LMFDB - Embedded newform 624.6.a.r.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [624,6,Mod(1,624)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(624, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("624.1"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	624.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-27,0,4,0,-242] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$100.079503563$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 238x + 778 $ x^3 - x^2 - 238*x + 778
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 3 $
Twist minimal:	no (minimal twist has level 156)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$14.0186$ of defining polynomial
Character	$\chi$	$=$	624.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000 q^{3} +26.3472 q^{5} +108.570 q^{7} +81.0000 q^{9} -285.612 q^{11} +169.000 q^{13} -237.125 q^{15} +964.281 q^{17} -1122.52 q^{19} -977.133 q^{21} -3199.74 q^{23} -2430.82 q^{25} -729.000 q^{27} -150.648 q^{29} +1018.47 q^{31} +2570.51 q^{33} +2860.53 q^{35} +11302.4 q^{37} -1521.00 q^{39} +3400.17 q^{41} -12816.8 q^{43} +2134.13 q^{45} -19597.8 q^{47} -5019.49 q^{49} -8678.53 q^{51} -9707.26 q^{53} -7525.09 q^{55} +10102.7 q^{57} +22700.9 q^{59} +41402.6 q^{61} +8794.20 q^{63} +4452.68 q^{65} +15356.0 q^{67} +28797.6 q^{69} -29876.9 q^{71} +37685.4 q^{73} +21877.4 q^{75} -31009.0 q^{77} -28054.2 q^{79} +6561.00 q^{81} +30521.3 q^{83} +25406.1 q^{85} +1355.83 q^{87} +6658.25 q^{89} +18348.4 q^{91} -9166.24 q^{93} -29575.2 q^{95} -68607.1 q^{97} -23134.6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 27 q^{3} + 4 q^{5} - 242 q^{7} + 243 q^{9} - 64 q^{11} + 507 q^{13} - 36 q^{15} + 622 q^{17} - 2330 q^{19} + 2178 q^{21} - 816 q^{23} + 6313 q^{25} - 2187 q^{27} + 6726 q^{29} - 5010 q^{31} + 576 q^{33}+ \cdots - 5184 q^{99}+O(q^{100}) $ 3 * q - 27 * q^3 + 4 * q^5 - 242 * q^7 + 243 * q^9 - 64 * q^11 + 507 * q^13 - 36 * q^15 + 622 * q^17 - 2330 * q^19 + 2178 * q^21 - 816 * q^23 + 6313 * q^25 - 2187 * q^27 + 6726 * q^29 - 5010 * q^31 + 576 * q^33 + 1136 * q^35 + 10670 * q^37 - 4563 * q^39 + 17584 * q^41 + 2092 * q^43 + 324 * q^45 - 4936 * q^47 + 24975 * q^49 - 5598 * q^51 + 39010 * q^53 - 48592 * q^55 + 20970 * q^57 - 37436 * q^59 + 46862 * q^61 - 19602 * q^63 + 676 * q^65 - 88734 * q^67 + 7344 * q^69 + 20476 * q^71 - 3246 * q^73 - 56817 * q^75 - 55088 * q^77 - 3000 * q^79 + 19683 * q^81 + 97956 * q^83 + 6664 * q^85 - 60534 * q^87 - 183380 * q^89 - 40898 * q^91 + 45090 * q^93 + 365984 * q^95 - 59254 * q^97 - 5184 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	0	0
$5$	26.3472	0.471314	0.235657	−	0.971836i	$-0.424276\pi$
$5$	26.3472	0.471314	0.235657	+	0.971836i	$0.424276\pi$
$6$	0	0
$7$	108.570	0.837464	0.418732	−	0.908110i	$-0.362475\pi$
$7$	108.570	0.837464	0.418732	+	0.908110i	$0.362475\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−285.612	−0.711696	−0.355848	−	0.934544i	$-0.615808\pi$
$11$	−285.612	−0.711696	−0.355848	+	0.934544i	$0.615808\pi$
$12$	0	0
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	0	0
$15$	−237.125	−0.272113
$16$	0	0
$17$	964.281	0.809248	0.404624	−	0.914483i	$-0.367402\pi$
$17$	964.281	0.809248	0.404624	+	0.914483i	$0.367402\pi$
$18$	0	0
$19$	−1122.52	−0.713361	−0.356680	−	0.934226i	$-0.616091\pi$
$19$	−1122.52	−0.713361	−0.356680	+	0.934226i	$0.616091\pi$
$20$	0	0
$21$	−977.133	−0.483510
$22$	0	0
$23$	−3199.74	−1.26123	−0.630615	−	0.776096i	$-0.717197\pi$
$23$	−3199.74	−1.26123	−0.630615	+	0.776096i	$0.717197\pi$
$24$	0	0
$25$	−2430.82	−0.777863
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	−150.648	−0.0332635	−0.0166317	−	0.999862i	$-0.505294\pi$
$29$	−150.648	−0.0332635	−0.0166317	+	0.999862i	$0.505294\pi$
$30$	0	0
$31$	1018.47	0.190346	0.0951732	−	0.995461i	$-0.469660\pi$
$31$	1018.47	0.190346	0.0951732	+	0.995461i	$0.469660\pi$
$32$	0	0
$33$	2570.51	0.410898
$34$	0	0
$35$	2860.53	0.394708
$36$	0	0
$37$	11302.4	1.35726	0.678632	−	0.734478i	$-0.262573\pi$
$37$	11302.4	1.35726	0.678632	+	0.734478i	$0.262573\pi$
$38$	0	0
$39$	−1521.00	−0.160128
$40$	0	0
$41$	3400.17	0.315894	0.157947	−	0.987448i	$-0.449512\pi$
$41$	3400.17	0.315894	0.157947	+	0.987448i	$0.449512\pi$
$42$	0	0
$43$	−12816.8	−1.05708	−0.528539	−	0.848909i	$-0.677260\pi$
$43$	−12816.8	−1.05708	−0.528539	+	0.848909i	$0.677260\pi$
$44$	0	0
$45$	2134.13	0.157105
$46$	0	0
$47$	−19597.8	−1.29409	−0.647043	−	0.762453i	$-0.723995\pi$
$47$	−19597.8	−1.29409	−0.647043	+	0.762453i	$0.723995\pi$
$48$	0	0
$49$	−5019.49	−0.298655
$50$	0	0
$51$	−8678.53	−0.467219
$52$	0	0
$53$	−9707.26	−0.474687	−0.237343	−	0.971426i	$-0.576277\pi$
$53$	−9707.26	−0.474687	−0.237343	+	0.971426i	$0.576277\pi$
$54$	0	0
$55$	−7525.09	−0.335432
$56$	0	0
$57$	10102.7	0.411859
$58$	0	0
$59$	22700.9	0.849010	0.424505	−	0.905426i	$-0.360448\pi$
