LMFDB - Embedded newform 294.6.a.w.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [294,6,Mod(1,294)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(294, 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("294.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 294 = 2 \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	294.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:	$47.1528430250$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{9601}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 2400 $ x^2 - x - 2400
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 42)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-48.4923$ of defining polynomial
Character	$\chi$	$=$	294.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} +75.4923 q^{5} +36.0000 q^{6} +64.0000 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q+4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} +75.4923 q^{5} +36.0000 q^{6} +64.0000 q^{8} +81.0000 q^{9} +301.969 q^{10} -149.462 q^{11} +144.000 q^{12} -349.416 q^{13} +679.431 q^{15} +256.000 q^{16} +1149.85 q^{17} +324.000 q^{18} +2795.20 q^{19} +1207.88 q^{20} -597.847 q^{22} +1813.97 q^{23} +576.000 q^{24} +2574.09 q^{25} -1397.66 q^{26} +729.000 q^{27} -759.033 q^{29} +2717.72 q^{30} -9031.74 q^{31} +1024.00 q^{32} -1345.16 q^{33} +4599.39 q^{34} +1296.00 q^{36} +7794.89 q^{37} +11180.8 q^{38} -3144.74 q^{39} +4831.51 q^{40} -7640.49 q^{41} +12188.8 q^{43} -2391.39 q^{44} +6114.88 q^{45} +7255.88 q^{46} -24598.8 q^{47} +2304.00 q^{48} +10296.4 q^{50} +10348.6 q^{51} -5590.65 q^{52} +13596.2 q^{53} +2916.00 q^{54} -11283.2 q^{55} +25156.8 q^{57} -3036.13 q^{58} +26358.8 q^{59} +10870.9 q^{60} -35321.8 q^{61} -36127.0 q^{62} +4096.00 q^{64} -26378.2 q^{65} -5380.62 q^{66} +54371.9 q^{67} +18397.6 q^{68} +16325.7 q^{69} -70145.7 q^{71} +5184.00 q^{72} +44468.8 q^{73} +31179.6 q^{74} +23166.8 q^{75} +44723.2 q^{76} -12579.0 q^{78} +61612.5 q^{79} +19326.0 q^{80} +6561.00 q^{81} -30562.0 q^{82} +87142.0 q^{83} +86804.6 q^{85} +48755.3 q^{86} -6831.30 q^{87} -9565.55 q^{88} -98569.4 q^{89} +24459.5 q^{90} +29023.5 q^{92} -81285.7 q^{93} -98395.2 q^{94} +211016. q^{95} +9216.00 q^{96} -32342.3 q^{97} -12106.4 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{2} + 18 q^{3} + 32 q^{4} + 53 q^{5} + 72 q^{6} + 128 q^{8} + 162 q^{9}+O(q^{10}) $ 2 * q + 8 * q^2 + 18 * q^3 + 32 * q^4 + 53 * q^5 + 72 * q^6 + 128 * q^8 + 162 * q^9	\( 2 q + 8 q^{2} + 18 q^{3} + 32 q^{4} + 53 q^{5} + 72 q^{6} + 128 q^{8} + 162 q^{9} + 212 q^{10} + 191 q^{11} + 288 q^{12} + 379 q^{13} + 477 q^{15} + 512 q^{16} + 340 q^{17} + 648 q^{18} + 1769 q^{19} + 848 q^{20} + 764 q^{22} + 3236 q^{23} + 1152 q^{24} - 45 q^{25} + 1516 q^{26} + 1458 q^{27} + 4459 q^{29} + 1908 q^{30} - 1994 q^{31} + 2048 q^{32} + 1719 q^{33} + 1360 q^{34} + 2592 q^{36} + 20587 q^{37} + 7076 q^{38} + 3411 q^{39} + 3392 q^{40} - 8814 q^{41} + 15853 q^{43} + 3056 q^{44} + 4293 q^{45} + 12944 q^{46} - 33912 q^{47} + 4608 q^{48} - 180 q^{50} + 3060 q^{51} + 6064 q^{52} + 49239 q^{53} + 5832 q^{54} - 18941 q^{55} + 15921 q^{57} + 17836 q^{58} + 56735 q^{59} + 7632 q^{60} - 67508 q^{61} - 7976 q^{62} + 8192 q^{64} - 42762 q^{65} + 6876 q^{66} + 75723 q^{67} + 5440 q^{68} + 29124 q^{69} - 8992 q^{71} + 10368 q^{72} + 3201 q^{73} + 82348 q^{74} - 405 q^{75} + 28304 q^{76} + 13644 q^{78} + 26612 q^{79} + 13568 q^{80} + 13122 q^{81} - 35256 q^{82} + 949 q^{83} + 105020 q^{85} + 63412 q^{86} + 40131 q^{87} + 12224 q^{88} - 176562 q^{89} + 17172 q^{90} + 51776 q^{92} - 17946 q^{93} - 135648 q^{94} + 234098 q^{95} + 18432 q^{96} + 129423 q^{97} + 15471 q^{99}+O(q^{100}) \) 2 * q + 8 * q^2 + 18 * q^3 + 32 * q^4 + 53 * q^5 + 72 * q^6 + 128 * q^8 + 162 * q^9 + 212 * q^10 + 191 * q^11 + 288 * q^12 + 379 * q^13 + 477 * q^15 + 512 * q^16 + 340 * q^17 + 648 * q^18 + 1769 * q^19 + 848 * q^20 + 764 * q^22 + 3236 * q^23 + 1152 * q^24 - 45 * q^25 + 1516 * q^26 + 1458 * q^27 + 4459 * q^29 + 1908 * q^30 - 1994 * q^31 + 2048 * q^32 + 1719 * q^33 + 1360 * q^34 + 2592 * q^36 + 20587 * q^37 + 7076 * q^38 + 3411 * q^39 + 3392 * q^40 - 8814 * q^41 + 15853 * q^43 + 3056 * q^44 + 4293 * q^45 + 12944 * q^46 - 33912 * q^47 + 4608 * q^48 - 180 * q^50 + 3060 * q^51 + 6064 * q^52 + 49239 * q^53 + 5832 * q^54 - 18941 * q^55 + 15921 * q^57 + 17836 * q^58 + 56735 * q^59 + 7632 * q^60 - 67508 * q^61 - 7976 * q^62 + 8192 * q^64 - 42762 * q^65 + 6876 * q^66 + 75723 * q^67 + 5440 * q^68 + 29124 * q^69 - 8992 * q^71 + 10368 * q^72 + 3201 * q^73 + 82348 * q^74 - 405 * q^75 + 28304 * q^76 + 13644 * q^78 + 26612 * q^79 + 13568 * q^80 + 13122 * q^81 - 35256 * q^82 + 949 * q^83 + 105020 * q^85 + 63412 * q^86 + 40131 * q^87 + 12224 * q^88 - 176562 * q^89 + 17172 * q^90 + 51776 * q^92 - 17946 * q^93 - 135648 * q^94 + 234098 * q^95 + 18432 * q^96 + 129423 * q^97 + 15471 * q^99

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 $

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	16.0000	0.500000
$5$	75.4923	1.35045	0.675224	−	0.737613i	$-0.264047\pi$
$5$	75.4923	1.35045	0.675224	+	0.737613i	$0.264047\pi$
$6$	36.0000	0.408248
$7$	0	0
$7$	0	0
$8$	64.0000	0.353553
$9$	81.0000	0.333333
$10$	301.969	0.954911
$11$	−149.462	−0.372433	−0.186217	−	0.982509i	$-0.559623\pi$
$11$	−149.462	−0.372433	−0.186217	+	0.982509i	$0.559623\pi$
$12$	144.000	0.288675
$13$	−349.416	−0.573435	−0.286717	−	0.958015i	$-0.592564\pi$
$13$	−349.416	−0.573435	−0.286717	+	0.958015i	$0.592564\pi$
$14$	0	0
$15$	679.431	0.779682
$16$	256.000	0.250000
$17$	1149.85	0.964979	0.482489	−	0.875902i	$-0.339733\pi$
$17$	1149.85	0.964979	0.482489	+	0.875902i	$0.339733\pi$
$18$	324.000	0.235702
$19$	2795.20	1.77635	0.888176	−	0.459503i	$-0.151972\pi$
$19$	2795.20	1.77635	0.888176	+	0.459503i	$0.151972\pi$
$20$	1207.88	0.675224
$21$	0	0
$22$	−597.847	−0.263350
$23$	1813.97	0.715007	0.357504	−	0.933912i	$-0.383628\pi$
$23$	1813.97	0.715007	0.357504	+	0.933912i	$0.383628\pi$
$24$	576.000	0.204124
$25$	2574.09	0.823710
$26$	−1397.66	−0.405480
$27$	729.000	0.192450
$28$	0	0
$29$	−759.033	−0.167597	−0.0837984	−	0.996483i	$-0.526705\pi$
$29$	−759.033	−0.167597	−0.0837984	+	0.996483i	$0.526705\pi$
