LMFDB - Embedded newform 84.6.i.c.37.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [84,6,Mod(25,84)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(84, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([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("84.25");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 84 = 2^{2} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	84.i (of order $3$, degree $2$, 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:	$13.4722408643$
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} - 2x^{7} + 703x^{6} + 2770x^{5} + 427565x^{4} + 718170x^{3} + 42175732x^{2} - 40929504x + 3559792896 $ x^8 - 2x^7 + 703x^6 + 2770x^5 + 427565x^4 + 718170x^3 + 42175732x^2 - 40929504*x + 3559792896
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{3}\cdot 7 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			37.1
Root			$4.59067 + 7.95128i$ of defining polynomial
Character	$\chi$	$=$	84.37
Dual form			84.6.i.c.25.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.50000 + 7.79423i) q^{3} +(-39.3359 + 68.1317i) q^{5} +(-100.606 - 81.7641i) q^{7} +(-40.5000 + 70.1481i) q^{9} +O(q^{10})$	\(q+(4.50000 + 7.79423i) q^{3} +(-39.3359 + 68.1317i) q^{5} +(-100.606 - 81.7641i) q^{7} +(-40.5000 + 70.1481i) q^{9} +(-345.759 - 598.873i) q^{11} +818.732 q^{13} -708.046 q^{15} +(-554.638 - 960.661i) q^{17} +(286.523 - 496.273i) q^{19} +(184.559 - 1152.09i) q^{21} +(-1258.80 + 2180.30i) q^{23} +(-1532.12 - 2653.71i) q^{25} -729.000 q^{27} -3258.19 q^{29} +(-5059.55 - 8763.40i) q^{31} +(3111.84 - 5389.86i) q^{33} +(9528.17 - 3638.23i) q^{35} +(-2434.81 + 4217.21i) q^{37} +(3684.30 + 6381.39i) q^{39} -13094.3 q^{41} -9303.64 q^{43} +(-3186.21 - 5518.67i) q^{45} +(-6452.90 + 11176.8i) q^{47} +(3436.28 + 16452.0i) q^{49} +(4991.74 - 8645.95i) q^{51} +(9770.12 + 16922.3i) q^{53} +54403.0 q^{55} +5157.42 q^{57} +(12560.2 + 21755.0i) q^{59} +(15681.1 - 27160.5i) q^{61} +(9810.15 - 3745.90i) q^{63} +(-32205.6 + 55781.7i) q^{65} +(-27971.9 - 48448.8i) q^{67} -22658.4 q^{69} +20501.0 q^{71} +(33825.0 + 58586.6i) q^{73} +(13789.1 - 23883.4i) q^{75} +(-14180.7 + 88521.1i) q^{77} +(-7039.95 + 12193.5i) q^{79} +(-3280.50 - 5681.99i) q^{81} -77129.1 q^{83} +87268.6 q^{85} +(-14661.9 - 25395.1i) q^{87} +(-160.396 + 277.815i) q^{89} +(-82369.7 - 66942.9i) q^{91} +(45536.0 - 78870.6i) q^{93} +(22541.3 + 39042.6i) q^{95} -112009. q^{97} +56013.0 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 36 q^{3} - 42 q^{7} - 324 q^{9}+O(q^{10}) $ 8 * q + 36 * q^3 - 42 * q^7 - 324 * q^9	$ 8 q + 36 q^{3} - 42 q^{7} - 324 q^{9} - 462 q^{11} - 1204 q^{13} + 228 q^{17} + 358 q^{19} + 1404 q^{21} - 2148 q^{23} - 5454 q^{25} - 5832 q^{27} - 11064 q^{29} + 830 q^{31} + 4158 q^{33} + 7692 q^{35} - 3914 q^{37} - 5418 q^{39} - 16632 q^{41} - 29036 q^{43} + 41700 q^{47} + 41876 q^{49} - 2052 q^{51} + 22164 q^{53} + 7784 q^{55} + 6444 q^{57} + 32886 q^{59} + 83732 q^{61} + 16038 q^{63} - 93192 q^{65} - 80034 q^{67} - 38664 q^{69} + 89544 q^{71} - 22470 q^{73} + 49086 q^{75} + 138732 q^{77} - 75286 q^{79} - 26244 q^{81} - 34836 q^{83} + 278504 q^{85} - 49788 q^{87} + 28944 q^{89} - 12058 q^{91} - 7470 q^{93} - 144120 q^{95} - 433356 q^{97} + 74844 q^{99}+O(q^{100}) $ 8 * q + 36 * q^3 - 42 * q^7 - 324 * q^9 - 462 * q^11 - 1204 * q^13 + 228 * q^17 + 358 * q^19 + 1404 * q^21 - 2148 * q^23 - 5454 * q^25 - 5832 * q^27 - 11064 * q^29 + 830 * q^31 + 4158 * q^33 + 7692 * q^35 - 3914 * q^37 - 5418 * q^39 - 16632 * q^41 - 29036 * q^43 + 41700 * q^47 + 41876 * q^49 - 2052 * q^51 + 22164 * q^53 + 7784 * q^55 + 6444 * q^57 + 32886 * q^59 + 83732 * q^61 + 16038 * q^63 - 93192 * q^65 - 80034 * q^67 - 38664 * q^69 + 89544 * q^71 - 22470 * q^73 + 49086 * q^75 + 138732 * q^77 - 75286 * q^79 - 26244 * q^81 - 34836 * q^83 + 278504 * q^85 - 49788 * q^87 + 28944 * q^89 - 12058 * q^91 - 7470 * q^93 - 144120 * q^95 - 433356 * q^97 + 74844 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/84\mathbb{Z}\right)^\times$.

$n$	$29$	$43$	$73$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$5$	−39.3359	+	68.1317i	−0.703662	+	1.21878i	0.263511	+	0.964656i	$0.415120\pi$
$5$	−39.3359	+	68.1317i	−0.703662	+	1.21878i	−0.967172	+	0.254121i	$0.918214\pi$
$6$		0			0
$7$	−100.606	−	81.7641i	−0.776033	−	0.630692i
$7$	−100.606	−	81.7641i	−0.776033	−	0.630692i
$8$		0			0
$9$	−40.5000	+	70.1481i	−0.166667	+	0.288675i
$10$		0			0
$11$	−345.759	−	598.873i	−0.861574	−	1.49229i	−0.870410	−	0.492328i	$-0.836146\pi$
$11$	−345.759	−	598.873i	−0.861574	−	1.49229i	0.00883597	−	0.999961i	$-0.497187\pi$
$12$		0			0
$13$	818.732			1.34364			0.671821	−	0.740714i	$-0.265512\pi$
$13$	818.732			1.34364			0.671821	+	0.740714i	$0.265512\pi$
$14$		0			0
$15$	−708.046			−0.812518
$16$		0			0
$17$	−554.638	−	960.661i	−0.465465	−	0.806209i	0.533757	−	0.845638i	$-0.320780\pi$
$17$	−554.638	−	960.661i	−0.465465	−	0.806209i	−0.999222	+	0.0394286i	$0.987446\pi$
$18$		0			0
$19$	286.523	−	496.273i	0.182086	−	0.315382i	−0.760505	−	0.649332i	$-0.775049\pi$
$19$	286.523	−	496.273i	0.182086	−	0.315382i	0.942591	+	0.333951i	$0.108382\pi$
$20$		0			0
$21$	184.559	−	1152.09i	0.0913245	−	0.570082i
$22$		0			0
$23$	−1258.80	+	2180.30i	−0.496177	+	0.859403i	−0.999990	−	0.00440926i	$-0.998596\pi$
$23$	−1258.80	+	2180.30i	−0.496177	+	0.859403i	0.503814	+	0.863812i	$0.331930\pi$
$24$		0			0
$25$	−1532.12	−	2653.71i	−0.490279	−	0.849189i
$26$		0			0
$27$	−729.000			−0.192450
$28$		0			0
$29$	−3258.19			−0.719419			−0.359710	−	0.933064i	$-0.617124\pi$
$29$	−3258.19			−0.719419			−0.359710	+	0.933064i	$0.617124\pi$
