LMFDB - Embedded newform 325.6.a.i.1.3

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			1.3
Root			$5.30592$ 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)

$f(q)$	$=$	\(q-4.30592 q^{2} -19.9083 q^{3} -13.4591 q^{4} +85.7235 q^{6} -125.783 q^{7} +195.743 q^{8} +153.341 q^{9} -709.332 q^{11} +267.947 q^{12} -169.000 q^{13} +541.610 q^{14} -412.163 q^{16} +1927.07 q^{17} -660.272 q^{18} -2166.48 q^{19} +2504.12 q^{21} +3054.33 q^{22} +305.642 q^{23} -3896.91 q^{24} +727.700 q^{26} +1784.97 q^{27} +1692.92 q^{28} +4514.28 q^{29} +4078.41 q^{31} -4489.03 q^{32} +14121.6 q^{33} -8297.79 q^{34} -2063.82 q^{36} +12350.6 q^{37} +9328.69 q^{38} +3364.50 q^{39} +14430.4 q^{41} -10782.5 q^{42} -17133.7 q^{43} +9546.95 q^{44} -1316.07 q^{46} +18349.2 q^{47} +8205.47 q^{48} -985.723 q^{49} -38364.6 q^{51} +2274.58 q^{52} +11938.3 q^{53} -7685.93 q^{54} -24621.1 q^{56} +43131.0 q^{57} -19438.1 q^{58} -39854.6 q^{59} +4490.25 q^{61} -17561.3 q^{62} -19287.6 q^{63} +32518.6 q^{64} -60806.5 q^{66} +30488.3 q^{67} -25936.5 q^{68} -6084.82 q^{69} -14561.2 q^{71} +30015.3 q^{72} -59007.3 q^{73} -53180.6 q^{74} +29158.8 q^{76} +89221.7 q^{77} -14487.3 q^{78} -43350.2 q^{79} -72797.4 q^{81} -62136.2 q^{82} +83849.9 q^{83} -33703.1 q^{84} +73776.2 q^{86} -89871.6 q^{87} -138847. q^{88} +124592. q^{89} +21257.3 q^{91} -4113.66 q^{92} -81194.2 q^{93} -79010.1 q^{94} +89369.1 q^{96} -93842.4 q^{97} +4244.44 q^{98} -108769. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−4.30592	−0.761186	−0.380593	−	0.924743i	$-0.624280\pi$
$2$	−4.30592	−0.761186	−0.380593	+	0.924743i	$0.624280\pi$
$3$	−19.9083	−1.27712	−0.638559	−	0.769573i	$-0.720469\pi$
$3$	−19.9083	−1.27712	−0.638559	+	0.769573i	$0.720469\pi$
$4$	−13.4591	−0.420596
$5$	0	0
$5$	0	0
$6$	85.7235	0.972125
$7$	−125.783	−0.970232	−0.485116	−	0.874450i	$-0.661223\pi$
$7$	−125.783	−0.970232	−0.485116	+	0.874450i	$0.661223\pi$
$8$	195.743	1.08134
$9$	153.341	0.631031
$10$	0	0
$11$	−709.332	−1.76754	−0.883768	−	0.467926i	$-0.845001\pi$
$11$	−709.332	−1.76754	−0.883768	+	0.467926i	$0.845001\pi$
$12$	267.947	0.537151
$13$	−169.000	−0.277350
$13$	−169.000	−0.277350
$14$	541.610	0.738527
$15$	0	0
$16$	−412.163	−0.402503
$17$	1927.07	1.61724	0.808620	−	0.588332i	$-0.200215\pi$
$17$	1927.07	1.61724	0.808620	+	0.588332i	$0.200215\pi$
$18$	−660.272	−0.480332
$19$	−2166.48	−1.37680	−0.688400	−	0.725331i	$-0.741687\pi$
$19$	−2166.48	−1.37680	−0.688400	+	0.725331i	$0.741687\pi$
$20$	0	0
$21$	2504.12	1.23910
$22$	3054.33	1.34542
$23$	305.642	0.120474	0.0602371	−	0.998184i	$-0.480814\pi$
$23$	305.642	0.120474	0.0602371	+	0.998184i	$0.480814\pi$
$24$	−3896.91	−1.38100
$25$	0	0
$26$	727.700	0.211115
$27$	1784.97	0.471217
$28$	1692.92	0.408076
$29$	4514.28	0.996766	0.498383	−	0.866957i	$-0.333927\pi$
$29$	4514.28	0.996766	0.498383	+	0.866957i	$0.333927\pi$
$30$	0	0
$31$	4078.41	0.762231	0.381116	−	0.924527i	$-0.375540\pi$
$31$	4078.41	0.762231	0.381116	+	0.924527i	$0.375540\pi$
$32$	−4489.03	−0.774958
$33$	14121.6	2.25735
$34$	−8297.79	−1.23102
$35$	0	0
$36$	−2063.82	−0.265409
$37$	12350.6	1.48314	0.741572	−	0.670873i	$-0.234080\pi$
$37$	12350.6	1.48314	0.741572	+	0.670873i	$0.234080\pi$
$38$	9328.69	1.04800
$39$	3364.50	0.354209
$40$	0	0
$41$	14430.4	1.34066	0.670331	−	0.742062i	$-0.266152\pi$
$41$	14430.4	1.34066	0.670331	+	0.742062i	$0.266152\pi$
$42$	−10782.5	−0.943187
$43$	−17133.7	−1.41312	−0.706561	−	0.707652i	$-0.749754\pi$
$43$	−17133.7	−1.41312	−0.706561	+	0.707652i	$0.749754\pi$
$44$	9546.95	0.743418
$45$	0	0
$46$	−1316.07	−0.0917032
$47$	18349.2	1.21164	0.605818	−	0.795603i	$-0.292846\pi$
$47$	18349.2	1.21164	0.605818	+	0.795603i	$0.292846\pi$
$48$	8205.47	0.514044
$49$	−985.723	−0.0586496
$50$	0	0
$51$	−38364.6	−2.06541
$52$	2274.58	0.116652
$53$	11938.3	0.583785	0.291893	−	0.956451i	$-0.405715\pi$
$53$	11938.3	0.583785	0.291893	+	0.956451i	$0.405715\pi$
$54$	−7685.93	−0.358684
$55$	0	0
$56$	−24621.1	−1.04915
$57$	43131.0	1.75834
$58$	−19438.1	−0.758724
$59$	−39854.6	−1.49056	−0.745278	−	0.666754i	$-0.767683\pi$
$59$	−39854.6	−1.49056	−0.745278	+	0.666754i	$0.767683\pi$
$60$	0	0
$61$	4490.25	0.154506	0.0772530	−	0.997012i	$-0.475385\pi$
$61$	4490.25	0.154506	0.0772530	+	0.997012i	$0.475385\pi$
$62$	−17561.3	−0.580200
$63$	−19287.6	−0.612247
$64$	32518.6	0.992390
$65$	0	0
$66$	−60806.5	−1.71826
$67$	30488.3	0.829749	0.414874	−	0.909879i	$-0.363826\pi$
$67$	30488.3	0.829749	0.414874	+	0.909879i	$0.363826\pi$
$68$	−25936.5	−0.680204
$69$	−6084.82	−0.153860
$70$	0	0
$71$	−14561.2	−0.342808	−0.171404	−	0.985201i	$-0.554830\pi$
$71$	−14561.2	−0.342808	−0.171404	+	0.985201i	$0.554830\pi$
$72$	30015.3	0.682358
$73$	−59007.3	−1.29598	−0.647990	−	0.761649i	$-0.724390\pi$
$73$	−59007.3	−1.29598	−0.647990	+	0.761649i	$0.724390\pi$
$74$	−53180.6	−1.12895
$75$	0	0
$76$	29158.8	0.579076
$77$	89221.7	1.71492
$78$	−14487.3	−0.269619
$79$	−43350.2	−0.781489	−0.390745	−	0.920499i	$-0.627782\pi$
$79$	−43350.2	−0.781489	−0.390745	+	0.920499i	$0.627782\pi$
$80$	0	0
$81$	−72797.4	−1.23283
$82$	−62136.2	−1.02049
$83$	83849.9	1.33600	0.668002	−	0.744160i	$-0.267150\pi$
$83$	83849.9	1.33600	0.668002	+	0.744160i	$0.267150\pi$
$84$	−33703.1	−0.521161
$85$	0	0
$86$	73776.2	1.07565
$87$	−89871.6	−1.27299