$30$	2717.72	0.551318
$31$	−9031.74	−1.68798	−0.843990	−	0.536359i	$-0.819799\pi$
$31$	−9031.74	−1.68798	−0.843990	+	0.536359i	$0.819799\pi$
$32$	1024.00	0.176777
$33$	−1345.16	−0.215024
$34$	4599.39	0.682343
$35$	0	0
$36$	1296.00	0.166667
$37$	7794.89	0.936064	0.468032	−	0.883711i	$-0.344963\pi$
$37$	7794.89	0.936064	0.468032	+	0.883711i	$0.344963\pi$
$38$	11180.8	1.25607
$39$	−3144.74	−0.331073
$40$	4831.51	0.477456
$41$	−7640.49	−0.709842	−0.354921	−	0.934896i	$-0.615492\pi$
$41$	−7640.49	−0.709842	−0.354921	+	0.934896i	$0.615492\pi$
$42$	0	0
$43$	12188.8	1.00529	0.502645	−	0.864493i	$-0.332360\pi$
$43$	12188.8	1.00529	0.502645	+	0.864493i	$0.332360\pi$
$44$	−2391.39	−0.186217
$45$	6114.88	0.450149
$46$	7255.88	0.505586
$47$	−24598.8	−1.62431	−0.812156	−	0.583441i	$-0.801706\pi$
$47$	−24598.8	−1.62431	−0.812156	+	0.583441i	$0.801706\pi$
$48$	2304.00	0.144338
$49$	0	0
$50$	10296.4	0.582451
$51$	10348.6	0.557131
$52$	−5590.65	−0.286717
$53$	13596.2	0.664858	0.332429	−	0.943128i	$-0.392132\pi$
$53$	13596.2	0.664858	0.332429	+	0.943128i	$0.392132\pi$
$54$	2916.00	0.136083
$55$	−11283.2	−0.502952
$56$	0	0
$57$	25156.8	1.02558
$58$	−3036.13	−0.118509
$59$	26358.8	0.985816	0.492908	−	0.870081i	$-0.335934\pi$
$59$	26358.8	0.985816	0.492908	+	0.870081i	$0.335934\pi$
$60$	10870.9	0.389841
$61$	−35321.8	−1.21540	−0.607698	−	0.794168i	$-0.707907\pi$
$61$	−35321.8	−1.21540	−0.607698	+	0.794168i	$0.707907\pi$
$62$	−36127.0	−1.19358
$63$	0	0
$64$	4096.00	0.125000
$65$	−26378.2	−0.774394
$66$	−5380.62	−0.152045
$67$	54371.9	1.47975	0.739874	−	0.672746i	$-0.234885\pi$
$67$	54371.9	1.47975	0.739874	+	0.672746i	$0.234885\pi$
$68$	18397.6	0.482489
$69$	16325.7	0.412810
$70$	0	0
$71$	−70145.7	−1.65141	−0.825706	−	0.564101i	$-0.809223\pi$
$71$	−70145.7	−1.65141	−0.825706	+	0.564101i	$0.809223\pi$
$72$	5184.00	0.117851
$73$	44468.8	0.976671	0.488335	−	0.872656i	$-0.337604\pi$
$73$	44468.8	0.976671	0.488335	+	0.872656i	$0.337604\pi$
$74$	31179.6	0.661897
$75$	23166.8	0.475569
$76$	44723.2	0.888176
$77$	0	0
$78$	−12579.0	−0.234104
$79$	61612.5	1.11071	0.555355	−	0.831613i	$-0.312582\pi$
$79$	61612.5	1.11071	0.555355	+	0.831613i	$0.312582\pi$
$80$	19326.0	0.337612
$81$	6561.00	0.111111
$82$	−30562.0	−0.501934
$83$	87142.0	1.38846	0.694228	−	0.719755i	$-0.255746\pi$
$83$	87142.0	1.38846	0.694228	+	0.719755i	$0.255746\pi$
$84$	0	0
$85$	86804.6	1.30315
$86$	48755.3	0.710847
$87$	−6831.30	−0.0967621
$88$	−9565.55	−0.131675
$89$	−98569.4	−1.31907	−0.659534	−	0.751675i	$-0.729246\pi$
$89$	−98569.4	−1.31907	−0.659534	+	0.751675i	$0.729246\pi$
$90$	24459.5	0.318304
$91$	0	0
$92$	29023.5	0.357504
$93$	−81285.7	−0.974556
$94$	−98395.2	−1.14856
$95$	211016.	2.39887
$96$	9216.00	0.102062
$97$	−32342.3	−0.349013	−0.174507	−	0.984656i	$-0.555833\pi$
$97$	−32342.3	−0.349013	−0.174507	+	0.984656i	$0.555833\pi$
$98$	0	0
$99$	−12106.4	−0.124144
$100$	41185.5	0.411855
$101$	−31346.4	−0.305763	−0.152881	−	0.988245i	$-0.548855\pi$
$101$	−31346.4	−0.305763	−0.152881	+	0.988245i	$0.548855\pi$
$102$	41394.5	0.393951
$103$	−99332.9	−0.922572	−0.461286	−	0.887252i	$-0.652612\pi$
$103$	−99332.9	−0.922572	−0.461286	+	0.887252i	$0.652612\pi$
$104$	−22362.6	−0.202740
$105$	0	0
$106$	54384.9	0.470125
$107$	−145268.	−1.22662	−0.613310	−	0.789842i	$-0.710163\pi$
$107$	−145268.	−1.22662	−0.613310	+	0.789842i	$0.710163\pi$
$108$	11664.0	0.0962250
$109$	−180851.	−1.45799	−0.728994	−	0.684520i	$-0.760012\pi$
$109$	−180851.	−1.45799	−0.728994	+	0.684520i	$0.760012\pi$
$110$	−45132.9	−0.355640
$111$	70154.0	0.540437
$112$	0	0
$113$	197832.	1.45748	0.728738	−	0.684793i	$-0.240107\pi$
$113$	197832.	1.45748	0.728738	+	0.684793i	$0.240107\pi$
$114$	100627.	0.725193
$115$	136941.	0.965580
$116$	−12144.5	−0.0837984
$117$	−28302.7	−0.191145
$118$	105435.	0.697077
$119$	0	0
$120$	43483.6	0.275659
$121$	−138712.	−0.861294
$122$	−141287.	−0.859415
$123$	−68764.5	−0.409828
$124$	−144508.	−0.843990
$125$	−41589.2	−0.238070
$126$	0	0
$127$	−33517.2	−0.184399	−0.0921996	−	0.995741i	$-0.529390\pi$
$127$	−33517.2	−0.184399	−0.0921996	+	0.995741i	$0.529390\pi$
$128$	16384.0	0.0883883
$129$	109700.	0.580404
$130$	−105513.	−0.547579
$131$	10808.9	0.0550305	0.0275153	−	0.999621i	$-0.491241\pi$
$131$	10808.9	0.0550305	0.0275153	+	0.999621i	$0.491241\pi$
$132$	−21522.5	−0.107512
$133$	0	0
$134$	217488.	1.04634
$135$	55033.9	0.259894
$136$	73590.2	0.341171
$137$	−18932.0	−0.0861778	−0.0430889	−	0.999071i	$-0.513720\pi$
$137$	−18932.0	−0.0861778	−0.0430889	+	0.999071i	$0.513720\pi$
$138$	65302.9	0.291900
$139$	168897.	0.741457	0.370729	−	0.928741i	$-0.379108\pi$
$139$	168897.	0.741457	0.370729	+	0.928741i	$0.379108\pi$
$140$	0	0
$141$	−221389.	−0.937797
$142$	−280583.	−1.16772
$143$	52224.3	0.213566
$144$	20736.0	0.0833333
$145$	−57301.2	−0.226331
$146$	177875.	0.690611
$147$	0	0
$148$	124718.	0.468032
$149$	264002.	0.974186	0.487093	−	0.873350i	$-0.338057\pi$
$149$	264002.	0.974186	0.487093	+	0.873350i	$0.338057\pi$
$150$	92667.4	0.336278
$151$	279175.	0.996400	0.498200	−	0.867062i	$-0.333994\pi$
$151$	279175.	0.996400	0.498200	+	0.867062i	$0.333994\pi$
$152$	178893.	0.628035
$153$	93137.6	0.321660
$154$	0	0
$155$	−681828.	−2.27953
$156$	−50315.9	−0.165536
$157$	−188602.	−0.610658	−0.305329	−	0.952247i	$-0.598766\pi$
$157$	−188602.	−0.610658	−0.305329	+	0.952247i	$0.598766\pi$
$158$	246450.	0.785391
$159$	122366.	0.383856
$160$	77304.2	0.238728
$161$	0	0
$162$	26244.0	0.0785674
$163$	89717.4	0.264489	0.132244	−	0.991217i	$-0.457782\pi$
$163$	89717.4	0.264489	0.132244	+	0.991217i	$0.457782\pi$
$164$	−122248.	−0.354921
$165$	−101549.	−0.290379
$166$	348568.	0.981787
$167$	−529411.	−1.46893	−0.734467	−	0.678645i	$-0.762568\pi$
$167$	−529411.	−1.46893	−0.734467	+	0.678645i	$0.762568\pi$
$168$	0	0
$169$	−249202.	−0.671172
$170$	347219.	0.921469
$171$	226411.	0.592117
$172$	195021.	0.502645
$173$	−33838.3	−0.0859594	−0.0429797	−	0.999076i	$-0.513685\pi$
$173$	−33838.3	−0.0859594	−0.0429797	+	0.999076i	$0.513685\pi$
$174$	−27325.2	−0.0684211
$175$	0	0
$176$	−38262.2	−0.0931083
$177$	237229.	0.569161
$178$	−394278.	−0.932722
$179$	−247744.	−0.577923	−0.288962	−	0.957341i	$-0.593310\pi$
