LMFDB - Embedded newform 84.6.i.c.37.4

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			37.4
Root			$-11.2416 - 19.4709i$ of defining polynomial
Character	$\chi$	$=$	84.37
Dual form			84.6.i.c.25.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.50000 + 7.79423i) q^{3} +(46.4128 - 80.3893i) q^{5} +(-118.369 - 52.8745i) q^{7} +(-40.5000 + 70.1481i) q^{9} +O(q^{10})$	\(q+(4.50000 + 7.79423i) q^{3} +(46.4128 - 80.3893i) q^{5} +(-118.369 - 52.8745i) q^{7} +(-40.5000 + 70.1481i) q^{9} +(-70.3812 - 121.904i) q^{11} -1111.24 q^{13} +835.430 q^{15} +(-27.4435 - 47.5335i) q^{17} +(855.929 - 1482.51i) q^{19} +(-120.546 - 1160.53i) q^{21} +(1643.95 - 2847.41i) q^{23} +(-2745.79 - 4755.85i) q^{25} -729.000 q^{27} -3790.72 q^{29} +(2423.66 + 4197.90i) q^{31} +(633.431 - 1097.13i) q^{33} +(-9744.39 + 7061.57i) q^{35} +(5683.35 - 9843.86i) q^{37} +(-5000.59 - 8661.28i) q^{39} -10385.6 q^{41} +7137.16 q^{43} +(3759.43 + 6511.53i) q^{45} +(8207.53 - 14215.9i) q^{47} +(11215.6 + 12517.4i) q^{49} +(246.991 - 427.801i) q^{51} +(10487.2 + 18164.3i) q^{53} -13066.3 q^{55} +15406.7 q^{57} +(18106.0 + 31360.5i) q^{59} +(-2474.35 + 4285.70i) q^{61} +(8503.00 - 6161.96i) q^{63} +(-51575.9 + 89332.0i) q^{65} +(11482.8 + 19888.8i) q^{67} +29591.2 q^{69} -26341.8 q^{71} +(-27693.7 - 47966.9i) q^{73} +(24712.1 - 42802.6i) q^{75} +(1885.36 + 18151.0i) q^{77} +(24978.3 - 43263.7i) q^{79} +(-3280.50 - 5681.99i) q^{81} +44858.9 q^{83} -5094.91 q^{85} +(-17058.2 - 29545.8i) q^{87} +(-63972.5 + 110804. i) q^{89} +(131537. + 58756.4i) q^{91} +(-21812.9 + 37781.1i) q^{93} +(-79452.1 - 137615. i) q^{95} -65685.9 q^{97} +11401.8 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	46.4128	−	80.3893i	0.830257	−	1.43805i	−0.0675775	−	0.997714i	$-0.521527\pi$
$5$	46.4128	−	80.3893i	0.830257	−	1.43805i	0.897834	−	0.440333i	$-0.145140\pi$
$6$		0			0
$7$	−118.369	−	52.8745i	−0.913049	−	0.407851i
$7$	−118.369	−	52.8745i	−0.913049	−	0.407851i
$8$		0			0
$9$	−40.5000	+	70.1481i	−0.166667	+	0.288675i
$10$		0			0
$11$	−70.3812	−	121.904i	−0.175378	−	0.303763i	0.764914	−	0.644132i	$-0.222781\pi$
$11$	−70.3812	−	121.904i	−0.175378	−	0.303763i	−0.940292	+	0.340369i	$0.889448\pi$
$12$		0			0
$13$	−1111.24			−1.82369			−0.911844	−	0.410537i	$-0.865341\pi$
$13$	−1111.24			−1.82369			−0.911844	+	0.410537i	$0.865341\pi$
$14$		0			0
$15$	835.430			0.958698
$16$		0			0
$17$	−27.4435	−	47.5335i	−0.0230312	−	0.0398912i	0.854280	−	0.519813i	$-0.173998\pi$
$17$	−27.4435	−	47.5335i	−0.0230312	−	0.0398912i	−0.877311	+	0.479922i	$0.840665\pi$
$18$		0			0
$19$	855.929	−	1482.51i	0.543943	−	0.942138i	−0.454729	−	0.890630i	$-0.650264\pi$
$19$	855.929	−	1482.51i	0.543943	−	0.942138i	0.998673	−	0.0515079i	$-0.0164027\pi$
$20$		0			0
$21$	−120.546	−	1160.53i	−0.0596490	−	0.574261i
$22$		0			0
$23$	1643.95	−	2847.41i	0.647993	−	1.12236i	−0.335609	−	0.942001i	$-0.608942\pi$
$23$	1643.95	−	2847.41i	0.647993	−	1.12236i	0.983602	−	0.180355i	$-0.0577247\pi$
$24$		0			0
$25$	−2745.79	−	4755.85i	−0.878653	−	1.52187i
$26$		0			0
$27$	−729.000			−0.192450
$28$		0			0
$29$	−3790.72			−0.837003			−0.418501	−	0.908216i	$-0.637445\pi$
$29$	−3790.72			−0.837003			−0.418501	+	0.908216i	$0.637445\pi$
$30$		0			0
$31$	2423.66	+	4197.90i	0.452968	+	0.784563i	0.998569	−	0.0534817i	$-0.0170319\pi$
$31$	2423.66	+	4197.90i	0.452968	+	0.784563i	−0.545601	+	0.838045i	$0.683699\pi$
$32$		0			0
$33$	633.431	−	1097.13i	0.101254	−	0.175378i
$34$		0			0
$35$	−9744.39	+	7061.57i	−1.34457	+	0.974386i
$36$		0			0
$37$	5683.35	−	9843.86i	0.682496	−	1.18212i	−0.291720	−	0.956504i	$-0.594228\pi$
$37$	5683.35	−	9843.86i	0.682496	−	1.18212i	0.974217	−	0.225615i	$-0.0724391\pi$
$38$		0			0
$39$	−5000.59	−	8661.28i	−0.526453	−	0.911844i
$40$		0			0
$41$	−10385.6			−0.964881			−0.482440	−	0.875929i	$-0.660249\pi$
$41$	−10385.6			−0.964881			−0.482440	+	0.875929i	$0.660249\pi$
$42$		0			0
$43$	7137.16			0.588646			0.294323	−	0.955706i	$-0.404906\pi$
$43$	7137.16			0.588646			0.294323	+	0.955706i	$0.404906\pi$
$44$		0			0
$45$	3759.43	+	6511.53i	0.276752	+	0.479349i
$46$		0			0
$47$	8207.53	−	14215.9i	0.541961	−	0.938704i	−0.456831	−	0.889554i	$-0.651015\pi$
$47$	8207.53	−	14215.9i	0.541961	−	0.938704i	0.998791	−	0.0491499i	$-0.0156512\pi$
$48$		0			0
$49$	11215.6	+	12517.4i	0.667315	+	0.744775i
$50$		0			0
$51$	246.991	−	427.801i	0.0132971	−	0.0230312i
$52$		0			0
$53$	10487.2	+	18164.3i	0.512824	+	0.888238i	0.999889	+	0.0148720i	$0.00473407\pi$
$53$	10487.2	+	18164.3i	0.512824	+	0.888238i	−0.487065	+	0.873366i	$0.661933\pi$
$54$		0			0
$55$	−13066.3			−0.582435
$56$		0			0
$57$	15406.7			0.628092
$58$		0			0
$59$	18106.0	+	31360.5i	0.677161	+	1.17288i	0.975832	+	0.218521i	$0.0701231\pi$
$59$	18106.0	+	31360.5i	0.677161	+	1.17288i	−0.298672	+	0.954356i	$0.596544\pi$
$60$		0			0
$61$	−2474.35	+	4285.70i	−0.0851405	+	0.147468i	−0.905451	−	0.424451i	$-0.860467\pi$
$61$	−2474.35	+	4285.70i	−0.0851405	+	0.147468i	0.820311	+	0.571918i	$0.193801\pi$
$62$		0			0
$63$	8503.00	−	6161.96i	0.269911	−	0.195599i
$64$		0			0
$65$	−51575.9	+	89332.0i	−1.51413	+	2.62255i
