LMFDB - Embedded newform 588.6.i.o.361.2

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:	$8$
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^{4}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.2
Root			$13.1471 - 22.7714i$ of defining polynomial
Character	$\chi$	$=$	588.361
Dual form			588.6.i.o.373.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.50000 + 7.79423i) q^{3} +(-15.9808 - 27.6796i) q^{5} +(-40.5000 - 70.1481i) q^{9} +O(q^{10})$	\(q+(-4.50000 + 7.79423i) q^{3} +(-15.9808 - 27.6796i) q^{5} +(-40.5000 - 70.1481i) q^{9} +(-130.442 + 225.932i) q^{11} -769.735 q^{13} +287.654 q^{15} +(-776.659 + 1345.21i) q^{17} +(375.024 + 649.561i) q^{19} +(-377.427 - 653.723i) q^{23} +(1051.73 - 1821.65i) q^{25} +729.000 q^{27} +6008.93 q^{29} +(-3210.02 + 5559.92i) q^{31} +(-1173.98 - 2033.39i) q^{33} +(2387.86 + 4135.90i) q^{37} +(3463.81 - 5999.49i) q^{39} +5423.27 q^{41} -11896.4 q^{43} +(-1294.44 + 2242.04i) q^{45} +(-8714.00 - 15093.1i) q^{47} +(-6989.93 - 12106.9i) q^{51} +(-18825.3 + 32606.4i) q^{53} +8338.26 q^{55} -6750.44 q^{57} +(11039.0 - 19120.2i) q^{59} +(-4086.69 - 7078.36i) q^{61} +(12301.0 + 21305.9i) q^{65} +(-6500.87 + 11259.8i) q^{67} +6793.69 q^{69} -12349.6 q^{71} +(21800.2 - 37759.0i) q^{73} +(9465.55 + 16394.8i) q^{75} +(-38374.7 - 66467.0i) q^{79} +(-3280.50 + 5681.99i) q^{81} -21893.6 q^{83} +49646.5 q^{85} +(-27040.2 + 46835.0i) q^{87} +(-68483.5 - 118617. i) q^{89} +(-28890.2 - 50039.3i) q^{93} +(11986.4 - 20761.0i) q^{95} +93050.1 q^{97} +21131.6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 36 q^{3} - 324 q^{9}+O(q^{10}) $ 8 * q - 36 * q^3 - 324 * q^9	$ 8 q - 36 q^{3} - 324 q^{9} - 462 q^{11} + 1204 q^{13} - 228 q^{17} - 358 q^{19} - 2148 q^{23} - 5454 q^{25} + 5832 q^{27} - 11064 q^{29} - 830 q^{31} - 4158 q^{33} - 3914 q^{37} - 5418 q^{39} + 16632 q^{41} - 29036 q^{43} - 41700 q^{47} - 2052 q^{51} + 22164 q^{53} - 7784 q^{55} + 6444 q^{57} - 32886 q^{59} - 83732 q^{61} - 93192 q^{65} - 80034 q^{67} + 38664 q^{69} + 89544 q^{71} + 22470 q^{73} - 49086 q^{75} - 75286 q^{79} - 26244 q^{81} + 34836 q^{83} + 278504 q^{85} + 49788 q^{87} - 28944 q^{89} - 7470 q^{93} - 144120 q^{95} + 433356 q^{97} + 74844 q^{99}+O(q^{100}) $ 8 * q - 36 * q^3 - 324 * q^9 - 462 * q^11 + 1204 * q^13 - 228 * q^17 - 358 * q^19 - 2148 * q^23 - 5454 * q^25 + 5832 * q^27 - 11064 * q^29 - 830 * q^31 - 4158 * q^33 - 3914 * q^37 - 5418 * q^39 + 16632 * q^41 - 29036 * q^43 - 41700 * q^47 - 2052 * q^51 + 22164 * q^53 - 7784 * q^55 + 6444 * q^57 - 32886 * q^59 - 83732 * q^61 - 93192 * q^65 - 80034 * q^67 + 38664 * q^69 + 89544 * q^71 + 22470 * q^73 - 49086 * q^75 - 75286 * q^79 - 26244 * q^81 + 34836 * q^83 + 278504 * q^85 + 49788 * q^87 - 28944 * q^89 - 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 $\left(\mathbb{Z}/588\mathbb{Z}\right)^\times$.

$n$	$197$	$295$	$493$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$5$	−15.9808	−	27.6796i	−0.285873	−	0.495147i	0.686947	−	0.726707i	$-0.258950\pi$
$5$	−15.9808	−	27.6796i	−0.285873	−	0.495147i	−0.972821	+	0.231560i	$0.925617\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−40.5000	−	70.1481i	−0.166667	−	0.288675i
$10$		0			0
$11$	−130.442	+	225.932i	−0.325039	+	0.562984i	−0.981520	−	0.191359i	$-0.938711\pi$
$11$	−130.442	+	225.932i	−0.325039	+	0.562984i	0.656481	+	0.754342i	$0.272044\pi$
$12$		0			0
$13$	−769.735			−1.26323			−0.631616	−	0.775281i	$-0.717608\pi$
$13$	−769.735			−1.26323			−0.631616	+	0.775281i	$0.717608\pi$
$14$		0			0
$15$	287.654			0.330098
$16$		0			0
$17$	−776.659	+	1345.21i	−0.651791	+	1.12893i	0.330898	+	0.943667i	$0.392649\pi$
$17$	−776.659	+	1345.21i	−0.651791	+	1.12893i	−0.982688	+	0.185268i	$0.940685\pi$
$18$		0			0
$19$	375.024	+	649.561i	0.238328	+	0.412797i	0.960235	−	0.279194i	$-0.0900673\pi$
$19$	375.024	+	649.561i	0.238328	+	0.412797i	−0.721906	+	0.691991i	$0.756734\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−377.427	−	653.723i	−0.148769	−	0.257676i	0.782004	−	0.623274i	$-0.214198\pi$
$23$	−377.427	−	653.723i	−0.148769	−	0.257676i	−0.930773	+	0.365598i	$0.880865\pi$
$24$		0			0
$25$	1051.73	−	1821.65i	0.336553	−	0.582927i
$26$		0			0
$27$	729.000			0.192450
$28$		0			0
$29$	6008.93			1.32679			0.663395	−	0.748269i	$-0.269115\pi$
$29$	6008.93			1.32679			0.663395	+	0.748269i	$0.269115\pi$
$30$		0			0
$31$	−3210.02	+	5559.92i	−0.599934	+	1.03912i	0.392896	+	0.919583i	$0.371473\pi$
$31$	−3210.02	+	5559.92i	−0.599934	+	1.03912i	−0.992830	+	0.119533i	$0.961860\pi$
$32$		0			0
$33$	−1173.98	−	2033.39i	−0.187661	−	0.325039i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	2387.86	+	4135.90i	0.286751	+	0.496668i	0.973032	−	0.230669i	$-0.0740914\pi$
$37$	2387.86	+	4135.90i	0.286751	+	0.496668i	−0.686281	+	0.727336i	$0.740758\pi$
$38$		0			0
$39$	3463.81	−	5999.49i	0.364664	−	0.631616i
$40$		0			0
$41$	5423.27			0.503850			0.251925	−	0.967747i	$-0.418936\pi$
$41$	5423.27			0.503850			0.251925	+	0.967747i	$0.418936\pi$
$42$		0			0
$43$	−11896.4			−0.981171			−0.490585	−	0.871393i	$-0.663217\pi$
$43$	−11896.4			−0.981171			−0.490585	+	0.871393i	$0.663217\pi$
$44$		0			0
$45$	−1294.44	+	2242.04i	−0.0952911	+	0.165049i