$59$	22700.9	0.849010	0.424505	+	0.905426i	$0.360448\pi$
$60$	0	0
$61$	41402.6	1.42463	0.712317	−	0.701858i	$-0.247646\pi$
$61$	41402.6	1.42463	0.712317	+	0.701858i	$0.247646\pi$
$62$	0	0
$63$	8794.20	0.279155
$64$	0	0
$65$	4452.68	0.130719
$66$	0	0
$67$	15356.0	0.417918	0.208959	−	0.977924i	$-0.432992\pi$
$67$	15356.0	0.417918	0.208959	+	0.977924i	$0.432992\pi$
$68$	0	0
$69$	28797.6	0.728172
$70$	0	0
$71$	−29876.9	−0.703380	−0.351690	−	0.936116i	$-0.614393\pi$
$71$	−29876.9	−0.703380	−0.351690	+	0.936116i	$0.614393\pi$
$72$	0	0
$73$	37685.4	0.827686	0.413843	−	0.910348i	$-0.364186\pi$
$73$	37685.4	0.827686	0.413843	+	0.910348i	$0.364186\pi$
$74$	0	0
$75$	21877.4	0.449100
$76$	0	0
$77$	−31009.0	−0.596020
$78$	0	0
$79$	−28054.2	−0.505743	−0.252872	−	0.967500i	$-0.581375\pi$
$79$	−28054.2	−0.505743	−0.252872	+	0.967500i	$0.581375\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	30521.3	0.486305	0.243152	−	0.969988i	$-0.421818\pi$
$83$	30521.3	0.486305	0.243152	+	0.969988i	$0.421818\pi$
$84$	0	0
$85$	25406.1	0.381409
$86$	0	0
$87$	1355.83	0.0192047
$88$	0	0
$89$	6658.25	0.0891015	0.0445507	−	0.999007i	$-0.485814\pi$
$89$	6658.25	0.0891015	0.0445507	+	0.999007i	$0.485814\pi$
$90$	0	0
$91$	18348.4	0.232271
$92$	0	0
$93$	−9166.24	−0.109897
$94$	0	0
$95$	−29575.2	−0.336217
$96$	0	0
$97$	−68607.1	−0.740354	−0.370177	−	0.928961i	$-0.620703\pi$
$97$	−68607.1	−0.740354	−0.370177	+	0.928961i	$0.620703\pi$
$98$	0	0
$99$	−23134.6	−0.237232
$100$	0	0
$101$	−193596.	−1.88839	−0.944196	−	0.329383i	$-0.893159\pi$
$101$	−193596.	−1.88839	−0.944196	+	0.329383i	$0.893159\pi$
$102$	0	0
$103$	32133.9	0.298450	0.149225	−	0.988803i	$-0.452322\pi$
$103$	32133.9	0.298450	0.149225	+	0.988803i	$0.452322\pi$
$104$	0	0
$105$	−25744.7	−0.227885
$106$	0	0
$107$	188835.	1.59449	0.797247	−	0.603653i	$-0.206289\pi$
$107$	188835.	1.59449	0.797247	+	0.603653i	$0.206289\pi$
$108$	0	0
$109$	−143157.	−1.15410	−0.577052	−	0.816707i	$-0.695797\pi$
$109$	−143157.	−1.15410	−0.577052	+	0.816707i	$0.695797\pi$
$110$	0	0
$111$	−101721.	−0.783617
$112$	0	0
$113$	−125402.	−0.923862	−0.461931	−	0.886916i	$-0.652843\pi$
$113$	−125402.	−0.923862	−0.461931	+	0.886916i	$0.652843\pi$
$114$	0	0
$115$	−84304.2	−0.594435
$116$	0	0
$117$	13689.0	0.0924500
$118$	0	0
$119$	104692.	0.677715
$120$	0	0
$121$	−79476.8	−0.493488
$122$	0	0
$123$	−30601.6	−0.182382
$124$	0	0
$125$	−146381.	−0.837931
$126$	0	0
$127$	−96499.1	−0.530901	−0.265451	−	0.964124i	$-0.585521\pi$
$127$	−96499.1	−0.530901	−0.265451	+	0.964124i	$0.585521\pi$
$128$	0	0
$129$	115351.	0.610305
$130$	0	0
$131$	−10012.2	−0.0509744	−0.0254872	−	0.999675i	$-0.508114\pi$
$131$	−10012.2	−0.0509744	−0.0254872	+	0.999675i	$0.508114\pi$
$132$	0	0
$133$	−121872.	−0.597414
$134$	0	0
$135$	−19207.1	−0.0907044
$136$	0	0
$137$	−189227.	−0.861356	−0.430678	−	0.902506i	$-0.641726\pi$
$137$	−189227.	−0.861356	−0.430678	+	0.902506i	$0.641726\pi$
$138$	0	0
$139$	−63574.5	−0.279091	−0.139545	−	0.990216i	$-0.544564\pi$
$139$	−63574.5	−0.279091	−0.139545	+	0.990216i	$0.544564\pi$
$140$	0	0
$141$	176380.	0.747141
$142$	0	0
$143$	−48268.4	−0.197389
$144$	0	0
$145$	−3969.15	−0.0156775
$146$	0	0
$147$	45175.4	0.172428
$148$	0	0
$149$	−2413.19	−0.00890484	−0.00445242	−	0.999990i	$-0.501417\pi$
$149$	−2413.19	−0.00890484	−0.00445242	+	0.999990i	$0.501417\pi$
$150$	0	0
$151$	−452236.	−1.61407	−0.807036	−	0.590502i	$-0.798930\pi$
$151$	−452236.	−1.61407	−0.807036	+	0.590502i	$0.798930\pi$
$152$	0	0
$153$	78106.8	0.269749
$154$	0	0
$155$	26833.9	0.0897128
$156$	0	0
$157$	−195398.	−0.632661	−0.316330	−	0.948649i	$-0.602451\pi$
$157$	−195398.	−0.632661	−0.316330	+	0.948649i	$0.602451\pi$
$158$	0	0
$159$	87365.3	0.274060
$160$	0	0
$161$	−347396.	−1.05623
$162$	0	0
$163$	−335838.	−0.990060	−0.495030	−	0.868876i	$-0.664843\pi$
$163$	−335838.	−0.990060	−0.495030	+	0.868876i	$0.664843\pi$
$164$	0	0
$165$	67725.8	0.193662
$166$	0	0
$167$	266301.	0.738893	0.369446	−	0.929252i	$-0.379547\pi$
$167$	266301.	0.738893	0.369446	+	0.929252i	$0.379547\pi$
$168$	0	0
$169$	28561.0	0.0769231
$170$	0	0
$171$	−90924.0	−0.237787
$172$	0	0
$173$	−413083.	−1.04936	−0.524678	−	0.851301i	$-0.675814\pi$
$173$	−413083.	−1.04936	−0.524678	+	0.851301i	$0.675814\pi$
$174$	0	0
$175$	−263915.	−0.651432
$176$	0	0
$177$	−204308.	−0.490176
$178$	0	0
$179$	864.231	0.00201603	0.00100802	−	0.999999i	$-0.499679\pi$
$179$	864.231	0.00201603	0.00100802	+	0.999999i	$0.499679\pi$
$180$	0	0
$181$	334650.	0.759266	0.379633	−	0.925137i	$-0.376050\pi$
$181$	334650.	0.759266	0.379633	+	0.925137i	$0.376050\pi$
$182$	0	0
$183$	−372624.	−0.822513
$184$	0	0
$185$	297786.	0.639697
$186$	0	0
$187$	−275410.	−0.575939
$188$	0	0
$189$	−79147.8	−0.161170
$190$	0	0
$191$	−811587.	−1.60973	−0.804863	−	0.593461i	$-0.797761\pi$
$191$	−811587.	−1.60973	−0.804863	+	0.593461i	$0.797761\pi$