$30$		0			0
$31$	−5059.55	−	8763.40i	−0.945601	−	1.63783i	−0.754544	−	0.656249i	$-0.772142\pi$
$31$	−5059.55	−	8763.40i	−0.945601	−	1.63783i	−0.191056	−	0.981579i	$-0.561191\pi$
$32$		0			0
$33$	3111.84	−	5389.86i	0.497430	−	0.861574i
$34$		0			0
$35$	9528.17	−	3638.23i	1.31474	−	0.502018i
$36$		0			0
$37$	−2434.81	+	4217.21i	−0.292389	+	0.506432i	−0.974374	−	0.224934i	$-0.927783\pi$
$37$	−2434.81	+	4217.21i	−0.292389	+	0.506432i	0.681985	+	0.731366i	$0.261117\pi$
$38$		0			0
$39$	3684.30	+	6381.39i	0.387876	+	0.671821i
$40$		0			0
$41$	−13094.3			−1.21653			−0.608265	−	0.793734i	$-0.708134\pi$
$41$	−13094.3			−1.21653			−0.608265	+	0.793734i	$0.708134\pi$
$42$		0			0
$43$	−9303.64			−0.767329			−0.383664	−	0.923473i	$-0.625338\pi$
$43$	−9303.64			−0.767329			−0.383664	+	0.923473i	$0.625338\pi$
$44$		0			0
$45$	−3186.21	−	5518.67i	−0.234554	−	0.406259i
$46$		0			0
$47$	−6452.90	+	11176.8i	−0.426099	+	0.738025i	−0.996522	−	0.0833256i	$-0.973446\pi$
$47$	−6452.90	+	11176.8i	−0.426099	+	0.738025i	0.570423	+	0.821351i	$0.306779\pi$
$48$		0			0
$49$	3436.28	+	16452.0i	0.204455	+	0.978876i
$50$		0			0
$51$	4991.74	−	8645.95i	0.268736	−	0.465465i
$52$		0			0
$53$	9770.12	+	16922.3i	0.477760	+	0.827505i	0.999675	−	0.0254926i	$-0.00811542\pi$
$53$	9770.12	+	16922.3i	0.477760	+	0.827505i	−0.521915	+	0.852998i	$0.674782\pi$
$54$		0			0
$55$	54403.0			2.42503
$56$		0			0
$57$	5157.42			0.210254
$58$		0			0
$59$	12560.2	+	21755.0i	0.469751	+	0.813632i	0.999402	−	0.0345835i	$-0.0110105\pi$
$59$	12560.2	+	21755.0i	0.469751	+	0.813632i	−0.529651	+	0.848216i	$0.677677\pi$
$60$		0			0
$61$	15681.1	−	27160.5i	0.539575	−	0.934571i	−0.459352	−	0.888254i	$-0.651918\pi$
$61$	15681.1	−	27160.5i	0.539575	−	0.934571i	0.998927	−	0.0463168i	$-0.0147484\pi$
$62$		0			0
$63$	9810.15	−	3745.90i	0.311404	−	0.118906i
$64$		0			0
$65$	−32205.6	+	55781.7i	−0.945469	+	1.63760i
$66$		0			0
$67$	−27971.9	−	48448.8i	−0.761264	−	1.31855i	−0.942200	−	0.335052i	$-0.891246\pi$
$67$	−27971.9	−	48448.8i	−0.761264	−	1.31855i	0.180936	−	0.983495i	$-0.442087\pi$
$68$		0			0
$69$	−22658.4			−0.572935
$70$		0			0
$71$	20501.0			0.482647			0.241323	−	0.970445i	$-0.422419\pi$
$71$	20501.0			0.482647			0.241323	+	0.970445i	$0.422419\pi$
$72$		0			0
$73$	33825.0	+	58586.6i	0.742900	+	1.28674i	0.951170	+	0.308669i	$0.0998835\pi$
$73$	33825.0	+	58586.6i	0.742900	+	1.28674i	−0.208270	+	0.978071i	$0.566783\pi$
$74$		0			0
$75$	13789.1	−	23883.4i	0.283063	−	0.490279i
$76$		0			0
$77$	−14180.7	+	88521.1i	−0.272565	+	1.70145i
$78$		0			0
$79$	−7039.95	+	12193.5i	−0.126912	+	0.219818i	−0.922479	−	0.386048i	$-0.873840\pi$
$79$	−7039.95	+	12193.5i	−0.126912	+	0.219818i	0.795567	+	0.605866i	$0.207173\pi$
$80$		0			0
$81$	−3280.50	−	5681.99i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	−77129.1			−1.22892			−0.614459	−	0.788949i	$-0.710626\pi$
$83$	−77129.1			−1.22892			−0.614459	+	0.788949i	$0.710626\pi$
$84$		0			0
$85$	87268.6			1.31012
$86$		0			0
$87$	−14661.9	−	25395.1i	−0.207678	−	0.359710i
$88$		0			0
$89$	−160.396	+	277.815i	−0.00214644	+	0.00371775i	−0.867097	−	0.498140i	$-0.834017\pi$
$89$	−160.396	+	277.815i	−0.00214644	+	0.00371775i	0.864950	+	0.501858i	$0.167350\pi$
$90$		0			0
$91$	−82369.7	−	66942.9i	−1.04271	−	0.847424i
$92$		0			0
$93$	45536.0	−	78870.6i	0.545943	−	0.945601i
$94$		0			0
$95$	22541.3	+	39042.6i	0.256253	+	0.443844i
$96$		0			0
$97$	−112009.			−1.20871			−0.604355	−	0.796715i	$-0.706569\pi$
$97$	−112009.			−1.20871			−0.604355	+	0.796715i	$0.706569\pi$
$98$		0			0
$99$	56013.0			0.574382
$100$		0			0
$101$	−33627.0	−	58243.7i	−0.328008	−	0.568127i	0.654108	−	0.756401i	$-0.273044\pi$
$101$	−33627.0	−	58243.7i	−0.328008	−	0.568127i	−0.982117	+	0.188274i	$0.939711\pi$
$102$		0			0
$103$	58317.6	−	101009.i	0.541635	−	0.938140i	−0.457175	−	0.889377i	$-0.651139\pi$
$103$	58317.6	−	101009.i	0.541635	−	0.938140i	0.998810	−	0.0487630i	$-0.0155279\pi$
$104$		0			0
$105$	71233.9	+	57892.7i	0.630541	+	0.512449i
$106$		0			0
$107$	80322.6	−	139123.i	0.678232	−	1.17473i	−0.297281	−	0.954790i	$-0.596080\pi$
$107$	80322.6	−	139123.i	0.678232	−	1.17473i	0.975513	−	0.219942i	$-0.0705869\pi$
$108$		0			0
$109$	−59668.3	−	103349.i	−0.481036	−	0.833179i	0.518727	−	0.854940i	$-0.326406\pi$
$109$	−59668.3	−	103349.i	−0.481036	−	0.833179i	−0.999763	+	0.0217610i	$0.993073\pi$
$110$		0			0
$111$	−43826.6			−0.337622
$112$		0			0
$113$	−110893.			−0.816972			−0.408486	−	0.912765i	$-0.633943\pi$
$113$	−110893.			−0.816972			−0.408486	+	0.912765i	$0.633943\pi$
$114$		0			0
$115$	−99031.8	−	171528.i	−0.698281	−	1.20946i
$116$		0			0
$117$	−33158.7	+	57432.5i	−0.223940	+	0.387876i
$118$		0			0
$119$	−22747.4	+	141998.i	−0.147253	+	0.919210i
$120$		0			0
$121$	−158574.	+	274658.i	−0.984618	+	1.70541i
$122$		0			0
$123$	−58924.4	−	102060.i	−0.351182	−	0.608265i
$124$		0			0
$125$	−4779.64			−0.0273603
$126$		0			0
$127$	315184.			1.73402			0.867012	−	0.498287i	$-0.166038\pi$
$127$	315184.			1.73402			0.867012	+	0.498287i	$0.166038\pi$
$128$		0			0
$129$	−41866.4	−	72514.7i	−0.221509	−	0.383664i
$130$		0			0
$131$	−42832.3	+	74187.6i	−0.218068	+	0.377705i	−0.954217	−	0.299114i	$-0.903309\pi$
$131$	−42832.3	+	74187.6i	−0.218068	+	0.377705i	0.736149	+	0.676819i	$0.236642\pi$
$132$		0			0
$133$	−69403.3	+	26500.9i	−0.340213	+	0.129907i
$134$		0			0
$135$	28675.9	−	49668.0i	0.135420	−	0.234554i
$136$		0			0
$137$	16316.1	+	28260.3i	0.0742703	+	0.128640i	0.900769	−	0.434299i	$-0.143004\pi$
$137$	16316.1	+	28260.3i	0.0742703	+	0.128640i	−0.826498	+	0.562939i	$0.809671\pi$
