LMFDB - Embedded newform 588.6.i.m.373.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [588,6,Mod(361,588)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(588, 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("588.361");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 588 = 2^{2} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	588.i (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:	$94.3056860500$
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:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			373.2
Root			$-20.7872 + 36.0044i$ of defining polynomial
Character	$\chi$	$=$	588.373
Dual form			588.6.i.m.361.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.50000 + 7.79423i) q^{3} +(32.7872 - 56.7890i) q^{5} +(-40.5000 + 70.1481i) q^{9} +O(q^{10})$	\(q+(4.50000 + 7.79423i) q^{3} +(32.7872 - 56.7890i) q^{5} +(-40.5000 + 70.1481i) q^{9} +(122.787 + 212.674i) q^{11} -434.872 q^{13} +590.169 q^{15} +(-551.149 - 954.618i) q^{17} +(-1438.65 + 2491.81i) q^{19} +(2114.72 - 3662.80i) q^{23} +(-587.497 - 1017.57i) q^{25} -729.000 q^{27} +4969.60 q^{29} +(-4391.32 - 7606.00i) q^{31} +(-1105.08 + 1914.06i) q^{33} +(1220.03 - 2113.15i) q^{37} +(-1956.92 - 3389.49i) q^{39} +3668.55 q^{41} -7198.06 q^{43} +(2655.76 + 4599.91i) q^{45} +(1636.66 - 2834.78i) q^{47} +(4960.34 - 8591.56i) q^{51} +(1510.84 + 2616.84i) q^{53} +16103.4 q^{55} -25895.6 q^{57} +(-25743.7 - 44589.5i) q^{59} +(-6656.65 + 11529.6i) q^{61} +(-14258.2 + 24695.9i) q^{65} +(-15447.9 - 26756.5i) q^{67} +38064.9 q^{69} -41882.8 q^{71} +(17308.7 + 29979.5i) q^{73} +(5287.47 - 9158.17i) q^{75} +(-38771.7 + 67154.6i) q^{79} +(-3280.50 - 5681.99i) q^{81} -100908. q^{83} -72282.4 q^{85} +(22363.2 + 38734.2i) q^{87} +(20391.8 - 35319.7i) q^{89} +(39521.9 - 68454.0i) q^{93} +(94338.2 + 163399. i) q^{95} -140147. q^{97} -19891.5 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 18 q^{3} + 47 q^{5} - 162 q^{9}+O(q^{10}) $ 4 * q + 18 * q^3 + 47 * q^5 - 162 * q^9	$ 4 q + 18 q^{3} + 47 q^{5} - 162 q^{9} + 407 q^{11} - 898 q^{13} + 846 q^{15} - 1868 q^{17} - 1463 q^{19} + 44 q^{23} + 1605 q^{25} - 2916 q^{27} + 1534 q^{29} - 11170 q^{31} - 3663 q^{33} + 3113 q^{37} - 4041 q^{39} + 15684 q^{41} - 25258 q^{43} + 3807 q^{45} + 9576 q^{47} + 16812 q^{51} - 13395 q^{53} + 26210 q^{55} - 26334 q^{57} - 47521 q^{59} - 63652 q^{61} - 28254 q^{65} - 44541 q^{67} + 792 q^{69} - 251680 q^{71} + 6039 q^{73} - 14445 q^{75} - 17588 q^{79} - 13122 q^{81} - 78650 q^{83} - 116120 q^{85} + 6903 q^{87} + 83082 q^{89} + 100530 q^{93} + 214946 q^{95} - 369570 q^{97} - 65934 q^{99}+O(q^{100}) $ 4 * q + 18 * q^3 + 47 * q^5 - 162 * q^9 + 407 * q^11 - 898 * q^13 + 846 * q^15 - 1868 * q^17 - 1463 * q^19 + 44 * q^23 + 1605 * q^25 - 2916 * q^27 + 1534 * q^29 - 11170 * q^31 - 3663 * q^33 + 3113 * q^37 - 4041 * q^39 + 15684 * q^41 - 25258 * q^43 + 3807 * q^45 + 9576 * q^47 + 16812 * q^51 - 13395 * q^53 + 26210 * q^55 - 26334 * q^57 - 47521 * q^59 - 63652 * q^61 - 28254 * q^65 - 44541 * q^67 + 792 * q^69 - 251680 * q^71 + 6039 * q^73 - 14445 * q^75 - 17588 * q^79 - 13122 * q^81 - 78650 * q^83 - 116120 * q^85 + 6903 * q^87 + 83082 * q^89 + 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 $\left(\mathbb{Z}/588\mathbb{Z}\right)^\times$.

