.02283e6	0.185535
$768$	2.74911e7	1.68186
$769$	2.20361e7	1.34375	0.671877	−	0.740663i	$-0.265488\pi$
$769$	2.20361e7	1.34375	0.671877	+	0.740663i	$0.265488\pi$
$770$	0	0
$771$	−3.13239e7	−1.89776
$772$	1.27815e6	0.0771859
$773$	−9.09786e6	−0.547634	−0.273817	−	0.961782i	$-0.588286\pi$
$773$	−9.09786e6	−0.547634	−0.273817	+	0.961782i	$0.588286\pi$
$774$	−1.60302e7	−0.961806
$775$	0	0
$776$	−2.65646e7	−1.58361
$777$	1.41208e6	0.0839088
$778$	−9.42950e6	−0.558521
$779$	−641341.	−0.0378656
$780$	0	0
$781$	1.97907e7	1.16100
$782$	1.86848e7	1.09263
$783$	4.26360e6	0.248526
$784$	−2.37598e6	−0.138055
$785$	0	0
$786$	1.83594e6	0.105999
$787$	1.65992e7	0.955325	0.477662	−	0.878544i	$-0.341484\pi$
$787$	1.65992e7	0.955325	0.477662	+	0.878544i	$0.341484\pi$
$788$	5.90374e6	0.338697
$789$	−9.41298e6	−0.538313
$790$	0	0
$791$	−2.29971e6	−0.130687
$792$	6.49867e7	3.68139
$793$	1.24925e6	0.0705449
$794$	−3.97841e6	−0.223954
$795$	0	0
$796$	1.54911e6	0.0866564
$797$	2.41936e7	1.34913	0.674567	−	0.738213i	$-0.264330\pi$
$797$	2.41936e7	1.34913	0.674567	+	0.738213i	$0.264330\pi$
$798$	38897.0	0.00216226
$799$	−2.76024e7	−1.52961
$800$	0	0
$801$	−2.93986e7	−1.61900
$802$	−3.11707e6	−0.171124
$803$	6.46329e6	0.353724
$804$	−1.50059e7	−0.818697
$805$	0	0
$806$	2.73963e6	0.148544
$807$	−3.35647e7	−1.81426
$808$	1.90989e7	1.02916
$809$	−1.25176e7	−0.672433	−0.336216	−	0.941785i	$-0.609147\pi$
$809$	−1.25176e7	−0.672433	−0.336216	+	0.941785i	$0.609147\pi$
$810$	0	0
$811$	−1.21034e7	−0.646183	−0.323091	−	0.946368i	$-0.604722\pi$
$811$	−1.21034e7	−0.646183	−0.323091	+	0.946368i	$0.604722\pi$
$812$	−93053.3	−0.00495270
$813$	3.11822e7	1.65455
$814$	−7.89166e6	−0.417453
$815$	0	0
$816$	−8.82576e6	−0.464010
$817$	−280083.	−0.0146802
$818$	1.83423e6	0.0958450
$819$	−1.23661e6	−0.0644205
$820$	0	0
$821$	9.24362e6	0.478613	0.239306	−	0.970944i	$-0.423080\pi$
$821$	9.24362e6	0.478613	0.239306	+	0.970944i	$0.423080\pi$
$822$	7.58615e6	0.391599
$823$	−2.93437e7	−1.51013	−0.755067	−	0.655648i	$-0.772396\pi$
$823$	−2.93437e7	−1.51013	−0.755067	+	0.655648i	$0.772396\pi$
$824$	−1.79851e7	−0.922774
$825$	0	0
$826$	−663846.	−0.0338545
$827$	−1.74083e6	−0.0885099	−0.0442550	−	0.999020i	$-0.514091\pi$
$827$	−1.74083e6	−0.0885099	−0.0442550	+	0.999020i	$0.514091\pi$
$828$	−3.82136e7	−1.93706
$829$	2.00227e7	1.01190	0.505948	−	0.862564i	$-0.331143\pi$
