LMFDB - Embedded newform 84.6.i.c.37.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 0, 4]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("84.25");

S:= CuspForms(chi, 6);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$13.4722408643$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 703x^{6} + 2770x^{5} + 427565x^{4} + 718170x^{3} + 42175732x^{2} - 40929504x + 3559792896 $ x^8 - 2x^7 + 703x^6 + 2770x^5 + 427565x^4 + 718170x^3 + 42175732x^2 - 40929504*x + 3559792896
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{3}\cdot 7 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			37.3
Root			$13.1471 + 22.7714i$ of defining polynomial
Character	$\chi$	$=$	84.37
Dual form			84.6.i.c.25.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.50000 + 7.79423i) q^{3} +(15.9808 - 27.6796i) q^{5} +(85.7043 + 97.2716i) q^{7} +(-40.5000 + 70.1481i) q^{9} +O(q^{10})$	\(q+(4.50000 + 7.79423i) q^{3} +(15.9808 - 27.6796i) q^{5} +(85.7043 + 97.2716i) q^{7} +(-40.5000 + 70.1481i) q^{9} +(-130.442 - 225.932i) q^{11} +769.735 q^{13} +287.654 q^{15} +(776.659 + 1345.21i) q^{17} +(-375.024 + 649.561i) q^{19} +(-372.488 + 1105.72i) q^{21} +(-377.427 + 653.723i) q^{23} +(1051.73 + 1821.65i) q^{25} -729.000 q^{27} +6008.93 q^{29} +(3210.02 + 5559.92i) q^{31} +(1173.98 - 2033.39i) q^{33} +(4062.06 - 817.779i) q^{35} +(2387.86 - 4135.90i) q^{37} +(3463.81 + 5999.49i) q^{39} -5423.27 q^{41} -11896.4 q^{43} +(1294.44 + 2242.04i) q^{45} +(8714.00 - 15093.1i) q^{47} +(-2116.54 + 16673.2i) q^{49} +(-6989.93 + 12106.9i) q^{51} +(-18825.3 - 32606.4i) q^{53} -8338.26 q^{55} -6750.44 q^{57} +(-11039.0 - 19120.2i) q^{59} +(4086.69 - 7078.36i) q^{61} +(-10294.4 + 2072.49i) q^{63} +(12301.0 - 21305.9i) q^{65} +(-6500.87 - 11259.8i) q^{67} -6793.69 q^{69} -12349.6 q^{71} +(-21800.2 - 37759.0i) q^{73} +(-9465.55 + 16394.8i) q^{75} +(10797.3 - 32051.6i) q^{77} +(-38374.7 + 66467.0i) q^{79} +(-3280.50 - 5681.99i) q^{81} +21893.6 q^{83} +49646.5 q^{85} +(27040.2 + 46835.0i) q^{87} +(68483.5 - 118617. i) q^{89} +(65969.7 + 74873.4i) q^{91} +(-28890.2 + 50039.3i) q^{93} +(11986.4 + 20761.0i) q^{95} -93050.1 q^{97} +21131.6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 36 q^{3} - 42 q^{7} - 324 q^{9}+O(q^{10}) $ 8 * q + 36 * q^3 - 42 * q^7 - 324 * q^9	$ 8 q + 36 q^{3} - 42 q^{7} - 324 q^{9} - 462 q^{11} - 1204 q^{13} + 228 q^{17} + 358 q^{19} + 1404 q^{21} - 2148 q^{23} - 5454 q^{25} - 5832 q^{27} - 11064 q^{29} + 830 q^{31} + 4158 q^{33} + 7692 q^{35} - 3914 q^{37} - 5418 q^{39} - 16632 q^{41} - 29036 q^{43} + 41700 q^{47} + 41876 q^{49} - 2052 q^{51} + 22164 q^{53} + 7784 q^{55} + 6444 q^{57} + 32886 q^{59} + 83732 q^{61} + 16038 q^{63} - 93192 q^{65} - 80034 q^{67} - 38664 q^{69} + 89544 q^{71} - 22470 q^{73} + 49086 q^{75} + 138732 q^{77} - 75286 q^{79} - 26244 q^{81} - 34836 q^{83} + 278504 q^{85} - 49788 q^{87} + 28944 q^{89} - 12058 q^{91} - 7470 q^{93} - 144120 q^{95} - 433356 q^{97} + 74844 q^{99}+O(q^{100}) $ 8 * q + 36 * q^3 - 42 * q^7 - 324 * q^9 - 462 * q^11 - 1204 * q^13 + 228 * q^17 + 358 * q^19 + 1404 * q^21 - 2148 * q^23 - 5454 * q^25 - 5832 * q^27 - 11064 * q^29 + 830 * q^31 + 4158 * q^33 + 7692 * q^35 - 3914 * q^37 - 5418 * q^39 - 16632 * q^41 - 29036 * q^43 + 41700 * q^47 + 41876 * q^49 - 2052 * q^51 + 22164 * q^53 + 7784 * q^55 + 6444 * q^57 + 32886 * q^59 + 83732 * q^61 + 16038 * q^63 - 93192 * q^65 - 80034 * q^67 - 38664 * q^69 + 89544 * q^71 - 22470 * q^73 + 49086 * q^75 + 138732 * q^77 - 75286 * q^79 - 26244 * q^81 - 34836 * q^83 + 278504 * q^85 - 49788 * q^87 + 28944 * q^89 - 12058 * q^91 - 7470 * q^93 - 144120 * q^95 - 433356 * q^97 + 74844 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	4.50000	+	7.79423i	0.288675	+	0.500000i
$3$	4.50000	+	7.79423i	0.288675	+	0.500000i
$4$		0			0
$5$	15.9808	−	27.6796i	0.285873	−	0.495147i	−0.686947	−	0.726707i	$-0.741050\pi$
$5$	15.9808	−	27.6796i	0.285873	−	0.495147i	0.972821	+	0.231560i	$0.0743831\pi$
$6$		0			0
$7$	85.7043	+	97.2716i	0.661086	+	0.750311i
$7$	85.7043	+	97.2716i	0.661086	+	0.750311i
$8$		0			0
$9$	−40.5000	+	70.1481i	−0.166667	+	0.288675i
$10$		0			0
