LMFDB - Embedded newform 336.6.q.g.289.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,6,Mod(193,336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))

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

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

chi := DirichletCharacter("336.193");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	336.q (of order $3$, degree $2$, not minimal)

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:	no
Analytic conductor:	$53.8889634572$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{7081})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 1771x^{2} + 1770x + 3132900 $ x^4 - x^3 + 1771x^2 + 1770x + 3132900
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			289.2
Root			$21.2872 - 36.8705i$ of defining polynomial
Character	$\chi$	$=$	336.289
Dual form			336.6.q.g.193.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.50000 + 7.79423i) q^{3} +(9.28717 - 16.0858i) q^{5} +(127.649 - 22.6454i) q^{7} +(-40.5000 + 70.1481i) q^{9} +O(q^{10})$	\(q+(4.50000 + 7.79423i) q^{3} +(9.28717 - 16.0858i) q^{5} +(127.649 - 22.6454i) q^{7} +(-40.5000 + 70.1481i) q^{9} +(-80.7128 - 139.799i) q^{11} +14.1283 q^{13} +167.169 q^{15} +(382.851 + 663.118i) q^{17} +(707.146 - 1224.81i) q^{19} +(750.923 + 893.019i) q^{21} +(2092.72 - 3624.69i) q^{23} +(1390.00 + 2407.55i) q^{25} -729.000 q^{27} -4202.60 q^{29} +(-1193.68 - 2067.51i) q^{31} +(726.415 - 1258.19i) q^{33} +(821.224 - 2263.65i) q^{35} +(336.469 - 582.782i) q^{37} +(63.5774 + 110.119i) q^{39} -4173.45 q^{41} +5430.94 q^{43} +(752.261 + 1302.95i) q^{45} +(3151.34 - 5458.28i) q^{47} +(15781.4 - 5781.32i) q^{49} +(-3445.66 + 5968.06i) q^{51} +(-8208.34 - 14217.3i) q^{53} -2998.37 q^{55} +12728.6 q^{57} +(1983.24 + 3435.08i) q^{59} +(25169.4 - 43594.6i) q^{61} +(-3581.24 + 9871.45i) q^{63} +(131.212 - 227.266i) q^{65} +(6822.63 + 11817.1i) q^{67} +37668.9 q^{69} +83957.2 q^{71} +(14289.2 + 24749.6i) q^{73} +(-12510.0 + 21667.9i) q^{75} +(-13468.7 - 16017.3i) q^{77} +(-29977.7 + 51923.0i) q^{79} +(-3280.50 - 5681.99i) q^{81} +61583.0 q^{83} +14222.4 q^{85} +(-18911.7 - 32756.0i) q^{87} +(-21149.2 + 36631.4i) q^{89} +(1803.46 - 319.941i) q^{91} +(10743.1 - 18607.6i) q^{93} +(-13134.8 - 22750.1i) q^{95} +44638.1 q^{97} +13075.5 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 18 q^{3} - 47 q^{5} + 174 q^{7} - 162 q^{9}+O(q^{10}) $ 4 * q + 18 * q^3 - 47 * q^5 + 174 * q^7 - 162 * q^9	$ 4 q + 18 q^{3} - 47 q^{5} + 174 q^{7} - 162 q^{9} - 407 q^{11} + 898 q^{13} - 846 q^{15} + 1868 q^{17} - 1463 q^{19} - 783 q^{21} - 44 q^{23} + 1605 q^{25} - 2916 q^{27} + 1534 q^{29} - 11170 q^{31} + 3663 q^{33} - 9674 q^{35} + 3113 q^{37} + 4041 q^{39} - 15684 q^{41} + 25258 q^{43} - 3807 q^{45} + 9576 q^{47} + 4558 q^{49} - 16812 q^{51} - 13395 q^{53} + 26210 q^{55} - 26334 q^{57} - 47521 q^{59} + 63652 q^{61} - 21141 q^{63} - 28254 q^{65} + 44541 q^{67} - 792 q^{69} + 251680 q^{71} - 6039 q^{73} - 14445 q^{75} + 35407 q^{77} + 17588 q^{79} - 13122 q^{81} - 78650 q^{83} - 116120 q^{85} + 6903 q^{87} - 83082 q^{89} - 31747 q^{91} + 100530 q^{93} - 214946 q^{95} + 369570 q^{97} + 65934 q^{99}+O(q^{100}) $ 4 * q + 18 * q^3 - 47 * q^5 + 174 * q^7 - 162 * q^9 - 407 * q^11 + 898 * q^13 - 846 * q^15 + 1868 * q^17 - 1463 * q^19 - 783 * q^21 - 44 * q^23 + 1605 * q^25 - 2916 * q^27 + 1534 * q^29 - 11170 * q^31 + 3663 * q^33 - 9674 * q^35 + 3113 * q^37 + 4041 * q^39 - 15684 * q^41 + 25258 * q^43 - 3807 * q^45 + 9576 * q^47 + 4558 * q^49 - 16812 * q^51 - 13395 * q^53 + 26210 * q^55 - 26334 * q^57 - 47521 * q^59 + 63652 * q^61 - 21141 * q^63 - 28254 * q^65 + 44541 * q^67 - 792 * q^69 + 251680 * q^71 - 6039 * q^73 - 14445 * q^75 + 35407 * q^77 + 17588 * q^79 - 13122 * q^81 - 78650 * q^83 - 116120 * q^85 + 6903 * q^87 - 83082 * q^89 - 31747 * q^91 + 100530 * q^93 - 214946 * q^95 + 369570 * q^97 + 65934 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$.

$n$	$85$	$113$	$127$	$241$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$3$	4.50000	+	7.79423i	0.288675	+	0.500000i
$3$	4.50000	+	7.79423i	0.288675	+	0.500000i
$4$		0			0
$5$	9.28717	−	16.0858i	0.166134	−	0.287752i	−0.770923	−	0.636928i	$-0.780205\pi$
$5$	9.28717	−	16.0858i	0.166134	−	0.287752i	0.937057	+	0.349175i	$0.113538\pi$
$6$		0			0
$7$	127.649	−	22.6454i	0.984626	−	0.174677i
$7$	127.649	−	22.6454i	0.984626	−	0.174677i
$8$		0			0
$9$	−40.5000	+	70.1481i	−0.166667	+	0.288675i
$10$		0			0
$11$	−80.7128	−	139.799i	−0.201123	−	0.348355i	0.747768	−	0.663960i	$-0.231126\pi$
$11$	−80.7128	−	139.799i	−0.201123	−	0.348355i	−0.948890	+	0.315606i	$0.897792\pi$
$12$		0			0
$13$	14.1283			0.0231863			0.0115932	−	0.999933i	$-0.496310\pi$
$13$	14.1283			0.0231863			0.0115932	+	0.999933i	$0.496310\pi$
$14$		0			0
$15$	167.169			0.191835
$16$		0			0
$17$	382.851	+	663.118i	0.321298	+	0.556504i	0.980756	−	0.195237i	$-0.0625476\pi$
$17$	382.851	+	663.118i	0.321298	+	0.556504i	−0.659458	+	0.751741i	$0.729214\pi$
$18$		0			0
$19$	707.146	−	1224.81i	0.449392	−	0.778369i	−0.548955	−	0.835852i	$-0.684974\pi$
$19$	707.146	−	1224.81i	0.449392	−	0.778369i	0.998346	+	0.0574830i	$0.0183075\pi$
$20$		0			0
$21$	750.923	+	893.019i	0.371575	+	0.441888i
$22$		0			0
$23$	2092.72	−	3624.69i	0.824880	−	1.42873i	−0.0771305	−	0.997021i	$-0.524576\pi$
$23$	2092.72	−	3624.69i	0.824880	−	1.42873i	0.902011	−	0.431713i	$-0.142091\pi$
$24$		0			0
$25$	1390.00	+	2407.55i	0.444799	+	0.770415i
$26$		0			0
$27$	−729.000			−0.192450
$28$		0			0
$29$	−4202.60			−0.927947			−0.463974	−	0.885849i	$-0.653577\pi$
$29$	−4202.60			−0.927947			−0.463974	+	0.885849i	$0.653577\pi$
