LMFDB - Embedded newform 84.6.i.b.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:	$4$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			37.1
Root			$-20.7872 + 36.0044i$ of defining polynomial
Character	$\chi$	$=$	84.37
Dual form			84.6.i.b.25.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.50000 - 7.79423i) q^{3} +(-32.7872 + 56.7890i) q^{5} +(40.6487 - 123.104i) q^{7} +(-40.5000 + 70.1481i) q^{9} +O(q^{10})$	\(q+(-4.50000 - 7.79423i) q^{3} +(-32.7872 + 56.7890i) q^{5} +(40.6487 - 123.104i) q^{7} +(-40.5000 + 70.1481i) q^{9} +(122.787 + 212.674i) q^{11} +434.872 q^{13} +590.169 q^{15} +(551.149 + 954.618i) q^{17} +(1438.65 - 2491.81i) q^{19} +(-1142.42 + 237.145i) q^{21} +(2114.72 - 3662.80i) q^{23} +(-587.497 - 1017.57i) q^{25} +729.000 q^{27} +4969.60 q^{29} +(4391.32 + 7606.00i) q^{31} +(1105.08 - 1914.06i) q^{33} +(5658.22 + 6344.64i) q^{35} +(1220.03 - 2113.15i) q^{37} +(-1956.92 - 3389.49i) q^{39} -3668.55 q^{41} -7198.06 q^{43} +(-2655.76 - 4599.91i) q^{45} +(-1636.66 + 2834.78i) q^{47} +(-13502.4 - 10008.1i) q^{49} +(4960.34 - 8591.56i) q^{51} +(1510.84 + 2616.84i) q^{53} -16103.4 q^{55} -25895.6 q^{57} +(25743.7 + 44589.5i) q^{59} +(6656.65 - 11529.6i) q^{61} +(6989.26 + 7837.15i) q^{63} +(-14258.2 + 24695.9i) q^{65} +(-15447.9 - 26756.5i) q^{67} -38064.9 q^{69} -41882.8 q^{71} +(-17308.7 - 29979.5i) q^{73} +(-5287.47 + 9158.17i) q^{75} +(31172.2 - 6470.74i) q^{77} +(-38771.7 + 67154.6i) q^{79} +(-3280.50 - 5681.99i) q^{81} +100908. q^{83} -72282.4 q^{85} +(-22363.2 - 38734.2i) q^{87} +(-20391.8 + 35319.7i) q^{89} +(17677.0 - 53534.6i) q^{91} +(39521.9 - 68454.0i) q^{93} +(94338.2 + 163399. i) q^{95} +140147. q^{97} -19891.5 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/84\mathbb{Z}\right)^\times$.

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

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
$5$	−32.7872	+	56.7890i	−0.586515	+	1.01587i	0.408170	+	0.912906i	$0.366167\pi$
$5$	−32.7872	+	56.7890i	−0.586515	+	1.01587i	−0.994685	+	0.102967i	$0.967166\pi$
$6$		0			0
$7$	40.6487	−	123.104i	0.313546	−	0.949573i
$7$	40.6487	−	123.104i	0.313546	−	0.949573i
$8$		0			0
$9$	−40.5000	+	70.1481i	−0.166667	+	0.288675i
$10$		0			0
$11$	122.787	+	212.674i	0.305965	+	0.529946i	0.977476	−	0.211048i	$-0.0676877\pi$
$11$	122.787	+	212.674i	0.305965	+	0.529946i	−0.671511	+	0.740995i	$0.734354\pi$
$12$		0			0
$13$	434.872			0.713679			0.356839	−	0.934166i	$-0.383854\pi$
$13$	434.872			0.713679			0.356839	+	0.934166i	$0.383854\pi$
$14$		0			0
$15$	590.169			0.677249
$16$		0			0
$17$	551.149	+	954.618i	0.462537	+	0.801138i	0.999087	−	0.0427312i	$-0.0136059\pi$
$17$	551.149	+	954.618i	0.462537	+	0.801138i	−0.536550	+	0.843869i	$0.680273\pi$
$18$		0			0
$19$	1438.65	−	2491.81i	0.914260	−	1.58355i	0.106279	−	0.994336i	$-0.466106\pi$
$19$	1438.65	−	2491.81i	0.914260	−	1.58355i	0.807981	−	0.589209i	$-0.200560\pi$
$20$		0			0
$21$	−1142.42	+	237.145i	−0.565299	+	0.117345i
$22$		0			0
$23$	2114.72	−	3662.80i	0.833552	−	1.44375i	−0.0616521	−	0.998098i	$-0.519637\pi$
$23$	2114.72	−	3662.80i	0.833552	−	1.44375i	0.895204	−	0.445657i	$-0.147030\pi$
$24$		0			0
$25$	−587.497	−	1017.57i	−0.187999	−	0.325624i
$26$		0			0
$27$	729.000			0.192450
$28$		0			0
$29$	4969.60			1.09730			0.548652	−	0.836051i	$-0.315141\pi$
$29$	4969.60			1.09730			0.548652	+	0.836051i	$0.315141\pi$
$30$		0			0
$31$	4391.32	+	7606.00i	0.820713	+	1.42152i	0.905152	+	0.425088i	$0.139757\pi$
$31$	4391.32	+	7606.00i	0.820713	+	1.42152i	−0.0844390	+	0.996429i	$0.526910\pi$
$32$		0			0
$33$	1105.08	−	1914.06i	0.176649	−	0.305965i
$34$		0			0
$35$	5658.22	+	6344.64i	0.780746	+	0.875462i
$36$		0			0
$37$	1220.03	−	2113.15i	0.146510	−	0.253762i	−0.783425	−	0.621486i	$-0.786529\pi$
$37$	1220.03	−	2113.15i	0.146510	−	0.253762i	0.929935	+	0.367723i	$0.119863\pi$
$38$		0			0
$39$	−1956.92	−	3389.49i	−0.206021	−	0.356839i
$40$		0			0
$41$	−3668.55			−0.340828			−0.170414	−	0.985373i	$-0.554511\pi$
$41$	−3668.55			−0.340828			−0.170414	+	0.985373i	$0.554511\pi$
$42$		0			0
$43$	−7198.06			−0.593669			−0.296835	−	0.954929i	$-0.595931\pi$
$43$	−7198.06			−0.593669			−0.296835	+	0.954929i	$0.595931\pi$
$44$		0			0
$45$	−2655.76	−	4599.91i	−0.195505	−	0.338624i
$46$		0			0
$47$	−1636.66	+	2834.78i	−0.108072	+	0.187187i	−0.914989	−	0.403478i	$-0.867801\pi$
$47$	−1636.66	+	2834.78i	−0.108072	+	0.187187i	0.806917	+	0.590665i	$0.201134\pi$
$48$		0			0
$49$	−13502.4	−	10008.1i	−0.803378	−	0.595470i
$50$		0			0
$51$	4960.34	−	8591.56i	0.267046	−	0.462537i
$52$		0			0
$53$	1510.84	+	2616.84i	0.0738801	+	0.127964i	0.900599	−	0.434651i	$-0.143128\pi$
$53$	1510.84	+	2616.84i	0.0738801	+	0.127964i	−0.826719	+	0.562616i	$0.809795\pi$
$54$		0			0
$55$	−16103.4			−0.717811
$56$		0			0
$57$	−25895.6			−1.05570
$58$		0			0
$59$	25743.7	+	44589.5i	0.962812	+	1.66764i	0.715381	+	0.698735i	$0.246253\pi$
$59$	25743.7	+	44589.5i	0.962812	+	1.66764i	0.247432	+	0.968905i	$0.420413\pi$
$60$		0			0
$61$	6656.65	−	11529.6i	0.229050	−	0.396727i	−0.728477	−	0.685071i	$-0.759771\pi$
$61$	6656.65	−	11529.6i	0.229050	−	0.396727i	0.957527	+	0.288344i	$0.0931046\pi$