$829$	2.00227e7	1.01190	0.505948	+	0.862564i	$0.331143\pi$
$830$	0	0
$831$	6.45447e7	3.24234
$832$	2.46444e6	0.123427
$833$	−3.49039e7	−1.74286
$834$	−2.71301e7	−1.35063
$835$	0	0
$836$	459038.	0.0227160
$837$	−5.82123e7	−2.87211
$838$	−1.30332e7	−0.641121
$839$	1.34556e7	0.659931	0.329966	−	0.943993i	$-0.392963\pi$
$839$	1.34556e7	0.659931	0.329966	+	0.943993i	$0.392963\pi$
$840$	0	0
$841$	−2.03741e7	−0.993317
$842$	−1.19804e7	−0.582360
$843$	6.25897e7	3.03343
$844$	−1.04437e7	−0.504661
$845$	0	0
$846$	−2.67333e7	−1.28418
$847$	−2.25681e6	−0.108090
$848$	5.45720e6	0.260604
$849$	1.15666e7	0.550727
$850$	0	0
$851$	1.14785e7	0.543326
$852$	−2.13084e7	−1.00566
$853$	2.46799e7	1.16137	0.580685	−	0.814129i	$-0.302785\pi$
$853$	2.46799e7	1.16137	0.580685	+	0.814129i	$0.302785\pi$
$854$	−274348.	−0.0128723
$855$	0	0
$856$	383973.	0.0179109
$857$	1.53970e6	0.0716118	0.0358059	−	0.999359i	$-0.488600\pi$
$857$	1.53970e6	0.0716118	0.0358059	+	0.999359i	$0.488600\pi$
$858$	9.56723e6	0.443678
$859$	2.43418e7	1.12556	0.562781	−	0.826606i	$-0.309732\pi$
$859$	2.43418e7	1.12556	0.562781	+	0.826606i	$0.309732\pi$
$860$	0	0
$861$	−6.19875e6	−0.284968
$862$	−4.92858e6	−0.225920
$863$	−2.27132e7	−1.03813	−0.519064	−	0.854735i	$-0.673720\pi$
$863$	−2.27132e7	−1.03813	−0.519064	+	0.854735i	$0.673720\pi$
$864$	−6.87403e7	−3.13276
$865$	0	0
$866$	359681.	0.0162976
$867$	−8.76474e7	−3.95996
$868$	1.27049e6	0.0572361
$869$	1.05945e7	0.475917
$870$	0	0
$871$	−3.94733e6	−0.176302
$872$	8.09821e6	0.360660
$873$	9.75003e7	4.32982
$874$	316184.	0.0140011
$875$	0	0
$876$	−6.95897e6	−0.306397
$877$	−1.86650e7	−0.819464	−0.409732	−	0.912206i	$-0.634378\pi$
$877$	−1.86650e7	−0.819464	−0.409732	+	0.912206i	$0.634378\pi$
$878$	−526817.	−0.0230634
$879$	7.23886e7	3.16008
$880$	0	0
$881$	2.91491e7	1.26528	0.632639	−	0.774447i	$-0.281972\pi$
$881$	2.91491e7	1.26528	0.632639	+	0.774447i	$0.281972\pi$
$882$	−3.38049e7	−1.46322
$883$	−837238.	−0.0361366	−0.0180683	−	0.999837i	$-0.505752\pi$
$883$	−837238.	−0.0361366	−0.0180683	+	0.999837i	$0.505752\pi$
$884$	−7.68295e6	−0.330672
$885$	0	0
$886$	1.48267e7	0.634543
$887$	2.00532e7	0.855806	0.427903	−	0.903825i	$-0.359252\pi$
$887$	2.00532e7	0.855806	0.427903	+	0.903825i	$0.359252\pi$
$888$	2.10176e7	0.894437
$889$	−1.01164e6	−0.0429309
$890$	0	0
$891$	−1.11612e8	−4.70997
$892$	2.92480e7	1.23079
$893$	−467088.	−0.0196006