$179$	−247744.	−0.577923	−0.288962	+	0.957341i	$0.593310\pi$
$180$	97838.1	0.225075
$181$	−369470.	−0.838268	−0.419134	−	0.907924i	$-0.637666\pi$
$181$	−369470.	−0.838268	−0.419134	+	0.907924i	$0.637666\pi$
$182$	0	0
$183$	−317896.	−0.701709
$184$	116094.	0.252793
$185$	588455.	1.26411
$186$	−325143.	−0.689115
$187$	−171858.	−0.359390
$188$	−393581.	−0.812156
$189$	0	0
$190$	844065.	1.69626
$191$	−485225.	−0.962408	−0.481204	−	0.876609i	$-0.659800\pi$
$191$	−485225.	−0.962408	−0.481204	+	0.876609i	$0.659800\pi$
$192$	36864.0	0.0721688
$193$	−330817.	−0.639285	−0.319643	−	0.947538i	$-0.603563\pi$
$193$	−330817.	−0.639285	−0.319643	+	0.947538i	$0.603563\pi$
$194$	−129369.	−0.246790
$195$	−237404.	−0.447097
$196$	0	0
$197$	−161963.	−0.297337	−0.148669	−	0.988887i	$-0.547499\pi$
$197$	−161963.	−0.297337	−0.148669	+	0.988887i	$0.547499\pi$
$198$	−48425.6	−0.0877833
$199$	−55396.2	−0.0991625	−0.0495813	−	0.998770i	$-0.515789\pi$
$199$	−55396.2	−0.0991625	−0.0495813	+	0.998770i	$0.515789\pi$
$200$	164742.	0.291226
$201$	489347.	0.854333
$202$	−125386.	−0.216207
$203$	0	0
$204$	165578.	0.278565
$205$	−576799.	−0.958605
$206$	−397332.	−0.652357
$207$	146932.	0.238336
$208$	−89450.4	−0.143359
$209$	−417776.	−0.661572
$210$	0	0
$211$	481748.	0.744926	0.372463	−	0.928047i	$-0.378513\pi$
$211$	481748.	0.744926	0.372463	+	0.928047i	$0.378513\pi$
$212$	217540.	0.332429
$213$	−631312.	−0.953443
$214$	−581072.	−0.867352
$215$	920164.	1.35759
$216$	46656.0	0.0680414
$217$	0	0
$218$	−723403.	−1.03095
$219$	400219.	0.563881
$220$	−180531.	−0.251476
$221$	−401775.	−0.553353
$222$	280616.	0.382147
$223$	−638779.	−0.860178	−0.430089	−	0.902787i	$-0.641518\pi$
$223$	−638779.	−0.860178	−0.430089	+	0.902787i	$0.641518\pi$
$224$	0	0
$225$	208502.	0.274570
$226$	791330.	1.03059
$227$	525682.	0.677109	0.338554	−	0.940947i	$-0.390062\pi$
$227$	525682.	0.677109	0.338554	+	0.940947i	$0.390062\pi$
$228$	402509.	0.512789
$229$	−932686.	−1.17529	−0.587647	−	0.809117i	$-0.699946\pi$
$229$	−932686.	−1.17529	−0.587647	+	0.809117i	$0.699946\pi$
$230$	547763.	0.682768
$231$	0	0
$232$	−48578.1	−0.0592544
$233$	−707793.	−0.854115	−0.427058	−	0.904224i	$-0.640450\pi$
$233$	−707793.	−0.854115	−0.427058	+	0.904224i	$0.640450\pi$
$234$	−113211.	−0.135160
$235$	−1.85702e6	−2.19355
$236$	421741.	0.492908
$237$	554512.	0.641269
$238$	0	0
$239$	−500614.	−0.566902	−0.283451	−	0.958987i	$-0.591479\pi$
$239$	−500614.	−0.566902	−0.283451	+	0.958987i	$0.591479\pi$
$240$	173934.	0.194920
$241$	−1.20822e6	−1.33999	−0.669997	−	0.742364i	$-0.733705\pi$
$241$	−1.20822e6	−1.33999	−0.669997	+	0.742364i	$0.733705\pi$
$242$	−554849.	−0.609027
$243$	59049.0	0.0641500
$244$	−565148.	−0.607698
$245$	0	0
$246$	−275058.	−0.289792
$247$	−976688.	−1.01862
$248$	−578032.	−0.596791
$249$	784278.	0.801625
$250$	−166357.	−0.168341
$251$	−97826.7	−0.0980106	−0.0490053	−	0.998799i	$-0.515605\pi$
$251$	−97826.7	−0.0980106	−0.0490053	+	0.998799i	$0.515605\pi$
$252$	0	0
$253$	−271119.	−0.266292
$254$	−134069.	−0.130390
$255$	781242.	0.752376
$256$	65536.0	0.0625000
$257$	1.10791e6	1.04634	0.523170	−	0.852228i	$-0.324749\pi$
$257$	1.10791e6	1.04634	0.523170	+	0.852228i	$0.324749\pi$
$258$	438798.	0.410408
$259$	0	0
$260$	−422052.	−0.387197
$261$	−61481.7	−0.0558656
$262$	43235.6	0.0389125
$263$	1.50685e6	1.34333	0.671663	−	0.740856i	$-0.265580\pi$
$263$	1.50685e6	1.34333	0.671663	+	0.740856i	$0.265580\pi$
$264$	−86090.0	−0.0760226
$265$	1.02641e6	0.897856
$266$	0	0
$267$	−887125.	−0.761564
$268$	869951.	0.739874
$269$	2.13502e6	1.79896	0.899481	−	0.436960i	$-0.143945\pi$
$269$	2.13502e6	1.79896	0.899481	+	0.436960i	$0.143945\pi$
$270$	220136.	0.183773
$271$	−1.22860e6	−1.01622	−0.508108	−	0.861293i	$-0.669655\pi$
$271$	−1.22860e6	−1.01622	−0.508108	+	0.861293i	$0.669655\pi$
$272$	294361.	0.241245
$273$	0	0
$274$	−75728.1	−0.0609369
$275$	−384729.	−0.306777
$276$	261212.	0.206405
$277$	−1.84682e6	−1.44619	−0.723097	−	0.690747i	$-0.757282\pi$
$277$	−1.84682e6	−1.44619	−0.723097	+	0.690747i	$0.757282\pi$
$278$	675590.	0.524289
$279$	−731571.	−0.562660
$280$	0	0
$281$	−2.28326e6	−1.72500	−0.862500	−	0.506056i	$-0.831103\pi$
$281$	−2.28326e6	−1.72500	−0.862500	+	0.506056i	$0.831103\pi$
$282$	−885557.	−0.663122
$283$	−1.33693e6	−0.992296	−0.496148	−	0.868238i	$-0.665253\pi$
$283$	−1.33693e6	−0.992296	−0.496148	+	0.868238i	$0.665253\pi$
$284$	−1.12233e6	−0.825706
$285$	1.89915e6	1.38499
$286$	208897.	0.151014
$287$	0	0
$288$	82944.0	0.0589256
$289$	−97709.0	−0.0688161
$290$	−229205.	−0.160040
$291$	−291081.	−0.201503
$292$	711501.	0.488335
$293$	2.23033e6	1.51775	0.758875	−	0.651236i	$-0.225749\pi$
$293$	2.23033e6	1.51775	0.758875	+	0.651236i	$0.225749\pi$
$294$	0	0
$295$	1.98989e6	1.33129
$296$	498873.	0.330949
$297$	−108958.	−0.0716748
$298$	1.05601e6	0.688853
$299$	−633830.	−0.410010
$300$	370670.	0.237785
$301$	0	0
$302$	1.11670e6	0.704561
$303$	−282118.	−0.176532
$304$	715572.	0.444088
$305$	−2.66652e6	−1.64133
$306$	372550.	0.227448
$307$	−1.77782e6	−1.07657	−0.538284	−	0.842763i	$-0.680927\pi$
$307$	−1.77782e6	−1.07657	−0.538284	+	0.842763i	$0.680927\pi$
$308$	0	0
$309$	−893996.	−0.532647
$310$	−2.72731e6	−1.61187
$311$	−1.40272e6	−0.822373	−0.411187	−	0.911551i	$-0.634886\pi$
$311$	−1.40272e6	−0.822373	−0.411187	+	0.911551i	$0.634886\pi$
$312$	−201264.	−0.117052
$313$	−1.21894e6	−0.703271	−0.351636	−	0.936137i	$-0.614374\pi$
$313$	−1.21894e6	−0.703271	−0.351636	+	0.936137i	$0.614374\pi$
$314$	−754409.	−0.431800
$315$	0	0
$316$	985799.	0.555355
$317$	2.68001e6	1.49792	0.748961	−	0.662615i	$-0.230553\pi$
$317$	2.68001e6	1.49792	0.748961	+	0.662615i	$0.230553\pi$
$318$	489464.	0.271427
$319$	113446.	0.0624186
$320$	309217.	0.168806
$321$	−1.30741e6	−0.708190
$322$	0	0
$323$	3.21405e6	1.71414
$324$	104976.	0.0555556
$325$	−899429.	−0.472344
$326$	358869.	0.187022
$327$	−1.62766e6	−0.841770
$328$	−488992.	−0.250967
$329$	0	0
$330$	−406196.	−0.205329
$331$	142560.	0.0715202	0.0357601	−	0.999360i	$-0.488615\pi$
$331$	142560.	0.0715202	0.0357601	+	0.999360i	$0.488615\pi$
$332$	1.39427e6	0.694228
$333$	631386.	0.312021
$334$	−2.11765e6	−1.03869
