LMFDB - Embedded newform 336.6.q.j.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:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + 98x^{6} + 83x^{5} + 9122x^{4} - 91x^{3} + 28567x^{2} + 2058x + 86436 $ x^8 - x^7 + 98x^6 + 83x^5 + 9122x^4 - 91x^3 + 28567x^2 + 2058x + 86436
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{8}\cdot 3\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 21)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			289.2
Root			$-4.61193 - 7.98809i$ of defining polynomial
Character	$\chi$	$=$	336.289
Dual form			336.6.q.j.193.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.50000 + 7.79423i) q^{3} +(-11.8764 + 20.5705i) q^{5} +(-30.6840 + 125.958i) q^{7} +(-40.5000 + 70.1481i) q^{9} +O(q^{10})$	\(q+(4.50000 + 7.79423i) q^{3} +(-11.8764 + 20.5705i) q^{5} +(-30.6840 + 125.958i) q^{7} +(-40.5000 + 70.1481i) q^{9} +(232.763 + 403.157i) q^{11} -1019.30 q^{13} -213.775 q^{15} +(-280.878 - 486.496i) q^{17} +(-693.789 + 1201.68i) q^{19} +(-1119.83 + 327.654i) q^{21} +(2056.81 - 3562.50i) q^{23} +(1280.40 + 2217.72i) q^{25} -729.000 q^{27} -2381.37 q^{29} +(1475.33 + 2555.34i) q^{31} +(-2094.87 + 3628.42i) q^{33} +(-2226.61 - 2127.12i) q^{35} +(4954.48 - 8581.40i) q^{37} +(-4586.85 - 7944.66i) q^{39} -4477.13 q^{41} -5181.48 q^{43} +(-961.989 - 1666.21i) q^{45} +(-1560.80 + 2703.38i) q^{47} +(-14924.0 - 7729.82i) q^{49} +(2527.91 - 4378.46i) q^{51} +(-570.499 - 988.133i) q^{53} -11057.6 q^{55} -12488.2 q^{57} +(13748.5 + 23813.2i) q^{59} +(10551.8 - 18276.2i) q^{61} +(-7593.03 - 7253.74i) q^{63} +(12105.6 - 20967.6i) q^{65} +(-27794.2 - 48141.0i) q^{67} +37022.6 q^{69} +6076.90 q^{71} +(8389.82 + 14531.6i) q^{73} +(-11523.6 + 19959.5i) q^{75} +(-57923.1 + 16947.9i) q^{77} +(-2422.63 + 4196.12i) q^{79} +(-3280.50 - 5681.99i) q^{81} -60145.4 q^{83} +13343.3 q^{85} +(-10716.2 - 18561.0i) q^{87} +(31248.7 - 54124.4i) q^{89} +(31276.2 - 128389. i) q^{91} +(-13277.9 + 22998.1i) q^{93} +(-16479.4 - 28543.2i) q^{95} -63653.8 q^{97} -37707.6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 36 q^{3} - 258 q^{7} - 324 q^{9}+O(q^{10}) $ 8 * q + 36 * q^3 - 258 * q^7 - 324 * q^9	$ 8 q + 36 q^{3} - 258 q^{7} - 324 q^{9} + 402 q^{11} + 924 q^{13} - 276 q^{17} + 510 q^{19} - 3564 q^{21} + 6900 q^{23} - 2814 q^{25} - 5832 q^{27} + 1080 q^{29} - 6410 q^{31} - 3618 q^{33} + 33108 q^{35} - 15250 q^{37} + 4158 q^{39} + 8616 q^{41} - 58396 q^{43} - 15060 q^{47} - 64252 q^{49} + 2484 q^{51} - 13692 q^{53} - 146248 q^{55} + 9180 q^{57} + 34830 q^{59} + 5364 q^{61} - 11178 q^{63} - 66864 q^{65} - 5994 q^{67} + 124200 q^{69} - 178536 q^{71} - 59638 q^{73} + 25326 q^{75} - 75660 q^{77} - 44062 q^{79} - 26244 q^{81} + 416892 q^{83} + 72648 q^{85} + 4860 q^{87} + 77520 q^{89} - 104722 q^{91} + 57690 q^{93} - 221376 q^{95} - 377260 q^{97} - 65124 q^{99}+O(q^{100}) $ 8 * q + 36 * q^3 - 258 * q^7 - 324 * q^9 + 402 * q^11 + 924 * q^13 - 276 * q^17 + 510 * q^19 - 3564 * q^21 + 6900 * q^23 - 2814 * q^25 - 5832 * q^27 + 1080 * q^29 - 6410 * q^31 - 3618 * q^33 + 33108 * q^35 - 15250 * q^37 + 4158 * q^39 + 8616 * q^41 - 58396 * q^43 - 15060 * q^47 - 64252 * q^49 + 2484 * q^51 - 13692 * q^53 - 146248 * q^55 + 9180 * q^57 + 34830 * q^59 + 5364 * q^61 - 11178 * q^63 - 66864 * q^65 - 5994 * q^67 + 124200 * q^69 - 178536 * q^71 - 59638 * q^73 + 25326 * q^75 - 75660 * q^77 - 44062 * q^79 - 26244 * q^81 + 416892 * q^83 + 72648 * q^85 + 4860 * q^87 + 77520 * q^89 - 104722 * q^91 + 57690 * q^93 - 221376 * q^95 - 377260 * q^97 - 65124 * 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
$5$	−11.8764	+	20.5705i	−0.212452	+	0.367977i	−0.952481	−	0.304597i	$-0.901478\pi$
$5$	−11.8764	+	20.5705i	−0.212452	+	0.367977i	0.740030	+	0.672574i	$0.234811\pi$
$6$		0			0
$7$	−30.6840	+	125.958i	−0.236683	+	0.971587i
$7$	−30.6840	+	125.958i	−0.236683	+	0.971587i
$8$		0			0
$9$	−40.5000	+	70.1481i	−0.166667	+	0.288675i
$10$		0			0
$11$	232.763	+	403.157i	0.580006	+	1.00460i	0.995478	+	0.0949934i	$0.0302830\pi$
$11$	232.763	+	403.157i	0.580006	+	1.00460i	−0.415472	+	0.909606i	$0.636384\pi$
$12$		0			0
$13$	−1019.30			−1.67280			−0.836399	−	0.548121i	$-0.815343\pi$
$13$	−1019.30			−1.67280			−0.836399	+	0.548121i	$0.815343\pi$
$14$		0			0
$15$	−213.775			−0.245318
$16$		0			0
$17$	−280.878	−	486.496i	−0.235720	−	0.408279i	0.723762	−	0.690050i	$-0.242411\pi$
$17$	−280.878	−	486.496i	−0.235720	−	0.408279i	−0.959482	+	0.281771i	$0.909078\pi$
$18$		0			0
$19$	−693.789	+	1201.68i	−0.440903	+	0.763667i	−0.997757	−	0.0669438i	$-0.978675\pi$
$19$	−693.789	+	1201.68i	−0.440903	+	0.763667i	0.556853	+	0.830611i	$0.312009\pi$
$20$		0			0
$21$	−1119.83	+	327.654i	−0.554118	+	0.162131i
$22$		0			0
$23$	2056.81	−	3562.50i	0.810727	−	1.40422i	−0.101630	−	0.994822i	$-0.532406\pi$
$23$	2056.81	−	3562.50i	0.810727	−	1.40422i	0.912356	−	0.409397i	$-0.134261\pi$
$24$		0			0
$25$	1280.40	+	2217.72i	0.409729	+	0.709671i
$26$		0			0
$27$	−729.000			−0.192450
$28$		0			0
$29$	−2381.37			−0.525814			−0.262907	−	0.964821i	$-0.584681\pi$
$29$	−2381.37			−0.525814			−0.262907	+	0.964821i	$0.584681\pi$
$30$		0			0
$31$	1475.33	+	2555.34i	0.275730	+	0.477579i	0.970319	−	0.241828i	$-0.0777471\pi$
$31$	1475.33	+	2555.34i	0.275730	+	0.477579i	−0.694589	+	0.719407i	$0.744414\pi$
$32$		0			0
$33$	−2094.87	+	3628.42i	−0.334866	+	0.580006i
$34$		0			0
$35$	−2226.61	−	2127.12i	−0.307238	−	0.293509i
$36$		0			0
$37$	4954.48	−	8581.40i	0.594968	−	1.03051i	−0.398584	−	0.917132i	$-0.630498\pi$