$66$		0			0
$67$	11482.8	+	19888.8i	0.312507	+	0.541279i	0.978905	−	0.204318i	$-0.0654978\pi$
$67$	11482.8	+	19888.8i	0.312507	+	0.541279i	−0.666397	+	0.745597i	$0.732164\pi$
$68$		0			0
$69$	29591.2			0.748238
$70$		0			0
$71$	−26341.8			−0.620154			−0.310077	−	0.950711i	$-0.600355\pi$
$71$	−26341.8			−0.620154			−0.310077	+	0.950711i	$0.600355\pi$
$72$		0			0
$73$	−27693.7	−	47966.9i	−0.608238	−	1.05350i	−0.991531	−	0.129872i	$-0.958543\pi$
$73$	−27693.7	−	47966.9i	−0.608238	−	1.05350i	0.383293	−	0.923627i	$-0.374790\pi$
$74$		0			0
$75$	24712.1	−	42802.6i	0.507291	−	0.878653i
$76$		0			0
$77$	1885.36	+	18151.0i	0.0362383	+	0.348879i
$78$		0			0
$79$	24978.3	−	43263.7i	0.450293	−	0.779930i	−0.548111	−	0.836406i	$-0.684653\pi$
$79$	24978.3	−	43263.7i	0.450293	−	0.779930i	0.998404	+	0.0564752i	$0.0179862\pi$
$80$		0			0
$81$	−3280.50	−	5681.99i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	44858.9			0.714749			0.357374	−	0.933961i	$-0.383672\pi$
$83$	44858.9			0.714749			0.357374	+	0.933961i	$0.383672\pi$
$84$		0			0
$85$	−5094.91			−0.0764873
$86$		0			0
$87$	−17058.2	−	29545.8i	−0.241622	−	0.418501i
$88$		0			0
$89$	−63972.5	+	110804.i	−0.856087	+	1.48279i	0.0195454	+	0.999809i	$0.493778\pi$
$89$	−63972.5	+	110804.i	−0.856087	+	1.48279i	−0.875633	+	0.482978i	$0.839555\pi$
$90$		0			0
$91$	131537.	+	58756.4i	1.66512	+	0.743793i
$92$		0			0
$93$	−21812.9	+	37781.1i	−0.261521	+	0.452968i
$94$		0			0
$95$	−79452.1	−	137615.i	−0.903226	−	1.56443i
$96$		0			0
$97$	−65685.9			−0.708831			−0.354415	−	0.935088i	$-0.615320\pi$
$97$	−65685.9			−0.708831			−0.354415	+	0.935088i	$0.615320\pi$
$98$		0			0
$99$	11401.8			0.116919
$100$		0			0
$101$	−87891.4	−	152232.i	−0.857320	−	1.48492i	−0.874476	−	0.485068i	$-0.838795\pi$
$101$	−87891.4	−	152232.i	−0.857320	−	1.48492i	0.0171565	−	0.999853i	$-0.494539\pi$
$102$		0			0
$103$	77054.7	−	133463.i	0.715659	−	1.23956i	−0.247046	−	0.969004i	$-0.579460\pi$
$103$	77054.7	−	133463.i	0.715659	−	1.23956i	0.962705	−	0.270554i	$-0.0872068\pi$
$104$		0			0
$105$	−98889.2	−	44173.0i	−0.875338	−	0.391006i
$106$		0			0
$107$	−91681.0	+	158796.i	−0.774141	+	1.34085i	0.161135	+	0.986932i	$0.448485\pi$
$107$	−91681.0	+	158796.i	−0.774141	+	1.34085i	−0.935276	+	0.353919i	$0.884849\pi$
$108$		0			0
$109$	67322.3	+	116606.i	0.542741	+	0.940055i	0.998745	+	0.0500775i	$0.0159468\pi$
$109$	67322.3	+	116606.i	0.542741	+	0.940055i	−0.456004	+	0.889978i	$0.650720\pi$
$110$		0			0
$111$	102300.			0.788079
$112$		0			0
$113$	176955.			1.30367			0.651833	−	0.758362i	$-0.274000\pi$
$113$	176955.			1.30367			0.651833	+	0.758362i	$0.274000\pi$
$114$		0			0
$115$	−152601.	−	264313.i	−1.07600	−	1.86369i
$116$		0			0
$117$	45005.3	−	77951.5i	0.303948	−	0.526453i
$118$		0			0
$119$	735.153	+	7077.57i	0.00475894	+	0.0458159i
$120$		0			0
$121$	70618.5	−	122315.i	0.438485	−	0.759479i
$122$		0			0
$123$	−46735.4	−	80948.0i	−0.278537	−	0.482440i
$124$		0			0
$125$	−219679.			−1.25752
$126$		0			0
$127$	144432.			0.794608			0.397304	−	0.917687i	$-0.369946\pi$
$127$	144432.			0.794608			0.397304	+	0.917687i	$0.369946\pi$
$128$		0			0
$129$	32117.2	+	55628.6i	0.169927	+	0.294323i
$130$		0			0
$131$	118754.	−	205688.i	0.604602	−	1.04720i	−0.387512	−	0.921865i	$-0.626665\pi$
$131$	118754.	−	205688.i	0.604602	−	1.04720i	0.992114	−	0.125337i	$-0.0400013\pi$
$132$		0			0
$133$	−179703.	+	130227.i	−0.880899	+	0.638370i
$134$		0			0
$135$	−33834.9	+	58603.8i	−0.159783	+	0.276752i
$136$		0			0
$137$	48101.6	+	83314.5i	0.218957	+	0.379244i	0.954489	−	0.298245i	$-0.0964013\pi$
$137$	48101.6	+	83314.5i	0.218957	+	0.379244i	−0.735533	+	0.677489i	$0.763068\pi$
$138$		0			0
$139$	−391373.			−1.71812			−0.859060	−	0.511874i	$-0.828951\pi$
$139$	−391373.			−1.71812			−0.859060	+	0.511874i	$0.828951\pi$
$140$		0			0
$141$	147736.			0.625802
$142$		0			0
$143$	78210.6	+	135465.i	0.319835	+	0.553970i
$144$		0			0
$145$	−175938.	+	304733.i	−0.694927	+	1.20365i
$146$		0			0
$147$	−47093.7	+	143745.i	−0.179750	+	0.548656i
$148$		0			0
$149$	40339.4	−	69869.9i	0.148855	−	0.257825i	−0.781949	−	0.623342i	$-0.785775\pi$
$149$	40339.4	−	69869.9i	0.148855	−	0.257825i	0.930805	+	0.365517i	$0.119108\pi$
$150$		0			0
$151$	−47841.3	−	82863.6i	−0.170750	−	0.295748i	0.767932	−	0.640531i	$-0.221286\pi$
$151$	−47841.3	−	82863.6i	−0.170750	−	0.295748i	−0.938682	+	0.344783i	$0.887952\pi$
$152$		0			0
$153$	4445.84			0.0153541
$154$		0			0
$155$	449955.			1.50432
$156$		0			0
$157$	97812.7	+	169417.i	0.316699	+	0.548538i	0.979797	−	0.199994i	$-0.0640923\pi$
$157$	97812.7	+	169417.i	0.316699	+	0.548538i	−0.663098	+	0.748532i	$0.730759\pi$
$158$		0			0
$159$	−94384.5	+	163479.i	−0.296079	+	0.512824i
$160$		0			0
$161$	−345149.	+	250123.i	−1.04940	+	0.760481i
$162$		0			0
$163$	−18822.8	+	32602.1i	−0.0554902	+	0.0961118i	−0.892436	−	0.451174i	$-0.851005\pi$
$163$	−18822.8	+	32602.1i	−0.0554902	+	0.0961118i	0.836946	+	0.547286i	$0.184339\pi$
$164$		0			0
$165$	−58798.5	−	101842.i	−0.168134	−	0.291217i
$166$		0			0
$167$	646778.			1.79458			0.897292	−	0.441437i	$-0.145531\pi$