$11$	−130.442	−	225.932i	−0.325039	−	0.562984i	0.656481	−	0.754342i	$-0.272044\pi$
$11$	−130.442	−	225.932i	−0.325039	−	0.562984i	−0.981520	+	0.191359i	$0.938711\pi$
$12$		0			0
$13$	769.735			1.26323			0.631616	−	0.775281i	$-0.282392\pi$
$13$	769.735			1.26323			0.631616	+	0.775281i	$0.282392\pi$
$14$		0			0
$15$	287.654			0.330098
$16$		0			0
$17$	776.659	+	1345.21i	0.651791	+	1.12893i	0.982688	+	0.185268i	$0.0593152\pi$
$17$	776.659	+	1345.21i	0.651791	+	1.12893i	−0.330898	+	0.943667i	$0.607351\pi$
$18$		0			0
$19$	−375.024	+	649.561i	−0.238328	+	0.412797i	−0.960235	−	0.279194i	$-0.909933\pi$
$19$	−375.024	+	649.561i	−0.238328	+	0.412797i	0.721906	+	0.691991i	$0.243266\pi$
$20$		0			0
$21$	−372.488	+	1105.72i	−0.184316	+	0.547139i
$22$		0			0
$23$	−377.427	+	653.723i	−0.148769	+	0.257676i	−0.930773	−	0.365598i	$-0.880865\pi$
$23$	−377.427	+	653.723i	−0.148769	+	0.257676i	0.782004	+	0.623274i	$0.214198\pi$
$24$		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	0.992830	+	0.119533i	$0.0381399\pi$
$31$	3210.02	+	5559.92i	0.599934	+	1.03912i	−0.392896	+	0.919583i	$0.628527\pi$
$32$		0			0
$33$	1173.98	−	2033.39i	0.187661	−	0.325039i
$34$		0			0
$35$	4062.06	−	817.779i	0.560501	−	0.112841i
$36$		0			0
$37$	2387.86	−	4135.90i	0.286751	−	0.496668i	−0.686281	−	0.727336i	$-0.740758\pi$
$37$	2387.86	−	4135.90i	0.286751	−	0.496668i	0.973032	+	0.230669i	$0.0740914\pi$
$38$		0			0
$39$	3463.81	+	5999.49i	0.364664	+	0.631616i
$40$		0			0
$41$	−5423.27			−0.503850			−0.251925	−	0.967747i	$-0.581064\pi$
$41$	−5423.27			−0.503850			−0.251925	+	0.967747i	$0.581064\pi$
$42$		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	−0.420594	−	0.907249i	$-0.638178\pi$
$47$	8714.00	−	15093.1i	0.575404	−	0.996629i	0.995998	−	0.0893798i	$-0.0284885\pi$
$48$		0			0
$49$	−2116.54	+	16673.2i	−0.125932	+	0.992039i
$50$		0			0
$51$	−6989.93	+	12106.9i	−0.376311	+	0.651791i
$52$		0			0
$53$	−18825.3	−	32606.4i	−0.920561	−	1.59446i	−0.798549	−	0.601930i	$-0.794399\pi$
$53$	−18825.3	−	32606.4i	−0.920561	−	1.59446i	−0.122012	−	0.992529i	$-0.538935\pi$
$54$		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	$-0.302136\pi$
$59$	−11039.0	−	19120.2i	−0.412859	−	0.715092i	−0.995201	+	0.0978516i	$0.968803\pi$
$60$		0			0
$61$	4086.69	−	7078.36i	0.140620	−	0.243561i	−0.787110	−	0.616812i	$-0.788424\pi$
$61$	4086.69	−	7078.36i	0.140620	−	0.243561i	0.927730	+	0.373251i	$0.121757\pi$
$62$		0			0
$63$	−10294.4	+	2072.49i	−0.326777	+	0.0657872i
$64$		0			0
$65$	12301.0	−	21305.9i	0.361124	−	0.625485i
$66$		0			0
$67$	−6500.87	−	11259.8i	−0.176923	−	0.306440i	0.763902	−	0.645332i	$-0.223281\pi$
$67$	−6500.87	−	11259.8i	−0.176923	−	0.306440i	−0.940825	+	0.338892i	$0.889948\pi$
$68$		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	$-0.325594\pi$
$73$	−21800.2	−	37759.0i	−0.478798	−	0.829303i	−0.999704	+	0.0243110i	$0.992261\pi$
$74$		0			0
$75$	−9465.55	+	16394.8i	−0.194309	+	0.336553i
$76$		0			0
$77$	10797.3	−	32051.6i	0.207534	−	0.616060i
$78$		0			0
$79$	−38374.7	+	66467.0i	−0.691796	+	1.19823i	0.279453	+	0.960159i	$0.409847\pi$
$79$	−38374.7	+	66467.0i	−0.691796	+	1.19823i	−0.971249	+	0.238066i	$0.923487\pi$
$80$		0			0
$81$	−3280.50	−	5681.99i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	21893.6			0.348836			0.174418	−	0.984672i	$-0.444196\pi$
$83$	21893.6			0.348836			0.174418	+	0.984672i	$0.444196\pi$
$84$		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	0.111696	−	0.993742i	$-0.464372\pi$
$89$	68483.5	−	118617.i	0.916454	−	1.58735i	0.804758	−	0.593603i	$-0.202295\pi$
$90$		0			0
$91$	65969.7	+	74873.4i	0.835104	+	0.947816i
$92$		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	$-0.667425\pi$
$97$	−93050.1			−1.00412			−0.502062	+	0.864832i	$0.667425\pi$
$98$		0			0
$99$	21131.6			0.216692
$100$		0			0
$101$	−61150.8	−	105916.i	−0.596484	−	1.03314i	−0.993336	−	0.115258i	$-0.963231\pi$
$101$	−61150.8	−	105916.i	−0.596484	−	1.03314i	0.396852	−	0.917883i	$-0.370103\pi$
$102$		0			0
$103$	−37400.6	+	64779.8i	−0.347365	+	0.601654i	−0.985781	−	0.168038i	$-0.946257\pi$
$103$	−37400.6	+	64779.8i	−0.347365	+	0.601654i	0.638415	+	0.769692i	$0.279590\pi$