$62$		0			0
$63$	6989.26	+	7837.15i	0.221860	+	0.248775i
$64$		0			0
$65$	−14258.2	+	24695.9i	−0.418583	+	0.725007i
$66$		0			0
$67$	−15447.9	−	26756.5i	−0.420418	−	0.728186i	0.575562	−	0.817758i	$-0.304783\pi$
$67$	−15447.9	−	26756.5i	−0.420418	−	0.728186i	−0.995980	+	0.0895723i	$0.971450\pi$
$68$		0			0
$69$	−38064.9			−0.962503
$70$		0			0
$71$	−41882.8			−0.986030			−0.493015	−	0.870021i	$-0.664105\pi$
$71$	−41882.8			−0.986030			−0.493015	+	0.870021i	$0.664105\pi$
$72$		0			0
$73$	−17308.7	−	29979.5i	−0.380151	−	0.658441i	0.610932	−	0.791683i	$-0.290795\pi$
$73$	−17308.7	−	29979.5i	−0.380151	−	0.658441i	−0.991084	+	0.133242i	$0.957461\pi$
$74$		0			0
$75$	−5287.47	+	9158.17i	−0.108541	+	0.187999i
$76$		0			0
$77$	31172.2	−	6470.74i	0.599157	−	0.124373i
$78$		0			0
$79$	−38771.7	+	67154.6i	−0.698952	+	1.21062i	0.269878	+	0.962895i	$0.413017\pi$
$79$	−38771.7	+	67154.6i	−0.698952	+	1.21062i	−0.968830	+	0.247726i	$0.920317\pi$
$80$		0			0
$81$	−3280.50	−	5681.99i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	100908.			1.60779			0.803897	−	0.594768i	$-0.202756\pi$
$83$	100908.			1.60779			0.803897	+	0.594768i	$0.202756\pi$
$84$		0			0
$85$	−72282.4			−1.08514
$86$		0			0
$87$	−22363.2	−	38734.2i	−0.316764	−	0.548652i
$88$		0			0
$89$	−20391.8	+	35319.7i	−0.272886	+	0.472652i	−0.969600	−	0.244697i	$-0.921312\pi$
$89$	−20391.8	+	35319.7i	−0.272886	+	0.472652i	0.696714	+	0.717349i	$0.254645\pi$
$90$		0			0
$91$	17677.0	−	53534.6i	0.223771	−	0.677690i
$92$		0			0
$93$	39521.9	−	68454.0i	0.473839	−	0.820713i
$94$		0			0
$95$	94338.2	+	163399.i	1.07245	+	1.85755i
$96$		0			0
$97$	140147.			1.51236			0.756178	−	0.654366i	$-0.227064\pi$
$97$	140147.			1.51236			0.756178	+	0.654366i	$0.227064\pi$
$98$		0			0
$99$	−19891.5			−0.203976
$100$		0			0
$101$	−4288.23	−	7427.43i	−0.0418287	−	0.0724494i	0.844353	−	0.535787i	$-0.179985\pi$
$101$	−4288.23	−	7427.43i	−0.0418287	−	0.0724494i	−0.886182	+	0.463338i	$0.846652\pi$
$102$		0			0
$103$	17630.1	−	30536.2i	0.163742	−	0.283610i	−0.772466	−	0.635057i	$-0.780977\pi$
$103$	17630.1	−	30536.2i	0.163742	−	0.283610i	0.936208	+	0.351447i	$0.114310\pi$
$104$		0			0
$105$	23989.6	−	72652.4i	0.212349	−	0.643097i
$106$		0			0
$107$	96314.3	−	166821.i	0.813264	−	1.40861i	−0.0973041	−	0.995255i	$-0.531022\pi$
$107$	96314.3	−	166821.i	0.813264	−	1.40861i	0.910568	−	0.413360i	$-0.135645\pi$
$108$		0			0
$109$	−61646.1	−	106774.i	−0.496980	−	0.860795i	0.503014	−	0.864279i	$-0.332225\pi$
$109$	−61646.1	−	106774.i	−0.496980	−	0.860795i	−0.999994	+	0.00348322i	$0.998891\pi$
$110$		0			0
$111$	−21960.6			−0.169175
$112$		0			0
$113$	28400.9			0.209236			0.104618	−	0.994512i	$-0.466638\pi$
$113$	28400.9			0.209236			0.104618	+	0.994512i	$0.466638\pi$
$114$		0			0
$115$	138671.	+	240186.i	0.977781	+	1.69357i
$116$		0			0
$117$	−17612.3	+	30505.4i	−0.118946	+	0.206021i
$118$		0			0
$119$	139921.	−	29044.9i	0.905765	−	0.188019i
$120$		0			0
$121$	50372.1	−	87247.1i	0.312771	−	0.541736i
$122$		0			0
$123$	16508.5	+	28593.5i	0.0983886	+	0.170414i
$124$		0			0
$125$	−127870.			−0.731973
$126$		0			0
$127$	−47198.8			−0.259670			−0.129835	−	0.991536i	$-0.541445\pi$
$127$	−47198.8			−0.259670			−0.129835	+	0.991536i	$0.541445\pi$
$128$		0			0
$129$	32391.3	+	56103.3i	0.171377	+	0.296835i
$130$		0			0
$131$	−39904.7	+	69116.9i	−0.203163	+	0.351889i	−0.949546	−	0.313628i	$-0.898456\pi$
$131$	−39904.7	+	69116.9i	−0.203163	+	0.351889i	0.746383	+	0.665517i	$0.231789\pi$
$132$		0			0
$133$	−248273.	−	278392.i	−1.21703	−	1.36467i
$134$		0			0
$135$	−23901.8	+	41399.2i	−0.112875	+	0.195505i
$136$		0			0
$137$	43032.1	+	74533.9i	0.195881	+	0.339275i	0.947189	−	0.320676i	$-0.103910\pi$
$137$	43032.1	+	74533.9i	0.195881	+	0.339275i	−0.751308	+	0.659952i	$0.770577\pi$
$138$		0			0
$139$	270587.			1.18787			0.593935	−	0.804513i	$-0.297573\pi$
$139$	270587.			1.18787			0.593935	+	0.804513i	$0.297573\pi$
$140$		0			0
$141$	29459.9			0.124791
$142$		0			0
$143$	53396.7	+	92485.7i	0.218360	+	0.378211i
$144$		0			0
$145$	−162939.	+	282219.i	−0.643585	+	1.11472i
$146$		0			0
$147$	−17244.4	+	150277.i	−0.0658197	+	0.573586i
$148$		0			0
$149$	86273.9	−	149431.i	0.318356	−	0.551410i	−0.661789	−	0.749690i	$-0.730202\pi$
$149$	86273.9	−	149431.i	0.318356	−	0.551410i	0.980145	+	0.198281i	$0.0635358\pi$
$150$		0			0
$151$	8949.75	+	15501.4i	0.0319425	+	0.0553260i	0.881555	−	0.472082i	$-0.156497\pi$
$151$	8949.75	+	15501.4i	0.0319425	+	0.0553260i	−0.849612	+	0.527408i	$0.823164\pi$
$152$		0			0
$153$	−89286.1			−0.308358
$154$		0			0
$155$	−575916.			−1.92544
$156$		0			0
$157$	89509.3	+	155035.i	0.289814	+	0.501972i	0.973765	−	0.227555i	$-0.0730733\pi$
$157$	89509.3	+	155035.i	0.289814	+	0.501972i	−0.683951	+	0.729528i	$0.739740\pi$
$158$		0			0
$159$	13597.5	−	23551.6i	0.0426547	−	0.0738801i
$160$		0			0
$161$	−364946.	−	409219.i	−1.10959	−	1.24420i
$162$		0			0
$163$	118645.	−	205498.i	0.349767	−	0.605814i	−0.636441	−	0.771325i	$-0.719594\pi$
$163$	118645.	−	205498.i	0.349767	−	0.605814i	0.986208	+	0.165511i	$0.0529275\pi$
$164$		0			0
$165$	72465.2	+	125513.i	0.207214	+	0.358906i