$167$	646778.			1.79458			0.897292	+	0.441437i	$0.145531\pi$
$168$		0			0
$169$	863568.			2.32584
$170$		0			0
$171$	69330.3	+	120084.i	0.181314	+	0.314046i
$172$		0			0
$173$	193454.	−	335072.i	0.491431	−	0.851184i	−0.508520	−	0.861050i	$-0.669807\pi$
$173$	193454.	−	335072.i	0.491431	−	0.851184i	0.999951	+	0.00986616i	$0.00314055\pi$
$174$		0			0
$175$	73554.0	+	708129.i	0.181556	+	1.74790i
$176$		0			0
$177$	−162954.	+	282244.i	−0.390959	+	0.677161i
$178$		0			0
$179$	−13705.3	−	23738.2i	−0.0319709	−	0.0553752i	0.849597	−	0.527432i	$-0.176845\pi$
$179$	−13705.3	−	23738.2i	−0.0319709	−	0.0553752i	−0.881568	+	0.472057i	$0.843512\pi$
$180$		0			0
$181$	−196155.			−0.445044			−0.222522	−	0.974928i	$-0.571429\pi$
$181$	−196155.			−0.445044			−0.222522	+	0.974928i	$0.571429\pi$
$182$		0			0
$183$	−44538.3			−0.0983118
$184$		0			0
$185$	−527560.	−	913761.i	−1.13329	−	1.96292i
$186$		0			0
$187$	−3863.01	+	6690.93i	−0.00807833	+	0.0139921i
$188$		0			0
$189$	86291.2	+	38545.5i	0.175716	+	0.0784909i
$190$		0			0
$191$	−133491.	+	231213.i	−0.264770	+	0.458595i	−0.967503	−	0.252858i	$-0.918629\pi$
$191$	−133491.	+	231213.i	−0.264770	+	0.458595i	0.702733	+	0.711453i	$0.251963\pi$
$192$		0			0
$193$	50681.9	+	87783.6i	0.0979398	+	0.169637i	0.910832	−	0.412778i	$-0.135441\pi$
$193$	50681.9	+	87783.6i	0.0979398	+	0.169637i	−0.812892	+	0.582415i	$0.802108\pi$
$194$		0			0
$195$	−928366.			−1.74837
$196$		0			0
$197$	362366.			0.665245			0.332623	−	0.943060i	$-0.392066\pi$
$197$	362366.			0.665245			0.332623	+	0.943060i	$0.392066\pi$
$198$		0			0
$199$	−112706.	−	195213.i	−0.201751	−	0.349443i	0.747342	−	0.664440i	$-0.231330\pi$
$199$	−112706.	−	195213.i	−0.201751	−	0.349443i	−0.949093	+	0.314997i	$0.897997\pi$
$200$		0			0
$201$	−103345.	+	178999.i	−0.180426	+	0.312507i
$202$		0			0
$203$	448705.	+	200433.i	0.764224	+	0.341372i
$204$		0			0
$205$	−482026.	+	834894.i	−0.801099	+	1.38754i
$206$		0			0
$207$	133160.	+	230640.i	0.215998	+	0.374119i
$208$		0			0
$209$	−240965.			−0.381583
$210$		0			0
$211$	−327801.			−0.506878			−0.253439	−	0.967351i	$-0.581562\pi$
$211$	−327801.			−0.506878			−0.253439	+	0.967351i	$0.581562\pi$
$212$		0			0
$213$	−118538.	−	205314.i	−0.179023	−	0.310077i
$214$		0			0
$215$	331255.	−	573751.i	0.488727	−	0.846501i
$216$		0			0
$217$	−64924.7	−	625053.i	−0.0935968	−	0.901088i
$218$		0			0
$219$	249243.	−	431702.i	0.351166	−	0.608238i
$220$		0			0
$221$	30496.4	+	52821.2i	0.0420017	+	0.0727492i
$222$		0			0
$223$	109690.			0.147708			0.0738538	−	0.997269i	$-0.476470\pi$
$223$	109690.			0.147708			0.0738538	+	0.997269i	$0.476470\pi$
$224$		0			0
$225$	444818.			0.585769
$226$		0			0
$227$	−608733.	−	1.05436e6i	−0.784083	−	1.35807i	−0.929545	−	0.368708i	$-0.879800\pi$
$227$	−608733.	−	1.05436e6i	−0.784083	−	1.35807i	0.145463	−	0.989364i	$-0.453533\pi$
$228$		0			0
$229$	668013.	−	1.15703e6i	0.841776	−	1.45800i	−0.0466161	−	0.998913i	$-0.514844\pi$
$229$	668013.	−	1.15703e6i	0.841776	−	1.45800i	0.888392	−	0.459086i	$-0.151823\pi$
$230$		0			0
$231$	−132989.	+	96374.6i	−0.163978	+	0.118832i
$232$		0			0
$233$	−23394.2	+	40520.0i	−0.0282305	+	0.0488967i	−0.879795	−	0.475352i	$-0.842321\pi$
$233$	−23394.2	+	40520.0i	−0.0282305	+	0.0488967i	0.851565	+	0.524249i	$0.175654\pi$
$234$		0			0
$235$	−761868.	−	1.31959e6i	−0.899933	−	1.55873i
$236$		0			0
$237$	449609.			0.519954
$238$		0			0
$239$	86350.2			0.0977841			0.0488921	−	0.998804i	$-0.484431\pi$
$239$	86350.2			0.0977841			0.0488921	+	0.998804i	$0.484431\pi$
$240$		0			0
$241$	−412649.	−	714730.i	−0.457655	−	0.792682i	0.541181	−	0.840906i	$-0.317977\pi$
$241$	−412649.	−	714730.i	−0.457655	−	0.792682i	−0.998837	+	0.0482236i	$0.984644\pi$
$242$		0			0
$243$	29524.5	−	51137.9i	0.0320750	−	0.0555556i
$244$		0			0
$245$	1.52681e6	−	320642.i	1.62507	−	0.341276i
$246$		0			0
$247$	−951145.	+	1.64743e6i	−0.991983	+	1.71817i
$248$		0			0
$249$	201865.	+	349641.i	0.206330	+	0.357374i
$250$		0			0
$251$	−130188.			−0.130432			−0.0652162	−	0.997871i	$-0.520774\pi$
$251$	−130188.			−0.130432			−0.0652162	+	0.997871i	$0.520774\pi$
$252$		0			0
$253$	−462814.			−0.454574
$254$		0			0
$255$	−22927.1	−	39710.9i	−0.0220800	−	0.0382436i
$256$		0			0
$257$	−51395.1	+	89018.9i	−0.0485388	+	0.0840716i	−0.889274	−	0.457375i	$-0.848790\pi$
$257$	−51395.1	+	89018.9i	−0.0485388	+	0.0840716i	0.840735	+	0.541446i	$0.182123\pi$
$258$		0			0
$259$	−1.19322e6	+	864706.i	−1.10528	+	0.800975i
$260$		0			0
$261$	153524.	−	265912.i	0.139500	−	0.241622i
$262$		0			0
$263$	17750.4	+	30744.5i	0.0158241	+	0.0274081i	0.873829	−	0.486233i	$-0.161630\pi$
$263$	17750.4	+	30744.5i	0.0158241	+	0.0274081i	−0.858005	+	0.513641i	$0.828296\pi$
$264$		0			0
$265$	1.94695e6			1.70310
$266$		0			0
$267$	−1.15150e6			−0.988524
$268$		0			0
$269$	−523171.	−	906158.i	−0.440821	−	0.763525i	0.556929	−	0.830560i	$-0.311979\pi$
$269$	−523171.	−	906158.i	−0.440821	−	0.763525i	−0.997751	+	0.0670350i	$0.978646\pi$
$270$		0			0
$271$	−88639.2	+	153528.i	−0.0733167	+	0.126988i	−0.900353	−	0.435160i	$-0.856692\pi$
$271$	−88639.2	+	153528.i	−0.0733167	+	0.126988i	0.827036	+	0.562148i	$0.190025\pi$