$335$	4.10466e6	1.99832
$336$	0	0
$337$	−1.21206e6	−0.581367	−0.290683	−	0.956819i	$-0.593883\pi$
$337$	−1.21206e6	−0.581367	−0.290683	+	0.956819i	$0.593883\pi$
$338$	−996806.	−0.474591
$339$	1.78049e6	0.841474
$340$	1.38887e6	0.651577
$341$	1.34990e6	0.628660
$342$	905645.	0.418690
$343$	0	0
$344$	780085.	0.355423
$345$	1.23247e6	0.557478
$346$	−135353.	−0.0607825
$347$	3.46853e6	1.54640	0.773199	−	0.634163i	$-0.218655\pi$
$347$	3.46853e6	1.54640	0.773199	+	0.634163i	$0.218655\pi$
$348$	−109301.	−0.0483810
$349$	−1.01692e6	−0.446911	−0.223456	−	0.974714i	$-0.571734\pi$
$349$	−1.01692e6	−0.446911	−0.223456	+	0.974714i	$0.571734\pi$
$350$	0	0
$351$	−254724.	−0.110358
$352$	−153049.	−0.0658375
$353$	−1.56385e6	−0.667974	−0.333987	−	0.942578i	$-0.608394\pi$
$353$	−1.56385e6	−0.667974	−0.333987	+	0.942578i	$0.608394\pi$
$354$	948917.	0.402458
$355$	−5.29547e6	−2.23015
$356$	−1.57711e6	−0.659534
$357$	0	0
$358$	−990975.	−0.408653
$359$	49553.1	0.0202925	0.0101462	−	0.999949i	$-0.496770\pi$
$359$	49553.1	0.0202925	0.0101462	+	0.999949i	$0.496770\pi$
$360$	391352.	0.159152
$361$	5.33705e6	2.15543
$362$	−1.47788e6	−0.592745
$363$	−1.24841e6	−0.497268
$364$	0	0
$365$	3.35705e6	1.31894
$366$	−1.27158e6	−0.496183
$367$	−3.54831e6	−1.37517	−0.687585	−	0.726104i	$-0.741329\pi$
$367$	−3.54831e6	−1.37517	−0.687585	+	0.726104i	$0.741329\pi$
$368$	464376.	0.178752
$369$	−618880.	−0.236614
$370$	2.35382e6	0.893858
$371$	0	0
$372$	−1.30057e6	−0.487278
$373$	−2.25573e6	−0.839490	−0.419745	−	0.907642i	$-0.637880\pi$
$373$	−2.25573e6	−0.839490	−0.419745	+	0.907642i	$0.637880\pi$
$374$	−687432.	−0.254127
$375$	−374302.	−0.137450
$376$	−1.57432e6	−0.574281
$377$	265218.	0.0961059
$378$	0	0
$379$	4.39503e6	1.57168	0.785840	−	0.618430i	$-0.212231\pi$
$379$	4.39503e6	1.57168	0.785840	+	0.618430i	$0.212231\pi$
$380$	3.37626e6	1.19944
$381$	−301655.	−0.106463
$382$	−1.94090e6	−0.680525
$383$	1.22781e6	0.427694	0.213847	−	0.976867i	$-0.431401\pi$
$383$	1.22781e6	0.427694	0.213847	+	0.976867i	$0.431401\pi$
$384$	147456.	0.0510310
$385$	0	0
$386$	−1.32327e6	−0.452043
$387$	987296.	0.335096
$388$	−517477.	−0.174507
$389$	2.05424e6	0.688300	0.344150	−	0.938915i	$-0.388167\pi$
$389$	2.05424e6	0.688300	0.344150	+	0.938915i	$0.388167\pi$
$390$	−949616.	−0.316145
$391$	2.08579e6	0.689967
$392$	0	0
$393$	97280.2	0.0317719
$394$	−647851.	−0.210249
$395$	4.65127e6	1.49996
$396$	−193702.	−0.0620722
$397$	3.80367e6	1.21123	0.605615	−	0.795758i	$-0.292927\pi$
$397$	3.80367e6	1.21123	0.605615	+	0.795758i	$0.292927\pi$
$398$	−221585.	−0.0701185
$399$	0	0
$400$	658968.	0.205928
$401$	1.18524e6	0.368084	0.184042	−	0.982918i	$-0.441082\pi$
$401$	1.18524e6	0.368084	0.184042	+	0.982918i	$0.441082\pi$
$402$	1.95739e6	0.604104
$403$	3.15583e6	0.967947
$404$	−501543.	−0.152881
$405$	495305.	0.150050
$406$	0	0
$407$	−1.16504e6	−0.348621
$408$	662312.	0.196975
$409$	4.30393e6	1.27220	0.636102	−	0.771605i	$-0.280546\pi$
$409$	4.30393e6	1.27220	0.636102	+	0.771605i	$0.280546\pi$
$410$	−2.30720e6	−0.677836
$411$	−170388.	−0.0497548
$412$	−1.58933e6	−0.461286
$413$	0	0
$414$	587726.	0.168529
$415$	6.57855e6	1.87504
$416$	−357802.	−0.101370
$417$	1.52008e6	0.428080
$418$	−1.67110e6	−0.467802
$419$	113725.	0.0316461	0.0158230	−	0.999875i	$-0.494963\pi$
$419$	113725.	0.0316461	0.0158230	+	0.999875i	$0.494963\pi$
$420$	0	0
$421$	443417.	0.121929	0.0609645	−	0.998140i	$-0.480582\pi$
$421$	443417.	0.121929	0.0609645	+	0.998140i	$0.480582\pi$
$422$	1.92699e6	0.526743
$423$	−1.99250e6	−0.541437
$424$	870158.	0.235063
$425$	2.95981e6	0.794863
$426$	−2.52525e6	−0.674186
$427$	0	0
$428$	−2.32429e6	−0.613310
$429$	470019.	0.123302
$430$	3.68065e6	0.959962
$431$	4.63310e6	1.20138	0.600688	−	0.799484i	$-0.294893\pi$
$431$	4.63310e6	1.20138	0.600688	+	0.799484i	$0.294893\pi$
$432$	186624.	0.0481125
$433$	−6.57955e6	−1.68646	−0.843231	−	0.537552i	$-0.819349\pi$
$433$	−6.57955e6	−1.68646	−0.843231	+	0.537552i	$0.819349\pi$
$434$	0	0
$435$	−515711.	−0.130672
$436$	−2.89361e6	−0.728994
$437$	5.07041e6	1.27010
$438$	1.60088e6	0.398724
$439$	1.82070e6	0.450896	0.225448	−	0.974255i	$-0.427615\pi$
$439$	1.82070e6	0.450896	0.225448	+	0.974255i	$0.427615\pi$
$440$	−722126.	−0.177820
$441$	0	0
$442$	−1.60710e6	−0.391279
$443$	−2.06990e6	−0.501119	−0.250559	−	0.968101i	$-0.580615\pi$
$443$	−2.06990e6	−0.501119	−0.250559	+	0.968101i	$0.580615\pi$
$444$	1.12246e6	0.270218
$445$	−7.44123e6	−1.78133
$446$	−2.55511e6	−0.608238
$447$	2.37602e6	0.562447
$448$	0	0
$449$	5.72581e6	1.34036	0.670180	−	0.742199i	$-0.266217\pi$
$449$	5.72581e6	1.34036	0.670180	+	0.742199i	$0.266217\pi$
$450$	834007.	0.194150
$451$	1.14196e6	0.264369
$452$	3.16532e6	0.728738
$453$	2.51257e6	0.575272
$454$	2.10273e6	0.478788
$455$	0	0
$456$	1.61004e6	0.362596
$457$	318676.	0.0713771	0.0356886	−	0.999363i	$-0.488638\pi$
$457$	318676.	0.0713771	0.0356886	+	0.999363i	$0.488638\pi$
$458$	−3.73074e6	−0.831059
$459$	838238.	0.185710
$460$	2.19105e6	0.482790
$461$	3.42470e6	0.750535	0.375267	−	0.926917i	$-0.377551\pi$
$461$	3.42470e6	0.750535	0.375267	+	0.926917i	$0.377551\pi$
$462$	0	0
$463$	3.82945e6	0.830201	0.415101	−	0.909775i	$-0.363746\pi$
$463$	3.82945e6	0.830201	0.415101	+	0.909775i	$0.363746\pi$
$464$	−194312.	−0.0418992
$465$	−6.13645e6	−1.31609
$466$	−2.83117e6	−0.603951
$467$	−2.84462e6	−0.603576	−0.301788	−	0.953375i	$-0.597583\pi$
$467$	−2.84462e6	−0.603576	−0.301788	+	0.953375i	$0.597583\pi$
$468$	−452843.	−0.0955725
$469$	0	0
$470$	−7.42809e6	−1.55107
$471$	−1.69742e6	−0.352563
$472$	1.68696e6	0.348539
$473$	−1.82176e6	−0.374403
$474$	2.21805e6	0.453446
$475$	7.19511e6	1.46320
$476$	0	0
$477$	1.10129e6	0.221619
$478$	−2.00246e6	−0.400860
$479$	−1.30365e6	−0.259611	−0.129806	−	0.991539i	$-0.541435\pi$
$479$	−1.30365e6	−0.259611	−0.129806	+	0.991539i	$0.541435\pi$
$480$	695737.	0.137830
$481$	−2.72366e6	−0.536772
$482$	−4.83287e6	−0.947519
$483$	0	0
$484$	−2.21940e6	−0.430647
$485$	−2.44160e6	−0.471324
$486$	236196.	0.0453609
$487$	6.23624e6	1.19152	0.595759	−	0.803164i	$-0.296852\pi$
$487$	6.23624e6	1.19152	0.595759	+	0.803164i	$0.296852\pi$
$488$	−2.26059e6	−0.429707
$489$	807456.	0.152703