$138$		0			0
$139$	206590.			0.906925			0.453462	−	0.891275i	$-0.350189\pi$
$139$	206590.			0.906925			0.453462	+	0.891275i	$0.350189\pi$
$140$		0			0
$141$	−116152.			−0.492017
$142$		0			0
$143$	−283084.	−	490317.i	−1.15765	−	2.00510i
$144$		0			0
$145$	128164.	−	221986.i	0.506228	−	0.876812i
$146$		0			0
$147$	−112767.	+	100817.i	−0.430417	+	0.384805i
$148$		0			0
$149$	−241855.	+	418906.i	−0.892463	+	1.54579i	−0.0555495	+	0.998456i	$0.517691\pi$
$149$	−241855.	+	418906.i	−0.892463	+	1.54579i	−0.836913	+	0.547335i	$0.815642\pi$
$150$		0			0
$151$	83610.4	+	144818.i	0.298413	+	0.516867i	0.975773	−	0.218785i	$-0.0702093\pi$
$151$	83610.4	+	144818.i	0.298413	+	0.516867i	−0.677360	+	0.735652i	$0.736876\pi$
$152$		0			0
$153$	89851.3			0.310310
$154$		0			0
$155$	796087.			2.66153
$156$		0			0
$157$	7054.26	+	12218.3i	0.0228403	+	0.0395606i	0.877220	−	0.480089i	$-0.159396\pi$
$157$	7054.26	+	12218.3i	0.0228403	+	0.0395606i	−0.854379	+	0.519650i	$0.826062\pi$
$158$		0			0
$159$	−87931.0	+	152301.i	−0.275835	+	0.477760i
$160$		0			0
$161$	304913.	−	116428.i	0.927068	−	0.353991i
$162$		0			0
$163$	−327188.	+	566706.i	−0.964558	+	1.67066i	−0.253762	+	0.967267i	$0.581668\pi$
$163$	−327188.	+	566706.i	−0.964558	+	1.67066i	−0.710797	+	0.703397i	$0.751665\pi$
$164$		0			0
$165$	244814.	+	424030.i	0.700044	+	1.21251i
$166$		0			0
$167$	182945.			0.507610			0.253805	−	0.967255i	$-0.418318\pi$
$167$	182945.			0.507610			0.253805	+	0.967255i	$0.418318\pi$
$168$		0			0
$169$	299030.			0.805374
$170$		0			0
$171$	23208.4	+	40198.1i	0.0606952	+	0.105127i
$172$		0			0
$173$	146004.	−	252887.i	0.370894	−	0.642407i	−0.618809	−	0.785541i	$-0.712385\pi$
$173$	146004.	−	252887.i	0.370894	−	0.642407i	0.989703	+	0.143134i	$0.0457180\pi$
$174$		0			0
$175$	−62837.2	+	392253.i	−0.155103	+	0.968214i
$176$		0			0
$177$	−113042.	+	195795.i	−0.271211	+	0.469751i
$178$		0			0
$179$	−97383.8	−	168674.i	−0.227172	−	0.393473i	0.729797	−	0.683664i	$-0.239615\pi$
$179$	−97383.8	−	168674.i	−0.227172	−	0.393473i	−0.956969	+	0.290191i	$0.906281\pi$
$180$		0			0
$181$	−256634.			−0.582261			−0.291130	−	0.956683i	$-0.594031\pi$
$181$	−256634.			−0.582261			−0.291130	+	0.956683i	$0.594031\pi$
$182$		0			0
$183$	282260.			0.623047
$184$		0			0
$185$	−191551.	−	331776.i	−0.411486	−	0.712714i
$186$		0			0
$187$	−383542.	+	664315.i	−0.802065	+	1.38922i
$188$		0			0
$189$	73342.0	+	59606.0i	0.149348	+	0.121377i
$190$		0			0
$191$	146975.	−	254567.i	0.291513	−	0.504916i	−0.682654	−	0.730741i	$-0.739175\pi$
$191$	146975.	−	254567.i	0.291513	−	0.504916i	0.974168	+	0.225825i	$0.0725079\pi$
$192$		0			0
$193$	34565.0	+	59868.4i	0.0667950	+	0.115692i	0.897489	−	0.441037i	$-0.145389\pi$
$193$	34565.0	+	59868.4i	0.0667950	+	0.115692i	−0.830694	+	0.556730i	$0.812056\pi$
$194$		0			0
$195$	−579700.			−1.09173
$196$		0			0
$197$	331748.			0.609037			0.304518	−	0.952506i	$-0.401505\pi$
$197$	331748.			0.609037			0.304518	+	0.952506i	$0.401505\pi$
$198$		0			0
$199$	−287214.	−	497470.i	−0.514130	−	0.890500i	−0.999866	−	0.0163939i	$-0.994781\pi$
$199$	−287214.	−	497470.i	−0.514130	−	0.890500i	0.485735	−	0.874106i	$-0.338552\pi$
$200$		0			0
$201$	251747.	−	436039.i	0.439516	−	0.761264i
$202$		0			0
$203$	327795.	+	266403.i	0.558293	+	0.453732i
$204$		0			0
$205$	515076.	−	892138.i	0.856025	−	1.48268i
$206$		0			0
$207$	−101963.	−	176604.i	−0.165392	−	0.286468i
$208$		0			0
$209$	−396272.			−0.627521
$210$		0			0
$211$	−383999.			−0.593777			−0.296889	−	0.954912i	$-0.595949\pi$
$211$	−383999.			−0.593777			−0.296889	+	0.954912i	$0.595949\pi$
$212$		0			0
$213$	92254.5	+	159790.i	0.139328	+	0.241323i
$214$		0			0
$215$	365967.	−	633873.i	0.539940	−	0.935203i
$216$		0			0
$217$	−207508.	+	1.29534e6i	−0.299148	+	1.86739i
$218$		0			0
$219$	−304425.	+	527279.i	−0.428913	+	0.742900i
$220$		0			0
$221$	−454100.	−	786524.i	−0.625418	−	1.08326i
$222$		0			0
$223$	−347165.			−0.467492			−0.233746	−	0.972298i	$-0.575098\pi$
$223$	−347165.			−0.467492			−0.233746	+	0.972298i	$0.575098\pi$
$224$		0			0
$225$	248204.			0.326853
$226$		0			0
$227$	−180641.	−	312879.i	−0.232676	−	0.403006i	0.725919	−	0.687780i	$-0.241415\pi$
$227$	−180641.	−	312879.i	−0.232676	−	0.403006i	−0.958595	+	0.284774i	$0.908081\pi$
$228$		0			0
$229$	376492.	−	652102.i	0.474424	−	0.821726i	−0.525147	−	0.851011i	$-0.675990\pi$
$229$	376492.	−	652102.i	0.474424	−	0.821726i	0.999571	+	0.0292852i	$0.00932309\pi$
$230$		0			0
$231$	−753767.	+	287818.i	−0.929410	+	0.354885i
$232$		0			0
$233$	−357199.	+	618686.i	−0.431042	+	0.746587i	−0.996963	−	0.0778721i	$-0.975187\pi$
$233$	−357199.	+	618686.i	−0.431042	+	0.746587i	0.565921	+	0.824460i	$0.308521\pi$
$234$		0			0
$235$	−507661.	−	879295.i	−0.599659	−	1.03864i
$236$		0			0
$237$	−126719.			−0.146545
$238$		0			0
$239$	343587.			0.389083			0.194541	−	0.980894i	$-0.437678\pi$
$239$	343587.			0.389083			0.194541	+	0.980894i	$0.437678\pi$
$240$		0			0
$241$	38152.5	+	66082.0i	0.0423136	+	0.0732893i	0.886407	−	0.462908i	$-0.153194\pi$
$241$	38152.5	+	66082.0i	0.0423136	+	0.0732893i	−0.844093	+	0.536197i	$0.819860\pi$
$242$		0			0
$243$	29524.5	−	51137.9i	0.0320750	−	0.0555556i
$244$		0			0
$245$	−1.25607e6	−	413033.i	−1.33690	−	0.439612i
$246$		0			0
$247$	234586.	−	406314.i	0.244658	−	0.423760i
$248$		0			0
$249$	−347081.	−	601162.i	−0.354758	−	0.614459i
$250$		0			0
$251$	196249.			0.196618			0.0983090	−	0.995156i	$-0.468657\pi$
$251$	196249.			0.196618			0.0983090	+	0.995156i	$0.468657\pi$
$252$		0			0
$253$	1.74096e6			1.70997
$254$		0			0
$255$	392709.	+	680192.i	0.378199	+	0.655060i