$166$		0			0
$167$	−94040.0			−0.260928			−0.130464	−	0.991453i	$-0.541647\pi$
$167$	−94040.0			−0.260928			−0.130464	+	0.991453i	$0.541647\pi$
$168$		0			0
$169$	−182180.			−0.490663
$170$		0			0
$171$	116530.	+	201836.i	0.304753	+	0.527848i
$172$		0			0
$173$	−244357.	+	423238.i	−0.620739	+	1.07515i	0.368609	+	0.929584i	$0.379834\pi$
$173$	−244357.	+	423238.i	−0.620739	+	1.07515i	−0.989348	+	0.145567i	$0.953499\pi$
$174$		0			0
$175$	−149149.	+	30960.4i	−0.368150	+	0.0764207i
$176$		0			0
$177$	231694.	−	401305.i	0.555880	−	0.962812i
$178$		0			0
$179$	−406498.	−	704075.i	−0.948257	−	1.64243i	−0.749095	−	0.662462i	$-0.769512\pi$
$179$	−406498.	−	704075.i	−0.948257	−	1.64243i	−0.199161	−	0.979967i	$-0.563822\pi$
$180$		0			0
$181$	−332961.			−0.755434			−0.377717	−	0.925921i	$-0.623291\pi$
$181$	−332961.			−0.755434			−0.377717	+	0.925921i	$0.623291\pi$
$182$		0			0
$183$	−119820.			−0.264484
$184$		0			0
$185$	80002.7	+	138569.i	0.171860	+	0.297671i
$186$		0			0
$187$	−135348.	+	234430.i	−0.283040	+	0.490240i
$188$		0			0
$189$	29632.9	−	89743.1i	0.0603420	−	0.182745i
$190$		0			0
$191$	72143.9	−	124957.i	0.143092	−	0.247843i	−0.785567	−	0.618776i	$-0.787629\pi$
$191$	72143.9	−	124957.i	0.143092	−	0.247843i	0.928660	+	0.370933i	$0.120962\pi$
$192$		0			0
$193$	447310.	+	774763.i	0.864400	+	1.49719i	0.867641	+	0.497190i	$0.165635\pi$
$193$	447310.	+	774763.i	0.864400	+	1.49719i	−0.00324119	+	0.999995i	$0.501032\pi$
$194$		0			0
$195$	256648.			0.483338
$196$		0			0
$197$	−599462.			−1.10052			−0.550258	−	0.834995i	$-0.685470\pi$
$197$	−599462.			−1.10052			−0.550258	+	0.834995i	$0.685470\pi$
$198$		0			0
$199$	389129.	+	673992.i	0.696564	+	1.20648i	0.969651	+	0.244495i	$0.0786222\pi$
$199$	389129.	+	673992.i	0.696564	+	1.20648i	−0.273086	+	0.961990i	$0.588044\pi$
$200$		0			0
$201$	−139031.	+	240808.i	−0.242729	+	0.420418i
$202$		0			0
$203$	202008.	−	611780.i	0.344055	−	1.04197i
$204$		0			0
$205$	120282.	−	208334.i	0.199901	−	0.346238i
$206$		0			0
$207$	171292.	+	296687.i	0.277851	+	0.481251i
$208$		0			0
$209$	706589.			1.11893
$210$		0			0
$211$	−810532.			−1.25333			−0.626663	−	0.779291i	$-0.715580\pi$
$211$	−810532.			−1.25333			−0.626663	+	0.779291i	$0.715580\pi$
$212$		0			0
$213$	188473.	+	326444.i	0.284642	+	0.493015i
$214$		0			0
$215$	236004.	−	408771.i	0.348196	−	0.603093i
$216$		0			0
$217$	1.11483e6	−	231418.i	1.60717	−	0.333616i
$218$		0			0
$219$	−155778.	+	269815.i	−0.219480	+	0.380151i
$220$		0			0
$221$	239679.	+	415136.i	0.330103	+	0.571755i
$222$		0			0
$223$	220486.			0.296905			0.148453	−	0.988920i	$-0.452571\pi$
$223$	220486.			0.296905			0.148453	+	0.988920i	$0.452571\pi$
$224$		0			0
$225$	95174.5			0.125333
$226$		0			0
$227$	−671890.	−	1.16375e6i	−0.865433	−	1.49897i	−0.866617	−	0.498974i	$-0.833710\pi$
$227$	−671890.	−	1.16375e6i	−0.865433	−	1.49897i	0.00118391	−	0.999999i	$-0.499623\pi$
$228$		0			0
$229$	521292.	−	902904.i	0.656889	−	1.13777i	−0.324528	−	0.945876i	$-0.605205\pi$
$229$	521292.	−	902904.i	0.656889	−	1.13777i	0.981417	−	0.191889i	$-0.0614614\pi$
$230$		0			0
$231$	−190709.	−	213845.i	−0.235148	−	0.263675i
$232$		0			0
$233$	−316256.	+	547771.i	−0.381635	+	0.661011i	−0.991296	−	0.131651i	$-0.957972\pi$
$233$	−316256.	+	547771.i	−0.381635	+	0.661011i	0.609661	+	0.792662i	$0.291306\pi$
$234$		0			0
$235$	−107323.	−	185889.i	−0.126772	−	0.219575i
$236$		0			0
$237$	697891.			0.807081
$238$		0			0
$239$	−684919.			−0.775612			−0.387806	−	0.921741i	$-0.626767\pi$
$239$	−684919.			−0.775612			−0.387806	+	0.921741i	$0.626767\pi$
$240$		0			0
$241$	3866.44	+	6696.87i	0.00428814	+	0.00742728i	0.868162	−	0.496282i	$-0.165302\pi$
$241$	3866.44	+	6696.87i	0.00428814	+	0.00742728i	−0.863873	+	0.503709i	$0.831968\pi$
$242$		0			0
$243$	−29524.5	+	51137.9i	−0.0320750	+	0.0555556i
$244$		0			0
$245$	1.01105e6	−	438651.i	1.07611	−	0.466878i
$246$		0			0
$247$	625626.	−	1.08362e6i	0.652488	−	1.13014i
$248$		0			0
$249$	−454086.	−	786500.i	−0.464130	−	0.803897i
$250$		0			0
$251$	1.09614e6			1.09820			0.549098	−	0.835758i	$-0.314971\pi$
$251$	1.09614e6			1.09820			0.549098	+	0.835758i	$0.314971\pi$
$252$		0			0
$253$	1.03864e6			1.02015
$254$		0			0
$255$	325271.	+	563386.i	0.313253	+	0.542570i
$256$		0			0
$257$	−357460.	+	619139.i	−0.337594	+	0.584730i	−0.983980	−	0.178281i	$-0.942946\pi$
$257$	−357460.	+	619139.i	−0.337594	+	0.584730i	0.646386	+	0.763011i	$0.276280\pi$
$258$		0			0
$259$	−210546.	−	236088.i	−0.195028	−	0.218688i
$260$		0			0
$261$	−201269.	+	348608.i	−0.182884	+	0.316764i
$262$		0			0
$263$	298256.	+	516595.i	0.265889	+	0.460533i	0.967796	−	0.251735i	$-0.0810012\pi$
$263$	298256.	+	516595.i	0.265889	+	0.460533i	−0.701907	+	0.712268i	$0.747668\pi$
$264$		0			0
$265$	−198144.			−0.173327
$266$		0			0
$267$	367053.			0.315102
$268$		0			0
$269$	1.01654e6	+	1.76070e6i	0.856531	+	1.48356i	0.875217	+	0.483730i	$0.160718\pi$
$269$	1.01654e6	+	1.76070e6i	0.856531	+	1.48356i	−0.0186864	+	0.999825i	$0.505948\pi$
$270$		0			0
$271$	−897277.	+	1.55413e6i	−0.742170	+	1.28548i	0.209336	+	0.977844i	$0.432870\pi$
