LMFDB - Embedded newform 10.6.b.a.9.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [10,6,Mod(9,10)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(10, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

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

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

chi := DirichletCharacter("10.9");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 10 = 2 \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	10.b (of order $2$, degree $1$, 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:	$1.60383819813$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			9.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	10.9
Dual form			10.6.b.a.9.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.00000i q^{2} +14.0000i q^{3} -16.0000 q^{4} +(55.0000 + 10.0000i) q^{5} -56.0000 q^{6} -158.000i q^{7} -64.0000i q^{8} +47.0000 q^{9} +(-40.0000 + 220.000i) q^{10} -148.000 q^{11} -224.000i q^{12} +684.000i q^{13} +632.000 q^{14} +(-140.000 + 770.000i) q^{15} +256.000 q^{16} -2048.00i q^{17} +188.000i q^{18} -2220.00 q^{19} +(-880.000 - 160.000i) q^{20} +2212.00 q^{21} -592.000i q^{22} -1246.00i q^{23} +896.000 q^{24} +(2925.00 + 1100.00i) q^{25} -2736.00 q^{26} +4060.00i q^{27} +2528.00i q^{28} +270.000 q^{29} +(-3080.00 - 560.000i) q^{30} -2048.00 q^{31} +1024.00i q^{32} -2072.00i q^{33} +8192.00 q^{34} +(1580.00 - 8690.00i) q^{35} -752.000 q^{36} +4372.00i q^{37} -8880.00i q^{38} -9576.00 q^{39} +(640.000 - 3520.00i) q^{40} -2398.00 q^{41} +8848.00i q^{42} +2294.00i q^{43} +2368.00 q^{44} +(2585.00 + 470.000i) q^{45} +4984.00 q^{46} +10682.0i q^{47} +3584.00i q^{48} -8157.00 q^{49} +(-4400.00 + 11700.0i) q^{50} +28672.0 q^{51} -10944.0i q^{52} +2964.00i q^{53} -16240.0 q^{54} +(-8140.00 - 1480.00i) q^{55} -10112.0 q^{56} -31080.0i q^{57} +1080.00i q^{58} +39740.0 q^{59} +(2240.00 - 12320.0i) q^{60} -42298.0 q^{61} -8192.00i q^{62} -7426.00i q^{63} -4096.00 q^{64} +(-6840.00 + 37620.0i) q^{65} +8288.00 q^{66} -32098.0i q^{67} +32768.0i q^{68} +17444.0 q^{69} +(34760.0 + 6320.00i) q^{70} -4248.00 q^{71} -3008.00i q^{72} +30104.0i q^{73} -17488.0 q^{74} +(-15400.0 + 40950.0i) q^{75} +35520.0 q^{76} +23384.0i q^{77} -38304.0i q^{78} -35280.0 q^{79} +(14080.0 + 2560.00i) q^{80} -45419.0 q^{81} -9592.00i q^{82} -27826.0i q^{83} -35392.0 q^{84} +(20480.0 - 112640. i) q^{85} -9176.00 q^{86} +3780.00i q^{87} +9472.00i q^{88} +85210.0 q^{89} +(-1880.00 + 10340.0i) q^{90} +108072. q^{91} +19936.0i q^{92} -28672.0i q^{93} -42728.0 q^{94} +(-122100. - 22200.0i) q^{95} -14336.0 q^{96} +97232.0i q^{97} -32628.0i q^{98} -6956.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 32 q^{4} + 110 q^{5} - 112 q^{6} + 94 q^{9} - 80 q^{10} - 296 q^{11} + 1264 q^{14} - 280 q^{15} + 512 q^{16} - 4440 q^{19} - 1760 q^{20} + 4424 q^{21} + 1792 q^{24} + 5850 q^{25} - 5472 q^{26} + 540 q^{29}+ \cdots - 13912 q^{99}+O(q^{100}) $ 2 * q - 32 * q^4 + 110 * q^5 - 112 * q^6 + 94 * q^9 - 80 * q^10 - 296 * q^11 + 1264 * q^14 - 280 * q^15 + 512 * q^16 - 4440 * q^19 - 1760 * q^20 + 4424 * q^21 + 1792 * q^24 + 5850 * q^25 - 5472 * q^26 + 540 * q^29 - 6160 * q^30 - 4096 * q^31 + 16384 * q^34 + 3160 * q^35 - 1504 * q^36 - 19152 * q^39 + 1280 * q^40 - 4796 * q^41 + 4736 * q^44 + 5170 * q^45 + 9968 * q^46 - 16314 * q^49 - 8800 * q^50 + 57344 * q^51 - 32480 * q^54 - 16280 * q^55 - 20224 * q^56 + 79480 * q^59 + 4480 * q^60 - 84596 * q^61 - 8192 * q^64 - 13680 * q^65 + 16576 * q^66 + 34888 * q^69 + 69520 * q^70 - 8496 * q^71 - 34976 * q^74 - 30800 * q^75 + 71040 * q^76 - 70560 * q^79 + 28160 * q^80 - 90838 * q^81 - 70784 * q^84 + 40960 * q^85 - 18352 * q^86 + 170420 * q^89 - 3760 * q^90 + 216144 * q^91 - 85456 * q^94 - 244200 * q^95 - 28672 * q^96 - 13912 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/10\mathbb{Z}\right)^\times$.