$490$	0	0
$491$	−3.93928e6	−0.737417	−0.368709	−	0.929545i	$-0.620200\pi$
$491$	−3.93928e6	−0.737417	−0.368709	+	0.929545i	$0.620200\pi$
$492$	−1.10023e6	−0.204914
$493$	−872772.	−0.161727
$494$	−3.90675e6	−0.720275
$495$	−913941.	−0.167651
$496$	−2.31213e6	−0.421995
$497$	0	0
$498$	3.13711e6	0.566835
$499$	7.97571e6	1.43390	0.716948	−	0.697126i	$-0.245538\pi$
$499$	7.97571e6	1.43390	0.716948	+	0.697126i	$0.245538\pi$
$500$	−665427.	−0.119035
$501$	−4.76470e6	−0.848089
$502$	−391307.	−0.0693040
$503$	2.70777e6	0.477190	0.238595	−	0.971119i	$-0.423313\pi$
$503$	2.70777e6	0.477190	0.238595	+	0.971119i	$0.423313\pi$
$504$	0	0
$505$	−2.36641e6	−0.412917
$506$	−1.08448e6	−0.188297
$507$	−2.24281e6	−0.387502
$508$	−536276.	−0.0921996
$509$	−3.90049e6	−0.667306	−0.333653	−	0.942696i	$-0.608281\pi$
$509$	−3.90049e6	−0.667306	−0.333653	+	0.942696i	$0.608281\pi$
$510$	3.12497e6	0.532010
$511$	0	0
$512$	262144.	0.0441942
$513$	2.03770e6	0.341859
$514$	4.43165e6	0.739874
$515$	−7.49887e6	−1.24589
$516$	1.75519e6	0.290202
$517$	3.67658e6	0.604947
$518$	0	0
$519$	−304545.	−0.0496287
$520$	−1.68821e6	−0.273790
$521$	−7.00063e6	−1.12991	−0.564954	−	0.825122i	$-0.691106\pi$
$521$	−7.00063e6	−1.12991	−0.564954	+	0.825122i	$0.691106\pi$
$522$	−245927.	−0.0395029
$523$	2.11208e6	0.337641	0.168821	−	0.985647i	$-0.446004\pi$
$523$	2.11208e6	0.337641	0.168821	+	0.985647i	$0.446004\pi$
$524$	172943.	0.0275153
$525$	0	0
$526$	6.02741e6	0.949875
$527$	−1.03851e7	−1.62887
$528$	−344360.	−0.0537561
$529$	−3.14586e6	−0.488765
$530$	4.10564e6	0.634880
$531$	2.13506e6	0.328605
$532$	0	0
$533$	2.66971e6	0.407048
$534$	−3.54850e6	−0.538507
$535$	−1.09666e7	−1.65649
$536$	3.47980e6	0.523170
$537$	−2.22969e6	−0.333664
$538$	8.54009e6	1.27206
$539$	0	0
$540$	880543.	0.129947
$541$	5.70567e6	0.838134	0.419067	−	0.907955i	$-0.362357\pi$
$541$	5.70567e6	0.838134	0.419067	+	0.907955i	$0.362357\pi$
$542$	−4.91439e6	−0.718574
$543$	−3.32523e6	−0.483974
$544$	1.17744e6	0.170586
$545$	−1.36528e7	−1.96894
$546$	0	0
$547$	2.28054e6	0.325888	0.162944	−	0.986635i	$-0.447901\pi$
$547$	2.28054e6	0.325888	0.162944	+	0.986635i	$0.447901\pi$
$548$	−302912.	−0.0430889
$549$	−2.86106e6	−0.405132
$550$	−1.53891e6	−0.216924
$551$	−2.12165e6	−0.297711
$552$	1.04485e6	0.145950
$553$	0	0
$554$	−7.38730e6	−1.02261
$555$	5.29609e6	0.729832
$556$	2.70236e6	0.370729
$557$	−6.73941e6	−0.920415	−0.460208	−	0.887811i	$-0.652225\pi$
$557$	−6.73941e6	−0.920415	−0.460208	+	0.887811i	$0.652225\pi$
$558$	−2.92629e6	−0.397861
$559$	−4.25897e6	−0.576468
$560$	0	0
$561$	−1.54672e6	−0.207494
$562$	−9.13304e6	−1.21976
$563$	9.66820e6	1.28551	0.642754	−	0.766073i	$-0.277792\pi$
$563$	9.66820e6	1.28551	0.642754	+	0.766073i	$0.277792\pi$
$564$	−3.54223e6	−0.468898
$565$	1.49348e7	1.96825
$566$	−5.34770e6	−0.701659
$567$	0	0
$568$	−4.48933e6	−0.583862
$569$	−1.39587e7	−1.80744	−0.903719	−	0.428125i	$-0.859174\pi$
$569$	−1.39587e7	−1.80744	−0.903719	+	0.428125i	$0.859174\pi$
$570$	7.59659e6	0.979335
$571$	−2.35841e6	−0.302711	−0.151355	−	0.988479i	$-0.548364\pi$
$571$	−2.35841e6	−0.302711	−0.151355	+	0.988479i	$0.548364\pi$
$572$	835589.	0.106783
$573$	−4.36702e6	−0.555647
$574$	0	0
$575$	4.66933e6	0.588959
$576$	331776.	0.0416667
$577$	−5.01098e6	−0.626589	−0.313295	−	0.949656i	$-0.601433\pi$
$577$	−5.01098e6	−0.626589	−0.313295	+	0.949656i	$0.601433\pi$
$578$	−390836.	−0.0486603
$579$	−2.97735e6	−0.369091
$580$	−916819.	−0.113165
$581$	0	0
$582$	−1.16432e6	−0.142484
$583$	−2.03211e6	−0.247615
$584$	2.84600e6	0.345305
$585$	−2.13664e6	−0.258131
$586$	8.92132e6	1.07321
$587$	−3.95106e6	−0.473280	−0.236640	−	0.971597i	$-0.576046\pi$
$587$	−3.95106e6	−0.473280	−0.236640	+	0.971597i	$0.576046\pi$
$588$	0	0
$589$	−2.52455e7	−2.99845
$590$	7.95955e6	0.941367
$591$	−1.45766e6	−0.171668
$592$	1.99549e6	0.234016
$593$	2.53680e6	0.296244	0.148122	−	0.988969i	$-0.452677\pi$
$593$	2.53680e6	0.296244	0.148122	+	0.988969i	$0.452677\pi$
$594$	−435830.	−0.0506817
$595$	0	0
$596$	4.22403e6	0.487093
$597$	−498566.	−0.0572515
$598$	−2.53532e6	−0.289921
$599$	7.17714e6	0.817305	0.408652	−	0.912690i	$-0.365999\pi$
$599$	7.17714e6	0.817305	0.408652	+	0.912690i	$0.365999\pi$
$600$	1.48268e6	0.168139
$601$	−1.12527e7	−1.27078	−0.635392	−	0.772190i	$-0.719162\pi$
$601$	−1.12527e7	−1.27078	−0.635392	+	0.772190i	$0.719162\pi$
$602$	0	0
$603$	4.40413e6	0.493249
$604$	4.46680e6	0.498200
$605$	−1.04717e7	−1.16313
$606$	−1.12847e6	−0.124827
$607$	1.40370e7	1.54633	0.773167	−	0.634202i	$-0.218671\pi$
$607$	1.40370e7	1.54633	0.773167	+	0.634202i	$0.218671\pi$
$608$	2.86229e6	0.314018
$609$	0	0
$610$	−1.06661e7	−1.16059
$611$	8.59521e6	0.931437
$612$	1.49020e6	0.160830
$613$	−7.62872e6	−0.819975	−0.409988	−	0.912091i	$-0.634467\pi$
$613$	−7.62872e6	−0.819975	−0.409988	+	0.912091i	$0.634467\pi$
$614$	−7.11128e6	−0.761249
$615$	−5.19119e6	−0.553451
$616$	0	0
$617$	−4.41080e6	−0.466450	−0.233225	−	0.972423i	$-0.574928\pi$
$617$	−4.41080e6	−0.466450	−0.233225	+	0.972423i	$0.574928\pi$
$618$	−3.57598e6	−0.376638
$619$	9.04355e6	0.948664	0.474332	−	0.880346i	$-0.342690\pi$
$619$	9.04355e6	0.948664	0.474332	+	0.880346i	$0.342690\pi$
$620$	−1.09092e7	−1.13977
$621$	1.32238e6	0.137603
$622$	−5.61087e6	−0.581506
$623$	0	0
$624$	−805054.	−0.0827682
$625$	−1.11837e7	−1.14521
$626$	−4.87578e6	−0.497288
$627$	−3.75998e6	−0.381959
$628$	−3.01764e6	−0.305329
$629$	8.96293e6	0.903282
$630$	0	0
$631$	−7.33039e6	−0.732916	−0.366458	−	0.930435i	$-0.619430\pi$
$631$	−7.33039e6	−0.732916	−0.366458	+	0.930435i	$0.619430\pi$
$632$	3.94320e6	0.392695
$633$	4.33573e6	0.430083
$634$	1.07201e7	1.05919
$635$	−2.53030e6	−0.249022
$636$	1.95786e6	0.191928
$637$	0	0
$638$	453786.	0.0441366
$639$	−5.68181e6	−0.550471
$640$	1.23687e6	0.119364
$641$	3.68824e6	0.354547	0.177273	−	0.984162i	$-0.443272\pi$
$641$	3.68824e6	0.354547	0.177273	+	0.984162i	$0.443272\pi$
$642$	−5.22965e6	−0.500766
$643$	1.17584e7	1.12155	0.560776	−	0.827968i	$-0.310503\pi$
$643$	1.17584e7	1.12155	0.560776	+	0.827968i	$0.310503\pi$
$644$	0	0
$645$	8.28147e6	0.783806
$646$	1.28562e7	1.21208
$647$	9.94050e6	0.933572	0.466786	−	0.884370i	$-0.345412\pi$