$256$		0			0
$257$	854554.	−	1.48013e6i	0.807062	−	1.39787i	−0.107828	−	0.994170i	$-0.534390\pi$
$257$	854554.	−	1.48013e6i	0.807062	−	1.39787i	0.914890	−	0.403703i	$-0.132277\pi$
$258$		0			0
$259$	589774.	−	225199.i	0.546306	−	0.208601i
$260$		0			0
$261$	131957.	−	228556.i	0.119903	−	0.207678i
$262$		0			0
$263$	395762.	+	685481.i	0.352814	+	0.611091i	0.986741	−	0.162301i	$-0.0518917\pi$
$263$	395762.	+	685481.i	0.352814	+	0.611091i	−0.633928	+	0.773392i	$0.718558\pi$
$264$		0			0
$265$	−1.53726e6			−1.34473
$266$		0			0
$267$	−2887.13			−0.00247850
$268$		0			0
$269$	−579502.	−	1.00373e6i	−0.488286	−	0.845736i	0.511623	−	0.859210i	$-0.329044\pi$
$269$	−579502.	−	1.00373e6i	−0.488286	−	0.845736i	−0.999909	+	0.0134740i	$0.995711\pi$
$270$		0			0
$271$	−111567.	+	193240.i	−0.0922811	+	0.159836i	−0.908471	−	0.417949i	$-0.862749\pi$
$271$	−111567.	+	193240.i	−0.0922811	+	0.159836i	0.816190	+	0.577784i	$0.196083\pi$
$272$		0			0
$273$	151105.	−	943251.i	0.122707	−	0.765986i
$274$		0			0
$275$	−1.05949e6	+	1.83509e6i	−0.844823	+	1.46328i
$276$		0			0
$277$	234523.	+	406205.i	0.183648	+	0.318087i	0.943120	−	0.332452i	$-0.107876\pi$
$277$	234523.	+	406205.i	0.183648	+	0.318087i	−0.759472	+	0.650540i	$0.774543\pi$
$278$		0			0
$279$	819647.			0.630400
$280$		0			0
$281$	−1.51447e6			−1.14418			−0.572090	−	0.820191i	$-0.693867\pi$
$281$	−1.51447e6			−1.14418			−0.572090	+	0.820191i	$0.693867\pi$
$282$		0			0
$283$	535303.	+	927171.i	0.397314	+	0.688167i	0.993393	−	0.114758i	$-0.0366092\pi$
$283$	535303.	+	927171.i	0.397314	+	0.688167i	−0.596080	+	0.802925i	$0.703276\pi$
$284$		0			0
$285$	−202871.	+	351384.i	−0.147948	+	0.256253i
$286$		0			0
$287$	1.31737e6	+	1.07064e6i	0.944068	+	0.767256i
$288$		0			0
$289$	94682.6	−	163995.i	0.0666846	−	0.115501i
$290$		0			0
$291$	−504039.	−	873021.i	−0.348925	−	0.604355i
$292$		0			0
$293$	1.45450e6			0.989794			0.494897	−	0.868952i	$-0.335206\pi$
$293$	1.45450e6			0.989794			0.494897	+	0.868952i	$0.335206\pi$
$294$		0			0
$295$	−1.97627e6			−1.32218
$296$		0			0
$297$	252059.	+	436578.i	0.165810	+	0.287191i
$298$		0			0
$299$	−1.03062e6	+	1.78508e6i	−0.666684	+	1.15473i
$300$		0			0
$301$	936005.	+	760703.i	0.595473	+	0.483948i
$302$		0			0
$303$	302643.	−	524193.i	0.189376	−	0.328008i
$304$		0			0
$305$	1.23366e6	+	2.13676e6i	0.759356	+	1.31524i
$306$		0			0
$307$	−2.23869e6			−1.35565			−0.677824	−	0.735224i	$-0.737077\pi$
$307$	−2.23869e6			−1.35565			−0.677824	+	0.735224i	$0.737077\pi$
$308$		0			0
$309$	1.04972e6			0.625426
$310$		0			0
$311$	750695.	+	1.30024e6i	0.440111	+	0.762295i	0.997697	−	0.0678240i	$-0.0216056\pi$
$311$	750695.	+	1.30024e6i	0.440111	+	0.762295i	−0.557586	+	0.830119i	$0.688272\pi$
$312$		0			0
$313$	890067.	−	1.54164e6i	0.513525	−	0.889452i	−0.486352	−	0.873763i	$-0.661673\pi$
$313$	890067.	−	1.54164e6i	0.513525	−	0.889452i	0.999877	−	0.0156888i	$-0.00499410\pi$
$314$		0			0
$315$	−130676.	+	815731.i	−0.0742028	+	0.463202i
$316$		0			0
$317$	−735500.	+	1.27392e6i	−0.411088	+	0.712025i	−0.995009	−	0.0997854i	$-0.968184\pi$
$317$	−735500.	+	1.27392e6i	−0.411088	+	0.712025i	0.583921	+	0.811810i	$0.301518\pi$
$318$		0			0
$319$	1.12655e6	+	1.95124e6i	0.619832	+	1.07358i
$320$		0			0
$321$	1.44581e6			0.783155
$322$		0			0
$323$	−635666.			−0.339018
$324$		0			0
$325$	−1.25440e6	−	2.17268e6i	−0.658760	−	1.14101i
$326$		0			0
$327$	537015.	−	930137.i	0.277726	−	0.481036i
$328$		0			0
$329$	1.56306e6	−	596837.i	0.796134	−	0.303995i
$330$		0			0
$331$	914873.	−	1.58461e6i	0.458977	−	0.794971i	−0.539930	−	0.841710i	$-0.681549\pi$
$331$	914873.	−	1.58461e6i	0.458977	−	0.794971i	0.998907	+	0.0467386i	$0.0148828\pi$
$332$		0			0
$333$	−197220.	−	341594.i	−0.0974629	−	0.168811i
$334$		0			0
$335$	4.40120e6			2.14269
$336$		0			0
$337$	3.18627e6			1.52830			0.764148	−	0.645041i	$-0.223160\pi$
$337$	3.18627e6			1.52830			0.764148	+	0.645041i	$0.223160\pi$
$338$		0			0
$339$	−499018.	−	864324.i	−0.235840	−	0.408486i
$340$		0			0
$341$	−3.49878e6	+	6.06006e6i	−1.62941	+	2.82222i
$342$		0			0
$343$	999468.	−	1.93614e6i	0.458705	−	0.888589i
$344$		0			0
$345$	891286.	−	1.54375e6i	0.403153	−	0.698281i
$346$		0			0
$347$	−1.23895e6	−	2.14592e6i	−0.552370	−	0.956732i	−0.998103	−	0.0615664i	$-0.980390\pi$
$347$	−1.23895e6	−	2.14592e6i	−0.552370	−	0.956732i	0.445733	−	0.895166i	$-0.352943\pi$
$348$		0			0
$349$	1.81720e6			0.798618			0.399309	−	0.916816i	$-0.369250\pi$
$349$	1.81720e6			0.798618			0.399309	+	0.916816i	$0.369250\pi$
$350$		0			0
$351$	−596856.			−0.258584
$352$		0			0
$353$	−1.57678e6	−	2.73106e6i	−0.673493	−	1.16652i	−0.976907	−	0.213666i	$-0.931460\pi$
$353$	−1.57678e6	−	2.73106e6i	−0.673493	−	1.16652i	0.303413	−	0.952859i	$-0.401874\pi$
$354$		0			0
$355$	−806425.	+	1.39677e6i	−0.339620	+	0.588239i
$356$		0			0
$357$	−1.20913e6	+	461692.i	−0.502113	+	0.191726i
$358$		0			0
$359$	−1.35105e6	+	2.34008e6i	−0.553266	+	0.958285i	0.444770	+	0.895645i	$0.353285\pi$
$359$	−1.35105e6	+	2.34008e6i	−0.553266	+	0.958285i	−0.998036	+	0.0626404i	$0.980048\pi$
$360$		0			0
$361$	1.07386e6	+	1.85998e6i	0.433690	+	0.751173i
$362$		0			0
$363$	−2.85433e6			−1.13694
$364$		0			0
$365$	−5.32214e6			−2.09100
$366$		0			0
$367$	1.80573e6	+	3.12762e6i	0.699824	+	1.21213i	0.968527	+	0.248908i	$0.0800716\pi$
$367$	1.80573e6	+	3.12762e6i	0.699824	+	1.21213i	−0.268703	+	0.963223i	$0.586595\pi$
$368$		0			0
$369$	530319.	−	918540.i	0.202755	−	0.351182i
$370$		0			0
$371$	400703.	−	2.50134e6i	0.151143	−	0.943491i
$372$		0			0
$373$	216759.	−	375438.i	0.0806688	−	0.139722i	−0.822869	−	0.568232i	$-0.807628\pi$