$104$		0			0
$105$	24653.2	+	27980.6i	0.218223	+	0.247676i
$106$		0			0
$107$	−35386.2	+	61290.7i	−0.298796	+	0.517530i	−0.975861	−	0.218394i	$-0.929918\pi$
$107$	−35386.2	+	61290.7i	−0.298796	+	0.517530i	0.677065	+	0.735923i	$0.263252\pi$
$108$		0			0
$109$	71131.0	+	123202.i	0.573446	+	0.993238i	0.996209	+	0.0869972i	$0.0277271\pi$
$109$	71131.0	+	123202.i	0.573446	+	0.993238i	−0.422762	+	0.906241i	$0.638940\pi$
$110$		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
$119$	−64288.0	+	190837.i	−0.416162	+	1.23537i
$120$		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	$-0.877737\pi$
$131$	−27311.1	+	47304.1i	−0.139047	+	0.240836i	0.788089	+	0.615561i	$0.211070\pi$
$132$		0			0
$133$	−95325.1	+	19191.0i	−0.467281	+	0.0940737i
$134$		0			0
$135$	−11650.0	+	20178.4i	−0.0550163	+	0.0952911i
$136$		0			0
$137$	−36471.4	−	63170.4i	−0.166017	−	0.287549i	0.770999	−	0.636836i	$-0.219757\pi$
$137$	−36471.4	−	63170.4i	−0.166017	−	0.287549i	−0.937016	+	0.349287i	$0.886424\pi$
$138$		0			0
$139$	−110668.			−0.485831			−0.242915	−	0.970047i	$-0.578104\pi$
$139$	−110668.			−0.485831			−0.242915	+	0.970047i	$0.578104\pi$
$140$		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
$147$	−139479.	+	58532.6i	−0.532373	+	0.223411i
$148$		0			0
$149$	−163513.	+	283212.i	−0.603373	+	1.04507i	0.388933	+	0.921266i	$0.372844\pi$
$149$	−163513.	+	283212.i	−0.603373	+	1.04507i	−0.992306	+	0.123807i	$0.960490\pi$
$150$		0			0
$151$	−87245.1	−	151113.i	−0.311386	−	0.539336i	0.667277	−	0.744810i	$-0.267460\pi$
$151$	−87245.1	−	151113.i	−0.311386	−	0.539336i	−0.978663	+	0.205474i	$0.934126\pi$
$152$		0			0
$153$	−125819.			−0.434527
$154$		0			0
$155$	205195.			0.686020
$156$		0			0
$157$	−220146.	−	381304.i	−0.712790	−	1.23459i	−0.963806	−	0.266606i	$-0.914098\pi$
$157$	−220146.	−	381304.i	−0.712790	−	1.23459i	0.251016	−	0.967983i	$-0.419235\pi$
$158$		0			0
$159$	169428.	−	293458.i	0.531486	−	0.920561i
$160$		0			0
$161$	−95935.8	+	19313.9i	−0.291686	+	0.0587227i
$162$		0			0
$163$	−28246.0	+	48923.5i	−0.0832698	+	0.144228i	−0.904653	−	0.426150i	$-0.859870\pi$
$163$	−28246.0	+	48923.5i	−0.0832698	+	0.144228i	0.821383	+	0.570377i	$0.193203\pi$
$164$		0			0
$165$	−37522.2	−	64990.3i	−0.107295	−	0.185840i
$166$		0			0
$167$	−688014.			−1.90900			−0.954501	−	0.298209i	$-0.903611\pi$
$167$	−688014.			−1.90900			−0.954501	+	0.298209i	$0.903611\pi$
$168$		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	$-0.950178\pi$
$173$	−141279.	+	244703.i	−0.358891	+	0.621618i	0.628885	+	0.777499i	$0.283512\pi$
$174$		0			0
$175$	−87056.9	+	258426.i	−0.214886	+	0.637884i
$176$		0			0
$177$	99351.4	−	172082.i	0.238364	−	0.412859i
$178$		0			0
$179$	−75890.9	−	131447.i	−0.177034	−	0.306632i	0.763829	−	0.645419i	$-0.223317\pi$
$179$	−75890.9	−	131447.i	−0.177034	−	0.306632i	−0.940863	+	0.338786i	$0.889984\pi$
$180$		0			0
$181$	−322258.			−0.731151			−0.365575	−	0.930782i	$-0.619128\pi$
$181$	−322258.			−0.731151			−0.365575	+	0.930782i	$0.619128\pi$
$182$		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
$189$	−62478.5	−	70911.0i	−0.127226	−	0.144397i
$190$		0			0
$191$	−269260.	+	466373.i	−0.534059	+	0.925017i	0.465150	+	0.885232i	$0.346001\pi$
$191$	−269260.	+	466373.i	−0.534059	+	0.925017i	−0.999208	+	0.0397847i	$0.987333\pi$
$192$		0			0
$193$	−151720.	−	262786.i	−0.293190	−	0.507820i	0.681372	−	0.731937i	$-0.261383\pi$
$193$	−151720.	−	262786.i	−0.293190	−	0.507820i	−0.974562	+	0.224117i	$0.928050\pi$
$194$		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	−0.945970	−	0.324254i	$-0.894887\pi$
$199$	−421102.	−	729370.i	−0.753797	−	1.30562i	0.192173	−	0.981361i	$-0.438447\pi$
$200$		0			0
$201$	58507.8	−	101339.i	0.102147	−	0.176923i
$202$		0			0
$203$	514991.	+	584498.i	0.877122	+	0.995505i
$204$		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
$217$	−265710.	+	788753.i	−0.383052	+	1.13708i
$218$		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	$-0.277094\pi$
$223$	957124.			1.28886			0.644431	+	0.764663i	$0.277094\pi$
$224$		0			0
$225$	−170380.			−0.224369
$226$		0			0
