LMFDB - Embedded newform 847.6.a.o.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [847,6,Mod(1,847)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(847, 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("847.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 847 = 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	847.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:	$135.845095382$
Analytic rank:	$0$
Dimension:	$26$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	847.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.958902 q^{2} +6.46272 q^{3} -31.0805 q^{4} -36.7648 q^{5} +6.19712 q^{6} +49.0000 q^{7} -60.4880 q^{8} -201.233 q^{9} +O(q^{10})$	\(q+0.958902 q^{2} +6.46272 q^{3} -31.0805 q^{4} -36.7648 q^{5} +6.19712 q^{6} +49.0000 q^{7} -60.4880 q^{8} -201.233 q^{9} -35.2538 q^{10} -200.865 q^{12} -768.337 q^{13} +46.9862 q^{14} -237.601 q^{15} +936.574 q^{16} -70.2689 q^{17} -192.963 q^{18} +1488.04 q^{19} +1142.67 q^{20} +316.673 q^{21} +150.392 q^{23} -390.917 q^{24} -1773.35 q^{25} -736.760 q^{26} -2870.96 q^{27} -1522.94 q^{28} -8102.39 q^{29} -227.836 q^{30} -9745.96 q^{31} +2833.70 q^{32} -67.3810 q^{34} -1801.48 q^{35} +6254.43 q^{36} +12883.1 q^{37} +1426.88 q^{38} -4965.55 q^{39} +2223.83 q^{40} -17175.2 q^{41} +303.659 q^{42} -19864.8 q^{43} +7398.30 q^{45} +144.211 q^{46} -26273.5 q^{47} +6052.82 q^{48} +2401.00 q^{49} -1700.47 q^{50} -454.129 q^{51} +23880.3 q^{52} +31176.8 q^{53} -2752.97 q^{54} -2963.91 q^{56} +9616.76 q^{57} -7769.40 q^{58} +36215.8 q^{59} +7384.75 q^{60} +19912.8 q^{61} -9345.41 q^{62} -9860.43 q^{63} -27253.1 q^{64} +28247.8 q^{65} -20332.1 q^{67} +2183.99 q^{68} +971.943 q^{69} -1727.44 q^{70} +56424.5 q^{71} +12172.2 q^{72} +44396.6 q^{73} +12353.7 q^{74} -11460.7 q^{75} -46248.9 q^{76} -4761.48 q^{78} -84575.0 q^{79} -34433.0 q^{80} +30345.5 q^{81} -16469.4 q^{82} +1445.80 q^{83} -9842.37 q^{84} +2583.42 q^{85} -19048.3 q^{86} -52363.5 q^{87} -82708.8 q^{89} +7094.24 q^{90} -37648.5 q^{91} -4674.27 q^{92} -62985.4 q^{93} -25193.7 q^{94} -54707.3 q^{95} +18313.4 q^{96} -78699.7 q^{97} +2302.32 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 26 q + 16 q^{2} - 2 q^{3} + 428 q^{4} + 156 q^{5} + 554 q^{6} + 1274 q^{7} + 768 q^{8} + 2368 q^{9}+O(q^{10}) $ 26 * q + 16 * q^2 - 2 * q^3 + 428 * q^4 + 156 * q^5 + 554 * q^6 + 1274 * q^7 + 768 * q^8 + 2368 * q^9	\( 26 q + 16 q^{2} - 2 q^{3} + 428 q^{4} + 156 q^{5} + 554 q^{6} + 1274 q^{7} + 768 q^{8} + 2368 q^{9} + 1132 q^{10} + 658 q^{12} + 1908 q^{13} + 784 q^{14} + 3114 q^{15} + 8772 q^{16} + 4398 q^{17} + 7020 q^{18} + 2922 q^{19} + 6982 q^{20} - 98 q^{21} + 6970 q^{23} + 19924 q^{24} + 10324 q^{25} + 5802 q^{26} - 1346 q^{27} + 20972 q^{28} + 21068 q^{29} - 19072 q^{30} + 1638 q^{31} + 49772 q^{32} + 55400 q^{34} + 7644 q^{35} + 68030 q^{36} + 21362 q^{37} + 20214 q^{38} - 28716 q^{39} + 66894 q^{40} + 30882 q^{41} + 27146 q^{42} + 16068 q^{43} + 32932 q^{45} + 15088 q^{46} - 26000 q^{47} - 10392 q^{48} + 62426 q^{49} + 20744 q^{50} + 41844 q^{51} - 42060 q^{52} + 86508 q^{53} + 160368 q^{54} + 37632 q^{56} + 93764 q^{57} - 52790 q^{58} - 45476 q^{59} - 46376 q^{60} + 101496 q^{61} + 116668 q^{62} + 116032 q^{63} + 121294 q^{64} + 288440 q^{65} + 41942 q^{67} + 255520 q^{68} + 1366 q^{69} + 55468 q^{70} + 86814 q^{71} + 204604 q^{72} + 149254 q^{73} - 161724 q^{74} - 336580 q^{75} - 11930 q^{76} - 617620 q^{78} + 11620 q^{79} + 161660 q^{80} + 195662 q^{81} - 356068 q^{82} + 216994 q^{83} + 32242 q^{84} + 360392 q^{85} + 100438 q^{86} + 49730 q^{87} - 403424 q^{89} + 463182 q^{90} + 93492 q^{91} - 78454 q^{92} + 82690 q^{93} - 467466 q^{94} + 250384 q^{95} + 520006 q^{96} - 360716 q^{97} + 38416 q^{98}+O(q^{100}) \) 26 * q + 16 * q^2 - 2 * q^3 + 428 * q^4 + 156 * q^5 + 554 * q^6 + 1274 * q^7 + 768 * q^8 + 2368 * q^9 + 1132 * q^10 + 658 * q^12 + 1908 * q^13 + 784 * q^14 + 3114 * q^15 + 8772 * q^16 + 4398 * q^17 + 7020 * q^18 + 2922 * q^19 + 6982 * q^20 - 98 * q^21 + 6970 * q^23 + 19924 * q^24 + 10324 * q^25 + 5802 * q^26 - 1346 * q^27 + 20972 * q^28 + 21068 * q^29 - 19072 * q^30 + 1638 * q^31 + 49772 * q^32 + 55400 * q^34 + 7644 * q^35 + 68030 * q^36 + 21362 * q^37 + 20214 * q^38 - 28716 * q^39 + 66894 * q^40 + 30882 * q^41 + 27146 * q^42 + 16068 * q^43 + 32932 * q^45 + 15088 * q^46 - 26000 * q^47 - 10392 * q^48 + 62426 * q^49 + 20744 * q^50 + 41844 * q^51 - 42060 * q^52 + 86508 * q^53 + 160368 * q^54 + 37632 * q^56 + 93764 * q^57 - 52790 * q^58 - 45476 * q^59 - 46376 * q^60 + 101496 * q^61 + 116668 * q^62 + 116032 * q^63 + 121294 * q^64 + 288440 * q^65 + 41942 * q^67 + 255520 * q^68 + 1366 * q^69 + 55468 * q^70 + 86814 * q^71 + 204604 * q^72 + 149254 * q^73 - 161724 * q^74 - 336580 * q^75 - 11930 * q^76 - 617620 * q^78 + 11620 * q^79 + 161660 * q^80 + 195662 * q^81 - 356068 * q^82 + 216994 * q^83 + 32242 * q^84 + 360392 * q^85 + 100438 * q^86 + 49730 * q^87 - 403424 * q^89 + 463182 * q^90 + 93492 * q^91 - 78454 * q^92 + 82690 * q^93 - 467466 * q^94 + 250384 * q^95 + 520006 * q^96 - 360716 * q^97 + 38416 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.958902	0.169512	0.0847558	−	0.996402i	$-0.472989\pi$
$2$	0.958902	0.169512	0.0847558	+	0.996402i	$0.472989\pi$
$3$	6.46272	0.414584	0.207292	−	0.978279i	$-0.433535\pi$
$3$	6.46272	0.414584	0.207292	+	0.978279i	$0.433535\pi$
$4$	−31.0805	−0.971266
$5$	−36.7648	−0.657669	−0.328834	−	0.944388i	$-0.606656\pi$
$5$	−36.7648	−0.657669	−0.328834	+	0.944388i	$0.606656\pi$
$6$	6.19712	0.0702767
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	−60.4880	−0.334152
$9$	−201.233	−0.828120
$10$	−35.2538	−0.111482
$11$	0	0
$11$	0	0
$12$	−200.865	−0.402671
$13$	−768.337	−1.26094	−0.630469	−	0.776215i	$-0.717137\pi$
$13$	−768.337	−1.26094	−0.630469	+	0.776215i	$0.717137\pi$
$14$	46.9862	0.0640693
$15$	−237.601	−0.272659
$16$	936.574	0.914623
$17$	−70.2689	−0.0589714	−0.0294857	−	0.999565i	$-0.509387\pi$
$17$	−70.2689	−0.0589714	−0.0294857	+	0.999565i	$0.509387\pi$
$18$	−192.963	−0.140376
$19$	1488.04	0.945647	0.472824	−	0.881157i	$-0.343235\pi$
$19$	1488.04	0.945647	0.472824	+	0.881157i	$0.343235\pi$
$20$	1142.67	0.638771
$21$	316.673	0.156698
$22$	0	0
$23$	150.392	0.0592797	0.0296398	−	0.999561i	$-0.490564\pi$
$23$	150.392	0.0592797	0.0296398	+	0.999561i	$0.490564\pi$