$88$	−138847.	−1.91130
$89$	124592.	1.66731	0.833656	−	0.552284i	$-0.186244\pi$
$89$	124592.	1.66731	0.833656	+	0.552284i	$0.186244\pi$
$90$	0	0
$91$	21257.3	0.269094
$92$	−4113.66	−0.0506709
$93$	−81194.2	−0.973459
$94$	−79010.1	−0.922281
$95$	0	0
$96$	89369.1	0.989713
$97$	−93842.4	−1.01267	−0.506337	−	0.862335i	$-0.669001\pi$
$97$	−93842.4	−1.01267	−0.506337	+	0.862335i	$0.669001\pi$
$98$	4244.44	0.0446432
$99$	−108769.	−1.11537
$100$	0	0
$101$	−117081.	−1.14205	−0.571023	−	0.820934i	$-0.693453\pi$
$101$	−117081.	−1.14205	−0.571023	+	0.820934i	$0.693453\pi$
$102$	165195.	1.57216
$103$	−116464.	−1.08168	−0.540842	−	0.841124i	$-0.681894\pi$
$103$	−116464.	−1.08168	−0.540842	+	0.841124i	$0.681894\pi$
$104$	−33080.6	−0.299909
$105$	0	0
$106$	−51405.4	−0.444369
$107$	−43117.6	−0.364079	−0.182039	−	0.983291i	$-0.558270\pi$
$107$	−43117.6	−0.364079	−0.182039	+	0.983291i	$0.558270\pi$
$108$	−24024.0	−0.198192
$109$	32460.7	0.261693	0.130846	−	0.991403i	$-0.458231\pi$
$109$	32460.7	0.261693	0.130846	+	0.991403i	$0.458231\pi$
$110$	0	0
$111$	−245879.	−1.89415
$112$	51843.0	0.390522
$113$	−73976.7	−0.545003	−0.272501	−	0.962155i	$-0.587851\pi$
$113$	−73976.7	−0.545003	−0.272501	+	0.962155i	$0.587851\pi$
$114$	−185718.	−1.33842
$115$	0	0
$116$	−60757.9	−0.419236
$117$	−25914.5	−0.175017
$118$	171611.	1.13459
$119$	−242392.	−1.56910
$120$	0	0
$121$	342101.	2.12418
$122$	−19334.6	−0.117608
$123$	−287285.	−1.71218
$124$	−54891.6	−0.320591
$125$	0	0
$126$	83050.7	0.466034
$127$	250540.	1.37838	0.689189	−	0.724582i	$-0.257967\pi$
$127$	250540.	1.37838	0.689189	+	0.724582i	$0.257967\pi$
$128$	3626.46	0.0195640
$129$	341103.	1.80472
$130$	0	0
$131$	98431.4	0.501136	0.250568	−	0.968099i	$-0.419383\pi$
$131$	98431.4	0.501136	0.250568	+	0.968099i	$0.419383\pi$
$132$	−190064.	−0.949433
$133$	272506.	1.33582
$134$	−131280.	−0.631593
$135$	0	0
$136$	377210.	1.74878
$137$	302938.	1.37896	0.689480	−	0.724305i	$-0.257839\pi$
$137$	302938.	1.37896	0.689480	+	0.724305i	$0.257839\pi$
$138$	26200.7	0.117116
$139$	−215720.	−0.947006	−0.473503	−	0.880792i	$-0.657011\pi$
$139$	−215720.	−0.947006	−0.473503	+	0.880792i	$0.657011\pi$
$140$	0	0
$141$	−365301.	−1.54740
$142$	62699.3	0.260941
$143$	119877.	0.490226
$144$	−63201.4	−0.253992
$145$	0	0
$146$	254080.	0.986482
$147$	19624.1	0.0749024
$148$	−166227.	−0.623804
$149$	−193167.	−0.712798	−0.356399	−	0.934334i	$-0.615996\pi$
$149$	−193167.	−0.712798	−0.356399	+	0.934334i	$0.615996\pi$
$150$	0	0
$151$	−54010.3	−0.192768	−0.0963839	−	0.995344i	$-0.530728\pi$
$151$	−54010.3	−0.192768	−0.0963839	+	0.995344i	$0.530728\pi$
$152$	−424074.	−1.48879
$153$	295497.	1.02053
$154$	−384181.	−1.30537
$155$	0	0
$156$	−45283.1	−0.148979
$157$	−386541.	−1.25155	−0.625773	−	0.780006i	$-0.715216\pi$
$157$	−386541.	−1.25155	−0.625773	+	0.780006i	$0.715216\pi$
$158$	186662.	0.594859
$159$	−237671.	−0.745563
$160$	0	0
$161$	−38444.5	−0.116888
$162$	313460.	0.938414
$163$	−106816.	−0.314896	−0.157448	−	0.987527i	$-0.550327\pi$
$163$	−106816.	−0.314896	−0.157448	+	0.987527i	$0.550327\pi$
$164$	−194220.	−0.563877
$165$	0	0
$166$	−361051.	−1.01695
$167$	371554.	1.03093	0.515467	−	0.856909i	$-0.327619\pi$
$167$	371554.	1.03093	0.515467	+	0.856909i	$0.327619\pi$
$168$	490164.	1.33989
$169$	28561.0	0.0769231
$170$	0	0
$171$	−332209.	−0.868803
$172$	230603.	0.594353
$173$	−274680.	−0.697769	−0.348884	−	0.937166i	$-0.613439\pi$
$173$	−274680.	−0.697769	−0.348884	+	0.937166i	$0.613439\pi$
$174$	386980.	0.968981
$175$	0	0
$176$	292361.	0.711439
$177$	793437.	1.90362
$178$	−536485.	−1.26913
$179$	317330.	0.740251	0.370125	−	0.928982i	$-0.379315\pi$
$179$	317330.	0.740251	0.370125	+	0.928982i	$0.379315\pi$
$180$	0	0
$181$	−241524.	−0.547979	−0.273989	−	0.961733i	$-0.588343\pi$
$181$	−241524.	−0.547979	−0.273989	+	0.961733i	$0.588343\pi$
$182$	−91532.1	−0.204831
$183$	−89393.2	−0.197322
$184$	59827.3	0.130273
$185$	0	0
$186$	349616.	0.740984
$187$	−1.36693e6	−2.85853
$188$	−246963.	−0.509609
$189$	−224518.	−0.457190
$190$	0	0
$191$	208827.	0.414193	0.207096	−	0.978321i	$-0.433599\pi$
$191$	208827.	0.414193	0.207096	+	0.978321i	$0.433599\pi$
$192$	−647391.	−1.26740
$193$	410293.	0.792868	0.396434	−	0.918063i	$-0.370248\pi$
$193$	410293.	0.792868	0.396434	+	0.918063i	$0.370248\pi$
$194$	404078.	0.770834
$195$	0	0
$196$	13266.9	0.0246678
$197$	−466919.	−0.857187	−0.428594	−	0.903497i	$-0.640991\pi$
$197$	−466919.	−0.857187	−0.428594	+	0.903497i	$0.640991\pi$
$198$	468352.	0.849004
$199$	655337.	1.17309	0.586546	−	0.809916i	$-0.300487\pi$
$199$	655337.	1.17309	0.586546	+	0.809916i	$0.300487\pi$
$200$	0	0
$201$	−606971.	−1.05969
$202$	504142.	0.869309
$203$	−567818.	−0.967094
$204$	516352.	0.868701
$205$	0	0
$206$	501486.	0.823362
$207$	46867.3	0.0760229
$208$	69655.6	0.111634
$209$	1.53676e6	2.43354
$210$	0	0
$211$	−508536.	−0.786350	−0.393175	−	0.919464i	$-0.628623\pi$
$211$	−508536.	−0.786350	−0.393175	+	0.919464i	$0.628623\pi$
$212$	−160679.	−0.245538
$213$	289889.	0.437806
$214$	185661.	0.277132
$215$	0	0
$216$	349395.	0.509545
$217$	−512993.	−0.739541
$218$	−139773.	−0.199197
$219$	1.17473e6	1.65512
$220$	0	0
$221$	−325674.	−0.448542
$222$	1.05874e6	1.44180
$223$	−97604.2	−0.131434	−0.0657168	−	0.997838i	$-0.520933\pi$