$37$	4954.48	−	8581.40i	0.594968	−	1.03051i	0.993551	−	0.113382i	$-0.0361685\pi$
$38$		0			0
$39$	−4586.85	−	7944.66i	−0.482895	−	0.836399i
$40$		0			0
$41$	−4477.13			−0.415949			−0.207974	−	0.978134i	$-0.566687\pi$
$41$	−4477.13			−0.415949			−0.207974	+	0.978134i	$0.566687\pi$
$42$		0			0
$43$	−5181.48			−0.427349			−0.213675	−	0.976905i	$-0.568543\pi$
$43$	−5181.48			−0.427349			−0.213675	+	0.976905i	$0.568543\pi$
$44$		0			0
$45$	−961.989	−	1666.21i	−0.0708172	−	0.122659i
$46$		0			0
$47$	−1560.80	+	2703.38i	−0.103063	+	0.178510i	−0.912945	−	0.408082i	$-0.866198\pi$
$47$	−1560.80	+	2703.38i	−0.103063	+	0.178510i	0.809882	+	0.586592i	$0.199531\pi$
$48$		0			0
$49$	−14924.0	−	7729.82i	−0.887962	−	0.459917i
$50$		0			0
$51$	2527.91	−	4378.46i	0.136093	−	0.235720i
$52$		0			0
$53$	−570.499	−	988.133i	−0.0278975	−	0.0483198i	0.851740	−	0.523965i	$-0.175548\pi$
$53$	−570.499	−	988.133i	−0.0278975	−	0.0483198i	−0.879637	+	0.475645i	$0.842215\pi$
$54$		0			0
$55$	−11057.6			−0.492893
$56$		0			0
$57$	−12488.2			−0.509111
$58$		0			0
$59$	13748.5	+	23813.2i	0.514194	+	0.890610i	0.999864	+	0.0164678i	$0.00524209\pi$
$59$	13748.5	+	23813.2i	0.514194	+	0.890610i	−0.485671	+	0.874142i	$0.661425\pi$
$60$		0			0
$61$	10551.8	−	18276.2i	0.363078	−	0.628870i	−0.625387	−	0.780314i	$-0.715059\pi$
$61$	10551.8	−	18276.2i	0.363078	−	0.628870i	0.988466	+	0.151444i	$0.0483924\pi$
$62$		0			0
$63$	−7593.03	−	7253.74i	−0.241026	−	0.230256i
$64$		0			0
$65$	12105.6	−	20967.6i	0.355389	−	0.615551i
$66$		0			0
$67$	−27794.2	−	48141.0i	−0.756428	−	1.31017i	−0.944661	−	0.328047i	$-0.893610\pi$
$67$	−27794.2	−	48141.0i	−0.756428	−	1.31017i	0.188234	−	0.982124i	$-0.439724\pi$
$68$		0			0
$69$	37022.6			0.936146
$70$		0			0
$71$	6076.90			0.143066			0.0715330	−	0.997438i	$-0.477211\pi$
$71$	6076.90			0.143066			0.0715330	+	0.997438i	$0.477211\pi$
$72$		0			0
$73$	8389.82	+	14531.6i	0.184266	+	0.319158i	0.943329	−	0.331859i	$-0.107676\pi$
$73$	8389.82	+	14531.6i	0.184266	+	0.319158i	−0.759063	+	0.651017i	$0.774343\pi$
$74$		0			0
$75$	−11523.6	+	19959.5i	−0.236557	+	0.409729i
$76$		0			0
$77$	−57923.1	+	16947.9i	−1.11333	+	0.325754i
$78$		0			0
$79$	−2422.63	+	4196.12i	−0.0436737	+	0.0756450i	−0.887036	−	0.461700i	$-0.847240\pi$
$79$	−2422.63	+	4196.12i	−0.0436737	+	0.0756450i	0.843362	+	0.537345i	$0.180573\pi$
$80$		0			0
$81$	−3280.50	−	5681.99i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	−60145.4			−0.958313			−0.479156	−	0.877730i	$-0.659057\pi$
$83$	−60145.4			−0.958313			−0.479156	+	0.877730i	$0.659057\pi$
$84$		0			0
$85$	13343.3			0.200316
$86$		0			0
$87$	−10716.2	−	18561.0i	−0.151790	−	0.262907i
$88$		0			0
$89$	31248.7	−	54124.4i	0.418174	−	0.724299i	−0.577582	−	0.816333i	$-0.696004\pi$
$89$	31248.7	−	54124.4i	0.418174	−	0.724299i	0.995756	+	0.0920340i	$0.0293368\pi$
$90$		0			0
$91$	31276.2	−	128389.i	0.395923	−	1.62527i
$92$		0			0
$93$	−13277.9	+	22998.1i	−0.159193	+	0.275730i
$94$		0			0
$95$	−16479.4	−	28543.2i	−0.187341	−	0.324485i
$96$		0			0
$97$	−63653.8			−0.686903			−0.343451	−	0.939170i	$-0.611596\pi$
$97$	−63653.8			−0.686903			−0.343451	+	0.939170i	$0.611596\pi$
$98$		0			0
$99$	−37707.6			−0.386670
$100$		0			0
$101$	−92311.4	−	159888.i	−0.900434	−	1.55960i	−0.826932	−	0.562302i	$-0.809916\pi$
$101$	−92311.4	−	159888.i	−0.900434	−	1.55960i	−0.0735022	−	0.997295i	$-0.523418\pi$
$102$		0			0
$103$	26021.9	−	45071.2i	0.241683	−	0.418607i	−0.719511	−	0.694481i	$-0.755634\pi$
$103$	26021.9	−	45071.2i	0.241683	−	0.418607i	0.961194	+	0.275874i	$0.0889674\pi$
$104$		0			0
$105$	6559.49	−	26926.8i	0.0580627	−	0.238348i
$106$		0			0
$107$	24088.9	−	41723.2i	0.203403	−	0.352304i	−0.746220	−	0.665700i	$-0.768133\pi$
$107$	24088.9	−	41723.2i	0.203403	−	0.352304i	0.949623	+	0.313395i	$0.101466\pi$
$108$		0			0
$109$	18217.8	+	31554.1i	0.146869	+	0.254384i	0.930069	−	0.367386i	$-0.119747\pi$
$109$	18217.8	+	31554.1i	0.146869	+	0.254384i	−0.783200	+	0.621770i	$0.786414\pi$
$110$		0			0
$111$	89180.6			0.687010
$112$		0			0
$113$	−96711.1			−0.712492			−0.356246	−	0.934392i	$-0.615944\pi$
$113$	−96711.1			−0.712492			−0.356246	+	0.934392i	$0.615944\pi$
$114$		0			0
$115$	48855.0	+	84619.4i	0.344480	+	0.596658i
$116$		0			0
$117$	41281.6	−	71501.9i	0.278800	−	0.482895i
$118$		0			0
$119$	69896.6	−	20451.3i	0.452469	−	0.132390i
$120$		0			0
$121$	−27831.7	+	48206.0i	−0.172813	+	0.299321i
$122$		0			0
$123$	−20147.1	−	34895.7i	−0.120074	−	0.207974i
$124$		0			0
$125$	−135054.			−0.773093
$126$		0			0
$127$	−23322.9			−0.128314			−0.0641568	−	0.997940i	$-0.520436\pi$
$127$	−23322.9			−0.128314			−0.0641568	+	0.997940i	$0.520436\pi$
$128$		0			0
$129$	−23316.7	−	40385.7i	−0.123365	−	0.213675i
$130$		0			0
$131$	−169379.	+	293373.i	−0.862345	+	1.49363i	0.00731374	+	0.999973i	$0.497672\pi$
$131$	−169379.	+	293373.i	−0.862345	+	1.49363i	−0.869659	+	0.493653i	$0.835661\pi$
$132$		0			0
$133$	−130073.	−	124261.i	−0.637614	−	0.609123i
$134$		0			0
$135$	8657.90	−	14995.9i	0.0408863	−	0.0708172i
$136$		0			0
$137$	31438.4	+	54452.9i	0.143106	+	0.247867i	0.928665	−	0.370920i	$-0.120958\pi$
$137$	31438.4	+	54452.9i	0.143106	+	0.247867i	−0.785559	+	0.618787i	$0.787624\pi$
$138$		0			0
$139$	−211927.			−0.930356			−0.465178	−	0.885217i	$-0.654010\pi$
$139$	−211927.			−0.930356			−0.465178	+	0.885217i	$0.654010\pi$
$140$		0			0
$141$	−28094.3			−0.119007
$142$		0			0
$143$	−237255.	−	410938.i	−0.970233	−	1.68049i
$144$		0			0
$145$	28282.2	−	48986.1i	0.111710	−	0.193488i
$146$		0			0