$24$	−390.917	−0.138534
$25$	−1773.35	−0.567472
$26$	−736.760	−0.213743
$27$	−2870.96	−0.757909
$28$	−1522.94	−0.367104
$29$	−8102.39	−1.78903	−0.894517	−	0.447035i	$-0.852480\pi$
$29$	−8102.39	−1.78903	−0.894517	+	0.447035i	$0.852480\pi$
$30$	−227.836	−0.0462188
$31$	−9745.96	−1.82146	−0.910731	−	0.413000i	$-0.864481\pi$
$31$	−9745.96	−1.82146	−0.910731	+	0.413000i	$0.864481\pi$
$32$	2833.70	0.489191
$33$	0	0
$34$	−67.3810	−0.00999632
$35$	−1801.48	−0.248575
$36$	6254.43	0.804325
$37$	12883.1	1.54710	0.773548	−	0.633738i	$-0.218480\pi$
$37$	12883.1	1.54710	0.773548	+	0.633738i	$0.218480\pi$
$38$	1426.88	0.160298
$39$	−4965.55	−0.522764
$40$	2223.83	0.219761
$41$	−17175.2	−1.59567	−0.797835	−	0.602877i	$-0.794021\pi$
$41$	−17175.2	−1.59567	−0.797835	+	0.602877i	$0.794021\pi$
$42$	303.659	0.0265621
$43$	−19864.8	−1.63837	−0.819185	−	0.573529i	$-0.805574\pi$
$43$	−19864.8	−1.63837	−0.819185	+	0.573529i	$0.805574\pi$
$44$	0	0
$45$	7398.30	0.544629
$46$	144.211	0.0100486
$47$	−26273.5	−1.73490	−0.867448	−	0.497528i	$-0.834241\pi$
$47$	−26273.5	−1.73490	−0.867448	+	0.497528i	$0.834241\pi$
$48$	6052.82	0.379188
$49$	2401.00	0.142857
$50$	−1700.47	−0.0961930
$51$	−454.129	−0.0244486
$52$	23880.3	1.22471
$53$	31176.8	1.52455	0.762275	−	0.647253i	$-0.224082\pi$
$53$	31176.8	1.52455	0.762275	+	0.647253i	$0.224082\pi$
$54$	−2752.97	−0.128474
$55$	0	0
$56$	−2963.91	−0.126298
$57$	9616.76	0.392050
$58$	−7769.40	−0.303262
$59$	36215.8	1.35446	0.677232	−	0.735769i	$-0.263179\pi$
$59$	36215.8	1.35446	0.677232	+	0.735769i	$0.263179\pi$
$60$	7384.75	0.264824
$61$	19912.8	0.685184	0.342592	−	0.939484i	$-0.388695\pi$
$61$	19912.8	0.685184	0.342592	+	0.939484i	$0.388695\pi$
$62$	−9345.41	−0.308759
$63$	−9860.43	−0.313000
$64$	−27253.1	−0.831700
$65$	28247.8	0.829279
$66$	0	0
$67$	−20332.1	−0.553345	−0.276673	−	0.960964i	$-0.589232\pi$
$67$	−20332.1	−0.553345	−0.276673	+	0.960964i	$0.589232\pi$
$68$	2183.99	0.0572769
$69$	971.943	0.0245764
$70$	−1727.44	−0.0421364
$71$	56424.5	1.32838	0.664189	−	0.747565i	$-0.268777\pi$
$71$	56424.5	1.32838	0.664189	+	0.747565i	$0.268777\pi$
$72$	12172.2	0.276718
$73$	44396.6	0.975086	0.487543	−	0.873099i	$-0.337893\pi$
$73$	44396.6	0.975086	0.487543	+	0.873099i	$0.337893\pi$
$74$	12353.7	0.262250
$75$	−11460.7	−0.235265
$76$	−46248.9	−0.918475
$77$	0	0
$78$	−4761.48	−0.0886146
$79$	−84575.0	−1.52467	−0.762333	−	0.647185i	$-0.775946\pi$
$79$	−84575.0	−1.52467	−0.762333	+	0.647185i	$0.775946\pi$
$80$	−34433.0	−0.601519
$81$	30345.5	0.513903
$82$	−16469.4	−0.270484
$83$	1445.80	0.0230363	0.0115181	−	0.999934i	$-0.496334\pi$
$83$	1445.80	0.0230363	0.0115181	+	0.999934i	$0.496334\pi$
$84$	−9842.37	−0.152195
$85$	2583.42	0.0387836
$86$	−19048.3	−0.277723
$87$	−52363.5	−0.741704
$88$	0	0
$89$	−82708.8	−1.10682	−0.553410	−	0.832909i	$-0.686674\pi$
$89$	−82708.8	−1.10682	−0.553410	+	0.832909i	$0.686674\pi$
$90$	7094.24	0.0923208
$91$	−37648.5	−0.476590
$92$	−4674.27	−0.0575763
$93$	−62985.4	−0.755149
$94$	−25193.7	−0.294085
$95$	−54707.3	−0.621923
$96$	18313.4	0.202811
$97$	−78699.7	−0.849266	−0.424633	−	0.905365i	$-0.639597\pi$
$97$	−78699.7	−0.849266	−0.424633	+	0.905365i	$0.639597\pi$
$98$	2302.32	0.0242159
$99$	0	0
$100$	55116.6	0.551166
$101$	100297.	0.978323	0.489162	−	0.872193i	$-0.337303\pi$
$101$	100297.	0.978323	0.489162	+	0.872193i	$0.337303\pi$
$102$	−435.465	−0.00414431
$103$	27620.5	0.256530	0.128265	−	0.991740i	$-0.459059\pi$
$103$	27620.5	0.256530	0.128265	+	0.991740i	$0.459059\pi$
$104$	46475.2	0.421345
$105$	−11642.4	−0.103055
$106$	29895.5	0.258429
$107$	36239.4	0.306000	0.153000	−	0.988226i	$-0.451107\pi$
$107$	36239.4	0.306000	0.153000	+	0.988226i	$0.451107\pi$
$108$	89230.8	0.736131
$109$	176023.	1.41907	0.709535	−	0.704670i	$-0.248905\pi$
$109$	176023.	1.41907	0.709535	+	0.704670i	$0.248905\pi$
$110$	0	0
$111$	83260.1	0.641401
$112$	45892.1	0.345695
$113$	−30850.6	−0.227283	−0.113642	−	0.993522i	$-0.536252\pi$
$113$	−30850.6	−0.227283	−0.113642	+	0.993522i	$0.536252\pi$
$114$	9221.53	0.0664570
$115$	−5529.14	−0.0389864
$116$	251827.	1.73763
$117$	154615.	1.04421
$118$	34727.4	0.229597
$119$	−3443.18	−0.0222891
$120$	14372.0	0.0911096
$121$	0	0
$122$	19094.4	0.116147
$123$	−110999.	−0.661539
$124$	302909.	1.76912
$125$	180087.	1.03088
$126$	−9455.18	−0.0530571
$127$	92424.0	0.508482	0.254241	−	0.967141i	$-0.418174\pi$
$127$	92424.0	0.508482	0.254241	+	0.967141i	$0.418174\pi$
$128$	−116811.	−0.630174
$129$	−128380.	−0.679242
$130$	27086.8	0.140572
$131$	−185602.	−0.944942	−0.472471	−	0.881346i	$-0.656638\pi$
$131$	−185602.	−0.944942	−0.472471	+	0.881346i	$0.656638\pi$
$132$	0	0
$133$	72913.7	0.357421
$134$	−19496.5	−0.0937984
$135$	105550.	0.498453
$136$	4250.43	0.0197054
$137$	−393164.	−1.78967	−0.894833	−	0.446401i	$-0.852706\pi$
$137$	−393164.	−1.78967	−0.894833	+	0.446401i	$0.852706\pi$
$138$	931.998	0.00416598
$139$	149117.	0.654620	0.327310	−	0.944917i	$-0.393858\pi$
$139$	149117.	0.654620	0.327310	+	0.944917i	$0.393858\pi$
$140$	55990.8	0.241433
$141$	−169798.	−0.719260
$142$	54105.5	0.225175
$143$	0	0
$144$	−188470.	−0.757418
$145$	297883.	1.17659
$146$	42572.0	0.165288
$147$	15517.0	0.0592263
$148$	−400414.	−1.50264
$149$	251343.	0.927472	0.463736	−	0.885973i	$-0.346509\pi$
$149$	251343.	0.927472	0.463736	+	0.885973i	$0.346509\pi$
$150$	−10989.7	−0.0398801
$151$	330654.	1.18013	0.590067	−	0.807354i	$-0.299101\pi$
$151$	330654.	1.18013	0.590067	+	0.807354i	$0.299101\pi$
$152$	−90008.3	−0.315990
$153$	14140.4	0.0488354
$154$	0	0
$155$	358308.	1.19792
$156$	154332.	0.507743
$157$	−228784.	−0.740759	−0.370379	−	0.928881i	$-0.620772\pi$
$157$	−228784.	−0.740759	−0.370379	+	0.928881i	$0.620772\pi$
$158$	−81099.2	−0.258448
$159$	201487.	0.632054
$160$	−104180.	−0.321726
$161$	7369.22	0.0224056
$162$	29098.3	0.0871125
$163$	502508.	1.48141	0.740703	−	0.671833i	$-0.234493\pi$
$163$	502508.	1.48141	0.740703	+	0.671833i	$0.234493\pi$
$164$	533815.	1.54982
$165$	0	0
$166$	1386.38	0.00390491
$167$	55733.7	0.154642	0.0773208	−	0.997006i	$-0.475363\pi$
$167$	55733.7	0.154642	0.0773208	+	0.997006i	$0.475363\pi$
$168$	−19154.9	−0.0523610
$169$	219049.	0.589964
$170$	2477.25	0.00657427
$171$	−299442.	−0.783110
$172$	617407.	1.59129