$272$		0			0
$273$	133955.	+	1.28963e6i	0.108781	+	1.04727i
$274$		0			0
$275$	−386504.	+	669445.i	−0.308193	+	0.533805i
$276$		0			0
$277$	−72793.0	−	126081.i	−0.0570020	−	0.0987304i	0.836116	−	0.548552i	$-0.184821\pi$
$277$	−72793.0	−	126081.i	−0.0570020	−	0.0987304i	−0.893118	+	0.449822i	$0.851487\pi$
$278$		0			0
$279$	−392633.			−0.301979
$280$		0			0
$281$	−1.03073e6			−0.778714			−0.389357	−	0.921087i	$-0.627303\pi$
$281$	−1.03073e6			−0.778714			−0.389357	+	0.921087i	$0.627303\pi$
$282$		0			0
$283$	558631.	+	967577.i	0.414628	+	0.718157i	0.995389	−	0.0959167i	$-0.0305783\pi$
$283$	558631.	+	967577.i	0.414628	+	0.718157i	−0.580761	+	0.814074i	$0.697245\pi$
$284$		0			0
$285$	715069.	−	1.23854e6i	0.521478	−	0.903226i
$286$		0			0
$287$	1.22934e6	+	549136.i	0.880983	+	0.393527i
$288$		0			0
$289$	708422.	−	1.22702e6i	0.498939	−	0.864188i
$290$		0			0
$291$	−295586.	−	511971.i	−0.204622	−	0.354415i
$292$		0			0
$293$	−2.02032e6			−1.37484			−0.687418	−	0.726262i	$-0.741256\pi$
$293$	−2.02032e6			−1.37484			−0.687418	+	0.726262i	$0.741256\pi$
$294$		0			0
$295$	3.36139e6			2.24887
$296$		0			0
$297$	51307.9	+	88867.9i	0.0337515	+	0.0584593i
$298$		0			0
$299$	−1.82683e6	+	3.16417e6i	−1.18174	+	2.04683i
$300$		0			0
$301$	−844820.	−	377374.i	−0.537462	−	0.240080i
$302$		0			0
$303$	791022.	−	1.37009e6i	0.494974	−	0.857320i
$304$		0			0
$305$	229683.	+	397822.i	0.141377	+	0.244872i
$306$		0			0
$307$	−535150.			−0.324063			−0.162031	−	0.986786i	$-0.551805\pi$
$307$	−535150.			−0.324063			−0.162031	+	0.986786i	$0.551805\pi$
$308$		0			0
$309$	1.38698e6			0.826372
$310$		0			0
$311$	1.15009e6	+	1.99202e6i	0.674268	+	1.16787i	0.976682	+	0.214689i	$0.0688739\pi$
$311$	1.15009e6	+	1.99202e6i	0.674268	+	1.16787i	−0.302415	+	0.953176i	$0.597793\pi$
$312$		0			0
$313$	1.37863e6	−	2.38786e6i	0.795404	−	1.37768i	−0.127178	−	0.991880i	$-0.540592\pi$
$313$	1.37863e6	−	2.38786e6i	0.795404	−	1.37768i	0.922582	−	0.385801i	$-0.126075\pi$
$314$		0			0
$315$	−100707.	−	969544.i	−0.0571853	−	0.550543i
$316$		0			0
$317$	−1.27404e6	+	2.20671e6i	−0.712092	+	1.23338i	0.251978	+	0.967733i	$0.418919\pi$
$317$	−1.27404e6	+	2.20671e6i	−0.712092	+	1.23338i	−0.964070	+	0.265647i	$0.914414\pi$
$318$		0			0
$319$	266795.	+	462103.i	0.146792	+	0.254251i
$320$		0			0
$321$	−1.65026e6			−0.893901
$322$		0			0
$323$	−93958.7			−0.0501107
$324$		0			0
$325$	3.05124e6	+	5.28490e6i	1.60239	+	2.77542i
$326$		0			0
$327$	−605901.	+	1.04945e6i	−0.313352	+	0.542741i
$328$		0			0
$329$	−1.72318e6	+	1.24875e6i	−0.877688	+	0.636043i
$330$		0			0
$331$	244667.	−	423776.i	0.122746	−	0.212602i	−0.798104	−	0.602520i	$-0.794163\pi$
$331$	244667.	−	423776.i	0.122746	−	0.212602i	0.920849	+	0.389918i	$0.127497\pi$
$332$		0			0
$333$	460352.	+	797352.i	0.227499	+	0.394039i
$334$		0			0
$335$	2.13179e6			1.03785
$336$		0			0
$337$	−1.47140e6			−0.705757			−0.352878	−	0.935669i	$-0.614797\pi$
$337$	−1.47140e6			−0.705757			−0.352878	+	0.935669i	$0.614797\pi$
$338$		0			0
$339$	796297.	+	1.37923e6i	0.376336	+	0.651833i
$340$		0			0
$341$	341160.	−	590907.i	0.158881	−	0.275190i
$342$		0			0
$343$	−665725.	−	2.07470e6i	−0.305534	−	0.952181i
$344$		0			0
$345$	1.37341e6	−	2.37881e6i	0.621229	−	1.07600i
$346$		0			0
$347$	1.20963e6	+	2.09513e6i	0.539296	+	0.934088i	0.998942	+	0.0459859i	$0.0146429\pi$
$347$	1.20963e6	+	2.09513e6i	0.539296	+	0.934088i	−0.459646	+	0.888102i	$0.652024\pi$
$348$		0			0
$349$	−2.58571e6			−1.13636			−0.568180	−	0.822905i	$-0.692352\pi$
$349$	−2.58571e6			−1.13636			−0.568180	+	0.822905i	$0.692352\pi$
$350$		0			0
$351$	810096.			0.350969
$352$		0			0
$353$	−502198.	−	869832.i	−0.214505	−	0.371534i	0.738614	−	0.674128i	$-0.235481\pi$
$353$	−502198.	−	869832.i	−0.214505	−	0.371534i	−0.953119	+	0.302594i	$0.902147\pi$
$354$		0			0
$355$	−1.22260e6	+	2.11760e6i	−0.514887	+	0.891811i
$356$		0			0
$357$	−51856.0	+	37579.0i	−0.0215342	+	0.0156054i
$358$		0			0
$359$	1.13383e6	−	1.96384e6i	0.464313	−	0.804213i	−0.534858	−	0.844942i	$-0.679635\pi$
$359$	1.13383e6	−	1.96384e6i	0.464313	−	0.804213i	0.999170	+	0.0407292i	$0.0129681\pi$
$360$		0			0
$361$	−227180.	−	393487.i	−0.0917490	−	0.158914i
$362$		0			0
$363$	1.27113e6			0.506319
$364$		0			0
$365$	−5.14136e6			−2.01998
$366$		0			0
$367$	−2.39312e6	−	4.14500e6i	−0.927467	−	1.60642i	−0.787545	−	0.616258i	$-0.788648\pi$
$367$	−2.39312e6	−	4.14500e6i	−0.927467	−	1.60642i	−0.139923	−	0.990162i	$-0.544685\pi$
$368$		0			0
$369$	420618.	−	728532.i	0.160813	−	0.278537i
$370$		0			0
$371$	−280929.	−	2.70460e6i	−0.105965	−	1.02016i
$372$		0			0
$373$	2.31351e6	−	4.00711e6i	0.860991	−	1.49128i	−0.00998166	−	0.999950i	$-0.503177\pi$
$373$	2.31351e6	−	4.00711e6i	0.860991	−	1.49128i	0.870973	−	0.491331i	$-0.163489\pi$
$374$		0			0
$375$	−988557.	−	1.71223e6i	−0.363014	−	0.628759i
$376$		0			0
$377$	4.21241e6			1.52643
$378$		0			0
$379$	−497760.			−0.178001			−0.0890004	−	0.996032i	$-0.528367\pi$
$379$	−497760.			−0.178001			−0.0890004	+	0.996032i	$0.528367\pi$
$380$		0			0
$381$	649942.	+	1.12573e6i	0.229384	+	0.397304i