$147$	−6909.92	−	151105.i	−0.0263742	−	0.576748i
$148$		0			0
$149$	−70136.4	+	121480.i	−0.258808	+	0.448269i	−0.965923	−	0.258830i	$-0.916663\pi$
$149$	−70136.4	+	121480.i	−0.258808	+	0.448269i	0.707115	+	0.707099i	$0.249996\pi$
$150$		0			0
$151$	−81995.7	−	142021.i	−0.292650	−	0.506885i	0.681785	−	0.731552i	$-0.261204\pi$
$151$	−81995.7	−	142021.i	−0.292650	−	0.506885i	−0.974436	+	0.224667i	$0.927870\pi$
$152$		0			0
$153$	45502.3			0.157147
$154$		0			0
$155$	−70086.4			−0.234317
$156$		0			0
$157$	−278272.	−	481981.i	−0.900990	−	1.56056i	−0.826212	−	0.563360i	$-0.809508\pi$
$157$	−278272.	−	481981.i	−0.900990	−	1.56056i	−0.0747781	−	0.997200i	$-0.523825\pi$
$158$		0			0
$159$	5134.49	−	8893.19i	0.0161066	−	0.0278975i
$160$		0			0
$161$	385615.	+	368384.i	1.17244	+	1.12005i
$162$		0			0
$163$	−9863.32	+	17083.8i	−0.0290773	+	0.0503634i	−0.880198	−	0.474607i	$-0.842590\pi$
$163$	−9863.32	+	17083.8i	−0.0290773	+	0.0503634i	0.851121	+	0.524970i	$0.175924\pi$
$164$		0			0
$165$	−49759.0	−	86185.1i	−0.142286	−	0.246446i
$166$		0			0
$167$	−94776.2			−0.262971			−0.131486	−	0.991318i	$-0.541975\pi$
$167$	−94776.2			−0.262971			−0.131486	+	0.991318i	$0.541975\pi$
$168$		0			0
$169$	667679.			1.79825
$170$		0			0
$171$	−56196.9	−	97335.9i	−0.146968	−	0.254556i
$172$		0			0
$173$	−169420.	+	293445.i	−0.430379	+	0.745437i	−0.996906	−	0.0786055i	$-0.974953\pi$
$173$	−169420.	+	293445.i	−0.430379	+	0.745437i	0.566527	+	0.824043i	$0.308287\pi$
$174$		0			0
$175$	−318628.	+	93228.6i	−0.786483	+	0.230120i
$176$		0			0
$177$	−123737.	+	214319.i	−0.296870	+	0.514194i
$178$		0			0
$179$	388096.	+	672203.i	0.905330	+	1.56808i	0.820473	+	0.571685i	$0.193710\pi$
$179$	388096.	+	672203.i	0.905330	+	1.56808i	0.0848573	+	0.996393i	$0.472957\pi$
$180$		0			0
$181$	−132697.			−0.301067			−0.150534	−	0.988605i	$-0.548099\pi$
$181$	−132697.			−0.301067			−0.150534	+	0.988605i	$0.548099\pi$
$182$		0			0
$183$	189932.			0.419247
$184$		0			0
$185$	117683.	+	203833.i	0.252804	+	0.437869i
$186$		0			0
$187$	130756.	−	226476.i	0.273438	−	0.473608i
$188$		0			0
$189$	22368.7	−	91823.6i	0.0455497	−	0.186982i
$190$		0			0
$191$	3318.71	−	5748.17i	0.00658242	−	0.0114011i	−0.862715	−	0.505690i	$-0.831238\pi$
$191$	3318.71	−	5748.17i	0.00658242	−	0.0114011i	0.869298	+	0.494289i	$0.164571\pi$
$192$		0			0
$193$	−226295.	−	391954.i	−0.437302	−	0.757430i	0.560178	−	0.828372i	$-0.310733\pi$
$193$	−226295.	−	391954.i	−0.437302	−	0.757430i	−0.997480	+	0.0709425i	$0.977399\pi$
$194$		0			0
$195$	217901.			0.410368
$196$		0			0
$197$	816952.			1.49979			0.749896	−	0.661556i	$-0.230104\pi$
$197$	816952.			1.49979			0.749896	+	0.661556i	$0.230104\pi$
$198$		0			0
$199$	403208.	+	698378.i	0.721767	+	1.25014i	0.960291	+	0.279000i	$0.0900031\pi$
$199$	403208.	+	698378.i	0.721767	+	1.25014i	−0.238524	+	0.971137i	$0.576664\pi$
$200$		0			0
$201$	250148.	−	433269.i	0.436724	−	0.756428i
$202$		0			0
$203$	73070.2	−	299954.i	0.124451	−	0.510874i
$204$		0			0
$205$	53172.2	−	92096.9i	0.0883690	−	0.153060i
$206$		0			0
$207$	166602.	+	288562.i	0.270242	+	0.468073i
$208$		0			0
$209$	−645954.			−1.02291
$210$		0			0
$211$	−68773.9			−0.106345			−0.0531726	−	0.998585i	$-0.516933\pi$
$211$	−68773.9			−0.106345			−0.0531726	+	0.998585i	$0.516933\pi$
$212$		0			0
$213$	27346.1	+	47364.8i	0.0412996	+	0.0715330i
$214$		0			0
$215$	61537.4	−	106586.i	0.0907911	−	0.157255i
$216$		0			0
$217$	−367136.	+	107421.i	−0.529270	+	0.154861i
$218$		0			0
$219$	−75508.4	+	130784.i	−0.106386	+	0.184266i
$220$		0			0
$221$	286299.	+	495885.i	0.394312	+	0.682968i
$222$		0			0
$223$	620227.			0.835196			0.417598	−	0.908632i	$-0.362872\pi$
$223$	620227.			0.835196			0.417598	+	0.908632i	$0.362872\pi$
$224$		0			0
$225$	−207425.			−0.273152
$226$		0			0
$227$	501116.	+	867959.i	0.645467	+	1.11798i	0.984193	+	0.177096i	$0.0566705\pi$
$227$	501116.	+	867959.i	0.645467	+	1.11798i	−0.338727	+	0.940885i	$0.609996\pi$
$228$		0			0
$229$	−442931.	+	767178.i	−0.558145	+	0.966735i	0.439506	+	0.898239i	$0.355153\pi$
$229$	−442931.	+	767178.i	−0.558145	+	0.966735i	−0.997651	+	0.0684960i	$0.978180\pi$
$230$		0			0
$231$	−392750.	−	375200.i	−0.484269	−	0.462629i
$232$		0			0
$233$	−298082.	+	516293.i	−0.359704	+	0.623026i	−0.987911	−	0.155020i	$-0.950456\pi$
$233$	−298082.	+	516293.i	−0.359704	+	0.623026i	0.628207	+	0.778046i	$0.283789\pi$
$234$		0			0
$235$	−37073.3	−	64212.9i	−0.0437917	−	0.0758495i
$236$		0			0
$237$	−43607.4			−0.0504300
$238$		0			0
$239$	−743111.			−0.841509			−0.420754	−	0.907175i	$-0.638235\pi$
$239$	−743111.			−0.841509			−0.420754	+	0.907175i	$0.638235\pi$
$240$		0			0
$241$	587419.	+	1.01744e6i	0.651487	+	1.12841i	0.982762	+	0.184874i	$0.0591877\pi$
$241$	587419.	+	1.01744e6i	0.651487	+	1.12841i	−0.331276	+	0.943534i	$0.607479\pi$
$242$		0			0
$243$	29524.5	−	51137.9i	0.0320750	−	0.0555556i
$244$		0			0
$245$	336250.	−	215192.i	0.357888	−	0.229040i
$246$		0			0
$247$	707179.	−	1.22487e6i	0.737542	−	1.27746i
$248$		0			0
$249$	−270654.	−	468787.i	−0.276641	−	0.479156i
$250$		0			0
$251$	−352992.			−0.353655			−0.176828	−	0.984242i	$-0.556584\pi$
$251$	−352992.			−0.353655			−0.176828	+	0.984242i	$0.556584\pi$
$252$		0			0
$253$	1.91500e6			1.88090
$254$		0			0
$255$	60044.9	+	104001.i	0.0578263	+	0.100158i
$256$		0			0
$257$	−5006.34	+	8671.23i	−0.00472811	+	0.00818932i	−0.868380	−	0.495900i	$-0.834838\pi$
$257$	−5006.34	+	8671.23i	−0.00472811	+	0.00818932i	0.863652	+	0.504089i	$0.168172\pi$
$258$		0			0