$271$	−897277.	+	1.55413e6i	−0.742170	+	1.28548i	−0.951505	+	0.307632i	$0.900463\pi$
$272$		0			0
$273$	−496807.	+	103127.i	−0.403442	+	0.0837467i
$274$		0			0
$275$	144274.	−	249890.i	0.115042	−	0.199259i
$276$		0			0
$277$	−211790.	−	366831.i	−0.165846	−	0.287254i	0.771109	−	0.636703i	$-0.219702\pi$
$277$	−211790.	−	366831.i	−0.165846	−	0.287254i	−0.936956	+	0.349449i	$0.886369\pi$
$278$		0			0
$279$	−711395.			−0.547142
$280$		0			0
$281$	−1.63799e6			−1.23750			−0.618749	−	0.785589i	$-0.712360\pi$
$281$	−1.63799e6			−1.23750			−0.618749	+	0.785589i	$0.712360\pi$
$282$		0			0
$283$	147018.	+	254642.i	0.109120	+	0.189001i	0.915414	−	0.402514i	$-0.131863\pi$
$283$	147018.	+	254642.i	0.109120	+	0.189001i	−0.806294	+	0.591515i	$0.798530\pi$
$284$		0			0
$285$	849044.	−	1.47059e6i	0.619182	−	1.07245i
$286$		0			0
$287$	−149122.	+	451615.i	−0.106865	+	0.323641i
$288$		0			0
$289$	102399.	−	177360.i	0.0721191	−	0.124914i
$290$		0			0
$291$	−630661.	−	1.09234e6i	−0.436580	−	0.756178i
$292$		0			0
$293$	−356961.			−0.242913			−0.121457	−	0.992597i	$-0.538757\pi$
$293$	−356961.			−0.242913			−0.121457	+	0.992597i	$0.538757\pi$
$294$		0			0
$295$	−3.37626e6			−2.25881
$296$		0			0
$297$	89511.8	+	155039.i	0.0588829	+	0.101988i
$298$		0			0
$299$	919631.	−	1.59285e6i	0.594888	−	1.03038i
$300$		0			0
$301$	−292592.	+	886113.i	−0.186143	+	0.563732i
$302$		0			0
$303$	−38594.0	+	66846.8i	−0.0241498	+	0.0418287i
$304$		0			0
$305$	436505.	+	756049.i	0.268683	+	0.465372i
$306$		0			0
$307$	−2.04097e6			−1.23592			−0.617960	−	0.786210i	$-0.712041\pi$
$307$	−2.04097e6			−1.23592			−0.617960	+	0.786210i	$0.712041\pi$
$308$		0			0
$309$	−317341.			−0.189073
$310$		0			0
$311$	−1.02337e6	−	1.77253e6i	−0.599972	−	1.03918i	−0.992824	−	0.119581i	$-0.961845\pi$
$311$	−1.02337e6	−	1.77253e6i	−0.599972	−	1.03918i	0.392852	−	0.919602i	$-0.371488\pi$
$312$		0			0
$313$	−319272.	+	552996.i	−0.184205	+	0.319052i	−0.943308	−	0.331918i	$-0.892304\pi$
$313$	−319272.	+	552996.i	−0.184205	+	0.319052i	0.759104	+	0.650970i	$0.225638\pi$
$314$		0			0
$315$	−674222.	+	139955.i	−0.382848	+	0.0794718i
$316$		0			0
$317$	−1.74954e6	+	3.03030e6i	−0.977860	+	1.69370i	−0.307705	+	0.951482i	$0.599561\pi$
$317$	−1.74954e6	+	3.03030e6i	−0.977860	+	1.69370i	−0.670155	+	0.742221i	$0.733772\pi$
$318$		0			0
$319$	610203.	+	1.05690e6i	0.335736	+	0.581512i
$320$		0			0
$321$	−1.73366e6			−0.939076
$322$		0			0
$323$	3.17163e6			1.69152
$324$		0			0
$325$	−255486.	−	442514.i	−0.134171	−	0.232391i
$326$		0			0
$327$	−554815.	+	960967.i	−0.286932	+	0.496980i
$328$		0			0
$329$	282446.	+	316710.i	0.143862	+	0.161314i
$330$		0			0
$331$	1.47089e6	−	2.54766e6i	0.737923	−	1.27812i	−0.215506	−	0.976503i	$-0.569140\pi$
$331$	1.47089e6	−	2.54766e6i	0.737923	−	1.27812i	0.953429	−	0.301618i	$-0.0975266\pi$
$332$		0			0
$333$	98822.5	+	171166.i	0.0488366	+	0.0845874i
$334$		0			0
$335$	2.02597e6			0.986326
$336$		0			0
$337$	2.77854e6			1.33273			0.666364	−	0.745627i	$-0.267850\pi$
$337$	2.77854e6			1.33273			0.666364	+	0.745627i	$0.267850\pi$
$338$		0			0
$339$	−127804.	−	221363.i	−0.0604012	−	0.104618i
$340$		0			0
$341$	−1.07840e6	+	1.86784e6i	−0.502218	+	0.869868i
$342$		0			0
$343$	−1.78089e6	+	1.25539e6i	−0.817338	+	0.576159i
$344$		0			0
$345$	1.24804e6	−	2.16167e6i	0.564522	−	0.977781i
$346$		0			0
$347$	86001.9	+	148960.i	0.0383429	+	0.0664118i	0.884560	−	0.466426i	$-0.154459\pi$
$347$	86001.9	+	148960.i	0.0383429	+	0.0664118i	−0.846217	+	0.532838i	$0.821125\pi$
$348$		0			0
$349$	−3.88321e6			−1.70658			−0.853290	−	0.521436i	$-0.825397\pi$
$349$	−3.88321e6			−1.70658			−0.853290	+	0.521436i	$0.825397\pi$
$350$		0			0
$351$	317021.			0.137348
$352$		0			0
$353$	385467.	+	667649.i	0.164646	+	0.285175i	0.936529	−	0.350589i	$-0.114019\pi$
$353$	385467.	+	667649.i	0.164646	+	0.285175i	−0.771884	+	0.635764i	$0.780685\pi$
$354$		0			0
$355$	1.37322e6	−	2.37849e6i	0.578321	−	1.00168i
$356$		0			0
$357$	−856027.	−	959875.i	−0.355481	−	0.398606i
$358$		0			0
$359$	−1.57017e6	+	2.71961e6i	−0.642998	+	1.11370i	0.341763	+	0.939786i	$0.388976\pi$
$359$	−1.57017e6	+	2.71961e6i	−0.642998	+	1.11370i	−0.984760	+	0.173918i	$0.944357\pi$
$360$		0			0
$361$	−2.90135e6	−	5.02529e6i	−1.17174	−	2.02952i
$362$		0			0
$363$	−906698.			−0.361157
$364$		0			0
$365$	2.27001e6			0.891857
$366$		0			0
$367$	182842.	+	316691.i	0.0708615	+	0.122736i	0.899279	−	0.437375i	$-0.144092\pi$
$367$	182842.	+	316691.i	0.0708615	+	0.122736i	−0.828418	+	0.560111i	$0.810758\pi$
$368$		0			0
$369$	148576.	−	257342.i	0.0568047	−	0.0983886i
$370$		0			0
$371$	383559.	−	79619.2i	0.144676	−	0.0300319i
$372$		0			0
$373$	77129.9	−	133593.i	0.0287045	−	0.0497177i	−0.851316	−	0.524653i	$-0.824195\pi$
$373$	77129.9	−	133593.i	0.0287045	−	0.0497177i	0.880021	+	0.474935i	$0.157528\pi$
$374$		0			0
$375$	575417.	+	996651.i	0.211302	+	0.365986i
$376$		0			0
$377$	2.16114e6			0.783122
$378$		0			0
$379$	4.06013e6			1.45192			0.725959	−	0.687738i	$-0.241396\pi$
$379$	4.06013e6			1.45192			0.725959	+	0.687738i	$0.241396\pi$
$380$		0			0
$381$	212395.	+	367878.i	0.0749602	+	0.129835i
$382$		0			0