$227$	−235740.	−	408313.i	−0.303647	−	0.525931i	0.673312	−	0.739358i	$-0.264871\pi$
$227$	−235740.	−	408313.i	−0.303647	−	0.525931i	−0.976959	+	0.213427i	$0.931538\pi$
$228$		0			0
$229$	134098.	−	232265.i	0.168980	−	0.292681i	−0.769082	−	0.639150i	$-0.779286\pi$
$229$	134098.	−	232265.i	0.168980	−	0.292681i	0.938061	+	0.346469i	$0.112619\pi$
$230$		0			0
$231$	298406.	−	60075.5i	0.367940	−	0.0740742i
$232$		0			0
$233$	−773610.	+	1.33993e6i	−0.933538	+	1.61694i	−0.156318	+	0.987707i	$0.549962\pi$
$233$	−773610.	+	1.33993e6i	−0.933538	+	1.61694i	−0.777220	+	0.629229i	$0.783371\pi$
$234$		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	0.963917	+	0.266203i	$0.0857692\pi$
$241$	642430.	+	1.11272e6i	0.712497	+	1.23408i	−0.251420	+	0.967878i	$0.580897\pi$
$242$		0			0
$243$	29524.5	−	51137.9i	0.0320750	−	0.0555556i
$244$		0			0
$245$	427683.	+	325036.i	0.455204	+	0.345952i
$246$		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	$-0.384293\pi$
$251$	709769.			0.711103			0.355552	+	0.934657i	$0.384293\pi$
$252$		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	−0.0129024	−	0.999917i	$-0.504107\pi$
$257$	910080.	−	1.57631e6i	0.859502	−	1.48870i	0.872405	−	0.488785i	$-0.162560\pi$
$258$		0			0
$259$	606956.	−	122193.i	0.562222	−	0.113187i
$260$		0			0
$261$	−243362.	+	421515.i	−0.221132	+	0.383011i
$262$		0			0
$263$	−179594.	−	311065.i	−0.160104	−	0.277308i	0.774802	−	0.632204i	$-0.217850\pi$
$263$	−179594.	−	311065.i	−0.160104	−	0.277308i	−0.934906	+	0.354896i	$0.884516\pi$
$264$		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	$-0.328839\pi$
$269$	−578834.	−	1.00257e6i	−0.487723	−	0.844761i	−0.999900	+	0.0141188i	$0.995506\pi$
$270$		0			0
$271$	390088.	−	675652.i	0.322655	−	0.558856i	−0.658380	−	0.752686i	$-0.728758\pi$
$271$	390088.	−	675652.i	0.322655	−	0.558856i	0.981035	+	0.193830i	$0.0620912\pi$
$272$		0			0
$273$	−286717.	+	851113.i	−0.232834	+	0.691163i
$274$		0			0
$275$	274379.	−	475238.i	0.218786	−	0.378948i
$276$		0			0
$277$	163165.	+	282611.i	0.127770	+	0.221304i	0.922812	−	0.385250i	$-0.125885\pi$
$277$	163165.	+	282611.i	0.127770	+	0.221304i	−0.795042	+	0.606554i	$0.792551\pi$
$278$		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	$-0.159103\pi$
$283$	32009.6	+	55442.3i	0.0237582	+	0.0411505i	−0.853902	+	0.520434i	$0.825770\pi$
$284$		0			0
$285$	−107877.	+	186849.i	−0.0786717	+	0.136263i
$286$		0			0
$287$	−464797.	−	527530.i	−0.333088	−	0.378044i
$288$		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	$-0.456759\pi$
$293$	398020.			0.270855			0.135427	+	0.990787i	$0.456759\pi$
$294$		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
$301$	−1.01957e6	−	1.15718e6i	−0.648638	−	0.736183i
$302$		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	$-0.561960\pi$
$307$	−638841.			−0.386854			−0.193427	+	0.981115i	$0.561960\pi$
$308$		0			0
$309$	−673212.			−0.401103
$310$		0			0
$311$	−1.24094e6	−	2.14937e6i	−0.727529	−	1.26012i	−0.957924	−	0.287020i	$-0.907335\pi$
$311$	−1.24094e6	−	2.14937e6i	−0.727529	−	1.26012i	0.230395	−	0.973097i	$-0.425998\pi$
$312$		0			0
$313$	−738267.	+	1.27872e6i	−0.425944	+	0.737757i	−0.996508	−	0.0834961i	$-0.973391\pi$
$313$	−738267.	+	1.27872e6i	−0.425944	+	0.737757i	0.570564	+	0.821253i	$0.306725\pi$
$314$		0			0
$315$	−107148.	+	318066.i	−0.0608424	+	0.180609i
$316$		0			0
$317$	1.11563e6	−	1.93233e6i	0.623551	−	1.08002i	−0.365269	−	0.930902i	$-0.619023\pi$
$317$	1.11563e6	−	1.93233e6i	0.623551	−	1.08002i	0.988819	−	0.149119i	$-0.0476438\pi$
$318$		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
$329$	2.21496e6	−	445918.i	1.12817	−	0.227125i
$330$		0			0
$331$	1.20561e6	−	2.08818e6i	0.604836	−	1.04761i	−0.387242	−	0.921978i	$-0.626572\pi$
$331$	1.20561e6	−	2.08818e6i	0.604836	−	1.04761i	0.992077	−	0.125628i	$-0.0400946\pi$
$332$		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
$343$	−1.80323e6	+	1.22309e6i	−0.827589	+	0.561335i
$344$		0			0
$345$	−108569.	+	188046.i	−0.0491085	+	0.0850583i
$346$		0			0
$347$	−1.10366e6	−	1.91159e6i	−0.492051	−	0.852258i	0.507907	−	0.861412i	$-0.330419\pi$
$347$	−1.10366e6	−	1.91159e6i	−0.492051	−	0.852258i	−0.999958	+	0.00915399i	$0.997086\pi$