$259$	928875.	+	887369.i	0.860415	+	0.821968i
$260$		0			0
$261$	96445.6	−	167049.i	0.0876357	−	0.151790i
$262$		0			0
$263$	840900.	+	1.45648e6i	0.749644	+	1.29842i	0.947993	+	0.318290i	$0.103109\pi$
$263$	840900.	+	1.45648e6i	0.749644	+	1.29842i	−0.198349	+	0.980131i	$0.563558\pi$
$264$		0			0
$265$	27101.9			0.0237075
$266$		0			0
$267$	562477.			0.482866
$268$		0			0
$269$	958587.	+	1.66032e6i	0.807702	+	1.39898i	0.914452	+	0.404694i	$0.132622\pi$
$269$	958587.	+	1.66032e6i	0.807702	+	1.39898i	−0.106750	+	0.994286i	$0.534045\pi$
$270$		0			0
$271$	1.14500e6	−	1.98320e6i	0.947069	−	1.64037i	0.195515	−	0.980701i	$-0.437362\pi$
$271$	1.14500e6	−	1.98320e6i	0.947069	−	1.64037i	0.751554	−	0.659672i	$-0.229305\pi$
$272$		0			0
$273$	1.14144e6	−	333977.i	0.926928	−	0.271213i
$274$		0			0
$275$	−596060.	+	1.03241e6i	−0.475290	+	0.823226i
$276$		0			0
$277$	197401.	+	341908.i	0.154579	+	0.267738i	0.932906	−	0.360121i	$-0.117265\pi$
$277$	197401.	+	341908.i	0.154579	+	0.267738i	−0.778327	+	0.627859i	$0.783931\pi$
$278$		0			0
$279$	−239003.			−0.183820
$280$		0			0
$281$	−1.77699e6			−1.34252			−0.671259	−	0.741223i	$-0.734246\pi$
$281$	−1.77699e6			−1.34252			−0.671259	+	0.741223i	$0.734246\pi$
$282$		0			0
$283$	607622.	+	1.05243e6i	0.450991	+	0.781139i	0.998448	−	0.0556949i	$-0.0177374\pi$
$283$	607622.	+	1.05243e6i	0.450991	+	0.781139i	−0.547457	+	0.836834i	$0.684404\pi$
$284$		0			0
$285$	148315.	−	256889.i	0.108162	−	0.187341i
$286$		0			0
$287$	137376.	−	563931.i	0.0984481	−	0.404130i
$288$		0			0
$289$	552143.	−	956340.i	0.388872	−	0.673547i
$290$		0			0
$291$	−286442.	−	496132.i	−0.198292	−	0.343451i
$292$		0			0
$293$	−1.48897e6			−1.01325			−0.506627	−	0.862165i	$-0.669108\pi$
$293$	−1.48897e6			−1.01325			−0.506627	+	0.862165i	$0.669108\pi$
$294$		0			0
$295$	−653133.			−0.436965
$296$		0			0
$297$	−169684.	−	293902.i	−0.111622	−	0.193335i
$298$		0			0
$299$	−2.09651e6	+	3.63125e6i	−1.35618	+	2.34898i
$300$		0			0
$301$	158989.	−	652651.i	0.101146	−	0.415207i
$302$		0			0
$303$	830803.	−	1.43899e6i	0.519866	−	0.900434i
$304$		0			0
$305$	250634.	+	434111.i	0.154273	+	0.267209i
$306$		0			0
$307$	−2.03109e6			−1.22994			−0.614968	−	0.788552i	$-0.710831\pi$
$307$	−2.03109e6			−1.22994			−0.614968	+	0.788552i	$0.710831\pi$
$308$		0			0
$309$	468394.			0.279071
$310$		0			0
$311$	144892.	+	250961.i	0.0849463	+	0.147131i	0.905368	−	0.424627i	$-0.139595\pi$
$311$	144892.	+	250961.i	0.0849463	+	0.147131i	−0.820422	+	0.571758i	$0.806261\pi$
$312$		0			0
$313$	−109105.	+	188976.i	−0.0629484	+	0.109030i	−0.895782	−	0.444493i	$-0.853384\pi$
$313$	−109105.	+	188976.i	−0.0629484	+	0.109030i	0.832834	+	0.553523i	$0.186717\pi$
$314$		0			0
$315$	239391.	−	70044.3i	0.135935	−	0.0397737i
$316$		0			0
$317$	−645315.	+	1.11772e6i	−0.360681	+	0.624718i	−0.988073	−	0.153985i	$-0.950789\pi$
$317$	−645315.	+	1.11772e6i	−0.360681	+	0.624718i	0.627392	+	0.778704i	$0.284122\pi$
$318$		0			0
$319$	−554296.	−	960068.i	−0.304975	−	0.528233i
$320$		0			0
$321$	433600.			0.234870
$322$		0			0
$323$	779481.			0.415719
$324$		0			0
$325$	−1.30511e6	−	2.26052e6i	−0.685393	−	1.18714i
$326$		0			0
$327$	−163960.	+	283987.i	−0.0847947	+	0.146869i
$328$		0			0
$329$	−292621.	−	279546.i	−0.149045	−	0.142385i
$330$		0			0
$331$	−1.74280e6	+	3.01862e6i	−0.874336	+	1.51439i	−0.0168673	+	0.999858i	$0.505369\pi$
$331$	−1.74280e6	+	3.01862e6i	−0.874336	+	1.51439i	−0.857469	+	0.514536i	$0.827964\pi$
$332$		0			0
$333$	401313.	+	695094.i	0.198323	+	0.343505i
$334$		0			0
$335$	1.32038e6			0.642817
$336$		0			0
$337$	249198.			0.119528			0.0597641	−	0.998213i	$-0.480965\pi$
$337$	249198.			0.119528			0.0597641	+	0.998213i	$0.480965\pi$
$338$		0			0
$339$	−435200.	−	753788.i	−0.205679	−	0.356246i
$340$		0			0
$341$	−686803.	+	1.18958e6i	−0.319850	+	0.553997i
$342$		0			0
$343$	1.43156e6	−	1.64262e6i	0.657015	−	0.753878i
$344$		0			0
$345$	−439695.	+	761574.i	−0.198886	+	0.344480i
$346$		0			0
$347$	−753007.	−	1.30425e6i	−0.335719	−	0.581482i	0.647904	−	0.761722i	$-0.275646\pi$
$347$	−753007.	−	1.30425e6i	−0.335719	−	0.581482i	−0.983623	+	0.180240i	$0.942312\pi$
$348$		0			0
$349$	1.54370e6			0.678423			0.339212	−	0.940710i	$-0.389840\pi$
$349$	1.54370e6			0.678423			0.339212	+	0.940710i	$0.389840\pi$
$350$		0			0
$351$	743070.			0.321930
$352$		0			0
$353$	838978.	+	1.45315e6i	0.358355	+	0.620689i	0.987686	−	0.156448i	$-0.0500043\pi$
$353$	838978.	+	1.45315e6i	0.358355	+	0.620689i	−0.629331	+	0.777137i	$0.716671\pi$
$354$		0			0
$355$	−72171.8	+	125005.i	−0.0303946	+	0.0526450i
$356$		0			0
$357$	473937.	+	452760.i	0.196811	+	0.188017i
$358$		0			0
$359$	−1.24987e6	+	2.16483e6i	−0.511832	+	0.886519i	0.488074	+	0.872802i	$0.337700\pi$
$359$	−1.24987e6	+	2.16483e6i	−0.511832	+	0.886519i	−0.999906	+	0.0137165i	$0.995634\pi$
$360$		0			0
$361$	275363.	+	476943.i	0.111208	+	0.192619i
$362$		0			0
$363$	−500971.			−0.199547
$364$		0			0
$365$	−398564.			−0.156591
$366$		0			0
$367$	−1.05371e6	−	1.82507e6i	−0.408370	−	0.707318i	0.586337	−	0.810067i	$-0.300569\pi$
$367$	−1.05371e6	−	1.82507e6i	−0.408370	−	0.707318i	−0.994707	+	0.102749i	$0.967236\pi$
$368$		0			0
$369$	181324.	−	314062.i	0.0693248	−	0.120074i
$370$		0			0
$371$	141969.	−	41539.1i	0.0535498	−	0.0156683i
$372$		0			0
$373$	−1.36612e6	+	2.36619e6i	−0.508414	+	0.880599i	0.491538	+	0.870856i	$0.336435\pi$
$373$	−1.36612e6	+	2.36619e6i	−0.508414	+	0.880599i	−0.999953	+	0.00974333i	$0.996899\pi$
$374$		0			0
$375$	−607742.	−	1.05264e6i	−0.223173	−	0.386547i