$173$	535743.	1.36095	0.680473	−	0.732773i	$-0.261774\pi$
$173$	535743.	1.36095	0.680473	+	0.732773i	$0.261774\pi$
$174$	−50211.5	−0.125727
$175$	−86894.1	−0.214484
$176$	0	0
$177$	234052.	0.561539
$178$	−79309.7	−0.187619
$179$	179658.	0.419097	0.209549	−	0.977798i	$-0.432801\pi$
$179$	179658.	0.419097	0.209549	+	0.977798i	$0.432801\pi$
$180$	−229943.	−0.528979
$181$	−249519.	−0.566118	−0.283059	−	0.959103i	$-0.591349\pi$
$181$	−249519.	−0.566118	−0.283059	+	0.959103i	$0.591349\pi$
$182$	−36101.2	−0.0807874
$183$	128691.	0.284066
$184$	−9096.93	−0.0198084
$185$	−473646.	−1.01748
$186$	−60396.8	−0.128006
$187$	0	0
$188$	816594.	1.68504
$189$	−140677.	−0.286463
$190$	−52458.9	−0.105423
$191$	92266.1	0.183003	0.0915016	−	0.995805i	$-0.470833\pi$
$191$	92266.1	0.183003	0.0915016	+	0.995805i	$0.470833\pi$
$192$	−176129.	−0.344809
$193$	−167484.	−0.323653	−0.161827	−	0.986819i	$-0.551739\pi$
$193$	−167484.	−0.323653	−0.161827	+	0.986819i	$0.551739\pi$
$194$	−75465.3	−0.143960
$195$	182558.	0.343806
$196$	−74624.3	−0.138752
$197$	−412582.	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$197$	−412582.	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$198$	0	0
$199$	−2287.30	−0.00409439	−0.00204720	−	0.999998i	$-0.500652\pi$
$199$	−2287.30	−0.00409439	−0.00204720	+	0.999998i	$0.500652\pi$
$200$	107266.	0.189622
$201$	−131401.	−0.229408
$202$	96174.5	0.165837
$203$	−397017.	−0.676191
$204$	14114.6	0.0237461
$205$	631444.	1.04942
$206$	26485.3	0.0434847
$207$	−30263.9	−0.0490907
$208$	−719605.	−1.15328
$209$	0	0
$210$	−11164.0	−0.0174691
$211$	270911.	0.418910	0.209455	−	0.977818i	$-0.432831\pi$
$211$	270911.	0.418910	0.209455	+	0.977818i	$0.432831\pi$
$212$	−968991.	−1.48074
$213$	364656.	0.550724
$214$	34750.0	0.0518705
$215$	730324.	1.07750
$216$	173658.	0.253257
$217$	−477552.	−0.688448
$218$	168789.	0.240549
$219$	286923.	0.404255
$220$	0	0
$221$	53990.3	0.0743592
$222$	79838.3	0.108725
$223$	186076.	0.250569	0.125284	−	0.992121i	$-0.460016\pi$
$223$	186076.	0.250569	0.125284	+	0.992121i	$0.460016\pi$
$224$	138851.	0.184897
$225$	356857.	0.469935
$226$	−29582.7	−0.0385271
$227$	1.22602e6	1.57919	0.789595	−	0.613629i	$-0.210291\pi$
$227$	1.22602e6	1.57919	0.789595	+	0.613629i	$0.210291\pi$
$228$	−298894.	−0.380785
$229$	−227726.	−0.286961	−0.143481	−	0.989653i	$-0.545829\pi$
$229$	−227726.	−0.286961	−0.143481	+	0.989653i	$0.545829\pi$
$230$	−5301.90	−0.00660864
$231$	0	0
$232$	490098.	0.597809
$233$	−259834.	−0.313550	−0.156775	−	0.987634i	$-0.550110\pi$
$233$	−259834.	−0.313550	−0.156775	+	0.987634i	$0.550110\pi$
$234$	148261.	0.177005
$235$	965940.	1.14099
$236$	−1.12560e6	−1.31555
$237$	−546585.	−0.632102
$238$	−3301.67	−0.00377825
$239$	900387.	1.01961	0.509806	−	0.860290i	$-0.329717\pi$
$239$	900387.	1.01961	0.509806	+	0.860290i	$0.329717\pi$
$240$	−222531.	−0.249380
$241$	−1.51367e6	−1.67876	−0.839382	−	0.543542i	$-0.817083\pi$
$241$	−1.51367e6	−1.67876	−0.839382	+	0.543542i	$0.817083\pi$
$242$	0	0
$243$	893757.	0.970965
$244$	−618899.	−0.665496
$245$	−88272.3	−0.0939527
$246$	−106437.	−0.112138
$247$	−1.14331e6	−1.19240
$248$	589514.	0.608646
$249$	9343.79	0.00955047
$250$	172686.	0.174746
$251$	508701.	0.509657	0.254829	−	0.966986i	$-0.417981\pi$
$251$	508701.	0.509657	0.254829	+	0.966986i	$0.417981\pi$
$252$	306467.	0.304006
$253$	0	0
$254$	88625.6	0.0861935
$255$	16696.0	0.0160791
$256$	760090.	0.724878
$257$	1.35386e6	1.27862	0.639311	−	0.768948i	$-0.279220\pi$
$257$	1.35386e6	1.27862	0.639311	+	0.768948i	$0.279220\pi$
$258$	−123104.	−0.115139
$259$	631273.	0.584747
$260$	−877955.	−0.805451
$261$	1.63047e6	1.48153
$262$	−177974.	−0.160178
$263$	−506445.	−0.451484	−0.225742	−	0.974187i	$-0.572481\pi$
$263$	−506445.	−0.451484	−0.225742	+	0.974187i	$0.572481\pi$
$264$	0	0
$265$	−1.14621e6	−1.00265
$266$	69917.1	0.0605870
$267$	−534524.	−0.458870
$268$	631933.	0.537445
$269$	−1.59809e6	−1.34655	−0.673274	−	0.739394i	$-0.735112\pi$
$269$	−1.59809e6	−1.34655	−0.673274	+	0.739394i	$0.735112\pi$
$270$	101212.	0.0844935
$271$	−364421.	−0.301426	−0.150713	−	0.988578i	$-0.548157\pi$
$271$	−364421.	−0.301426	−0.150713	+	0.988578i	$0.548157\pi$
$272$	−65812.1	−0.0539366
$273$	−243312.	−0.197586
$274$	−377005.	−0.303369
$275$	0	0
$276$	−30208.5	−0.0238702
$277$	−1.37996e6	−1.08061	−0.540303	−	0.841470i	$-0.681690\pi$
$277$	−1.37996e6	−1.08061	−0.540303	+	0.841470i	$0.681690\pi$
$278$	142988.	0.110966
$279$	1.96121e6	1.50839
$280$	108968.	0.0830620
$281$	664435.	0.501980	0.250990	−	0.967990i	$-0.419244\pi$
$281$	664435.	0.501980	0.250990	+	0.967990i	$0.419244\pi$
$282$	−162820.	−0.121923
$283$	1.54078e6	1.14360	0.571799	−	0.820393i	$-0.306246\pi$
$283$	1.54078e6	1.14360	0.571799	+	0.820393i	$0.306246\pi$
$284$	−1.75370e6	−1.29021
$285$	−353558.	−0.257839
$286$	0	0
$287$	−841586.	−0.603106
$288$	−570234.	−0.405109
$289$	−1.41492e6	−0.996522
$290$	285641.	0.199446
$291$	−508615.	−0.352092
$292$	−1.37987e6	−0.947068
$293$	1.85718e6	1.26382	0.631911	−	0.775041i	$-0.282271\pi$
$293$	1.85718e6	1.26382	0.631911	+	0.775041i	$0.282271\pi$
$294$	14879.3	0.0100395
$295$	−1.33147e6	−0.890789
$296$	−779275.	−0.516965
$297$	0	0
$298$	241013.	0.157217
$299$	−115552.	−0.0747480
$300$	356203.	0.228505
$301$	−973373.	−0.619246
$302$	317065.	0.200046
$303$	648189.	0.405597
$304$	1.39366e6	0.864911
$305$	−732089.	−0.450624
$306$	13559.3	0.00827816
$307$	2.24174e6	1.35750	0.678750	−	0.734370i	$-0.262522\pi$
$307$	2.24174e6	1.35750	0.678750	+	0.734370i	$0.262522\pi$
$308$	0	0
$309$	178503.	0.106353
$310$	343582.	0.203061
$311$	−631043.	−0.369963	−0.184981	−	0.982742i	$-0.559223\pi$
$311$	−631043.	−0.369963	−0.184981	+	0.982742i	$0.559223\pi$
$312$	300356.	0.174683
$313$	735763.	0.424499	0.212250	−	0.977215i	$-0.431921\pi$
$313$	735763.	0.424499	0.212250	+	0.977215i	$0.431921\pi$
$314$	−219382.	−0.125567
$315$	362517.	0.205850
$316$	2.62864e6	1.48086
$317$	1.00087e6	0.559408	0.279704	−	0.960086i	$-0.409764\pi$
$317$	1.00087e6	0.559408	0.279704	+	0.960086i	$0.409764\pi$
$318$	193206.	0.107140
$319$	0	0
$320$	1.00196e6	0.546983
$321$	234205.	0.126863
$322$	7066.36	0.00379801
$323$	−104563.	−0.0557661
$324$	−943153.	−0.499137
$325$	1.36253e6	0.715547
$326$	481856.	0.251115
$327$	1.13759e6	0.588324
$328$	1.03890e6	0.533196
$329$	−1.28740e6	−0.655729