$30$		0			0
$31$	−1193.68	−	2067.51i	−0.223091	−	0.386405i	0.732654	−	0.680601i	$-0.238281\pi$
$31$	−1193.68	−	2067.51i	−0.223091	−	0.386405i	−0.955745	+	0.294196i	$0.904948\pi$
$32$		0			0
$33$	726.415	−	1258.19i	0.116118	−	0.201123i
$34$		0			0
$35$	821.224	−	2263.65i	0.113316	−	0.312348i
$36$		0			0
$37$	336.469	−	582.782i	0.0404056	−	0.0699845i	−0.845115	−	0.534584i	$-0.820468\pi$
$37$	336.469	−	582.782i	0.0404056	−	0.0699845i	0.885521	+	0.464599i	$0.153802\pi$
$38$		0			0
$39$	63.5774	+	110.119i	0.00669331	+	0.0115932i
$40$		0			0
$41$	−4173.45			−0.387735			−0.193868	−	0.981028i	$-0.562103\pi$
$41$	−4173.45			−0.387735			−0.193868	+	0.981028i	$0.562103\pi$
$42$		0			0
$43$	5430.94			0.447923			0.223962	−	0.974598i	$-0.428101\pi$
$43$	5430.94			0.447923			0.223962	+	0.974598i	$0.428101\pi$
$44$		0			0
$45$	752.261	+	1302.95i	0.0553780	+	0.0959175i
$46$		0			0
$47$	3151.34	−	5458.28i	0.208090	−	0.360422i	−0.743023	−	0.669266i	$-0.766609\pi$
$47$	3151.34	−	5458.28i	0.208090	−	0.360422i	0.951113	+	0.308844i	$0.0999421\pi$
$48$		0			0
$49$	15781.4	−	5781.32i	0.938976	−	0.343983i
$50$		0			0
$51$	−3445.66	+	5968.06i	−0.185501	+	0.321298i
$52$		0			0
$53$	−8208.34	−	14217.3i	−0.401389	−	0.695226i	0.592505	−	0.805567i	$-0.298139\pi$
$53$	−8208.34	−	14217.3i	−0.401389	−	0.695226i	−0.993894	+	0.110341i	$0.964806\pi$
$54$		0			0
$55$	−2998.37			−0.133653
$56$		0			0
$57$	12728.6			0.518913
$58$		0			0
$59$	1983.24	+	3435.08i	0.0741731	+	0.128472i	0.900726	−	0.434387i	$-0.143035\pi$
$59$	1983.24	+	3435.08i	0.0741731	+	0.128472i	−0.826553	+	0.562858i	$0.809702\pi$
$60$		0			0
$61$	25169.4	−	43594.6i	0.866059	−	1.50006i	6.72450e−5	−	1.00000i	$-0.499979\pi$
$61$	25169.4	−	43594.6i	0.866059	−	1.50006i	0.865992	−	0.500058i	$-0.166688\pi$
$62$		0			0
$63$	−3581.24	+	9871.45i	−0.113679	+	0.313350i
$64$		0			0
$65$	131.212	−	227.266i	0.00385203	−	0.00667192i
$66$		0			0
$67$	6822.63	+	11817.1i	0.185680	+	0.321607i	0.943805	−	0.330502i	$-0.107218\pi$
$67$	6822.63	+	11817.1i	0.185680	+	0.321607i	−0.758126	+	0.652109i	$0.773885\pi$
$68$		0			0
$69$	37668.9			0.952490
$70$		0			0
$71$	83957.2			1.97657			0.988284	−	0.152624i	$-0.0487723\pi$
$71$	83957.2			1.97657			0.988284	+	0.152624i	$0.0487723\pi$
$72$		0			0
$73$	14289.2	+	24749.6i	0.313834	+	0.543576i	0.979189	−	0.202951i	$-0.0650532\pi$
$73$	14289.2	+	24749.6i	0.313834	+	0.543576i	−0.665355	+	0.746527i	$0.731720\pi$
$74$		0			0
$75$	−12510.0	+	21667.9i	−0.256805	+	0.444799i
$76$		0			0
$77$	−13468.7	−	16017.3i	−0.258880	−	0.307867i
$78$		0			0
$79$	−29977.7	+	51923.0i	−0.540420	+	0.936034i	0.458460	+	0.888715i	$0.348401\pi$
$79$	−29977.7	+	51923.0i	−0.540420	+	0.936034i	−0.998880	+	0.0473193i	$0.984932\pi$
$80$		0			0
$81$	−3280.50	−	5681.99i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	61583.0			0.981219			0.490610	−	0.871380i	$-0.336774\pi$
$83$	61583.0			0.981219			0.490610	+	0.871380i	$0.336774\pi$
$84$		0			0
$85$	14222.4			0.213514
$86$		0			0
$87$	−18911.7	−	32756.0i	−0.267875	−	0.463974i
$88$		0			0
$89$	−21149.2	+	36631.4i	−0.283021	+	0.490206i	−0.972127	−	0.234454i	$-0.924670\pi$
$89$	−21149.2	+	36631.4i	−0.283021	+	0.490206i	0.689107	+	0.724660i	$0.258003\pi$
$90$		0			0
$91$	1803.46	−	319.941i	0.0228298	−	0.00405011i
$92$		0			0
$93$	10743.1	−	18607.6i	0.128802	−	0.223091i
$94$		0			0
$95$	−13134.8	−	22750.1i	−0.149318	−	0.258627i
$96$		0			0
$97$	44638.1			0.481700			0.240850	−	0.970562i	$-0.422574\pi$
$97$	44638.1			0.481700			0.240850	+	0.970562i	$0.422574\pi$
$98$		0			0
$99$	13075.5			0.134082
$100$		0			0
$101$	55962.2	+	96929.4i	0.545873	+	0.945480i	0.998551	+	0.0538058i	$0.0171352\pi$
$101$	55962.2	+	96929.4i	0.545873	+	0.945480i	−0.452678	+	0.891674i	$0.649531\pi$
$102$		0			0
$103$	83474.6	−	144582.i	0.775285	−	1.34283i	−0.159350	−	0.987222i	$-0.550940\pi$
$103$	83474.6	−	144582.i	0.775285	−	1.34283i	0.934634	−	0.355610i	$-0.115727\pi$
$104$		0			0
$105$	21338.9	−	3785.61i	0.188886	−	0.0335091i
$106$		0			0
$107$	51492.8	−	89188.2i	0.434798	−	0.753092i	−0.562481	−	0.826810i	$-0.690153\pi$
$107$	51492.8	−	89188.2i	0.434798	−	0.753092i	0.997279	+	0.0737182i	$0.0234865\pi$
$108$		0			0
$109$	68910.6	+	119357.i	0.555546	+	0.962233i	0.997861	+	0.0653736i	$0.0208239\pi$
$109$	68910.6	+	119357.i	0.555546	+	0.962233i	−0.442315	+	0.896860i	$0.645843\pi$
$110$		0			0
$111$	6056.45			0.0466563
$112$		0			0
$113$	−129967.			−0.957495			−0.478748	−	0.877953i	$-0.658909\pi$
$113$	−129967.			−0.957495			−0.478748	+	0.877953i	$0.658909\pi$
$114$		0			0
$115$	−38870.8	−	67326.3i	−0.274081	−	0.474723i
$116$		0			0
$117$	−572.196	+	991.073i	−0.00386439	+	0.00669331i
$118$		0			0
$119$	63887.0	+	75976.3i	0.413567	+	0.491825i
$120$		0			0
$121$	67496.4	−	116907.i	0.419099	−	0.725901i
$122$		0			0
$123$	−18780.5	−	32528.8i	−0.111929	−	0.193868i
$124$		0			0
$125$	109681.			0.627853
$126$		0			0
$127$	243265.			1.33835			0.669177	−	0.743103i	$-0.266647\pi$
$127$	243265.			1.33835			0.669177	+	0.743103i	$0.266647\pi$
$128$		0			0
$129$	24439.2	+	42330.0i	0.129304	+	0.223962i
$130$		0			0
$131$	97167.8	−	168300.i	0.494703	−	0.856850i	−0.505279	−	0.862956i	$-0.668610\pi$
$131$	97167.8	−	168300.i	0.494703	−	0.856850i	0.999981	+	0.00610601i	$0.00194361\pi$
$132$		0			0
$133$	62529.8	−	172359.i	0.306519	−	0.844900i
$134$		0			0
$135$	−6770.35	+	11726.6i	−0.0319725	+	0.0553780i
$136$		0			0
$137$	1546.85	+	2679.22i	0.00704121	+	0.0121957i	0.869525	−	0.493890i	$-0.164425\pi$
$137$	1546.85	+	2679.22i	0.00704121	+	0.0121957i	−0.862483	+	0.506085i	$0.831092\pi$
$138$		0			0