$223$	−97604.2	−0.131434	−0.0657168	+	0.997838i	$0.520933\pi$
$224$	564643.	0.751889
$225$	0	0
$226$	318538.	0.414849
$227$	363749.	0.468530	0.234265	−	0.972173i	$-0.424732\pi$
$227$	363749.	0.468530	0.234265	+	0.972173i	$0.424732\pi$
$228$	−580502.	−0.739549
$229$	549995.	0.693059	0.346530	−	0.938039i	$-0.387360\pi$
$229$	549995.	0.693059	0.346530	+	0.938039i	$0.387360\pi$
$230$	0	0
$231$	−1.77625e6	−2.19016
$232$	883638.	1.07784
$233$	444960.	0.536947	0.268473	−	0.963287i	$-0.413481\pi$
$233$	444960.	0.536947	0.268473	+	0.963287i	$0.413481\pi$
$234$	111586.	0.133220
$235$	0	0
$236$	536406.	0.626922
$237$	863028.	0.998054
$238$	1.04372e6	1.19438
$239$	1.06065e6	1.20110	0.600549	−	0.799588i	$-0.294949\pi$
$239$	1.06065e6	1.20110	0.600549	+	0.799588i	$0.294949\pi$
$240$	0	0
$241$	642027.	0.712051	0.356025	−	0.934476i	$-0.384132\pi$
$241$	642027.	0.712051	0.356025	+	0.934476i	$0.384132\pi$
$242$	−1.47306e6	−1.61690
$243$	1.01553e6	1.10325
$244$	−60434.5	−0.0649846
$245$	0	0
$246$	1.23703e6	1.30329
$247$	366135.	0.381856
$248$	798320.	0.824229
$249$	−1.66931e6	−1.70623
$250$	0	0
$251$	386423.	0.387149	0.193575	−	0.981086i	$-0.437992\pi$
$251$	386423.	0.387149	0.193575	+	0.981086i	$0.437992\pi$
$252$	259593.	0.257508
$253$	−216802.	−0.212942
$254$	−1.07881e6	−1.04920
$255$	0	0
$256$	−1.05621e6	−1.00728
$257$	−432043.	−0.408032	−0.204016	−	0.978968i	$-0.565399\pi$
$257$	−432043.	−0.408032	−0.204016	+	0.978968i	$0.565399\pi$
$258$	−1.46876e6	−1.37373
$259$	−1.55349e6	−1.43899
$260$	0	0
$261$	692222.	0.628990
$262$	−423837.	−0.381457
$263$	−120603.	−0.107515	−0.0537575	−	0.998554i	$-0.517120\pi$
$263$	−120603.	−0.107515	−0.0537575	+	0.998554i	$0.517120\pi$
$264$	2.76421e6	2.44096
$265$	0	0
$266$	−1.17339e6	−1.01680
$267$	−2.48043e6	−2.12935
$268$	−410344.	−0.348989
$269$	−2.20375e6	−1.85687	−0.928436	−	0.371494i	$-0.878846\pi$
$269$	−2.20375e6	−1.85687	−0.928436	+	0.371494i	$0.878846\pi$
$270$	0	0
$271$	599069.	0.495511	0.247756	−	0.968823i	$-0.420307\pi$
$271$	599069.	0.495511	0.247756	+	0.968823i	$0.420307\pi$
$272$	−794266.	−0.650944
$273$	−423196.	−0.343665
$274$	−1.30442e6	−1.04965
$275$	0	0
$276$	81896.0	0.0647127
$277$	−67945.4	−0.0532060	−0.0266030	−	0.999646i	$-0.508469\pi$
$277$	−67945.4	−0.0532060	−0.0266030	+	0.999646i	$0.508469\pi$
$278$	928871.	0.720847
$279$	625386.	0.480991
$280$	0	0
$281$	79333.1	0.0599361	0.0299681	−	0.999551i	$-0.490459\pi$
$281$	79333.1	0.0599361	0.0299681	+	0.999551i	$0.490459\pi$
$282$	1.57296e6	1.17786
$283$	810998.	0.601941	0.300970	−	0.953633i	$-0.402689\pi$
$283$	810998.	0.601941	0.300970	+	0.953633i	$0.402689\pi$
$284$	195980.	0.144184
$285$	0	0
$286$	−516181.	−0.373153
$287$	−1.81510e6	−1.30075
$288$	−688351.	−0.489022
$289$	2.29373e6	1.61546
$290$	0	0
$291$	1.86824e6	1.29331
$292$	794183.	0.545084
$293$	2.31923e6	1.57825	0.789123	−	0.614236i	$-0.210536\pi$
$293$	2.31923e6	1.57825	0.789123	+	0.614236i	$0.210536\pi$
$294$	−84499.7	−0.0570147
$295$	0	0
$296$	2.41754e6	1.60378
$297$	−1.26614e6	−0.832893
$298$	831760.	0.542572
$299$	−51653.5	−0.0334135
$300$	0	0
$301$	2.15512e6	1.37106
$302$	232564.	0.146732
$303$	2.33089e6	1.45853
$304$	892944.	0.554167
$305$	0	0
$306$	−1.27239e6	−0.776812
$307$	804892.	0.487407	0.243703	−	0.969850i	$-0.421638\pi$
$307$	804892.	0.487407	0.243703	+	0.969850i	$0.421638\pi$
$308$	−1.20084e6	−0.721288
$309$	2.31861e6	1.38144
$310$	0	0
$311$	−2.49761e6	−1.46428	−0.732139	−	0.681156i	$-0.761478\pi$
$311$	−2.49761e6	−1.46428	−0.732139	+	0.681156i	$0.761478\pi$
$312$	658578.	0.383019
$313$	−146716.	−0.0846480	−0.0423240	−	0.999104i	$-0.513476\pi$
$313$	−146716.	−0.0846480	−0.0423240	+	0.999104i	$0.513476\pi$
$314$	1.66441e6	0.952659
$315$	0	0
$316$	583453.	0.328691
$317$	−2.37920e6	−1.32979	−0.664896	−	0.746936i	$-0.731524\pi$
$317$	−2.37920e6	−1.32979	−0.664896	+	0.746936i	$0.731524\pi$
$318$	1.02339e6	0.567512
$319$	−3.20212e6	−1.76182
$320$	0	0
$321$	858399.	0.464972
$322$	165539.	0.0889734
$323$	−4.17495e6	−2.22662
$324$	979785.	0.518523
$325$	0	0
$326$	459941.	0.239695
$327$	−646237.	−0.334212
$328$	2.82466e6	1.44971
$329$	−2.30801e6	−1.17557
$330$	0	0
$331$	2.55669e6	1.28265	0.641324	−	0.767270i	$-0.278385\pi$
$331$	2.55669e6	1.28265	0.641324	+	0.767270i	$0.278385\pi$
$332$	−1.12854e6	−0.561917
$333$	1.89385e6	0.935910
$334$	−1.59988e6	−0.784732
$335$	0	0
$336$	−1.03211e6	−0.498742
$337$	−2.38453e6	−1.14374	−0.571871	−	0.820344i	$-0.693782\pi$
$337$	−2.38453e6	−1.14374	−0.571871	+	0.820344i	$0.693782\pi$
$338$	−122981.	−0.0585528
$339$	1.47275e6	0.696033
$340$	0	0
$341$	−2.89295e6	−1.34727
$342$	1.43047e6	0.661321
$343$	2.23802e6	1.02714
$344$	−3.35380e6	−1.52806
$345$	0	0
$346$	1.18275e6	0.531132
$347$	427794.	0.190727	0.0953633	−	0.995443i	$-0.469599\pi$
$347$	427794.	0.190727	0.0953633	+	0.995443i	$0.469599\pi$
$348$	1.20959e6	0.535413
$349$	−3.91198e6	−1.71922	−0.859612	−	0.510947i	$-0.829295\pi$
$349$	−3.91198e6	−1.71922	−0.859612	+	0.510947i	$0.829295\pi$
$350$	0	0
$351$	−301660.	−0.130692
$352$	3.18422e6	1.36977
$353$	−1.85968e6	−0.794330	−0.397165	−	0.917747i	$-0.630006\pi$
$353$	−1.85968e6	−0.794330	−0.397165	+	0.917747i	$0.630006\pi$
$354$	−3.41648e6	−1.44901
$355$	0	0
$356$	−1.67690e6	−0.701264
$357$	4.82560e6	2.00392