$330$	0	0
$331$	2.18526e6	1.09631	0.548155	−	0.836377i	$-0.315330\pi$
$331$	2.18526e6	1.09631	0.548155	+	0.836377i	$0.315330\pi$
$332$	−44936.1	−0.0223743
$333$	−2.59251e6	−1.28118
$334$	53443.1	0.0262135
$335$	747507.	0.363918
$336$	296588.	0.143320
$337$	721350.	0.345996	0.172998	−	0.984922i	$-0.444655\pi$
$337$	721350.	0.345996	0.172998	+	0.984922i	$0.444655\pi$
$338$	210047.	0.100006
$339$	−199379.	−0.0942279
$340$	−80294.1	−0.0376692
$341$	0	0
$342$	−287136.	−0.132746
$343$	117649.	0.0539949
$344$	1.20158e6	0.547465
$345$	−35733.3	−0.0161631
$346$	513725.	0.230696
$347$	−1.17251e6	−0.522751	−0.261375	−	0.965237i	$-0.584176\pi$
$347$	−1.17251e6	−0.522751	−0.261375	+	0.965237i	$0.584176\pi$
$348$	1.62749e6	0.720392
$349$	2.24601e6	0.987070	0.493535	−	0.869726i	$-0.335704\pi$
$349$	2.24601e6	0.987070	0.493535	+	0.869726i	$0.335704\pi$
$350$	−83322.9	−0.0363575
$351$	2.20586e6	0.955676
$352$	0	0
$353$	−2.91111e6	−1.24343	−0.621715	−	0.783244i	$-0.713564\pi$
$353$	−2.91111e6	−1.24343	−0.621715	+	0.783244i	$0.713564\pi$
$354$	224433.	0.0951873
$355$	−2.07443e6	−0.873633
$356$	2.57063e6	1.07502
$357$	−22252.3	−0.00924069
$358$	172275.	0.0710418
$359$	−3.53764e6	−1.44870	−0.724349	−	0.689434i	$-0.757859\pi$
$359$	−3.53764e6	−1.44870	−0.724349	+	0.689434i	$0.757859\pi$
$360$	−447508.	−0.181989
$361$	−261850.	−0.105751
$362$	−239264.	−0.0959634
$363$	0	0
$364$	1.17014e6	0.462895
$365$	−1.63223e6	−0.641284
$366$	123402.	0.0481525
$367$	−774649.	−0.300220	−0.150110	−	0.988669i	$-0.547963\pi$
$367$	−774649.	−0.300220	−0.150110	+	0.988669i	$0.547963\pi$
$368$	140853.	0.0542186
$369$	3.45623e6	1.32141
$370$	−454180.	−0.172474
$371$	1.52766e6	0.576226
$372$	1.95762e6	0.733450
$373$	1.02837e6	0.382715	0.191358	−	0.981520i	$-0.438711\pi$
$373$	1.02837e6	0.382715	0.191358	+	0.981520i	$0.438711\pi$
$374$	0	0
$375$	1.16385e6	0.427385
$376$	1.58923e6	0.579719
$377$	6.22537e6	2.25586
$378$	−134895.	−0.0485587
$379$	−5.00513e6	−1.78985	−0.894927	−	0.446213i	$-0.852772\pi$
$379$	−5.00513e6	−1.78985	−0.894927	+	0.446213i	$0.852772\pi$
$380$	1.70033e6	0.604052
$381$	597311.	0.210808
$382$	88474.1	0.0310211
$383$	−1.68221e6	−0.585981	−0.292990	−	0.956115i	$-0.594650\pi$
$383$	−1.68221e6	−0.585981	−0.292990	+	0.956115i	$0.594650\pi$
$384$	−754920.	−0.261260
$385$	0	0
$386$	−160601.	−0.0548629
$387$	3.99745e6	1.35677
$388$	2.44603e6	0.824863
$389$	1.83112e6	0.613540	0.306770	−	0.951784i	$-0.400752\pi$
$389$	1.83112e6	0.613540	0.306770	+	0.951784i	$0.400752\pi$
$390$	175055.	0.0582790
$391$	−10567.9	−0.00349580
$392$	−145232.	−0.0477360
$393$	−1.19950e6	−0.391758
$394$	−395625.	−0.128394
$395$	3.10938e6	1.00272
$396$	0	0
$397$	−1.49640e6	−0.476510	−0.238255	−	0.971203i	$-0.576575\pi$
$397$	−1.49640e6	−0.476510	−0.238255	+	0.971203i	$0.576575\pi$
$398$	−2193.29	−0.000694047 0
$399$	471221.	0.148181
$400$	−1.66087e6	−0.519023
$401$	−598281.	−0.185799	−0.0928996	−	0.995675i	$-0.529614\pi$
$401$	−598281.	−0.185799	−0.0928996	+	0.995675i	$0.529614\pi$
$402$	−126001.	−0.0388873
$403$	7.48818e6	2.29675
$404$	−3.11727e6	−0.950212
$405$	−1.11565e6	−0.337978
$406$	−380701.	−0.114622
$407$	0	0
$408$	27469.3	0.00816955
$409$	425102.	0.125656	0.0628282	−	0.998024i	$-0.479988\pi$
$409$	425102.	0.125656	0.0628282	+	0.998024i	$0.479988\pi$
$410$	605493.	0.177889
$411$	−2.54091e6	−0.741967
$412$	−858458.	−0.249159
$413$	1.77457e6	0.511939
$414$	−29020.1	−0.00832144
$415$	−53154.5	−0.0151502
$416$	−2.17724e6	−0.616840
$417$	963700.	0.271395
$418$	0	0
$419$	5.86734e6	1.63270	0.816350	−	0.577557i	$-0.195994\pi$
$419$	5.86734e6	1.63270	0.816350	+	0.577557i	$0.195994\pi$
$420$	361853.	0.100094
$421$	−7.01271e6	−1.92833	−0.964163	−	0.265309i	$-0.914526\pi$
$421$	−7.01271e6	−1.92833	−0.964163	+	0.265309i	$0.914526\pi$
$422$	259777.	0.0710101
$423$	5.28710e6	1.43670
$424$	−1.88582e6	−0.509432
$425$	124611.	0.0334646
$426$	349669.	0.0933541
$427$	975726.	0.258975
$428$	−1.12634e6	−0.297207
$429$	0	0
$430$	700309.	0.182649
$431$	2.03472e6	0.527609	0.263805	−	0.964576i	$-0.415023\pi$
$431$	2.03472e6	0.527609	0.263805	+	0.964576i	$0.415023\pi$
$432$	−2.68886e6	−0.693201
$433$	2.32473e6	0.595871	0.297935	−	0.954586i	$-0.403702\pi$
$433$	2.32473e6	0.595871	0.297935	+	0.954586i	$0.403702\pi$
$434$	−457925.	−0.116700
$435$	1.92513e6	0.487796
$436$	−5.47089e6	−1.37829
$437$	223789.	0.0560576
$438$	275131.	0.0685259
$439$	−4.07058e6	−1.00808	−0.504040	−	0.863680i	$-0.668154\pi$
$439$	−4.07058e6	−1.00808	−0.504040	+	0.863680i	$0.668154\pi$
$440$	0	0
$441$	−483161.	−0.118303
$442$	51771.4	0.0126047
$443$	858857.	0.207927	0.103964	−	0.994581i	$-0.466847\pi$
$443$	858857.	0.207927	0.103964	+	0.994581i	$0.466847\pi$
$444$	−2.58777e6	−0.622971
$445$	3.04077e6	0.727921
$446$	178428.	0.0424743
$447$	1.62436e6	0.384515
$448$	−1.33540e6	−0.314353
$449$	−3.36786e6	−0.788385	−0.394193	−	0.919028i	$-0.628976\pi$
$449$	−3.36786e6	−0.788385	−0.394193	+	0.919028i	$0.628976\pi$
$450$	342191.	0.0796594
$451$	0	0
$452$	958852.	0.220752
$453$	2.13693e6	0.489265
$454$	1.17564e6	0.267691
$455$	1.38414e6	0.313438
$456$	−581699.	−0.131004
$457$	−2.37765e6	−0.532546	−0.266273	−	0.963898i	$-0.585792\pi$
$457$	−2.37765e6	−0.532546	−0.266273	+	0.963898i	$0.585792\pi$
$458$	−218367.	−0.0486432
$459$	201739.	0.0446949
$460$	171848.	0.0378661
$461$	6.76002e6	1.48148	0.740740	−	0.671792i	$-0.234475\pi$
$461$	6.76002e6	1.48148	0.740740	+	0.671792i	$0.234475\pi$
$462$	0	0
$463$	4.19894e6	0.910307	0.455153	−	0.890413i	$-0.349584\pi$
$463$	4.19894e6	0.910307	0.455153	+	0.890413i	$0.349584\pi$
$464$	−7.58849e6	−1.63629
$465$	2.31565e6	0.496638
$466$	−249156.	−0.0531503
$467$	−5.56498e6	−1.18079	−0.590393	−	0.807116i	$-0.701027\pi$
$467$	−5.56498e6	−1.18079	−0.590393	+	0.807116i	$0.701027\pi$
$468$	−4.80551e6	−1.01420
$469$	−996275.	−0.209145
$470$	926242.	0.193410
$471$	−1.47857e6	−0.307107
$472$	−2.19062e6	−0.452597
$473$	0	0
$474$	−524121.	−0.107148
$475$	−2.63881e6	−0.536628
$476$	107016.	0.0216486
$477$	−6.27381e6	−1.26251
$478$	863383.	0.172836
$479$	−7.86712e6	−1.56667	−0.783334	−	0.621601i	$-0.786482\pi$
$479$	−7.86712e6	−1.56667	−0.783334	+	0.621601i	$0.786482\pi$
$480$	−673289.	−0.133382
$481$	−9.89859e6	−1.95079
$482$	−1.45146e6	−0.284570
$483$	47625.2	0.00928900
$484$	0	0
$485$	2.89338e6	0.558536