$894$	−4.80600e7	−2.01113
$895$	0	0
$896$	1.66951e6	0.0694733
$897$	−1.39156e7	−0.577459
$898$	1.35186e6	0.0559425
$899$	−1.87161e6	−0.0772353
$900$	0	0
$901$	8.01680e7	3.28995
$902$	3.46428e7	1.41774
$903$	−2.70708e6	−0.110480
$904$	−3.42290e7	−1.39307
$905$	0	0
$906$	3.35208e7	1.35673
$907$	−2.26638e7	−0.914775	−0.457388	−	0.889267i	$-0.651215\pi$
$907$	−2.26638e7	−0.914775	−0.457388	+	0.889267i	$0.651215\pi$
$908$	1.56306e7	0.629162
$909$	−7.00989e7	−2.81386
$910$	0	0
$911$	−1.78322e7	−0.711885	−0.355942	−	0.934508i	$-0.615840\pi$
$911$	−1.78322e7	−0.711885	−0.355942	+	0.934508i	$0.615840\pi$
$912$	−149350.	−0.00594589
$913$	5.02682e7	1.99580
$914$	1.71385e7	0.678589
$915$	0	0
$916$	1.23436e7	0.486077
$917$	223962.	0.00879533
$918$	−7.73083e7	−3.02775
$919$	−1.61471e7	−0.630677	−0.315338	−	0.948979i	$-0.602118\pi$
$919$	−1.61471e7	−0.630677	−0.315338	+	0.948979i	$0.602118\pi$
$920$	0	0
$921$	4.70178e7	1.82647
$922$	−1.58792e7	−0.615178
$923$	−5.60521e6	−0.216565
$924$	4.43674e6	0.170956
$925$	0	0
$926$	−1.24338e7	−0.476514
$927$	6.60109e7	2.52299
$928$	−2.21010e6	−0.0842445
$929$	−4.41295e7	−1.67761	−0.838803	−	0.544435i	$-0.816744\pi$
$929$	−4.41295e7	−1.67761	−0.838803	+	0.544435i	$0.816744\pi$
$930$	0	0
$931$	−590644.	−0.0223332
$932$	−6.12708e6	−0.231054
$933$	−9.32536e7	−3.50721
$934$	−5.26022e6	−0.197305
$935$	0	0
$936$	−1.84058e7	−0.686698
$937$	2.76747e7	1.02976	0.514878	−	0.857263i	$-0.327837\pi$
$937$	2.76747e7	1.02976	0.514878	+	0.857263i	$0.327837\pi$
$938$	866876.	0.0321699
$939$	−4.70214e7	−1.74033
$940$	0	0
$941$	−9.84362e6	−0.362394	−0.181197	−	0.983447i	$-0.557997\pi$
$941$	−9.84362e6	−0.362394	−0.181197	+	0.983447i	$0.557997\pi$
$942$	−2.11356e7	−0.776044
$943$	−5.03882e7	−1.84522
$944$	2.54891e6	0.0930947
$945$	0	0
$946$	1.51290e7	0.549645
$947$	1.20978e7	0.438360	0.219180	−	0.975684i	$-0.429662\pi$
$947$	1.20978e7	0.438360	0.219180	+	0.975684i	$0.429662\pi$
$948$	−1.14070e7	−0.412240
$949$	−1.83056e6	−0.0659810
$950$	0	0
$951$	4.30541e7	1.54370
$952$	4.17353e6	0.149249
$953$	−2.22761e7	−0.794524	−0.397262	−	0.917705i	$-0.630040\pi$
$953$	−2.22761e7	−0.794524	−0.397262	+	0.917705i	$0.630040\pi$
$954$	7.76437e7	2.76207
$955$	0	0
$956$	3.11070e6	0.110081
$957$	−6.53596e6	−0.230690
$958$	8.45627e6	0.297690
$959$	925420.	0.0324932
$960$	0	0
$961$	−3.07551e6	−0.107426
$962$	2.23511e6	0.0778685