$358$	−1.36640e6	−0.563468
$359$	693152.	0.283852	0.141926	−	0.989877i	$-0.454670\pi$
$359$	693152.	0.283852	0.141926	+	0.989877i	$0.454670\pi$
$360$	0	0
$361$	2.21754e6	0.895578
$362$	1.03998e6	0.417114
$363$	−6.81066e6	−2.71283
$364$	−286103.	−0.113180
$365$	0	0
$366$	384920.	0.150199
$367$	−2.25998e6	−0.875870	−0.437935	−	0.899007i	$-0.644290\pi$
$367$	−2.25998e6	−0.875870	−0.437935	+	0.899007i	$0.644290\pi$
$368$	−125975.	−0.0484912
$369$	2.21277e6	0.846000
$370$	0	0
$371$	−1.50163e6	−0.566407
$372$	1.09280e6	0.409433
$373$	−802038.	−0.298485	−0.149243	−	0.988801i	$-0.547684\pi$
$373$	−802038.	−0.298485	−0.149243	+	0.988801i	$0.547684\pi$
$374$	5.88589e6	2.17587
$375$	0	0
$376$	3.59173e6	1.31019
$377$	−762913.	−0.276453
$378$	966756.	0.348007
$379$	1.44664e6	0.517325	0.258662	−	0.965968i	$-0.416718\pi$
$379$	1.44664e6	0.517325	0.258662	+	0.965968i	$0.416718\pi$
$380$	0	0
$381$	−4.98783e6	−1.76035
$382$	−899191.	−0.315278
$383$	−3.55257e6	−1.23750	−0.618751	−	0.785587i	$-0.712361\pi$
$383$	−3.55257e6	−1.23750	−0.618751	+	0.785587i	$0.712361\pi$
$384$	−72196.7	−0.0249856
$385$	0	0
$386$	−1.76669e6	−0.603520
$387$	−2.62729e6	−0.891724
$388$	1.26303e6	0.425927
$389$	−5.57943e6	−1.86946	−0.934730	−	0.355359i	$-0.884358\pi$
$389$	−5.57943e6	−1.86946	−0.934730	+	0.355359i	$0.884358\pi$
$390$	0	0
$391$	588993.	0.194836
$392$	−192948.	−0.0634200
$393$	−1.95960e6	−0.640009
$394$	2.01051e6	0.652479
$395$	0	0
$396$	1.46393e6	0.469120
$397$	3.43746e6	1.09462	0.547308	−	0.836931i	$-0.315653\pi$
$397$	3.43746e6	1.09462	0.547308	+	0.836931i	$0.315653\pi$
$398$	−2.82183e6	−0.892942
$399$	−5.42513e6	−1.70599
$400$	0	0
$401$	−4.49645e6	−1.39640	−0.698199	−	0.715904i	$-0.746015\pi$
$401$	−4.49645e6	−1.39640	−0.698199	+	0.715904i	$0.746015\pi$
$402$	2.61357e6	0.806619
$403$	−689251.	−0.211405
$404$	1.57580e6	0.480340
$405$	0	0
$406$	2.44498e6	0.736139
$407$	−8.76068e6	−2.62151
$408$	−7.50961e6	−2.23340
$409$	−4.00809e6	−1.18476	−0.592379	−	0.805659i	$-0.701811\pi$
$409$	−4.00809e6	−1.18476	−0.592379	+	0.805659i	$0.701811\pi$
$410$	0	0
$411$	−6.03097e6	−1.76109
$412$	1.56750e6	0.454952
$413$	5.01302e6	1.44619
$414$	−201807.	−0.0578676
$415$	0	0
$416$	758647.	0.214935
$417$	4.29461e6	1.20944
$418$	−6.61714e6	−1.85238
$419$	−30427.3	−0.00846697	−0.00423349	−	0.999991i	$-0.501348\pi$
$419$	−30427.3	−0.00846697	−0.00423349	+	0.999991i	$0.501348\pi$
$420$	0	0
$421$	1.28719e6	0.353946	0.176973	−	0.984216i	$-0.443369\pi$
$421$	1.28719e6	0.353946	0.176973	+	0.984216i	$0.443369\pi$
$422$	2.18972e6	0.598559
$423$	2.81367e6	0.764580
$424$	2.33684e6	0.631269
$425$	0	0
$426$	−1.24824e6	−0.333252
$427$	−564795.	−0.149907
$428$	580323.	0.153130
$429$	−2.38655e6	−0.626077
$430$	0	0
$431$	−3.27469e6	−0.849137	−0.424568	−	0.905396i	$-0.639574\pi$
$431$	−3.27469e6	−0.849137	−0.424568	+	0.905396i	$0.639574\pi$
$432$	−735699.	−0.189666
$433$	2.56449e6	0.657327	0.328664	−	0.944447i	$-0.393402\pi$
$433$	2.56449e6	0.657327	0.328664	+	0.944447i	$0.393402\pi$
$434$	2.20891e6	0.562928
$435$	0	0
$436$	−436890.	−0.110067
$437$	−662168.	−0.165869
$438$	−5.05831e6	−1.25985
$439$	973053.	0.240977	0.120488	−	0.992715i	$-0.461554\pi$
$439$	973053.	0.240977	0.120488	+	0.992715i	$0.461554\pi$
$440$	0	0
$441$	−151151.	−0.0370097
$442$	1.40233e6	0.341424
$443$	−1.98241e6	−0.479936	−0.239968	−	0.970781i	$-0.577137\pi$
$443$	−1.98241e6	−0.479936	−0.239968	+	0.970781i	$0.577137\pi$
$444$	3.30931e6	0.796672
$445$	0	0
$446$	420276.	0.100045
$447$	3.84562e6	0.910327
$448$	−4.09028e6	−0.962849
$449$	1.98450e6	0.464552	0.232276	−	0.972650i	$-0.425383\pi$
$449$	1.98450e6	0.464552	0.232276	+	0.972650i	$0.425383\pi$
$450$	0	0
$451$	−1.02360e7	−2.36967
$452$	995657.	0.229226
$453$	1.07525e6	0.246187
$454$	−1.56627e6	−0.356639
$455$	0	0
$456$	8.44258e6	1.90136
$457$	6.95085e6	1.55685	0.778427	−	0.627736i	$-0.216018\pi$
$457$	6.95085e6	1.55685	0.778427	+	0.627736i	$0.216018\pi$
$458$	−2.36824e6	−0.527547
$459$	3.43975e6	0.762071
$460$	0	0
$461$	−2.63571e6	−0.577625	−0.288812	−	0.957386i	$-0.593260\pi$
$461$	−2.63571e6	−0.577625	−0.288812	+	0.957386i	$0.593260\pi$
$462$	7.64840e6	1.66712
$463$	6.75578e6	1.46461	0.732306	−	0.680975i	$-0.238444\pi$
$463$	6.75578e6	1.46461	0.732306	+	0.680975i	$0.238444\pi$
$464$	−1.86062e6	−0.401202
$465$	0	0
$466$	−1.91596e6	−0.408716
$467$	7.34426e6	1.55832	0.779159	−	0.626826i	$-0.215646\pi$
$467$	7.34426e6	1.55832	0.779159	+	0.626826i	$0.215646\pi$
$468$	348786.	0.0736112
$469$	−3.83490e6	−0.805049
$470$	0	0
$471$	7.69538e6	1.59837
$472$	−7.80126e6	−1.61179
$473$	1.21535e7	2.49774
$474$	−3.71613e6	−0.759705
$475$	0	0
$476$	3.26236e6	0.659956
$477$	1.83063e6	0.368387
$478$	−4.56708e6	−0.914259
$479$	−6.14073e6	−1.22287	−0.611436	−	0.791294i	$-0.709408\pi$
$479$	−6.14073e6	−1.22287	−0.611436	+	0.791294i	$0.709408\pi$
$480$	0	0
$481$	−2.08725e6	−0.411350
$482$	−2.76452e6	−0.542003
$483$	765365.	0.149280
$484$	−4.60437e6	−0.893422
$485$	0	0
$486$	−4.37277e6	−0.839781
$487$	−1.73635e6	−0.331752	−0.165876	−	0.986147i	$-0.553045\pi$
$487$	−1.73635e6	−0.331752	−0.165876	+	0.986147i	$0.553045\pi$
$488$	878934.	0.167073
$489$	2.12653e6	0.402160
$490$	0	0
$491$	−1.66007e6	−0.310759	−0.155380	−	0.987855i	$-0.549660\pi$