$373$	216759.	−	375438.i	0.0806688	−	0.139722i	0.903537	+	0.428509i	$0.140961\pi$
$374$		0			0
$375$	−21508.4	−	37253.6i	−0.00789823	−	0.0136801i
$376$		0			0
$377$	−2.66759e6			−0.966642
$378$		0			0
$379$	−2.12163e6			−0.758704			−0.379352	−	0.925252i	$-0.623853\pi$
$379$	−2.12163e6			−0.758704			−0.379352	+	0.925252i	$0.623853\pi$
$380$		0			0
$381$	1.41833e6	+	2.45662e6i	0.500570	+	0.867012i
$382$		0			0
$383$	2.12339e6	−	3.67782e6i	0.739662	−	1.28113i	−0.212985	−	0.977055i	$-0.568319\pi$
$383$	2.12339e6	−	3.67782e6i	0.739662	−	1.28113i	0.952647	−	0.304077i	$-0.0983480\pi$
$384$		0			0
$385$	−5.47329e6	−	4.44821e6i	−1.88190	−	1.52944i
$386$		0			0
$387$	376797.	−	652632.i	0.127888	−	0.221509i
$388$		0			0
$389$	1.76088e6	+	3.04994e6i	0.590006	+	1.02192i	0.994231	+	0.107261i	$0.0342081\pi$
$389$	1.76088e6	+	3.04994e6i	0.590006	+	1.02192i	−0.404224	+	0.914660i	$0.632459\pi$
$390$		0			0
$391$	2.79271e6			0.923811
$392$		0			0
$393$	−770981.			−0.251804
$394$		0			0
$395$	−553845.	−	959288.i	−0.178606	−	0.309354i
$396$		0			0
$397$	367025.	−	635706.i	0.116874	−	0.202432i	−0.801653	−	0.597790i	$-0.796046\pi$
$397$	367025.	−	635706.i	0.116874	−	0.202432i	0.918527	+	0.395357i	$0.129379\pi$
$398$		0			0
$399$	−518869.	−	421691.i	−0.163164	−	0.132606i
$400$		0			0
$401$	111135.	−	192492.i	0.0345136	−	0.0597793i	−0.848253	−	0.529592i	$-0.822345\pi$
$401$	111135.	−	192492.i	0.0345136	−	0.0597793i	0.882766	+	0.469812i	$0.155678\pi$
$402$		0			0
$403$	−4.14242e6	−	7.17488e6i	−1.27055	−	2.20065i
$404$		0			0
$405$	516165.			0.156369
$406$		0			0
$407$	3.36743e6			1.00766
$408$		0			0
$409$	1.75379e6	+	3.03765e6i	0.518405	+	0.897904i	0.999771	+	0.0213846i	$0.00680745\pi$
$409$	1.75379e6	+	3.03765e6i	0.518405	+	0.897904i	−0.481366	+	0.876520i	$0.659859\pi$
$410$		0			0
$411$	−146845.	+	254343.i	−0.0428800	+	0.0742703i
$412$		0			0
$413$	515134.	−	3.21566e6i	0.148609	−	0.927674i
$414$		0			0
$415$	3.03394e6	−	5.25494e6i	0.864742	−	1.49778i
$416$		0			0
$417$	929653.	+	1.61021e6i	0.261807	+	0.453462i
$418$		0			0
$419$	−1.97040e6			−0.548302			−0.274151	−	0.961687i	$-0.588397\pi$
$419$	−1.97040e6			−0.548302			−0.274151	+	0.961687i	$0.588397\pi$
$420$		0			0
$421$	−7.19318e6			−1.97795			−0.988976	−	0.148079i	$-0.952691\pi$
$421$	−7.19318e6			−1.97795			−0.988976	+	0.148079i	$0.952691\pi$
$422$		0			0
$423$	−522685.	−	905317.i	−0.142033	−	0.246008i
$424$		0			0
$425$	−1.69955e6	+	2.94370e6i	−0.456416	+	0.790535i
$426$		0			0
$427$	−3.79837e6	+	1.45036e6i	−1.00815	+	0.384953i
$428$		0			0
$429$	2.54776e6	−	4.41285e6i	0.668367	−	1.15765i
$430$		0			0
$431$	−262091.	−	453955.i	−0.0679608	−	0.117712i	0.830043	−	0.557700i	$-0.188316\pi$
$431$	−262091.	−	453955.i	−0.0679608	−	0.117712i	−0.898004	+	0.439988i	$0.854983\pi$
$432$		0			0
$433$	1.49326e6			0.382750			0.191375	−	0.981517i	$-0.438705\pi$
$433$	1.49326e6			0.382750			0.191375	+	0.981517i	$0.438705\pi$
$434$		0			0
$435$	2.30695e6			0.584541
$436$		0			0
$437$	721349.	+	1.24941e6i	0.180693	+	0.312970i
$438$		0			0
$439$	−1.32835e6	+	2.30077e6i	−0.328966	+	0.569786i	−0.982307	−	0.187278i	$-0.940034\pi$
$439$	−1.32835e6	+	2.30077e6i	−0.328966	+	0.569786i	0.653341	+	0.757064i	$0.273367\pi$
$440$		0			0
$441$	−1.29324e6	−	425256.i	−0.316653	−	0.104125i
$442$		0			0
$443$	2.34399e6	−	4.05991e6i	0.567474	−	0.982895i	−0.429340	−	0.903143i	$-0.641254\pi$
$443$	2.34399e6	−	4.05991e6i	0.567474	−	0.982895i	0.996815	−	0.0797518i	$-0.0254128\pi$
$444$		0			0
$445$	−12618.7	−	21856.2i	−0.00302074	−	0.00523207i
$446$		0			0
$447$	−4.35340e6			−1.03053
$448$		0			0
$449$	−3.45899e6			−0.809716			−0.404858	−	0.914379i	$-0.632679\pi$
$449$	−3.45899e6			−0.809716			−0.404858	+	0.914379i	$0.632679\pi$
$450$		0			0
$451$	4.52748e6	+	7.84182e6i	1.04813	+	1.81541i
$452$		0			0
$453$	−752494.	+	1.30336e6i	−0.172289	+	0.298413i
$454$		0			0
$455$	7.80102e6	−	2.97873e6i	1.76654	−	0.674533i
$456$		0			0
$457$	−359140.	+	622048.i	−0.0804401	+	0.139326i	−0.903439	−	0.428717i	$-0.858966\pi$
$457$	−359140.	+	622048.i	−0.0804401	+	0.139326i	0.822999	+	0.568043i	$0.192299\pi$
$458$		0			0
$459$	404331.	+	700322.i	0.0895788	+	0.155155i
$460$		0			0
$461$	5.07998e6			1.11329			0.556647	−	0.830749i	$-0.312088\pi$
$461$	5.07998e6			1.11329			0.556647	+	0.830749i	$0.312088\pi$
$462$		0			0
$463$	3.40607e6			0.738416			0.369208	−	0.929347i	$-0.379629\pi$
$463$	3.40607e6			0.738416			0.369208	+	0.929347i	$0.379629\pi$
$464$		0			0
$465$	3.58239e6	+	6.20489e6i	0.768318	+	1.33077i
$466$		0			0
$467$	−921342.	+	1.59581e6i	−0.195492	+	0.338602i	−0.947062	−	0.321052i	$-0.895964\pi$
$467$	−921342.	+	1.59581e6i	−0.195492	+	0.338602i	0.751570	+	0.659654i	$0.229297\pi$
$468$		0			0
$469$	−1.14722e6	+	7.16135e6i	−0.240831	+	1.50336i
$470$		0			0
$471$	−63488.3	+	109965.i	−0.0131869	+	0.0228403i
$472$		0			0
$473$	3.21682e6	+	5.57170e6i	0.661110	+	1.14508i
$474$		0			0
$475$	−1.75595e6			−0.357091
$476$		0			0
$477$	−1.58276e6			−0.318507
$478$		0			0
$479$	−4.55714e6	−	7.89319e6i	−0.907514	−	1.57186i	−0.817507	−	0.575919i	$-0.804644\pi$
$479$	−4.55714e6	−	7.89319e6i	−0.907514	−	1.57186i	−0.0900071	−	0.995941i	$-0.528689\pi$
$480$		0			0
$481$	−1.99346e6	+	3.45277e6i	−0.392866	+	0.680464i
$482$		0			0
$483$	2.27957e6	+	1.85264e6i	0.444617	+	0.361346i
$484$		0			0
$485$	4.40596e6	−	7.63134e6i	0.850523	−	1.47315i
$486$		0			0
$487$	−2.19220e6	−	3.79701e6i	−0.418850	−	0.725469i	0.576974	−	0.816762i	$-0.304233\pi$
$487$	−2.19220e6	−	3.79701e6i	−0.418850	−	0.725469i	−0.995824	+	0.0912933i	$0.970900\pi$
$488$		0			0