$382$		0			0
$383$	−693745.	+	1.20160e6i	−0.241659	+	0.418566i	−0.961187	−	0.275898i	$-0.911025\pi$
$383$	−693745.	+	1.20160e6i	−0.241659	+	0.418566i	0.719528	+	0.694463i	$0.244358\pi$
$384$		0			0
$385$	1.54665e6	+	690877.i	0.531791	+	0.237547i
$386$		0			0
$387$	−289055.	+	500658.i	−0.0981077	+	0.169927i
$388$		0			0
$389$	293028.	+	507539.i	0.0981826	+	0.170057i	0.910932	−	0.412556i	$-0.135364\pi$
$389$	293028.	+	507539.i	0.0981826	+	0.170057i	−0.812750	+	0.582613i	$0.802030\pi$
$390$		0			0
$391$	−180463.			−0.0596962
$392$		0			0
$393$	2.13757e6			0.698135
$394$		0			0
$395$	−2.31862e6	−	4.01598e6i	−0.747718	−	1.29509i
$396$		0			0
$397$	−1.52668e6	+	2.64429e6i	−0.486151	+	0.842039i	−0.999873	−	0.0159181i	$-0.994933\pi$
$397$	−1.52668e6	+	2.64429e6i	−0.486151	+	0.842039i	0.513722	+	0.857957i	$0.328266\pi$
$398$		0			0
$399$	−1.82368e6	−	814623.i	−0.573478	−	0.256168i
$400$		0			0
$401$	−2.76158e6	+	4.78320e6i	−0.857624	+	1.48545i	0.0165657	+	0.999863i	$0.494727\pi$
$401$	−2.76158e6	+	4.78320e6i	−0.857624	+	1.48545i	−0.874189	+	0.485585i	$0.838607\pi$
$402$		0			0
$403$	−2.69327e6	−	4.66489e6i	−0.826072	−	1.43080i
$404$		0			0
$405$	−609028.			−0.184502
$406$		0			0
$407$	−1.60000e6			−0.478779
$408$		0			0
$409$	1.67464e6	+	2.90055e6i	0.495008	+	0.857379i	0.999983	−	0.00575475i	$-0.00183180\pi$
$409$	1.67464e6	+	2.90055e6i	0.495008	+	0.857379i	−0.504975	+	0.863134i	$0.668498\pi$
$410$		0			0
$411$	−432915.	+	749830.i	−0.126415	+	0.218957i
$412$		0			0
$413$	−485021.	−	4.66946e6i	−0.139922	−	1.34707i
$414$		0			0
$415$	2.08203e6	−	3.60618e6i	0.593425	−	1.02784i
$416$		0			0
$417$	−1.76118e6	−	3.05045e6i	−0.495979	−	0.859060i
$418$		0			0
$419$	2.97012e6			0.826493			0.413247	−	0.910619i	$-0.364395\pi$
$419$	2.97012e6			0.826493			0.413247	+	0.910619i	$0.364395\pi$
$420$		0			0
$421$	−5.41478e6			−1.48894			−0.744468	−	0.667658i	$-0.767297\pi$
$421$	−5.41478e6			−1.48894			−0.744468	+	0.667658i	$0.767297\pi$
$422$		0			0
$423$	664810.	+	1.15148e6i	0.180654	+	0.312901i
$424$		0			0
$425$	−150708.	+	261034.i	−0.0404729	+	0.0701011i
$426$		0			0
$427$	519491.	−	376465.i	0.137882	−	0.0999205i
$428$		0			0
$429$	−703895.	+	1.21918e6i	−0.184657	+	0.319835i
$430$		0			0
$431$	2.52975e6	+	4.38165e6i	0.655970	+	1.13617i	0.981650	+	0.190694i	$0.0610737\pi$
$431$	2.52975e6	+	4.38165e6i	0.655970	+	1.13617i	−0.325679	+	0.945480i	$0.605593\pi$
$432$		0			0
$433$	3.84174e6			0.984711			0.492355	−	0.870394i	$-0.336136\pi$
$433$	3.84174e6			0.984711			0.492355	+	0.870394i	$0.336136\pi$
$434$		0			0
$435$	−3.16688e6			−0.802433
$436$		0			0
$437$	−2.81422e6	−	4.87437e6i	−0.704943	−	1.22100i
$438$		0			0
$439$	762859.	−	1.32131e6i	0.188922	−	0.327223i	−0.755969	−	0.654607i	$-0.772834\pi$
$439$	762859.	−	1.32131e6i	0.188922	−	0.327223i	0.944891	+	0.327385i	$0.106167\pi$
$440$		0			0
$441$	−1.33230e6	+	279794.i	−0.326217	+	0.0685081i
$442$		0			0
$443$	735068.	−	1.27317e6i	0.177958	−	0.308233i	−0.763223	−	0.646135i	$-0.776384\pi$
$443$	735068.	−	1.27317e6i	0.177958	−	0.308233i	0.941181	+	0.337903i	$0.109717\pi$
$444$		0			0
$445$	5.93828e6	+	1.02854e7i	1.42154	+	2.46219i
$446$		0			0
$447$	726110.			0.171883
$448$		0			0
$449$	−6.05071e6			−1.41641			−0.708207	−	0.706004i	$-0.750496\pi$
$449$	−6.05071e6			−1.41641			−0.708207	+	0.706004i	$0.750496\pi$
$450$		0			0
$451$	730954.	+	1.26605e6i	0.169219	+	0.293095i
$452$		0			0
$453$	430572.	−	745772.i	0.0985826	−	0.170750i
$454$		0			0
$455$	1.08284e7	−	7.84712e6i	2.45208	−	1.77698i
$456$		0			0
$457$	2.87710e6	−	4.98328e6i	0.644413	−	1.11616i	−0.340024	−	0.940417i	$-0.610435\pi$
$457$	2.87710e6	−	4.98328e6i	0.644413	−	1.11616i	0.984437	−	0.175739i	$-0.0562315\pi$
$458$		0			0
$459$	20006.3	+	34651.9i	0.00443236	+	0.00767707i
$460$		0			0
$461$	2.83684e6			0.621703			0.310851	−	0.950458i	$-0.399386\pi$
$461$	2.83684e6			0.621703			0.310851	+	0.950458i	$0.399386\pi$
$462$		0			0
$463$	5.19089e6			1.12535			0.562677	−	0.826677i	$-0.309771\pi$
$463$	5.19089e6			1.12535			0.562677	+	0.826677i	$0.309771\pi$
$464$		0			0
$465$	2.02480e6	+	3.50705e6i	0.434260	+	0.752160i
$466$		0			0
$467$	−544520.	+	943136.i	−0.115537	+	0.200116i	−0.917994	−	0.396594i	$-0.870192\pi$
$467$	−544520.	+	943136.i	−0.115537	+	0.200116i	0.802457	+	0.596710i	$0.203526\pi$
$468$		0			0
$469$	−307600.	−	2.96137e6i	−0.0645734	−	0.621670i
$470$		0			0
$471$	−880315.	+	1.52475e6i	−0.182846	+	0.316699i
$472$		0			0
$473$	−502322.	−	870047.i	−0.103235	−	0.178809i
$474$		0			0
$475$	−9.40081e6			−1.91175
$476$		0			0
$477$	−1.69892e6			−0.341883
$478$		0			0
$479$	2.84705e6	+	4.93124e6i	0.566966	+	0.982013i	0.996864	+	0.0791359i	$0.0252161\pi$
$479$	2.84705e6	+	4.93124e6i	0.566966	+	0.982013i	−0.429898	+	0.902877i	$0.641451\pi$
$480$		0			0
$481$	−6.31559e6	+	1.09389e7i	−1.24466	+	2.15582i
$482$		0			0
$483$	−3.50269e6	−	1.56462e6i	−0.683177	−	0.305169i
$484$		0			0
$485$	−3.04866e6	+	5.28044e6i	−0.588512	+	1.01933i
$486$		0			0
$487$	2.05713e6	+	3.56305e6i	0.393042	+	0.680768i	0.992849	−	0.119376i	$-0.0380895\pi$
$487$	2.05713e6	+	3.56305e6i	0.393042	+	0.680768i	−0.599807	+	0.800144i	$0.704756\pi$