$n$	$197$	$295$	$493$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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	$-0.633833\pi$
$5$	32.7872	−	56.7890i	0.586515	−	1.01587i	0.994685	−	0.102967i	$-0.0328337\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		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	$-0.616146\pi$
$13$	−434.872			−0.713679			−0.356839	+	0.934166i	$0.616146\pi$
$14$		0			0
$15$	590.169			0.677249
$16$		0			0
$17$	−551.149	−	954.618i	−0.462537	−	0.801138i	0.536550	−	0.843869i	$-0.319727\pi$
$17$	−551.149	−	954.618i	−0.462537	−	0.801138i	−0.999087	+	0.0427312i	$0.986394\pi$
$18$		0			0
$19$	−1438.65	+	2491.81i	−0.914260	+	1.58355i	−0.106279	+	0.994336i	$0.533894\pi$
$19$	−1438.65	+	2491.81i	−0.914260	+	1.58355i	−0.807981	+	0.589209i	$0.799440\pi$
$20$		0			0
$21$		0			0
$22$		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	$-0.860243\pi$
$31$	−4391.32	−	7606.00i	−0.820713	−	1.42152i	0.0844390	−	0.996429i	$-0.473090\pi$
$32$		0			0
$33$	−1105.08	+	1914.06i	−0.176649	+	0.305965i
$34$		0			0
$35$		0			0
$36$		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	$-0.445489\pi$
$41$	3668.55			0.340828			0.170414	+	0.985373i	$0.445489\pi$
$42$		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	−0.806917	−	0.590665i	$-0.798866\pi$
$47$	1636.66	−	2834.78i	0.108072	−	0.187187i	0.914989	+	0.403478i	$0.132199\pi$
$48$		0			0
$49$		0			0
$50$		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	$-0.753747\pi$
$59$	−25743.7	−	44589.5i	−0.962812	−	1.66764i	−0.247432	−	0.968905i	$-0.579587\pi$
$60$		0			0
$61$	−6656.65	+	11529.6i	−0.229050	+	0.396727i	−0.957527	−	0.288344i	$-0.906895\pi$
$61$	−6656.65	+	11529.6i	−0.229050	+	0.396727i	0.728477	+	0.685071i	$0.240229\pi$
$62$		0			0
$63$		0			0
$64$		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	0.991084	−	0.133242i	$-0.0425386\pi$
$73$	17308.7	+	29979.5i	0.380151	+	0.658441i	−0.610932	+	0.791683i	$0.709205\pi$
$74$		0			0
$75$	5287.47	−	9158.17i	0.108541	−	0.187999i
$76$		0			0
$77$		0			0
$78$		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	$-0.797244\pi$
$83$	−100908.			−1.60779			−0.803897	+	0.594768i	$0.797244\pi$
$84$		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	−0.696714	−	0.717349i	$-0.745355\pi$
$89$	20391.8	−	35319.7i	0.272886	−	0.472652i	0.969600	+	0.244697i	$0.0786885\pi$
$90$		0			0
$91$		0			0
$92$		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	$-0.772936\pi$
$97$	−140147.			−1.51236			−0.756178	+	0.654366i	$0.772936\pi$
$98$		0			0
$99$	−19891.5			−0.203976
$100$		0			0
$101$	4288.23	+	7427.43i	0.0418287	+	0.0724494i	0.886182	−	0.463338i	$-0.153348\pi$
$101$	4288.23	+	7427.43i	0.0418287	+	0.0724494i	−0.844353	+	0.535787i	$0.820015\pi$
$102$		0			0
$103$	−17630.1	+	30536.2i	−0.163742	+	0.283610i	−0.936208	−	0.351447i	$-0.885690\pi$
$103$	−17630.1	+	30536.2i	−0.163742	+	0.283610i	0.772466	+	0.635057i	$0.219023\pi$
$104$		0			0
$105$		0			0
$106$		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