$139$	−22600.4			−0.0992155			−0.0496078	−	0.998769i	$-0.515797\pi$
$139$	−22600.4			−0.0992155			−0.0496078	+	0.998769i	$0.515797\pi$
$140$		0			0
$141$	56724.1			0.240281
$142$		0			0
$143$	−1140.34	−	1975.12i	−0.00466329	−	0.00807706i
$144$		0			0
$145$	−39030.3	+	67602.4i	−0.154164	+	0.267019i
$146$		0			0
$147$	116077.	+	96987.7i	0.443050	+	0.370189i
$148$		0			0
$149$	176397.	−	305529.i	0.650917	−	1.12742i	−0.331983	−	0.943285i	$-0.607718\pi$
$149$	176397.	−	305529.i	0.650917	−	1.12742i	0.982901	−	0.184137i	$-0.0589489\pi$
$150$		0			0
$151$	72548.2	+	125657.i	0.258931	+	0.448482i	0.965956	−	0.258707i	$-0.0832963\pi$
$151$	72548.2	+	125657.i	0.258931	+	0.448482i	−0.707025	+	0.707189i	$0.749963\pi$
$152$		0			0
$153$	−62021.9			−0.214199
$154$		0			0
$155$	−44343.5			−0.148252
$156$		0			0
$157$	108948.	+	188703.i	0.352751	+	0.610983i	0.986730	−	0.162367i	$-0.0519128\pi$
$157$	108948.	+	188703.i	0.352751	+	0.610983i	−0.633979	+	0.773350i	$0.718579\pi$
$158$		0			0
$159$	73875.0	−	127955.i	0.231742	−	0.401389i
$160$		0			0
$161$	185050.	−	510078.i	0.562632	−	1.55086i
$162$		0			0
$163$	−161055.	+	278956.i	−0.474795	+	0.822370i	−0.999583	−	0.0288633i	$-0.990811\pi$
$163$	−161055.	+	278956.i	−0.474795	+	0.822370i	0.524788	+	0.851233i	$0.324145\pi$
$164$		0			0
$165$	−13492.7	−	23370.0i	−0.0385823	−	0.0668266i
$166$		0			0
$167$	−707392.			−1.96277			−0.981384	−	0.192056i	$-0.938485\pi$
$167$	−707392.			−1.96277			−0.981384	+	0.192056i	$0.938485\pi$
$168$		0			0
$169$	−371093.			−0.999462
$170$		0			0
$171$	57278.8	+	99209.8i	0.149797	+	0.259456i
$172$		0			0
$173$	85253.7	−	147664.i	0.216570	−	0.375110i	−0.737187	−	0.675689i	$-0.763846\pi$
$173$	85253.7	−	147664.i	0.216570	−	0.375110i	0.953757	+	0.300579i	$0.0971798\pi$
$174$		0			0
$175$	231951.	+	275843.i	0.572534	+	0.680874i
$176$		0			0
$177$	−17849.2	+	30915.7i	−0.0428238	+	0.0741731i
$178$		0			0
$179$	−207703.	−	359752.i	−0.484519	−	0.839211i	0.515323	−	0.856996i	$-0.327672\pi$
$179$	−207703.	−	359752.i	−0.484519	−	0.839211i	−0.999842	+	0.0177851i	$0.994339\pi$
$180$		0			0
$181$	−1162.38			−0.00263726			−0.00131863	−	0.999999i	$-0.500420\pi$
$181$	−1162.38			−0.00263726			−0.00131863	+	0.999999i	$0.500420\pi$
$182$		0			0
$183$	453048.			1.00004
$184$		0			0
$185$	−6249.70	−	10824.8i	−0.0134255	−	0.0232536i
$186$		0			0
$187$	61802.0	−	107044.i	0.129241	−	0.223851i
$188$		0			0
$189$	−93055.9	+	16508.5i	−0.189491	+	0.0336166i
$190$		0			0
$191$	−372807.	+	645721.i	−0.739436	+	1.28074i	0.213313	+	0.976984i	$0.431575\pi$
$191$	−372807.	+	645721.i	−0.739436	+	1.28074i	−0.952749	+	0.303757i	$0.901759\pi$
$192$		0			0
$193$	99186.5	+	171796.i	0.191672	+	0.331986i	0.945805	−	0.324737i	$-0.105276\pi$
$193$	99186.5	+	171796.i	0.191672	+	0.331986i	−0.754132	+	0.656722i	$0.771942\pi$
$194$		0			0
$195$	2361.82			0.00444795
$196$		0			0
$197$	−469368.			−0.861684			−0.430842	−	0.902427i	$-0.641783\pi$
$197$	−469368.			−0.861684			−0.430842	+	0.902427i	$0.641783\pi$
$198$		0			0
$199$	−96796.7	−	167657.i	−0.173272	−	0.300116i	0.766290	−	0.642495i	$-0.222101\pi$
$199$	−96796.7	−	167657.i	−0.173272	−	0.300116i	−0.939562	+	0.342379i	$0.888767\pi$
$200$		0			0
$201$	−61403.7	+	106354.i	−0.107202	+	0.185680i
$202$		0			0
$203$	−536457.	+	95169.7i	−0.913681	+	0.162091i
$204$		0			0
$205$	−38759.5	+	67133.4i	−0.0644160	+	0.111572i
$206$		0			0
$207$	169510.	+	293600.i	0.274960	+	0.476245i
$208$		0			0
$209$	−228303.			−0.361531
$210$		0			0
$211$	298066.			0.460900			0.230450	−	0.973084i	$-0.425980\pi$
$211$	298066.			0.460900			0.230450	+	0.973084i	$0.425980\pi$
$212$		0			0
$213$	377807.	+	654381.i	0.570586	+	0.988284i
$214$		0			0
$215$	50438.0	−	87361.3i	0.0744153	−	0.128891i
$216$		0			0
$217$	−199191.	−	236883.i	−0.287157	−	0.341495i
$218$		0			0
$219$	−128602.	+	222746.i	−0.181192	+	0.313834i
$220$		0			0
$221$	5409.04	+	9368.73i	0.00744971	+	0.0129033i
$222$		0			0
$223$	187215.			0.252103			0.126051	−	0.992024i	$-0.459770\pi$
$223$	187215.			0.252103			0.126051	+	0.992024i	$0.459770\pi$
$224$		0			0
$225$	−225180.			−0.296533
$226$		0			0
$227$	−669482.	−	1.15958e6i	−0.862332	−	1.49360i	−0.869672	−	0.493630i	$-0.835670\pi$
$227$	−669482.	−	1.15958e6i	−0.862332	−	1.49360i	0.00734045	−	0.999973i	$-0.497663\pi$
$228$		0			0
$229$	−475828.	+	824158.i	−0.599600	+	1.03854i	0.393280	+	0.919419i	$0.371340\pi$
$229$	−475828.	+	824158.i	−0.599600	+	1.03854i	−0.992880	+	0.119119i	$0.961993\pi$
$230$		0			0
$231$	64233.8	−	177056.i	0.0792015	−	0.218314i
$232$		0			0
$233$	−371709.	+	643820.i	−0.448553	+	0.776917i	−0.998292	−	0.0584197i	$-0.981394\pi$
$233$	−371709.	+	643820.i	−0.448553	+	0.776917i	0.549739	+	0.835336i	$0.314727\pi$
$234$		0			0
$235$	−58534.0	−	101384.i	−0.0691415	−	0.119757i
$236$		0			0
$237$	−539599.			−0.624023
$238$		0			0
$239$	625847.			0.708718			0.354359	−	0.935110i	$-0.384699\pi$
$239$	625847.			0.708718			0.354359	+	0.935110i	$0.384699\pi$
$240$		0			0
$241$	666411.	+	1.15426e6i	0.739094	+	1.28015i	0.952904	+	0.303273i	$0.0980793\pi$
$241$	666411.	+	1.15426e6i	0.739094	+	1.28015i	−0.213810	+	0.976875i	$0.568587\pi$
$242$		0			0
$243$	29524.5	−	51137.9i	0.0320750	−	0.0555556i
$244$		0			0
$245$	53566.9	−	307549.i	0.0570140	−	0.327340i
$246$		0			0
$247$	9990.77	−	17304.5i	0.0104197	−	0.0180475i
$248$		0			0
$249$	277124.	+	479992.i	0.283254	+	0.490610i
$250$		0			0
$251$	−1.78809e6			−1.79145			−0.895726	−	0.444606i	$-0.853344\pi$
$251$	−1.78809e6			−1.79145			−0.895726	+	0.444606i	$0.853344\pi$
$252$		0			0
$253$	−675636.			−0.663608
$254$		0			0
$255$	64000.9	+	110853.i	0.0616362	+	0.106757i
$256$		0			0