$488$		0			0
$489$	−338811.			−0.0640745
$490$		0			0
$491$	−609530.			−0.114101			−0.0570507	−	0.998371i	$-0.518170\pi$
$491$	−609530.			−0.114101			−0.0570507	+	0.998371i	$0.518170\pi$
$492$		0			0
$493$	104031.	+	180186.i	0.0192772	+	0.0333891i
$494$		0			0
$495$	529187.	−	916579.i	0.0970725	−	0.168134i
$496$		0			0
$497$	3.11806e6	+	1.39281e6i	0.566231	+	0.252930i
$498$		0			0
$499$	−497896.	+	862382.i	−0.0895133	+	0.155042i	−0.907306	−	0.420472i	$-0.861864\pi$
$499$	−497896.	+	862382.i	−0.0895133	+	0.155042i	0.817792	+	0.575514i	$0.195198\pi$
$500$		0			0
$501$	2.91050e6	+	5.04113e6i	0.518052	+	0.897292i
$502$		0			0
$503$	1.02730e7			1.81042			0.905209	−	0.424966i	$-0.139714\pi$
$503$	1.02730e7			1.81042			0.905209	+	0.424966i	$0.139714\pi$
$504$		0			0
$505$	−1.63171e7			−2.84718
$506$		0			0
$507$	3.88605e6	+	6.73084e6i	0.671412	+	1.16292i
$508$		0			0
$509$	1.39213e6	−	2.41125e6i	0.238170	−	0.412522i	−0.722020	−	0.691873i	$-0.756786\pi$
$509$	1.39213e6	−	2.41125e6i	0.238170	−	0.412522i	0.960189	+	0.279351i	$0.0901193\pi$
$510$		0			0
$511$	741856.	+	7.14210e6i	0.125680	+	1.20997i
$512$		0			0
$513$	−623972.	+	1.08075e6i	−0.104682	+	0.181314i
$514$		0			0
$515$	−7.15264e6	−	1.23887e7i	−1.18836	−	2.05830i
$516$		0			0
$517$	−2.31062e6			−0.380192
$518$		0			0
$519$	3.48218e6			0.567456
$520$		0			0
$521$	−522678.	−	905306.i	−0.0843607	−	0.146117i	0.820758	−	0.571276i	$-0.193551\pi$
$521$	−522678.	−	905306.i	−0.0843607	−	0.146117i	−0.905119	+	0.425159i	$0.860218\pi$
$522$		0			0
$523$	1.81578e6	−	3.14503e6i	0.290275	−	0.502771i	−0.683600	−	0.729857i	$-0.739587\pi$
$523$	1.81578e6	−	3.14503e6i	0.290275	−	0.502771i	0.973875	+	0.227086i	$0.0729199\pi$
$524$		0			0
$525$	−5.18833e6	+	3.75988e6i	−0.821540	+	0.595354i
$526$		0			0
$527$	133027.	−	230410.i	0.0208648	−	0.0361389i
$528$		0			0
$529$	−2.18700e6	−	3.78800e6i	−0.339789	−	0.588532i
$530$		0			0
$531$	−2.93317e6			−0.451440
$532$		0			0
$533$	1.15410e7			1.75964
$534$		0			0
$535$	8.51034e6	+	1.47403e7i	1.28547	+	2.22650i
$536$		0			0
$537$	123347.	−	213644.i	0.0184584	−	0.0319709i
$538$		0			0
$539$	736558.	−	2.24821e6i	0.109203	−	0.333323i
$540$		0			0
$541$	4.72442e6	−	8.18294e6i	0.693994	−	1.20203i	−0.276525	−	0.961007i	$-0.589183\pi$
$541$	4.72442e6	−	8.18294e6i	0.693994	−	1.20203i	0.970519	−	0.241025i	$-0.0774837\pi$
$542$		0			0
$543$	−882698.	−	1.52888e6i	−0.128473	−	0.222522i
$544$		0			0
$545$	1.24985e7			1.80246
$546$		0			0
$547$	2.69014e6			0.384421			0.192210	−	0.981354i	$-0.438434\pi$
$547$	2.69014e6			0.384421			0.192210	+	0.981354i	$0.438434\pi$
$548$		0			0
$549$	−200422.	−	347142.i	−0.0283802	−	0.0491559i
$550$		0			0
$551$	−3.24459e6	+	5.61979e6i	−0.455282	+	0.788572i
$552$		0			0
$553$	−5.24421e6	+	3.80038e6i	−0.729235	+	0.528462i
$554$		0			0
$555$	4.74804e6	−	8.22385e6i	0.654308	−	1.13329i
$556$		0			0
$557$	4.44490e6	+	7.69880e6i	0.607050	+	1.05144i	0.991724	+	0.128389i	$0.0409805\pi$
$557$	4.44490e6	+	7.69880e6i	0.607050	+	1.05144i	−0.384674	+	0.923052i	$0.625686\pi$
$558$		0			0
$559$	−7.93112e6			−1.07351
$560$		0			0
$561$	−69534.1			−0.00932805
$562$		0			0
$563$	−629963.	−	1.09113e6i	−0.0837615	−	0.145079i	0.821101	−	0.570782i	$-0.193360\pi$
$563$	−629963.	−	1.09113e6i	−0.0837615	−	0.145079i	−0.904863	+	0.425703i	$0.860027\pi$
$564$		0			0
$565$	8.21297e6	−	1.42253e7i	1.08238	−	1.87473i
$566$		0			0
$567$	87877.7	+	846028.i	0.0114794	+	0.110517i
$568$		0			0
$569$	2.91019e6	−	5.04059e6i	0.376826	−	0.652681i	−0.613773	−	0.789483i	$-0.710349\pi$
$569$	2.91019e6	−	5.04059e6i	0.376826	−	0.652681i	0.990598	+	0.136801i	$0.0436822\pi$
$570$		0			0
$571$	−2.32980e6	−	4.03534e6i	−0.299040	−	0.517952i	0.676877	−	0.736096i	$-0.263333\pi$
$571$	−2.32980e6	−	4.03534e6i	−0.299040	−	0.517952i	−0.975917	+	0.218144i	$0.930000\pi$
$572$		0			0
$573$	−2.40284e6			−0.305730
$574$		0			0
$575$	−1.80558e7			−2.27744
$576$		0			0
$577$	762391.	+	1.32050e6i	0.0953319	+	0.165120i	0.909747	−	0.415163i	$-0.136275\pi$
$577$	762391.	+	1.32050e6i	0.0953319	+	0.165120i	−0.814415	+	0.580283i	$0.802942\pi$
$578$		0			0
$579$	−456137.	+	790052.i	−0.0565456	+	0.0979398i
$580$		0			0
$581$	−5.30992e6	−	2.37189e6i	−0.652600	−	0.291511i
$582$		0			0
$583$	1.47620e6	−	2.55685e6i	0.179876	−	0.311554i
$584$		0			0
$585$	−4.17764e6	−	7.23589e6i	−0.504710	−	0.874183i
$586$		0			0
$587$	−1.50087e7			−1.79783			−0.898914	−	0.438124i	$-0.855643\pi$
$587$	−1.50087e7			−1.79783			−0.898914	+	0.438124i	$0.855643\pi$
$588$		0			0
$589$	8.29792e6			0.985556
$590$		0			0
$591$	1.63065e6	+	2.82436e6i	0.192040	+	0.332623i
$592$		0			0
$593$	3.53973e6	−	6.13099e6i	0.413364	−	0.715968i	−0.581891	−	0.813267i	$-0.697687\pi$
$593$	3.53973e6	−	6.13099e6i	0.413364	−	0.715968i	0.995255	+	0.0972987i	$0.0310202\pi$
$594$		0			0
$595$	603081.	+	269391.i	0.0698366	+	0.0311954i
$596$		0			0
$597$	1.01436e6	−	1.75692e6i	0.116481	−	0.201751i
$598$		0			0
$599$	1.08761e6	+	1.88379e6i	0.123853	+	0.214519i	0.921284	−	0.388891i	$-0.127142\pi$
$599$	1.08761e6	+	1.88379e6i	0.123853	+	0.214519i	−0.797431	+	0.603410i	$0.793808\pi$