$383$	−1.02152e6	+	1.76932e6i	−0.355834	+	0.616323i	−0.987260	−	0.159113i	$-0.949137\pi$
$383$	−1.02152e6	+	1.76932e6i	−0.355834	+	0.616323i	0.631426	+	0.775436i	$0.282470\pi$
$384$		0			0
$385$	−654581.	+	1.98240e6i	−0.225067	+	0.681614i
$386$		0			0
$387$	291521.	−	504930.i	0.0989448	−	0.171377i
$388$		0			0
$389$	1.06183e6	+	1.83915e6i	0.355780	+	0.616229i	0.987251	−	0.159170i	$-0.0508819\pi$
$389$	1.06183e6	+	1.83915e6i	0.355780	+	0.616229i	−0.631471	+	0.775399i	$0.717549\pi$
$390$		0			0
$391$	4.66209e6			1.54219
$392$		0			0
$393$	718284.			0.234593
$394$		0			0
$395$	−2.54243e6	−	4.40362e6i	−0.819892	−	1.42009i
$396$		0			0
$397$	1.83182e6	−	3.17280e6i	0.583319	−	1.01034i	−0.411764	−	0.911291i	$-0.635087\pi$
$397$	1.83182e6	−	3.17280e6i	0.583319	−	1.01034i	0.995083	−	0.0990473i	$-0.0315795\pi$
$398$		0			0
$399$	−1.05262e6	+	3.18786e6i	−0.331010	+	1.00246i
$400$		0			0
$401$	32044.9	−	55503.3i	0.00995170	−	0.0172369i	−0.861007	−	0.508594i	$-0.830166\pi$
$401$	32044.9	−	55503.3i	0.00995170	−	0.0172369i	0.870958	+	0.491357i	$0.163499\pi$
$402$		0			0
$403$	1.90966e6	+	3.30763e6i	0.585725	+	1.01451i
$404$		0			0
$405$	430233.			0.130337
$406$		0			0
$407$	599216.			0.179307
$408$		0			0
$409$	−1.01287e6	−	1.75434e6i	−0.299395	−	0.518568i	0.676603	−	0.736348i	$-0.263452\pi$
$409$	−1.01287e6	−	1.75434e6i	−0.299395	−	0.518568i	−0.975998	+	0.217781i	$0.930118\pi$
$410$		0			0
$411$	387289.	−	670805.i	0.113092	−	0.195881i
$412$		0			0
$413$	6.53561e6	−	1.35666e6i	1.88543	−	0.391379i
$414$		0			0
$415$	−3.30849e6	+	5.73047e6i	−0.942995	+	1.63332i
$416$		0			0
$417$	−1.21764e6	−	2.10901e6i	−0.342909	−	0.593935i
$418$		0			0
$419$	2.38986e6			0.665025			0.332513	−	0.943099i	$-0.392104\pi$
$419$	2.38986e6			0.665025			0.332513	+	0.943099i	$0.392104\pi$
$420$		0			0
$421$	3.46875e6			0.953822			0.476911	−	0.878952i	$-0.341756\pi$
$421$	3.46875e6			0.953822			0.476911	+	0.878952i	$0.341756\pi$
$422$		0			0
$423$	−132570.	−	229617.i	−0.0360241	−	0.0623956i
$424$		0			0
$425$	647596.	−	1.12167e6i	0.173913	−	0.301226i
$426$		0			0
$427$	−1.14877e6	−	1.28813e6i	−0.304903	−	0.341892i
$428$		0			0
$429$	480570.	−	832372.i	0.126070	−	0.218360i
$430$		0			0
$431$	802201.	+	1.38945e6i	0.208013	+	0.360289i	0.951088	−	0.308919i	$-0.0999671\pi$
$431$	802201.	+	1.38945e6i	0.208013	+	0.360289i	−0.743076	+	0.669207i	$0.766634\pi$
$432$		0			0
$433$	741661.			0.190102			0.0950508	−	0.995472i	$-0.469699\pi$
$433$	741661.			0.190102			0.0950508	+	0.995472i	$0.469699\pi$
$434$		0			0
$435$	2.93291e6			0.743147
$436$		0			0
$437$	−6.08466e6	−	1.05389e7i	−1.52417	−	2.63993i
$438$		0			0
$439$	−1.43394e6	+	2.48365e6i	−0.355115	+	0.615077i	−0.987138	−	0.159873i	$-0.948892\pi$
$439$	−1.43394e6	+	2.48365e6i	−0.355115	+	0.615077i	0.632023	+	0.774950i	$0.282225\pi$
$440$		0			0
$441$	1.24889e6	−	541839.i	0.305794	−	0.132670i
$442$		0			0
$443$	89148.9	−	154410.i	0.0215827	−	0.0373824i	−0.855032	−	0.518575i	$-0.826463\pi$
$443$	89148.9	−	154410.i	0.0215827	−	0.0373824i	0.876615	+	0.481192i	$0.159796\pi$
$444$		0			0
$445$	−1.33718e6	−	2.31607e6i	−0.320103	−	0.554435i
$446$		0			0
$447$	−1.55293e6			−0.367606
$448$		0			0
$449$	−2.77890e6			−0.650515			−0.325258	−	0.945625i	$-0.605451\pi$
$449$	−2.77890e6			−0.650515			−0.325258	+	0.945625i	$0.605451\pi$
$450$		0			0
$451$	−450451.	−	780205.i	−0.104281	−	0.180621i
$452$		0			0
$453$	80547.7	−	139513.i	0.0184420	−	0.0319425i
$454$		0			0
$455$	2.46060e6	+	2.75911e6i	0.557202	+	0.624798i
$456$		0			0
$457$	−3.20669e6	+	5.55416e6i	−0.718236	+	1.24402i	0.243462	+	0.969910i	$0.421717\pi$
$457$	−3.20669e6	+	5.55416e6i	−0.718236	+	1.24402i	−0.961698	+	0.274111i	$0.911617\pi$
$458$		0			0
$459$	401787.	+	695916.i	0.0890153	+	0.154179i
$460$		0			0
$461$	−6.88393e6			−1.50864			−0.754318	−	0.656510i	$-0.772032\pi$
$461$	−6.88393e6			−1.50864			−0.754318	+	0.656510i	$0.772032\pi$
$462$		0			0
$463$	6.70530e6			1.45367			0.726835	−	0.686812i	$-0.240991\pi$
$463$	6.70530e6			1.45367			0.726835	+	0.686812i	$0.240991\pi$
$464$		0			0
$465$	2.59162e6	+	4.48882e6i	0.555827	+	0.962721i
$466$		0			0
$467$	1.62836e6	−	2.82041e6i	0.345509	−	0.598439i	−0.639937	−	0.768427i	$-0.721040\pi$
$467$	1.62836e6	−	2.82041e6i	0.345509	−	0.598439i	0.985446	+	0.169988i	$0.0543730\pi$
$468$		0			0
$469$	−3.92178e6	+	814084.i	−0.823286	+	0.170898i
$470$		0			0
$471$	805584.	−	1.39531e6i	0.167324	−	0.289814i
$472$		0			0
$473$	−883830.	−	1.53084e6i	−0.181642	−	0.314613i
$474$		0			0
$475$	−3.38080e6			−0.687520
$476$		0			0
$477$	−244755.			−0.0492534
$478$		0			0
$479$	−4.67916e6	−	8.10454e6i	−0.931814	−	1.61395i	−0.780219	−	0.625506i	$-0.784892\pi$
$479$	−4.67916e6	−	8.10454e6i	−0.931814	−	1.61395i	−0.151595	−	0.988443i	$-0.548441\pi$
$480$		0			0
$481$	530557.	−	918951.i	0.104561	−	0.181105i
$482$		0			0
$483$	−1.54729e6	+	4.68596e6i	−0.301789	+	0.913967i
$484$		0			0
$485$	−4.59502e6	+	7.95881e6i	−0.887019	+	1.53636i
$486$		0			0
$487$	1.31470e6	+	2.27712e6i	0.251191	+	0.435075i	0.963854	−	0.266431i	$-0.0858445\pi$
$487$	1.31470e6	+	2.27712e6i	0.251191	+	0.435075i	−0.712663	+	0.701506i	$0.752511\pi$