$192$	0	0
$193$	−29461.9	−0.0569334	−0.0284667	−	0.999595i	$-0.509062\pi$
$193$	−29461.9	−0.0569334	−0.0284667	+	0.999595i	$0.509062\pi$
$194$	0	0
$195$	−40074.1	−0.0754706
$196$	0	0
$197$	−334514.	−0.614113	−0.307056	−	0.951691i	$-0.599344\pi$
$197$	−334514.	−0.614113	−0.307056	+	0.951691i	$0.599344\pi$
$198$	0	0
$199$	473775.	0.848086	0.424043	−	0.905642i	$-0.360611\pi$
$199$	473775.	0.848086	0.424043	+	0.905642i	$0.360611\pi$
$200$	0	0
$201$	−138204.	−0.241285
$202$	0	0
$203$	−16355.9	−0.0278570
$204$	0	0
$205$	89585.2	0.148885
$206$	0	0
$207$	−259179.	−0.420410
$208$	0	0
$209$	320605.	0.507696
$210$	0	0
$211$	179756.	0.277956	0.138978	−	0.990295i	$-0.455618\pi$
$211$	179756.	0.277956	0.138978	+	0.990295i	$0.455618\pi$
$212$	0	0
$213$	268892.	0.406097
$214$	0	0
$215$	−337686.	−0.498216
$216$	0	0
$217$	110576.	0.159408
$218$	0	0
$219$	−339168.	−0.477865
$220$	0	0
$221$	162964.	0.224445
$222$	0	0
$223$	−926508.	−1.24763	−0.623817	−	0.781571i	$-0.714419\pi$
$223$	−926508.	−1.24763	−0.623817	+	0.781571i	$0.714419\pi$
$224$	0	0
$225$	−196897.	−0.259288
$226$	0	0
$227$	−583675.	−0.751807	−0.375903	−	0.926659i	$-0.622668\pi$
$227$	−583675.	−0.751807	−0.375903	+	0.926659i	$0.622668\pi$
$228$	0	0
$229$	−373754.	−0.470974	−0.235487	−	0.971877i	$-0.575669\pi$
$229$	−373754.	−0.470974	−0.235487	+	0.971877i	$0.575669\pi$
$230$	0	0
$231$	279081.	0.344112
$232$	0	0
$233$	−1.39688e6	−1.68566	−0.842829	−	0.538182i	$-0.819111\pi$
$233$	−1.39688e6	−1.68566	−0.842829	+	0.538182i	$0.819111\pi$
$234$	0	0
$235$	−516349.	−0.609921
$236$	0	0
$237$	252488.	0.291991
$238$	0	0
$239$	584527.	0.661926	0.330963	−	0.943644i	$-0.392626\pi$
$239$	584527.	0.661926	0.330963	+	0.943644i	$0.392626\pi$
$240$	0	0
$241$	1.12255e6	1.24498	0.622490	−	0.782628i	$-0.286121\pi$
$241$	1.12255e6	1.24498	0.622490	+	0.782628i	$0.286121\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	−132250.	−0.140760
$246$	0	0
$247$	−189706.	−0.197851
$248$	0	0
$249$	−274692.	−0.280768
$250$	0	0
$251$	−727271.	−0.728638	−0.364319	−	0.931274i	$-0.618698\pi$
$251$	−727271.	−0.728638	−0.364319	+	0.931274i	$0.618698\pi$
$252$	0	0
$253$	913883.	0.897613
$254$	0	0
$255$	−228655.	−0.220207
$256$	0	0
$257$	650466.	0.614316	0.307158	−	0.951659i	$-0.400622\pi$
$257$	650466.	0.614316	0.307158	+	0.951659i	$0.400622\pi$
$258$	0	0
$259$	1.22710e6	1.13666
$260$	0	0
$261$	−12202.5	−0.0110878
$262$	0	0
$263$	−979886.	−0.873547	−0.436774	−	0.899571i	$-0.643879\pi$
$263$	−979886.	−0.873547	−0.436774	+	0.899571i	$0.643879\pi$
$264$	0	0
$265$	−255759.	−0.223726
$266$	0	0
$267$	−59924.2	−0.0514428
$268$	0	0
$269$	−242796.	−0.204579	−0.102289	−	0.994755i	$-0.532617\pi$
$269$	−242796.	−0.204579	−0.102289	+	0.994755i	$0.532617\pi$
$270$	0	0
$271$	917731.	0.759088	0.379544	−	0.925174i	$-0.376081\pi$
$271$	917731.	0.759088	0.379544	+	0.925174i	$0.376081\pi$
$272$	0	0
$273$	−165135.	−0.134102
$274$	0	0
$275$	694272.	0.553603
$276$	0	0
$277$	−1.70782e6	−1.33734	−0.668670	−	0.743560i	$-0.733136\pi$
$277$	−1.70782e6	−1.33734	−0.668670	+	0.743560i	$0.733136\pi$
$278$	0	0
$279$	82496.2	0.0634488
$280$	0	0
$281$	−78457.8	−0.0592748	−0.0296374	−	0.999561i	$-0.509435\pi$
$281$	−78457.8	−0.0592748	−0.0296374	+	0.999561i	$0.509435\pi$
$282$	0	0
$283$	1.12505e6	0.835038	0.417519	−	0.908668i	$-0.362900\pi$
$283$	1.12505e6	0.835038	0.417519	+	0.908668i	$0.362900\pi$
$284$	0	0
$285$	266177.	0.194115
$286$	0	0
$287$	369158.	0.264550
$288$	0	0
$289$	−490019.	−0.345118
$290$	0	0
$291$	617464.	0.427444
$292$	0	0
$293$	−1.15641e6	−0.786940	−0.393470	−	0.919338i	$-0.628725\pi$
$293$	−1.15641e6	−0.786940	−0.393470	+	0.919338i	$0.628725\pi$
$294$	0	0
$295$	598106.	0.400150
$296$	0	0
$297$	208211.	0.136966
$298$	0	0
$299$	−540755.	−0.349802
$300$	0	0
$301$	−1.39152e6	−0.885265
$302$	0	0
$303$	1.74236e6	1.09026
$304$	0	0
$305$	1.09084e6	0.671449
$306$	0	0
$307$	89309.1	0.0540816	0.0270408	−	0.999634i	$-0.491392\pi$
$307$	89309.1	0.0540816	0.0270408	+	0.999634i	$0.491392\pi$
$308$	0	0
$309$	−289205.	−0.172310
$310$	0	0
$311$	−234215.	−0.137314	−0.0686569	−	0.997640i	$-0.521871\pi$
$311$	−234215.	−0.137314	−0.0686569	+	0.997640i	$0.521871\pi$
$312$	0	0
$313$	−480239.	−0.277075	−0.138537	−	0.990357i	$-0.544240\pi$
$313$	−480239.	−0.277075	−0.138537	+	0.990357i	$0.544240\pi$
$314$	0	0
$315$	231703.	0.131569
$316$	0	0
$317$	2.71795e6	1.51912	0.759561	−	0.650436i	$-0.225414\pi$
$317$	2.71795e6	1.51912	0.759561	+	0.650436i	$0.225414\pi$
$318$	0	0
$319$	43026.8	0.0236735
$320$	0	0
$321$	−1.69951e6	−0.920582
$322$	0	0
$323$	−1.08242e6	−0.577286
$324$	0	0
$325$	−410809.	−0.215740
$326$	0	0
$327$	1.28841e6	0.666323
$328$	0	0
$329$	−2.12774e6	−1.08375
$330$	0	0
$331$	2.99899e6	1.50454	0.752272	−	0.658853i	$-0.228958\pi$
$331$	2.99899e6	1.50454	0.752272	+	0.658853i	$0.228958\pi$
$332$	0	0
$333$	915491.	0.452422
$334$	0	0