$600$		0			0
$601$	2.69785e6			0.304671			0.152336	−	0.988329i	$-0.451321\pi$
$601$	2.69785e6			0.304671			0.152336	+	0.988329i	$0.451321\pi$
$602$		0			0
$603$	−1.86021e6			−0.208338
$604$		0			0
$605$	−6.55520e6	−	1.13539e7i	−0.728111	−	1.26112i
$606$		0			0
$607$	−2.49417e6	+	4.32003e6i	−0.274760	+	0.475899i	−0.970075	−	0.242807i	$-0.921932\pi$
$607$	−2.49417e6	+	4.32003e6i	−0.274760	+	0.475899i	0.695314	+	0.718706i	$0.255265\pi$
$608$		0			0
$609$	456955.	+	4.39926e6i	0.0499263	+	0.480658i
$610$		0			0
$611$	−9.12056e6	+	1.57973e7i	−0.988367	+	1.71190i
$612$		0			0
$613$	−556820.	−	964441.i	−0.0598500	−	0.103663i	0.834548	−	0.550935i	$-0.185729\pi$
$613$	−556820.	−	964441.i	−0.0598500	−	0.103663i	−0.894398	+	0.447272i	$0.852396\pi$
$614$		0			0
$615$	−8.67647e6			−0.925029
$616$		0			0
$617$	−5.14757e6			−0.544364			−0.272182	−	0.962246i	$-0.587745\pi$
$617$	−5.14757e6			−0.544364			−0.272182	+	0.962246i	$0.587745\pi$
$618$		0			0
$619$	−121588.	−	210597.i	−0.0127545	−	0.0220915i	0.859578	−	0.511005i	$-0.170727\pi$
$619$	−121588.	−	210597.i	−0.0127545	−	0.0220915i	−0.872332	+	0.488914i	$0.837393\pi$
$620$		0			0
$621$	−1.19844e6	+	2.07576e6i	−0.124706	+	0.215998i
$622$		0			0
$623$	1.34311e7	−	9.73322e6i	1.38641	−	1.00470i
$624$		0			0
$625$	−1.61533e6	+	2.79783e6i	−0.165410	+	0.286498i
$626$		0			0
$627$	−1.08434e6	−	1.87814e6i	−0.110153	−	0.190791i
$628$		0			0
$629$	−623884.			−0.0628749
$630$		0			0
$631$	9.94255e6			0.994087			0.497044	−	0.867726i	$-0.334419\pi$
$631$	9.94255e6			0.994087			0.497044	+	0.867726i	$0.334419\pi$
$632$		0			0
$633$	−1.47510e6	−	2.55495e6i	−0.146323	−	0.253439i
$634$		0			0
$635$	6.70347e6	−	1.16108e7i	0.659729	−	1.14268i
$636$		0			0
$637$	−1.24632e7	−	1.39099e7i	−1.21697	−	1.35824i
$638$		0			0
$639$	1.06684e6	−	1.84783e6i	0.103359	−	0.179023i
$640$		0			0
$641$	5.77767e6	+	1.00072e7i	0.555402	+	0.961985i	0.997872	+	0.0652017i	$0.0207691\pi$
$641$	5.77767e6	+	1.00072e7i	0.555402	+	0.961985i	−0.442470	+	0.896783i	$0.645898\pi$
$642$		0			0
$643$	1.35139e6			0.128900			0.0644500	−	0.997921i	$-0.479471\pi$
$643$	1.35139e6			0.128900			0.0644500	+	0.997921i	$0.479471\pi$
$644$		0			0
$645$	5.96260e6			0.564334
$646$		0			0
$647$	−8.77678e6	−	1.52018e7i	−0.824279	−	1.42769i	−0.902469	−	0.430755i	$-0.858247\pi$
$647$	−8.77678e6	−	1.52018e7i	−0.824279	−	1.42769i	0.0781895	−	0.996939i	$-0.475086\pi$
$648$		0			0
$649$	2.54864e6	−	4.41437e6i	0.237518	−	0.411393i
$650$		0			0
$651$	4.57964e6	−	3.31877e6i	0.423525	−	0.306920i
$652$		0			0
$653$	3.99936e6	−	6.92710e6i	0.367035	−	0.635724i	−0.622065	−	0.782965i	$-0.713706\pi$
$653$	3.99936e6	−	6.92710e6i	0.367035	−	0.635724i	0.989101	+	0.147242i	$0.0470395\pi$
$654$		0			0
$655$	−1.10234e7	−	1.90931e7i	−1.00395	−	1.73889i
$656$		0			0
$657$	4.48638e6			0.405492
$658$		0			0
$659$	229419.			0.0205786			0.0102893	−	0.999947i	$-0.496725\pi$
$659$	229419.			0.0205786			0.0102893	+	0.999947i	$0.496725\pi$
$660$		0			0
$661$	1.64279e6	+	2.84539e6i	0.146244	+	0.253302i	0.929836	−	0.367973i	$-0.119948\pi$
$661$	1.64279e6	+	2.84539e6i	0.146244	+	0.253302i	−0.783593	+	0.621275i	$0.786615\pi$
$662$		0			0
$663$	−274467.	+	475391.i	−0.0242497	+	0.0420017i
$664$		0			0
$665$	2.12835e6	+	2.04904e7i	0.186634	+	1.79678i
$666$		0			0
$667$	−6.23177e6	+	1.07937e7i	−0.542372	+	0.939416i
$668$		0			0
$669$	493603.	+	854945.i	0.0426395	+	0.0738538i
$670$		0			0
$671$	696590.			0.0597271
$672$		0			0
$673$	8.22188e6			0.699734			0.349867	−	0.936799i	$-0.386227\pi$
$673$	8.22188e6			0.699734			0.349867	+	0.936799i	$0.386227\pi$
$674$		0			0
$675$	2.00168e6	+	3.46701e6i	0.169097	+	0.292884i
$676$		0			0
$677$	−1.07702e7	+	1.86546e7i	−0.903138	+	1.56428i	−0.0797405	+	0.996816i	$0.525409\pi$
$677$	−1.07702e7	+	1.86546e7i	−0.903138	+	1.56428i	−0.823397	+	0.567465i	$0.807924\pi$
$678$		0			0
$679$	7.77519e6	+	3.47311e6i	0.647197	+	0.289097i
$680$		0			0
$681$	5.47859e6	−	9.48920e6i	0.452690	−	0.784083i
$682$		0			0
$683$	8.08706e6	+	1.40072e7i	0.663344	+	1.14895i	0.979731	+	0.200316i	$0.0641968\pi$
$683$	8.08706e6	+	1.40072e7i	0.663344	+	1.14895i	−0.316387	+	0.948630i	$0.602470\pi$
$684$		0			0
$685$	8.93012e6			0.727162
$686$		0			0
$687$	1.20242e7			0.971999
$688$		0			0
$689$	−1.16538e7	−	2.01850e7i	−0.935231	−	1.61987i
$690$		0			0
$691$	−5.32318e6	+	9.22002e6i	−0.424108	+	0.734576i	−0.996337	−	0.0855179i	$-0.972746\pi$
$691$	−5.32318e6	+	9.22002e6i	−0.424108	+	0.734576i	0.572229	+	0.820094i	$0.306079\pi$
$692$		0			0
$693$	−1.34962e6	−	602862.i	−0.106752	−	0.0476854i
$694$		0			0
$695$	−1.81647e7	+	3.14622e7i	−1.42648	+	2.47074i
$696$		0			0
$697$	285018.	+	493666.i	0.0222224	+	0.0384903i
$698$		0			0
$699$	−421096.			−0.0325978
$700$		0			0
$701$	−4.55461e6			−0.350071			−0.175035	−	0.984562i	$-0.556004\pi$
$701$	−4.55461e6			−0.350071			−0.175035	+	0.984562i	$0.556004\pi$
$702$		0			0
$703$	−9.72909e6	−	1.68513e7i	−0.742479	−	1.28601i
$704$		0			0
$705$	6.85682e6	−	1.18764e7i	0.519577	−	0.899933i
$706$		0			0
$707$	2.35442e6	+	2.26668e7i	0.177148	+	1.70546i
$708$		0			0