$488$		0			0
$489$	−2.13560e6			−0.403876
$490$		0			0
$491$	−4.81856e6			−0.902015			−0.451008	−	0.892520i	$-0.648935\pi$
$491$	−4.81856e6			−0.902015			−0.451008	+	0.892520i	$0.648935\pi$
$492$		0			0
$493$	2.73899e6	+	4.74407e6i	0.507543	+	0.879091i
$494$		0			0
$495$	652187.	−	1.12962e6i	0.119635	−	0.207214i
$496$		0			0
$497$	−1.70248e6	+	5.15596e6i	−0.309166	+	0.936308i
$498$		0			0
$499$	2.09549e6	−	3.62949e6i	0.376733	−	0.652521i	−0.613851	−	0.789422i	$-0.710381\pi$
$499$	2.09549e6	−	3.62949e6i	0.376733	−	0.652521i	0.990585	+	0.136900i	$0.0437139\pi$
$500$		0			0
$501$	423180.	+	732969.i	0.0753236	+	0.130464i
$502$		0			0
$503$	−1.75338e6			−0.308999			−0.154500	−	0.987993i	$-0.549377\pi$
$503$	−1.75338e6			−0.308999			−0.154500	+	0.987993i	$0.549377\pi$
$504$		0			0
$505$	562395.			0.0981326
$506$		0			0
$507$	819808.	+	1.41995e6i	0.141642	+	0.245331i
$508$		0			0
$509$	−2.80923e6	+	4.86573e6i	−0.480610	+	0.832441i	−0.999753	−	0.0222464i	$-0.992918\pi$
$509$	−2.80923e6	+	4.86573e6i	−0.480610	+	0.832441i	0.519142	+	0.854688i	$0.326251\pi$
$510$		0			0
$511$	−4.39418e6	+	912146.i	−0.744433	+	0.154530i
$512$		0			0
$513$	1.04877e6	−	1.81653e6i	0.175949	−	0.304753i
$514$		0			0
$515$	1.15608e6	+	2.00239e6i	0.192075	+	0.332683i
$516$		0			0
$517$	−803844.			−0.132265
$518$		0			0
$519$	4.39842e6			0.716768
$520$		0			0
$521$	−5.53589e6	−	9.58845e6i	−0.893498	−	1.54758i	−0.835653	−	0.549258i	$-0.814911\pi$
$521$	−5.53589e6	−	9.58845e6i	−0.893498	−	1.54758i	−0.0578446	−	0.998326i	$-0.518423\pi$
$522$		0			0
$523$	3.79894e6	−	6.57996e6i	0.607307	−	1.05189i	−0.384375	−	0.923177i	$-0.625583\pi$
$523$	3.79894e6	−	6.57996e6i	0.607307	−	1.05189i	0.991682	−	0.128710i	$-0.0410836\pi$
$524$		0			0
$525$	912482.	+	1.02318e6i	0.144486	+	0.162014i
$526$		0			0
$527$	−4.84055e6	+	8.38407e6i	−0.759220	+	1.31501i
$528$		0			0
$529$	−5.72588e6	−	9.91752e6i	−0.889618	−	1.54086i
$530$		0			0
$531$	−4.17049e6			−0.641875
$532$		0			0
$533$	−1.59535e6			−0.243242
$534$		0			0
$535$	6.31575e6	+	1.09392e7i	0.953982	+	1.65235i
$536$		0			0
$537$	−3.65848e6	+	6.33668e6i	−0.547476	+	0.948257i
$538$		0			0
$539$	470532.	−	4.10046e6i	0.0697618	−	0.607940i
$540$		0			0
$541$	1.67539e6	−	2.90187e6i	0.246107	−	0.426270i	−0.716335	−	0.697756i	$-0.754182\pi$
$541$	1.67539e6	−	2.90187e6i	0.246107	−	0.426270i	0.962442	+	0.271486i	$0.0875152\pi$
$542$		0			0
$543$	1.49832e6	+	2.59517e6i	0.218075	+	0.377717i
$544$		0			0
$545$	8.08480e6			1.16595
$546$		0			0
$547$	−1.00856e7			−1.44123			−0.720615	−	0.693335i	$-0.756140\pi$
$547$	−1.00856e7			−1.44123			−0.720615	+	0.693335i	$0.756140\pi$
$548$		0			0
$549$	539188.	+	933901.i	0.0763501	+	0.132242i
$550$		0			0
$551$	7.14950e6	−	1.23833e7i	1.00322	−	1.73763i
$552$		0			0
$553$	6.69101e6	+	7.50272e6i	0.930419	+	1.04329i
$554$		0			0
$555$	720024.	−	1.24712e6i	0.0992235	−	0.171860i
$556$		0			0
$557$	−6.78337e6	−	1.17491e7i	−0.926419	−	1.60460i	−0.789263	−	0.614055i	$-0.789537\pi$
$557$	−6.78337e6	−	1.17491e7i	−0.926419	−	1.60460i	−0.137155	−	0.990550i	$-0.543796\pi$
$558$		0			0
$559$	−3.13023e6			−0.423689
$560$		0			0
$561$	2.43626e6			0.326826
$562$		0			0
$563$	1.98494e6	+	3.43802e6i	0.263922	+	0.457127i	0.967281	−	0.253708i	$-0.0816504\pi$
$563$	1.98494e6	+	3.43802e6i	0.263922	+	0.457127i	−0.703358	+	0.710835i	$0.748317\pi$
$564$		0			0
$565$	−931185.	+	1.61286e6i	−0.122720	+	0.212557i
$566$		0			0
$567$	−832826.	+	172878.i	−0.108792	+	0.0225831i
$568$		0			0
$569$	2.08724e6	−	3.61521e6i	0.270267	−	0.468116i	−0.698663	−	0.715451i	$-0.746221\pi$
$569$	2.08724e6	−	3.61521e6i	0.270267	−	0.468116i	0.968930	+	0.247335i	$0.0795548\pi$
$570$		0			0
$571$	−1.92833e6	−	3.33996e6i	−0.247509	−	0.428698i	0.715325	−	0.698792i	$-0.246279\pi$
$571$	−1.92833e6	−	3.33996e6i	−0.247509	−	0.428698i	−0.962834	+	0.270094i	$0.912945\pi$
$572$		0			0
$573$	−1.29859e6			−0.165229
$574$		0			0
$575$	−4.96956e6			−0.626828
$576$		0			0
$577$	1.12288e6	+	1.94488e6i	0.140408	+	0.243194i	0.927650	−	0.373450i	$-0.121825\pi$
$577$	1.12288e6	+	1.94488e6i	0.140408	+	0.243194i	−0.787242	+	0.616644i	$0.788492\pi$
$578$		0			0
$579$	4.02579e6	−	6.97287e6i	0.499062	−	0.864400i
$580$		0			0
$581$	4.10178e6	−	1.24222e7i	0.504118	−	1.52672i
$582$		0			0
$583$	−371023.	+	642630.i	−0.0452094	+	0.0783050i
$584$		0			0
$585$	−1.15492e6	−	2.00037e6i	−0.139528	−	0.241669i
$586$		0			0
$587$	6.40082e6			0.766726			0.383363	−	0.923598i	$-0.374766\pi$
$587$	6.40082e6			0.766726			0.383363	+	0.923598i	$0.374766\pi$
$588$		0			0
$589$	2.52702e7			3.00138
$590$		0			0
$591$	2.69758e6	+	4.67234e6i	0.317691	+	0.550258i
$592$		0			0
$593$	−5.56031e6	+	9.63074e6i	−0.649325	+	1.12466i	0.333959	+	0.942588i	$0.391615\pi$
$593$	−5.56031e6	+	9.63074e6i	−0.649325	+	1.12466i	−0.983284	+	0.182077i	$0.941718\pi$
$594$		0			0
$595$	−2.93818e6	+	8.89828e6i	−0.340241	+	1.03042i
$596$		0			0
$597$	3.50216e6	−	6.06592e6i	0.402162	−	0.696564i
$598$		0			0
$599$	6.56707e6	+	1.13745e7i	0.747833	+	1.29528i	0.948860	+	0.315698i	$0.102239\pi$
$599$	6.56707e6	+	1.13745e7i	0.747833	+	1.29528i	−0.201027	+	0.979586i	$0.564428\pi$