$486$	857025.	0.164590
$487$	6.57826e6	1.25686	0.628432	−	0.777864i	$-0.283697\pi$
$487$	6.57826e6	1.25686	0.628432	+	0.777864i	$0.283697\pi$
$488$	−1.20448e6	−0.228956
$489$	3.24757e6	0.614167
$490$	−84644.5	−0.0159261
$491$	−3.37633e6	−0.632035	−0.316018	−	0.948753i	$-0.602346\pi$
$491$	−3.37633e6	−0.632035	−0.316018	+	0.948753i	$0.602346\pi$
$492$	3.44990e6	0.642530
$493$	569347.	0.105502
$494$	−1.09633e6	−0.202126
$495$	0	0
$496$	−9.12781e6	−1.66595
$497$	2.76480e6	0.502080
$498$	8959.78	0.00161891
$499$	−3.80002e6	−0.683178	−0.341589	−	0.939849i	$-0.610965\pi$
$499$	−3.80002e6	−0.683178	−0.341589	+	0.939849i	$0.610965\pi$
$500$	−5.59719e6	−1.00126
$501$	360191.	0.0641119
$502$	487794.	0.0863927
$503$	−7.37630e6	−1.29993	−0.649963	−	0.759966i	$-0.725216\pi$
$503$	−7.37630e6	−1.29993	−0.649963	+	0.759966i	$0.725216\pi$
$504$	596438.	0.104590
$505$	−3.68738e6	−0.643413
$506$	0	0
$507$	1.41566e6	0.244589
$508$	−2.87259e6	−0.493871
$509$	−99425.5	−0.0170100	−0.00850498	−	0.999964i	$-0.502707\pi$
$509$	−99425.5	−0.0170100	−0.00850498	+	0.999964i	$0.502707\pi$
$510$	16009.8	0.00272559
$511$	2.17544e6	0.368548
$512$	4.46682e6	0.753049
$513$	−4.27208e6	−0.716715
$514$	1.29822e6	0.216741
$515$	−1.01546e6	−0.168712
$516$	3.99013e6	0.659724
$517$	0	0
$518$	605329.	0.0991214
$519$	3.46236e6	0.564226
$520$	−1.70865e6	−0.277106
$521$	4.16961e6	0.672978	0.336489	−	0.941687i	$-0.390761\pi$
$521$	4.16961e6	0.672978	0.336489	+	0.941687i	$0.390761\pi$
$522$	1.56346e6	0.251137
$523$	4.09039e6	0.653898	0.326949	−	0.945042i	$-0.393979\pi$
$523$	4.09039e6	0.653898	0.326949	+	0.945042i	$0.393979\pi$
$524$	5.76861e6	0.917790
$525$	−561573.	−0.0889217
$526$	−485631.	−0.0765318
$527$	684838.	0.107414
$528$	0	0
$529$	−6.41373e6	−0.996486
$530$	−1.09910e6	−0.169961
$531$	−7.28781e6	−1.12166
$532$	−2.26620e6	−0.347151
$533$	1.31964e7	2.01204
$534$	−512556.	−0.0777837
$535$	−1.33233e6	−0.201247
$536$	1.22985e6	0.184902
$537$	1.16108e6	0.173751
$538$	−1.53241e6	−0.228255
$539$	0	0
$540$	−3.28055e6	−0.484131
$541$	1.24753e7	1.83256	0.916278	−	0.400542i	$-0.131178\pi$
$541$	1.24753e7	1.83256	0.916278	+	0.400542i	$0.131178\pi$
$542$	−349444.	−0.0510951
$543$	−1.61257e6	−0.234703
$544$	−199121.	−0.0288483
$545$	−6.47146e6	−0.933278
$546$	−233312.	−0.0334932
$547$	−1.49733e6	−0.213968	−0.106984	−	0.994261i	$-0.534119\pi$
$547$	−1.49733e6	−0.213968	−0.106984	+	0.994261i	$0.534119\pi$
$548$	1.22197e7	1.73824
$549$	−4.00711e6	−0.567415
$550$	0	0
$551$	−1.20566e7	−1.69179
$552$	−58790.9	−0.00821226
$553$	−4.14418e6	−0.576269
$554$	−1.32325e6	−0.183175
$555$	−3.06104e6	−0.421829
$556$	−4.63462e6	−0.635810
$557$	−1.38390e7	−1.89002	−0.945010	−	0.327041i	$-0.893948\pi$
$557$	−1.38390e7	−1.89002	−0.945010	+	0.327041i	$0.893948\pi$
$558$	1.88061e6	0.255689
$559$	1.52628e7	2.06588
$560$	−1.68722e6	−0.227353
$561$	0	0
$562$	637128.	0.0850914
$563$	−5.00282e6	−0.665187	−0.332594	−	0.943070i	$-0.607924\pi$
$563$	−5.00282e6	−0.665187	−0.332594	+	0.943070i	$0.607924\pi$
$564$	5.27742e6	0.698592
$565$	1.13422e6	0.149477
$566$	1.47745e6	0.193853
$567$	1.48693e6	0.194237
$568$	−3.41300e6	−0.443880
$569$	−1.41678e6	−0.183451	−0.0917256	−	0.995784i	$-0.529238\pi$
$569$	−1.41678e6	−0.183451	−0.0917256	+	0.995784i	$0.529238\pi$
$570$	−339028.	−0.0437067
$571$	2.41819e6	0.310385	0.155192	−	0.987884i	$-0.450400\pi$
$571$	2.41819e6	0.310385	0.155192	+	0.987884i	$0.450400\pi$
$572$	0	0
$573$	596290.	0.0758702
$574$	−806998.	−0.102233
$575$	−266698.	−0.0336395
$576$	5.48424e6	0.688747
$577$	−1.20706e6	−0.150934	−0.0754671	−	0.997148i	$-0.524045\pi$
$577$	−1.20706e6	−0.150934	−0.0754671	+	0.997148i	$0.524045\pi$
$578$	−1.35677e6	−0.168922
$579$	−1.08240e6	−0.134181
$580$	−9.25835e6	−1.14278
$581$	70844.1	0.00870689
$582$	−487711.	−0.0596837
$583$	0	0
$584$	−2.68547e6	−0.325827
$585$	−5.68439e6	−0.686743
$586$	1.78086e6	0.214232
$587$	7.12904e6	0.853956	0.426978	−	0.904262i	$-0.359578\pi$
$587$	7.12904e6	0.853956	0.426978	+	0.904262i	$0.359578\pi$
$588$	−482276.	−0.0575245
$589$	−1.45023e7	−1.72246
$590$	−1.27674e6	−0.150999
$591$	−2.66640e6	−0.314020
$592$	1.20660e7	1.41501
$593$	−143219.	−0.0167249	−0.00836246	−	0.999965i	$-0.502662\pi$
$593$	−143219.	−0.0167249	−0.00836246	+	0.999965i	$0.502662\pi$
$594$	0	0
$595$	126588.	0.0146588
$596$	−7.81186e6	−0.900822
$597$	−14782.2	−0.00169747
$598$	−110803.	−0.0126706
$599$	1.38830e7	1.58094	0.790472	−	0.612498i	$-0.209835\pi$
$599$	1.38830e7	1.58094	0.790472	+	0.612498i	$0.209835\pi$
$600$	693233.	0.0786142
$601$	179233.	0.0202410	0.0101205	−	0.999949i	$-0.496778\pi$
$601$	179233.	0.0202410	0.0101205	+	0.999949i	$0.496778\pi$
$602$	−933369.	−0.104969
$603$	4.09150e6	0.458236
$604$	−1.02769e7	−1.14622
$605$	0	0
$606$	621549.	0.0687534
$607$	−6.55746e6	−0.722377	−0.361189	−	0.932493i	$-0.617629\pi$
$607$	−6.55746e6	−0.722377	−0.361189	+	0.932493i	$0.617629\pi$
$608$	4.21664e6	0.462603
$609$	−2.56581e6	−0.280338
$610$	−702002.	−0.0763860
$611$	2.01869e7	2.18760
$612$	−439492.	−0.0474321
$613$	9.21833e6	0.990834	0.495417	−	0.868655i	$-0.335015\pi$
$613$	9.21833e6	0.990834	0.495417	+	0.868655i	$0.335015\pi$
$614$	2.14961e6	0.230112
$615$	4.08085e6	0.435073
$616$	0	0
$617$	−937221.	−0.0991127	−0.0495563	−	0.998771i	$-0.515781\pi$
$617$	−937221.	−0.0991127	−0.0495563	+	0.998771i	$0.515781\pi$
$618$	171167.	0.0180281
$619$	1.78675e6	0.187430	0.0937148	−	0.995599i	$-0.470126\pi$
$619$	1.78675e6	0.187430	0.0937148	+	0.995599i	$0.470126\pi$
$620$	−1.11364e7	−1.16350
$621$	−431769.	−0.0449286
$622$	−605109.	−0.0627130
$623$	−4.05273e6	−0.418339
$624$	−4.65061e6	−0.478132
$625$	−1.07914e6	−0.110504
$626$	705524.	0.0719575
$627$	0	0
$628$	7.11073e6	0.719474
$629$	−905284.	−0.0912343
$630$	347618.	0.0348940
$631$	−1.14205e7	−1.14186	−0.570930	−	0.820999i	$-0.693417\pi$
$631$	−1.14205e7	−1.14186	−0.570930	+	0.820999i	$0.693417\pi$
$632$	5.11578e6	0.509470
$633$	1.75083e6	0.173673
$634$	959734.	0.0948261
$635$	−3.39795e6	−0.334413
$636$	−6.26232e6	−0.613893
$637$	−1.84478e6	−0.180134
$638$	0	0
$639$	−1.13545e7	−1.10006
$640$	4.29455e6	0.414446
$641$	6.83506e6	0.657048	0.328524	−	0.944496i	$-0.393449\pi$
$641$	6.83506e6	0.657048	0.328524	+	0.944496i	$0.393449\pi$
$642$	224580.	0.0215047
$643$	1.43138e7	1.36530	0.682649	−	0.730747i	$-0.260828\pi$