$963$	−1.40930e6	−0.0489708
$964$	8.03309e6	0.278413
$965$	0	0
$966$	3.05602e6	0.105369
$967$	−1.59432e7	−0.548288	−0.274144	−	0.961689i	$-0.588395\pi$
$967$	−1.59432e7	−0.548288	−0.274144	+	0.961689i	$0.588395\pi$
$968$	−3.35905e7	−1.15220
$969$	−2.19399e6	−0.0750629
$970$	0	0
$971$	1.41319e6	0.0481008	0.0240504	−	0.999711i	$-0.492344\pi$
$971$	1.41319e6	0.0481008	0.0240504	+	0.999711i	$0.492344\pi$
$972$	5.94038e7	2.01673
$973$	−3.30955e6	−0.112069
$974$	−1.11326e7	−0.376010
$975$	0	0
$976$	1.05339e6	0.0353969
$977$	−3.55699e7	−1.19219	−0.596096	−	0.802913i	$-0.703282\pi$
$977$	−3.55699e7	−1.19219	−0.596096	+	0.802913i	$0.703282\pi$
$978$	4.60923e7	1.54092
$979$	2.77458e7	0.925211
$980$	0	0
$981$	−2.97229e7	−0.986095
$982$	−8.79548e6	−0.291059
$983$	4.80377e7	1.58562	0.792808	−	0.609471i	$-0.208618\pi$
$983$	4.80377e7	1.58562	0.792808	+	0.609471i	$0.208618\pi$
$984$	−9.22627e7	−3.03765
$985$	0	0
$986$	−2.48557e6	−0.0814206
$987$	−4.51455e6	−0.147510
$988$	−130011.	−0.00423728
$989$	−2.20052e7	−0.715377
$990$	0	0
$991$	−4.57999e7	−1.48143	−0.740714	−	0.671821i	$-0.765512\pi$
$991$	−4.57999e7	−1.48143	−0.740714	+	0.671821i	$0.765512\pi$
$992$	3.01752e7	0.973577
$993$	−9.52409e7	−3.06514
$994$	1.23096e6	0.0395166
$995$	0	0
$996$	−5.41233e7	−1.72877
$997$	−1.03492e7	−0.329738	−0.164869	−	0.986315i	$-0.552720\pi$
$997$	−1.03492e7	−0.329738	−0.164869	+	0.986315i	$0.552720\pi$
$998$	−3.22498e7	−1.02494
$999$	−4.74922e7	−1.50560

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.a.h.1.6	✓	9
5.2	odd	4		325.6.b.h.274.12		18
5.3	odd	4		325.6.b.h.274.7		18
5.4	even	2		325.6.a.i.1.4	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.6.a.h.1.6	✓	9	1.1	even	1	trivial
325.6.a.i.1.4	yes	9	5.4	even	2
325.6.b.h.274.7		18	5.3	odd	4
325.6.b.h.274.12		18	5.2	odd	4

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:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	325.a (trivial)

\(f(q)\)	\(=\)	\(q+3.20685 q^{2} -29.5845 q^{3} -21.7161 q^{4} -94.8732 q^{6} -11.5734 q^{7} -172.260 q^{8} +632.245 q^{9} -596.700 q^{11} +642.461 q^{12} +169.000 q^{13} -37.1142 q^{14} +142.504 q^{16} +2093.43 q^{17} +2027.52 q^{18} +35.4251 q^{19} +342.394 q^{21} -1913.53 q^{22} +2783.24 q^{23} +5096.22 q^{24} +541.958 q^{26} -11515.6 q^{27} +251.329 q^{28} -370.244 q^{29} +5055.06 q^{31} +5969.30 q^{32} +17653.1 q^{33} +6713.33 q^{34} -13729.9 