$257$	−254462.	+	440741.i	−0.240320	+	0.416247i	−0.960805	−	0.277224i	$-0.910586\pi$
$257$	−254462.	+	440741.i	−0.240320	+	0.416247i	0.720485	+	0.693470i	$0.243919\pi$
$258$		0			0
$259$	29752.5	−	82010.9i	0.0275597	−	0.0759665i
$260$		0			0
$261$	170205.	−	294804.i	0.154658	−	0.267875i
$262$		0			0
$263$	−168247.	−	291412.i	−0.149988	−	0.259787i	0.781235	−	0.624237i	$-0.214590\pi$
$263$	−168247.	−	291412.i	−0.149988	−	0.259787i	−0.931223	+	0.364450i	$0.881257\pi$
$264$		0			0
$265$	−304929.			−0.266737
$266$		0			0
$267$	−380685.			−0.326804
$268$		0			0
$269$	−747765.	−	1.29517e6i	−0.630064	−	1.09130i	−0.987538	−	0.157379i	$-0.949695\pi$
$269$	−747765.	−	1.29517e6i	−0.630064	−	1.09130i	0.357475	−	0.933923i	$-0.383638\pi$
$270$		0			0
$271$	−888484.	+	1.53890e6i	−0.734897	+	1.27288i	0.219871	+	0.975529i	$0.429436\pi$
$271$	−888484.	+	1.53890e6i	−0.734897	+	1.27288i	−0.954768	+	0.297350i	$0.903897\pi$
$272$		0			0
$273$	10609.3	+	12616.8i	0.00861546	+	0.0102458i
$274$		0			0
$275$	224381.	−	388640.i	0.178918	−	0.309896i
$276$		0			0
$277$	−440632.	−	763198.i	−0.345046	−	0.597637i	0.640316	−	0.768112i	$-0.278803\pi$
$277$	−440632.	−	763198.i	−0.345046	−	0.597637i	−0.985362	+	0.170474i	$0.945470\pi$
$278$		0			0
$279$	193375.			0.148727
$280$		0			0
$281$	−1.13932e6			−0.860756			−0.430378	−	0.902649i	$-0.641620\pi$
$281$	−1.13932e6			−0.860756			−0.430378	+	0.902649i	$0.641620\pi$
$282$		0			0
$283$	−448733.	−	777228.i	−0.333059	−	0.576876i	0.650051	−	0.759891i	$-0.274748\pi$
$283$	−448733.	−	777228.i	−0.333059	−	0.576876i	−0.983110	+	0.183015i	$0.941414\pi$
$284$		0			0
$285$	118213.	−	204751.i	0.0862090	−	0.149318i
$286$		0			0
$287$	−532735.	+	94509.4i	−0.381774	+	0.0677283i
$288$		0			0
$289$	416778.	−	721881.i	0.293535	−	0.508418i
$290$		0			0
$291$	200872.	+	347920.i	0.139055	+	0.240850i
$292$		0			0
$293$	−1.84614e6			−1.25631			−0.628153	−	0.778090i	$-0.716189\pi$
$293$	−1.84614e6			−1.25631			−0.628153	+	0.778090i	$0.716189\pi$
$294$		0			0
$295$	73674.9			0.0492907
$296$		0			0
$297$	58839.7	+	101913.i	0.0387061	+	0.0670409i
$298$		0			0
$299$	29566.5	−	51210.8i	0.0191259	−	0.0331271i
$300$		0			0
$301$	693252.	−	122986.i	0.441037	−	0.0782418i
$302$		0			0
$303$	−503660.	+	872365.i	−0.315160	+	0.545873i
$304$		0			0
$305$	−467504.	−	809741.i	−0.287764	−	0.498421i
$306$		0			0
$307$	−1.38565e6			−0.839089			−0.419544	−	0.907735i	$-0.637810\pi$
$307$	−1.38565e6			−0.839089			−0.419544	+	0.907735i	$0.637810\pi$
$308$		0			0
$309$	1.50254e6			0.895221
$310$		0			0
$311$	1.15624e6	+	2.00267e6i	0.677871	+	1.17411i	0.975621	+	0.219463i	$0.0704305\pi$
$311$	1.15624e6	+	2.00267e6i	0.677871	+	1.17411i	−0.297750	+	0.954644i	$0.596236\pi$
$312$		0			0
$313$	584400.	−	1.01221e6i	0.337171	−	0.583996i	−0.646729	−	0.762720i	$-0.723863\pi$
$313$	584400.	−	1.01221e6i	0.337171	−	0.583996i	0.983899	+	0.178724i	$0.0571968\pi$
$314$		0			0
$315$	125531.	+	149285.i	0.0712811	+	0.0847696i
$316$		0			0
$317$	−717166.	+	1.24217e6i	−0.400840	+	0.694276i	−0.993828	−	0.110936i	$-0.964615\pi$
$317$	−717166.	+	1.24217e6i	−0.400840	+	0.694276i	0.592987	+	0.805212i	$0.297948\pi$
$318$		0			0
$319$	339204.	+	587519.i	0.186631	+	0.323255i
$320$		0			0
$321$	926871.			0.502061
$322$		0			0
$323$	1.08293e6			0.577554
$324$		0			0
$325$	19638.3	+	34014.5i	0.0103132	+	0.0178631i
$326$		0			0
$327$	−620195.	+	1.07421e6i	−0.320744	+	0.555546i
$328$		0			0
$329$	278659.	−	768105.i	0.141933	−	0.391229i
$330$		0			0
$331$	−1.11709e6	+	1.93486e6i	−0.560426	+	0.970686i	0.437033	+	0.899445i	$0.356029\pi$
$331$	−1.11709e6	+	1.93486e6i	−0.560426	+	0.970686i	−0.997459	+	0.0712406i	$0.977304\pi$
$332$		0			0
$333$	27254.0	+	47205.4i	0.0134685	+	0.0233282i
$334$		0			0
$335$	253452.			0.123391
$336$		0			0
$337$	3.08787e6			1.48110			0.740549	−	0.672002i	$-0.234565\pi$
$337$	3.08787e6			1.48110			0.740549	+	0.672002i	$0.234565\pi$
$338$		0			0
$339$	−584851.	−	1.01299e6i	−0.276405	−	0.478748i
$340$		0			0
$341$	−192690.	+	333749.i	−0.0897373	+	0.155429i
$342$		0			0
$343$	1.88355e6	−	1.09535e6i	0.864454	−	0.502711i
$344$		0			0
$345$	349838.	−	605936.i	0.158241	−	0.274081i
$346$		0			0
$347$	−1.55877e6	−	2.69987e6i	−0.694959	−	1.20370i	−0.970195	−	0.242327i	$-0.922089\pi$
$347$	−1.55877e6	−	2.69987e6i	−0.694959	−	1.20370i	0.275236	−	0.961377i	$-0.411244\pi$
$348$		0			0
$349$	613026.			0.269411			0.134706	−	0.990886i	$-0.456991\pi$
$349$	613026.			0.269411			0.134706	+	0.990886i	$0.456991\pi$
$350$		0			0
$351$	−10299.5			−0.00446221
$352$		0			0
$353$	−1.89185e6	−	3.27678e6i	−0.808071	−	1.39962i	−0.914198	−	0.405268i	$-0.867178\pi$
$353$	−1.89185e6	−	3.27678e6i	−0.808071	−	1.39962i	0.106127	−	0.994353i	$-0.466155\pi$
$354$		0			0
$355$	779724.	−	1.35052e6i	0.328375	−	0.568762i
$356$		0			0
$357$	−304685.	+	839844.i	−0.126526	+	0.348761i
$358$		0			0
$359$	1.86107e6	−	3.22346e6i	0.762125	−	1.32004i	−0.179629	−	0.983734i	$-0.557490\pi$
$359$	1.86107e6	−	3.22346e6i	0.762125	−	1.32004i	0.941753	−	0.336304i	$-0.109177\pi$
$360$		0			0
$361$	237940.	+	412123.i	0.0960945	+	0.166441i
$362$		0			0
$363$	1.21493e6			0.483934
$364$		0			0
$365$	530824.			0.208554
$366$		0			0
$367$	2.06898e6	+	3.58357e6i	0.801845	+	1.38884i	0.918400	+	0.395652i	$0.129481\pi$
$367$	2.06898e6	+	3.58357e6i	0.801845	+	1.38884i	−0.116555	+	0.993184i	$0.537185\pi$
$368$		0			0
$369$	169025.	−	292759.i	0.0646225	−	0.111929i
$370$		0			0
$371$	−1.36974e6	−	1.62893e6i	−0.516658	−	0.614424i
$372$		0			0
$373$	−1.47209e6	+	2.54973e6i	−0.547851	+	0.948905i	0.450571	+	0.892741i	$0.351220\pi$