$46$		0			0
$47$	−8714.00	−	15093.1i	−0.575404	−	0.996629i	−0.995998	−	0.0893798i	$-0.971512\pi$
$47$	−8714.00	−	15093.1i	−0.575404	−	0.996629i	0.420594	−	0.907249i	$-0.361822\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−6989.93	−	12106.9i	−0.376311	−	0.651791i
$52$		0			0
$53$	−18825.3	+	32606.4i	−0.920561	+	1.59446i	−0.122012	+	0.992529i	$0.538935\pi$
$53$	−18825.3	+	32606.4i	−0.920561	+	1.59446i	−0.798549	+	0.601930i	$0.794399\pi$
$54$		0			0
$55$	8338.26			0.371679
$56$		0			0
$57$	−6750.44			−0.275198
$58$		0			0
$59$	11039.0	−	19120.2i	0.412859	−	0.715092i	−0.582342	−	0.812944i	$-0.697864\pi$
$59$	11039.0	−	19120.2i	0.412859	−	0.715092i	0.995201	+	0.0978516i	$0.0311971\pi$
$60$		0			0
$61$	−4086.69	−	7078.36i	−0.140620	−	0.243561i	0.787110	−	0.616812i	$-0.211576\pi$
$61$	−4086.69	−	7078.36i	−0.140620	−	0.243561i	−0.927730	+	0.373251i	$0.878243\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	12301.0	+	21305.9i	0.361124	+	0.625485i
$66$		0			0
$67$	−6500.87	+	11259.8i	−0.176923	+	0.306440i	−0.940825	−	0.338892i	$-0.889948\pi$
$67$	−6500.87	+	11259.8i	−0.176923	+	0.306440i	0.763902	+	0.645332i	$0.223281\pi$
$68$		0			0
$69$	6793.69			0.171784
$70$		0			0
$71$	−12349.6			−0.290742			−0.145371	−	0.989377i	$-0.546438\pi$
$71$	−12349.6			−0.290742			−0.145371	+	0.989377i	$0.546438\pi$
$72$		0			0
$73$	21800.2	−	37759.0i	0.478798	−	0.829303i	−0.520906	−	0.853614i	$-0.674406\pi$
$73$	21800.2	−	37759.0i	0.478798	−	0.829303i	0.999704	+	0.0243110i	$0.00773921\pi$
$74$		0			0
$75$	9465.55	+	16394.8i	0.194309	+	0.336553i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−38374.7	−	66467.0i	−0.691796	−	1.19823i	−0.971249	−	0.238066i	$-0.923487\pi$
$79$	−38374.7	−	66467.0i	−0.691796	−	1.19823i	0.279453	−	0.960159i	$-0.409847\pi$
$80$		0			0
$81$	−3280.50	+	5681.99i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	−21893.6			−0.348836			−0.174418	−	0.984672i	$-0.555804\pi$
$83$	−21893.6			−0.348836			−0.174418	+	0.984672i	$0.555804\pi$
$84$		0			0
$85$	49646.5			0.745318
$86$		0			0
$87$	−27040.2	+	46835.0i	−0.383011	+	0.663395i
$88$		0			0
$89$	−68483.5	−	118617.i	−0.916454	−	1.58735i	−0.804758	−	0.593603i	$-0.797705\pi$
$89$	−68483.5	−	118617.i	−0.916454	−	1.58735i	−0.111696	−	0.993742i	$-0.535628\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−28890.2	−	50039.3i	−0.346372	−	0.599934i
$94$		0			0
$95$	11986.4	−	20761.0i	0.136263	−	0.236015i
$96$		0			0
$97$	93050.1			1.00412			0.502062	−	0.864832i	$-0.332575\pi$
$97$	93050.1			1.00412			0.502062	+	0.864832i	$0.332575\pi$
$98$		0			0
$99$	21131.6			0.216692
$100$		0			0
$101$	61150.8	−	105916.i	0.596484	−	1.03314i	−0.396852	−	0.917883i	$-0.629897\pi$
$101$	61150.8	−	105916.i	0.596484	−	1.03314i	0.993336	−	0.115258i	$-0.0367693\pi$
$102$		0			0
$103$	37400.6	+	64779.8i	0.347365	+	0.601654i	0.985781	−	0.168038i	$-0.0537432\pi$
$103$	37400.6	+	64779.8i	0.347365	+	0.601654i	−0.638415	+	0.769692i	$0.720410\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−35386.2	−	61290.7i	−0.298796	−	0.517530i	0.677065	−	0.735923i	$-0.263252\pi$
$107$	−35386.2	−	61290.7i	−0.298796	−	0.517530i	−0.975861	+	0.218394i	$0.929918\pi$
$108$		0			0
$109$	71131.0	−	123202.i	0.573446	−	0.993238i	−0.422762	−	0.906241i	$-0.638940\pi$
$109$	71131.0	−	123202.i	0.573446	−	0.993238i	0.996209	−	0.0869972i	$-0.0277271\pi$
$110$		0			0
$111$	−42981.5			−0.331112
$112$		0			0
$113$	198899.			1.46533			0.732667	−	0.680588i	$-0.238275\pi$
$113$	198899.			1.46533			0.732667	+	0.680588i	$0.238275\pi$
$114$		0			0
$115$	−12063.2	+	20894.0i	−0.0850583	+	0.147325i
$116$		0			0
$117$	31174.3	+	53995.4i	0.210539	+	0.364664i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	46495.4	+	80532.3i	0.288700	+	0.500042i
$122$		0			0
$123$	−24404.7	+	42270.2i	−0.145449	+	0.251925i
$124$		0			0
$125$	−167110.			−0.956592
$126$		0			0
$127$	219619.			1.20826			0.604131	−	0.796885i	$-0.293521\pi$
$127$	219619.			1.20826			0.604131	+	0.796885i	$0.293521\pi$
$128$		0			0
$129$	53533.8	−	92723.3i	0.283240	−	0.490585i
$130$		0			0
$131$	27311.1	+	47304.1i	0.139047	+	0.240836i	0.927136	−	0.374725i	$-0.122263\pi$
$131$	27311.1	+	47304.1i	0.139047	+	0.240836i	−0.788089	+	0.615561i	$0.788930\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−11650.0	−	20178.4i	−0.0550163	−	0.0952911i
$136$		0			0
$137$	−36471.4	+	63170.4i	−0.166017	+	0.287549i	−0.937016	−	0.349287i	$-0.886424\pi$
$137$	−36471.4	+	63170.4i	−0.166017	+	0.287549i	0.770999	+	0.636836i	$0.219757\pi$
$138$		0			0
$139$	110668.			0.485831			0.242915	−	0.970047i	$-0.421896\pi$
$139$	110668.			0.485831			0.242915	+	0.970047i	$0.421896\pi$
$140$		0			0
$141$	156852.			0.664419
$142$		0			0
$143$	100406.	−	173908.i	0.410599	−	0.711179i
$144$		0			0
$145$	−96027.5	−	166325.i	−0.379294	−	0.656956i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−163513.	−	283212.i	−0.603373	−	1.04507i	−0.992306	−	0.123807i	$-0.960490\pi$
$149$	−163513.	−	283212.i	−0.603373	−	1.04507i	0.388933	−	0.921266i	$-0.372844\pi$
$150$		0			0
$151$	−87245.1	+	151113.i	−0.311386	+	0.539336i	−0.978663	−	0.205474i	$-0.934126\pi$
$151$	−87245.1	+	151113.i	−0.311386	+	0.539336i	0.667277	+	0.744810i	$0.267460\pi$