$489$	−5.88939e6			−1.11378
$490$		0			0
$491$	−9.50592e6			−1.77947			−0.889735	−	0.456478i	$-0.849111\pi$
$491$	−9.50592e6			−1.77947			−0.889735	+	0.456478i	$0.849111\pi$
$492$		0			0
$493$	1.80712e6	+	3.13002e6i	0.334864	+	0.580002i
$494$		0			0
$495$	−2.20332e6	+	3.81627e6i	−0.404171	+	0.700044i
$496$		0			0
$497$	−2.06253e6	−	1.67625e6i	−0.374550	−	0.304401i
$498$		0			0
$499$	−1.68783e6	+	2.92341e6i	−0.303444	+	0.525580i	−0.976914	−	0.213635i	$-0.931470\pi$
$499$	−1.68783e6	+	2.92341e6i	−0.303444	+	0.525580i	0.673470	+	0.739215i	$0.264803\pi$
$500$		0			0
$501$	823255.	+	1.42592e6i	0.146535	+	0.253805i
$502$		0			0
$503$	7.82645e6			1.37926			0.689628	−	0.724164i	$-0.257774\pi$
$503$	7.82645e6			1.37926			0.689628	+	0.724164i	$0.257774\pi$
$504$		0			0
$505$	5.29099e6			0.923227
$506$		0			0
$507$	1.34563e6	+	2.33070e6i	0.232491	+	0.402687i
$508$		0			0
$509$	3.71219e6	−	6.42969e6i	0.635090	−	1.10001i	−0.351406	−	0.936223i	$-0.614296\pi$
$509$	3.71219e6	−	6.42969e6i	0.635090	−	1.10001i	0.986496	−	0.163785i	$-0.0523703\pi$
$510$		0			0
$511$	1.38727e6	−	8.65985e6i	0.235022	−	1.46709i
$512$		0			0
$513$	−208875.	+	361783.i	−0.0350424	+	0.0606952i
$514$		0			0
$515$	4.58795e6	+	7.94656e6i	0.762256	+	1.32027i
$516$		0			0
$517$	8.92461e6			1.46846
$518$		0			0
$519$	2.62807e6			0.428272
$520$		0			0
$521$	530710.	+	919217.i	0.0856570	+	0.148362i	0.905671	−	0.423981i	$-0.139368\pi$
$521$	530710.	+	919217.i	0.0856570	+	0.148362i	−0.820014	+	0.572344i	$0.806034\pi$
$522$		0			0
$523$	956760.	−	1.65716e6i	0.152950	−	0.264917i	−0.779361	−	0.626575i	$-0.784456\pi$
$523$	956760.	−	1.65716e6i	0.152950	−	0.264917i	0.932311	+	0.361659i	$0.117789\pi$
$524$		0			0
$525$	−3.34008e6	+	1.27537e6i	−0.528881	+	0.201948i
$526$		0			0
$527$	−5.61243e6	+	9.72102e6i	−0.880288	+	1.52470i
$528$		0			0
$529$	49029.1	+	84921.0i	0.00761755	+	0.0131940i
$530$		0			0
$531$	−2.03476e6			−0.313167
$532$		0			0
$533$	−1.07207e7			−1.63458
$534$		0			0
$535$	6.31912e6	+	1.09450e7i	0.954492	+	1.65323i
$536$		0			0
$537$	876454.	−	1.51806e6i	0.131158	−	0.227172i
$538$		0			0
$539$	8.66451e6	−	7.74632e6i	1.28461	−	1.14848i
$540$		0			0
$541$	−5.83140e6	+	1.01003e7i	−0.856603	+	1.48368i	0.0185472	+	0.999828i	$0.494096\pi$
$541$	−5.83140e6	+	1.01003e7i	−0.856603	+	1.48368i	−0.875150	+	0.483852i	$0.839237\pi$
$542$		0			0
$543$	−1.15485e6	−	2.00026e6i	−0.168084	−	0.291130i
$544$		0			0
$545$	9.38842e6			1.35395
$546$		0			0
$547$	−6.36659e6			−0.909785			−0.454893	−	0.890546i	$-0.650322\pi$
$547$	−6.36659e6			−0.909785			−0.454893	+	0.890546i	$0.650322\pi$
$548$		0			0
$549$	1.27017e6	+	2.20000e6i	0.179858	+	0.311524i
$550$		0			0
$551$	−933548.	+	1.61695e6i	−0.130996	+	0.226891i
$552$		0			0
$553$	1.70526e6	−	651134.i	0.237125	−	0.0905435i
$554$		0			0
$555$	1.72396e6	−	2.98598e6i	0.237571	−	0.411486i
$556$		0			0
$557$	−4.86221e6	−	8.42159e6i	−0.664042	−	1.15015i	−0.979544	−	0.201230i	$-0.935506\pi$
$557$	−4.86221e6	−	8.42159e6i	−0.664042	−	1.15015i	0.315502	−	0.948925i	$-0.397827\pi$
$558$		0			0
$559$	−7.61719e6			−1.03102
$560$		0			0
$561$	−6.90376e6			−0.926145
$562$		0			0
$563$	−3.15133e6	−	5.45826e6i	−0.419008	−	0.725744i	0.576832	−	0.816863i	$-0.304289\pi$
$563$	−3.15133e6	−	5.45826e6i	−0.419008	−	0.725744i	−0.995840	+	0.0911193i	$0.970956\pi$
$564$		0			0
$565$	4.36207e6	−	7.55532e6i	0.574872	−	0.995708i
$566$		0			0
$567$	−134544.	+	839872.i	−0.0175754	+	0.109712i
$568$		0			0
$569$	−4.08244e6	+	7.07099e6i	−0.528614	+	0.915586i	0.470829	+	0.882224i	$0.343955\pi$
$569$	−4.08244e6	+	7.07099e6i	−0.528614	+	0.915586i	−0.999443	+	0.0333620i	$0.989379\pi$
$570$		0			0
$571$	6.43857e6	+	1.11519e7i	0.826416	+	1.43139i	0.900832	+	0.434168i	$0.142957\pi$
$571$	6.43857e6	+	1.11519e7i	0.826416	+	1.43139i	−0.0744158	+	0.997227i	$0.523709\pi$
$572$		0			0
$573$	2.64554e6			0.336611
$574$		0			0
$575$	7.71453e6			0.973061
$576$		0			0
$577$	−5.70887e6	−	9.88805e6i	−0.713856	−	1.23643i	−0.963399	−	0.268071i	$-0.913614\pi$
$577$	−5.70887e6	−	9.88805e6i	−0.713856	−	1.23643i	0.249543	−	0.968364i	$-0.419720\pi$
$578$		0			0
$579$	−311085.	+	538816.i	−0.0385641	+	0.0667950i
$580$		0			0
$581$	7.75968e6	+	6.30639e6i	0.953681	+	0.775069i
$582$		0			0
$583$	6.75622e6	−	1.17021e7i	0.823251	−	1.42591i
$584$		0			0
$585$	−2.60865e6	−	4.51831e6i	−0.315156	−	0.545867i
$586$		0			0
$587$	2.12781e6			0.254882			0.127441	−	0.991846i	$-0.459324\pi$
$587$	2.12781e6			0.254882			0.127441	+	0.991846i	$0.459324\pi$
$588$		0			0
$589$	−5.79871e6			−0.688721
$590$		0			0
$591$	1.49287e6	+	2.58572e6i	0.175814	+	0.304518i
$592$		0			0
$593$	4.28689e6	−	7.42511e6i	0.500617	−	0.867094i	−0.499383	−	0.866382i	$-0.666440\pi$
$593$	4.28689e6	−	7.42511e6i	0.500617	−	0.867094i	1.00000		0.000712836i	$-0.000226903\pi$
$594$		0			0
$595$	−8.77978e6	−	7.13544e6i	−1.01670	−	0.826282i
$596$		0			0
$597$	2.58493e6	−	4.47723e6i	0.296833	−	0.514130i
$598$		0			0
$599$	−5.64372e6	−	9.77521e6i	−0.642686	−	1.11316i	−0.984831	−	0.173517i	$-0.944487\pi$
$599$	−5.64372e6	−	9.77521e6i	−0.642686	−	1.11316i	0.342145	−	0.939647i	$-0.388846\pi$
$600$		0			0
$601$	−1.58260e7			−1.78725			−0.893625	−	0.448814i	$-0.851847\pi$
$601$	−1.58260e7			−1.78725			−0.893625	+	0.448814i	$0.851847\pi$
$602$		0			0
$603$	4.53145e6			0.507509
$604$		0			0
$605$	−1.24753e7	−	2.16078e7i	−1.38568	−	2.40006i
$606$		0			0
$607$	−4.87183e6	+	8.43826e6i	−0.536686	+	0.929568i	0.462393	+	0.886675i	$0.346991\pi$
$607$	−4.87183e6	+	8.43826e6i	−0.536686	+	0.929568i	−0.999080	+	0.0428932i	$0.986342\pi$
$608$		0			0
$609$	−601329.	+	3.75372e6i	−0.0657006	+	0.410128i