$373$	−1.47209e6	+	2.54973e6i	−0.547851	+	0.948905i	−0.998422	+	0.0561644i	$0.982113\pi$
$374$		0			0
$375$	493566.	+	854882.i	0.181245	+	0.313926i
$376$		0			0
$377$	−59375.7			−0.0215157
$378$		0			0
$379$	−2.97504e6			−1.06388			−0.531942	−	0.846781i	$-0.678538\pi$
$379$	−2.97504e6			−1.06388			−0.531942	+	0.846781i	$0.678538\pi$
$380$		0			0
$381$	1.09469e6	+	1.89606e6i	0.386349	+	0.669177i
$382$		0			0
$383$	1.03052e6	−	1.78491e6i	0.358971	−	0.621756i	−0.628818	−	0.777552i	$-0.716461\pi$
$383$	1.03052e6	−	1.78491e6i	0.358971	−	0.621756i	0.987789	+	0.155796i	$0.0497944\pi$
$384$		0			0
$385$	−382739.	+	67899.5i	−0.131598	+	0.0233461i
$386$		0			0
$387$	−219953.	+	380970.i	−0.0746539	+	0.129304i
$388$		0			0
$389$	−1.04542e6	−	1.81072e6i	−0.350281	−	0.606705i	0.636018	−	0.771675i	$-0.280581\pi$
$389$	−1.04542e6	−	1.81072e6i	−0.350281	−	0.606705i	−0.986299	+	0.164970i	$0.947247\pi$
$390$		0			0
$391$	3.20480e6			1.06013
$392$		0			0
$393$	1.74902e6			0.571233
$394$		0			0
$395$	556817.	+	964435.i	0.179564	+	0.311014i
$396$		0			0
$397$	−585647.	+	1.01437e6i	−0.186492	+	0.323013i	−0.944078	−	0.329722i	$-0.893045\pi$
$397$	−585647.	+	1.01437e6i	−0.186492	+	0.323013i	0.757586	+	0.652735i	$0.226378\pi$
$398$		0			0
$399$	1.62479e6	−	288245.i	0.510935	−	0.0906420i
$400$		0			0
$401$	−1.54574e6	+	2.67731e6i	−0.480039	+	0.831452i	−0.999738	−	0.0228979i	$-0.992711\pi$
$401$	−1.54574e6	+	2.67731e6i	−0.480039	+	0.831452i	0.519699	+	0.854349i	$0.326044\pi$
$402$		0			0
$403$	−16864.6	−	29210.4i	−0.00517266	−	0.00895930i
$404$		0			0
$405$	−121866.			−0.0369187
$406$		0			0
$407$	−108630.			−0.0325059
$408$		0			0
$409$	2.17426e6	+	3.76593e6i	0.642693	+	1.11318i	0.984829	+	0.173527i	$0.0555163\pi$
$409$	2.17426e6	+	3.76593e6i	0.642693	+	1.11318i	−0.342136	+	0.939650i	$0.611150\pi$
$410$		0			0
$411$	−13921.7	+	24113.0i	−0.00406524	+	0.00704121i
$412$		0			0
$413$	330947.	+	393572.i	0.0954737	+	0.113540i
$414$		0			0
$415$	571932.	−	990616.i	0.163014	−	0.282348i
$416$		0			0
$417$	−101702.	−	176153.i	−0.0286411	−	0.0496078i
$418$		0			0
$419$	−3.13660e6			−0.872818			−0.436409	−	0.899748i	$-0.643750\pi$
$419$	−3.13660e6			−0.872818			−0.436409	+	0.899748i	$0.643750\pi$
$420$		0			0
$421$	6.02560e6			1.65690			0.828448	−	0.560066i	$-0.189224\pi$
$421$	6.02560e6			1.65690			0.828448	+	0.560066i	$0.189224\pi$
$422$		0			0
$423$	255258.	+	442120.i	0.0693632	+	0.120141i
$424$		0			0
$425$	−1.06432e6	+	1.84346e6i	−0.285826	+	0.495065i
$426$		0			0
$427$	2.22562e6	−	6.13476e6i	0.590719	−	1.62828i
$428$		0			0
$429$	10263.0	−	17776.1i	0.00269235	−	0.00466329i
$430$		0			0
$431$	−1.14931e6	−	1.99067e6i	−0.298020	−	0.516186i	0.677663	−	0.735373i	$-0.262993\pi$
$431$	−1.14931e6	−	1.99067e6i	−0.298020	−	0.516186i	−0.975683	+	0.219187i	$0.929660\pi$
$432$		0			0
$433$	−5.62982e6			−1.44303			−0.721515	−	0.692399i	$-0.756554\pi$
$433$	−5.62982e6			−1.44303			−0.721515	+	0.692399i	$0.756554\pi$
$434$		0			0
$435$	−702545.			−0.178013
$436$		0			0
$437$	−2.95971e6	−	5.12637e6i	−0.741388	−	1.28412i
$438$		0			0
$439$	−2.72482e6	+	4.71952e6i	−0.674801	+	1.16879i	0.301725	+	0.953395i	$0.402437\pi$
$439$	−2.72482e6	+	4.71952e6i	−0.674801	+	1.16879i	−0.976527	+	0.215396i	$0.930896\pi$
$440$		0			0
$441$	−233597.	+	1.34118e6i	−0.0571968	+	0.328389i
$442$		0			0
$443$	2.85467e6	−	4.94443e6i	0.691108	−	1.19703i	−0.280367	−	0.959893i	$-0.590456\pi$
$443$	2.85467e6	−	4.94443e6i	0.691108	−	1.19703i	0.971475	−	0.237142i	$-0.0762105\pi$
$444$		0			0
$445$	392832.	+	680405.i	0.0940387	+	0.162880i
$446$		0			0
$447$	3.17515e6			0.751615
$448$		0			0
$449$	5.38234e6			1.25996			0.629978	−	0.776613i	$-0.283064\pi$
$449$	5.38234e6			1.25996			0.629978	+	0.776613i	$0.283064\pi$
$450$		0			0
$451$	336851.	+	583442.i	0.0779823	+	0.135069i
$452$		0			0
$453$	−652934.	+	1.13092e6i	−0.149494	+	0.258931i
$454$		0			0
$455$	11602.5	−	31981.5i	0.00262738	−	0.00724220i
$456$		0			0
$457$	3.19525e6	−	5.53434e6i	0.715673	−	1.23958i	−0.247026	−	0.969009i	$-0.579453\pi$
$457$	3.19525e6	−	5.53434e6i	0.715673	−	1.23958i	0.962699	−	0.270573i	$-0.0872133\pi$
$458$		0			0
$459$	−279099.	−	483413.i	−0.0618338	−	0.107099i
$460$		0			0
$461$	7.34511e6			1.60970			0.804851	−	0.593476i	$-0.202245\pi$
$461$	7.34511e6			1.60970			0.804851	+	0.593476i	$0.202245\pi$
$462$		0			0
$463$	4.63416e6			1.00466			0.502329	−	0.864677i	$-0.332477\pi$
$463$	4.63416e6			1.00466			0.502329	+	0.864677i	$0.332477\pi$
$464$		0			0
$465$	−199546.	−	345623.i	−0.0427966	−	0.0741259i
$466$		0			0
$467$	−3.81598e6	+	6.60946e6i	−0.809680	+	1.40241i	0.103406	+	0.994639i	$0.467026\pi$
$467$	−3.81598e6	+	6.60946e6i	−0.809680	+	1.40241i	−0.913086	+	0.407768i	$0.866307\pi$
$468$		0			0
$469$	1.13850e6	+	1.35394e6i	0.239002	+	0.284229i
$470$		0			0
$471$	−980529.	+	1.69833e6i	−0.203661	+	0.352751i
$472$		0			0
$473$	−438346.	−	759238.i	−0.0900875	−	0.156036i
$474$		0			0
$475$	3.93172e6			0.799556
$476$		0			0
$477$	1.32975e6			0.267593
$478$		0			0
$479$	4.80631e6	+	8.32477e6i	0.957135	+	1.65781i	0.729406	+	0.684081i	$0.239796\pi$
$479$	4.80631e6	+	8.32477e6i	0.957135	+	1.65781i	0.227728	+	0.973725i	$0.426870\pi$
$480$		0			0
$481$	4753.74	−	8233.72i	0.000936856	−	0.00162268i
$482$		0			0
$483$	4.80839e6	−	853028.i	0.937846	−	0.166378i
$484$		0			0
$485$	414562.	−	718042.i	0.0800267	−	0.138610i
$486$		0			0
$487$	1.40112e6	+	2.42680e6i	0.267702	+	0.463673i	0.968268	−	0.249914i	$-0.0804024\pi$
$487$	1.40112e6	+	2.42680e6i	0.267702	+	0.463673i	−0.700566	+	0.713588i	$0.747069\pi$
$488$		0			0
$489$	−2.89900e6			−0.548246
$490$		0			0
$491$	−4.82008e6			−0.902299			−0.451149	−	0.892448i	$-0.648986\pi$