$335$	404588.	0.196970
$336$	0	0
$337$	−2.05193e6	−0.984209	−0.492105	−	0.870536i	$-0.663772\pi$
$337$	−2.05193e6	−0.984209	−0.492105	+	0.870536i	$0.663772\pi$
$338$	0	0
$339$	1.12861e6	0.533392
$340$	0	0
$341$	−290888.	−0.135469
$342$	0	0
$343$	−2.36971e6	−1.08758
$344$	0	0
$345$	758738.	0.343197
$346$	0	0
$347$	−1.17162e6	−0.522353	−0.261176	−	0.965291i	$-0.584110\pi$
$347$	−1.17162e6	−0.522353	−0.261176	+	0.965291i	$0.584110\pi$
$348$	0	0
$349$	4.07956e6	1.79288	0.896438	−	0.443170i	$-0.146146\pi$
$349$	4.07956e6	1.79288	0.896438	+	0.443170i	$0.146146\pi$
$350$	0	0
$351$	−123201.	−0.0533761
$352$	0	0
$353$	−1.76727e6	−0.754858	−0.377429	−	0.926039i	$-0.623192\pi$
$353$	−1.76727e6	−0.754858	−0.377429	+	0.926039i	$0.623192\pi$
$354$	0	0
$355$	−787175.	−0.331513
$356$	0	0
$357$	−942231.	−0.391279
$358$	0	0
$359$	−2.98058e6	−1.22058	−0.610289	−	0.792179i	$-0.708947\pi$
$359$	−2.98058e6	−1.22058	−0.610289	+	0.792179i	$0.708947\pi$
$360$	0	0
$361$	−1.21605e6	−0.491116
$362$	0	0
$363$	715291.	0.284916
$364$	0	0
$365$	992905.	0.390100
$366$	0	0
$367$	3.39556e6	1.31597	0.657986	−	0.753031i	$-0.271409\pi$
$367$	3.39556e6	1.31597	0.657986	+	0.753031i	$0.271409\pi$
$368$	0	0
$369$	275414.	0.105298
$370$	0	0
$371$	−1.05392e6	−0.397533
$372$	0	0
$373$	1.93570e6	0.720387	0.360194	−	0.932878i	$-0.382711\pi$
$373$	1.93570e6	0.720387	0.360194	+	0.932878i	$0.382711\pi$
$374$	0	0
$375$	1.31743e6	0.483780
$376$	0	0
$377$	−25459.5	−0.00922563
$378$	0	0
$379$	2.83891e6	1.01520	0.507601	−	0.861592i	$-0.330532\pi$
$379$	2.83891e6	1.01520	0.507601	+	0.861592i	$0.330532\pi$
$380$	0	0
$381$	868492.	0.306516
$382$	0	0
$383$	−3.66320e6	−1.27604	−0.638019	−	0.770021i	$-0.720246\pi$
$383$	−3.66320e6	−1.27604	−0.638019	+	0.770021i	$0.720246\pi$
$384$	0	0
$385$	−817001.	−0.280912
$386$	0	0
$387$	−1.03816e6	−0.352360
$388$	0	0
$389$	−1.96777e6	−0.659327	−0.329664	−	0.944098i	$-0.606935\pi$
$389$	−1.96777e6	−0.659327	−0.329664	+	0.944098i	$0.606935\pi$
$390$	0	0
$391$	−3.08545e6	−1.02065
$392$	0	0
$393$	90109.9	0.0294301
$394$	0	0
$395$	−739151.	−0.238364
$396$	0	0
$397$	2.79713e6	0.890711	0.445356	−	0.895354i	$-0.353077\pi$
$397$	2.79713e6	0.890711	0.445356	+	0.895354i	$0.353077\pi$
$398$	0	0
$399$	1.09685e6	0.344917
$400$	0	0
$401$	580264.	0.180204	0.0901020	−	0.995933i	$-0.471281\pi$
$401$	580264.	0.180204	0.0901020	+	0.995933i	$0.471281\pi$
$402$	0	0
$403$	172122.	0.0527926
$404$	0	0
$405$	172864.	0.0523682
$406$	0	0
$407$	−3.22809e6	−0.965960
$408$	0	0
$409$	1.36264e6	0.402784	0.201392	−	0.979511i	$-0.435453\pi$
$409$	1.36264e6	0.402784	0.201392	+	0.979511i	$0.435453\pi$
$410$	0	0
$411$	1.70305e6	0.497304
$412$	0	0
$413$	2.46464e6	0.711015
$414$	0	0
$415$	804153.	0.229202
$416$	0	0
$417$	572170.	0.161133
$418$	0	0
$419$	385118.	0.107166	0.0535832	−	0.998563i	$-0.482936\pi$
$419$	385118.	0.107166	0.0535832	+	0.998563i	$0.482936\pi$
$420$	0	0
$421$	6.63054e6	1.82324	0.911619	−	0.411035i	$-0.134833\pi$
$421$	6.63054e6	1.82324	0.911619	+	0.411035i	$0.134833\pi$
$422$	0	0
$423$	−1.58742e6	−0.431362
$424$	0	0
$425$	−2.34400e6	−0.629484
$426$	0	0
$427$	4.49509e6	1.19308
$428$	0	0
$429$	434416.	0.113963
$430$	0	0
$431$	2.87522e6	0.745552	0.372776	−	0.927921i	$-0.378406\pi$
$431$	2.87522e6	0.745552	0.372776	+	0.927921i	$0.378406\pi$
$432$	0	0
$433$	−1.63890e6	−0.420082	−0.210041	−	0.977693i	$-0.567360\pi$
$433$	−1.63890e6	−0.420082	−0.210041	+	0.977693i	$0.567360\pi$
$434$	0	0
$435$	35722.4	0.00905143
$436$	0	0
$437$	3.59176e6	0.899713
$438$	0	0
$439$	1.95804e6	0.484909	0.242455	−	0.970163i	$-0.422047\pi$
$439$	1.95804e6	0.484909	0.242455	+	0.970163i	$0.422047\pi$
$440$	0	0
$441$	−406579.	−0.0995515
$442$	0	0
$443$	−7.54536e6	−1.82671	−0.913357	−	0.407159i	$-0.866520\pi$
$443$	−7.54536e6	−1.82671	−0.913357	+	0.407159i	$0.866520\pi$
$444$	0	0
$445$	175426.	0.0419947
$446$	0	0
$447$	21718.7	0.00514121
$448$	0	0
$449$	5.88112e6	1.37671	0.688357	−	0.725372i	$-0.258332\pi$
$449$	5.88112e6	1.37671	0.688357	+	0.725372i	$0.258332\pi$
$450$	0	0
$451$	−971131.	−0.224821
$452$	0	0
$453$	4.07012e6	0.931885
$454$	0	0
$455$	483429.	0.109472
$456$	0	0
$457$	−3.27148e6	−0.732747	−0.366374	−	0.930468i	$-0.619401\pi$
$457$	−3.27148e6	−0.732747	−0.366374	+	0.930468i	$0.619401\pi$
$458$	0	0
$459$	−702961.	−0.155740
$460$	0	0
$461$	−1.39477e6	−0.305668	−0.152834	−	0.988252i	$-0.548840\pi$
$461$	−1.39477e6	−0.305668	−0.152834	+	0.988252i	$0.548840\pi$
$462$	0	0
$463$	−3.81656e6	−0.827407	−0.413704	−	0.910412i	$-0.635765\pi$
$463$	−3.81656e6	−0.827407	−0.413704	+	0.910412i	$0.635765\pi$
$464$	0	0
$465$	−241505.	−0.0517957
$466$	0	0
$467$	3.70467e6	0.786063	0.393031	−	0.919525i	$-0.371426\pi$
$467$	3.70467e6	0.786063	0.393031	+	0.919525i	$0.371426\pi$
$468$	0	0
$469$	1.66721e6	0.349991
$470$	0	0
$471$	1.75858e6	0.365267
$472$	0	0
$473$	3.66062e6	0.752319