$647$	9.94050e6	0.933572	0.466786	+	0.884370i	$0.345412\pi$
$648$	419904.	0.0392837
$649$	−3.93963e6	−0.367151
$650$	−3.59772e6	−0.333998
$651$	0	0
$652$	1.43548e6	0.132244
$653$	−1.02620e6	−0.0941778	−0.0470889	−	0.998891i	$-0.514994\pi$
$653$	−1.02620e6	−0.0941778	−0.0470889	+	0.998891i	$0.514994\pi$
$654$	−6.51063e6	−0.595221
$655$	815990.	0.0743159
$656$	−1.95597e6	−0.177461
$657$	3.60197e6	0.325557
$658$	0	0
$659$	1.00207e7	0.898846	0.449423	−	0.893319i	$-0.351630\pi$
$659$	1.00207e7	0.898846	0.449423	+	0.893319i	$0.351630\pi$
$660$	−1.62478e6	−0.145190
$661$	−2.62826e6	−0.233972	−0.116986	−	0.993134i	$-0.537323\pi$
$661$	−2.62826e6	−0.233972	−0.116986	+	0.993134i	$0.537323\pi$
$662$	570241.	0.0505724
$663$	−3.61597e6	−0.319478
$664$	5.57709e6	0.490893
$665$	0	0
$666$	2.52554e6	0.220632
$667$	−1.37686e6	−0.119833
$668$	−8.47058e6	−0.734467
$669$	−5.74901e6	−0.496624
$670$	1.64187e7	1.41303
$671$	5.27925e6	0.452654
$672$	0	0
$673$	−1.50220e7	−1.27847	−0.639233	−	0.769013i	$-0.720748\pi$
$673$	−1.50220e7	−1.27847	−0.639233	+	0.769013i	$0.720748\pi$
$674$	−4.84825e6	−0.411088
$675$	1.87651e6	0.158523
$676$	−3.98723e6	−0.335586
$677$	6.03310e6	0.505905	0.252953	−	0.967479i	$-0.418598\pi$
$677$	6.03310e6	0.505905	0.252953	+	0.967479i	$0.418598\pi$
$678$	7.12197e6	0.595012
$679$	0	0
$680$	5.55550e6	0.460734
$681$	4.73114e6	0.390929
$682$	5.39960e6	0.444530
$683$	1.10720e7	0.908187	0.454093	−	0.890954i	$-0.349963\pi$
$683$	1.10720e7	0.908187	0.454093	+	0.890954i	$0.349963\pi$
$684$	3.62258e6	0.296059
$685$	−1.42922e6	−0.116379
$686$	0	0
$687$	−8.39417e6	−0.678557
$688$	3.12034e6	0.251322
$689$	−4.75073e6	−0.381253
$690$	4.92987e6	0.394196
$691$	1.05548e7	0.840919	0.420460	−	0.907311i	$-0.361869\pi$
$691$	1.05548e7	0.840919	0.420460	+	0.907311i	$0.361869\pi$
$692$	−541413.	−0.0429797
$693$	0	0
$694$	1.38741e7	1.09347
$695$	1.27505e7	1.00130
$696$	−437203.	−0.0342106
$697$	−8.78540e6	−0.684983
$698$	−4.06766e6	−0.316014
$699$	−6.37014e6	−0.493124
$700$	0	0
$701$	−7.20675e6	−0.553917	−0.276958	−	0.960882i	$-0.589326\pi$
$701$	−7.20675e6	−0.553917	−0.276958	+	0.960882i	$0.589326\pi$
$702$	−1.01890e6	−0.0780346
$703$	2.17883e7	1.66278
$704$	−612195.	−0.0465541
$705$	−1.67132e7	−1.26645
$706$	−6.25542e6	−0.472329
$707$	0	0
$708$	3.79567e6	0.284581
$709$	2.69373e6	0.201252	0.100626	−	0.994924i	$-0.467916\pi$
$709$	2.69373e6	0.201252	0.100626	+	0.994924i	$0.467916\pi$
$710$	−2.11819e7	−1.57695
$711$	4.99061e6	0.370237
$712$	−6.30844e6	−0.466361
$713$	−1.63833e7	−1.20692
$714$	0	0
$715$	3.94253e6	0.288410
$716$	−3.96390e6	−0.288962
$717$	−4.50552e6	−0.327301
$718$	198213.	0.0143489
$719$	−6.92550e6	−0.499608	−0.249804	−	0.968296i	$-0.580366\pi$
$719$	−6.92550e6	−0.499608	−0.249804	+	0.968296i	$0.580366\pi$
$720$	1.56541e6	0.112537
$721$	0	0
$722$	2.13482e7	1.52412
$723$	−1.08740e7	−0.773646
$724$	−5.91152e6	−0.419134
$725$	−1.95382e6	−0.138051
$726$	−4.99364e6	−0.351622
$727$	−4.11366e6	−0.288664	−0.144332	−	0.989529i	$-0.546103\pi$
$727$	−4.11366e6	−0.288664	−0.144332	+	0.989529i	$0.546103\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	1.34282e7	0.932634
$731$	1.40153e7	0.970083
$732$	−5.08633e6	−0.350855
$733$	−7.92038e6	−0.544485	−0.272243	−	0.962229i	$-0.587765\pi$
$733$	−7.92038e6	−0.544485	−0.272243	+	0.962229i	$0.587765\pi$
$734$	−1.41932e7	−0.972392
$735$	0	0
$736$	1.85750e6	0.126397
$737$	−8.12652e6	−0.551107
$738$	−2.47552e6	−0.167311
$739$	1.60410e7	1.08049	0.540245	−	0.841508i	$-0.318332\pi$
$739$	1.60410e7	1.08049	0.540245	+	0.841508i	$0.318332\pi$
$740$	9.41527e6	0.632053
$741$	−8.79019e6	−0.588102
$742$	0	0
$743$	1.53453e7	1.01977	0.509887	−	0.860241i	$-0.329687\pi$
$743$	1.53453e7	1.01977	0.509887	+	0.860241i	$0.329687\pi$
$744$	−5.20229e6	−0.344558
$745$	1.99301e7	1.31559
$746$	−9.02293e6	−0.593609
$747$	7.05850e6	0.462819
$748$	−2.74973e6	−0.179695
$749$	0	0
$750$	−1.49721e6	−0.0971918
$751$	2.24976e7	1.45558	0.727790	−	0.685800i	$-0.240547\pi$
$751$	2.24976e7	1.45558	0.727790	+	0.685800i	$0.240547\pi$
$752$	−6.29729e6	−0.406078
$753$	−880440.	−0.0565864
$754$	1.06087e6	0.0679571
$755$	2.10756e7	1.34559
$756$	0	0
$757$	2.30349e7	1.46099	0.730494	−	0.682919i	$-0.239290\pi$
$757$	2.30349e7	1.46099	0.730494	+	0.682919i	$0.239290\pi$
$758$	1.75801e7	1.11135
$759$	−2.44007e6	−0.153744
$760$	1.35050e7	0.848129
$761$	−5.40735e6	−0.338472	−0.169236	−	0.985576i	$-0.554130\pi$
$761$	−5.40735e6	−0.338472	−0.169236	+	0.985576i	$0.554130\pi$
$762$	−1.20662e6	−0.0752806
$763$	0	0
$764$	−7.76359e6	−0.481204
$765$	7.03118e6	0.434385
$766$	4.91123e6	0.302426
$767$	−9.21019e6	−0.565301
$768$	589824.	0.0360844
$769$	7.93100e6	0.483629	0.241814	−	0.970323i	$-0.422258\pi$
$769$	7.93100e6	0.483629	0.241814	+	0.970323i	$0.422258\pi$
$770$	0	0
$771$	9.97122e6	0.604105
$772$	−5.29307e6	−0.319643
$773$	−269579.	−0.0162270	−0.00811349	−	0.999967i	$-0.502583\pi$
$773$	−269579.	−0.0162270	−0.00811349	+	0.999967i	$0.502583\pi$
$774$	3.94918e6	0.236949
$775$	−2.32486e7	−1.39041
$776$	−2.06991e6	−0.123395
$777$	0	0
$778$	8.21698e6	0.486702
$779$	−2.13567e7	−1.26093
$780$	−3.79846e6	−0.223548
$781$	1.04841e7	0.615041
$782$	8.34315e6	0.487880
$783$	−553335.	−0.0322540
$784$	0	0
$785$	−1.42380e7	−0.824662
$786$	389121.	0.0224661
$787$	8.98621e6	0.517177	0.258589	−	0.965988i	$-0.416743\pi$
$787$	8.98621e6	0.517177	0.258589	+	0.965988i	$0.416743\pi$
$788$	−2.59140e6	−0.148669
$789$	1.35617e7	0.775570
$790$	1.86051e7	1.06063
$791$	0	0
$792$	−774810.	−0.0438917
$793$	1.23420e7	0.696950
$794$	1.52147e7	0.856469
$795$	9.23770e6	0.518377
$796$	−886340.	−0.0495813
$797$	2.65558e7	1.48086	0.740429	−	0.672135i	$-0.234622\pi$
$797$	2.65558e7	1.48086	0.740429	+	0.672135i	$0.234622\pi$
$798$	0	0
$799$	−2.82849e7	−1.56743
$800$	2.63587e6	0.145613
$801$	−7.98412e6	−0.439689
$802$	4.74098e6	0.260275
$803$	−6.64638e6	−0.363745
$804$	7.82956e6	0.427166
$805$	0	0
$806$	1.26233e7	0.684442
$807$	1.92152e7	1.03863
$808$	−2.00617e6	−0.108103
$809$	2.52557e7	1.35671	0.678356	−	0.734733i	$-0.262693\pi$
$809$	2.52557e7	1.35671	0.678356	+	0.734733i	$0.262693\pi$
$810$	1.98122e6	0.106101
$811$	4.76866e6	0.254592	0.127296	−	0.991865i	$-0.459370\pi$