$376$		0			0
$377$	2.42733e6			0.879581
$378$		0			0
$379$	2.04295e6			0.730567			0.365283	−	0.930896i	$-0.380972\pi$
$379$	2.04295e6			0.730567			0.365283	+	0.930896i	$0.380972\pi$
$380$		0			0
$381$	−104953.	−	181784.i	−0.0370410	−	0.0641568i
$382$		0			0
$383$	754497.	−	1.30683e6i	0.262821	−	0.455220i	−0.704169	−	0.710032i	$-0.748680\pi$
$383$	754497.	−	1.30683e6i	0.262821	−	0.455220i	0.966991	+	0.254812i	$0.0820137\pi$
$384$		0			0
$385$	339290.	−	1.39279e6i	0.116659	−	0.478888i
$386$		0			0
$387$	209850.	−	363471.i	0.0712249	−	0.123365i
$388$		0			0
$389$	−2.25033e6	−	3.89768e6i	−0.754000	−	1.30597i	−0.945870	−	0.324546i	$-0.894788\pi$
$389$	−2.25033e6	−	3.89768e6i	−0.754000	−	1.30597i	0.191870	−	0.981420i	$-0.438545\pi$
$390$		0			0
$391$	−2.31085e6			−0.764417
$392$		0			0
$393$	−3.04882e6			−0.995751
$394$		0			0
$395$	−57544.3	−	99669.7i	−0.0185571	−	0.0321418i
$396$		0			0
$397$	1.76068e6	−	3.04958e6i	0.560665	−	0.971101i	−0.436773	−	0.899572i	$-0.643879\pi$
$397$	1.76068e6	−	3.04958e6i	0.560665	−	0.971101i	0.997439	−	0.0715292i	$-0.0227879\pi$
$398$		0			0
$399$	383189.	−	1.57299e6i	0.120498	−	0.494646i
$400$		0			0
$401$	453554.	−	785578.i	0.140853	−	0.243965i	−0.786965	−	0.616998i	$-0.788349\pi$
$401$	453554.	−	785578.i	0.140853	−	0.243965i	0.927818	+	0.373033i	$0.121682\pi$
$402$		0			0
$403$	−1.50380e6	−	2.60466e6i	−0.461241	−	0.798893i
$404$		0			0
$405$	155842.			0.0472115
$406$		0			0
$407$	4.61287e6			1.38034
$408$		0			0
$409$	−2.15927e6	−	3.73996e6i	−0.638260	−	1.10550i	−0.985814	−	0.167839i	$-0.946321\pi$
$409$	−2.15927e6	−	3.73996e6i	−0.638260	−	1.10550i	0.347554	−	0.937660i	$-0.387012\pi$
$410$		0			0
$411$	−282945.	+	490076.i	−0.0826224	+	0.143106i
$412$		0			0
$413$	−3.42133e6	+	1.00106e6i	−0.987006	+	0.288791i
$414$		0			0
$415$	714311.	−	1.23722e6i	0.203595	−	0.352637i
$416$		0			0
$417$	−953671.	−	1.65181e6i	−0.268571	−	0.465178i
$418$		0			0
$419$	1.52041e6			0.423083			0.211541	−	0.977369i	$-0.432152\pi$
$419$	1.52041e6			0.423083			0.211541	+	0.977369i	$0.432152\pi$
$420$		0			0
$421$	−4.42050e6			−1.21553			−0.607766	−	0.794116i	$-0.707934\pi$
$421$	−4.42050e6			−1.21553			−0.607766	+	0.794116i	$0.707934\pi$
$422$		0			0
$423$	−126425.	−	218974.i	−0.0343542	−	0.0595033i
$424$		0			0
$425$	719274.	−	1.24582e6i	0.193162	−	0.334567i
$426$		0			0
$427$	1.97827e6	+	1.88987e6i	0.525068	+	0.501605i
$428$		0			0
$429$	2.13530e6	−	3.69844e6i	0.560164	−	0.970233i
$430$		0			0
$431$	−3.42393e6	−	5.93041e6i	−0.887833	−	1.53777i	−0.842432	−	0.538803i	$-0.818877\pi$
$431$	−3.42393e6	−	5.93041e6i	−0.887833	−	1.53777i	−0.0454009	−	0.998969i	$-0.514457\pi$
$432$		0			0
$433$	−3.99328e6			−1.02355			−0.511777	−	0.859119i	$-0.671012\pi$
$433$	−3.99328e6			−1.02355			−0.511777	+	0.859119i	$0.671012\pi$
$434$		0			0
$435$	509079.			0.128992
$436$		0			0
$437$	2.85398e6	+	4.94324e6i	0.714904	+	1.23825i
$438$		0			0
$439$	742783.	−	1.28654e6i	0.183950	−	0.318611i	−0.759272	−	0.650773i	$-0.774445\pi$
$439$	742783.	−	1.28654e6i	0.183950	−	0.318611i	0.943222	+	0.332162i	$0.107778\pi$
$440$		0			0
$441$	1.14665e6	−	733830.i	0.280760	−	0.179680i
$442$		0			0
$443$	−2.81097e6	+	4.86874e6i	−0.680528	+	1.17871i	0.294291	+	0.955716i	$0.404916\pi$
$443$	−2.81097e6	+	4.86874e6i	−0.680528	+	1.17871i	−0.974820	+	0.222994i	$0.928417\pi$
$444$		0			0
$445$	742245.	+	1.28561e6i	0.177684	+	0.307757i
$446$		0			0
$447$	−1.26246e6			−0.298846
$448$		0			0
$449$	1.94883e6			0.456202			0.228101	−	0.973637i	$-0.426748\pi$
$449$	1.94883e6			0.456202			0.228101	+	0.973637i	$0.426748\pi$
$450$		0			0
$451$	−1.04211e6	−	1.80499e6i	−0.241253	−	0.417862i
$452$		0			0
$453$	737961.	−	1.27819e6i	0.168962	−	0.292650i
$454$		0			0
$455$	2.26959e6	+	2.16817e6i	0.513947	+	0.490982i
$456$		0			0
$457$	1.33656e6	−	2.31499e6i	0.299363	−	0.518511i	−0.676628	−	0.736325i	$-0.736559\pi$
$457$	1.33656e6	−	2.31499e6i	0.299363	−	0.518511i	0.975990	+	0.217814i	$0.0698926\pi$
$458$		0			0
$459$	204760.	+	354655.i	0.0453643	+	0.0785733i
$460$		0			0
$461$	−4.65262e6			−1.01964			−0.509819	−	0.860282i	$-0.670287\pi$
$461$	−4.65262e6			−1.01964			−0.509819	+	0.860282i	$0.670287\pi$
$462$		0			0
$463$	5.18586e6			1.12426			0.562132	−	0.827047i	$-0.309981\pi$
$463$	5.18586e6			1.12426			0.562132	+	0.827047i	$0.309981\pi$
$464$		0			0
$465$	−315389.	−	546269.i	−0.0676416	−	0.117159i
$466$		0			0
$467$	−2.52990e6	+	4.38192e6i	−0.536799	+	0.929763i	0.462275	+	0.886737i	$0.347033\pi$
$467$	−2.52990e6	+	4.38192e6i	−0.536799	+	0.929763i	−0.999074	+	0.0430263i	$0.986300\pi$
$468$		0			0
$469$	6.91660e6	−	2.02375e6i	1.45198	−	0.424840i
$470$		0			0
$471$	2.50444e6	−	4.33783e6i	0.520187	−	0.900990i
$472$		0			0
$473$	−1.20606e6	−	2.08895e6i	−0.247865	−	0.429315i
$474$		0			0
$475$	−3.55331e6			−0.722603
$476$		0			0
$477$	92420.8			0.0185983
$478$		0			0
$479$	−2.77543e6	−	4.80718e6i	−0.552702	−	0.957308i	−0.998078	−	0.0619642i	$-0.980264\pi$
$479$	−2.77543e6	−	4.80718e6i	−0.552702	−	0.957308i	0.445377	−	0.895343i	$-0.353070\pi$
$480$		0			0
$481$	−5.05010e6	+	8.74702e6i	−0.995261	+	1.72384i
$482$		0			0
$483$	−1.13600e6	+	4.66330e6i	−0.221570	+	0.909548i
$484$		0			0
$485$	755979.	−	1.30939e6i	0.145934	−	0.252764i
$486$		0			0
$487$	−3.43905e6	−	5.95661e6i	−0.657077	−	1.13809i	−0.981369	−	0.192133i	$-0.938459\pi$
$487$	−3.43905e6	−	5.95661e6i	−0.657077	−	1.13809i	0.324292	−	0.945957i	$-0.394874\pi$
$488$		0			0
$489$	−177540.			−0.0335756
$490$		0			0
$491$	−7.44459e6			−1.39360			−0.696798	−	0.717267i	$-0.745393\pi$