$491$	−4.82008e6			−0.902299			−0.451149	+	0.892448i	$0.648986\pi$
$492$		0			0
$493$	−1.60897e6	−	2.78682e6i	−0.298148	−	0.516407i
$494$		0			0
$495$	121434.	−	210330.i	0.0222755	−	0.0385823i
$496$		0			0
$497$	1.07170e7	−	1.90125e6i	1.94618	−	0.345261i
$498$		0			0
$499$	2.29829e6	−	3.98076e6i	0.413194	−	0.715673i	−0.582043	−	0.813158i	$-0.697747\pi$
$499$	2.29829e6	−	3.98076e6i	0.413194	−	0.715673i	0.995237	+	0.0974853i	$0.0310799\pi$
$500$		0			0
$501$	−3.18326e6	−	5.51358e6i	−0.566602	−	0.981384i
$502$		0			0
$503$	−1.80055e6			−0.317311			−0.158656	−	0.987334i	$-0.550716\pi$
$503$	−1.80055e6			−0.317311			−0.158656	+	0.987334i	$0.550716\pi$
$504$		0			0
$505$	2.07892e6			0.362752
$506$		0			0
$507$	−1.66992e6	−	2.89239e6i	−0.288520	−	0.499731i
$508$		0			0
$509$	2.41602e6	−	4.18467e6i	0.413339	−	0.715924i	−0.581913	−	0.813251i	$-0.697696\pi$
$509$	2.41602e6	−	4.18467e6i	0.413339	−	0.715924i	0.995253	+	0.0973263i	$0.0310291\pi$
$510$		0			0
$511$	2.38446e6	+	2.83566e6i	0.403959	+	0.480400i
$512$		0			0
$513$	−515509.	+	892888.i	−0.0864854	+	0.149797i
$514$		0			0
$515$	−1.55048e6	−	2.68552e6i	−0.257602	−	0.446180i
$516$		0			0
$517$	−1.01741e6			−0.167406
$518$		0			0
$519$	1.53457e6			0.250073
$520$		0			0
$521$	−1.40562e6	−	2.43461e6i	−0.226869	−	0.392948i	0.730010	−	0.683437i	$-0.239516\pi$
$521$	−1.40562e6	−	2.43461e6i	−0.226869	−	0.392948i	−0.956878	+	0.290489i	$0.906182\pi$
$522$		0			0
$523$	−1.22715e6	+	2.12548e6i	−0.196174	+	0.339784i	−0.947285	−	0.320392i	$-0.896185\pi$
$523$	−1.22715e6	+	2.12548e6i	−0.196174	+	0.339784i	0.751110	+	0.660177i	$0.229519\pi$
$524$		0			0
$525$	−1.10620e6	+	3.04917e6i	−0.175161	+	0.482818i
$526$		0			0
$527$	914000.	−	1.58309e6i	0.143357	−	0.248302i
$528$		0			0
$529$	−5.54076e6	−	9.59687e6i	−0.860855	−	1.49104i
$530$		0			0
$531$	−321286.			−0.0494487
$532$		0			0
$533$	−58963.7			−0.00899015
$534$		0			0
$535$	−956445.	−	1.65661e6i	−0.144469	−	0.250228i
$536$		0			0
$537$	1.86933e6	−	3.23777e6i	0.279737	−	0.484519i
$538$		0			0
$539$	−2.08198e6	−	1.73959e6i	−0.308677	−	0.257914i
$540$		0			0
$541$	579232.	−	1.00326e6i	0.0850862	−	0.147374i	−0.820342	−	0.571874i	$-0.806217\pi$
$541$	579232.	−	1.00326e6i	0.0850862	−	0.147374i	0.905428	+	0.424500i	$0.139550\pi$
$542$		0			0
$543$	−5230.72	−	9059.87i	−0.000761310	−	0.00131863i
$544$		0			0
$545$	2.55994e6			0.369180
$546$		0			0
$547$	−4.63638e6			−0.662538			−0.331269	−	0.943536i	$-0.607477\pi$
$547$	−4.63638e6			−0.662538			−0.331269	+	0.943536i	$0.607477\pi$
$548$		0			0
$549$	2.03872e6	+	3.53116e6i	0.288686	+	0.500019i
$550$		0			0
$551$	−2.97185e6	+	5.14740e6i	−0.417012	+	0.722285i
$552$		0			0
$553$	−2.65080e6	+	7.30676e6i	−0.368608	+	1.01604i
$554$		0			0
$555$	56247.3	−	97423.1i	0.00775120	−	0.0134255i
$556$		0			0
$557$	−135916.	−	235414.i	−0.0185624	−	0.0321510i	0.856595	−	0.515989i	$-0.172576\pi$
$557$	−135916.	−	235414.i	−0.0185624	−	0.0321510i	−0.875157	+	0.483838i	$0.839242\pi$
$558$		0			0
$559$	76730.0			0.0103857
$560$		0			0
$561$	1.11244e6			0.149234
$562$		0			0
$563$	1.69122e6	+	2.92928e6i	0.224869	+	0.389485i	0.956280	−	0.292452i	$-0.0944712\pi$
$563$	1.69122e6	+	2.92928e6i	0.224869	+	0.389485i	−0.731411	+	0.681937i	$0.761138\pi$
$564$		0			0
$565$	−1.20702e6	+	2.09063e6i	−0.159072	+	0.275522i
$566$		0			0
$567$	−547423.	−	651011.i	−0.0715097	−	0.0850414i
$568$		0			0
$569$	−6.56130e6	+	1.13645e7i	−0.849590	+	1.47153i	0.0319843	+	0.999488i	$0.489817\pi$
$569$	−6.56130e6	+	1.13645e7i	−0.849590	+	1.47153i	−0.881574	+	0.472045i	$0.843516\pi$
$570$		0			0
$571$	−7.35811e6	−	1.27446e7i	−0.944443	−	1.63582i	−0.756862	−	0.653575i	$-0.773269\pi$
$571$	−7.35811e6	−	1.27446e7i	−0.944443	−	1.63582i	−0.187581	−	0.982249i	$-0.560065\pi$
$572$		0			0
$573$	−6.71053e6			−0.853828
$574$		0			0
$575$	1.16355e7			1.46762
$576$		0			0
$577$	1.94148e6	+	3.36273e6i	0.242769	+	0.420487i	0.961502	−	0.274798i	$-0.0886111\pi$
$577$	1.94148e6	+	3.36273e6i	0.242769	+	0.420487i	−0.718733	+	0.695286i	$0.755278\pi$
$578$		0			0
$579$	−892678.	+	1.54616e6i	−0.110662	+	0.191672i
$580$		0			0
$581$	7.86099e6	−	1.39457e6i	0.966134	−	0.171396i
$582$		0			0
$583$	−1.32504e6	+	2.29503e6i	−0.161457	+	0.279651i
$584$		0			0
$585$	10628.2	+	18408.5i	0.00128401	+	0.00222397i
$586$		0			0
$587$	−4.97913e6			−0.596428			−0.298214	−	0.954499i	$-0.596391\pi$
$587$	−4.97913e6			−0.596428			−0.298214	+	0.954499i	$0.596391\pi$
$588$		0			0
$589$	−3.37641e6			−0.401021
$590$		0			0
$591$	−2.11216e6	−	3.65836e6i	−0.248747	−	0.430842i
$592$		0			0
$593$	−7.66344e6	+	1.32735e7i	−0.894926	+	1.55006i	−0.0610292	+	0.998136i	$0.519438\pi$
$593$	−7.66344e6	+	1.32735e7i	−0.894926	+	1.55006i	−0.833896	+	0.551921i	$0.813895\pi$
$594$		0			0
$595$	1.81547e6	−	322073.i	0.210231	−	0.0372959i
$596$		0			0
$597$	871171.	−	1.50891e6i	0.100039	−	0.173272i
$598$		0			0
$599$	−3.31480e6	−	5.74141e6i	−0.377477	−	0.653810i	0.613217	−	0.789914i	$-0.289875\pi$
$599$	−3.31480e6	−	5.74141e6i	−0.377477	−	0.653810i	−0.990694	+	0.136104i	$0.956542\pi$
$600$		0			0
$601$	−1.45010e6			−0.163762			−0.0818808	−	0.996642i	$-0.526093\pi$
$601$	−1.45010e6			−0.163762			−0.0818808	+	0.996642i	$0.526093\pi$
$602$		0			0
$603$	−1.10527e6			−0.123787
$604$		0			0
$605$	−1.25370e6	−	2.17147e6i	−0.139253	−	0.241194i
$606$		0			0
$607$	1.95555e6	−	3.38711e6i	0.215425	−	0.373128i	−0.737979	−	0.674824i	$-0.764220\pi$
$607$	1.95555e6	−	3.38711e6i	0.215425	−	0.373128i	0.953404	+	0.301696i	$0.0975528\pi$
$608$		0			0
$609$	−3.15583e6	−	3.75300e6i	−0.344802	−	0.410049i
$610$		0			0
$611$	44523.1	−	77116.2i	0.00482483	−	0.00835685i