$474$	0	0
$475$	2.72864e6	0.554897
$476$	0	0
$477$	−786288.	−0.158229
$478$	0	0
$479$	1.53152e6	0.304988	0.152494	−	0.988304i	$-0.451270\pi$
$479$	1.53152e6	0.304988	0.152494	+	0.988304i	$0.451270\pi$
$480$	0	0
$481$	1.91010e6	0.376437
$482$	0	0
$483$	3.12657e6	0.609818
$484$	0	0
$485$	−1.80761e6	−0.348939
$486$	0	0
$487$	818721.	0.156428	0.0782138	−	0.996937i	$-0.475078\pi$
$487$	818721.	0.156428	0.0782138	+	0.996937i	$0.475078\pi$
$488$	0	0
$489$	3.02255e6	0.571611
$490$	0	0
$491$	−6.11842e6	−1.14534	−0.572672	−	0.819785i	$-0.694093\pi$
$491$	−6.11842e6	−1.14534	−0.572672	+	0.819785i	$0.694093\pi$
$492$	0	0
$493$	−145267.	−0.0269184
$494$	0	0
$495$	−609532.	−0.111811
$496$	0	0
$497$	−3.24375e6	−0.589055
$498$	0	0
$499$	−1.70315e6	−0.306198	−0.153099	−	0.988211i	$-0.548925\pi$
$499$	−1.70315e6	−0.306198	−0.153099	+	0.988211i	$0.548925\pi$
$500$	0	0
$501$	−2.39671e6	−0.426600
$502$	0	0
$503$	8.17306e6	1.44034	0.720170	−	0.693798i	$-0.244064\pi$
$503$	8.17306e6	1.44034	0.720170	+	0.693798i	$0.244064\pi$
$504$	0	0
$505$	−5.10071e6	−0.890025
$506$	0	0
$507$	−257049.	−0.0444116
$508$	0	0
$509$	−5.84045e6	−0.999198	−0.499599	−	0.866257i	$-0.666519\pi$
$509$	−5.84045e6	−0.999198	−0.499599	+	0.866257i	$0.666519\pi$
$510$	0	0
$511$	4.09151e6	0.693157
$512$	0	0
$513$	818316.	0.137286
$514$	0	0
$515$	846640.	0.140663
$516$	0	0
$517$	5.59738e6	0.920997
$518$	0	0
$519$	3.71775e6	0.605845
$520$	0	0
$521$	4.06221e6	0.655644	0.327822	−	0.944740i	$-0.393685\pi$
$521$	4.06221e6	0.655644	0.327822	+	0.944740i	$0.393685\pi$
$522$	0	0
$523$	−2.72472e6	−0.435580	−0.217790	−	0.975996i	$-0.569885\pi$
$523$	−2.72472e6	−0.435580	−0.217790	+	0.975996i	$0.569885\pi$
$524$	0	0
$525$	2.37524e6	0.376105
$526$	0	0
$527$	982093.	0.154037
$528$	0	0
$529$	3.80197e6	0.590703
$530$	0	0
$531$	1.83877e6	0.283003
$532$	0	0
$533$	574630.	0.0876133
$534$	0	0
$535$	4.97528e6	0.751507
$536$	0	0
$537$	−7778.08	−0.00116396
$538$	0	0
$539$	1.43363e6	0.212551
$540$	0	0
$541$	1.11651e7	1.64010	0.820050	−	0.572291i	$-0.193945\pi$
$541$	1.11651e7	1.64010	0.820050	+	0.572291i	$0.193945\pi$
$542$	0	0
$543$	−3.01185e6	−0.438362
$544$	0	0
$545$	−3.77178e6	−0.543945
$546$	0	0
$547$	−8.72639e6	−1.24700	−0.623500	−	0.781824i	$-0.714290\pi$
$547$	−8.72639e6	−1.24700	−0.623500	+	0.781824i	$0.714290\pi$
$548$	0	0
$549$	3.35361e6	0.474878
$550$	0	0
$551$	169105.	0.0237289
$552$	0	0
$553$	−3.04585e6	−0.423542
$554$	0	0
$555$	−2.68007e6	−0.369329
$556$	0	0
$557$	−4.63674e6	−0.633250	−0.316625	−	0.948551i	$-0.602550\pi$
$557$	−4.63674e6	−0.633250	−0.316625	+	0.948551i	$0.602550\pi$
$558$	0	0
$559$	−2.16603e6	−0.293181
$560$	0	0
$561$	2.47869e6	0.332518
$562$	0	0
$563$	9.08393e6	1.20782	0.603911	−	0.797052i	$-0.293608\pi$
$563$	9.08393e6	1.20782	0.603911	+	0.797052i	$0.293608\pi$
$564$	0	0
$565$	−3.30399e6	−0.435429
$566$	0	0
$567$	712330.	0.0930515
$568$	0	0
$569$	−4.73177e6	−0.612694	−0.306347	−	0.951920i	$-0.599107\pi$
$569$	−4.73177e6	−0.612694	−0.306347	+	0.951920i	$0.599107\pi$
$570$	0	0
$571$	1.38030e7	1.77167	0.885834	−	0.464002i	$-0.153587\pi$
$571$	1.38030e7	1.77167	0.885834	+	0.464002i	$0.153587\pi$
$572$	0	0
$573$	7.30428e6	0.929375
$574$	0	0
$575$	7.77799e6	0.981065
$576$	0	0
$577$	1.20728e7	1.50962	0.754811	−	0.655942i	$-0.227728\pi$
$577$	1.20728e7	1.50962	0.754811	+	0.655942i	$0.227728\pi$
$578$	0	0
$579$	265157.	0.0328705
$580$	0	0
$581$	3.31371e6	0.407263
$582$	0	0
$583$	2.77251e6	0.337833
$584$	0	0
$585$	360667.	0.0435730
$586$	0	0
$587$	1.57937e7	1.89186	0.945928	−	0.324378i	$-0.105155\pi$
$587$	1.57937e7	1.89186	0.945928	+	0.324378i	$0.105155\pi$
$588$	0	0
$589$	−1.14325e6	−0.135786
$590$	0	0
$591$	3.01062e6	0.354558
$592$	0	0
$593$	−1.30721e7	−1.52654	−0.763269	−	0.646080i	$-0.776407\pi$
$593$	−1.30721e7	−1.52654	−0.763269	+	0.646080i	$0.776407\pi$
$594$	0	0
$595$	2.75835e6	0.319417
$596$	0	0
$597$	−4.26398e6	−0.489643
$598$	0	0
$599$	−3.75229e6	−0.427296	−0.213648	−	0.976911i	$-0.568535\pi$
$599$	−3.75229e6	−0.427296	−0.213648	+	0.976911i	$0.568535\pi$
$600$	0	0
$601$	−1.07459e7	−1.21355	−0.606775	−	0.794873i	$-0.707537\pi$
$601$	−1.07459e7	−1.21355	−0.606775	+	0.794873i	$0.707537\pi$
$602$	0	0
$603$	1.24384e6	0.139306
$604$	0	0
$605$	−2.09399e6	−0.232588
$606$	0	0
$607$	1.30469e7	1.43726	0.718628	−	0.695394i	$-0.244770\pi$
$607$	1.30469e7	1.43726	0.718628	+	0.695394i	$0.244770\pi$
$608$	0	0
$609$	147203.	0.0160832
$610$	0	0
$611$	−3.31203e6	−0.358915
$612$	0	0
$613$	699071.	0.0751398	0.0375699	−	0.999294i	$-0.488038\pi$
$613$	699071.	0.0751398	0.0375699	+	0.999294i	$0.488038\pi$
$614$	0	0
$615$	−806267.	−0.0859589
$616$	0	0
$617$	−1.41603e7	−1.49748	−0.748738	−	0.662866i	$-0.769340\pi$
$617$	−1.41603e7	−1.49748	−0.748738	+	0.662866i	$0.769340\pi$
$618$	0	0
$619$	877757.	0.0920763	0.0460381	−	0.998940i	$-0.485340\pi$