$491$	−7.44459e6			−1.39360			−0.696798	+	0.717267i	$0.745393\pi$
$492$		0			0
$493$	668876.	+	1.15853e6i	0.123945	+	0.214679i
$494$		0			0
$495$	447831.	−	775666.i	0.0821488	−	0.142286i
$496$		0			0
$497$	−186464.	+	765436.i	−0.0338613	+	0.139001i
$498$		0			0
$499$	1.07665e6	−	1.86481e6i	0.193563	−	0.335261i	−0.752866	−	0.658174i	$-0.771329\pi$
$499$	1.07665e6	−	1.86481e6i	0.193563	−	0.335261i	0.946428	+	0.322914i	$0.104662\pi$
$500$		0			0
$501$	−426493.	−	738707.i	−0.0759132	−	0.131486i
$502$		0			0
$503$	−8.61012e6			−1.51736			−0.758681	−	0.651463i	$-0.774156\pi$
$503$	−8.61012e6			−1.51736			−0.758681	+	0.651463i	$0.774156\pi$
$504$		0			0
$505$	4.38531e6			0.765195
$506$		0			0
$507$	3.00456e6	+	5.20404e6i	0.519111	+	0.899127i
$508$		0			0
$509$	−1.67100e6	+	2.89426e6i	−0.285879	+	0.495157i	−0.972822	−	0.231554i	$-0.925619\pi$
$509$	−1.67100e6	+	2.89426e6i	−0.285879	+	0.495157i	0.686943	+	0.726712i	$0.258952\pi$
$510$		0			0
$511$	−2.08781e6	+	610879.i	−0.353703	+	0.103491i
$512$		0			0
$513$	505772.	−	876023.i	0.0848519	−	0.146968i
$514$		0			0
$515$	618093.	+	1.07057e6i	0.102692	+	0.177867i
$516$		0			0
$517$	−1.45318e6			−0.239108
$518$		0			0
$519$	−3.04957e6			−0.496958
$520$		0			0
$521$	1.34152e6	+	2.32359e6i	0.216523	+	0.375029i	0.953743	−	0.300624i	$-0.0971950\pi$
$521$	1.34152e6	+	2.32359e6i	0.216523	+	0.375029i	−0.737220	+	0.675653i	$0.763862\pi$
$522$		0			0
$523$	−2.71178e6	+	4.69694e6i	−0.433511	+	0.750863i	−0.997173	−	0.0751429i	$-0.976059\pi$
$523$	−2.71178e6	+	4.69694e6i	−0.433511	+	0.750863i	0.563662	+	0.826005i	$0.309392\pi$
$524$		0			0
$525$	−2.16047e6	−	2.06393e6i	−0.342098	−	0.326811i
$526$		0			0
$527$	828775.	−	1.43548e6i	0.129990	−	0.225149i
$528$		0			0
$529$	−5.24276e6	−	9.08072e6i	−0.814555	−	1.41085i
$530$		0			0
$531$	−2.22726e6			−0.342796
$532$		0			0
$533$	4.56353e6			0.695798
$534$		0			0
$535$	572179.	+	991044.i	0.0864266	+	0.149695i
$536$		0			0
$537$	−3.49287e6	+	6.04983e6i	−0.522693	+	0.905330i
$538$		0			0
$539$	−357417.	−	7.81593e6i	−0.0529911	−	1.15880i
$540$		0			0
$541$	−1.09228e6	+	1.89188e6i	−0.160450	+	0.277908i	−0.935030	−	0.354568i	$-0.884628\pi$
$541$	−1.09228e6	+	1.89188e6i	−0.160450	+	0.277908i	0.774580	+	0.632476i	$0.217961\pi$
$542$		0			0
$543$	−597135.	−	1.03427e6i	−0.0869106	−	0.150534i
$544$		0			0
$545$	−865447.			−0.124810
$546$		0			0
$547$	691437.			0.0988062			0.0494031	−	0.998779i	$-0.484268\pi$
$547$	691437.			0.0988062			0.0494031	+	0.998779i	$0.484268\pi$
$548$		0			0
$549$	854693.	+	1.48037e6i	0.121026	+	0.209623i
$550$		0			0
$551$	1.65217e6	−	2.86164e6i	0.231833	−	0.401547i
$552$		0			0
$553$	−454200.	−	433904.i	−0.0631589	−	0.0603367i
$554$		0			0
$555$	−1.05914e6	+	1.83449e6i	−0.145956	+	0.252804i
$556$		0			0
$557$	−2.73828e6	−	4.74284e6i	−0.373973	−	0.647740i	0.616200	−	0.787590i	$-0.288671\pi$
$557$	−2.73828e6	−	4.74284e6i	−0.373973	−	0.647740i	−0.990173	+	0.139850i	$0.955338\pi$
$558$		0			0
$559$	5.28149e6			0.714869
$560$		0			0
$561$	2.35361e6			0.315739
$562$		0			0
$563$	199488.	+	345523.i	0.0265244	+	0.0459416i	0.878983	−	0.476853i	$-0.158223\pi$
$563$	199488.	+	345523.i	0.0265244	+	0.0459416i	−0.852459	+	0.522795i	$0.824889\pi$
$564$		0			0
$565$	1.14858e6	−	1.98940e6i	0.151370	−	0.262181i
$566$		0			0
$567$	816353.	−	238860.i	0.106640	−	0.0312022i
$568$		0			0
$569$	2.11556e6	−	3.66426e6i	0.273934	−	0.474467i	−0.695932	−	0.718108i	$-0.745008\pi$
$569$	2.11556e6	−	3.66426e6i	0.273934	−	0.474467i	0.969866	+	0.243641i	$0.0783418\pi$
$570$		0			0
$571$	4.37551e6	+	7.57861e6i	0.561615	+	0.972746i	0.997356	+	0.0726734i	$0.0231531\pi$
$571$	4.37551e6	+	7.57861e6i	0.561615	+	0.972746i	−0.435741	+	0.900072i	$0.643514\pi$
$572$		0			0
$573$	59736.7			0.00760072
$574$		0			0
$575$	1.05342e7			1.32871
$576$		0			0
$577$	883558.	+	1.53037e6i	0.110483	+	0.191362i	0.915965	−	0.401258i	$-0.131427\pi$
$577$	883558.	+	1.53037e6i	0.110483	+	0.191362i	−0.805482	+	0.592620i	$0.798094\pi$
$578$		0			0
$579$	2.03665e6	−	3.52759e6i	0.252477	−	0.437302i
$580$		0			0
$581$	1.84550e6	−	7.57581e6i	0.226817	−	0.931084i
$582$		0			0
$583$	265582.	−	460001.i	0.0323614	−	0.0560516i
$584$		0			0
$585$	980555.	+	1.69837e6i	0.118463	+	0.205184i
$586$		0			0
$587$	8.65009e6			1.03616			0.518078	−	0.855333i	$-0.326648\pi$
$587$	8.65009e6			1.03616			0.518078	+	0.855333i	$0.326648\pi$
$588$		0			0
$589$	−4.09426e6			−0.486281
$590$		0			0
$591$	3.67628e6	+	6.36751e6i	0.432953	+	0.749896i
$592$		0			0
$593$	−7.72568e6	+	1.33813e7i	−0.902194	+	1.56265i	−0.0775549	+	0.996988i	$0.524711\pi$
$593$	−7.72568e6	+	1.33813e7i	−0.902194	+	1.56265i	−0.824640	+	0.565659i	$0.808622\pi$
$594$		0			0
$595$	−409427.	+	1.68070e6i	−0.0474115	+	0.194625i
$596$		0			0
$597$	−3.62888e6	+	6.28540e6i	−0.416712	+	0.721767i
$598$		0			0
$599$	−1.92400e6	−	3.33247e6i	−0.219098	−	0.379489i	0.735434	−	0.677596i	$-0.236978\pi$
$599$	−1.92400e6	−	3.33247e6i	−0.219098	−	0.379489i	−0.954533	+	0.298107i	$0.903645\pi$
$600$		0			0
$601$	1.27578e7			1.44075			0.720376	−	0.693583i	$-0.243969\pi$
$601$	1.27578e7			1.44075			0.720376	+	0.693583i	$0.243969\pi$
$602$		0			0
$603$	4.50266e6			0.504285
$604$		0			0
$605$	−661082.	−	1.14503e6i	−0.0734289	−	0.127183i
$606$		0			0
$607$	−6.81033e6	+	1.17958e7i	−0.750233	+	1.29944i	0.197476	+	0.980308i	$0.436726\pi$
$607$	−6.81033e6	+	1.17958e7i	−0.750233	+	1.29944i	−0.947709	+	0.319135i	$0.896608\pi$
$608$		0			0
$609$	2.66672e6	−	780266.i	0.291363	−	0.0852510i
$610$		0			0