$152$		0			0
$153$	125819.			0.434527
$154$		0			0
$155$	205195.			0.686020
$156$		0			0
$157$	220146.	−	381304.i	0.712790	−	1.23459i	−0.251016	−	0.967983i	$-0.580765\pi$
$157$	220146.	−	381304.i	0.712790	−	1.23459i	0.963806	−	0.266606i	$-0.0859022\pi$
$158$		0			0
$159$	−169428.	−	293458.i	−0.531486	−	0.920561i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−28246.0	−	48923.5i	−0.0832698	−	0.144228i	0.821383	−	0.570377i	$-0.193203\pi$
$163$	−28246.0	−	48923.5i	−0.0832698	−	0.144228i	−0.904653	+	0.426150i	$0.859870\pi$
$164$		0			0
$165$	−37522.2	+	64990.3i	−0.107295	+	0.185840i
$166$		0			0
$167$	688014.			1.90900			0.954501	−	0.298209i	$-0.0963891\pi$
$167$	688014.			1.90900			0.954501	+	0.298209i	$0.0963891\pi$
$168$		0			0
$169$	221200.			0.595755
$170$		0			0
$171$	30377.0	−	52614.5i	0.0794427	−	0.137599i
$172$		0			0
$173$	141279.	+	244703.i	0.358891	+	0.621618i	0.987776	−	0.155881i	$-0.0498216\pi$
$173$	141279.	+	244703.i	0.358891	+	0.621618i	−0.628885	+	0.777499i	$0.716488\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	99351.4	+	172082.i	0.238364	+	0.412859i
$178$		0			0
$179$	−75890.9	+	131447.i	−0.177034	+	0.306632i	−0.940863	−	0.338786i	$-0.889984\pi$
$179$	−75890.9	+	131447.i	−0.177034	+	0.306632i	0.763829	+	0.645419i	$0.223317\pi$
$180$		0			0
$181$	322258.			0.731151			0.365575	−	0.930782i	$-0.380872\pi$
$181$	322258.			0.731151			0.365575	+	0.930782i	$0.380872\pi$
$182$		0			0
$183$	73560.4			0.162374
$184$		0			0
$185$	76319.9	−	132190.i	0.163949	−	0.283968i
$186$		0			0
$187$	−202618.	−	350944.i	−0.423714	−	0.733895i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−269260.	−	466373.i	−0.534059	−	0.925017i	−0.999208	−	0.0397847i	$-0.987333\pi$
$191$	−269260.	−	466373.i	−0.534059	−	0.925017i	0.465150	−	0.885232i	$-0.346001\pi$
$192$		0			0
$193$	−151720.	+	262786.i	−0.293190	+	0.507820i	−0.974562	−	0.224117i	$-0.928050\pi$
$193$	−151720.	+	262786.i	−0.293190	+	0.507820i	0.681372	+	0.731937i	$0.261383\pi$
$194$		0			0
$195$	−221418.			−0.416990
$196$		0			0
$197$	656475.			1.20518			0.602591	−	0.798050i	$-0.294135\pi$
$197$	656475.			1.20518			0.602591	+	0.798050i	$0.294135\pi$
$198$		0			0
$199$	421102.	−	729370.i	0.753797	−	1.30562i	−0.192173	−	0.981361i	$-0.561553\pi$
$199$	421102.	−	729370.i	0.753797	−	1.30562i	0.945970	−	0.324254i	$-0.105113\pi$
$200$		0			0
$201$	−58507.8	−	101339.i	−0.102147	−	0.176923i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−86668.1	−	150114.i	−0.144037	−	0.249480i
$206$		0			0
$207$	−30571.6	+	52951.6i	−0.0495898	+	0.0858920i
$208$		0			0
$209$	−195675.			−0.309864
$210$		0			0
$211$	1.01133e6			1.56382			0.781908	−	0.623394i	$-0.214247\pi$
$211$	1.01133e6			1.56382			0.781908	+	0.623394i	$0.214247\pi$
$212$		0			0
$213$	55573.3	−	96255.8i	0.0839300	−	0.145371i
$214$		0			0
$215$	190114.	+	329287.i	0.280490	+	0.485824i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	196201.	+	339831.i	0.276434	+	0.478798i
$220$		0			0
$221$	597822.	−	1.03546e6i	0.823363	−	1.42611i
$222$		0			0
$223$	−957124.			−1.28886			−0.644431	−	0.764663i	$-0.722906\pi$
$223$	−957124.			−1.28886			−0.644431	+	0.764663i	$0.722906\pi$
$224$		0			0
$225$	−170380.			−0.224369
$226$		0			0
$227$	235740.	−	408313.i	0.303647	−	0.525931i	−0.673312	−	0.739358i	$-0.735129\pi$
$227$	235740.	−	408313.i	0.303647	−	0.525931i	0.976959	+	0.213427i	$0.0684624\pi$
$228$		0			0
$229$	−134098.	−	232265.i	−0.168980	−	0.292681i	0.769082	−	0.639150i	$-0.220714\pi$
$229$	−134098.	−	232265.i	−0.168980	−	0.292681i	−0.938061	+	0.346469i	$0.887381\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−773610.	−	1.33993e6i	−0.933538	−	1.61694i	−0.777220	−	0.629229i	$-0.783371\pi$
$233$	−773610.	−	1.33993e6i	−0.933538	−	1.61694i	−0.156318	−	0.987707i	$-0.549962\pi$
$234$		0			0
$235$	−278513.	+	482399.i	−0.328985	+	0.569819i
$236$		0			0
$237$	690745.			0.798817
$238$		0			0
$239$	−937542.			−1.06169			−0.530843	−	0.847470i	$-0.678125\pi$
$239$	−937542.			−1.06169			−0.530843	+	0.847470i	$0.678125\pi$
$240$		0			0
$241$	−642430.	+	1.11272e6i	−0.712497	+	1.23408i	0.251420	+	0.967878i	$0.419103\pi$
$241$	−642430.	+	1.11272e6i	−0.712497	+	1.23408i	−0.963917	+	0.266203i	$0.914231\pi$
$242$		0			0
$243$	−29524.5	−	51137.9i	−0.0320750	−	0.0555556i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−288670.	−	499990.i	−0.301064	−	0.521458i
$248$		0			0
$249$	98521.1	−	170644.i	0.100700	−	0.174418i
$250$		0			0
$251$	−709769.			−0.711103			−0.355552	−	0.934657i	$-0.615707\pi$
$251$	−709769.			−0.711103			−0.355552	+	0.934657i	$0.615707\pi$
$252$		0			0
$253$	196929.			0.193423
$254$		0			0
$255$	−223409.	+	386956.i	−0.215155	+	0.372659i
$256$		0			0
$257$	−910080.	−	1.57631e6i	−0.859502	−	1.48870i	−0.872405	−	0.488785i	$-0.837440\pi$
$257$	−910080.	−	1.57631e6i	−0.859502	−	1.48870i	0.0129024	−	0.999917i	$-0.495893\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−243362.	−	421515.i	−0.221132	−	0.383011i
$262$		0			0
$263$	−179594.	+	311065.i	−0.160104	+	0.277308i	−0.934906	−	0.354896i	$-0.884516\pi$
$263$	−179594.	+	311065.i	−0.160104	+	0.277308i	0.774802	+	0.632204i	$0.217850\pi$