$643$	1.43138e7	1.36530	0.682649	+	0.730747i	$0.260828\pi$
$644$	−229039.	−0.0217618
$645$	4.71988e6	0.446716
$646$	−100265.	−0.00945300
$647$	−1.91367e7	−1.79724	−0.898620	−	0.438728i	$-0.855429\pi$
$647$	−1.91367e7	−1.79724	−0.898620	+	0.438728i	$0.855429\pi$
$648$	−1.83554e6	−0.171722
$649$	0	0
$650$	1.30653e6	0.121293
$651$	−3.08629e6	−0.285419
$652$	−1.56182e7	−1.43884
$653$	−1.45179e7	−1.33236	−0.666178	−	0.745793i	$-0.732071\pi$
$653$	−1.45179e7	−1.33236	−0.666178	+	0.745793i	$0.732071\pi$
$654$	1.09084e6	0.0997277
$655$	6.82363e6	0.621459
$656$	−1.60859e7	−1.45944
$657$	−8.93408e6	−0.807489
$658$	−1.23449e6	−0.111154
$659$	6.68001e6	0.599189	0.299594	−	0.954067i	$-0.403149\pi$
$659$	6.68001e6	0.599189	0.299594	+	0.954067i	$0.403149\pi$
$660$	0	0
$661$	−1.45158e7	−1.29222	−0.646112	−	0.763243i	$-0.723606\pi$
$661$	−1.45158e7	−1.29222	−0.646112	+	0.763243i	$0.723606\pi$
$662$	2.09545e6	0.185837
$663$	348924.	0.0308281
$664$	−87453.4	−0.00769762
$665$	−2.68066e6	−0.235065
$666$	−2.48597e6	−0.217175
$667$	−1.21854e6	−0.106053
$668$	−1.73223e6	−0.150198
$669$	1.20255e6	0.103882
$670$	716786.	0.0616883
$671$	0	0
$672$	897357.	0.0766553
$673$	−1.22917e7	−1.04610	−0.523049	−	0.852302i	$-0.675206\pi$
$673$	−1.22917e7	−1.04610	−0.523049	+	0.852302i	$0.675206\pi$
$674$	691703.	0.0586503
$675$	5.09121e6	0.430092
$676$	−6.80817e6	−0.573012
$677$	−1.29834e7	−1.08872	−0.544362	−	0.838850i	$-0.683228\pi$
$677$	−1.29834e7	−1.08872	−0.544362	+	0.838850i	$0.683228\pi$
$678$	−191185.	−0.0159727
$679$	−3.85629e6	−0.320993
$680$	−156266.	−0.0129596
$681$	7.92345e6	0.654706
$682$	0	0
$683$	−1.34111e7	−1.10005	−0.550027	−	0.835147i	$-0.685382\pi$
$683$	−1.34111e7	−1.10005	−0.550027	+	0.835147i	$0.685382\pi$
$684$	9.30681e6	0.760608
$685$	1.44546e7	1.17701
$686$	112814.	0.00915276
$687$	−1.47173e6	−0.118970
$688$	−1.86048e7	−1.49849
$689$	−2.39543e7	−1.92236
$690$	−34264.7	−0.00273984
$691$	6.14724e6	0.489762	0.244881	−	0.969553i	$-0.421251\pi$
$691$	6.14724e6	0.489762	0.244881	+	0.969553i	$0.421251\pi$
$692$	−1.66512e7	−1.32184
$693$	0	0
$694$	−1.12433e6	−0.0886122
$695$	−5.48225e6	−0.430523
$696$	3.16737e6	0.247842
$697$	1.20688e6	0.0940988
$698$	2.15370e6	0.167320
$699$	−1.67924e6	−0.129993
$700$	2.70071e6	0.208321
$701$	1.01514e6	0.0780246	0.0390123	−	0.999239i	$-0.487579\pi$
$701$	1.01514e6	0.0780246	0.0390123	+	0.999239i	$0.487579\pi$
$702$	2.11521e6	0.161998
$703$	1.91706e7	1.46301
$704$	0	0
$705$	6.24260e6	0.473035
$706$	−2.79147e6	−0.210776
$707$	4.91453e6	0.369772
$708$	−7.27447e6	−0.545404
$709$	9.05949e6	0.676843	0.338422	−	0.940995i	$-0.390107\pi$
$709$	9.05949e6	0.676843	0.338422	+	0.940995i	$0.390107\pi$
$710$	−1.98918e6	−0.148091
$711$	1.70193e7	1.26261
$712$	5.00289e6	0.369846
$713$	−1.46572e6	−0.107976
$714$	−21337.8	−0.00156640
$715$	0	0
$716$	−5.58387e6	−0.407055
$717$	5.81895e6	0.422714
$718$	−3.39225e6	−0.245571
$719$	−2.76160e6	−0.199223	−0.0996114	−	0.995026i	$-0.531760\pi$
$719$	−2.76160e6	−0.199223	−0.0996114	+	0.995026i	$0.531760\pi$
$720$	6.92906e6	0.498130
$721$	1.35340e6	0.0969591
$722$	−251089.	−0.0179260
$723$	−9.78245e6	−0.695988
$724$	7.75517e6	0.549851
$725$	1.43684e7	1.01523
$726$	0	0
$727$	5.19162e6	0.364307	0.182153	−	0.983270i	$-0.441693\pi$
$727$	5.19162e6	0.364307	0.182153	+	0.983270i	$0.441693\pi$
$728$	2.27729e6	0.159253
$729$	−1.59785e6	−0.111357
$730$	−1.56515e6	−0.108705
$731$	1.39588e6	0.0966169
$732$	−3.99977e6	−0.275904
$733$	−1.40906e7	−0.968657	−0.484328	−	0.874886i	$-0.660936\pi$
$733$	−1.40906e7	−0.968657	−0.484328	+	0.874886i	$0.660936\pi$
$734$	−742812.	−0.0508908
$735$	−570479.	−0.0389513
$736$	426166.	0.0289991
$737$	0	0
$738$	3.31418e6	0.223993
$739$	−474381.	−0.0319533	−0.0159767	−	0.999872i	$-0.505086\pi$
$739$	−474381.	−0.0319533	−0.0159767	+	0.999872i	$0.505086\pi$
$740$	1.47211e7	0.988240
$741$	−7.38892e6	−0.494351
$742$	1.46488e6	0.0976769
$743$	2.78899e7	1.85342	0.926712	−	0.375773i	$-0.122623\pi$
$743$	2.78899e7	1.85342	0.926712	+	0.375773i	$0.122623\pi$
$744$	3.80986e6	0.252335
$745$	−9.24057e6	−0.609970
$746$	986102.	0.0648746
$747$	−290942.	−0.0190768
$748$	0	0
$749$	1.77573e6	0.115657
$750$	1.11602e6	0.0724467
$751$	5.28129e6	0.341696	0.170848	−	0.985297i	$-0.445349\pi$
$751$	5.28129e6	0.341696	0.170848	+	0.985297i	$0.445349\pi$
$752$	−2.46071e7	−1.58678
$753$	3.28759e6	0.211296
$754$	5.96952e6	0.382394
$755$	−1.21564e7	−0.776137
$756$	4.37231e6	0.278231
$757$	−9.63635e6	−0.611185	−0.305593	−	0.952162i	$-0.598855\pi$
$757$	−9.63635e6	−0.611185	−0.305593	+	0.952162i	$0.598855\pi$
$758$	−4.79943e6	−0.303401
$759$	0	0
$760$	3.30914e6	0.207817
$761$	1.94928e7	1.22015	0.610074	−	0.792344i	$-0.291140\pi$
$761$	1.94928e7	1.22015	0.610074	+	0.792344i	$0.291140\pi$
$762$	572763.	0.0357345
$763$	8.62514e6	0.536358
$764$	−2.86768e6	−0.177745
$765$	−519871.	−0.0321175
$766$	−1.61307e6	−0.0993304
$767$	−2.78259e7	−1.70790
$768$	4.91225e6	0.300523
$769$	1.45709e7	0.888529	0.444264	−	0.895896i	$-0.353465\pi$
$769$	1.45709e7	0.888529	0.444264	+	0.895896i	$0.353465\pi$
$770$	0	0
$771$	8.74965e6	0.530096
$772$	5.20549e6	0.314353
$773$	−1.16527e7	−0.701420	−0.350710	−	0.936484i	$-0.614060\pi$
$773$	−1.16527e7	−0.701420	−0.350710	+	0.936484i	$0.614060\pi$
$774$	3.83316e6	0.229988
$775$	1.72830e7	1.03363
$776$	4.76039e6	0.283784
$777$	4.07974e6	0.242427
$778$	1.75586e6	0.104002
$779$	−2.55573e7	−1.50894
$780$	−5.67398e6	−0.333927
$781$	0	0
$782$	−10133.6	−0.000592579 0
$783$	2.32616e7	1.35592
$784$	2.24871e6	0.130660
$785$	8.41121e6	0.487174
$786$	−1.15020e6	−0.0664074
$787$	−2.65309e7	−1.52691	−0.763457	−	0.645858i	$-0.776500\pi$
$787$	−2.65309e7	−1.52691	−0.763457	+	0.645858i	$0.776500\pi$
$788$	1.28232e7	0.735669
$789$	−3.27301e6	−0.187178
$790$	2.98159e6	0.169973
$791$	−1.51168e6	−0.0859050
$792$	0	0
$793$	−1.52997e7	−0.863974
$794$	−1.43490e6	−0.0807739
$795$	−7.40763e6	−0.415682
$796$	71090.3	0.00397675
$797$	−1.21394e7	−0.676943	−0.338472	−	0.940977i	$-0.609910\pi$
$797$	−1.21394e7	−0.676943	−0.338472	+	0.940977i	$0.609910\pi$
$798$	451855.	0.0251184
$799$	1.84621e6	0.102309
$800$	−5.02514e6	−0.277602
$801$	1.66438e7	0.916580
$802$	−573692.	−0.0314951
$803$	0	0
$804$	4.08401e6	0.222816
$805$	−270928.	−0.0147355
$806$	7.18043e6	0.389326