$619$	877757.	0.0920763	0.0460381	+	0.998940i	$0.485340\pi$
$620$	0	0
$621$	2.33261e6	0.242724
$622$	0	0
$623$	722888.	0.0746192
$624$	0	0
$625$	3.73960e6	0.382935
$626$	0	0
$627$	−2.88544e6	−0.293119
$628$	0	0
$629$	1.08986e7	1.09836
$630$	0	0
$631$	−1.48609e7	−1.48584	−0.742921	−	0.669379i	$-0.766560\pi$
$631$	−1.48609e7	−1.48584	−0.742921	+	0.669379i	$0.766560\pi$
$632$	0	0
$633$	−1.61780e6	−0.160478
$634$	0	0
$635$	−2.54248e6	−0.250221
$636$	0	0
$637$	−848293.	−0.0828319
$638$	0	0
$639$	−2.42003e6	−0.234460
$640$	0	0
$641$	−1.93523e7	−1.86032	−0.930160	−	0.367155i	$-0.880332\pi$
$641$	−1.93523e7	−1.86032	−0.930160	+	0.367155i	$0.880332\pi$
$642$	0	0
$643$	−4.09314e6	−0.390418	−0.195209	−	0.980762i	$-0.562538\pi$
$643$	−4.09314e6	−0.390418	−0.195209	+	0.980762i	$0.562538\pi$
$644$	0	0
$645$	3.03918e6	0.287645
$646$	0	0
$647$	−3.02321e6	−0.283928	−0.141964	−	0.989872i	$-0.545342\pi$
$647$	−3.02321e6	−0.283928	−0.141964	+	0.989872i	$0.545342\pi$
$648$	0	0
$649$	−6.48365e6	−0.604238
$650$	0	0
$651$	−995182.	−0.0920343
$652$	0	0
$653$	−1.67741e7	−1.53942	−0.769709	−	0.638395i	$-0.779599\pi$
$653$	−1.67741e7	−1.53942	−0.769709	+	0.638395i	$0.779599\pi$
$654$	0	0
$655$	−263794.	−0.0240249
$656$	0	0
$657$	3.05251e6	0.275895
$658$	0	0
$659$	−1.86700e6	−0.167468	−0.0837340	−	0.996488i	$-0.526685\pi$
$659$	−1.86700e6	−0.167468	−0.0837340	+	0.996488i	$0.526685\pi$
$660$	0	0
$661$	4.11497e6	0.366322	0.183161	−	0.983083i	$-0.441367\pi$
$661$	4.11497e6	0.366322	0.183161	+	0.983083i	$0.441367\pi$
$662$	0	0
$663$	−1.46667e6	−0.129583
$664$	0	0
$665$	−3.21099e6	−0.281569
$666$	0	0
$667$	482033.	0.0419529
$668$	0	0
$669$	8.33857e6	0.720321
$670$	0	0
$671$	−1.18251e7	−1.01391
$672$	0	0
$673$	1.07014e7	0.910755	0.455377	−	0.890298i	$-0.349504\pi$
$673$	1.07014e7	0.910755	0.455377	+	0.890298i	$0.349504\pi$
$674$	0	0
$675$	1.77207e6	0.149700
$676$	0	0
$677$	−3.73378e6	−0.313096	−0.156548	−	0.987670i	$-0.550037\pi$
$677$	−3.73378e6	−0.313096	−0.156548	+	0.987670i	$0.550037\pi$
$678$	0	0
$679$	−7.44869e6	−0.620020
$680$	0	0
$681$	5.25307e6	0.434056
$682$	0	0
$683$	−2.33008e7	−1.91126	−0.955628	−	0.294577i	$-0.904821\pi$
$683$	−2.33008e7	−1.91126	−0.955628	+	0.294577i	$0.904821\pi$
$684$	0	0
$685$	−4.98562e6	−0.405969
$686$	0	0
$687$	3.36378e6	0.271917
$688$	0	0
$689$	−1.64053e6	−0.131654
$690$	0	0
$691$	3.66731e6	0.292181	0.146091	−	0.989271i	$-0.453331\pi$
$691$	3.66731e6	0.292181	0.146091	+	0.989271i	$0.453331\pi$
$692$	0	0
$693$	−2.51173e6	−0.198673
$694$	0	0
$695$	−1.67501e6	−0.131539
$696$	0	0
$697$	3.27872e6	0.255637
$698$	0	0
$699$	1.25719e7	0.973215
$700$	0	0
$701$	6.26713e6	0.481697	0.240848	−	0.970563i	$-0.422574\pi$
$701$	6.26713e6	0.481697	0.240848	+	0.970563i	$0.422574\pi$
$702$	0	0
$703$	−1.26871e7	−0.968219
$704$	0	0
$705$	4.64714e6	0.352138
$706$	0	0
$707$	−2.10187e7	−1.58146
$708$	0	0
$709$	1.67388e7	1.25057	0.625284	−	0.780397i	$-0.284983\pi$
$709$	1.67388e7	1.25057	0.625284	+	0.780397i	$0.284983\pi$
$710$	0	0
$711$	−2.27239e6	−0.168581
$712$	0	0
$713$	−3.25884e6	−0.240071
$714$	0	0
$715$	−1.27174e6	−0.0930322
$716$	0	0
$717$	−5.26074e6	−0.382163
$718$	0	0
$719$	1.45447e6	0.104926	0.0524631	−	0.998623i	$-0.483293\pi$
$719$	1.45447e6	0.104926	0.0524631	+	0.998623i	$0.483293\pi$
$720$	0	0
$721$	3.48879e6	0.249941
$722$	0	0
$723$	−1.01029e7	−0.718790
$724$	0	0
$725$	366198.	0.0258744
$726$	0	0
$727$	2.27668e7	1.59759	0.798795	−	0.601603i	$-0.205471\pi$
$727$	2.27668e7	1.59759	0.798795	+	0.601603i	$0.205471\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−1.23590e7	−0.855438
$732$	0	0
$733$	−5.66897e6	−0.389712	−0.194856	−	0.980832i	$-0.562424\pi$
$733$	−5.66897e6	−0.389712	−0.194856	+	0.980832i	$0.562424\pi$
$734$	0	0
$735$	1.19025e6	0.0812678
$736$	0	0
$737$	−4.38586e6	−0.297431
$738$	0	0
$739$	−1.65419e7	−1.11423	−0.557113	−	0.830437i	$-0.688091\pi$
$739$	−1.65419e7	−1.11423	−0.557113	+	0.830437i	$0.688091\pi$
$740$	0	0
$741$	1.70735e6	0.114229
$742$	0	0
$743$	2.51152e7	1.66903	0.834514	−	0.550986i	$-0.185748\pi$
$743$	2.51152e7	1.66903	0.834514	+	0.550986i	$0.185748\pi$
$744$	0	0
$745$	−63580.9	−0.00419697
$746$	0	0
$747$	2.47223e6	0.162102
$748$	0	0
$749$	2.05019e7	1.33533
$750$	0	0
$751$	1.02904e7	0.665784	0.332892	−	0.942965i	$-0.391976\pi$
$751$	1.02904e7	0.665784	0.332892	+	0.942965i	$0.391976\pi$
$752$	0	0
$753$	6.54544e6	0.420679
$754$	0	0
$755$	−1.19152e7	−0.760734
$756$	0	0
$757$	1.14246e7	0.724606	0.362303	−	0.932060i	$-0.381991\pi$
$757$	1.14246e7	0.724606	0.362303	+	0.932060i	$0.381991\pi$
$758$	0	0
$759$	−8.22495e6	−0.518237
$760$	0	0
$761$	−1.12573e7	−0.704646	−0.352323	−	0.935878i	$-0.614608\pi$
$761$	−1.12573e7	−0.704646	−0.352323	+	0.935878i	$0.614608\pi$
$762$	0	0
$763$	−1.55426e7	−0.966521
$764$	0	0
$765$	2.05790e6	0.127136
$766$	0	0
$767$	3.83645e6	0.235473
$768$	0	0