$264$		0			0
$265$	1.20337e6			1.05265
$266$		0			0
$267$	1.23270e6			1.05823
$268$		0			0
$269$	578834.	−	1.00257e6i	0.487723	−	0.844761i	−0.512177	−	0.858880i	$-0.671161\pi$
$269$	578834.	−	1.00257e6i	0.487723	−	0.844761i	0.999900	+	0.0141188i	$0.00449429\pi$
$270$		0			0
$271$	−390088.	−	675652.i	−0.322655	−	0.558856i	0.658380	−	0.752686i	$-0.271242\pi$
$271$	−390088.	−	675652.i	−0.322655	−	0.558856i	−0.981035	+	0.193830i	$0.937909\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	274379.	+	475238.i	0.218786	+	0.378948i
$276$		0			0
$277$	163165.	−	282611.i	0.127770	−	0.221304i	−0.795042	−	0.606554i	$-0.792551\pi$
$277$	163165.	−	282611.i	0.127770	−	0.221304i	0.922812	+	0.385250i	$0.125885\pi$
$278$		0			0
$279$	520023.			0.399956
$280$		0			0
$281$	364094.			0.275073			0.137536	−	0.990497i	$-0.456082\pi$
$281$	364094.			0.275073			0.137536	+	0.990497i	$0.456082\pi$
$282$		0			0
$283$	−32009.6	+	55442.3i	−0.0237582	+	0.0411505i	−0.877660	−	0.479284i	$-0.840897\pi$
$283$	−32009.6	+	55442.3i	−0.0237582	+	0.0411505i	0.853902	+	0.520434i	$0.174230\pi$
$284$		0			0
$285$	107877.	+	186849.i	0.0786717	+	0.136263i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−496470.	−	859911.i	−0.349662	−	0.605632i
$290$		0			0
$291$	−418725.	+	725254.i	−0.289866	+	0.502062i
$292$		0			0
$293$	−398020.			−0.270855			−0.135427	−	0.990787i	$-0.543241\pi$
$293$	−398020.			−0.270855			−0.135427	+	0.990787i	$0.543241\pi$
$294$		0			0
$295$	−705651.			−0.472101
$296$		0			0
$297$	−95092.1	+	164704.i	−0.0625537	+	0.108346i
$298$		0			0
$299$	290519.	+	503194.i	0.187930	+	0.325505i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	550357.	+	953247.i	0.344380	+	0.596484i
$304$		0			0
$305$	−130617.	+	226236.i	−0.0803990	+	0.139255i
$306$		0			0
$307$	638841.			0.386854			0.193427	−	0.981115i	$-0.438040\pi$
$307$	638841.			0.386854			0.193427	+	0.981115i	$0.438040\pi$
$308$		0			0
$309$	−673212.			−0.401103
$310$		0			0
$311$	1.24094e6	−	2.14937e6i	0.727529	−	1.26012i	−0.230395	−	0.973097i	$-0.574002\pi$
$311$	1.24094e6	−	2.14937e6i	0.727529	−	1.26012i	0.957924	−	0.287020i	$-0.0926648\pi$
$312$		0			0
$313$	738267.	+	1.27872e6i	0.425944	+	0.737757i	0.996508	−	0.0834961i	$-0.0266086\pi$
$313$	738267.	+	1.27872e6i	0.425944	+	0.737757i	−0.570564	+	0.821253i	$0.693275\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.11563e6	+	1.93233e6i	0.623551	+	1.08002i	0.988819	+	0.149119i	$0.0476438\pi$
$317$	1.11563e6	+	1.93233e6i	0.623551	+	1.08002i	−0.365269	+	0.930902i	$0.619023\pi$
$318$		0			0
$319$	−783816.	+	1.35761e6i	−0.431258	+	0.746961i
$320$		0			0
$321$	636952.			0.345020
$322$		0			0
$323$	−1.16506e6			−0.621360
$324$		0			0
$325$	−809552.	+	1.40219e6i	−0.425145	+	0.736372i
$326$		0			0
$327$	640179.	+	1.10882e6i	0.331079	+	0.573446i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	1.20561e6	+	2.08818e6i	0.604836	+	1.04761i	0.992077	+	0.125628i	$0.0400946\pi$
$331$	1.20561e6	+	2.08818e6i	0.604836	+	1.04761i	−0.387242	+	0.921978i	$0.626572\pi$
$332$		0			0
$333$	193417.	−	335008.i	0.0955837	−	0.165556i
$334$		0			0
$335$	415556.			0.202310
$336$		0			0
$337$	−1.79586e6			−0.861388			−0.430694	−	0.902498i	$-0.641731\pi$
$337$	−1.79586e6			−0.861388			−0.430694	+	0.902498i	$0.641731\pi$
$338$		0			0
$339$	−895045.	+	1.55026e6i	−0.423005	+	0.732667i
$340$		0			0
$341$	−837442.	−	1.45049e6i	−0.390004	−	0.675506i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−108569.	−	188046.i	−0.0491085	−	0.0850583i
$346$		0			0
$347$	−1.10366e6	+	1.91159e6i	−0.492051	+	0.852258i	−0.999958	−	0.00915399i	$-0.997086\pi$
$347$	−1.10366e6	+	1.91159e6i	−0.492051	+	0.852258i	0.507907	+	0.861412i	$0.330419\pi$
$348$		0			0
$349$	−148188.			−0.0651252			−0.0325626	−	0.999470i	$-0.510367\pi$
$349$	−148188.			−0.0651252			−0.0325626	+	0.999470i	$0.510367\pi$
$350$		0			0
$351$	−561137.			−0.243109
$352$		0			0
$353$	985200.	−	1.70642e6i	0.420811	−	0.728867i	−0.575208	−	0.818007i	$-0.695079\pi$
$353$	985200.	−	1.70642e6i	0.420811	−	0.728867i	0.996019	+	0.0891407i	$0.0284121\pi$
$354$		0			0
$355$	197357.	+	341832.i	0.0831154	+	0.143960i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1.21697e6	−	2.10786e6i	−0.498362	−	0.863188i	0.501636	−	0.865079i	$-0.332732\pi$
$359$	−1.21697e6	−	2.10786e6i	−0.498362	−	0.863188i	−0.999998	+	0.00189067i	$0.999398\pi$
$360$		0			0
$361$	956763.	−	1.65716e6i	0.386399	−	0.669263i
$362$		0			0
$363$	−836917.			−0.333362
$364$		0			0
$365$	−1.39354e6			−0.547502
$366$		0			0
$367$	−2.15174e6	+	3.72692e6i	−0.833920	+	1.44439i	0.0609873	+	0.998139i	$0.480575\pi$
$367$	−2.15174e6	+	3.72692e6i	−0.833920	+	1.44439i	−0.894907	+	0.446253i	$0.852758\pi$
$368$		0			0
$369$	−219642.	−	380432.i	−0.0839750	−	0.145449i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	601526.	+	1.04187e6i	0.223863	+	0.387742i	0.955978	−	0.293439i	$-0.0947999\pi$
$373$	601526.	+	1.04187e6i	0.223863	+	0.387742i	−0.732115	+	0.681181i	$0.761467\pi$
$374$		0			0
$375$	751994.	−	1.30249e6i	0.276144	−	0.478296i
$376$		0			0
$377$	−4.62529e6			−1.67604
$378$		0			0
$379$	5.11488e6			1.82910			0.914550	−	0.404473i	$-0.132545\pi$