$610$		0			0
$611$	−5.28320e6	+	9.15077e6i	−0.572525	+	0.991642i
$612$		0			0
$613$	−3.07524e6	−	5.32647e6i	−0.330543	−	0.572517i	0.652075	−	0.758154i	$-0.273899\pi$
$613$	−3.07524e6	−	5.32647e6i	−0.330543	−	0.572517i	−0.982618	+	0.185637i	$0.940565\pi$
$614$		0			0
$615$	9.27137e6			0.988453
$616$		0			0
$617$	−3.36084e6			−0.355414			−0.177707	−	0.984083i	$-0.556868\pi$
$617$	−3.36084e6			−0.355414			−0.177707	+	0.984083i	$0.556868\pi$
$618$		0			0
$619$	−3.44329e6	−	5.96396e6i	−0.361200	−	0.625616i	0.626959	−	0.779052i	$-0.284299\pi$
$619$	−3.44329e6	−	5.96396e6i	−0.361200	−	0.625616i	−0.988159	+	0.153436i	$0.950966\pi$
$620$		0			0
$621$	917663.	−	1.58944e6i	0.0954892	−	0.165392i
$622$		0			0
$623$	38852.1	−	14835.3i	0.00401046	−	0.00153135i
$624$		0			0
$625$	4.97590e6	−	8.61850e6i	0.509532	−	0.882535i
$626$		0			0
$627$	−1.78323e6	−	3.08864e6i	−0.181150	−	0.313760i
$628$		0			0
$629$	5.40175e6			0.544387
$630$		0			0
$631$	3.59545e6			0.359484			0.179742	−	0.983714i	$-0.442474\pi$
$631$	3.59545e6			0.359484			0.179742	+	0.983714i	$0.442474\pi$
$632$		0			0
$633$	−1.72799e6	−	2.99297e6i	−0.171409	−	0.296889i
$634$		0			0
$635$	−1.23980e7	+	2.14740e7i	−1.22017	+	2.11339i
$636$		0			0
$637$	2.81339e6	+	1.34698e7i	0.274715	+	1.31526i
$638$		0			0
$639$	−830291.	+	1.43811e6i	−0.0804411	+	0.139328i
$640$		0			0
$641$	9.91433e6	+	1.71721e7i	0.953056	+	1.65074i	0.738756	+	0.673973i	$0.235413\pi$
$641$	9.91433e6	+	1.71721e7i	0.953056	+	1.65074i	0.214299	+	0.976768i	$0.431253\pi$
$642$		0			0
$643$	−8.10851e6			−0.773417			−0.386708	−	0.922202i	$-0.626388\pi$
$643$	−8.10851e6			−0.773417			−0.386708	+	0.922202i	$0.626388\pi$
$644$		0			0
$645$	6.58740e6			0.623469
$646$		0			0
$647$	−1.01373e6	−	1.75583e6i	−0.0952053	−	0.164900i	0.814489	−	0.580179i	$-0.197017\pi$
$647$	−1.01373e6	−	1.75583e6i	−0.0952053	−	0.164900i	−0.909694	+	0.415279i	$0.863684\pi$
$648$		0			0
$649$	8.68564e6	−	1.50440e7i	0.809450	−	1.40201i
$650$		0			0
$651$	−1.10300e7	+	4.21168e6i	−1.02005	+	0.389496i
$652$		0			0
$653$	−757133.	+	1.31139e6i	−0.0694847	+	0.120351i	−0.898675	−	0.438616i	$-0.855469\pi$
$653$	−757133.	+	1.31139e6i	−0.0694847	+	0.120351i	0.829190	+	0.558967i	$0.188802\pi$
$654$		0			0
$655$	−3.36969e6	−	5.83647e6i	−0.306893	−	0.531554i
$656$		0			0
$657$	−5.47965e6			−0.495267
$658$		0			0
$659$	−3.22358e6			−0.289151			−0.144576	−	0.989494i	$-0.546182\pi$
$659$	−3.22358e6			−0.289151			−0.144576	+	0.989494i	$0.546182\pi$
$660$		0			0
$661$	7.19538e6	+	1.24628e7i	0.640546	+	1.10946i	0.985311	+	0.170769i	$0.0546253\pi$
$661$	7.19538e6	+	1.24628e7i	0.640546	+	1.10946i	−0.344765	+	0.938689i	$0.612041\pi$
$662$		0			0
$663$	4.08690e6	−	7.07872e6i	0.361085	−	0.625418i
$664$		0			0
$665$	924488.	−	5.77100e6i	0.0810676	−	0.506054i
$666$		0			0
$667$	4.10141e6	−	7.10384e6i	0.356959	−	0.618271i
$668$		0			0
$669$	−1.56224e6	−	2.70589e6i	−0.134953	−	0.233746i
$670$		0			0
$671$	−2.16875e7			−1.85953
$672$		0			0
$673$	7.33478e6			0.624237			0.312118	−	0.950043i	$-0.398961\pi$
$673$	7.33478e6			0.624237			0.312118	+	0.950043i	$0.398961\pi$
$674$		0			0
$675$	1.11692e6	+	1.93456e6i	0.0943543	+	0.163426i
$676$		0			0
$677$	1.52642e6	−	2.64384e6i	0.127998	−	0.221699i	−0.794903	−	0.606737i	$-0.792478\pi$
$677$	1.52642e6	−	2.64384e6i	0.127998	−	0.221699i	0.922901	+	0.385038i	$0.125812\pi$
$678$		0			0
$679$	1.12688e7	+	9.15828e6i	0.937999	+	0.762324i
$680$		0			0
$681$	1.62577e6	−	2.81591e6i	0.134335	−	0.232676i
$682$		0			0
$683$	−7.98793e6	−	1.38355e7i	−0.655213	−	1.13486i	−0.981840	−	0.189709i	$-0.939245\pi$
$683$	−7.98793e6	−	1.38355e7i	−0.655213	−	1.13486i	0.326627	−	0.945153i	$-0.394088\pi$
$684$		0			0
$685$	−2.56724e6			−0.209045
$686$		0			0
$687$	6.77685e6			0.547817
$688$		0			0
$689$	7.99911e6	+	1.38549e7i	0.641939	+	1.11187i
$690$		0			0
$691$	1.23267e6	−	2.13504e6i	0.0982087	−	0.170103i	−0.812735	−	0.582634i	$-0.802022\pi$
$691$	1.23267e6	−	2.13504e6i	0.0982087	−	0.170103i	0.910943	+	0.412532i	$0.135355\pi$
$692$		0			0
$693$	−5.63527e6	−	4.57985e6i	−0.445740	−	0.362258i
$694$		0			0
$695$	−8.12638e6	+	1.40753e7i	−0.638168	+	1.10534i
$696$		0			0
$697$	7.26259e6	+	1.25792e7i	0.566252	+	0.980777i
$698$		0			0
$699$	−6.42958e6			−0.497725
$700$		0			0
$701$	1.74893e7			1.34424			0.672122	−	0.740440i	$-0.265383\pi$
$701$	1.74893e7			1.34424			0.672122	+	0.740440i	$0.265383\pi$
$702$		0			0
$703$	1.39526e6	+	2.41666e6i	0.106480	+	0.184428i
$704$		0			0
$705$	4.56895e6	−	7.91366e6i	0.346213	−	0.599659i
$706$		0			0
$707$	−1.37915e6	+	8.60916e6i	−0.103768	+	0.647757i
$708$		0			0
$709$	1.09216e7	−	1.89167e7i	0.815961	−	1.41329i	−0.0926752	−	0.995696i	$-0.529542\pi$
$709$	1.09216e7	−	1.89167e7i	0.815961	−	1.41329i	0.908636	−	0.417589i	$-0.137125\pi$
$710$		0			0
$711$	−570236.	−	987677.i	−0.0423039	−	0.0732725i
$712$		0			0
$713$	2.54758e7			1.87674
$714$		0			0
$715$	4.45415e7			3.25837
$716$		0			0
$717$	1.54614e6	+	2.67800e6i	0.112319	+	0.194541i
$718$		0			0
$719$	−9.55868e6	+	1.65561e7i	−0.689566	+	1.19436i	0.282413	+	0.959293i	$0.408865\pi$
$719$	−9.55868e6	+	1.65561e7i	−0.689566	+	1.19436i	−0.971978	+	0.235070i	$0.924468\pi$
$720$		0			0
$721$	−1.41260e7	+	5.39387e6i	−1.01200	+	0.386423i
$722$		0			0
$723$	−343372.	+	594738.i	−0.0244298	+	0.0423136i
$724$		0			0
$725$	4.99195e6	+	8.64632e6i	0.352716	+	0.610923i
$726$		0			0
$727$	−4.08900e6			−0.286933			−0.143467	−	0.989655i	$-0.545825\pi$
$727$	−4.08900e6			−0.286933			−0.143467	+	0.989655i	$0.545825\pi$
$728$		0			0
$729$	531441.			0.0370370
$730$		0			0