$611$	1.59092e6	−	2.75555e6i	0.172403	−	0.298611i
$612$		0			0
$613$	−7.67372e6	−	1.32913e7i	−0.824812	−	1.42862i	−0.902063	−	0.431605i	$-0.857948\pi$
$613$	−7.67372e6	−	1.32913e7i	−0.824812	−	1.42862i	0.0772508	−	0.997012i	$-0.475386\pi$
$614$		0			0
$615$	957099.			0.102040
$616$		0			0
$617$	1.18485e7			1.25300			0.626500	−	0.779421i	$-0.284487\pi$
$617$	1.18485e7			1.25300			0.626500	+	0.779421i	$0.284487\pi$
$618$		0			0
$619$	−437230.	−	757304.i	−0.0458652	−	0.0794408i	0.842181	−	0.539194i	$-0.181271\pi$
$619$	−437230.	−	757304.i	−0.0458652	−	0.0794408i	−0.888047	+	0.459753i	$0.847938\pi$
$620$		0			0
$621$	−1.49941e6	+	2.59706e6i	−0.156024	+	0.270242i
$622$		0			0
$623$	5.85858e6	+	5.59679e6i	0.604744	+	0.577722i
$624$		0			0
$625$	−2.39730e6	+	4.15225e6i	−0.245484	+	0.425190i
$626$		0			0
$627$	−2.90679e6	−	5.03471e6i	−0.295287	−	0.511453i
$628$		0			0
$629$	−5.56642e6			−0.560983
$630$		0			0
$631$	4.43859e6			0.443784			0.221892	−	0.975071i	$-0.428777\pi$
$631$	4.43859e6			0.443784			0.221892	+	0.975071i	$0.428777\pi$
$632$		0			0
$633$	−309483.	−	536040.i	−0.0306992	−	0.0531726i
$634$		0			0
$635$	276992.	−	479764.i	0.0272605	−	0.0472165i
$636$		0			0
$637$	1.52120e7	+	7.87900e6i	1.48538	+	0.769348i
$638$		0			0
$639$	−246115.	+	426283.i	−0.0238443	+	0.0412996i
$640$		0			0
$641$	−406276.	−	703690.i	−0.0390549	−	0.0676451i	0.845837	−	0.533441i	$-0.179101\pi$
$641$	−406276.	−	703690.i	−0.0390549	−	0.0676451i	−0.884892	+	0.465796i	$0.845768\pi$
$642$		0			0
$643$	1.17941e7			1.12496			0.562481	−	0.826810i	$-0.309847\pi$
$643$	1.17941e7			1.12496			0.562481	+	0.826810i	$0.309847\pi$
$644$		0			0
$645$	1.10767e6			0.104837
$646$		0			0
$647$	−1.21690e6	−	2.10773e6i	−0.114286	−	0.197950i	0.803208	−	0.595699i	$-0.203125\pi$
$647$	−1.21690e6	−	2.10773e6i	−0.114286	−	0.197950i	−0.917494	+	0.397749i	$0.869791\pi$
$648$		0			0
$649$	−6.40031e6	+	1.10857e7i	−0.596471	+	1.03312i
$650$		0			0
$651$	−2.48938e6	−	2.37814e6i	−0.230217	−	0.219930i
$652$		0			0
$653$	4.73332e6	−	8.19835e6i	0.434393	−	0.752391i	−0.562853	−	0.826557i	$-0.690296\pi$
$653$	4.73332e6	−	8.19835e6i	0.434393	−	0.752391i	0.997246	+	0.0741664i	$0.0236296\pi$
$654$		0			0
$655$	−4.02323e6	−	6.96843e6i	−0.366413	−	0.634647i
$656$		0			0
$657$	−1.35915e6			−0.122844
$658$		0			0
$659$	−1.40662e7			−1.26172			−0.630860	−	0.775896i	$-0.717298\pi$
$659$	−1.40662e7			−1.26172			−0.630860	+	0.775896i	$0.717298\pi$
$660$		0			0
$661$	8.58029e6	+	1.48615e7i	0.763832	+	1.32300i	0.940862	+	0.338791i	$0.110018\pi$
$661$	8.58029e6	+	1.48615e7i	0.763832	+	1.32300i	−0.177029	+	0.984206i	$0.556649\pi$
$662$		0			0
$663$	−2.57669e6	+	4.46296e6i	−0.227656	+	0.394312i
$664$		0			0
$665$	4.10091e6	−	1.19990e6i	0.359606	−	0.105218i
$666$		0			0
$667$	−4.89803e6	+	8.48364e6i	−0.426292	+	0.738359i
$668$		0			0
$669$	2.79102e6	+	4.83419e6i	0.241100	+	0.417598i
$670$		0			0
$671$	9.82424e6			0.842350
$672$		0			0
$673$	5.87113e6			0.499671			0.249836	−	0.968288i	$-0.419623\pi$
$673$	5.87113e6			0.499671			0.249836	+	0.968288i	$0.419623\pi$
$674$		0			0
$675$	−933413.	−	1.61672e6i	−0.0788523	−	0.136576i
$676$		0			0
$677$	−6.96368e6	+	1.20614e7i	−0.583938	+	1.01141i	0.411069	+	0.911604i	$0.365156\pi$
$677$	−6.96368e6	+	1.20614e7i	−0.583938	+	1.01141i	−0.995007	+	0.0998062i	$0.968178\pi$
$678$		0			0
$679$	1.95316e6	−	8.01773e6i	0.162578	−	0.667386i
$680$		0			0
$681$	−4.51005e6	+	7.81163e6i	−0.372660	+	0.645467i
$682$		0			0
$683$	9.15503e6	+	1.58570e7i	0.750945	+	1.30067i	0.947365	+	0.320155i	$0.103735\pi$
$683$	9.15503e6	+	1.58570e7i	0.750945	+	1.30067i	−0.196421	+	0.980520i	$0.562932\pi$
$684$		0			0
$685$	−1.49350e6			−0.121613
$686$		0			0
$687$	−7.97275e6			−0.644490
$688$		0			0
$689$	581509.	+	1.00720e6i	0.0466669	+	0.0808294i
$690$		0			0
$691$	−8.83583e6	+	1.53041e7i	−0.703967	+	1.21931i	0.263096	+	0.964770i	$0.415256\pi$
$691$	−8.83583e6	+	1.53041e7i	−0.703967	+	1.21931i	−0.967063	+	0.254537i	$0.918077\pi$
$692$		0			0
$693$	1.15702e6	−	4.74959e6i	0.0915184	−	0.375684i
$694$		0			0
$695$	2.51693e6	−	4.35945e6i	0.197656	−	0.342350i
$696$		0			0
$697$	1.25753e6	+	2.17810e6i	0.0980473	+	0.169823i
$698$		0			0
$699$	−5.36547e6			−0.415351
$700$		0			0
$701$	8.22993e6			0.632559			0.316279	−	0.948666i	$-0.397566\pi$
$701$	8.22993e6			0.632559			0.316279	+	0.948666i	$0.397566\pi$
$702$		0			0
$703$	6.87472e6	+	1.19074e7i	0.524646	+	0.908714i
$704$		0			0
$705$	333660.	−	577916.i	0.0252832	−	0.0437917i
$706$		0			0
$707$	2.29717e7	−	6.72137e6i	1.72840	−	0.505719i
$708$		0			0
$709$	−1.23183e7	+	2.13360e7i	−0.920314	+	1.59403i	−0.121386	+	0.992605i	$0.538734\pi$
$709$	−1.23183e7	+	2.13360e7i	−0.920314	+	1.59403i	−0.798929	+	0.601426i	$0.794600\pi$
$710$		0			0
$711$	−196233.	−	339886.i	−0.0145579	−	0.0252150i
$712$		0			0
$713$	1.21379e7			0.894167
$714$		0			0
$715$	1.12710e7			0.824510
$716$		0			0
$717$	−3.34400e6	−	5.79197e6i	−0.242923	−	0.420754i
$718$		0			0
$719$	−9.41269e6	+	1.63033e7i	−0.679034	+	1.17612i	0.296238	+	0.955114i	$0.404268\pi$
$719$	−9.41269e6	+	1.63033e7i	−0.679034	+	1.17612i	−0.975272	+	0.221007i	$0.929066\pi$
$720$		0			0
$721$	4.87863e6	+	4.66064e6i	0.349510	+	0.333893i
$722$		0			0
$723$	−5.28677e6	+	9.15696e6i	−0.376136	+	0.651487i
$724$		0			0
$725$	−3.04911e6	−	5.28122e6i	−0.215441	−	0.373155i
$726$		0			0
$727$	−6.77607e6			−0.475491			−0.237745	−	0.971328i	$-0.576408\pi$
$727$	−6.77607e6			−0.475491			−0.237745	+	0.971328i	$0.576408\pi$
$728$		0			0
$729$	531441.			0.0370370