q^{36} +4124.15 q^{37} +113.603 q^{38} -4999.79 q^{39} -18104.1 q^{41} +1098.01 q^{42} -7906.34 q^{43} +12958.0 q^{44} +8925.44 q^{46} -13185.2 q^{47} -4215.93 q^{48} -16673.1 q^{49} -61933.3 q^{51} -3670.02 q^{52} +38295.0 q^{53} -36929.0 q^{54} +1993.63 q^{56} -1048.03 q^{57} -1187.32 q^{58} +17886.6 q^{59} +7392.00 q^{61} +16210.8 q^{62} -7317.23 q^{63} +14582.5 q^{64} +56610.8 q^{66} -23357.0 q^{67} -45461.2 q^{68} -82340.9 q^{69} -33166.9 q^{71} -108910. q^{72} -10831.7 q^{73} +13225.5 q^{74} -769.295 q^{76} +6905.85 q^{77} -16033.6 q^{78} -17755.2 q^{79} +187049. q^{81} -58057.3 q^{82} -84243.8 q^{83} -7435.46 q^{84} -25354.5 q^{86} +10953.5 q^{87} +102787. q^{88} -46498.8 q^{89} -1955.91 q^{91} -60441.1 q^{92} -149552. q^{93} -42283.1 q^{94} -176599. q^{96} +154213. q^{97} -53468.0 q^{98} -377260. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 5 q^{2} - 11 q^{3} + 91 q^{4} - 83 q^{6} + 12 q^{7} - 639 q^{8} + 562 q^{9} - 1422 q^{11} + 1567 q^{12} + 1521 q^{13} - 342 q^{14} - 1061 q^{16} + 648 q^{17} + 418 q^{18} - 408 q^{19} - 3912 q^{21}+ \cdots - 757776 q^{99}+O(q^{100}) \) 9 * q - 5 * q^2 - 11 * q^3 + 91 * q^4 - 83 * q^6 + 12 * q^7 - 639 * q^8 + 562 * q^9 - 1422 * q^11 + 1567 * q^12 + 1521 * q^13 - 342 * q^14 - 1061 * q^16 + 648 * q^17 + 418 * q^18 - 408 * q^19 - 3912 * q^21 + 4345 * q^22 + 1839 * q^23 - 8469 * q^24 - 845 * q^26 - 7649 * q^27 - 2836 * q^28 - 8737 * q^29 + 748 * q^31 - 423 * q^32 - 356 * q^33 - 17789 * q^34 + 512 * q^36 - 15486 * q^37 - 3425 * q^38 - 1859 * q^39 - 28676 * q^41 + 6876 * q^42 + 28665 * q^43 - 30599 * q^44 - 12056 * q^46 - 29452 * q^47 + 64759 * q^48 - 40907 * q^49 - 31006 * q^51 + 15379 * q^52 + 75977 * q^53 - 102761 * q^54 - 23002 * q^56 - 38038 * q^57 + 142384 * q^58 - 88142 * q^59 + 28165 * q^61 - 137308 * q^62 + 41492 * q^63 - 100845 * q^64 + 42577 * q^66 - 94754 * q^67 + 89267 * q^68 - 181747 * q^69 - 70562 * q^71 - 263778 * q^72 + 60602 * q^73 - 135676 * q^74 + 46373 * q^76 - 140292 * q^77 - 14027 * q^78 - 164073 * q^79 - 69935 * q^81 - 72887 * q^82 - 22458 * q^83 - 345656 * q^84 - 294920 * q^86 - 87031 * q^87 + 430607 * q^88 - 252698 * q^89 + 2028 * q^91 - 237824 * q^92 + 56556 * q^93 - 501606 * q^94 - 319181 * q^96 + 137986 * q^97 + 378699 * q^98 - 757776 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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