$731$	5.16015e6	+	8.93764e6i	0.357165	+	0.618628i
$732$		0			0
$733$	1.81634e6	−	3.14599e6i	0.124864	−	0.216271i	−0.796816	−	0.604222i	$-0.793484\pi$
$733$	1.81634e6	−	3.14599e6i	0.124864	−	0.216271i	0.921680	+	0.387952i	$0.126817\pi$
$734$		0			0
$735$	−2.43304e6	−	1.16487e7i	−0.166124	−	0.795355i
$736$		0			0
$737$	−1.93431e7	+	3.35032e7i	−1.31177	+	2.27205i
$738$		0			0
$739$	−1.08978e6	−	1.88756e6i	−0.0734056	−	0.127142i	0.826986	−	0.562222i	$-0.190053\pi$
$739$	−1.08978e6	−	1.88756e6i	−0.0734056	−	0.127142i	−0.900392	+	0.435080i	$0.856720\pi$
$740$		0			0
$741$	4.22254e6			0.282507
$742$		0			0
$743$	1.21595e7			0.808060			0.404030	−	0.914746i	$-0.367609\pi$
$743$	1.21595e7			0.808060			0.404030	+	0.914746i	$0.367609\pi$
$744$		0			0
$745$	−1.90272e7	−	3.29561e7i	−1.25598	−	2.17543i
$746$		0			0
$747$	3.12373e6	−	5.41045e6i	0.204820	−	0.354758i
$748$		0			0
$749$	−1.94562e7	+	7.42914e6i	−1.26722	+	0.483876i
$750$		0			0
$751$	−1.16248e7	+	2.01347e7i	−0.752115	+	1.30270i	0.194681	+	0.980867i	$0.437633\pi$
$751$	−1.16248e7	+	2.01347e7i	−0.752115	+	1.30270i	−0.946796	+	0.321835i	$0.895701\pi$
$752$		0			0
$753$	883121.	+	1.52961e6i	0.0567587	+	0.0983090i
$754$		0			0
$755$	−1.31556e7			−0.839928
$756$		0			0
$757$	−1.49939e7			−0.950985			−0.475492	−	0.879720i	$-0.657730\pi$
$757$	−1.49939e7			−0.950985			−0.475492	+	0.879720i	$0.657730\pi$
$758$		0			0
$759$	7.83434e6	+	1.35695e7i	0.493626	+	0.854985i
$760$		0			0
$761$	6.36596e6	−	1.10262e7i	0.398476	−	0.690181i	−0.595062	−	0.803680i	$-0.702873\pi$
$761$	6.36596e6	−	1.10262e7i	0.398476	−	0.690181i	0.993538	+	0.113499i	$0.0362058\pi$
$762$		0			0
$763$	−2.44719e6	+	1.52762e7i	−0.152179	+	0.949960i
$764$		0			0
$765$	−3.53438e6	+	6.12173e6i	−0.218353	+	0.378199i
$766$		0			0
$767$	1.02835e7	+	1.78115e7i	0.631177	+	1.09323i
$768$		0			0
$769$	2.34653e6			0.143090			0.0715451	−	0.997437i	$-0.477207\pi$
$769$	2.34653e6			0.143090			0.0715451	+	0.997437i	$0.477207\pi$
$770$		0			0
$771$	1.53820e7			0.931915
$772$		0			0
$773$	−1.35931e7	−	2.35440e7i	−0.818222	−	1.41720i	−0.906991	−	0.421150i	$-0.861627\pi$
$773$	−1.35931e7	−	2.35440e7i	−0.818222	−	1.41720i	0.0887694	−	0.996052i	$-0.471707\pi$
$774$		0			0
$775$	−1.55037e7	+	2.68532e7i	−0.927217	+	1.60599i
$776$		0			0
$777$	4.40923e6	+	3.58344e6i	0.262006	+	0.212935i
$778$		0			0
$779$	−3.75182e6	+	6.49834e6i	−0.221513	+	0.383671i
$780$		0			0
$781$	−7.08842e6	−	1.22775e7i	−0.415836	−	0.720249i
$782$		0			0

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:	\( N \)	\(=\)	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	84.i (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(4.50000 + 7.79423i) q^{3} +(-39.3359 + 68.1317i) q^{5} +(-100.606 - 81.7641i) q^{7} +(-40.5000 + 70.1481i) q^{9} +O(q^{10})\)	\(q+(4.50000 + 7.79423i) q^{3} +(-39.3359 + 68.1317i) q^{5} +(-100.606 - 81.7641i) q^{7} +(-40.5000 + 70.1481i) q^{9} +(-345.759 - 598.873i) q^{11} +818.732 q^{13} -708.046 q^{15} +(-554.638 - 960.661i) q^{17} +(286.523 - 496.273i) q^{19} +(184.559 - 1152.09i) q^{21} +(-1258.80 + 2180.30i) q^{23} +(-1532.12 - 2653.71i) q^{25} -729.000 q^{27} -3258.19 q^{29} +(-5059.55 - 8763.40i) q^{31} +(3111.84 - 5389.86i) q^{33} +(9528.17 - 3638.23i) q^{35} +(-2434.81 + 4217.21i) q^{37} +(3684.30 + 6381.39i) q^{39} -13094.3 q^{41} -9303.64 q^{43} +(-3186.21 - 5518.67i) q^{45} +(-6452.90 + 11176.8i) q^{47} +(3436.28 + 16452.0i) q^{49} +(4991.74 - 8645.95i) q^{51} +(9770.12 + 16922.3i) q^{53} +54403.0 q^{55} +5157.42 q^{57} +(12560.2 + 21755.0i) q^{59} +(15681.1 - 27160.5i) q^{61} +(9810.15 - 3745.90i) q^{63} +(-32205.6 + 55781.7i) q^{65} +(-27971.9 - 48448.8i) q^{67} -22658.4 q^{69} +20501.0 q^{71} +(33825.0 + 58586.6i) q^{73} +(13789.1 - 23883.4i) q^{75} +(-14180.7 + 88521.1i) q^{77} +(-7039.95 + 12193.5i) q^{79} +(-3280.50 - 5681.99i) q^{81} -77129.1 q^{83} +87268.6 q^{85} +(-14661.9 - 25395.1i) q^{87} +(-160.396 + 277.815i) q^{89} +(-82369.7 - 66942.9i) q^{91} +(45536.0 - 78870.6i) q^{93} +(22541.3 + 39042.6i) q^{95} -112009. q^{97} +56013.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 36 q^{3} - 42 q^{7} - 324 q^{9}+O(q^{10}) \) 8 * q + 36 * q^3 - 42 * q^7 - 324 * q^9	\( 8 q + 36 q^{3} - 42 q^{7} - 324 q^{9} - 462 q^{11} - 1204 q^{13} + 228 q^{17} + 358 q^{19} + 1404 q^{21} - 2148 q^{23} - 5454 q^{25} - 5832 q^{27} - 11064 q^{29} + 830 q^{31} + 4158 q^{33} + 7692 q^{35} - 3914 q^{37} - 5418 q^{39} - 16632 q^{41} - 29036 q^{43} + 41700 q^{47} + 41876 q^{49} - 2052 q^{51} + 22164 q^{53} + 7784 q^{55} + 6444 q^{57} + 32886 q^{59} + 83732 q^{61} + 16038 q^{63} - 93192 q^{65} - 80034 q^{67} - 38664 q^{69} + 89544 q^{71} - 22470 q^{73} + 49086 q^{75} + 138732 q^{77} - 75286 q^{79} - 26244 q^{81} - 34836 q^{83} + 278504 q^{85} - 49788 q^{87} + 28944 q^{89} - 12058 q^{91} - 7470 q^{93} - 144120 q^{95} - 433356 q^{97} + 74844 q^{99}+O(q^{100}) \) 8 * q + 36 * q^3 - 42 * q^7 - 324 * q^9 - 462 * q^11 - 1204 * q^13 + 228 * q^17 + 358 * q^19 + 1404 * q^21 - 2148 * q^23 - 5454 * q^25 - 5832 * q^27 - 11064 * q^29 + 830 * q^31 + 4158 * q^33 + 7692 * q^35 - 3914 * q^37 - 5418 * q^39 - 16632 * q^41 - 29036 * q^43 + 41700 * q^47 + 41876 * q^49 - 2052 * q^51 + 22164 * q^53 + 7784 * q^55 + 6444 * q^57 + 32886 * q^59 + 83732 * q^61 + 16038 * q^63 - 93192 * q^65 - 80034 * q^67 - 38664 * q^69 + 89544 * q^71 - 22470 * q^73 + 49086 * q^75 + 138732 * q^77 - 75286 * q^79 - 26244 * q^81 - 34836 * q^83 + 278504 * q^85 - 49788 * q^87 + 28944 * q^89 - 12058 * q^91 - 7470 * q^93 - 144120 * q^95 - 433356 * q^97 + 74844 * q^99

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