$811$	4.76866e6	0.254592	0.127296	+	0.991865i	$0.459370\pi$
$812$	0	0
$813$	−1.10574e7	−0.586713
$814$	−4.66015e6	−0.246513
$815$	6.77297e6	0.357179
$816$	2.64925e6	0.139283
$817$	3.40702e7	1.78575
$818$	1.72157e7	0.899585
$819$	0	0
$820$	−9.22878e6	−0.479303
$821$	−5.95534e6	−0.308353	−0.154177	−	0.988043i	$-0.549273\pi$
$821$	−5.95534e6	−0.308353	−0.154177	+	0.988043i	$0.549273\pi$
$822$	−681553.	−0.0351819
$823$	1.06659e7	0.548907	0.274453	−	0.961600i	$-0.411503\pi$
$823$	1.06659e7	0.548907	0.274453	+	0.961600i	$0.411503\pi$
$824$	−6.35731e6	−0.326178
$825$	−3.46256e6	−0.177118
$826$	0	0
$827$	−8.52356e6	−0.433369	−0.216684	−	0.976242i	$-0.569524\pi$
$827$	−8.52356e6	−0.433369	−0.216684	+	0.976242i	$0.569524\pi$
$828$	2.35090e6	0.119168
$829$	−3.87458e6	−0.195811	−0.0979057	−	0.995196i	$-0.531214\pi$
$829$	−3.87458e6	−0.195811	−0.0979057	+	0.995196i	$0.531214\pi$
$830$	2.63142e7	1.32585
$831$	−1.66214e7	−0.834960
$832$	−1.43121e6	−0.0716794
$833$	0	0
$834$	6.08031e6	0.302699
$835$	−3.99665e7	−1.98372
$836$	−6.68441e6	−0.330786
$837$	−6.58414e6	−0.324852
$838$	454899.	0.0223772
$839$	−2.13769e7	−1.04843	−0.524215	−	0.851586i	$-0.675641\pi$
$839$	−2.13769e7	−1.04843	−0.524215	+	0.851586i	$0.675641\pi$
$840$	0	0
$841$	−1.99350e7	−0.971911
$842$	1.77367e6	0.0862168
$843$	−2.05493e7	−0.995930
$844$	7.70796e6	0.372463
$845$	−1.88128e7	−0.906383
$846$	−7.97001e6	−0.382854
$847$	0	0
$848$	3.48063e6	0.166214
$849$	−1.20323e7	−0.572902
$850$	1.18393e7	0.562053
$851$	1.41397e7	0.669293
$852$	−1.01010e7	−0.476722
$853$	85930.0	0.00404364	0.00202182	−	0.999998i	$-0.499356\pi$
$853$	85930.0	0.00404364	0.00202182	+	0.999998i	$0.499356\pi$
$854$	0	0
$855$	1.70923e7	0.799624
$856$	−9.29715e6	−0.433676
$857$	1.85664e7	0.863528	0.431764	−	0.901987i	$-0.357891\pi$
$857$	1.85664e7	0.863528	0.431764	+	0.901987i	$0.357891\pi$
$858$	1.88007e6	0.0871880
$859$	−3.92215e7	−1.81360	−0.906799	−	0.421564i	$-0.861481\pi$
$859$	−3.92215e7	−1.81360	−0.906799	+	0.421564i	$0.861481\pi$
$860$	1.47226e7	0.678795
$861$	0	0
$862$	1.85324e7	0.849501
$863$	1.83642e7	0.839352	0.419676	−	0.907674i	$-0.362144\pi$
$863$	1.83642e7	0.839352	0.419676	+	0.907674i	$0.362144\pi$
$864$	746496.	0.0340207
$865$	−2.55453e6	−0.116084
$866$	−2.63182e7	−1.19251
$867$	−879381.	−0.0397310
$868$	0	0
$869$	−9.20870e6	−0.413665
$870$	−2.06284e6	−0.0923992
$871$	−1.89984e7	−0.848539
$872$	−1.15744e7	−0.515477
$873$	−2.61973e6	−0.116338
$874$	2.02816e7	0.898100
$875$	0	0
$876$	6.40351e6	0.281941
$877$	−9.42578e6	−0.413826	−0.206913	−	0.978359i	$-0.566342\pi$
$877$	−9.42578e6	−0.413826	−0.206913	+	0.978359i	$0.566342\pi$
$878$	7.28278e6	0.318831
$879$	2.00730e7	0.876273
$880$	−2.88850e6	−0.125738
$881$	−1.32820e7	−0.576534	−0.288267	−	0.957550i	$-0.593079\pi$
$881$	−1.32820e7	−0.576534	−0.288267	+	0.957550i	$0.593079\pi$
$882$	0	0
$883$	1.88897e7	0.815309	0.407655	−	0.913136i	$-0.366347\pi$
$883$	1.88897e7	0.815309	0.407655	+	0.913136i	$0.366347\pi$
$884$	−6.42840e6	−0.276676
$885$	1.79090e7	0.768623
$886$	−8.27961e6	−0.354345
$887$	−4.17083e7	−1.77997	−0.889987	−	0.455985i	$-0.849287\pi$
$887$	−4.17083e7	−1.77997	−0.889987	+	0.455985i	$0.849287\pi$
$888$	4.48986e6	0.191073
$889$	0	0
$890$	−2.97649e7	−1.25959
$891$	−980618.	−0.0413815
$892$	−1.02205e7	−0.430089
$893$	−6.87586e7	−2.88535
$894$	9.50408e6	0.397710
$895$	−1.87028e7	−0.780455
$896$	0	0
$897$	−5.70447e6	−0.236719
$898$	2.29033e7	0.947777
$899$	6.85539e6	0.282900
$900$	3.33603e6	0.137285
$901$	1.56336e7	0.641573
$902$	4.56785e6	0.186937
$903$	0	0
$904$	1.26613e7	0.515295
$905$	−2.78922e7	−1.13204
$906$	1.00503e7	0.406779
$907$	4.60801e7	1.85993	0.929963	−	0.367654i	$-0.119839\pi$
$907$	4.60801e7	1.85993	0.929963	+	0.367654i	$0.119839\pi$
$908$	8.41091e6	0.338554
$909$	−2.53906e6	−0.101921
$910$	0	0
$911$	2.25527e7	0.900334	0.450167	−	0.892944i	$-0.351364\pi$
$911$	2.25527e7	0.900334	0.450167	+	0.892944i	$0.351364\pi$
$912$	6.44014e6	0.256394
$913$	−1.30244e7	−0.517107
$914$	1.27470e6	0.0504712
$915$	−2.39987e7	−0.947622
$916$	−1.49230e7	−0.587647
$917$	0	0
$918$	3.35295e6	0.131317
$919$	−3.25898e7	−1.27290	−0.636448	−	0.771319i	$-0.719597\pi$
$919$	−3.25898e7	−1.27290	−0.636448	+	0.771319i	$0.719597\pi$
$920$	8.76421e6	0.341384
$921$	−1.60004e7	−0.621557
$922$	1.36988e7	0.530708
$923$	2.45100e7	0.946977
$924$	0	0
$925$	2.00648e7	0.771046
$926$	1.53178e7	0.587041
$927$	−8.04597e6	−0.307524
$928$	−777250.	−0.0296272
$929$	4.56149e7	1.73407	0.867036	−	0.498246i	$-0.166022\pi$
$929$	4.56149e7	1.73407	0.867036	+	0.498246i	$0.166022\pi$
$930$	−2.45458e7	−0.930614
$931$	0	0
$932$	−1.13247e7	−0.427058
$933$	−1.26244e7	−0.474797
$934$	−1.13785e7	−0.426792
$935$	−1.29740e7	−0.485338
$936$	−1.81137e6	−0.0675800
$937$	−3.60574e6	−0.134167	−0.0670835	−	0.997747i	$-0.521369\pi$
$937$	−3.60574e6	−0.134167	−0.0670835	+	0.997747i	$0.521369\pi$
$938$	0	0
$939$	−1.09705e7	−0.406034
$940$	−2.97123e7	−1.09677
$941$	4.40826e6	0.162290	0.0811452	−	0.996702i	$-0.474142\pi$
$941$	4.40826e6	0.162290	0.0811452	+	0.996702i	$0.474142\pi$
$942$	−6.78968e6	−0.249300
$943$	−1.38596e7	−0.507542
$944$	6.74786e6	0.246454
$945$	0	0
$946$	−7.28706e6	−0.264743
$947$	7.36278e6	0.266788	0.133394	−	0.991063i	$-0.457412\pi$
$947$	7.36278e6	0.266788	0.133394	+	0.991063i	$0.457412\pi$
$948$	8.87219e6	0.320634
$949$	−1.55381e7	−0.560057
$950$	2.87804e7	1.03464
$951$	2.41201e7	0.864825
$952$	0	0
$953$	3.73903e7	1.33360	0.666802	−	0.745235i	$-0.267663\pi$
$953$	3.73903e7	1.33360	0.666802	+	0.745235i	$0.267663\pi$
$954$	4.40518e6	0.156708
$955$	−3.66307e7	−1.29968
$956$	−8.00982e6	−0.283451
$957$	1.02102e6	0.0360374
$958$	−5.21462e6	−0.183573
$959$	0	0
$960$	2.78295e6	0.0974602
$961$	5.29433e7	1.84928
$962$	−1.08946e7	−0.379555
$963$	−1.17667e7	−0.408874
$964$	−1.93315e7	−0.669997
$965$	−2.49742e7	−0.863322
$966$	0	0
$967$	1.30730e7	0.449582	0.224791	−	0.974407i	$-0.427830\pi$
$967$	1.30730e7	0.449582	0.224791	+	0.974407i	$0.427830\pi$
$968$	−8.87758e6	−0.304513
$969$	2.89265e7	0.989660
$970$	−9.76640e6	−0.333277
$971$	1.52454e7	0.518908	0.259454	−	0.965755i	$-0.416457\pi$
$971$	1.52454e7	0.518908	0.259454	+	0.965755i	$0.416457\pi$