$119$		0			0
$120$		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	−0.746383	−	0.665517i	$-0.768211\pi$
$131$	39904.7	−	69116.9i	0.203163	−	0.351889i	0.949546	+	0.313628i	$0.101544\pi$
$132$		0			0
$133$		0			0
$134$		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	$-0.702427\pi$
$139$	−270587.			−1.18787			−0.593935	+	0.804513i	$0.702427\pi$
$140$		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
$147$		0			0
$148$		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	0.683951	−	0.729528i	$-0.260260\pi$
$157$	−89509.3	−	155035.i	−0.289814	−	0.501972i	−0.973765	+	0.227555i	$0.926927\pi$
$158$		0			0
$159$	−13597.5	+	23551.6i	−0.0426547	+	0.0738801i
$160$		0			0
$161$		0			0
$162$		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	$-0.458353\pi$
$167$	94040.0			0.260928			0.130464	+	0.991453i	$0.458353\pi$
$168$		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	$-0.620166\pi$
$173$	244357.	−	423238.i	0.620739	−	1.07515i	0.989348	−	0.145567i	$-0.0465006\pi$
$174$		0			0
$175$		0			0
$176$		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	$-0.376709\pi$
$181$	332961.			0.755434			0.377717	+	0.925921i	$0.376709\pi$
$182$		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
$189$		0			0
$190$		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	$-0.921378\pi$
$199$	−389129.	−	673992.i	−0.696564	−	1.20648i	0.273086	−	0.961990i	$-0.411956\pi$
$200$		0			0
$201$	139031.	−	240808.i	0.242729	−	0.420418i
$202$		0			0
$203$		0			0
$204$		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
$217$		0			0
$218$		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	$-0.547429\pi$
$223$	−220486.			−0.296905			−0.148453	+	0.988920i	$0.547429\pi$
$224$		0			0
$225$	95174.5			0.125333
$226$		0			0
$227$	671890.	+	1.16375e6i	0.865433	+	1.49897i	0.866617	+	0.498974i	$0.166290\pi$
$227$	671890.	+	1.16375e6i	0.865433	+	1.49897i	−0.00118391	+	0.999999i	$0.500377\pi$
$228$		0			0
$229$	−521292.	+	902904.i	−0.656889	+	1.13777i	0.324528	+	0.945876i	$0.394795\pi$
$229$	−521292.	+	902904.i	−0.656889	+	1.13777i	−0.981417	+	0.191889i	$0.938539\pi$
$230$		0			0
$231$		0			0
$232$		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	0.863873	−	0.503709i	$-0.168032\pi$
$241$	−3866.44	−	6696.87i	−0.00428814	−	0.00742728i	−0.868162	+	0.496282i	$0.834698\pi$
$242$		0			0
$243$	29524.5	−	51137.9i	0.0320750	−	0.0555556i
$244$		0			0
$245$		0			0
$246$		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	$-0.685029\pi$
$251$	−1.09614e6			−1.09820			−0.549098	+	0.835758i	$0.685029\pi$
$252$		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	−0.646386	−	0.763011i	$-0.723720\pi$
$257$	357460.	−	619139.i	0.337594	−	0.584730i	0.983980	+	0.178281i	$0.0570536\pi$
$258$		0			0
$259$		0			0
$260$		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	$-0.839282\pi$
$269$	−1.01654e6	−	1.76070e6i	−0.856531	−	1.48356i	0.0186864	−	0.999825i	$-0.494052\pi$
$270$		0			0
$271$	897277.	−	1.55413e6i	0.742170	−	1.28548i	−0.209336	−	0.977844i	$-0.567130\pi$