$600$		0			0
$601$	−7.88546e6			−0.890514			−0.445257	−	0.895403i	$-0.646888\pi$
$601$	−7.88546e6			−0.890514			−0.445257	+	0.895403i	$0.646888\pi$
$602$		0			0
$603$	2.50255e6			0.280279
$604$		0			0
$605$	3.30312e6	+	5.72117e6i	0.366890	+	0.635472i
$606$		0			0
$607$	3.82967e6	−	6.63319e6i	0.421881	−	0.730719i	−0.574243	−	0.818685i	$-0.694703\pi$
$607$	3.82967e6	−	6.63319e6i	0.421881	−	0.730719i	0.996123	+	0.0879660i	$0.0280367\pi$
$608$		0			0
$609$	−5.67739e6	+	1.17851e6i	−0.620305	+	0.128763i
$610$		0			0
$611$	−711738.	+	1.23277e6i	−0.0771289	+	0.133591i
$612$		0			0
$613$	7.62066e6	+	1.31994e7i	0.819108	+	1.41874i	0.906340	+	0.422549i	$0.138864\pi$
$613$	7.62066e6	+	1.31994e7i	0.819108	+	1.41874i	−0.0872322	+	0.996188i	$0.527802\pi$
$614$		0			0
$615$	−2.16507e6			−0.230825
$616$		0			0
$617$	1.35844e7			1.43657			0.718285	−	0.695749i	$-0.244927\pi$
$617$	1.35844e7			1.43657			0.718285	+	0.695749i	$0.244927\pi$
$618$		0			0
$619$	−3.18709e6	−	5.52019e6i	−0.334324	−	0.579066i	0.649031	−	0.760762i	$-0.275175\pi$
$619$	−3.18709e6	−	5.52019e6i	−0.334324	−	0.579066i	−0.983355	+	0.181696i	$0.941841\pi$
$620$		0			0
$621$	1.54163e6	−	2.67018e6i	0.160417	−	0.277851i
$622$		0			0
$623$	3.51911e6	+	3.94602e6i	0.363256	+	0.407323i
$624$		0			0
$625$	6.02844e6	−	1.04416e7i	0.617312	−	1.06922i
$626$		0			0
$627$	−3.17965e6	−	5.50732e6i	−0.323006	−	0.559463i
$628$		0			0
$629$	2.68967e6			0.271065
$630$		0			0
$631$	−1.42736e7			−1.42712			−0.713561	−	0.700593i	$-0.752919\pi$
$631$	−1.42736e7			−1.42712			−0.713561	+	0.700593i	$0.752919\pi$
$632$		0			0
$633$	3.64739e6	+	6.31747e6i	0.361804	+	0.626663i
$634$		0			0
$635$	1.54751e6	−	2.68037e6i	0.152300	−	0.263792i
$636$		0			0
$637$	−5.87180e6	−	4.35222e6i	−0.573354	−	0.424974i
$638$		0			0
$639$	1.69625e6	−	2.93800e6i	0.164338	−	0.284642i
$640$		0			0
$641$	4.63480e6	+	8.02771e6i	0.445539	+	0.771697i	0.998090	−	0.0617828i	$-0.0196786\pi$
$641$	4.63480e6	+	8.02771e6i	0.445539	+	0.771697i	−0.552550	+	0.833480i	$0.686345\pi$
$642$		0			0
$643$	1.17375e7			1.11956			0.559780	−	0.828641i	$-0.310885\pi$
$643$	1.17375e7			1.11956			0.559780	+	0.828641i	$0.310885\pi$
$644$		0			0
$645$	−4.24807e6			−0.402062
$646$		0			0
$647$	−5.12724e6	−	8.88063e6i	−0.481529	−	0.834033i	0.518246	−	0.855232i	$-0.326585\pi$
$647$	−5.12724e6	−	8.88063e6i	−0.481529	−	0.834033i	−0.999775	+	0.0211984i	$0.993252\pi$
$648$		0			0
$649$	−6.32200e6	+	1.09500e7i	−0.589173	+	1.02048i
$650$		0			0
$651$	−6.82047e6	−	7.64789e6i	−0.630757	−	0.707276i
$652$		0			0
$653$	−4.14489e6	+	7.17915e6i	−0.380391	+	0.658856i	−0.991118	−	0.132985i	$-0.957544\pi$
$653$	−4.14489e6	+	7.17915e6i	−0.380391	+	0.658856i	0.610727	+	0.791841i	$0.290877\pi$
$654$		0			0
$655$	−2.61672e6	−	4.53230e6i	−0.238317	−	0.412776i
$656$		0			0
$657$	2.80400e6			0.253434
$658$		0			0
$659$	−2.06731e7			−1.85435			−0.927174	−	0.374631i	$-0.877770\pi$
$659$	−2.06731e7			−1.85435			−0.927174	+	0.374631i	$0.877770\pi$
$660$		0			0
$661$	−97171.2	−	168305.i	−0.00865035	−	0.0149829i	0.861668	−	0.507473i	$-0.169420\pi$
$661$	−97171.2	−	168305.i	−0.00865035	−	0.0149829i	−0.870318	+	0.492490i	$0.836087\pi$
$662$		0			0
$663$	2.15711e6	−	3.73623e6i	0.190585	−	0.330103i
$664$		0			0
$665$	2.39498e7	−	4.97151e6i	2.10014	−	0.435948i
$666$		0			0
$667$	1.05093e7	−	1.82026e7i	0.914659	−	1.58424i
$668$		0			0
$669$	−992186.	−	1.71852e6i	−0.0857092	−	0.148453i
$670$		0			0
$671$	3.26940e6			0.280325
$672$		0			0
$673$	−1.14437e7			−0.973929			−0.486965	−	0.873422i	$-0.661896\pi$
$673$	−1.14437e7			−0.973929			−0.486965	+	0.873422i	$0.661896\pi$
$674$		0			0
$675$	−428285.	−	741812.i	−0.0361804	−	0.0626663i
$676$		0			0
$677$	−4.51869e6	+	7.82660e6i	−0.378914	+	0.656299i	−0.990905	−	0.134567i	$-0.957036\pi$
$677$	−4.51869e6	+	7.82660e6i	−0.378914	+	0.656299i	0.611990	+	0.790865i	$0.290369\pi$
$678$		0			0
$679$	5.69679e6	−	1.72527e7i	0.474193	−	1.43609i
$680$		0			0
$681$	−6.04701e6	+	1.04737e7i	−0.499658	+	0.865433i
$682$		0			0
$683$	5.59212e6	+	9.68584e6i	0.458696	+	0.794485i	0.998892	−	0.0470541i	$-0.0149833\pi$
$683$	5.59212e6	+	9.68584e6i	0.458696	+	0.794485i	−0.540196	+	0.841539i	$0.681650\pi$
$684$		0			0
$685$	−5.64361e6			−0.459548
$686$		0			0
$687$	−9.38325e6			−0.758510
$688$		0			0
$689$	657020.	+	1.13799e6i	0.0527267	+	0.0913253i
$690$		0			0
$691$	−2.83465e6	+	4.90976e6i	−0.225842	+	0.391170i	−0.956572	−	0.291497i	$-0.905847\pi$
$691$	−2.83465e6	+	4.90976e6i	−0.225842	+	0.391170i	0.730730	+	0.682667i	$0.239180\pi$
$692$		0			0
$693$	−808564.	+	2.44873e6i	−0.0639560	+	0.193691i
$694$		0			0
$695$	−8.87177e6	+	1.53664e7i	−0.696704	+	1.20673i
$696$		0			0
$697$	−2.02192e6	−	3.50207e6i	−0.157646	−	0.273050i
$698$		0			0
$699$	5.69260e6			0.440674
$700$		0			0
$701$	−1.31822e6			−0.101320			−0.0506599	−	0.998716i	$-0.516132\pi$
$701$	−1.31822e6			−0.101320			−0.0506599	+	0.998716i	$0.516132\pi$
$702$		0			0
$703$	−3.51038e6	−	6.08016e6i	−0.267896	−	0.464009i
$704$		0			0
$705$	−965907.	+	1.67300e6i	−0.0731918	+	0.126772i
$706$		0			0
$707$	−1.08866e6	+	225984.i	−0.0819112	+	0.0170032i
$708$		0			0