$769$	−2.63365e7	−1.60599	−0.802994	−	0.595987i	$-0.796761\pi$
$769$	−2.63365e7	−1.60599	−0.802994	+	0.595987i	$0.796761\pi$
$770$	0	0
$771$	−5.85419e6	−0.354675
$772$	0	0
$773$	2.33227e7	1.40388	0.701941	−	0.712235i	$-0.252317\pi$
$773$	2.33227e7	1.40388	0.701941	+	0.712235i	$0.252317\pi$
$774$	0	0
$775$	−2.47572e6	−0.148063
$776$	0	0
$777$	−1.10439e7	−0.656251
$778$	0	0
$779$	−3.81676e6	−0.225347
$780$	0	0
$781$	8.53321e6	0.500593
$782$	0	0
$783$	109822.	0.00640156
$784$	0	0
$785$	−5.14820e6	−0.298182
$786$	0	0
$787$	5.48266e6	0.315540	0.157770	−	0.987476i	$-0.449570\pi$
$787$	5.48266e6	0.315540	0.157770	+	0.987476i	$0.449570\pi$
$788$	0	0
$789$	8.81898e6	0.504343
$790$	0	0
$791$	−1.36149e7	−0.773701
$792$	0	0
$793$	6.99704e6	0.395122
$794$	0	0
$795$	2.30184e6	0.129168
$796$	0	0
$797$	−1.67574e7	−0.934462	−0.467231	−	0.884135i	$-0.654748\pi$
$797$	−1.67574e7	−0.934462	−0.467231	+	0.884135i	$0.654748\pi$
$798$	0	0
$799$	−1.88978e7	−1.04724
$800$	0	0
$801$	539318.	0.0297005
$802$	0	0
$803$	−1.07634e7	−0.589061
$804$	0	0
$805$	−9.15293e6	−0.497818
$806$	0	0
$807$	2.18516e6	0.118114
$808$	0	0
$809$	−3.29903e7	−1.77221	−0.886104	−	0.463487i	$-0.846598\pi$
$809$	−3.29903e7	−1.77221	−0.886104	+	0.463487i	$0.846598\pi$
$810$	0	0
$811$	−1.12019e7	−0.598052	−0.299026	−	0.954245i	$-0.596662\pi$
$811$	−1.12019e7	−0.598052	−0.299026	+	0.954245i	$0.596662\pi$
$812$	0	0
$813$	−8.25958e6	−0.438260
$814$	0	0
$815$	−8.84841e6	−0.466629
$816$	0	0
$817$	1.43871e7	0.754079
$818$	0	0
$819$	1.48622e6	0.0774235
$820$	0	0
$821$	−7.15858e6	−0.370654	−0.185327	−	0.982677i	$-0.559334\pi$
$821$	−7.15858e6	−0.370654	−0.185327	+	0.982677i	$0.559334\pi$
$822$	0	0
$823$	1.87301e7	0.963918	0.481959	−	0.876194i	$-0.339925\pi$
$823$	1.87301e7	0.963918	0.481959	+	0.876194i	$0.339925\pi$
$824$	0	0
$825$	−6.24845e6	−0.319623
$826$	0	0
$827$	2.33449e7	1.18694	0.593468	−	0.804857i	$-0.297758\pi$
$827$	2.33449e7	1.18694	0.593468	+	0.804857i	$0.297758\pi$
$828$	0	0
$829$	1.66953e7	0.843740	0.421870	−	0.906656i	$-0.361374\pi$
$829$	1.66953e7	0.843740	0.421870	+	0.906656i	$0.361374\pi$
$830$	0	0
$831$	1.53703e7	0.772113
$832$	0	0
$833$	−4.84020e6	−0.241686
$834$	0	0
$835$	7.01629e6	0.348250
$836$	0	0
$837$	−742466.	−0.0366322
$838$	0	0
$839$	−2.95646e7	−1.45000	−0.724998	−	0.688751i	$-0.758159\pi$
$839$	−2.95646e7	−1.45000	−0.724998	+	0.688751i	$0.758159\pi$
$840$	0	0
$841$	−2.04885e7	−0.998894
$842$	0	0
$843$	706120.	0.0342223
$844$	0	0
$845$	752503.	0.0362549
$846$	0	0
$847$	−8.62882e6	−0.413278
$848$	0	0
$849$	−1.01255e7	−0.482110
$850$	0	0
$851$	−3.61645e7	−1.71182
$852$	0	0
$853$	1.25512e7	0.590626	0.295313	−	0.955401i	$-0.404576\pi$
$853$	1.25512e7	0.590626	0.295313	+	0.955401i	$0.404576\pi$
$854$	0	0
$855$	−2.39559e6	−0.112072
$856$	0	0
$857$	−2.22431e7	−1.03453	−0.517266	−	0.855825i	$-0.673050\pi$
$857$	−2.22431e7	−1.03453	−0.517266	+	0.855825i	$0.673050\pi$
$858$	0	0
$859$	1.58384e7	0.732368	0.366184	−	0.930542i	$-0.380664\pi$
$859$	1.58384e7	0.732368	0.366184	+	0.930542i	$0.380664\pi$
$860$	0	0
$861$	−3.32242e6	−0.152738
$862$	0	0
$863$	−3.55005e6	−0.162259	−0.0811293	−	0.996704i	$-0.525853\pi$
$863$	−3.55005e6	−0.162259	−0.0811293	+	0.996704i	$0.525853\pi$
$864$	0	0
$865$	−1.08836e7	−0.494575
$866$	0	0
$867$	4.41017e6	0.199254
$868$	0	0
$869$	8.01262e6	0.359936
$870$	0	0
$871$	2.59516e6	0.115910
$872$	0	0
$873$	−5.55717e6	−0.246785
$874$	0	0
$875$	−1.58926e7	−0.701737
$876$	0	0
$877$	−2.90118e7	−1.27372	−0.636861	−	0.770978i	$-0.719768\pi$
$877$	−2.90118e7	−1.27372	−0.636861	+	0.770978i	$0.719768\pi$
$878$	0	0
$879$	1.04077e7	0.454340
$880$	0	0
$881$	−1.87411e7	−0.813494	−0.406747	−	0.913541i	$-0.633337\pi$
$881$	−1.87411e7	−0.813494	−0.406747	+	0.913541i	$0.633337\pi$
$882$	0	0
$883$	−907252.	−0.0391585	−0.0195793	−	0.999808i	$-0.506233\pi$
$883$	−907252.	−0.0391585	−0.0195793	+	0.999808i	$0.506233\pi$
$884$	0	0
$885$	−5.38295e6	−0.231027
$886$	0	0
$887$	−4.40390e6	−0.187944	−0.0939720	−	0.995575i	$-0.529956\pi$
$887$	−4.40390e6	−0.187944	−0.0939720	+	0.995575i	$0.529956\pi$
$888$	0	0
$889$	−1.04769e7	−0.444611
$890$	0	0
$891$	−1.87390e6	−0.0790774
$892$	0	0
$893$	2.19989e7	0.923151
$894$	0	0
$895$	22770.1	0.000950183 0
$896$	0	0
$897$	4.86680e6	0.201959
$898$	0	0
$899$	−153430.	−0.00633158
$900$	0	0
$901$	−9.36053e6	−0.384139
$902$	0	0
$903$	1.25237e7	0.511108
$904$	0	0
$905$	8.81709e6	0.357852
$906$	0	0
$907$	4.22081e7	1.70364	0.851820	−	0.523835i	$-0.175499\pi$
$907$	4.22081e7	1.70364	0.851820	+	0.523835i	$0.175499\pi$
$908$	0	0
$909$	−1.56813e7	−0.629464
$910$	0	0
$911$	−1.78418e7	−0.712266	−0.356133	−	0.934435i	$-0.615905\pi$
$911$	−1.78418e7	−0.712266	−0.356133	+	0.934435i	$0.615905\pi$
$912$	0	0
$913$	−8.71726e6	−0.346101
$914$	0	0
$915$	−9.81760e6	−0.387661
$916$	0	0
$917$	−1.08703e6	−0.0426892
$918$	0	0