$348$		0			0
$349$	148188.			0.0651252			0.0325626	−	0.999470i	$-0.489633\pi$
$349$	148188.			0.0651252			0.0325626	+	0.999470i	$0.489633\pi$
$350$		0			0
$351$	−561137.			−0.243109
$352$		0			0
$353$	−985200.	−	1.70642e6i	−0.420811	−	0.728867i	0.575208	−	0.818007i	$-0.304921\pi$
$353$	−985200.	−	1.70642e6i	−0.420811	−	0.728867i	−0.996019	+	0.0891407i	$0.971588\pi$
$354$		0			0
$355$	−197357.	+	341832.i	−0.0831154	+	0.143960i
$356$		0			0
$357$	−1.77673e6	+	357693.i	−0.737819	+	0.148539i
$358$		0			0
$359$	−1.21697e6	+	2.10786e6i	−0.498362	+	0.863188i	−0.999998	−	0.00189067i	$-0.999398\pi$
$359$	−1.21697e6	+	2.10786e6i	−0.498362	+	0.863188i	0.501636	+	0.865079i	$0.332732\pi$
$360$		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	0.894907	+	0.446253i	$0.147242\pi$
$367$	2.15174e6	+	3.72692e6i	0.833920	+	1.44439i	−0.0609873	+	0.998139i	$0.519425\pi$
$368$		0			0
$369$	219642.	−	380432.i	0.0839750	−	0.145449i
$370$		0			0
$371$	1.55827e6	−	4.62568e6i	0.587769	−	1.74478i
$372$		0			0
$373$	601526.	−	1.04187e6i	0.223863	−	0.387742i	−0.732115	−	0.681181i	$-0.761467\pi$
$373$	601526.	−	1.04187e6i	0.223863	−	0.387742i	0.955978	+	0.293439i	$0.0947999\pi$
$374$		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	−0.0134808	+	0.999909i	$0.504291\pi$
$383$	−2.50528e6	+	4.33926e6i	−0.872687	+	1.51154i	−0.859206	+	0.511629i	$0.829042\pi$
$384$		0			0
$385$	−714625.	−	811076.i	−0.245712	−	0.278875i
$386$		0			0
$387$	481804.	−	834510.i	0.163528	−	0.283240i
$388$		0			0
$389$	2.04163e6	+	3.53620e6i	0.684073	+	1.18485i	0.973727	+	0.227718i	$0.0731265\pi$
$389$	2.04163e6	+	3.53620e6i	0.684073	+	1.18485i	−0.289654	+	0.957131i	$0.593540\pi$
$390$		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	$-0.778685\pi$
$397$	536498.	−	929242.i	0.170841	−	0.295905i	0.938714	+	0.344697i	$0.112018\pi$
$398$		0			0
$399$	−578542.	−	656626.i	−0.181929	−	0.206484i
$400$		0			0
$401$	949741.	−	1.64500e6i	0.294947	−	0.510863i	−0.680026	−	0.733188i	$-0.738031\pi$
$401$	949741.	−	1.64500e6i	0.294947	−	0.510863i	0.974973	+	0.222325i	$0.0713647\pi$
$402$		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	$-0.259025\pi$
$409$	−967887.	−	1.67643e6i	−0.286099	−	0.495538i	−0.972875	+	0.231331i	$0.925692\pi$
$410$		0			0
$411$	328243.	−	568533.i	0.0958497	−	0.166017i
$412$		0			0
$413$	913757.	−	2.71247e6i	0.263606	−	0.782509i
$414$		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	$-0.315609\pi$
$419$	3.93448e6			1.09484			0.547422	+	0.836856i	$0.315609\pi$
$420$		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
$427$	1.03877e6	−	209127.i	0.275708	−	0.0555060i
$428$		0			0
$429$	903651.	−	1.56517e6i	0.237060	−	0.410599i
$430$		0			0
$431$	1.06475e6	+	1.84421e6i	0.276093	+	0.478208i	0.970410	−	0.241462i	$-0.0776268\pi$
$431$	1.06475e6	+	1.84421e6i	0.276093	+	0.478208i	−0.694317	+	0.719669i	$0.744293\pi$
$432$		0			0
$433$	3.20879e6			0.822473			0.411237	−	0.911529i	$-0.365097\pi$
$433$	3.20879e6			0.822473			0.411237	+	0.911529i	$0.365097\pi$
$434$		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	0.000785585		1.00000i	$0.499750\pi$
$439$	−3.49538e6	+	6.05418e6i	−0.865632	+	1.49932i	−0.866418	+	0.499320i	$0.833583\pi$
$440$		0			0
$441$	−1.08387e6	−	823735.i	−0.265388	−	0.201693i
$442$		0			0
$443$	−3.73004e6	+	6.46062e6i	−0.903034	+	1.56410i	−0.0795000	+	0.996835i	$0.525332\pi$
$443$	−3.73004e6	+	6.46062e6i	−0.903034	+	1.56410i	−0.823534	+	0.567266i	$0.808001\pi$
$444$		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
$455$	3.12671e6	−	629474.i	0.708042	−	0.142544i
$456$		0			0
$457$	4.40636e6	−	7.63203e6i	0.986936	−	1.70942i	0.353942	−	0.935267i	$-0.384841\pi$
$457$	4.40636e6	−	7.63203e6i	0.986936	−	1.70942i	0.632994	−	0.774156i	$-0.281826\pi$
$458$		0			0
$459$	−566184.	−	980660.i	−0.125437	−	0.217264i
$460$		0			0
$461$	3.24339e6			0.710798			0.355399	−	0.934715i	$-0.384345\pi$
$461$	3.24339e6			0.710798			0.355399	+	0.934715i	$0.384345\pi$
$462$		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	$-0.907629\pi$
$467$	−1.09007e6	+	1.88805e6i	−0.231292	+	0.400610i	0.726896	+	0.686747i	$0.240962\pi$