$807$	−1.03280e7	−0.558257
$808$	−6.06674e6	−0.326909
$809$	9.25477e6	0.497158	0.248579	−	0.968612i	$-0.420036\pi$
$809$	9.25477e6	0.497158	0.248579	+	0.968612i	$0.420036\pi$
$810$	−1.06979e6	−0.0572912
$811$	1.51775e7	0.810304	0.405152	−	0.914249i	$-0.367219\pi$
$811$	1.51775e7	0.810304	0.405152	+	0.914249i	$0.367219\pi$
$812$	1.23395e7	0.656761
$813$	−2.35515e6	−0.124966
$814$	0	0
$815$	−1.84746e7	−0.974274
$816$	−425325.	−0.0223612
$817$	−2.95595e7	−1.54932
$818$	407631.	0.0213002
$819$	7.57614e6	0.394673
$820$	−1.96256e7	−1.01927
$821$	9.22700e6	0.477752	0.238876	−	0.971050i	$-0.423221\pi$
$821$	9.22700e6	0.477752	0.238876	+	0.971050i	$0.423221\pi$
$822$	−2.43648e6	−0.125772
$823$	−1.71945e7	−0.884892	−0.442446	−	0.896795i	$-0.645889\pi$
$823$	−1.71945e7	−0.884892	−0.442446	+	0.896795i	$0.645889\pi$
$824$	−1.67071e6	−0.0857200
$825$	0	0
$826$	1.70164e6	0.0867796
$827$	432670.	0.0219985	0.0109992	−	0.999940i	$-0.496499\pi$
$827$	432670.	0.0219985	0.0109992	+	0.999940i	$0.496499\pi$
$828$	940617.	0.0476801
$829$	−307440.	−0.0155373	−0.00776863	−	0.999970i	$-0.502473\pi$
$829$	−307440.	−0.0155373	−0.00776863	+	0.999970i	$0.502473\pi$
$830$	−50969.9	−0.00256814
$831$	−8.91831e6	−0.448002
$832$	2.09396e7	1.04872
$833$	−168716.	−0.00842448
$834$	924094.	0.0460046
$835$	−2.04904e6	−0.101703
$836$	0	0
$837$	2.79802e7	1.38050
$838$	5.62621e6	0.276761
$839$	−1.98679e7	−0.974424	−0.487212	−	0.873284i	$-0.661986\pi$
$839$	−1.98679e7	−0.974424	−0.487212	+	0.873284i	$0.661986\pi$
$840$	704228.	0.0344362
$841$	4.51377e7	2.20064
$842$	−6.72450e6	−0.326874
$843$	4.29406e6	0.208113
$844$	−8.42006e6	−0.406873
$845$	−8.05331e6	−0.388001
$846$	5.06981e6	0.243538
$847$	0	0
$848$	2.91994e7	1.39439
$849$	9.95761e6	0.474118
$850$	119490.	0.00567263
$851$	1.93752e6	0.0917113
$852$	−1.13337e7	−0.534900
$853$	2.40519e7	1.13182	0.565908	−	0.824468i	$-0.308526\pi$
$853$	2.40519e7	1.13182	0.565908	+	0.824468i	$0.308526\pi$
$854$	935625.	0.0438993
$855$	1.10089e7	0.515027
$856$	−2.19205e6	−0.102251
$857$	1.80084e7	0.837575	0.418787	−	0.908084i	$-0.362455\pi$
$857$	1.80084e7	0.837575	0.418787	+	0.908084i	$0.362455\pi$
$858$	0	0
$859$	2.72203e7	1.25866	0.629332	−	0.777137i	$-0.283329\pi$
$859$	2.72203e7	1.25866	0.629332	+	0.777137i	$0.283329\pi$
$860$	−2.26988e7	−1.04654
$861$	−5.43894e6	−0.250038
$862$	1.95110e6	0.0894359
$863$	−3.14434e7	−1.43715	−0.718577	−	0.695448i	$-0.755206\pi$
$863$	−3.14434e7	−1.43715	−0.718577	+	0.695448i	$0.755206\pi$
$864$	−8.13543e6	−0.370763
$865$	−1.96965e7	−0.895052
$866$	2.22918e6	0.101007
$867$	−9.14423e6	−0.413142
$868$	1.48426e7	0.668666
$869$	0	0
$870$	1.84602e6	0.0826870
$871$	1.56219e7	0.697734
$872$	−1.06473e7	−0.474186
$873$	1.58370e7	0.703295
$874$	214592.	0.00950242
$875$	8.82426e6	0.389635
$876$	−8.91772e6	−0.392639
$877$	−2.52942e7	−1.11051	−0.555254	−	0.831681i	$-0.687379\pi$
$877$	−2.52942e7	−1.11051	−0.555254	+	0.831681i	$0.687379\pi$
$878$	−3.90329e6	−0.170881
$879$	1.20025e7	0.523960
$880$	0	0
$881$	−2.26455e6	−0.0982972	−0.0491486	−	0.998791i	$-0.515651\pi$
$881$	−2.26455e6	−0.0982972	−0.0491486	+	0.998791i	$0.515651\pi$
$882$	−463304.	−0.0200537
$883$	3.60897e7	1.55769	0.778845	−	0.627217i	$-0.215806\pi$
$883$	3.60897e7	1.55769	0.778845	+	0.627217i	$0.215806\pi$
$884$	−1.67804e6	−0.0722225
$885$	−8.60489e6	−0.369307
$886$	823560.	0.0352461
$887$	−1.43415e6	−0.0612046	−0.0306023	−	0.999532i	$-0.509743\pi$
$887$	−1.43415e6	−0.0612046	−0.0306023	+	0.999532i	$0.509743\pi$
$888$	−5.03624e6	−0.214326
$889$	4.52878e6	0.192188
$890$	2.91580e6	0.123391
$891$	0	0
$892$	−5.78332e6	−0.243369
$893$	−3.90959e7	−1.64060
$894$	1.55760e6	0.0651797
$895$	−6.60511e6	−0.275627
$896$	−5.72376e6	−0.238183
$897$	−746780.	−0.0309893
$898$	−3.22945e6	−0.133640
$899$	7.89656e7	3.25866
$900$	−1.10913e7	−0.456432
$901$	−2.19076e6	−0.0899048
$902$	0	0
$903$	−6.29064e6	−0.256729
$904$	1.86609e6	0.0759472
$905$	9.17350e6	0.372318
$906$	2.04910e6	0.0829360
$907$	−6.28925e6	−0.253852	−0.126926	−	0.991912i	$-0.540511\pi$
$907$	−6.28925e6	−0.253852	−0.126926	+	0.991912i	$0.540511\pi$
$908$	−3.81054e7	−1.53381
$909$	−2.01830e7	−0.810169
$910$	1.32726e6	0.0531314
$911$	−2.42242e7	−0.967061	−0.483530	−	0.875328i	$-0.660646\pi$
$911$	−2.42242e7	−0.967061	−0.483530	+	0.875328i	$0.660646\pi$
$912$	9.00681e6	0.358578
$913$	0	0
$914$	−2.27993e6	−0.0902726
$915$	−4.73129e6	−0.186821
$916$	7.07783e6	0.278716
$917$	−9.09451e6	−0.357154
$918$	193448.	0.00757630
$919$	2.48196e7	0.969408	0.484704	−	0.874678i	$-0.338927\pi$
$919$	2.48196e7	0.969408	0.484704	+	0.874678i	$0.338927\pi$
$920$	334447.	0.0130274
$921$	1.44878e7	0.562797
$922$	6.48220e6	0.251128
$923$	−4.33530e7	−1.67500
$924$	0	0
$925$	−2.28463e7	−0.877933
$926$	4.02638e6	0.154307
$927$	−5.55815e6	−0.212437
$928$	−2.29598e7	−0.875180
$929$	1.09165e7	0.414997	0.207498	−	0.978235i	$-0.433468\pi$
$929$	1.09165e7	0.414997	0.207498	+	0.978235i	$0.433468\pi$
$930$	2.22048e6	0.0841858
$931$	3.57277e6	0.135092
$932$	8.07579e6	0.304541
$933$	−4.07826e6	−0.153381
$934$	−5.33627e6	−0.200157
$935$	0	0
$936$	−9.35235e6	−0.348924
$937$	−1.65501e7	−0.615816	−0.307908	−	0.951416i	$-0.599629\pi$
$937$	−1.65501e7	−0.615816	−0.307908	+	0.951416i	$0.599629\pi$
$938$	−955330.	−0.0354525
$939$	4.75503e6	0.175991
$940$	−3.00219e7	−1.10820
$941$	2.74702e7	1.01132	0.505659	−	0.862733i	$-0.331249\pi$
$941$	2.74702e7	1.01132	0.505659	+	0.862733i	$0.331249\pi$
$942$	−1.41780e6	−0.0520581
$943$	−2.58302e6	−0.0945907
$944$	3.39187e7	1.23882
$945$	5.17196e6	0.188398
$946$	0	0
$947$	−2.25439e7	−0.816870	−0.408435	−	0.912787i	$-0.633925\pi$
$947$	−2.25439e7	−0.816870	−0.408435	+	0.912787i	$0.633925\pi$
$948$	1.69881e7	0.613939
$949$	−3.41116e7	−1.22952
$950$	−2.53036e6	−0.0909646
$951$	6.46833e6	0.231921
$952$	208271.	0.00744794
$953$	4.22460e7	1.50679	0.753397	−	0.657566i	$-0.228414\pi$
$953$	4.22460e7	1.50679	0.753397	+	0.657566i	$0.228414\pi$
$954$	−6.01597e6	−0.214010
$955$	−3.39214e6	−0.120355
$956$	−2.79845e7	−0.990314
$957$	0	0
$958$	−7.54379e6	−0.265568
$959$	−1.92650e7	−0.676430
$960$	6.47536e6	0.226770
$961$	6.63545e7	2.31772
$962$	−9.49178e6	−0.330681
$963$	−7.29257e6	−0.253405
$964$	4.70457e7	1.63053
$965$	6.15751e6	0.212857
$966$	45667.9	0.00157459
$967$	−1.12060e7	−0.385376	−0.192688	−	0.981260i	$-0.561721\pi$