$379$	5.11488e6			1.82910			0.914550	+	0.404473i	$0.132545\pi$
$380$		0			0
$381$	−988286.	+	1.71176e6i	−0.348795	+	0.604131i
$382$		0			0
$383$	2.50528e6	+	4.33926e6i	0.872687	+	1.51154i	0.859206	+	0.511629i	$0.170958\pi$
$383$	2.50528e6	+	4.33926e6i	0.872687	+	1.51154i	0.0134808	+	0.999909i	$0.495709\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	481804.	+	834510.i	0.163528	+	0.283240i
$388$		0			0
$389$	2.04163e6	−	3.53620e6i	0.684073	−	1.18485i	−0.289654	−	0.957131i	$-0.593540\pi$
$389$	2.04163e6	−	3.53620e6i	0.684073	−	1.18485i	0.973727	−	0.227718i	$-0.0731265\pi$
$390$		0			0
$391$	1.17253e6			0.387866
$392$		0			0
$393$	−491599.			−0.160557
$394$		0			0
$395$	−1.22652e6	+	2.12439e6i	−0.395532	+	0.685081i
$396$		0			0
$397$	−536498.	−	929242.i	−0.170841	−	0.295905i	0.767873	−	0.640602i	$-0.221315\pi$
$397$	−536498.	−	929242.i	−0.170841	−	0.295905i	−0.938714	+	0.344697i	$0.887982\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	949741.	+	1.64500e6i	0.294947	+	0.510863i	0.974973	−	0.222325i	$-0.0713647\pi$
$401$	949741.	+	1.64500e6i	0.294947	+	0.510863i	−0.680026	+	0.733188i	$0.738031\pi$
$402$		0			0
$403$	2.47087e6	−	4.27967e6i	0.757856	−	1.31264i
$404$		0			0
$405$	209700.			0.0635274
$406$		0			0
$407$	−1.24591e6			−0.372821
$408$		0			0
$409$	967887.	−	1.67643e6i	0.286099	−	0.495538i	−0.686776	−	0.726869i	$-0.740975\pi$
$409$	967887.	−	1.67643e6i	0.286099	−	0.495538i	0.972875	+	0.231331i	$0.0743081\pi$
$410$		0			0
$411$	−328243.	−	568533.i	−0.0958497	−	0.166017i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	349877.	+	606004.i	0.0997229	+	0.172725i
$416$		0			0
$417$	−498006.	+	862572.i	−0.140247	+	0.242915i
$418$		0			0
$419$	−3.93448e6			−1.09484			−0.547422	−	0.836856i	$-0.684391\pi$
$419$	−3.93448e6			−1.09484			−0.547422	+	0.836856i	$0.684391\pi$
$420$		0			0
$421$	−5.16927e6			−1.42142			−0.710712	−	0.703483i	$-0.751627\pi$
$421$	−5.16927e6			−1.42142			−0.710712	+	0.703483i	$0.751627\pi$
$422$		0			0
$423$	−705834.	+	1.22254e6i	−0.191801	+	0.332210i
$424$		0			0
$425$	1.63367e6	+	2.82960e6i	0.438724	+	0.759893i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	903651.	+	1.56517e6i	0.237060	+	0.410599i
$430$		0			0
$431$	1.06475e6	−	1.84421e6i	0.276093	−	0.478208i	−0.694317	−	0.719669i	$-0.744293\pi$
$431$	1.06475e6	−	1.84421e6i	0.276093	−	0.478208i	0.970410	+	0.241462i	$0.0776268\pi$
$432$		0			0
$433$	−3.20879e6			−0.822473			−0.411237	−	0.911529i	$-0.634903\pi$
$433$	−3.20879e6			−0.822473			−0.411237	+	0.911529i	$0.634903\pi$
$434$		0			0
$435$	1.72849e6			0.437971
$436$		0			0
$437$	283089.	−	490324.i	0.0709119	−	0.122823i
$438$		0			0
$439$	3.49538e6	+	6.05418e6i	0.865632	+	1.49932i	0.866418	+	0.499320i	$0.166417\pi$
$439$	3.49538e6	+	6.05418e6i	0.865632	+	1.49932i	−0.000785585		1.00000i	$0.500250\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−3.73004e6	−	6.46062e6i	−0.903034	−	1.56410i	−0.823534	−	0.567266i	$-0.808001\pi$
$443$	−3.73004e6	−	6.46062e6i	−0.903034	−	1.56410i	−0.0795000	−	0.996835i	$-0.525332\pi$
$444$		0			0
$445$	−2.18884e6	+	3.79118e6i	−0.523979	+	0.907559i
$446$		0			0
$447$	2.94323e6			0.696715
$448$		0			0
$449$	−5.61620e6			−1.31470			−0.657350	−	0.753586i	$-0.728323\pi$
$449$	−5.61620e6			−1.31470			−0.657350	+	0.753586i	$0.728323\pi$
$450$		0			0
$451$	−707421.	+	1.22529e6i	−0.163771	+	0.283659i
$452$		0			0
$453$	−785206.	−	1.36002e6i	−0.179779	−	0.311386i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	4.40636e6	+	7.63203e6i	0.986936	+	1.70942i	0.632994	+	0.774156i	$0.281826\pi$
$457$	4.40636e6	+	7.63203e6i	0.986936	+	1.70942i	0.353942	+	0.935267i	$0.384841\pi$
$458$		0			0
$459$	−566184.	+	980660.i	−0.125437	+	0.217264i
$460$		0			0
$461$	−3.24339e6			−0.710798			−0.355399	−	0.934715i	$-0.615655\pi$
$461$	−3.24339e6			−0.710798			−0.355399	+	0.934715i	$0.615655\pi$
$462$		0			0
$463$	−5.70039e6			−1.23581			−0.617906	−	0.786252i	$-0.712019\pi$
$463$	−5.70039e6			−1.23581			−0.617906	+	0.786252i	$0.712019\pi$
$464$		0			0
$465$	−923376.	+	1.59933e6i	−0.198037	+	0.343010i
$466$		0			0
$467$	1.09007e6	+	1.88805e6i	0.231292	+	0.400610i	0.958189	−	0.286137i	$-0.0923712\pi$
$467$	1.09007e6	+	1.88805e6i	0.231292	+	0.400610i	−0.726896	+	0.686747i	$0.759038\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	1.98131e6	+	3.43174e6i	0.411530	+	0.712790i
$472$		0			0
$473$	1.55179e6	−	2.68778e6i	0.318918	−	0.552383i
$474$		0			0
$475$	1.57770e6			0.320840
$476$		0			0
$477$	3.04970e6			0.613707
$478$		0			0
$479$	1.96808e6	−	3.40881e6i	0.391926	−	0.678835i	−0.600778	−	0.799416i	$-0.705142\pi$
$479$	1.96808e6	−	3.40881e6i	0.391926	−	0.678835i	0.992704	+	0.120581i	$0.0384757\pi$
$480$		0			0
$481$	−1.83802e6	−	3.18355e6i	−0.362233	−	0.627406i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1.48701e6	−	2.57558e6i	−0.287052	−	0.497189i
$486$		0			0
$487$	−3.50088e6	+	6.06370e6i	−0.668890	+	1.15855i	0.309325	+	0.950956i	$0.399897\pi$
$487$	−3.50088e6	+	6.06370e6i	−0.668890	+	1.15855i	−0.978215	+	0.207595i	$0.933436\pi$
$488$		0			0
$489$	508428.			0.0961517
$490$		0			0
$491$	−9.37114e6			−1.75424			−0.877119	−	0.480273i	$-0.840537\pi$
$491$	−9.37114e6			−1.75424			−0.877119	+	0.480273i	$0.840537\pi$