$491$	−1.66007e6	−0.310759	−0.155380	+	0.987855i	$0.549660\pi$
$492$	3.86659e6	0.720138
$493$	8.69931e6	1.61201
$494$	−1.57655e6	−0.290663
$495$	0	0
$496$	−1.68097e6	−0.306801
$497$	1.83155e6	0.332603
$498$	7.18791e6	1.29876
$499$	7.82789e6	1.40732	0.703660	−	0.710536i	$-0.251548\pi$
$499$	7.82789e6	1.40732	0.703660	+	0.710536i	$0.251548\pi$
$500$	0	0
$501$	−7.39701e6	−1.31662
$502$	−1.66390e6	−0.294692
$503$	4.93710e6	0.870066	0.435033	−	0.900415i	$-0.356737\pi$
$503$	4.93710e6	0.870066	0.435033	+	0.900415i	$0.356737\pi$
$504$	−3.77541e6	−0.662045
$505$	0	0
$506$	933531.	0.162089
$507$	−568601.	−0.0982399
$508$	−3.37204e6	−0.579740
$509$	−8.26567e6	−1.41411	−0.707056	−	0.707158i	$-0.749977\pi$
$509$	−8.26567e6	−1.41411	−0.707056	+	0.707158i	$0.749977\pi$
$510$	0	0
$511$	7.42209e6	1.25740
$512$	4.43192e6	0.747165
$513$	−3.86710e6	−0.648772
$514$	1.86034e6	0.310588
$515$	0	0
$516$	−4.59092e6	−0.759059
$517$	−1.30157e7	−2.14161
$518$	6.68920e6	1.09534
$519$	5.46841e6	0.891133
$520$	0	0
$521$	7.91723e6	1.27785	0.638924	−	0.769270i	$-0.279380\pi$
$521$	7.91723e6	1.27785	0.638924	+	0.769270i	$0.279380\pi$
$522$	−2.98065e6	−0.478779
$523$	−6.02805e6	−0.963657	−0.481829	−	0.876265i	$-0.660027\pi$
$523$	−6.02805e6	−0.963657	−0.481829	+	0.876265i	$0.660027\pi$
$524$	−1.32479e6	−0.210776
$525$	0	0
$526$	519307.	0.0818389
$527$	7.85937e6	1.23271
$528$	−5.82041e6	−0.908592
$529$	−6.34293e6	−0.985486
$530$	0	0
$531$	−6.11132e6	−0.940587
$532$	−3.66767e6	−0.561838
$533$	−2.43874e6	−0.371833
$534$	1.06805e7	1.62083
$535$	0	0
$536$	5.96788e6	0.897239
$537$	−6.31750e6	−0.945388
$538$	9.48917e6	1.41342
$539$	699205.	0.103665
$540$	0	0
$541$	9.65945e6	1.41892	0.709462	−	0.704744i	$-0.248938\pi$
$541$	9.65945e6	1.41892	0.709462	+	0.704744i	$0.248938\pi$
$542$	−2.57954e6	−0.377176
$543$	4.80833e6	0.699834
$544$	−8.65067e6	−1.25329
$545$	0	0
$546$	1.82225e6	0.261593
$547$	1.23221e7	1.76083	0.880415	−	0.474203i	$-0.157264\pi$
$547$	1.23221e7	1.76083	0.880415	+	0.474203i	$0.157264\pi$
$548$	−4.07726e6	−0.579985
$549$	688537.	0.0974981
$550$	0	0
$551$	−9.78009e6	−1.37235
$552$	−1.19106e6	−0.166374
$553$	5.45270e6	0.758226
$554$	292567.	0.0404997
$555$	0	0
$556$	2.90338e6	0.398307
$557$	−6.45087e6	−0.881009	−0.440504	−	0.897750i	$-0.645200\pi$
$557$	−6.45087e6	−0.881009	−0.440504	+	0.897750i	$0.645200\pi$
$558$	−2.69286e6	−0.366124
$559$	2.89559e6	0.391929
$560$	0	0
$561$	2.72133e7	3.65068
$562$	−341602.	−0.0456226
$563$	−5.38104e6	−0.715476	−0.357738	−	0.933822i	$-0.616452\pi$
$563$	−5.38104e6	−0.715476	−0.357738	+	0.933822i	$0.616452\pi$
$564$	4.91661e6	0.650831
$565$	0	0
$566$	−3.49209e6	−0.458189
$567$	9.15665e6	1.19613
$568$	−2.85025e6	−0.370691
$569$	−1.19810e6	−0.155136	−0.0775680	−	0.996987i	$-0.524715\pi$
$569$	−1.19810e6	−0.155136	−0.0775680	+	0.996987i	$0.524715\pi$
$570$	0	0
$571$	−5.65494e6	−0.725834	−0.362917	−	0.931821i	$-0.618219\pi$
$571$	−5.65494e6	−0.725834	−0.362917	+	0.931821i	$0.618219\pi$
$572$	−1.61343e6	−0.206187
$573$	−4.15739e6	−0.528973
$574$	7.81566e6	0.990116
$575$	0	0
$576$	4.98643e6	0.626229
$577$	−5.03350e6	−0.629405	−0.314703	−	0.949190i	$-0.601905\pi$
$577$	−5.03350e6	−0.629405	−0.314703	+	0.949190i	$0.601905\pi$
$578$	−9.87661e6	−1.22967
$579$	−8.16823e6	−1.01259
$580$	0	0
$581$	−1.05469e7	−1.29623
$582$	−8.04451e6	−0.984446
$583$	−8.46823e6	−1.03186
$584$	−1.15503e7	−1.40139
$585$	0	0
$586$	−9.98641e6	−1.20134
$587$	7.95091e6	0.952405	0.476203	−	0.879336i	$-0.342013\pi$
$587$	7.95091e6	0.952405	0.476203	+	0.879336i	$0.342013\pi$
$588$	−264122.	−0.0315036
$589$	−8.83580e6	−1.04944
$590$	0	0
$591$	9.29556e6	1.09473
$592$	−5.09046e6	−0.596971
$593$	−3.79733e6	−0.443447	−0.221724	−	0.975110i	$-0.571168\pi$
$593$	−3.79733e6	−0.443447	−0.221724	+	0.975110i	$0.571168\pi$
$594$	5.45188e6	0.633986
$595$	0	0
$596$	2.59984e6	0.299800
$597$	−1.30467e7	−1.49818
$598$	222416.	0.0254339
$599$	1.28647e7	1.46498	0.732489	−	0.680779i	$-0.238359\pi$
$599$	1.28647e7	1.46498	0.732489	+	0.680779i	$0.238359\pi$
$600$	0	0
$601$	5.01991e6	0.566904	0.283452	−	0.958986i	$-0.408520\pi$
$601$	5.01991e6	0.566904	0.283452	+	0.958986i	$0.408520\pi$
$602$	−9.27977e6	−1.04363
$603$	4.67510e6	0.523597
$604$	726929.	0.0810773
$605$	0	0
$606$	−1.00366e7	−1.11021
$607$	1.26908e7	1.39803	0.699017	−	0.715105i	$-0.253621\pi$
$607$	1.26908e7	1.39803	0.699017	+	0.715105i	$0.253621\pi$
$608$	9.72541e6	1.06696
$609$	1.13043e7	1.23509
$610$	0	0
$611$	−3.10101e6	−0.336047
$612$	−3.97712e6	−0.429230
$613$	1.60182e7	1.72172	0.860861	−	0.508840i	$-0.169925\pi$
$613$	1.60182e7	1.72172	0.860861	+	0.508840i	$0.169925\pi$
$614$	−3.46580e6	−0.371007
$615$	0	0
$616$	1.74645e7	1.85441
$617$	4.75362e6	0.502703	0.251352	−	0.967896i	$-0.419125\pi$
$617$	4.75362e6	0.502703	0.251352	+	0.967896i	$0.419125\pi$
$618$	−9.98374e6	−1.05153
$619$	−1.08346e7	−1.13655	−0.568274	−	0.822839i	$-0.692389\pi$
$619$	−1.08346e7	−1.13655	−0.568274	+	0.822839i	$0.692389\pi$
$620$	0	0
$621$	545562.	0.0567695
$622$	1.07545e7	1.11459
$623$	−1.56716e7	−1.61768
$624$	−1.38673e6	−0.142570
$625$	0	0
$626$	631747.	0.0644329
$627$	−3.05942e7	−3.10792
$628$	5.20248e6	0.526395
$629$	2.38004e7	2.39860
$630$	0	0
$631$	−1.16256e7	−1.16236	−0.581182	−	0.813774i	$-0.697409\pi$
$631$	−1.16256e7	−1.16236	−0.581182	+	0.813774i	$0.697409\pi$