$730$		0			0
$731$	1.45537e6	+	2.52077e6i	0.100735	+	0.174478i
$732$		0			0
$733$	8.61136e6	−	1.49153e7i	0.591986	−	1.02535i	−0.401978	−	0.915649i	$-0.631677\pi$
$733$	8.61136e6	−	1.49153e7i	0.591986	−	1.02535i	0.993965	−	0.109701i	$-0.0349894\pi$
$734$		0			0
$735$	3.19038e6	+	1.65244e6i	0.217833	+	0.112826i
$736$		0			0
$737$	1.29389e7	−	2.24109e7i	0.877465	−	1.51981i
$738$		0			0
$739$	−8.82669e6	−	1.52883e7i	−0.594548	−	1.02979i	−0.993610	−	0.112864i	$-0.963998\pi$
$739$	−8.82669e6	−	1.52883e7i	−0.594548	−	1.02979i	0.399062	−	0.916924i	$-0.369336\pi$
$740$		0			0
$741$	1.27292e7			0.851641
$742$		0			0
$743$	1.36977e7			0.910281			0.455141	−	0.890420i	$-0.349589\pi$
$743$	1.36977e7			0.910281			0.455141	+	0.890420i	$0.349589\pi$
$744$		0			0
$745$	−1.66594e6	−	2.88549e6i	−0.109968	−	0.190471i
$746$		0			0
$747$	2.43589e6	−	4.21908e6i	0.159719	−	0.276641i
$748$		0			0
$749$	4.51624e6	+	4.31443e6i	0.294152	+	0.281008i
$750$		0			0
$751$	5.80446e6	−	1.00536e7i	0.375545	−	0.650463i	−0.614864	−	0.788633i	$-0.710789\pi$
$751$	5.80446e6	−	1.00536e7i	0.375545	−	0.650463i	0.990408	+	0.138171i	$0.0441223\pi$
$752$		0			0
$753$	−1.58846e6	−	2.75130e6i	−0.102092	−	0.176828i
$754$		0			0
$755$	3.89526e6			0.248696
$756$		0			0
$757$	6.25226e6			0.396550			0.198275	−	0.980146i	$-0.436466\pi$
$757$	6.25226e6			0.396550			0.198275	+	0.980146i	$0.436466\pi$
$758$		0			0
$759$	8.61748e6	+	1.49259e7i	0.542970	+	0.940452i
$760$		0			0
$761$	1.51063e7	−	2.61648e7i	0.945574	−	1.63778i	0.190977	−	0.981595i	$-0.438834\pi$
$761$	1.51063e7	−	2.61648e7i	0.945574	−	1.63778i	0.754597	−	0.656188i	$-0.227832\pi$
$762$		0			0
$763$	−4.53350e6	+	1.32647e6i	−0.281918	+	0.0824873i
$764$		0			0
$765$	−540404.	+	936007.i	−0.0333860	+	0.0578263i
$766$		0			0
$767$	−1.40139e7	−	2.42728e7i	−0.860142	−	1.48981i
$768$		0			0
$769$	−2.58756e7			−1.57788			−0.788940	−	0.614470i	$-0.789370\pi$
$769$	−2.58756e7			−1.57788			−0.788940	+	0.614470i	$0.789370\pi$
$770$		0			0
$771$	−90114.1			−0.00545955
$772$		0			0
$773$	−1.27705e7	−	2.21191e7i	−0.768701	−	1.33143i	−0.938267	−	0.345911i	$-0.887570\pi$
$773$	−1.27705e7	−	2.21191e7i	−0.768701	−	1.33143i	0.169566	−	0.985519i	$-0.445763\pi$
$774$		0			0
$775$	−3.77802e6	+	6.54373e6i	−0.225949	+	0.391355i
$776$		0			0
$777$	−2.73642e6	+	1.12330e7i	−0.162604	+	0.667489i
$778$		0			0
$779$	3.10618e6	−	5.38006e6i	0.183393	−	0.317646i
$780$		0			0
$781$	1.41448e6	+	2.44995e6i	0.0829791	+	0.143724i
$782$

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)

\(f(q)\)	\(=\)	\(q+(4.50000 + 7.79423i) q^{3} +(-11.8764 + 20.5705i) q^{5} +(-30.6840 + 125.958i) q^{7} +(-40.5000 + 70.1481i) q^{9} +O(q^{10})\)	\(q+(4.50000 + 7.79423i) q^{3} +(-11.8764 + 20.5705i) q^{5} +(-30.6840 + 125.958i) q^{7} +(-40.5000 + 70.1481i) q^{9} +(232.763 + 403.157i) q^{11} -1019.30 q^{13} -213.775 q^{15} +(-280.878 - 486.496i) q^{17} +(-693.789 + 1201.68i) q^{19} +(-1119.83 + 327.654i) q^{21} +(2056.81 - 3562.50i) q^{23} +(1280.40 + 2217.72i) q^{25} -729.000 q^{27} -2381.37 q^{29} +(1475.33 + 2555.34i) q^{31} +(-2094.87 + 3628.42i) q^{33} +(-2226.61 - 2127.12i) q^{35} +(4954.48 - 8581.40i) q^{37} +(-4586.85 - 7944.66i) q^{39} -4477.13 q^{41} -5181.48 q^{43} +(-961.989 - 1666.21i) q^{45} +(-1560.80 + 2703.38i) q^{47} +(-14924.0 - 7729.82i) q^{49} +(2527.91 - 4378.46i) q^{51} +(-570.499 - 988.133i) q^{53} -11057.6 q^{55} -12488.2 q^{57} +(13748.5 + 23813.2i) q^{59} +(10551.8 - 18276.2i) q^{61} +(-7593.03 - 7253.74i) q^{63} +(12105.6 - 20967.6i) q^{65} +(-27794.2 - 48141.0i) q^{67} +37022.6 q^{69} +6076.90 q^{71} +(8389.82 + 14531.6i) q^{73} +(-11523.6 + 19959.5i) q^{75} +(-57923.1 + 16947.9i) q^{77} +(-2422.63 + 4196.12i) q^{79} +(-3280.50 - 5681.99i) q^{81} -60145.4 q^{83} +13343.3 q^{85} +(-10716.2 - 18561.0i) q^{87} +(31248.7 - 54124.4i) q^{89} +(31276.2 - 128389. i) q^{91} +(-13277.9 + 22998.1i) q^{93} +(-16479.4 - 28543.2i) q^{95} -63653.8 q^{97} -37707.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 36 q^{3} - 258 q^{7} - 324 q^{9}+O(q^{10}) \) 8 * q + 36 * q^3 - 258 * q^7 - 324 * q^9	\( 8 q + 36 q^{3} - 258 q^{7} - 324 q^{9} + 402 q^{11} + 924 q^{13} - 276 q^{17} + 510 q^{19} - 3564 q^{21} + 6900 q^{23} - 2814 q^{25} - 5832 q^{27} + 1080 q^{29} - 6410 q^{31} - 3618 q^{33} + 33108 q^{35} - 15250 q^{37} + 4158 q^{39} + 8616 q^{41} - 58396 q^{43} - 15060 q^{47} - 64252 q^{49} + 2484 q^{51} - 13692 q^{53} - 146248 q^{55} + 9180 q^{57} + 34830 q^{59} + 5364 q^{61} - 11178 q^{63} - 66864 q^{65} - 5994 q^{67} + 124200 q^{69} - 178536 q^{71} - 59638 q^{73} + 25326 q^{75} - 75660 q^{77} - 44062 q^{79} - 26244 q^{81} + 416892 q^{83} + 72648 q^{85} + 4860 q^{87} + 77520 q^{89} - 104722 q^{91} + 57690 q^{93} - 221376 q^{95} - 377260 q^{97} - 65124 q^{99}+O(q^{100}) \) 8 * q + 36 * q^3 - 258 * q^7 - 324 * q^9 + 402 * q^11 + 924 * q^13 - 276 * q^17 + 510 * q^19 - 3564 * q^21 + 6900 * q^23 - 2814 * q^25 - 5832 * q^27 + 1080 * q^29 - 6410 * q^31 - 3618 * q^33 + 33108 * q^35 - 15250 * q^37 + 4158 * q^39 + 8616 * q^41 - 58396 * q^43 - 15060 * q^47 - 64252 * q^49 + 2484 * q^51 - 13692 * q^53 - 146248 * q^55 + 9180 * q^57 + 34830 * q^59 + 5364 * q^61 - 11178 * q^63 - 66864 * q^65 - 5994 * q^67 + 124200 * q^69 - 178536 * q^71 - 59638 * q^73 + 25326 * q^75 - 75660 * q^77 - 44062 * q^79 - 26244 * q^81 + 416892 * q^83 + 72648 * q^85 + 4860 * q^87 + 77520 * q^89 - 104722 * q^91 + 57690 * q^93 - 221376 * q^95 - 377260 * q^97 - 65124 * q^99

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