$468$		0			0
$469$	538110.	−	1.59737e6i	0.112964	−	0.335330i
$470$		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	$-0.294858\pi$
$479$	−1.96808e6	−	3.40881e6i	−0.391926	−	0.678835i	−0.992704	+	0.120581i	$0.961524\pi$
$480$		0			0
$481$	1.83802e6	−	3.18355e6i	0.362233	−	0.627406i
$482$		0			0
$483$	−582248.	−	660833.i	−0.113564	−	0.128891i
$484$		0			0
$485$	−1.48701e6	+	2.57558e6i	−0.287052	+	0.497189i
$486$		0			0
$487$	−3.50088e6	−	6.06370e6i	−0.668890	−	1.15855i	−0.978215	−	0.207595i	$-0.933436\pi$
$487$	−3.50088e6	−	6.06370e6i	−0.668890	−	1.15855i	0.309325	−	0.950956i	$-0.399897\pi$
$488$		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
$497$	−1.05842e6	−	1.20127e6i	−0.192205	−	0.218147i
$498$		0			0
$499$	4.87966e6	−	8.45182e6i	0.877280	−	1.51949i	0.0229668	−	0.999736i	$-0.492689\pi$
$499$	4.87966e6	−	8.45182e6i	0.877280	−	1.51949i	0.854314	−	0.519758i	$-0.173978\pi$
$500$		0			0
$501$	−3.09606e6	−	5.36254e6i	−0.551081	−	0.954501i
$502$		0			0
$503$	−1.69667e6			−0.299004			−0.149502	−	0.988761i	$-0.547767\pi$
$503$	−1.69667e6			−0.299004			−0.149502	+	0.988761i	$0.547767\pi$
$504$		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	$-0.729801\pi$
$509$	1.86783e6	−	3.23518e6i	0.319553	−	0.553482i	0.980395	+	0.197043i	$0.0631339\pi$
$510$		0			0
$511$	1.80451e6	−	5.35665e6i	0.305708	−	0.907488i
$512$		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	−0.994057	−	0.108863i	$-0.965279\pi$
$521$	−3.66359e6	−	6.34552e6i	−0.591306	−	1.02417i	0.402750	−	0.915310i	$-0.368054\pi$
$522$		0			0
$523$	4795.71	−	8306.42i	0.000766653	−	0.00132788i	−0.865642	−	0.500664i	$-0.833089\pi$
$523$	4795.71	−	8306.42i	0.000766653	−	0.00132788i	0.866408	+	0.499336i	$0.166423\pi$
$524$		0			0
$525$	−2.40599e6	+	484377.i	−0.380974	+	0.0766983i
$526$		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
$539$	4.04309e6	−	1.69669e6i	0.599434	−	0.251554i
$540$		0			0
$541$	1.87846e6	−	3.25359e6i	0.275936	−	0.477935i	−0.694435	−	0.719556i	$-0.744346\pi$
$541$	1.87846e6	−	3.25359e6i	0.275936	−	0.477935i	0.970371	+	0.241620i	$0.0776788\pi$
$542$		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
$553$	−9.75424e6	+	1.96374e6i	−1.35638	+	0.273068i
$554$		0			0
$555$	686879.	−	1.18971e6i	0.0946560	−	0.163949i
$556$		0			0
$557$	5.02378e6	+	8.70144e6i	0.686108	+	1.18837i	0.973087	+	0.230437i	$0.0740156\pi$
$557$	5.02378e6	+	8.70144e6i	0.686108	+	1.18837i	−0.286979	+	0.957937i	$0.592651\pi$
$558$		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	0.997080	+	0.0763704i	$0.0243331\pi$
$563$	4.24690e6	+	7.35585e6i	0.564678	+	0.978052i	−0.432401	+	0.901681i	$0.642334\pi$
$564$		0			0
$565$	3.17856e6	−	5.50543e6i	0.418899	−	0.725555i
$566$		0			0
$567$	271544.	−	806071.i	0.0354717	−	0.105297i
$568$		0			0
$569$	−1.90765e6	+	3.30415e6i	−0.247012	+	0.427838i	−0.962695	−	0.270587i	$-0.912782\pi$
$569$	−1.90765e6	+	3.30415e6i	−0.247012	+	0.427838i	0.715683	+	0.698425i	$0.246115\pi$
$570$		0			0
$571$	−6.10664e6	−	1.05770e7i	−0.783812	−	1.35760i	−0.929706	−	0.368302i	$-0.879939\pi$
$571$	−6.10664e6	−	1.05770e7i	−0.783812	−	1.35760i	0.145894	−	0.989300i	$-0.453394\pi$
$572$		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	$-0.120841\pi$
$577$	1.14735e6	+	1.98727e6i	0.143469	+	0.248495i	−0.785332	+	0.619075i	$0.787508\pi$
$578$		0			0
$579$	1.36548e6	−	2.36508e6i	0.169273	−	0.293190i
$580$		0			0
$581$	1.87637e6	+	2.12962e6i	0.230611	+	0.261735i
$582$		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	$-0.350986\pi$
$587$	7.53391e6			0.902454			0.451227	+	0.892409i	$0.350986\pi$
$588$		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	$-0.972166\pi$
$593$	−3.61760e6	+	6.26588e6i	−0.422459	+	0.731720i	0.573721	+	0.819051i	$0.305500\pi$
$594$		0			0
$595$	4.25492e6	+	4.82920e6i	0.492719	+	0.559220i
$596$		0			0
$597$	3.78992e6	−	6.56433e6i	0.435205	−	0.753797i
$598$		0			0
$599$	−1.66928e6	−	2.89128e6i	−0.190091	−	0.329247i	0.755189	−	0.655507i	$-0.227545\pi$
$599$	−1.66928e6	−	2.89128e6i	−0.190091	−	0.329247i	−0.945280	+	0.326259i	$0.894212\pi$
$600$		0			0
$601$	1.48297e7			1.67474			0.837370	−	0.546636i	$-0.184092\pi$