$632$	−8.48549e6	−0.845054
$633$	1.01241e7	1.00426
$634$	1.02447e7	1.01222
$635$	0	0
$636$	3.19884e6	0.313581
$637$	166587.	0.0162665
$638$	1.37881e7	1.34107
$639$	−2.23282e6	−0.216323
$640$	0	0
$641$	−1.49693e7	−1.43899	−0.719495	−	0.694498i	$-0.755627\pi$
$641$	−1.49693e7	−1.43899	−0.719495	+	0.694498i	$0.755627\pi$
$642$	−3.69619e6	−0.353930
$643$	1.67965e6	0.160210	0.0801051	−	0.996786i	$-0.474474\pi$
$643$	1.67965e6	0.160210	0.0801051	+	0.996786i	$0.474474\pi$
$644$	517427.	0.0491625
$645$	0	0
$646$	1.79770e7	1.69487
$647$	−6.56503e6	−0.616561	−0.308281	−	0.951295i	$-0.599754\pi$
$647$	−6.56503e6	−0.616561	−0.308281	+	0.951295i	$0.599754\pi$
$648$	−1.42496e7	−1.33311
$649$	2.82702e7	2.63461
$650$	0	0
$651$	1.02128e7	0.944482
$652$	1.43764e6	0.132444
$653$	1.78321e7	1.63652	0.818259	−	0.574850i	$-0.194940\pi$
$653$	1.78321e7	1.63652	0.818259	+	0.574850i	$0.194940\pi$
$654$	2.78264e6	0.254398
$655$	0	0
$656$	−5.94769e6	−0.539621
$657$	−9.04820e6	−0.817804
$658$	9.93810e6	0.894826
$659$	−1.18216e6	−0.106038	−0.0530190	−	0.998594i	$-0.516884\pi$
$659$	−1.18216e6	−0.106038	−0.0530190	+	0.998594i	$0.516884\pi$
$660$	0	0
$661$	1.20695e7	1.07445	0.537225	−	0.843439i	$-0.319472\pi$
$661$	1.20695e7	1.07445	0.537225	+	0.843439i	$0.319472\pi$
$662$	−1.10089e7	−0.976334
$663$	6.48362e6	0.572841
$664$	1.64130e7	1.44467
$665$	0	0
$666$	−8.15475e6	−0.712402
$667$	1.37975e6	0.120084
$668$	−5.00077e6	−0.433606
$669$	1.94313e6	0.167856
$670$	0	0
$671$	−3.18508e6	−0.273095
$672$	−1.12411e7	−0.960251
$673$	−1.89473e7	−1.61254	−0.806270	−	0.591548i	$-0.798517\pi$
$673$	−1.89473e7	−1.61254	−0.806270	+	0.591548i	$0.798517\pi$
$674$	1.02676e7	0.870600
$675$	0	0
$676$	−384404.	−0.0323535
$677$	−817954.	−0.0685894	−0.0342947	−	0.999412i	$-0.510918\pi$
$677$	−817954.	−0.0685894	−0.0342947	+	0.999412i	$0.510918\pi$
$678$	−6.34154e6	−0.529811
$679$	1.18038e7	0.982530
$680$	0	0
$681$	−7.24163e6	−0.598368
$682$	1.24568e7	1.02552
$683$	3.54928e6	0.291131	0.145565	−	0.989349i	$-0.453500\pi$
$683$	3.54928e6	0.291131	0.145565	+	0.989349i	$0.453500\pi$
$684$	4.47123e6	0.365415
$685$	0	0
$686$	−9.63671e6	−0.781841
$687$	−1.09495e7	−0.885118
$688$	7.06188e6	0.568786
$689$	−2.01757e6	−0.161913
$690$	0	0
$691$	2.39285e6	0.190643	0.0953214	−	0.995447i	$-0.469612\pi$
$691$	2.39285e6	0.190643	0.0953214	+	0.995447i	$0.469612\pi$
$692$	3.69693e6	0.293479
$693$	1.36813e7	1.08217
$694$	−1.84205e6	−0.145178
$695$	0	0
$696$	−1.75917e7	−1.37653
$697$	2.78084e7	2.16817
$698$	1.68446e7	1.30865
$699$	−8.85840e6	−0.685744
$700$	0	0
$701$	1.83475e7	1.41020	0.705101	−	0.709106i	$-0.250901\pi$
$701$	1.83475e7	1.41020	0.705101	+	0.709106i	$0.250901\pi$
$702$	1.29892e6	0.0994810
$703$	−2.67573e7	−2.04199
$704$	−2.30665e7	−1.75409
$705$	0	0
$706$	8.00762e6	0.604633
$707$	1.47268e7	1.10805
$708$	−1.06789e7	−0.800653
$709$	−3.07694e6	−0.229881	−0.114941	−	0.993372i	$-0.536668\pi$
$709$	−3.07694e6	−0.229881	−0.114941	+	0.993372i	$0.536668\pi$
$710$	0	0
$711$	−6.64733e6	−0.493144
$712$	2.43881e7	1.80293
$713$	1.24653e6	0.0918291
$714$	−2.07787e7	−1.52536
$715$	0	0
$716$	−4.27097e6	−0.311346
$717$	−2.11158e7	−1.53394
$718$	−2.98466e6	−0.216064
$719$	1.80795e7	1.30426	0.652129	−	0.758108i	$-0.273876\pi$
$719$	1.80795e7	1.30426	0.652129	+	0.758108i	$0.273876\pi$
$720$	0	0
$721$	1.46492e7	1.04948
$722$	−9.54855e6	−0.681702
$723$	−1.27817e7	−0.909373
$724$	3.25069e6	0.230478
$725$	0	0
$726$	2.93261e7	2.06497
$727$	−2.35372e7	−1.65165	−0.825826	−	0.563925i	$-0.809291\pi$
$727$	−2.35372e7	−1.65165	−0.825826	+	0.563925i	$0.809291\pi$
$728$	4.16096e6	0.290981
$729$	−2.52763e6	−0.176155
$730$	0	0
$731$	−3.30177e7	−2.28536
$732$	1.20315e6	0.0829930
$733$	4.96555e6	0.341356	0.170678	−	0.985327i	$-0.445404\pi$
$733$	4.96555e6	0.341356	0.170678	+	0.985327i	$0.445404\pi$
$734$	9.73130e6	0.666700
$735$	0	0
$736$	−1.37204e6	−0.0933624
$737$	−2.16264e7	−1.46661
$738$	−9.52800e6	−0.643963
$739$	−2.40104e7	−1.61729	−0.808646	−	0.588296i	$-0.799799\pi$
$739$	−2.40104e7	−1.61729	−0.808646	+	0.588296i	$0.799799\pi$
$740$	0	0
$741$	−7.28913e6	−0.487675
$742$	6.46591e6	0.431141
$743$	−1.15136e7	−0.765140	−0.382570	−	0.923927i	$-0.624961\pi$
$743$	−1.15136e7	−0.765140	−0.382570	+	0.923927i	$0.624961\pi$
$744$	−1.58932e7	−1.05264
$745$	0	0
$746$	3.45351e6	0.227203
$747$	1.28576e7	0.843060
$748$	1.83976e7	1.20228
$749$	5.42345e6	0.353241
$750$	0	0
$751$	−2.24902e7	−1.45510	−0.727552	−	0.686052i	$-0.759342\pi$
$751$	−2.24902e7	−1.45510	−0.727552	+	0.686052i	$0.759342\pi$
$752$	−7.56286e6	−0.487688
$753$	−7.69302e6	−0.494435
$754$	3.28504e6	0.210432
$755$	0	0
$756$	3.02180e6	0.192292
$757$	788552.	0.0500139	0.0250069	−	0.999687i	$-0.492039\pi$
$757$	788552.	0.0500139	0.0250069	+	0.999687i	$0.492039\pi$
$758$	−6.22913e6	−0.393780
$759$	4.31616e6	0.271952
$760$	0	0
$761$	−5.67255e6	−0.355072	−0.177536	−	0.984114i	$-0.556813\pi$
$761$	−5.67255e6	−0.355072	−0.177536	+	0.984114i	$0.556813\pi$
$762$	2.14772e7	1.33995
$763$	−4.08299e6	−0.253903
$764$	−2.81061e6	−0.174208
$765$	0	0
$766$	1.52971e7	0.941970
$767$	6.73543e6	0.413406
$768$	2.10274e7	1.28642
$769$	−2.85131e6	−0.173872	−0.0869358	−	0.996214i	$-0.527707\pi$
$769$	−2.85131e6	−0.173872	−0.0869358	+	0.996214i	$0.527707\pi$