$612$		0			0
$613$	−5.88765e6	−	1.01977e7i	−0.632835	−	1.09610i	−0.986969	−	0.160908i	$-0.948558\pi$
$613$	−5.88765e6	−	1.01977e7i	−0.632835	−	1.09610i	0.354134	−	0.935195i	$-0.384776\pi$
$614$		0			0
$615$	−697671.			−0.0743812
$616$		0			0
$617$	−4.61462e6			−0.488004			−0.244002	−	0.969775i	$-0.578460\pi$
$617$	−4.61462e6			−0.488004			−0.244002	+	0.969775i	$0.578460\pi$
$618$		0			0
$619$	−2.83733e6	−	4.91440e6i	−0.297634	−	0.515518i	0.677960	−	0.735099i	$-0.262864\pi$
$619$	−2.83733e6	−	4.91440e6i	−0.297634	−	0.515518i	−0.975594	+	0.219581i	$0.929531\pi$
$620$		0			0
$621$	−1.52559e6	+	2.64240e6i	−0.158748	+	0.274960i
$622$		0			0
$623$	−1.87013e6	+	5.15489e6i	−0.193042	+	0.532107i
$624$		0			0
$625$	−3.32511e6	+	5.75926e6i	−0.340491	+	0.589748i
$626$		0			0
$627$	−1.02736e6	−	1.77945e6i	−0.104365	−	0.180766i
$628$		0			0
$629$	515271.			0.0519289
$630$		0			0
$631$	−5.67894e6			−0.567798			−0.283899	−	0.958854i	$-0.591628\pi$
$631$	−5.67894e6			−0.567798			−0.283899	+	0.958854i	$0.591628\pi$
$632$		0			0
$633$	1.34130e6	+	2.32320e6i	0.133050	+	0.230450i
$634$		0			0
$635$	2.25925e6	−	3.91313e6i	0.222346	−	0.385114i
$636$		0			0
$637$	222964.	−	81680.2i	0.0217714	−	0.00797569i
$638$		0			0
$639$	−3.40027e6	+	5.88943e6i	−0.329428	+	0.570586i
$640$		0			0
$641$	1.05369e6	+	1.82504e6i	0.101290	+	0.175439i	0.912216	−	0.409709i	$-0.134370\pi$
$641$	1.05369e6	+	1.82504e6i	0.101290	+	0.175439i	−0.810926	+	0.585148i	$0.801036\pi$
$642$		0			0
$643$	2.30987e6			0.220323			0.110162	−	0.993914i	$-0.464863\pi$
$643$	2.30987e6			0.220323			0.110162	+	0.993914i	$0.464863\pi$
$644$		0			0
$645$	907885.			0.0859274
$646$		0			0
$647$	4.93033e6	+	8.53958e6i	0.463036	+	0.802003i	0.999111	−	0.0421683i	$-0.0134266\pi$
$647$	4.93033e6	+	8.53958e6i	0.463036	+	0.802003i	−0.536074	+	0.844171i	$0.680093\pi$
$648$		0			0
$649$	320147.	−	554510.i	0.0298358	−	0.0516771i
$650$		0			0
$651$	949964.	−	2.61851e6i	0.0878526	−	0.242160i
$652$		0			0
$653$	−5.01545e6	+	8.68701e6i	−0.460285	+	0.797237i	−0.998975	−	0.0452673i	$-0.985586\pi$
$653$	−5.01545e6	+	8.68701e6i	−0.460285	+	0.797237i	0.538690	+	0.842504i	$0.318919\pi$
$654$		0			0
$655$	−1.80483e6	−	3.12605e6i	−0.164374	−	0.284704i
$656$		0			0
$657$	−2.31484e6			−0.209223
$658$		0			0
$659$	1.42828e7			1.28115			0.640575	−	0.767895i	$-0.278696\pi$
$659$	1.42828e7			1.28115			0.640575	+	0.767895i	$0.278696\pi$
$660$		0			0
$661$	9.82097e6	+	1.70104e7i	0.874280	+	1.51430i	0.857527	+	0.514438i	$0.172000\pi$
$661$	9.82097e6	+	1.70104e7i	0.874280	+	1.51430i	0.0167532	+	0.999860i	$0.494667\pi$
$662$		0			0
$663$	−48681.4	+	84318.6i	−0.00430109	+	0.00744971i
$664$		0			0
$665$	−2.19182e6	−	2.60657e6i	−0.192199	−	0.228568i
$666$		0			0
$667$	−8.79486e6	+	1.52331e7i	−0.765445	+	1.32579i
$668$		0			0
$669$	842466.	+	1.45919e6i	0.0727758	+	0.126051i
$670$		0			0
$671$	−8.12596e6			−0.696736
$672$		0			0
$673$	−9.00150e6			−0.766086			−0.383043	−	0.923731i	$-0.625124\pi$
$673$	−9.00150e6			−0.766086			−0.383043	+	0.923731i	$0.625124\pi$
$674$		0			0
$675$	−1.01331e6	−	1.75510e6i	−0.0856016	−	0.148266i
$676$		0			0
$677$	−6.71181e6	+	1.16252e7i	−0.562818	+	0.974830i	0.434431	+	0.900705i	$0.356950\pi$
$677$	−6.71181e6	+	1.16252e7i	−0.562818	+	0.974830i	−0.997249	+	0.0741247i	$0.976384\pi$
$678$		0			0
$679$	5.69800e6	−	1.01085e6i	0.474294	−	0.0841418i
$680$		0			0
$681$	6.02534e6	−	1.04362e7i	0.497868	−	0.862332i
$682$		0			0
$683$	−4.00840e6	−	6.94276e6i	−0.328791	−	0.569483i	0.653481	−	0.756943i	$-0.273308\pi$
$683$	−4.00840e6	−	6.94276e6i	−0.328791	−	0.569483i	−0.982272	+	0.187460i	$0.939974\pi$
$684$		0			0
$685$	57463.5			0.00467913
$686$		0			0
$687$	−8.56491e6			−0.692358
$688$		0			0
$689$	−115970.	−	200866.i	−0.00930673	−	0.0161197i
$690$		0			0
$691$	−1.20987e7	+	2.09555e7i	−0.963923	+	1.66956i	−0.251442	+	0.967872i	$0.580905\pi$
$691$	−1.20987e7	+	2.09555e7i	−0.963923	+	1.66956i	−0.712481	+	0.701691i	$0.752428\pi$
$692$		0			0
$693$	1.66907e6	−	296100.i	0.132020	−	0.0234210i
$694$		0			0
$695$	−209894.	+	363547.i	−0.0164831	+	0.0285495i
$696$		0			0
$697$	−1.59781e6	−	2.76749e6i	−0.124578	−	0.215776i
$698$		0			0
$699$	−6.69077e6			−0.517944
$700$		0			0
$701$	−1.56268e6			−0.120109			−0.0600543	−	0.998195i	$-0.519127\pi$
$701$	−1.56268e6			−0.120109			−0.0600543	+	0.998195i	$0.519127\pi$
$702$		0			0
$703$	−475866.	−	824224.i	−0.0363158	−	0.0629009i
$704$		0			0
$705$	526806.	−	912455.i	0.0399189	−	0.0691415i
$706$		0			0
$707$	9.33851e6	+	1.11056e7i	0.702634	+	0.835592i
$708$		0			0
$709$	9.76950e6	−	1.69213e7i	0.729889	−	1.26420i	−0.227041	−	0.973885i	$-0.572905\pi$
$709$	9.76950e6	−	1.69213e7i	0.729889	−	1.26420i	0.956930	−	0.290320i	$-0.0937616\pi$
$710$		0			0
$711$	−2.42820e6	−	4.20576e6i	−0.180140	−	0.312011i
$712$		0			0
$713$	−9.99210e6			−0.736093
$714$		0			0
$715$	−42362.0			−0.00309892
$716$		0			0
$717$	2.81631e6	+	4.87799e6i	0.204589	+	0.354359i
$718$		0			0
$719$	−4.05095e6	+	7.01645e6i	−0.292237	+	0.506169i	−0.974338	−	0.225089i	$-0.927733\pi$
$719$	−4.05095e6	+	7.01645e6i	−0.292237	+	0.506169i	0.682102	+	0.731257i	$0.261066\pi$
$720$		0			0
$721$	7.38129e6	−	2.03460e7i	0.528804	−	1.45761i
$722$		0			0
$723$	−5.99770e6	+	1.03883e7i	−0.426716	+	0.739094i
$724$		0			0
$725$	−5.84161e6	−	1.01180e7i	−0.412750	−	0.714904i
$726$		0			0
$727$	−2.52759e7			−1.77366			−0.886829	−	0.462098i	$-0.847097\pi$
$727$	−2.52759e7			−1.77366			−0.886829	+	0.462098i	$0.847097\pi$
$728$		0			0
$729$	531441.			0.0370370
$730$		0			0
$731$	2.07924e6	+	3.60135e6i	0.143917	+	0.249271i
$732$		0			0