$972$	944784.	0.0320750
$973$	0	0
$974$	2.49450e7	0.842530
$975$	−8.09486e6	−0.272708
$976$	−9.04237e6	−0.303849
$977$	3.08690e6	0.103463	0.0517316	−	0.998661i	$-0.483526\pi$
$977$	3.08690e6	0.103463	0.0517316	+	0.998661i	$0.483526\pi$
$978$	3.22983e6	0.107977
$979$	1.47324e7	0.491264
$980$	0	0
$981$	−1.46489e7	−0.485996
$982$	−1.57571e7	−0.521433
$983$	−1.70201e7	−0.561796	−0.280898	−	0.959738i	$-0.590632\pi$
$983$	−1.70201e7	−0.561796	−0.280898	+	0.959738i	$0.590632\pi$
$984$	−4.40093e6	−0.144896
$985$	−1.22269e7	−0.401539
$986$	−3.49109e6	−0.114359
$987$	0	0
$988$	−1.56270e7	−0.509311
$989$	2.21102e7	0.718789
$990$	−3.65576e6	−0.118547
$991$	−4.34672e7	−1.40597	−0.702987	−	0.711203i	$-0.748151\pi$
$991$	−4.34672e7	−1.40597	−0.702987	+	0.711203i	$0.748151\pi$
$992$	−9.24851e6	−0.298396
$993$	1.28304e6	0.0412922
$994$	0	0
$995$	−4.18199e6	−0.133914
$996$	1.25484e7	0.400813
$997$	2.46672e7	0.785926	0.392963	−	0.919554i	$-0.371450\pi$
$997$	2.46672e7	0.785926	0.392963	+	0.919554i	$0.371450\pi$
$998$	3.19028e7	1.01392
$999$	5.68248e6	0.180146

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	294.6.a.w.1.2		2
3.2	odd	2		882.6.a.bb.1.1		2
7.2	even	3		294.6.e.s.67.1		4
7.3	odd	6		42.6.e.c.37.2	yes	4
7.4	even	3		294.6.e.s.79.1		4
7.5	odd	6		42.6.e.c.25.2	✓	4
7.6	odd	2		294.6.a.r.1.1		2
21.5	even	6		126.6.g.h.109.1		4
21.17	even	6		126.6.g.h.37.1		4
21.20	even	2		882.6.a.bh.1.2		2
28.3	even	6		336.6.q.f.289.2		4
28.19	even	6		336.6.q.f.193.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.6.e.c.25.2	✓	4	7.5	odd	6
42.6.e.c.37.2	yes	4	7.3	odd	6
126.6.g.h.37.1		4	21.17	even	6
126.6.g.h.109.1		4	21.5	even	6
294.6.a.r.1.1		2	7.6	odd	2
294.6.a.w.1.2		2	1.1	even	1	trivial
294.6.e.s.67.1		4	7.2	even	3
294.6.e.s.79.1		4	7.4	even	3
336.6.q.f.193.2		4	28.19	even	6
336.6.q.f.289.2		4	28.3	even	6
882.6.a.bb.1.1		2	3.2	odd	2
882.6.a.bh.1.2		2	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	294.a (trivial)

\(f(q)\)	\(=\)	\(q+4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} +75.4923 q^{5} +36.0000 q^{6} +64.0000 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q+4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} +75.4923 q^{5} +36.0000 q^{6} +64.0000 q^{8} +81.0000 q^{9} +301.969 q^{10} -149.462 q^{11} +144.000 q^{12} -349.416 q^{13} +679.431 q^{15} +256.000 q^{16} +1149.85 q^{17} +324.000 q^{18} +2795.20 q^{19} +1207.88 q^{20} -597.847 q^{22} +1813.97 q^{23} +576.000 q^{24} +2574.09 q^{25} -1397.66 q^{26} +729.000 q^{27} -759.033 q^{29} +2717.72 q^{30} -9031.74 q^{31} +1024.00 q^{32} -1345.16 q^{33} +4599.39 q^{34} +1296.00 q^{36} +7794.89 q^{37} +11180.8 q^{38} -3144.74 q^{39} +4831.51 q^{40} -7640.49 q^{41} +12188.8 q^{43} -2391.39 q^{44} +6114.88 q^{45} +7255.88 q^{46} -24598.8 q^{47} +2304.00 q^{48} +10296.4 q^{50} +10348.6 q^{51} -5590.65 q^{52} +13596.2 q^{53} +2916.00 q^{54} -11283.2 q^{55} +25156.8 q^{57} -3036.13 q^{58} +26358.8 q^{59} +10870.9 q^{60} -35321.8 q^{61} -36127.0 q^{62} +4096.00 q^{64} -26378.2 q^{65} -5380.62 q^{66} +54371.9 q^{67} +18397.6 q^{68} +16325.7 q^{69} -70145.7 q^{71} +5184.00 q^{72} +44468.8 q^{73} +31179.6 q^{74} +23166.8 q^{75} +44723.2 q^{76} -12579.0 q^{78} +61612.5 q^{79} +19326.0 q^{80} +6561.00 q^{81} -30562.0 q^{82} +87142.0 q^{83} +86804.6 q^{85} +48755.3 q^{86} -6831.30 q^{87} -9565.55 q^{88} -98569.4 q^{89} +24459.5 q^{90} +29023.5 q^{92} -81285.7 q^{93} -98395.2 q^{94} +211016. q^{95} +9216.00 q^{96} -32342.3 q^{97} -12106.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{2} + 18 q^{3} + 32 q^{4} + 53 q^{5} + 72 q^{6} + 128 q^{8} + 162 q^{9}+O(q^{10}) \) 2 * q + 8 * q^2 + 18 * q^3 + 32 * q^4 + 53 * q^5 + 72 * q^6 + 128 * q^8 + 162 * q^9	\( 2 q + 8 q^{2} + 18 q^{3} + 32 q^{4} + 53 q^{5} + 72 q^{6} + 128 q^{8} + 162 q^{9} + 212 q^{10} + 191 q^{11} + 288 q^{12} + 379 q^{13} + 477 q^{15} + 512 q^{16} + 340 q^{17} + 648 q^{18} + 1769 q^{19} + 848 q^{20} + 764 q^{22} + 3236 q^{23} + 1152 q^{24} - 45 q^{25} + 1516 q^{26} + 1458 q^{27} + 4459 q^{29} + 1908 q^{30} - 1994 q^{31} + 2048 q^{32} + 1719 q^{33} + 1360 q^{34} + 2592 q^{36} + 20587 q^{37} + 7076 q^{38} + 3411 q^{39} + 3392 q^{40} - 8814 q^{41} + 15853 q^{43} + 3056 q^{44} + 4293 q^{45} + 12944 q^{46} - 33912 q^{47} + 4608 q^{48} - 180 q^{50} + 3060 q^{51} + 6064 q^{52} + 49239 q^{53} + 5832 q^{54} - 18941 q^{55} + 15921 q^{57} + 17836 q^{58} + 56735 q^{59} + 7632 q^{60} - 67508 q^{61} - 7976 q^{62} + 8192 q^{64} - 42762 q^{65} + 6876 q^{66} + 75723 q^{67} + 5440 q^{68} + 29124 q^{69} - 8992 q^{71} + 10368 q^{72} + 3201 q^{73} + 82348 q^{74} - 405 q^{75} + 28304 q^{76} + 13644 q^{78} + 26612 q^{79} + 13568 q^{80} + 13122 q^{81} - 35256 q^{82} + 949 q^{83} + 105020 q^{85} + 63412 q^{86} + 40131 q^{87} + 12224 q^{88} - 176562 q^{89} + 17172 q^{90} + 51776 q^{92} - 17946 q^{93} - 135648 q^{94} + 234098 q^{95} + 18432 q^{96} + 129423 q^{97} + 15471 q^{99}+O(q^{100}) \) 2 * q + 8 * q^2 + 18 * q^3 + 32 * q^4 + 53 * q^5 + 72 * q^6 + 128 * q^8 + 162 * q^9 + 212 * q^10 + 191 * q^11 + 288 * q^12 + 379 * q^13 + 477 * q^15 + 512 * q^16 + 340 * q^17 + 648 * q^18 + 1769 * q^19 + 848 * q^20 + 764 * q^22 + 3236 * q^23 + 1152 * q^24 - 45 * q^25 + 1516 * q^26 + 1458 * q^27 + 4459 * q^29 + 1908 * q^30 - 1994 * q^31 + 2048 * q^32 + 1719 * q^33 + 1360 * q^34 + 2592 * q^36 + 20587 * q^37 + 7076 * q^38 + 3411 * q^39 + 3392 * q^40 - 8814 * q^41 + 15853 * q^43 + 3056 * q^44 + 4293 * q^45 + 12944 * q^46 - 33912 * q^47 + 4608 * q^48 - 180 * q^50 + 3060 * q^51 + 6064 * q^52 + 49239 * q^53 + 5832 * q^54 - 18941 * q^55 + 15921 * q^57 + 17836 * q^58 + 56735 * q^59 + 7632 * q^60 - 67508 * q^61 - 7976 * q^62 + 8192 * q^64 - 42762 * q^65 + 6876 * q^66 + 75723 * q^67 + 5440 * q^68 + 29124 * q^69 - 8992 * q^71 + 10368 * q^72 + 3201 * q^73 + 82348 * q^74 - 405 * q^75 + 28304 * q^76 + 13644 * q^78 + 26612 * q^79 + 13568 * q^80 + 13122 * q^81 - 35256 * q^82 + 949 * q^83 + 105020 * q^85 + 63412 * q^86 + 40131 * q^87 + 12224 * q^88 - 176562 * q^89 + 17172 * q^90 + 51776 * q^92 - 17946 * q^93 - 135648 * q^94 + 234098 * q^95 + 18432 * q^96 + 129423 * q^97 + 15471 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more