$919$	−2.46819e6	−0.0964030	−0.0482015	−	0.998838i	$-0.515349\pi$
$919$	−2.46819e6	−0.0964030	−0.0482015	+	0.998838i	$0.515349\pi$
$920$	0	0
$921$	−803782.	−0.0312240
$922$	0	0
$923$	−5.04920e6	−0.195083
$924$	0	0
$925$	−2.74740e7	−1.05577
$926$	0	0
$927$	2.60285e6	0.0994832
$928$	0	0
$929$	3.76239e7	1.43029	0.715146	−	0.698975i	$-0.246360\pi$
$929$	3.76239e7	1.43029	0.715146	+	0.698975i	$0.246360\pi$
$930$	0	0
$931$	5.63447e6	0.213049
$932$	0	0
$933$	2.10794e6	0.0792782
$934$	0	0
$935$	−7.25630e6	−0.271448
$936$	0	0
$937$	−2.72154e7	−1.01266	−0.506332	−	0.862339i	$-0.668999\pi$
$937$	−2.72154e7	−1.01266	−0.506332	+	0.862339i	$0.668999\pi$
$938$	0	0
$939$	4.32215e6	0.159969
$940$	0	0
$941$	5.51883e6	0.203176	0.101588	−	0.994827i	$-0.467608\pi$
$941$	5.51883e6	0.203176	0.101588	+	0.994827i	$0.467608\pi$
$942$	0	0
$943$	−1.08797e7	−0.398415
$944$	0	0
$945$	−2.08532e6	−0.0759616
$946$	0	0
$947$	4.03933e7	1.46364	0.731819	−	0.681499i	$-0.238671\pi$
$947$	4.03933e7	1.46364	0.731819	+	0.681499i	$0.238671\pi$
$948$	0	0
$949$	6.36883e6	0.229559
$950$	0	0
$951$	−2.44615e7	−0.877065
$952$	0	0
$953$	−3.42332e6	−0.122100	−0.0610499	−	0.998135i	$-0.519445\pi$
$953$	−3.42332e6	−0.122100	−0.0610499	+	0.998135i	$0.519445\pi$
$954$	0	0
$955$	−2.13831e7	−0.758685
$956$	0	0
$957$	−387241.	−0.0136679
$958$	0	0
$959$	−2.05445e7	−0.721354
$960$	0	0
$961$	−2.75919e7	−0.963768
$962$	0	0
$963$	1.52956e7	0.531498
$964$	0	0
$965$	−776239.	−0.0268335
$966$	0	0
$967$	1.18244e7	0.406642	0.203321	−	0.979112i	$-0.434827\pi$
$967$	1.18244e7	0.406642	0.203321	+	0.979112i	$0.434827\pi$
$968$	0	0
$969$	9.74181e6	0.333296
$970$	0	0
$971$	3.02651e6	0.103013	0.0515067	−	0.998673i	$-0.483598\pi$
$971$	3.02651e6	0.103013	0.0515067	+	0.998673i	$0.483598\pi$
$972$	0	0
$973$	−6.90230e6	−0.233729
$974$	0	0
$975$	3.69728e6	0.124558
$976$	0	0
$977$	−7.03356e6	−0.235743	−0.117871	−	0.993029i	$-0.537607\pi$
$977$	−7.03356e6	−0.235743	−0.117871	+	0.993029i	$0.537607\pi$
$978$	0	0
$979$	−1.90168e6	−0.0634132
$980$	0	0
$981$	−1.15957e7	−0.384702
$982$	0	0
$983$	2.26337e7	0.747086	0.373543	−	0.927613i	$-0.378143\pi$
$983$	2.26337e7	0.747086	0.373543	+	0.927613i	$0.378143\pi$
$984$	0	0
$985$	−8.81351e6	−0.289440
$986$	0	0
$987$	1.91497e7	0.625704
$988$	0	0
$989$	4.10103e7	1.33322
$990$	0	0
$991$	−5.45429e6	−0.176423	−0.0882113	−	0.996102i	$-0.528115\pi$
$991$	−5.45429e6	−0.176423	−0.0882113	+	0.996102i	$0.528115\pi$
$992$	0	0
$993$	−2.69909e7	−0.868649
$994$	0	0
$995$	1.24827e7	0.399715
$996$	0	0
$997$	−6.16465e6	−0.196413	−0.0982066	−	0.995166i	$-0.531311\pi$
$997$	−6.16465e6	−0.196413	−0.0982066	+	0.995166i	$0.531311\pi$
$998$	0	0
$999$	−8.23942e6	−0.261206

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.6.a.r.1.2		3
4.3	odd	2		156.6.a.d.1.2	✓	3
12.11	even	2		468.6.a.f.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.6.a.d.1.2	✓	3	4.3	odd	2
468.6.a.f.1.2		3	12.11	even	2
624.6.a.r.1.2		3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	624.a (trivial)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} +26.3472 q^{5} +108.570 q^{7} +81.0000 q^{9} -285.612 q^{11} +169.000 q^{13} -237.125 q^{15} +964.281 q^{17} -1122.52 q^{19} -977.133 q^{21} -3199.74 q^{23} -2430.82 q^{25} -729.000 q^{27} -150.648 q^{29} +1018.47 q^{31} +2570.51 q^{33} +2860.53 q^{35} +11302.4 q^{37} -1521.00 q^{39} +3400.17 q^{41} -12816.8 q^{43} +2134.13 q^{45} -19597.8 q^{47} -5019.49 q^{49} -8678.53 q^{51} -9707.26 q^{53} -7525.09 q^{55} +10102.7 q^{57} +22700.9 q^{59} +41402.6 q^{61} +8794.20 q^{63} +4452.68 q^{65} +15356.0 q^{67} +28797.6 q^{69} -29876.9 q^{71} +37685.4 q^{73} +21877.4 q^{75} -31009.0 q^{77} -28054.2 q^{79} +6561.00 q^{81} +30521.3 q^{83} +25406.1 q^{85} +1355.83 q^{87} +6658.25 q^{89} +18348.4 q^{91} -9166.24 q^{93} -29575.2 q^{95} -68607.1 q^{97} -23134.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 27 q^{3} + 4 q^{5} - 242 q^{7} + 243 q^{9} - 64 q^{11} + 507 q^{13} - 36 q^{15} + 622 q^{17} - 2330 q^{19} + 2178 q^{21} - 816 q^{23} + 6313 q^{25} - 2187 q^{27} + 6726 q^{29} - 5010 q^{31} + 576 q^{33}+ \cdots - 5184 q^{99}+O(q^{100}) \) 3 * q - 27 * q^3 + 4 * q^5 - 242 * q^7 + 243 * q^9 - 64 * q^11 + 507 * q^13 - 36 * q^15 + 622 * q^17 - 2330 * q^19 + 2178 * q^21 - 816 * q^23 + 6313 * q^25 - 2187 * q^27 + 6726 * q^29 - 5010 * q^31 + 576 * q^33 + 1136 * q^35 + 10670 * q^37 - 4563 * q^39 + 17584 * q^41 + 2092 * q^43 + 324 * q^45 - 4936 * q^47 + 24975 * q^49 - 5598 * q^51 + 39010 * q^53 - 48592 * q^55 + 20970 * q^57 - 37436 * q^59 + 46862 * q^61 - 19602 * q^63 + 676 * q^65 - 88734 * q^67 + 7344 * q^69 + 20476 * q^71 - 3246 * q^73 - 56817 * q^75 - 55088 * q^77 - 3000 * q^79 + 19683 * q^81 + 97956 * q^83 + 6664 * q^85 - 60534 * q^87 - 183380 * q^89 - 40898 * q^91 + 45090 * q^93 + 365984 * q^95 - 59254 * q^97 - 5184 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more