$271$	897277.	−	1.55413e6i	0.742170	−	1.28548i	0.951505	−	0.307632i	$-0.0995365\pi$
$272$		0			0
$273$		0			0
$274$		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	0.806294	−	0.591515i	$-0.201470\pi$
$283$	−147018.	−	254642.i	−0.109120	−	0.189001i	−0.915414	+	0.402514i	$0.868137\pi$
$284$		0			0
$285$	−849044.	+	1.47059e6i	−0.619182	+	1.07245i
$286$		0			0
$287$		0			0
$288$		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	$-0.461243\pi$
$293$	356961.			0.242913			0.121457	+	0.992597i	$0.461243\pi$
$294$		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
$301$		0			0
$302$		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	$-0.287959\pi$
$307$	2.04097e6			1.23592			0.617960	+	0.786210i	$0.287959\pi$
$308$		0			0
$309$	−317341.			−0.189073
$310$		0			0
$311$	1.02337e6	+	1.77253e6i	0.599972	+	1.03918i	0.992824	+	0.119581i	$0.0381552\pi$
$311$	1.02337e6	+	1.77253e6i	0.599972	+	1.03918i	−0.392852	+	0.919602i	$0.628512\pi$
$312$		0			0
$313$	319272.	−	552996.i	0.184205	−	0.319052i	−0.759104	−	0.650970i	$-0.774362\pi$
$313$	319272.	−	552996.i	0.184205	−	0.319052i	0.943308	+	0.331918i	$0.107696\pi$
$314$		0			0
$315$		0			0
$316$		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
$329$		0			0
$330$		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
$343$		0			0
$344$		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	$-0.174603\pi$
$349$	3.88321e6			1.70658			0.853290	+	0.521436i	$0.174603\pi$
$350$		0			0
$351$	317021.			0.137348
$352$		0			0
$353$	−385467.	−	667649.i	−0.164646	−	0.285175i	0.771884	−	0.635764i	$-0.219315\pi$
$353$	−385467.	−	667649.i	−0.164646	−	0.285175i	−0.936529	+	0.350589i	$0.885981\pi$
$354$		0			0
$355$	−1.37322e6	+	2.37849e6i	−0.578321	+	1.00168i
$356$		0			0
$357$		0			0
$358$		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	0.828418	−	0.560111i	$-0.189242\pi$
$367$	−182842.	−	316691.i	−0.0708615	−	0.122736i	−0.899279	+	0.437375i	$0.855908\pi$
$368$		0			0
$369$	−148576.	+	257342.i	−0.0568047	+	0.0983886i
$370$		0			0
$371$		0			0
$372$		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	−0.631426	−	0.775436i	$-0.717530\pi$
$383$	1.02152e6	−	1.76932e6i	0.355834	−	0.616323i	0.987260	+	0.159113i	$0.0508633\pi$
$384$		0			0
$385$		0			0
$386$		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	$0.364913\pi$
$397$	−1.83182e6	+	3.17280e6i	−0.583319	+	1.01034i	−0.995083	+	0.0990473i	$0.968420\pi$
$398$		0			0
$399$		0			0
$400$		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	0.975998	−	0.217781i	$-0.0698818\pi$
$409$	1.01287e6	+	1.75434e6i	0.299395	+	0.518568i	−0.676603	+	0.736348i	$0.736548\pi$
$410$		0			0
$411$	−387289.	+	670805.i	−0.113092	+	0.195881i
$412$		0			0
$413$		0			0
$414$		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	$-0.607896\pi$
$419$	−2.38986e6			−0.665025			−0.332513	+	0.943099i	$0.607896\pi$
$420$		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
$427$		0			0