$709$	−6.55174e6	+	1.13479e7i	−0.489487	+	0.847816i	−0.999927	−	0.0120971i	$-0.996149\pi$
$709$	−6.55174e6	+	1.13479e7i	−0.489487	+	0.847816i	0.510440	+	0.859913i	$0.329483\pi$
$710$		0			0
$711$	2.02324e6	+	3.50436e6i	0.150098	+	0.259977i
$712$		0			0
$713$	1.59375e7			1.17408
$714$		0			0
$715$	1.45199e7			1.06218
$716$		0			0
$717$	388576.	+	673033.i	0.0282278	+	0.0488921i
$718$		0			0
$719$	−8.95942e6	+	1.55182e7i	−0.646335	+	1.11949i	0.337656	+	0.941269i	$0.390366\pi$
$719$	−8.95942e6	+	1.55182e7i	−0.646335	+	1.11949i	−0.983991	+	0.178216i	$0.942968\pi$
$720$		0			0
$721$	−1.61777e7	+	1.17236e7i	−1.15899	+	0.839894i
$722$		0			0
$723$	3.71384e6	−	6.43257e6i	0.264227	−	0.457655i
$724$		0			0
$725$	1.04085e7	+	1.80281e7i	0.735435	+	1.27381i
$726$		0			0
$727$	1.41466e7			0.992693			0.496347	−	0.868125i	$-0.334675\pi$
$727$	1.41466e7			0.992693			0.496347	+	0.868125i	$0.334675\pi$
$728$		0			0
$729$	531441.			0.0370370
$730$		0			0
$731$	−195868.	−	339254.i	−0.0135572	−	0.0234818i
$732$		0			0
$733$	6.22241e6	−	1.07775e7i	0.427758	−	0.740899i	−0.568915	−	0.822396i	$-0.692637\pi$
$733$	6.22241e6	−	1.07775e7i	0.427758	−	0.740899i	0.996674	+	0.0814969i	$0.0259701\pi$
$734$		0			0
$735$	9.36982e6	+	1.04574e7i	0.639754	+	0.714015i
$736$		0			0
$737$	1.61634e6	−	2.79959e6i	0.109614	−	0.189857i
$738$		0			0
$739$	883322.	+	1.52996e6i	0.0594988	+	0.103055i	0.894241	−	0.447587i	$-0.147716\pi$
$739$	883322.	+	1.52996e6i	0.0594988	+	0.103055i	−0.834742	+	0.550642i	$0.814383\pi$
$740$		0			0
$741$	−1.71206e7			−1.14544
$742$		0			0
$743$	5.73250e6			0.380953			0.190477	−	0.981692i	$-0.438997\pi$
$743$	5.73250e6			0.380953			0.190477	+	0.981692i	$0.438997\pi$
$744$		0			0
$745$	−3.74453e6	−	6.48571e6i	−0.247176	−	0.428122i
$746$		0			0
$747$	−1.81679e6	+	3.14677e6i	−0.119125	+	0.206330i
$748$		0			0
$749$	1.92485e7	−	1.39490e7i	1.25370	−	0.908528i
$750$		0			0
$751$	−1.31566e7	+	2.27879e7i	−0.851224	+	1.47436i	0.0288804	+	0.999583i	$0.490806\pi$
$751$	−1.31566e7	+	2.27879e7i	−0.851224	+	1.47436i	−0.880104	+	0.474780i	$0.842528\pi$
$752$		0			0
$753$	−585844.	−	1.01471e6i	−0.0376526	−	0.0652162i
$754$		0			0
$755$	−8.88179e6			−0.567066
$756$		0			0
$757$	1.63104e7			1.03449			0.517245	−	0.855838i	$-0.326958\pi$
$757$	1.63104e7			1.03449			0.517245	+	0.855838i	$0.326958\pi$
$758$		0			0
$759$	−2.08266e6	−	3.60728e6i	−0.131224	−	0.227287i
$760$		0			0
$761$	−8.42080e6	+	1.45853e7i	−0.527098	+	0.912961i	0.472403	+	0.881383i	$0.343387\pi$
$761$	−8.42080e6	+	1.45853e7i	−0.527098	+	0.912961i	−0.999501	+	0.0315785i	$0.989947\pi$
$762$		0			0
$763$	−1.80342e6	−	1.73622e7i	−0.112147	−	1.07967i
$764$		0			0
$765$	206344.	−	357398.i	0.0127479	−	0.0220800i
$766$		0			0
$767$	−2.01201e7	−	3.48491e7i	−1.23493	−	2.13896i
$768$		0			0
$769$	1.19890e7			0.731084			0.365542	−	0.930795i	$-0.380884\pi$
$769$	1.19890e7			0.731084			0.365542	+	0.930795i	$0.380884\pi$
$770$		0			0
$771$	−925112.			−0.0560477
$772$		0			0
$773$	−8.48868e6	−	1.47028e7i	−0.510965	−	0.885018i	−0.999919	−	0.0127081i	$-0.995955\pi$
$773$	−8.48868e6	−	1.47028e7i	−0.510965	−	0.885018i	0.488954	−	0.872310i	$-0.337379\pi$
$774$		0			0
$775$	1.33097e7	−	2.30531e7i	0.796003	−	1.37872i
$776$		0			0
$777$	−1.21092e7	−	5.40908e6i	−0.719554	−	0.321419i
$778$		0			0
$779$	−8.88937e6	+	1.53968e7i	−0.524841	+	0.909050i
$780$		0			0
$781$	1.85397e6	+	3.21117e6i	0.108761	+	0.188380i
$782$		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)

\(f(q)\)	\(=\)	\(q+(4.50000 + 7.79423i) q^{3} +(46.4128 - 80.3893i) q^{5} +(-118.369 - 52.8745i) q^{7} +(-40.5000 + 70.1481i) q^{9} +O(q^{10})\)	\(q+(4.50000 + 7.79423i) q^{3} +(46.4128 - 80.3893i) q^{5} +(-118.369 - 52.8745i) q^{7} +(-40.5000 + 70.1481i) q^{9} +(-70.3812 - 121.904i) q^{11} -1111.24 q^{13} +835.430 q^{15} +(-27.4435 - 47.5335i) q^{17} +(855.929 - 1482.51i) q^{19} +(-120.546 - 1160.53i) q^{21} +(1643.95 - 2847.41i) q^{23} +(-2745.79 - 4755.85i) q^{25} -729.000 q^{27} -3790.72 q^{29} +(2423.66 + 4197.90i) q^{31} +(633.431 - 1097.13i) q^{33} +(-9744.39 + 7061.57i) q^{35} +(5683.35 - 9843.86i) q^{37} +(-5000.59 - 8661.28i) q^{39} -10385.6 q^{41} +7137.16 q^{43} +(3759.43 + 6511.53i) q^{45} +(8207.53 - 14215.9i) q^{47} +(11215.6 + 12517.4i) q^{49} +(246.991 - 427.801i) q^{51} +(10487.2 + 18164.3i) q^{53} -13066.3 q^{55} +15406.7 q^{57} +(18106.0 + 31360.5i) q^{59} +(-2474.35 + 4285.70i) q^{61} +(8503.00 - 6161.96i) q^{63} +(-51575.9 + 89332.0i) q^{65} +(11482.8 + 19888.8i) q^{67} +29591.2 q^{69} -26341.8 q^{71} +(-27693.7 - 47966.9i) q^{73} +(24712.1 - 42802.6i) q^{75} +(1885.36 + 18151.0i) q^{77} +(24978.3 - 43263.7i) q^{79} +(-3280.50 - 5681.99i) q^{81} +44858.9 q^{83} -5094.91 q^{85} +(-17058.2 - 29545.8i) q^{87} +(-63972.5 + 110804. i) q^{89} +(131537. + 58756.4i) q^{91} +(-21812.9 + 37781.1i) q^{93} +(-79452.1 - 137615. i) q^{95} -65685.9 q^{97} +11401.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 36 q^{3} - 42 q^{7} - 324 q^{9}+O(q^{10}) \) 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

Introduction

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