$709$	−4.13018e6	+	7.15368e6i	−0.308570	+	0.534458i	−0.978050	−	0.208372i	$-0.933184\pi$
$709$	−4.13018e6	+	7.15368e6i	−0.308570	+	0.534458i	0.669480	+	0.742830i	$0.266517\pi$
$710$		0			0
$711$	−3.14051e6	−	5.43952e6i	−0.232984	−	0.403540i
$712$		0			0
$713$	3.71456e7			2.73643
$714$		0			0
$715$	−7.00290e6			−0.512287
$716$		0			0
$717$	3.08214e6	+	5.33842e6i	0.223900	+	0.387806i
$718$		0			0
$719$	−1.13372e7	+	1.96365e7i	−0.817865	+	1.41658i	0.0893870	+	0.995997i	$0.471509\pi$
$719$	−1.13372e7	+	1.96365e7i	−0.817865	+	1.41658i	−0.907252	+	0.420587i	$0.861824\pi$
$720$		0			0
$721$	−3.04250e6	−	3.41159e6i	−0.217968	−	0.244410i
$722$		0			0
$723$	34798.0	−	60271.9i	0.00247576	−	0.00428814i
$724$		0			0
$725$	−2.91963e6	−	5.05694e6i	−0.206292	−	0.357308i
$726$		0			0
$727$	−1.93477e7			−1.35767			−0.678833	−	0.734293i	$-0.737514\pi$
$727$	−1.93477e7			−1.35767			−0.678833	+	0.734293i	$0.737514\pi$
$728$		0			0
$729$	531441.			0.0370370
$730$		0			0
$731$	−3.96720e6	−	6.87140e6i	−0.274594	−	0.475611i
$732$		0			0
$733$	−7.42922e6	+	1.28678e7i	−0.510720	+	0.884593i	0.489203	+	0.872170i	$0.337288\pi$
$733$	−7.42922e6	+	1.28678e7i	−0.510720	+	0.884593i	−0.999923	+	0.0124232i	$0.996045\pi$
$734$		0			0
$735$	−7.96868e6	−	5.90645e6i	−0.544087	−	0.403281i
$736$		0			0
$737$	3.79360e6	−	6.57071e6i	0.257266	−	0.445598i
$738$		0			0
$739$	8.84089e6	+	1.53129e7i	0.595504	+	1.03144i	0.993476	+	0.114045i	$0.0363810\pi$
$739$	8.84089e6	+	1.53129e7i	0.595504	+	1.03144i	−0.397971	+	0.917398i	$0.630286\pi$
$740$		0			0
$741$	−1.12613e7			−0.753428
$742$		0			0
$743$	−8.28756e6			−0.550750			−0.275375	−	0.961337i	$-0.588802\pi$
$743$	−8.28756e6			−0.550750			−0.275375	+	0.961337i	$0.588802\pi$
$744$		0			0
$745$	5.65735e6	+	9.79882e6i	0.373442	+	0.646820i
$746$		0			0
$747$	−4.08678e6	+	7.07850e6i	−0.267966	+	0.464130i
$748$		0			0
$749$	−1.66214e7	−	1.86378e7i	−1.08259	−	1.21392i
$750$		0			0
$751$	1.15682e7	−	2.00368e7i	0.748457	−	1.29637i	−0.200104	−	0.979775i	$-0.564128\pi$
$751$	1.15682e7	−	2.00368e7i	0.748457	−	1.29637i	0.948562	−	0.316592i	$-0.102539\pi$
$752$		0			0
$753$	−4.93261e6	−	8.54354e6i	−0.317022	−	0.549098i
$754$		0			0
$755$	−1.17375e6			−0.0749389
$756$		0			0
$757$	−1.95475e7			−1.23980			−0.619900	−	0.784681i	$-0.712827\pi$
$757$	−1.95475e7			−1.23980			−0.619900	+	0.784681i	$0.712827\pi$
$758$		0			0
$759$	−4.67388e6	−	8.09540e6i	−0.294492	−	0.510075i
$760$		0			0
$761$	−3.91233e6	+	6.77635e6i	−0.244891	+	0.424164i	−0.962101	−	0.272693i	$-0.912086\pi$
$761$	−3.91233e6	+	6.77635e6i	−0.244891	+	0.424164i	0.717210	+	0.696857i	$0.245419\pi$
$762$		0			0
$763$	−1.56502e7	+	3.24867e6i	−0.973214	+	0.202020i
$764$		0			0
$765$	2.92744e6	−	5.07047e6i	0.180857	−	0.313253i
$766$		0			0
$767$	1.11952e7	+	1.93907e7i	0.687139	+	1.19016i
$768$		0			0
$769$	−8.27325e6			−0.504499			−0.252250	−	0.967662i	$-0.581170\pi$
$769$	−8.27325e6			−0.504499			−0.252250	+	0.967662i	$0.581170\pi$
$770$		0			0
$771$	6.43428e6			0.389820
$772$		0			0
$773$	9.41844e6	+	1.63132e7i	0.566931	+	0.981953i	0.996867	+	0.0790941i	$0.0252028\pi$
$773$	9.41844e6	+	1.63132e7i	0.566931	+	0.981953i	−0.429936	+	0.902859i	$0.641464\pi$
$774$		0			0
$775$	5.15978e6	−	8.93700e6i	0.308587	−	0.534488i
$776$		0			0
$777$	−892667.	+	2.70344e6i	−0.0530441	+	0.160644i
$778$		0			0
$779$	−5.27775e6	+	9.14133e6i	−0.311605	+	0.539717i
$780$		0			0
$781$	−5.14267e6	−	8.90737e6i	−0.301690	−	0.522543i
$782$		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)

\(f(q)\)	\(=\)	\(q+(-4.50000 - 7.79423i) q^{3} +(-32.7872 + 56.7890i) q^{5} +(40.6487 - 123.104i) q^{7} +(-40.5000 + 70.1481i) q^{9} +O(q^{10})\)	\(q+(-4.50000 - 7.79423i) q^{3} +(-32.7872 + 56.7890i) q^{5} +(40.6487 - 123.104i) q^{7} +(-40.5000 + 70.1481i) q^{9} +(122.787 + 212.674i) q^{11} +434.872 q^{13} +590.169 q^{15} +(551.149 + 954.618i) q^{17} +(1438.65 - 2491.81i) q^{19} +(-1142.42 + 237.145i) q^{21} +(2114.72 - 3662.80i) q^{23} +(-587.497 - 1017.57i) q^{25} +729.000 q^{27} +4969.60 q^{29} +(4391.32 + 7606.00i) q^{31} +(1105.08 - 1914.06i) q^{33} +(5658.22 + 6344.64i) q^{35} +(1220.03 - 2113.15i) q^{37} +(-1956.92 - 3389.49i) q^{39} -3668.55 q^{41} -7198.06 q^{43} +(-2655.76 - 4599.91i) q^{45} +(-1636.66 + 2834.78i) q^{47} +(-13502.4 - 10008.1i) q^{49} +(4960.34 - 8591.56i) q^{51} +(1510.84 + 2616.84i) q^{53} -16103.4 q^{55} -25895.6 q^{57} +(25743.7 + 44589.5i) q^{59} +(6656.65 - 11529.6i) q^{61} +(6989.26 + 7837.15i) q^{63} +(-14258.2 + 24695.9i) q^{65} +(-15447.9 - 26756.5i) q^{67} -38064.9 q^{69} -41882.8 q^{71} +(-17308.7 - 29979.5i) q^{73} +(-5287.47 + 9158.17i) q^{75} +(31172.2 - 6470.74i) q^{77} +(-38771.7 + 67154.6i) q^{79} +(-3280.50 - 5681.99i) q^{81} +100908. q^{83} -72282.4 q^{85} +(-22363.2 - 38734.2i) q^{87} +(-20391.8 + 35319.7i) q^{89} +(17677.0 - 53534.6i) q^{91} +(39521.9 - 68454.0i) q^{93} +(94338.2 + 163399. i) q^{95} +140147. q^{97} -19891.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 18 q^{3} - 47 q^{5} - 174 q^{7} - 162 q^{9}+O(q^{10}) \) 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

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