$428$		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	$-0.530301\pi$
$433$	−741661.			−0.190102			−0.0950508	+	0.995472i	$0.530301\pi$
$434$		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	−0.632023	−	0.774950i	$-0.717775\pi$
$439$	1.43394e6	−	2.48365e6i	0.355115	−	0.615077i	0.987138	+	0.159873i	$0.0511084\pi$
$440$		0			0
$441$		0			0
$442$		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
$455$		0			0
$456$		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	$-0.227968\pi$
$461$	6.88393e6			1.50864			0.754318	+	0.656510i	$0.227968\pi$
$462$		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	−0.985446	−	0.169988i	$-0.945627\pi$
$467$	−1.62836e6	+	2.82041e6i	−0.345509	+	0.598439i	0.639937	+	0.768427i	$0.278960\pi$
$468$		0			0
$469$		0			0
$470$		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	$0.215108\pi$
$479$	4.67916e6	+	8.10454e6i	0.931814	+	1.61395i	0.151595	+	0.988443i	$0.451559\pi$
$480$		0			0
$481$	−530557.	+	918951.i	−0.104561	+	0.181105i
$482$		0			0
$483$		0			0
$484$		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
$497$		0			0
$498$		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	$-0.450623\pi$
$503$	1.75338e6			0.308999			0.154500	+	0.987993i	$0.450623\pi$
$504$		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	−0.519142	−	0.854688i	$-0.673749\pi$
$509$	2.80923e6	−	4.86573e6i	0.480610	−	0.832441i	0.999753	+	0.0222464i	$0.00708185\pi$
$510$		0			0
$511$		0			0
$512$		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	$0.185089\pi$
$521$	5.53589e6	+	9.58845e6i	0.893498	+	1.54758i	0.0578446	+	0.998326i	$0.481577\pi$
$522$		0			0
$523$	−3.79894e6	+	6.57996e6i	−0.607307	+	1.05189i	0.384375	+	0.923177i	$0.374417\pi$
$523$	−3.79894e6	+	6.57996e6i	−0.607307	+	1.05189i	−0.991682	+	0.128710i	$0.958916\pi$
$524$		0			0
$525$		0			0
$526$		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
$539$		0			0
$540$		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
$553$		0			0
$554$		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	0.703358	−	0.710835i	$-0.251683\pi$
$563$	−1.98494e6	−	3.43802e6i	−0.263922	−	0.457127i	−0.967281	+	0.253708i	$0.918350\pi$
$564$		0			0
$565$	931185.	−	1.61286e6i	0.122720	−	0.212557i
$566$		0			0
$567$		0			0
$568$		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	0.787242	−	0.616644i	$-0.211508\pi$
$577$	−1.12288e6	−	1.94488e6i	−0.140408	−	0.243194i	−0.927650	+	0.373450i	$0.878175\pi$
$578$		0			0
$579$	−4.02579e6	+	6.97287e6i	−0.499062	+	0.864400i
$580$		0			0
$581$		0			0
$582$		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	$-0.625234\pi$
$587$	−6.40082e6			−0.766726			−0.383363	+	0.923598i	$0.625234\pi$
$588$		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	$-0.608385\pi$
$593$	5.56031e6	−	9.63074e6i	0.649325	−	1.12466i	0.983284	−	0.182077i	$-0.0582820\pi$
$594$		0			0
$595$		0			0
$596$		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	$-0.353112\pi$
$601$	7.88546e6			0.890514			0.445257	+	0.895403i	$0.353112\pi$