$770$	0	0
$771$	8.60124e6	0.521105
$772$	−5.52216e6	−0.333477
$773$	1.94976e7	1.17363	0.586817	−	0.809720i	$-0.300381\pi$
$773$	1.94976e7	1.17363	0.586817	+	0.809720i	$0.300381\pi$
$774$	1.13129e7	0.678768
$775$	0	0
$776$	−1.83690e7	−1.09504
$777$	3.09274e7	1.83777
$778$	2.40246e7	1.42301
$779$	−3.12632e7	−1.84582
$780$	0	0
$781$	1.03287e7	0.605925
$782$	−2.53616e6	−0.148306
$783$	8.05784e6	0.469693
$784$	406279.	0.0236066
$785$	0	0
$786$	8.43788e6	0.487166
$787$	8.39703e6	0.483269	0.241634	−	0.970367i	$-0.422317\pi$
$787$	8.39703e6	0.483269	0.241634	+	0.970367i	$0.422317\pi$
$788$	6.28429e6	0.360529
$789$	2.40100e6	0.137309
$790$	0	0
$791$	9.30499e6	0.528779
$792$	−2.12909e7	−1.20609
$793$	−758851.	−0.0428523
$794$	−1.48014e7	−0.833206
$795$	0	0
$796$	−8.82023e6	−0.493398
$797$	1.20402e6	0.0671411	0.0335705	−	0.999436i	$-0.489312\pi$
$797$	1.20402e6	0.0671411	0.0335705	+	0.999436i	$0.489312\pi$
$798$	2.33602e7	1.29858
$799$	3.53601e7	1.95951
$800$	0	0
$801$	1.91051e7	1.05213
$802$	1.93614e7	1.06292
$803$	4.18558e7	2.29069
$804$	8.16926e6	0.445700
$805$	0	0
$806$	2.96786e6	0.160918
$807$	4.38729e7	2.37144
$808$	−2.29178e7	−1.23494
$809$	8.10437e6	0.435360	0.217680	−	0.976020i	$-0.430151\pi$
$809$	8.10437e6	0.435360	0.217680	+	0.976020i	$0.430151\pi$
$810$	0	0
$811$	−4.53486e6	−0.242109	−0.121055	−	0.992646i	$-0.538628\pi$
$811$	−4.53486e6	−0.242109	−0.121055	+	0.992646i	$0.538628\pi$
$812$	7.64230e6	0.406756
$813$	−1.19264e7	−0.632827
$814$	3.77228e7	1.99546
$815$	0	0
$816$	1.58125e7	0.831333
$817$	3.71198e7	1.94559
$818$	1.72585e7	0.901821
$819$	3.25960e6	0.169807
$820$	0	0
$821$	1.59739e6	0.0827092	0.0413546	−	0.999145i	$-0.486833\pi$
$821$	1.59739e6	0.0827092	0.0413546	+	0.999145i	$0.486833\pi$
$822$	2.59689e7	1.34052
$823$	−2.00928e7	−1.03405	−0.517024	−	0.855971i	$-0.672960\pi$
$823$	−2.00928e7	−1.03405	−0.517024	+	0.855971i	$0.672960\pi$
$824$	−2.27971e7	−1.16967
$825$	0	0
$826$	−2.15856e7	−1.10082
$827$	−1.57246e7	−0.799498	−0.399749	−	0.916625i	$-0.630903\pi$
$827$	−1.57246e7	−0.799498	−0.399749	+	0.916625i	$0.630903\pi$
$828$	−630791.	−0.0319749
$829$	−2.66412e7	−1.34638	−0.673190	−	0.739470i	$-0.735076\pi$
$829$	−2.66412e7	−1.34638	−0.673190	+	0.739470i	$0.735076\pi$
$830$	0	0
$831$	1.35268e6	0.0679503
$832$	−5.49565e6	−0.275240
$833$	−1.89955e6	−0.0948504
$834$	−1.84922e7	−0.920607
$835$	0	0
$836$	−2.06833e7	−1.02354
$837$	7.27983e6	0.359176
$838$	131017.	0.00644494
$839$	−2.06873e7	−1.01461	−0.507306	−	0.861766i	$-0.669359\pi$
$839$	−2.06873e7	−1.01461	−0.507306	+	0.861766i	$0.669359\pi$
$840$	0	0
$841$	−132454.	−0.00645768
$842$	−5.54253e6	−0.269419
$843$	−1.57939e6	−0.0765455
$844$	6.84442e6	0.330735
$845$	0	0
$846$	−1.21155e7	−0.581988
$847$	−4.30304e7	−2.06095
$848$	−4.92054e6	−0.234976
$849$	−1.61456e7	−0.768750
$850$	0	0
$851$	3.77486e6	0.178681
$852$	−3.90163e6	−0.184140
$853$	706242.	0.0332339	0.0166169	−	0.999862i	$-0.494710\pi$
$853$	706242.	0.0332339	0.0166169	+	0.999862i	$0.494710\pi$
$854$	2.43196e6	0.114107
$855$	0	0
$856$	−8.43997e6	−0.393692
$857$	−1.54355e7	−0.717908	−0.358954	−	0.933355i	$-0.616867\pi$
$857$	−1.54355e7	−0.717908	−0.358954	+	0.933355i	$0.616867\pi$
$858$	1.02763e7	0.476561
$859$	2.09391e7	0.968222	0.484111	−	0.875007i	$-0.339143\pi$
$859$	2.09391e7	0.968222	0.484111	+	0.875007i	$0.339143\pi$
$860$	0	0
$861$	3.61355e7	1.66122
$862$	1.41006e7	0.646351
$863$	−1.50017e7	−0.685665	−0.342833	−	0.939397i	$-0.611386\pi$
$863$	−1.50017e7	−0.685665	−0.342833	+	0.939397i	$0.611386\pi$
$864$	−8.01278e6	−0.365173
$865$	0	0
$866$	−1.10425e7	−0.500348
$867$	−4.56642e7	−2.06314
$868$	6.90441e6	0.311048
$869$	3.07497e7	1.38131
$870$	0	0
$871$	−5.15253e6	−0.230131
$872$	6.35395e6	0.282978
$873$	−1.43898e7	−0.639029
$874$	2.85124e6	0.126257
$875$	0	0
$876$	−1.58108e7	−0.696136
$877$	2.64018e7	1.15914	0.579569	−	0.814923i	$-0.303221\pi$
$877$	2.64018e7	1.15914	0.579569	+	0.814923i	$0.303221\pi$
$878$	−4.18989e6	−0.183428
$879$	−4.61719e7	−2.01561
$880$	0	0
$881$	−3.15138e7	−1.36792	−0.683960	−	0.729520i	$-0.739744\pi$
$881$	−3.15138e7	−1.36792	−0.683960	+	0.729520i	$0.739744\pi$
$882$	650845.	0.0281713
$883$	−1.04657e7	−0.451718	−0.225859	−	0.974160i	$-0.572519\pi$
$883$	−1.04657e7	−0.451718	−0.225859	+	0.974160i	$0.572519\pi$
$884$	4.38327e6	0.188655
$885$	0	0
$886$	8.53607e6	0.365320
$887$	2.29824e7	0.980815	0.490408	−	0.871493i	$-0.336848\pi$
$887$	2.29824e7	0.980815	0.490408	+	0.871493i	$0.336848\pi$
$888$	−4.81292e7	−2.04822
$889$	−3.15136e7	−1.33735
$890$	0	0
$891$	5.16376e7	2.17907
$892$	1.31366e6	0.0552804
$893$	−3.97532e7	−1.66818
$894$	−1.65589e7	−0.692928
$895$	0	0
$896$	−456146.	−0.0189816
$897$	1.02833e6	0.0426730
$898$	−8.54508e6	−0.353610
$899$	1.84111e7	0.759766
$900$	0	0
$901$	2.30059e7	0.944121
$902$	4.40752e7	1.80376
$903$	−4.29048e7	−1.75100
$904$	−1.44804e7	−0.589332
$905$	0	0
$906$	−4.62996e6	−0.187394
$907$	−3.45508e7	−1.39457	−0.697284	−	0.716795i	$-0.745608\pi$
$907$	−3.45508e7	−1.39457	−0.697284	+	0.716795i	$0.745608\pi$
$908$	−4.89572e6	−0.197062
$909$	−1.79533e7	−0.720666
$910$	0	0
$911$	−2.24713e7	−0.897081	−0.448540	−	0.893763i	$-0.648056\pi$
$911$	−2.24713e7	−0.897081	−0.448540	+	0.893763i	$0.648056\pi$
$912$	−1.77770e7	−0.707736
$913$	−5.94775e7	−2.36143