$967$	−1.12060e7	−0.385376	−0.192688	+	0.981260i	$0.561721\pi$
$968$	0	0
$969$	−675759.	−0.0231197
$970$	2.77447e6	0.0946783
$971$	2.38526e7	0.811872	0.405936	−	0.913902i	$-0.366946\pi$
$971$	2.38526e7	0.811872	0.405936	+	0.913902i	$0.366946\pi$
$972$	−2.77784e7	−0.943065
$973$	7.30672e6	0.247423
$974$	6.30790e6	0.213053
$975$	8.80566e6	0.296654
$976$	1.86498e7	0.626685
$977$	4.60048e7	1.54194	0.770969	−	0.636873i	$-0.219773\pi$
$977$	4.60048e7	1.54194	0.770969	+	0.636873i	$0.219773\pi$
$978$	3.11410e6	0.104108
$979$	0	0
$980$	2.74355e6	0.0912530
$981$	−3.54217e7	−1.17516
$982$	−3.23757e6	−0.107137
$983$	4.17649e7	1.37857	0.689283	−	0.724492i	$-0.257926\pi$
$983$	4.17649e7	1.37857	0.689283	+	0.724492i	$0.257926\pi$
$984$	6.71409e6	0.221055
$985$	1.51685e7	0.498140
$986$	545948.	0.0178838
$987$	−8.32012e6	−0.271855
$988$	3.55348e7	1.15814
$989$	−2.98750e6	−0.0971220
$990$	0	0
$991$	4.50508e7	1.45720	0.728598	−	0.684941i	$-0.240172\pi$
$991$	4.50508e7	1.45720	0.728598	+	0.684941i	$0.240172\pi$
$992$	−2.76171e7	−0.891044
$993$	1.41227e7	0.454513
$994$	2.65117e6	0.0851083
$995$	84092.0	0.00269276
$996$	−290410.	−0.00927604
$997$	−2.44970e7	−0.780505	−0.390252	−	0.920708i	$-0.627612\pi$
$997$	−2.44970e7	−0.780505	−0.390252	+	0.920708i	$0.627612\pi$
$998$	−3.64384e6	−0.115807
$999$	−3.69869e7	−1.17256

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.6.a.o.1.14	yes	26
11.10	odd	2		847.6.a.n.1.13	✓	26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
847.6.a.n.1.13	✓	26	11.10	odd	2
847.6.a.o.1.14	yes	26	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 847 = 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	847.a (trivial)

\(f(q)\)	\(=\)	\(q+0.958902 q^{2} +6.46272 q^{3} -31.0805 q^{4} -36.7648 q^{5} +6.19712 q^{6} +49.0000 q^{7} -60.4880 q^{8} -201.233 q^{9} +O(q^{10})\)	\(q+0.958902 q^{2} +6.46272 q^{3} -31.0805 q^{4} -36.7648 q^{5} +6.19712 q^{6} +49.0000 q^{7} -60.4880 q^{8} -201.233 q^{9} -35.2538 q^{10} -200.865 q^{12} -768.337 q^{13} +46.9862 q^{14} -237.601 q^{15} +936.574 q^{16} -70.2689 q^{17} -192.963 q^{18} +1488.04 q^{19} +1142.67 q^{20} +316.673 q^{21} +150.392 q^{23} -390.917 q^{24} -1773.35 q^{25} -736.760 q^{26} -2870.96 q^{27} -1522.94 q^{28} -8102.39 q^{29} -227.836 q^{30} -9745.96 q^{31} +2833.70 q^{32} -67.3810 q^{34} -1801.48 q^{35} +6254.43 q^{36} +12883.1 q^{37} +1426.88 q^{38} -4965.55 q^{39} +2223.83 q^{40} -17175.2 q^{41} +303.659 q^{42} -19864.8 q^{43} +7398.30 q^{45} +144.211 q^{46} -26273.5 q^{47} +6052.82 q^{48} +2401.00 q^{49} -1700.47 q^{50} -454.129 q^{51} +23880.3 q^{52} +31176.8 q^{53} -2752.97 q^{54} -2963.91 q^{56} +9616.76 q^{57} -7769.40 q^{58} +36215.8 q^{59} +7384.75 q^{60} +19912.8 q^{61} -9345.41 q^{62} -9860.43 q^{63} -27253.1 q^{64} +28247.8 q^{65} -20332.1 q^{67} +2183.99 q^{68} +971.943 q^{69} -1727.44 q^{70} +56424.5 q^{71} +12172.2 q^{72} +44396.6 q^{73} +12353.7 q^{74} -11460.7 q^{75} -46248.9 q^{76} -4761.48 q^{78} -84575.0 q^{79} -34433.0 q^{80} +30345.5 q^{81} -16469.4 q^{82} +1445.80 q^{83} -9842.37 q^{84} +2583.42 q^{85} -19048.3 q^{86} -52363.5 q^{87} -82708.8 q^{89} +7094.24 q^{90} -37648.5 q^{91} -4674.27 q^{92} -62985.4 q^{93} -25193.7 q^{94} -54707.3 q^{95} +18313.4 q^{96} -78699.7 q^{97} +2302.32 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q + 16 q^{2} - 2 q^{3} + 428 q^{4} + 156 q^{5} + 554 q^{6} + 1274 q^{7} + 768 q^{8} + 2368 q^{9}+O(q^{10}) \) 26 * q + 16 * q^2 - 2 * q^3 + 428 * q^4 + 156 * q^5 + 554 * q^6 + 1274 * q^7 + 768 * q^8 + 2368 * q^9	\( 26 q + 16 q^{2} - 2 q^{3} + 428 q^{4} + 156 q^{5} + 554 q^{6} + 1274 q^{7} + 768 q^{8} + 2368 q^{9} + 1132 q^{10} + 658 q^{12} + 1908 q^{13} + 784 q^{14} + 3114 q^{15} + 8772 q^{16} + 4398 q^{17} + 7020 q^{18} + 2922 q^{19} + 6982 q^{20} - 98 q^{21} + 6970 q^{23} + 19924 q^{24} + 10324 q^{25} + 5802 q^{26} - 1346 q^{27} + 20972 q^{28} + 21068 q^{29} - 19072 q^{30} + 1638 q^{31} + 49772 q^{32} + 55400 q^{34} + 7644 q^{35} + 68030 q^{36} + 21362 q^{37} + 20214 q^{38} - 28716 q^{39} + 66894 q^{40} + 30882 q^{41} + 27146 q^{42} + 16068 q^{43} + 32932 q^{45} + 15088 q^{46} - 26000 q^{47} - 10392 q^{48} + 62426 q^{49} + 20744 q^{50} + 41844 q^{51} - 42060 q^{52} + 86508 q^{53} + 160368 q^{54} + 37632 q^{56} + 93764 q^{57} - 52790 q^{58} - 45476 q^{59} - 46376 q^{60} + 101496 q^{61} + 116668 q^{62} + 116032 q^{63} + 121294 q^{64} + 288440 q^{65} + 41942 q^{67} + 255520 q^{68} + 1366 q^{69} + 55468 q^{70} + 86814 q^{71} + 204604 q^{72} + 149254 q^{73} - 161724 q^{74} - 336580 q^{75} - 11930 q^{76} - 617620 q^{78} + 11620 q^{79} + 161660 q^{80} + 195662 q^{81} - 356068 q^{82} + 216994 q^{83} + 32242 q^{84} + 360392 q^{85} + 100438 q^{86} + 49730 q^{87} - 403424 q^{89} + 463182 q^{90} + 93492 q^{91} - 78454 q^{92} + 82690 q^{93} - 467466 q^{94} + 250384 q^{95} + 520006 q^{96} - 360716 q^{97} + 38416 q^{98}+O(q^{100}) \) 26 * q + 16 * q^2 - 2 * q^3 + 428 * q^4 + 156 * q^5 + 554 * q^6 + 1274 * q^7 + 768 * q^8 + 2368 * q^9 + 1132 * q^10 + 658 * q^12 + 1908 * q^13 + 784 * q^14 + 3114 * q^15 + 8772 * q^16 + 4398 * q^17 + 7020 * q^18 + 2922 * q^19 + 6982 * q^20 - 98 * q^21 + 6970 * q^23 + 19924 * q^24 + 10324 * q^25 + 5802 * q^26 - 1346 * q^27 + 20972 * q^28 + 21068 * q^29 - 19072 * q^30 + 1638 * q^31 + 49772 * q^32 + 55400 * q^34 + 7644 * q^35 + 68030 * q^36 + 21362 * q^37 + 20214 * q^38 - 28716 * q^39 + 66894 * q^40 + 30882 * q^41 + 27146 * q^42 + 16068 * q^43 + 32932 * q^45 + 15088 * q^46 - 26000 * q^47 - 10392 * q^48 + 62426 * q^49 + 20744 * q^50 + 41844 * q^51 - 42060 * q^52 + 86508 * q^53 + 160368 * q^54 + 37632 * q^56 + 93764 * q^57 - 52790 * q^58 - 45476 * q^59 - 46376 * q^60 + 101496 * q^61 + 116668 * q^62 + 116032 * q^63 + 121294 * q^64 + 288440 * q^65 + 41942 * q^67 + 255520 * q^68 + 1366 * q^69 + 55468 * q^70 + 86814 * q^71 + 204604 * q^72 + 149254 * q^73 - 161724 * q^74 - 336580 * q^75 - 11930 * q^76 - 617620 * q^78 + 11620 * q^79 + 161660 * q^80 + 195662 * q^81 - 356068 * q^82 + 216994 * q^83 + 32242 * q^84 + 360392 * q^85 + 100438 * q^86 + 49730 * q^87 - 403424 * q^89 + 463182 * q^90 + 93492 * q^91 - 78454 * q^92 + 82690 * q^93 - 467466 * q^94 + 250384 * q^95 + 520006 * q^96 - 360716 * q^97 + 38416 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more