$492$		0			0
$493$	−4.66689e6	+	8.08329e6i	−0.864789	+	1.49786i
$494$		0			0
$495$	−337699.	−	584912.i	−0.0619466	−	0.107295i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	4.87966e6	+	8.45182e6i	0.877280	+	1.51949i	0.854314	+	0.519758i	$0.173978\pi$
$499$	4.87966e6	+	8.45182e6i	0.877280	+	1.51949i	0.0229668	+	0.999736i	$0.492689\pi$
$500$		0			0
$501$	−3.09606e6	+	5.36254e6i	−0.551081	+	0.954501i
$502$		0			0
$503$	1.69667e6			0.299004			0.149502	−	0.988761i	$-0.452233\pi$
$503$	1.69667e6			0.299004			0.149502	+	0.988761i	$0.452233\pi$
$504$		0			0
$505$	−3.90895e6			−0.682075
$506$		0			0
$507$	−995398.	+	1.72408e6i	−0.171980	+	0.297877i
$508$		0			0
$509$	−1.86783e6	−	3.23518e6i	−0.319553	−	0.553482i	0.660842	−	0.750525i	$-0.270199\pi$
$509$	−1.86783e6	−	3.23518e6i	−0.319553	−	0.553482i	−0.980395	+	0.197043i	$0.936866\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	273393.	+	473530.i	0.0458663	+	0.0794427i
$514$		0			0
$515$	1.19538e6	−	2.07047e6i	0.198605	−	0.343993i
$516$		0			0
$517$	4.54668e6			0.748114
$518$		0			0
$519$	−2.54302e6			−0.414412
$520$		0			0
$521$	3.66359e6	−	6.34552e6i	0.591306	−	1.02417i	−0.402750	−	0.915310i	$-0.631946\pi$
$521$	3.66359e6	−	6.34552e6i	0.591306	−	1.02417i	0.994057	−	0.108863i	$-0.0347209\pi$
$522$		0			0
$523$	−4795.71	−	8306.42i	−0.000766653	−	0.00132788i	0.865642	−	0.500664i	$-0.166911\pi$
$523$	−4795.71	−	8306.42i	−0.000766653	−	0.00132788i	−0.866408	+	0.499336i	$0.833577\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−4.98618e6	−	8.63632e6i	−0.782063	−	1.35457i
$528$		0			0
$529$	2.93327e6	−	5.08057e6i	0.455735	−	0.789357i
$530$		0			0
$531$	−1.78832e6			−0.275239
$532$		0			0
$533$	−4.17448e6			−0.636479
$534$		0			0
$535$	−1.13100e6	+	1.95895e6i	−0.170836	+	0.295896i
$536$		0			0
$537$	−683018.	−	1.18302e6i	−0.102211	−	0.177034i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	1.87846e6	+	3.25359e6i	0.275936	+	0.477935i	0.970371	−	0.241620i	$-0.0776788\pi$
$541$	1.87846e6	+	3.25359e6i	0.275936	+	0.477935i	−0.694435	+	0.719556i	$0.744346\pi$
$542$		0			0
$543$	−1.45016e6	+	2.51175e6i	−0.211065	+	0.365575i
$544$		0			0
$545$	−4.54692e6			−0.655731
$546$		0			0
$547$	1.03391e7			1.47746			0.738729	−	0.674002i	$-0.235426\pi$
$547$	1.03391e7			1.47746			0.738729	+	0.674002i	$0.235426\pi$
$548$		0			0
$549$	−331022.	+	573347.i	−0.0468734	+	0.0811870i
$550$		0			0
$551$	2.25350e6	+	3.90317e6i	0.316212	+	0.547694i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	686879.	+	1.18971e6i	0.0946560	+	0.163949i
$556$		0			0
$557$	5.02378e6	−	8.70144e6i	0.686108	−	1.18837i	−0.286979	−	0.957937i	$-0.592651\pi$
$557$	5.02378e6	−	8.70144e6i	0.686108	−	1.18837i	0.973087	−	0.230437i	$-0.0740156\pi$
$558$		0			0
$559$	9.15708e6			1.23945
$560$		0			0
$561$	3.64712e6			0.489263
$562$		0			0
$563$	−4.24690e6	+	7.35585e6i	−0.564678	+	0.978052i	0.432401	+	0.901681i	$0.357666\pi$
$563$	−4.24690e6	+	7.35585e6i	−0.564678	+	0.978052i	−0.997080	+	0.0763704i	$0.975667\pi$
$564$		0			0
$565$	−3.17856e6	−	5.50543e6i	−0.418899	−	0.725555i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1.90765e6	−	3.30415e6i	−0.247012	−	0.427838i	0.715683	−	0.698425i	$-0.246115\pi$
$569$	−1.90765e6	−	3.30415e6i	−0.247012	−	0.427838i	−0.962695	+	0.270587i	$0.912782\pi$
$570$		0			0
$571$	−6.10664e6	+	1.05770e7i	−0.783812	+	1.35760i	0.145894	+	0.989300i	$0.453394\pi$
$571$	−6.10664e6	+	1.05770e7i	−0.783812	+	1.35760i	−0.929706	+	0.368302i	$0.879939\pi$
$572$		0			0
$573$	4.84669e6			0.616678
$574$		0			0
$575$	−1.58780e6			−0.200275
$576$		0			0
$577$	−1.14735e6	+	1.98727e6i	−0.143469	+	0.248495i	−0.928801	−	0.370580i	$-0.879159\pi$
$577$	−1.14735e6	+	1.98727e6i	−0.143469	+	0.248495i	0.785332	+	0.619075i	$0.212492\pi$
$578$		0			0
$579$	−1.36548e6	−	2.36508e6i	−0.169273	−	0.293190i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−4.91122e6	−	8.50648e6i	−0.598436	−	1.03652i
$584$		0			0
$585$	996380.	−	1.72578e6i	0.120375	−	0.208495i
$586$		0			0
$587$	−7.53391e6			−0.902454			−0.451227	−	0.892409i	$-0.649014\pi$
$587$	−7.53391e6			−0.902454			−0.451227	+	0.892409i	$0.649014\pi$
$588$		0			0
$589$	−4.81534e6			−0.571925
$590$		0			0
$591$	−2.95414e6	+	5.11671e6i	−0.347906	+	0.602591i
$592$		0			0
$593$	3.61760e6	+	6.26588e6i	0.422459	+	0.731720i	0.996179	−	0.0873310i	$-0.0278338\pi$
$593$	3.61760e6	+	6.26588e6i	0.422459	+	0.731720i	−0.573721	+	0.819051i	$0.694500\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	3.78992e6	+	6.56433e6i	0.435205	+	0.753797i
$598$		0			0
$599$	−1.66928e6	+	2.89128e6i	−0.190091	+	0.329247i	−0.945280	−	0.326259i	$-0.894212\pi$
$599$	−1.66928e6	+	2.89128e6i	−0.190091	+	0.329247i	0.755189	+	0.655507i	$0.227545\pi$
$600$		0			0
$601$	−1.48297e7			−1.67474			−0.837370	−	0.546636i	$-0.815908\pi$
$601$	−1.48297e7			−1.67474			−0.837370	+	0.546636i	$0.815908\pi$
$602$		0			0
$603$	1.05314e6			0.117949
$604$		0			0
$605$	1.48607e6	−	2.57394e6i	0.165063	−	0.285897i
$606$		0			0
$607$	−3.18737e6	−	5.52068e6i	−0.351124	−	0.608164i	0.635323	−	0.772247i	$-0.280867\pi$