$601$	1.48297e7			1.67474			0.837370	+	0.546636i	$0.184092\pi$
$602$		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	$-0.719133\pi$
$607$	3.18737e6	−	5.52068e6i	0.351124	−	0.608164i	0.986447	+	0.164083i	$0.0524663\pi$
$608$		0			0
$609$	−2.23825e6	+	6.64420e6i	−0.244549	+	0.725938i
$610$		0			0
$611$	6.70747e6	−	1.16177e7i	0.726869	−	1.25897i
$612$		0			0
$613$	7.74344e6	+	1.34120e7i	0.832306	+	1.44160i	0.896205	+	0.443639i	$0.146313\pi$
$613$	7.74344e6	+	1.34120e7i	0.832306	+	1.44160i	−0.0638996	+	0.997956i	$0.520354\pi$
$614$		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	0.991135	+	0.132861i	$0.0424163\pi$
$619$	5.82108e6	+	1.00824e7i	0.610628	+	1.05764i	−0.380507	+	0.924778i	$0.624250\pi$
$620$		0			0
$621$	275144.	−	476564.i	0.0286307	−	0.0495898i
$622$		0			0
$623$	1.74074e7	−	3.50448e6i	1.79686	−	0.361746i
$624$		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
$637$	−1.62917e6	+	1.28339e7i	−0.159081	+	1.25318i
$638$		0			0
$639$	500160.	−	866302.i	0.0484570	−	0.0839300i
$640$		0			0
$641$	−6.15689e6	−	1.06641e7i	−0.591857	−	1.02513i	−0.993982	−	0.109541i	$-0.965062\pi$
$641$	−6.15689e6	−	1.06641e7i	−0.591857	−	1.02513i	0.402126	−	0.915585i	$-0.368272\pi$
$642$		0			0
$643$	−1.67944e7			−1.60191			−0.800955	−	0.598724i	$-0.795675\pi$
$643$	−1.67944e7			−1.60191			−0.800955	+	0.598724i	$0.795675\pi$
$644$		0			0
$645$	−3.42205e6			−0.323882
$646$		0			0
$647$	−9.95539e6	−	1.72432e7i	−0.934970	−	1.61942i	−0.774687	−	0.632345i	$-0.782093\pi$
$647$	−9.95539e6	−	1.72432e7i	−0.934970	−	1.61942i	−0.160283	−	0.987071i	$-0.551241\pi$
$648$		0			0
$649$	−2.87990e6	+	4.98814e6i	−0.268390	+	0.464865i
$650$		0			0
$651$	−7.34341e6	+	1.47839e6i	−0.679118	+	0.136721i
$652$		0			0
$653$	−1.05495e7	+	1.82723e7i	−0.968163	+	1.67691i	−0.267299	+	0.963614i	$0.586131\pi$
$653$	−1.05495e7	+	1.82723e7i	−0.968163	+	1.67691i	−0.700864	+	0.713295i	$0.747202\pi$
$654$		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	−0.918333	−	0.395810i	$-0.870464\pi$
$661$	−9.00844e6	−	1.56031e7i	−0.801947	−	1.38901i	0.116385	−	0.993204i	$-0.462869\pi$
$662$		0			0
$663$	−5.38040e6	+	9.31912e6i	−0.475369	+	0.823363i
$664$		0			0
$665$	−992173.	+	2.94524e6i	−0.0870028	+	0.258266i
$666$		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	$-0.930712\pi$
$677$	−3.59160e6	+	6.22084e6i	−0.301173	+	0.521648i	0.675229	+	0.737609i	$0.264045\pi$
$678$		0			0
$679$	−7.97480e6	−	9.05113e6i	−0.663812	−	0.753405i
$680$		0			0
$681$	2.12166e6	−	3.67482e6i	0.175310	−	0.303647i
$682$		0			0
$683$	−825461.	−	1.42974e6i	−0.0677087	−	0.117275i	0.830184	−	0.557490i	$-0.188236\pi$
$683$	−825461.	−	1.42974e6i	−0.0677087	−	0.117275i	−0.897892	+	0.440215i	$0.854902\pi$
$684$		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	$-0.843193\pi$
$691$	−388722.	+	673286.i	−0.0309702	+	0.0536419i	0.850125	+	0.526581i	$0.176526\pi$
$692$		0			0
$693$	1.81107e6	+	2.05550e6i	0.143252	+	0.162587i
$694$		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
$707$	5.06176e6	−	1.50257e7i	0.380849	−	1.13054i
$708$		0			0
$709$	−2.49950e6	+	4.32927e6i	−0.186740	+	0.323444i	−0.944162	−	0.329483i	$-0.893126\pi$
$709$	−2.49950e6	+	4.32927e6i	−0.186740	+	0.323444i	0.757421	+	0.652927i	$0.226459\pi$
$710$		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	−0.327828	−	0.944738i	$-0.606317\pi$
$719$	9.06918e6	−	1.57083e7i	0.654253	−	1.13320i	0.982081	−	0.188462i	$-0.0603501\pi$
$720$		0			0
$721$	−9.50664e6	+	1.91389e6i	−0.681065	+	0.137113i
$722$		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	$-0.364227\pi$
$727$	1.17918e7			0.827454			0.413727	+	0.910401i	$0.364227\pi$
$728$		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	$-0.778747\pi$
$733$	2.48235e6	−	4.29955e6i	0.170648	−	0.295572i	0.938647	+	0.344880i	$0.112080\pi$
$734$		0			0
$735$	−608831.	+	4.79612e6i	−0.0415698	+	0.327470i
$736$		0			0
$737$	−1.69597e6	+	2.93751e6i	−0.115014	+	0.199210i
$738$		0			0
$739$	−9.31461e6	−	1.61334e7i	−0.627413	−	1.08671i	−0.988069	−	0.154013i	$-0.950780\pi$
$739$	−9.31461e6	−	1.61334e7i	−0.627413	−	1.08671i	0.360656	−	0.932699i	$-0.382553\pi$