$n$	$7$
$\chi(n)$	$-1$

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 $

$2$			4.00000i			0.707107i
$2$			4.00000i			0.707107i
$3$			14.0000i			0.898100i	0.893507	+	0.449050i	$0.148238\pi$
$3$			14.0000i			0.898100i	−0.893507	+	0.449050i	$0.851762\pi$
$4$	−16.0000			−0.500000
$5$	55.0000	+	10.0000i	0.983870	+	0.178885i
$5$	55.0000	+	10.0000i	0.983870	+	0.178885i
$6$	−56.0000			−0.635053
$7$		−	158.000i		−	1.21874i	−0.792885	−	0.609371i	$-0.791422\pi$
$7$		−	158.000i		−	1.21874i	0.792885	−	0.609371i	$-0.208578\pi$
$8$		−	64.0000i		−	0.353553i
$9$	47.0000			0.193416
$10$	−40.0000	+	220.000i	−0.126491	+	0.695701i
$11$	−148.000			−0.368791			−0.184395	−	0.982852i	$-0.559033\pi$
$11$	−148.000			−0.368791			−0.184395	+	0.982852i	$0.559033\pi$
$12$		−	224.000i		−	0.449050i
$13$			684.000i			1.12253i	0.827636	+	0.561265i	$0.189685\pi$
$13$			684.000i			1.12253i	−0.827636	+	0.561265i	$0.810315\pi$
$14$	632.000			0.861781
$15$	−140.000	+	770.000i	−0.160657	+	0.883614i
$16$	256.000			0.250000
$17$		−	2048.00i		−	1.71873i	−0.511363	−	0.859365i	$-0.670859\pi$
$17$		−	2048.00i		−	1.71873i	0.511363	−	0.859365i	$-0.329141\pi$
$18$			188.000i			0.136766i
$19$	−2220.00			−1.41081			−0.705406	−	0.708804i	$-0.749235\pi$
$19$	−2220.00			−1.41081			−0.705406	+	0.708804i	$0.749235\pi$
$20$	−880.000	−	160.000i	−0.491935	−	0.0894427i
$21$	2212.00			1.09455
$22$		−	592.000i		−	0.260774i
$23$		−	1246.00i		−	0.491132i	−0.969380	−	0.245566i	$-0.921026\pi$
$23$		−	1246.00i		−	0.491132i	0.969380	−	0.245566i	$-0.0789738\pi$
$24$	896.000			0.317526
$25$	2925.00	+	1100.00i	0.936000	+	0.352000i
$26$	−2736.00			−0.793748
$27$			4060.00i			1.07181i
$28$			2528.00i			0.609371i
$29$	270.000			0.0596168			0.0298084	−	0.999556i	$-0.490510\pi$
$29$	270.000			0.0596168			0.0298084	+	0.999556i	$0.490510\pi$
$30$	−3080.00	−	560.000i	−0.624809	−	0.113602i
$31$	−2048.00			−0.382759			−0.191380	−	0.981516i	$-0.561296\pi$
$31$	−2048.00			−0.382759			−0.191380	+	0.981516i	$0.561296\pi$
$32$			1024.00i			0.176777i
$33$		−	2072.00i		−	0.331211i
$34$	8192.00			1.21533
$35$	1580.00	−	8690.00i	0.218015	−	1.19908i
$36$	−752.000			−0.0967078
$37$			4372.00i			0.525020i	0.964929	+	0.262510i	$0.0845503\pi$
$37$			4372.00i			0.525020i	−0.964929	+	0.262510i	$0.915450\pi$
$38$		−	8880.00i		−	0.997594i
$39$	−9576.00			−1.00814
$40$	640.000	−	3520.00i	0.0632456	−	0.347851i
$41$	−2398.00			−0.222787			−0.111393	−	0.993776i	$-0.535531\pi$
$41$	−2398.00			−0.222787			−0.111393	+	0.993776i	$0.535531\pi$
$42$			8848.00i			0.773966i
$43$			2294.00i			0.189200i	0.995515	+	0.0946002i	$0.0301573\pi$
$43$			2294.00i			0.189200i	−0.995515	+	0.0946002i	$0.969843\pi$
$44$	2368.00			0.184395
$45$	2585.00	+	470.000i	0.190296	+	0.0345992i
$46$	4984.00			0.347283
$47$			10682.0i			0.705355i	0.935745	+	0.352678i	$0.114729\pi$
$47$			10682.0i			0.705355i	−0.935745	+	0.352678i	$0.885271\pi$
$48$			3584.00i			0.224525i
$49$	−8157.00			−0.485333
$50$	−4400.00	+	11700.0i	−0.248902	+	0.661852i
$51$	28672.0			1.54359
$52$		−	10944.0i		−	0.561265i
$53$			2964.00i			0.144940i	0.997371	+	0.0724700i	$0.0230882\pi$
$53$			2964.00i			0.144940i	−0.997371	+	0.0724700i	$0.976912\pi$
$54$	−16240.0			−0.757882