$914$	−2.99298e7	−1.18506
$915$	0	0
$916$	−7.40242e6	−0.291498
$917$	−1.23810e7	−0.486218
$918$	−1.48113e7	−0.580078
$919$	1.60273e7	0.625995	0.312997	−	0.949754i	$-0.398667\pi$
$919$	1.60273e7	0.625995	0.312997	+	0.949754i	$0.398667\pi$
$920$	0	0
$921$	−1.60240e7	−0.622476
$922$	1.13492e7	0.439680
$923$	2.46084e6	0.0950779
$924$	2.39067e7	0.921170
$925$	0	0
$926$	−2.90898e7	−1.11484
$927$	−1.78587e7	−0.682576
$928$	−2.02647e7	−0.772451
$929$	−2.25541e6	−0.0857404	−0.0428702	−	0.999081i	$-0.513650\pi$
$929$	−2.25541e6	−0.0857404	−0.0428702	+	0.999081i	$0.513650\pi$
$930$	0	0
$931$	2.13555e6	0.0807487
$932$	−5.98875e6	−0.225838
$933$	4.97231e7	1.87006
$934$	−3.16238e7	−1.18617
$935$	0	0
$936$	−5.07259e6	−0.189252
$937$	1.45299e7	0.540645	0.270323	−	0.962770i	$-0.412870\pi$
$937$	1.45299e7	0.540645	0.270323	+	0.962770i	$0.412870\pi$
$938$	1.65128e7	0.612792
$939$	2.92087e6	0.108106
$940$	0	0
$941$	−2.81974e7	−1.03809	−0.519046	−	0.854746i	$-0.673713\pi$
$941$	−2.81974e7	−1.03809	−0.519046	+	0.854746i	$0.673713\pi$
$942$	−3.31357e7	−1.21666
$943$	4.41055e6	0.161515
$944$	1.64266e7	0.599954
$945$	0	0
$946$	−5.23319e7	−1.90125
$947$	−1.11561e7	−0.404238	−0.202119	−	0.979361i	$-0.564783\pi$
$947$	−1.11561e7	−0.404238	−0.202119	+	0.979361i	$0.564783\pi$
$948$	−1.16155e7	−0.419777
$949$	9.97223e6	0.359440
$950$	0	0
$951$	4.73659e7	1.69830
$952$	−4.74465e7	−1.69672
$953$	−2.78329e7	−0.992719	−0.496359	−	0.868117i	$-0.665330\pi$
$953$	−2.78329e7	−0.992719	−0.496359	+	0.868117i	$0.665330\pi$
$954$	−7.88253e6	−0.280411
$955$	0	0
$956$	−1.42754e7	−0.505177
$957$	6.37488e7	2.25005
$958$	2.64415e7	0.930833
$959$	−3.81043e7	−1.33791
$960$	0	0
$961$	−1.19957e7	−0.419004
$962$	8.98753e6	0.313114
$963$	−6.61168e6	−0.229745
$964$	−8.64109e6	−0.299486
$965$	0	0
$966$	−3.29560e6	−0.113630
$967$	2.30836e7	0.793847	0.396924	−	0.917852i	$-0.370078\pi$
$967$	2.30836e7	0.793847	0.396924	+	0.917852i	$0.370078\pi$
$968$	6.69640e7	2.29696
$969$	8.31162e7	2.84365
$970$	0	0
$971$	1.79497e7	0.610955	0.305477	−	0.952199i	$-0.401184\pi$
$971$	1.79497e7	0.610955	0.305477	+	0.952199i	$0.401184\pi$
$972$	−1.36680e7	−0.464024
$973$	2.71338e7	0.918815
$974$	7.47657e6	0.252525
$975$	0	0
$976$	−1.85072e6	−0.0621892
$977$	−5.63392e7	−1.88832	−0.944158	−	0.329494i	$-0.893122\pi$
$977$	−5.63392e7	−1.88832	−0.944158	+	0.329494i	$0.893122\pi$
$978$	−9.15665e6	−0.306118
$979$	−8.83775e7	−2.94703
$980$	0	0
$981$	4.97754e6	0.165136
$982$	7.14815e6	0.236545
$983$	−4.50208e6	−0.148604	−0.0743018	−	0.997236i	$-0.523673\pi$
$983$	−4.50208e6	−0.148604	−0.0743018	+	0.997236i	$0.523673\pi$
$984$	−5.62341e7	−1.85145
$985$	0	0
$986$	−3.74585e7	−1.22704
$987$	4.59486e7	1.50134
$988$	−4.92784e6	−0.160607
$989$	−5.23678e6	−0.170245
$990$	0	0
$991$	4.18623e7	1.35406	0.677032	−	0.735954i	$-0.263266\pi$
$991$	4.18623e7	1.35406	0.677032	+	0.735954i	$0.263266\pi$
$992$	−1.83081e7	−0.590697
$993$	−5.08993e7	−1.63809
$994$	−7.88649e6	−0.253173
$995$	0	0
$996$	2.24674e7	0.717635
$997$	−5.44586e7	−1.73512	−0.867558	−	0.497337i	$-0.834311\pi$
$997$	−5.44586e7	−1.73512	−0.867558	+	0.497337i	$0.834311\pi$
$998$	−3.37062e7	−1.07123
$999$	2.20454e7	0.698883

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.a.i.1.3	yes	9
5.2	odd	4		325.6.b.h.274.6		18
5.3	odd	4		325.6.b.h.274.13		18
5.4	even	2		325.6.a.h.1.7	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.6.a.h.1.7	✓	9	5.4	even	2
325.6.a.i.1.3	yes	9	1.1	even	1	trivial
325.6.b.h.274.6		18	5.2	odd	4
325.6.b.h.274.13		18	5.3	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)

\(f(q)\)	\(=\)	\(q-4.30592 q^{2} -19.9083 q^{3} -13.4591 q^{4} +85.7235 q^{6} -125.783 q^{7} +195.743 q^{8} +153.341 q^{9} -709.332 q^{11} +267.947 q^{12} -169.000 q^{13} +541.610 q^{14} -412.163 q^{16} +1927.07 q^{17} -660.272 q^{18} -2166.48 q^{19} +2504.12 q^{21} +3054.33 q^{22} +305.642 q^{23} -3896.91 q^{24} +727.700 q^{26} +1784.97 q^{27} +1692.92 q^{28} +4514.28 q^{29} +4078.41 q^{31} -4489.03 q^{32} +14121.6 q^{33} -8297.79 q^{34} -2063.82 q^{36} +12350.6 q^{37} +9328.69 q^{38} +3364.50 q^{39} +14430.4 q^{41} -10782.5 q^{42} -17133.7 q^{43} +9546.95 q^{44} -1316.07 q^{46} +18349.2 q^{47} +8205.47 q^{48} -985.723 q^{49} -38364.6 q^{51} +2274.58 q^{52} +11938.3 q^{53} -7685.93 q^{54} -24621.1 q^{56} +43131.0 q^{57} -19438.1 q^{58} -39854.6 q^{59} +4490.25 q^{61} -17561.3 q^{62} -19287.6 q^{63} +32518.6 q^{64} -60806.5 q^{66} +30488.3 q^{67} -25936.5 q^{68} -6084.82 q^{69} -14561.2 q^{71} +30015.3 q^{72} -59007.3 q^{73} -53180.6 q^{74} +29158.8 q^{76} +89221.7 q^{77} -14487.3 q^{78} -43350.2 q^{79} -72797.4 q^{81} -62136.2 q^{82} +83849.9 q^{83} -33703.1 q^{84} +73776.2 q^{86} -89871.6 q^{87} -138847. q^{88} +124592. q^{89} +21257.3 q^{91} -4113.66 q^{92} -81194.2 q^{93} -79010.1 q^{94} +89369.1 q^{96} -93842.4 q^{97} +4244.44 q^{98} -108769. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 5 q^{2} + 11 q^{3} + 91 q^{4} - 83 q^{6} - 12 q^{7} + 639 q^{8} + 562 q^{9} - 1422 q^{11} - 1567 q^{12} - 1521 q^{13} - 342 q^{14} - 1061 q^{16} - 648 q^{17} - 418 q^{18} - 408 q^{19} - 3912 q^{21}+ \cdots - 757776 q^{99}+O(q^{100}) \) 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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