$740$		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
$749$	−8.99460e6	+	1.81081e6i	−0.585838	+	0.117942i
$750$		0			0
$751$	−2.33030e6	+	4.03619e6i	−0.150769	+	0.261139i	−0.931510	−	0.363715i	$-0.881508\pi$
$751$	−2.33030e6	+	4.03619e6i	−0.150769	+	0.261139i	0.780742	+	0.624854i	$0.214842\pi$
$752$		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	−0.444202	−	0.895926i	$-0.646513\pi$
$761$	8.84728e6	−	1.53239e7i	0.553794	−	0.959199i	0.997996	−	0.0632726i	$-0.0201538\pi$
$762$		0			0
$763$	−5.88787e6	+	1.74780e7i	−0.366140	+	1.08688i
$764$		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	$-0.577465\pi$
$769$	−7.90324e6			−0.481936			−0.240968	+	0.970533i	$0.577465\pi$
$770$		0			0
$771$	1.63814e7			0.992468
$772$		0			0
$773$	−1.11230e7	−	1.92656e7i	−0.669534	−	1.15967i	−0.978035	−	0.208442i	$-0.933161\pi$
$773$	−1.11230e7	−	1.92656e7i	−0.669534	−	1.15967i	0.308501	−	0.951224i	$-0.400173\pi$
$774$		0			0
$775$	−6.75214e6	+	1.16950e7i	−0.403819	+	0.699436i
$776$		0			0
$777$	3.68370e6	+	4.18088e6i	0.218893	+	0.248437i
$778$		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

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)

\(f(q)\)	\(=\)	\(q+(4.50000 + 7.79423i) q^{3} +(15.9808 - 27.6796i) q^{5} +(85.7043 + 97.2716i) q^{7} +(-40.5000 + 70.1481i) q^{9} +O(q^{10})\)	\(q+(4.50000 + 7.79423i) q^{3} +(15.9808 - 27.6796i) q^{5} +(85.7043 + 97.2716i) q^{7} +(-40.5000 + 70.1481i) q^{9} +(-130.442 - 225.932i) q^{11} +769.735 q^{13} +287.654 q^{15} +(776.659 + 1345.21i) q^{17} +(-375.024 + 649.561i) q^{19} +(-372.488 + 1105.72i) q^{21} +(-377.427 + 653.723i) q^{23} +(1051.73 + 1821.65i) q^{25} -729.000 q^{27} +6008.93 q^{29} +(3210.02 + 5559.92i) q^{31} +(1173.98 - 2033.39i) q^{33} +(4062.06 - 817.779i) q^{35} +(2387.86 - 4135.90i) q^{37} +(3463.81 + 5999.49i) q^{39} -5423.27 q^{41} -11896.4 q^{43} +(1294.44 + 2242.04i) q^{45} +(8714.00 - 15093.1i) q^{47} +(-2116.54 + 16673.2i) q^{49} +(-6989.93 + 12106.9i) q^{51} +(-18825.3 - 32606.4i) q^{53} -8338.26 q^{55} -6750.44 q^{57} +(-11039.0 - 19120.2i) q^{59} +(4086.69 - 7078.36i) q^{61} +(-10294.4 + 2072.49i) q^{63} +(12301.0 - 21305.9i) q^{65} +(-6500.87 - 11259.8i) q^{67} -6793.69 q^{69} -12349.6 q^{71} +(-21800.2 - 37759.0i) q^{73} +(-9465.55 + 16394.8i) q^{75} +(10797.3 - 32051.6i) q^{77} +(-38374.7 + 66467.0i) q^{79} +(-3280.50 - 5681.99i) q^{81} +21893.6 q^{83} +49646.5 q^{85} +(27040.2 + 46835.0i) q^{87} +(68483.5 - 118617. i) q^{89} +(65969.7 + 74873.4i) q^{91} +(-28890.2 + 50039.3i) q^{93} +(11986.4 + 20761.0i) q^{95} -93050.1 q^{97} +21131.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 36 q^{3} - 42 q^{7} - 324 q^{9}+O(q^{10}) \) 8 * q + 36 * q^3 - 42 * q^7 - 324 * q^9	\( 8 q + 36 q^{3} - 42 q^{7} - 324 q^{9} - 462 q^{11} - 1204 q^{13} + 228 q^{17} + 358 q^{19} + 1404 q^{21} - 2148 q^{23} - 5454 q^{25} - 5832 q^{27} - 11064 q^{29} + 830 q^{31} + 4158 q^{33} + 7692 q^{35} - 3914 q^{37} - 5418 q^{39} - 16632 q^{41} - 29036 q^{43} + 41700 q^{47} + 41876 q^{49} - 2052 q^{51} + 22164 q^{53} + 7784 q^{55} + 6444 q^{57} + 32886 q^{59} + 83732 q^{61} + 16038 q^{63} - 93192 q^{65} - 80034 q^{67} - 38664 q^{69} + 89544 q^{71} - 22470 q^{73} + 49086 q^{75} + 138732 q^{77} - 75286 q^{79} - 26244 q^{81} - 34836 q^{83} + 278504 q^{85} - 49788 q^{87} + 28944 q^{89} - 12058 q^{91} - 7470 q^{93} - 144120 q^{95} - 433356 q^{97} + 74844 q^{99}+O(q^{100}) \) 8 * q + 36 * q^3 - 42 * q^7 - 324 * q^9 - 462 * q^11 - 1204 * q^13 + 228 * q^17 + 358 * q^19 + 1404 * q^21 - 2148 * q^23 - 5454 * q^25 - 5832 * q^27 - 11064 * q^29 + 830 * q^31 + 4158 * q^33 + 7692 * q^35 - 3914 * q^37 - 5418 * q^39 - 16632 * q^41 - 29036 * q^43 + 41700 * q^47 + 41876 * q^49 - 2052 * q^51 + 22164 * q^53 + 7784 * q^55 + 6444 * q^57 + 32886 * q^59 + 83732 * q^61 + 16038 * q^63 - 93192 * q^65 - 80034 * q^67 - 38664 * q^69 + 89544 * q^71 - 22470 * q^73 + 49086 * q^75 + 138732 * q^77 - 75286 * q^79 - 26244 * q^81 - 34836 * q^83 + 278504 * q^85 - 49788 * q^87 + 28944 * q^89 - 12058 * q^91 - 7470 * q^93 - 144120 * q^95 - 433356 * q^97 + 74844 * q^99

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