$607$	−3.18737e6	−	5.52068e6i	−0.351124	−	0.608164i	−0.986447	+	0.164083i	$0.947534\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	6.70747e6	+	1.16177e7i	0.726869	+	1.25897i
$612$		0			0
$613$	7.74344e6	−	1.34120e7i	0.832306	−	1.44160i	−0.0638996	−	0.997956i	$-0.520354\pi$
$613$	7.74344e6	−	1.34120e7i	0.832306	−	1.44160i	0.896205	−	0.443639i	$-0.146313\pi$
$614$		0			0
$615$	1.56003e6			0.166320
$616$		0			0
$617$	−4.55324e6			−0.481513			−0.240756	−	0.970586i	$-0.577396\pi$
$617$	−4.55324e6			−0.481513			−0.240756	+	0.970586i	$0.577396\pi$
$618$		0			0
$619$	−5.82108e6	+	1.00824e7i	−0.610628	+	1.05764i	0.380507	+	0.924778i	$0.375750\pi$
$619$	−5.82108e6	+	1.00824e7i	−0.610628	+	1.05764i	−0.991135	+	0.132861i	$0.957584\pi$
$620$		0			0
$621$	−275144.	−	476564.i	−0.0286307	−	0.0495898i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−616103.	−	1.06712e6i	−0.0630890	−	0.109273i
$626$		0			0
$627$	880539.	−	1.52514e6i	0.0894499	−	0.154932i
$628$		0			0
$629$	−7.41822e6			−0.747607
$630$		0			0
$631$	1.07436e7			1.07417			0.537087	−	0.843527i	$-0.319525\pi$
$631$	1.07436e7			1.07417			0.537087	+	0.843527i	$0.319525\pi$
$632$		0			0
$633$	−4.55097e6	+	7.88252e6i	−0.451435	+	0.781908i
$634$		0			0
$635$	−3.50969e6	−	6.07896e6i	−0.345409	−	0.598267i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	500160.	+	866302.i	0.0484570	+	0.0839300i
$640$		0			0
$641$	−6.15689e6	+	1.06641e7i	−0.591857	+	1.02513i	0.402126	+	0.915585i	$0.368272\pi$
$641$	−6.15689e6	+	1.06641e7i	−0.591857	+	1.02513i	−0.993982	+	0.109541i	$0.965062\pi$
$642$		0			0
$643$	1.67944e7			1.60191			0.800955	−	0.598724i	$-0.204325\pi$
$643$	1.67944e7			1.60191			0.800955	+	0.598724i	$0.204325\pi$
$644$		0			0
$645$	−3.42205e6			−0.323882
$646$		0			0
$647$	9.95539e6	−	1.72432e7i	0.934970	−	1.61942i	0.160283	−	0.987071i	$-0.448759\pi$
$647$	9.95539e6	−	1.72432e7i	0.934970	−	1.61942i	0.774687	−	0.632345i	$-0.217907\pi$
$648$		0			0
$649$	2.87990e6	+	4.98814e6i	0.268390	+	0.464865i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.05495e7	−	1.82723e7i	−0.968163	−	1.67691i	−0.700864	−	0.713295i	$-0.747202\pi$
$653$	−1.05495e7	−	1.82723e7i	−0.968163	−	1.67691i	−0.267299	−	0.963614i	$-0.586131\pi$
$654$		0			0
$655$	872905.	−	1.51192e6i	0.0794994	−	0.137697i
$656$		0			0
$657$	−3.53163e6			−0.319199
$658$		0			0
$659$	1.75070e7			1.57035			0.785177	−	0.619271i	$-0.212572\pi$
$659$	1.75070e7			1.57035			0.785177	+	0.619271i	$0.212572\pi$
$660$		0			0
$661$	9.00844e6	−	1.56031e7i	0.801947	−	1.38901i	−0.116385	−	0.993204i	$-0.537131\pi$
$661$	9.00844e6	−	1.56031e7i	0.801947	−	1.38901i	0.918333	−	0.395810i	$-0.129536\pi$
$662$		0			0
$663$	5.38040e6	+	9.31912e6i	0.475369	+	0.823363i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−2.26793e6	−	3.92818e6i	−0.197386	−	0.341882i
$668$		0			0
$669$	4.30706e6	−	7.46004e6i	0.372062	−	0.644431i
$670$		0			0
$671$	2.13230e6			0.182828
$672$		0			0
$673$	1.48593e7			1.26462			0.632312	−	0.774714i	$-0.282106\pi$
$673$	1.48593e7			1.26462			0.632312	+	0.774714i	$0.282106\pi$
$674$		0			0
$675$	766710.	−	1.32798e6i	0.0647697	−	0.112184i
$676$		0			0
$677$	3.59160e6	+	6.22084e6i	0.301173	+	0.521648i	0.976402	−	0.215961i	$-0.0692884\pi$
$677$	3.59160e6	+	6.22084e6i	0.301173	+	0.521648i	−0.675229	+	0.737609i	$0.735955\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	2.12166e6	+	3.67482e6i	0.175310	+	0.303647i
$682$		0			0
$683$	−825461.	+	1.42974e6i	−0.0677087	+	0.117275i	−0.897892	−	0.440215i	$-0.854902\pi$
$683$	−825461.	+	1.42974e6i	−0.0677087	+	0.117275i	0.830184	+	0.557490i	$0.188236\pi$
$684$		0			0
$685$	2.33137e6			0.189839
$686$		0			0
$687$	2.41377e6			0.195121
$688$		0			0
$689$	1.44905e7	−	2.50983e7i	1.16288	−	2.01417i
$690$		0			0
$691$	388722.	+	673286.i	0.0309702	+	0.0536419i	0.881095	−	0.472939i	$-0.156807\pi$
$691$	388722.	+	673286.i	0.0309702	+	0.0536419i	−0.850125	+	0.526581i	$0.823474\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.76856e6	−	3.06324e6i	−0.138886	−	0.240558i
$696$		0			0
$697$	−4.21203e6	+	7.29545e6i	−0.328405	+	0.568814i
$698$		0			0
$699$	1.39250e7			1.07796
$700$		0			0
$701$	1.65671e6			0.127336			0.0636681	−	0.997971i	$-0.479720\pi$
$701$	1.65671e6			0.127336			0.0636681	+	0.997971i	$0.479720\pi$
$702$		0			0
$703$	−1.79101e6	+	3.10213e6i	−0.136682	+	0.236740i
$704$		0			0
$705$	−2.50662e6	−	4.34159e6i	−0.189940	−	0.328985i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−2.49950e6	−	4.32927e6i	−0.186740	−	0.323444i	0.757421	−	0.652927i	$-0.226459\pi$
$709$	−2.49950e6	−	4.32927e6i	−0.186740	−	0.323444i	−0.944162	+	0.329483i	$0.893126\pi$
$710$		0			0
$711$	−3.10835e6	+	5.38383e6i	−0.230599	+	0.399408i
$712$		0			0
$713$	4.84619e6			0.357007
$714$		0			0
$715$	−6.41825e6			−0.469517
$716$		0			0
$717$	4.21894e6	−	7.30742e6i	0.306482	−	0.530843i
$718$		0			0
$719$	−9.06918e6	−	1.57083e7i	−0.654253	−	1.13320i	−0.982081	−	0.188462i	$-0.939650\pi$
$719$	−9.06918e6	−	1.57083e7i	−0.654253	−	1.13320i	0.327828	−	0.944738i	$-0.393683\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−5.78187e6	−	1.00145e7i	−0.411360	−	0.712497i
$724$		0			0
$725$	6.31976e6	−	1.09461e7i	0.446535	−	0.773422i
$726$		0			0
$727$	−1.17918e7			−0.827454			−0.413727	−	0.910401i	$-0.635773\pi$