$602$		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	−0.996123	−	0.0879660i	$-0.971963\pi$
$607$	−3.82967e6	+	6.63319e6i	−0.421881	+	0.730719i	0.574243	+	0.818685i	$0.305297\pi$
$608$		0			0
$609$		0			0
$610$		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	0.983355	−	0.181696i	$-0.0581588\pi$
$619$	3.18709e6	+	5.52019e6i	0.334324	+	0.579066i	−0.649031	+	0.760762i	$0.724825\pi$
$620$		0			0
$621$	−1.54163e6	+	2.67018e6i	−0.160417	+	0.277851i
$622$		0			0
$623$		0			0
$624$		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
$637$		0			0
$638$		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	$-0.689115\pi$
$643$	−1.17375e7			−1.11956			−0.559780	+	0.828641i	$0.689115\pi$
$644$		0			0
$645$	−4.24807e6			−0.402062
$646$		0			0
$647$	5.12724e6	+	8.88063e6i	0.481529	+	0.834033i	0.999775	−	0.0211984i	$-0.00674818\pi$
$647$	5.12724e6	+	8.88063e6i	0.481529	+	0.834033i	−0.518246	+	0.855232i	$0.673415\pi$
$648$		0			0
$649$	6.32200e6	−	1.09500e7i	0.589173	−	1.02048i
$650$		0			0
$651$		0			0
$652$		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	0.870318	−	0.492490i	$-0.163913\pi$
$661$	97171.2	+	168305.i	0.00865035	+	0.0149829i	−0.861668	+	0.507473i	$0.830580\pi$
$662$		0			0
$663$	−2.15711e6	+	3.73623e6i	−0.190585	+	0.330103i
$664$		0			0
$665$		0			0
$666$		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	−0.611990	−	0.790865i	$-0.709631\pi$
$677$	4.51869e6	−	7.82660e6i	0.378914	−	0.656299i	0.990905	+	0.134567i	$0.0429642\pi$
$678$		0			0
$679$		0			0
$680$		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	−0.730730	−	0.682667i	$-0.760820\pi$
$691$	2.83465e6	−	4.90976e6i	0.225842	−	0.391170i	0.956572	+	0.291497i	$0.0941534\pi$
$692$		0			0
$693$		0			0
$694$		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
$707$		0			0
$708$		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	$-0.528491\pi$
$719$	1.13372e7	−	1.96365e7i	0.817865	−	1.41658i	0.907252	−	0.420587i	$-0.138176\pi$
$720$		0			0
$721$		0			0
$722$		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	$-0.262486\pi$
$727$	1.93477e7			1.35767			0.678833	+	0.734293i	$0.262486\pi$
$728$		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	$-0.662712\pi$
$733$	7.42922e6	−	1.28678e7i	0.510720	−	0.884593i	0.999923	−	0.0124232i	$-0.00395452\pi$
$734$		0			0
$735$		0			0
$736$		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
$749$		0			0
$750$		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	−0.717210	−	0.696857i	$-0.754581\pi$
$761$	3.91233e6	−	6.77635e6i	0.244891	−	0.424164i	0.962101	+	0.272693i	$0.0879144\pi$
$762$		0			0
$763$		0			0
$764$		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	$-0.418830\pi$
$769$	8.27325e6			0.504499			0.252250	+	0.967662i	$0.418830\pi$
$770$		0			0
$771$	6.43428e6			0.389820
$772$		0			0
$773$	−9.41844e6	−	1.63132e7i	−0.566931	−	0.981953i	−0.996867	−	0.0790941i	$-0.974797\pi$
$773$	−9.41844e6	−	1.63132e7i	−0.566931	−	0.981953i	0.429936	−	0.902859i	$-0.358536\pi$
$774$		0			0