$733$	−1.37207e7	+	2.37650e7i	−0.943228	+	1.63372i	−0.183969	+	0.982932i	$0.558894\pi$
$733$	−1.37207e7	+	2.37650e7i	−0.943228	+	1.63372i	−0.759260	+	0.650787i	$0.774439\pi$
$734$		0			0
$735$	2.63816e6	−	966457.i	0.180128	−	0.0659879i
$736$		0			0
$737$	1.10135e6	−	1.90759e6i	0.0746888	−	0.129365i
$738$		0			0
$739$	−3.26785e6	−	5.66008e6i	−0.220115	−	0.381251i	0.734727	−	0.678362i	$-0.237310\pi$
$739$	−3.26785e6	−	5.66008e6i	−0.220115	−	0.381251i	−0.954843	+	0.297111i	$0.903977\pi$
$740$		0			0
$741$	179834.			0.0120317
$742$		0			0
$743$	2.46784e7			1.64000			0.820001	−	0.572362i	$-0.193973\pi$
$743$	2.46784e7			1.64000			0.820001	+	0.572362i	$0.193973\pi$
$744$		0			0
$745$	−3.27646e6	−	5.67499e6i	−0.216279	−	0.374606i
$746$		0			0
$747$	−2.49411e6	+	4.31993e6i	−0.163537	+	0.283254i
$748$		0			0
$749$	4.55329e6	−	1.25508e7i	0.296565	−	0.817463i
$750$		0			0
$751$	−1.63911e6	+	2.83902e6i	−0.106049	+	0.183683i	−0.914166	−	0.405339i	$-0.867153\pi$
$751$	−1.63911e6	+	2.83902e6i	−0.106049	+	0.183683i	0.808117	+	0.589022i	$0.200487\pi$
$752$		0			0
$753$	−8.04641e6	−	1.39368e7i	−0.517148	−	0.895726i
$754$		0			0
$755$	2.69507e6			0.172069
$756$		0			0
$757$	3.68090e6			0.233461			0.116731	−	0.993164i	$-0.462759\pi$
$757$	3.68090e6			0.233461			0.116731	+	0.993164i	$0.462759\pi$
$758$		0			0
$759$	−3.04036e6	−	5.26606e6i	−0.191567	−	0.331804i
$760$		0			0
$761$	−1.05099e7	+	1.82037e7i	−0.657866	+	1.13946i	0.323301	+	0.946296i	$0.395208\pi$
$761$	−1.05099e7	+	1.82037e7i	−0.657866	+	1.13946i	−0.981167	+	0.193162i	$0.938126\pi$
$762$		0			0
$763$	1.14992e7	+	1.36752e7i	0.715084	+	0.850399i
$764$		0			0
$765$	−576008.	+	997675.i	−0.0355857	+	0.0616362i
$766$		0			0
$767$	28019.9	+	48531.9i	0.00171980	+	0.00297878i
$768$		0			0
$769$	−4.15418e6			−0.253320			−0.126660	−	0.991946i	$-0.540426\pi$
$769$	−4.15418e6			−0.253320			−0.126660	+	0.991946i	$0.540426\pi$
$770$		0			0
$771$	−4.58032e6			−0.277498
$772$		0			0
$773$	−731741.	−	1.26741e6i	−0.0440462	−	0.0762903i	0.843162	−	0.537660i	$-0.180692\pi$
$773$	−731741.	−	1.26741e6i	−0.0440462	−	0.0762903i	−0.887208	+	0.461370i	$0.847358\pi$
$774$		0			0
$775$	3.31841e6	−	5.74765e6i	0.198461	−	0.343745i
$776$		0			0
$777$	773098.	−	137151.i	0.0459390	−	0.00814978i
$778$		0			0
$779$	−2.95123e6	+	5.11169e6i	−0.174245	+	0.301801i
$780$		0			0
$781$	−6.77642e6	−	1.17371e7i	−0.397533	−	0.688547i
$782$		0			0
$783$	3.06370e6			0.178584

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:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	336.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(4.50000 + 7.79423i) q^{3} +(9.28717 - 16.0858i) q^{5} +(127.649 - 22.6454i) q^{7} +(-40.5000 + 70.1481i) q^{9} +O(q^{10})\)	\(q+(4.50000 + 7.79423i) q^{3} +(9.28717 - 16.0858i) q^{5} +(127.649 - 22.6454i) q^{7} +(-40.5000 + 70.1481i) q^{9} +(-80.7128 - 139.799i) q^{11} +14.1283 q^{13} +167.169 q^{15} +(382.851 + 663.118i) q^{17} +(707.146 - 1224.81i) q^{19} +(750.923 + 893.019i) q^{21} +(2092.72 - 3624.69i) q^{23} +(1390.00 + 2407.55i) q^{25} -729.000 q^{27} -4202.60 q^{29} +(-1193.68 - 2067.51i) q^{31} +(726.415 - 1258.19i) q^{33} +(821.224 - 2263.65i) q^{35} +(336.469 - 582.782i) q^{37} +(63.5774 + 110.119i) q^{39} -4173.45 q^{41} +5430.94 q^{43} +(752.261 + 1302.95i) q^{45} +(3151.34 - 5458.28i) q^{47} +(15781.4 - 5781.32i) q^{49} +(-3445.66 + 5968.06i) q^{51} +(-8208.34 - 14217.3i) q^{53} -2998.37 q^{55} +12728.6 q^{57} +(1983.24 + 3435.08i) q^{59} +(25169.4 - 43594.6i) q^{61} +(-3581.24 + 9871.45i) q^{63} +(131.212 - 227.266i) q^{65} +(6822.63 + 11817.1i) q^{67} +37668.9 q^{69} +83957.2 q^{71} +(14289.2 + 24749.6i) q^{73} +(-12510.0 + 21667.9i) q^{75} +(-13468.7 - 16017.3i) q^{77} +(-29977.7 + 51923.0i) q^{79} +(-3280.50 - 5681.99i) q^{81} +61583.0 q^{83} +14222.4 q^{85} +(-18911.7 - 32756.0i) q^{87} +(-21149.2 + 36631.4i) q^{89} +(1803.46 - 319.941i) q^{91} +(10743.1 - 18607.6i) q^{93} +(-13134.8 - 22750.1i) q^{95} +44638.1 q^{97} +13075.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 18 q^{3} - 47 q^{5} + 174 q^{7} - 162 q^{9}+O(q^{10}) \) 4 * q + 18 * q^3 - 47 * q^5 + 174 * q^7 - 162 * q^9	\( 4 q + 18 q^{3} - 47 q^{5} + 174 q^{7} - 162 q^{9} - 407 q^{11} + 898 q^{13} - 846 q^{15} + 1868 q^{17} - 1463 q^{19} - 783 q^{21} - 44 q^{23} + 1605 q^{25} - 2916 q^{27} + 1534 q^{29} - 11170 q^{31} + 3663 q^{33} - 9674 q^{35} + 3113 q^{37} + 4041 q^{39} - 15684 q^{41} + 25258 q^{43} - 3807 q^{45} + 9576 q^{47} + 4558 q^{49} - 16812 q^{51} - 13395 q^{53} + 26210 q^{55} - 26334 q^{57} - 47521 q^{59} + 63652 q^{61} - 21141 q^{63} - 28254 q^{65} + 44541 q^{67} - 792 q^{69} + 251680 q^{71} - 6039 q^{73} - 14445 q^{75} + 35407 q^{77} + 17588 q^{79} - 13122 q^{81} - 78650 q^{83} - 116120 q^{85} + 6903 q^{87} - 83082 q^{89} - 31747 q^{91} + 100530 q^{93} - 214946 q^{95} + 369570 q^{97} + 65934 q^{99}+O(q^{100}) \) 4 * q + 18 * q^3 - 47 * q^5 + 174 * q^7 - 162 * q^9 - 407 * q^11 + 898 * q^13 - 846 * q^15 + 1868 * q^17 - 1463 * q^19 - 783 * q^21 - 44 * q^23 + 1605 * q^25 - 2916 * q^27 + 1534 * q^29 - 11170 * q^31 + 3663 * q^33 - 9674 * q^35 + 3113 * q^37 + 4041 * q^39 - 15684 * q^41 + 25258 * q^43 - 3807 * q^45 + 9576 * q^47 + 4558 * q^49 - 16812 * q^51 - 13395 * q^53 + 26210 * q^55 - 26334 * q^57 - 47521 * q^59 + 63652 * q^61 - 21141 * q^63 - 28254 * q^65 + 44541 * q^67 - 792 * q^69 + 251680 * q^71 - 6039 * q^73 - 14445 * q^75 + 35407 * q^77 + 17588 * q^79 - 13122 * q^81 - 78650 * q^83 - 116120 * q^85 + 6903 * q^87 - 83082 * q^89 - 31747 * q^91 + 100530 * q^93 - 214946 * q^95 + 369570 * q^97 + 65934 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more