$727$	−1.17918e7			−0.827454			−0.413727	+	0.910401i	$0.635773\pi$
$728$		0			0
$729$	531441.			0.0370370
$730$		0			0
$731$	9.23945e6	−	1.60032e7i	0.639518	−	1.10768i
$732$		0			0
$733$	−2.48235e6	−	4.29955e6i	−0.170648	−	0.295572i	0.767998	−	0.640452i	$-0.221253\pi$
$733$	−2.48235e6	−	4.29955e6i	−0.170648	−	0.295572i	−0.938647	+	0.344880i	$0.887920\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1.69597e6	−	2.93751e6i	−0.115014	−	0.199210i
$738$		0			0
$739$	−9.31461e6	+	1.61334e7i	−0.627413	+	1.08671i	0.360656	+	0.932699i	$0.382553\pi$
$739$	−9.31461e6	+	1.61334e7i	−0.627413	+	1.08671i	−0.988069	+	0.154013i	$0.950780\pi$
$740$		0			0
$741$	5.19605e6			0.347639
$742$		0			0
$743$	1.77727e7			1.18109			0.590544	−	0.807006i	$-0.298913\pi$
$743$	1.77727e7			1.18109			0.590544	+	0.807006i	$0.298913\pi$
$744$		0			0
$745$	−5.22613e6	+	9.05192e6i	−0.344976	+	0.597516i
$746$		0			0
$747$	886690.	+	1.53579e6i	0.0581394	+	0.100700i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−2.33030e6	−	4.03619e6i	−0.150769	−	0.261139i	0.780742	−	0.624854i	$-0.214842\pi$
$751$	−2.33030e6	−	4.03619e6i	−0.150769	−	0.261139i	−0.931510	+	0.363715i	$0.881508\pi$
$752$		0			0
$753$	3.19396e6	−	5.53210e6i	0.205278	−	0.355552i
$754$		0			0
$755$	5.57699e6			0.356067
$756$		0			0
$757$	−2.36520e6			−0.150013			−0.0750064	−	0.997183i	$-0.523898\pi$
$757$	−2.36520e6			−0.150013			−0.0750064	+	0.997183i	$0.523898\pi$
$758$		0			0
$759$	−886181.	+	1.53491e6i	−0.0558365	+	0.0967116i
$760$		0			0
$761$	−8.84728e6	−	1.53239e7i	−0.553794	−	0.959199i	−0.997996	−	0.0632726i	$-0.979846\pi$
$761$	−8.84728e6	−	1.53239e7i	−0.553794	−	0.959199i	0.444202	−	0.895926i	$-0.353487\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−2.01068e6	−	3.48261e6i	−0.124220	−	0.215155i
$766$		0			0
$767$	−8.49714e6	+	1.47175e7i	−0.521536	+	0.903327i
$768$		0			0
$769$	7.90324e6			0.481936			0.240968	−	0.970533i	$-0.422535\pi$
$769$	7.90324e6			0.481936			0.240968	+	0.970533i	$0.422535\pi$
$770$		0			0
$771$	1.63814e7			0.992468
$772$		0			0
$773$	1.11230e7	−	1.92656e7i	0.669534	−	1.15967i	−0.308501	−	0.951224i	$-0.599827\pi$
$773$	1.11230e7	−	1.92656e7i	0.669534	−	1.15967i	0.978035	−	0.208442i	$-0.0668393\pi$
$774$		0			0
$775$	6.75214e6	+	1.16950e7i	0.403819	+	0.699436i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	2.03386e6	+	3.52274e6i	0.120082	+	0.207988i
$780$		0			0
$781$	1.61091e6	−	2.79017e6i	0.0945024	−	0.163683i
$782$		0			0
$783$	4.38051e6			0.255341
$784$		0			0
$785$	−1.40724e7			−0.815070

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)

\(f(q)\)	\(=\)	\(q+(-4.50000 + 7.79423i) q^{3} +(-15.9808 - 27.6796i) q^{5} +(-40.5000 - 70.1481i) q^{9} +O(q^{10})\)	\(q+(-4.50000 + 7.79423i) q^{3} +(-15.9808 - 27.6796i) q^{5} +(-40.5000 - 70.1481i) q^{9} +(-130.442 + 225.932i) q^{11} -769.735 q^{13} +287.654 q^{15} +(-776.659 + 1345.21i) q^{17} +(375.024 + 649.561i) q^{19} +(-377.427 - 653.723i) q^{23} +(1051.73 - 1821.65i) q^{25} +729.000 q^{27} +6008.93 q^{29} +(-3210.02 + 5559.92i) q^{31} +(-1173.98 - 2033.39i) q^{33} +(2387.86 + 4135.90i) q^{37} +(3463.81 - 5999.49i) q^{39} +5423.27 q^{41} -11896.4 q^{43} +(-1294.44 + 2242.04i) q^{45} +(-8714.00 - 15093.1i) q^{47} +(-6989.93 - 12106.9i) q^{51} +(-18825.3 + 32606.4i) q^{53} +8338.26 q^{55} -6750.44 q^{57} +(11039.0 - 19120.2i) q^{59} +(-4086.69 - 7078.36i) q^{61} +(12301.0 + 21305.9i) q^{65} +(-6500.87 + 11259.8i) q^{67} +6793.69 q^{69} -12349.6 q^{71} +(21800.2 - 37759.0i) q^{73} +(9465.55 + 16394.8i) q^{75} +(-38374.7 - 66467.0i) q^{79} +(-3280.50 + 5681.99i) q^{81} -21893.6 q^{83} +49646.5 q^{85} +(-27040.2 + 46835.0i) q^{87} +(-68483.5 - 118617. i) q^{89} +(-28890.2 - 50039.3i) q^{93} +(11986.4 - 20761.0i) q^{95} +93050.1 q^{97} +21131.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 36 q^{3} - 324 q^{9}+O(q^{10}) \) 8 * q - 36 * q^3 - 324 * q^9	\( 8 q - 36 q^{3} - 324 q^{9} - 462 q^{11} + 1204 q^{13} - 228 q^{17} - 358 q^{19} - 2148 q^{23} - 5454 q^{25} + 5832 q^{27} - 11064 q^{29} - 830 q^{31} - 4158 q^{33} - 3914 q^{37} - 5418 q^{39} + 16632 q^{41} - 29036 q^{43} - 41700 q^{47} - 2052 q^{51} + 22164 q^{53} - 7784 q^{55} + 6444 q^{57} - 32886 q^{59} - 83732 q^{61} - 93192 q^{65} - 80034 q^{67} + 38664 q^{69} + 89544 q^{71} + 22470 q^{73} - 49086 q^{75} - 75286 q^{79} - 26244 q^{81} + 34836 q^{83} + 278504 q^{85} + 49788 q^{87} - 28944 q^{89} - 7470 q^{93} - 144120 q^{95} + 433356 q^{97} + 74844 q^{99}+O(q^{100}) \) 8 * q - 36 * q^3 - 324 * q^9 - 462 * q^11 + 1204 * q^13 - 228 * q^17 - 358 * q^19 - 2148 * q^23 - 5454 * q^25 + 5832 * q^27 - 11064 * q^29 - 830 * q^31 - 4158 * q^33 - 3914 * q^37 - 5418 * q^39 + 16632 * q^41 - 29036 * q^43 - 41700 * q^47 - 2052 * q^51 + 22164 * q^53 - 7784 * q^55 + 6444 * q^57 - 32886 * q^59 - 83732 * q^61 - 93192 * q^65 - 80034 * q^67 + 38664 * q^69 + 89544 * q^71 + 22470 * q^73 - 49086 * q^75 - 75286 * q^79 - 26244 * q^81 + 34836 * q^83 + 278504 * q^85 + 49788 * q^87 - 28944 * q^89 - 7470 * q^93 - 144120 * q^95 + 433356 * q^97 + 74844 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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