$775$	−5.15978e6	+	8.93700e6i	−0.308587	+	0.534488i
$776$		0			0
$777$		0			0
$778$		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
$783$	−3.62284e6			−0.211176
$784$		0			0
$785$	−1.17390e7			−0.679920
$786$		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:	\( N \)	\(=\)	\( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	588.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(4.50000 + 7.79423i) q^{3} +(32.7872 - 56.7890i) q^{5} +(-40.5000 + 70.1481i) q^{9} +O(q^{10})\)	\(q+(4.50000 + 7.79423i) q^{3} +(32.7872 - 56.7890i) q^{5} +(-40.5000 + 70.1481i) q^{9} +(122.787 + 212.674i) q^{11} -434.872 q^{13} +590.169 q^{15} +(-551.149 - 954.618i) q^{17} +(-1438.65 + 2491.81i) q^{19} +(2114.72 - 3662.80i) q^{23} +(-587.497 - 1017.57i) q^{25} -729.000 q^{27} +4969.60 q^{29} +(-4391.32 - 7606.00i) q^{31} +(-1105.08 + 1914.06i) q^{33} +(1220.03 - 2113.15i) q^{37} +(-1956.92 - 3389.49i) q^{39} +3668.55 q^{41} -7198.06 q^{43} +(2655.76 + 4599.91i) q^{45} +(1636.66 - 2834.78i) q^{47} +(4960.34 - 8591.56i) q^{51} +(1510.84 + 2616.84i) q^{53} +16103.4 q^{55} -25895.6 q^{57} +(-25743.7 - 44589.5i) q^{59} +(-6656.65 + 11529.6i) q^{61} +(-14258.2 + 24695.9i) q^{65} +(-15447.9 - 26756.5i) q^{67} +38064.9 q^{69} -41882.8 q^{71} +(17308.7 + 29979.5i) q^{73} +(5287.47 - 9158.17i) q^{75} +(-38771.7 + 67154.6i) q^{79} +(-3280.50 - 5681.99i) q^{81} -100908. q^{83} -72282.4 q^{85} +(22363.2 + 38734.2i) q^{87} +(20391.8 - 35319.7i) q^{89} +(39521.9 - 68454.0i) q^{93} +(94338.2 + 163399. i) q^{95} -140147. q^{97} -19891.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 18 q^{3} + 47 q^{5} - 162 q^{9}+O(q^{10}) \) 4 * q + 18 * q^3 + 47 * q^5 - 162 * q^9	\( 4 q + 18 q^{3} + 47 q^{5} - 162 q^{9} + 407 q^{11} - 898 q^{13} + 846 q^{15} - 1868 q^{17} - 1463 q^{19} + 44 q^{23} + 1605 q^{25} - 2916 q^{27} + 1534 q^{29} - 11170 q^{31} - 3663 q^{33} + 3113 q^{37} - 4041 q^{39} + 15684 q^{41} - 25258 q^{43} + 3807 q^{45} + 9576 q^{47} + 16812 q^{51} - 13395 q^{53} + 26210 q^{55} - 26334 q^{57} - 47521 q^{59} - 63652 q^{61} - 28254 q^{65} - 44541 q^{67} + 792 q^{69} - 251680 q^{71} + 6039 q^{73} - 14445 q^{75} - 17588 q^{79} - 13122 q^{81} - 78650 q^{83} - 116120 q^{85} + 6903 q^{87} + 83082 q^{89} + 100530 q^{93} + 214946 q^{95} - 369570 q^{97} - 65934 q^{99}+O(q^{100}) \) 4 * q + 18 * q^3 + 47 * q^5 - 162 * q^9 + 407 * q^11 - 898 * q^13 + 846 * q^15 - 1868 * q^17 - 1463 * q^19 + 44 * q^23 + 1605 * q^25 - 2916 * q^27 + 1534 * q^29 - 11170 * q^31 - 3663 * q^33 + 3113 * q^37 - 4041 * q^39 + 15684 * q^41 - 25258 * q^43 + 3807 * q^45 + 9576 * q^47 + 16812 * q^51 - 13395 * q^53 + 26210 * q^55 - 26334 * q^57 - 47521 * q^59 - 63652 * q^61 - 28254 * q^65 - 44541 * q^67 + 792 * q^69 - 251680 * q^71 + 6039 * q^73 - 14445 * q^75 - 17588 * q^79 - 13122 * q^81 - 78650 * q^83 - 116120 * q^85 + 6903 * q^87 + 83082 * q^89 + 100530 * q^93 + 214946 * q^95 - 369570 * q^97 - 65934 * q^99

Introduction

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