$55$	−8140.00	−	1480.00i	−0.362842	−	0.0659713i
$56$	−10112.0			−0.430891
$57$		−	31080.0i		−	1.26705i
$58$			1080.00i			0.0421555i
$59$	39740.0			1.48627			0.743135	−	0.669141i	$-0.233338\pi$
$59$	39740.0			1.48627			0.743135	+	0.669141i	$0.233338\pi$
$60$	2240.00	−	12320.0i	0.0803285	−	0.441807i
$61$	−42298.0			−1.45544			−0.727722	−	0.685873i	$-0.759421\pi$
$61$	−42298.0			−1.45544			−0.727722	+	0.685873i	$0.759421\pi$
$62$		−	8192.00i		−	0.270652i
$63$		−	7426.00i		−	0.235724i
$64$	−4096.00			−0.125000
$65$	−6840.00	+	37620.0i	−0.200804	+	1.10442i
$66$	8288.00			0.234202
$67$		−	32098.0i		−	0.873556i	−0.899569	−	0.436778i	$-0.856119\pi$
$67$		−	32098.0i		−	0.873556i	0.899569	−	0.436778i	$-0.143881\pi$
$68$			32768.0i			0.859365i
$69$	17444.0			0.441086
$70$	34760.0	+	6320.00i	0.847881	+	0.154160i
$71$	−4248.00			−0.100009			−0.0500044	−	0.998749i	$-0.515924\pi$
$71$	−4248.00			−0.100009			−0.0500044	+	0.998749i	$0.515924\pi$
$72$		−	3008.00i		−	0.0683828i
$73$			30104.0i			0.661176i	0.943775	+	0.330588i	$0.107247\pi$
$73$			30104.0i			0.661176i	−0.943775	+	0.330588i	$0.892753\pi$
$74$	−17488.0			−0.371245
$75$	−15400.0	+	40950.0i	−0.316131	+	0.840622i
$76$	35520.0			0.705406
$77$			23384.0i			0.449461i
$78$		−	38304.0i		−	0.712866i
$79$	−35280.0			−0.636005			−0.318003	−	0.948090i	$-0.603012\pi$
$79$	−35280.0			−0.636005			−0.318003	+	0.948090i	$0.603012\pi$
$80$	14080.0	+	2560.00i	0.245967	+	0.0447214i
$81$	−45419.0			−0.769175
$82$		−	9592.00i		−	0.157534i
$83$		−	27826.0i		−	0.443359i	−0.975120	−	0.221680i	$-0.928846\pi$
$83$		−	27826.0i		−	0.443359i	0.975120	−	0.221680i	$-0.0711539\pi$
$84$	−35392.0			−0.547277
$85$	20480.0	−	112640.i	0.307456	−	1.69101i
$86$	−9176.00			−0.133785
$87$			3780.00i			0.0535419i
$88$			9472.00i			0.130387i
$89$	85210.0			1.14029			0.570145	−	0.821544i	$-0.306887\pi$
$89$	85210.0			1.14029			0.570145	+	0.821544i	$0.306887\pi$
$90$	−1880.00	+	10340.0i	−0.0244654	+	0.134559i
$91$	108072.			1.36807
$92$			19936.0i			0.245566i
$93$		−	28672.0i		−	0.343756i
$94$	−42728.0			−0.498762
$95$	−122100.	−	22200.0i	−1.38805	−	0.252374i
$96$	−14336.0			−0.158763
$97$			97232.0i			1.04925i	0.851333	+	0.524626i	$0.175795\pi$
$97$			97232.0i			1.04925i	−0.851333	+	0.524626i	$0.824205\pi$
$98$		−	32628.0i		−	0.343183i
$99$	−6956.00			−0.0713299
$100$	−46800.0	−	17600.0i	−0.468000	−	0.176000i
$101$	−4298.00			−0.0419240			−0.0209620	−	0.999780i	$-0.506673\pi$
$101$	−4298.00			−0.0419240			−0.0209620	+	0.999780i	$0.506673\pi$
$102$			114688.i			1.09148i
$103$			124114.i			1.15273i	0.817192	+	0.576365i	$0.195529\pi$
$103$			124114.i			1.15273i	−0.817192	+	0.576365i	$0.804471\pi$
$104$	43776.0			0.396874
$105$	121660.	+	22120.0i	1.07690	+	0.195800i
$106$	−11856.0			−0.102488
$107$			42342.0i			0.357530i	0.983892	+	0.178765i	$0.0572101\pi$
$107$			42342.0i			0.357530i	−0.983892	+	0.178765i	$0.942790\pi$
$108$		−	64960.0i		−	0.535904i
$109$	35990.0			0.290145			0.145073	−	0.989421i	$-0.453658\pi$
$109$	35990.0			0.290145			0.145073	+	0.989421i	$0.453658\pi$
$110$	5920.00	−	32560.0i	0.0466487	−	0.256568i
$111$	−61208.0			−0.471521
$112$		−	40448.0i		−	0.304686i
$113$		−	228816.i		−	1.68574i	−0.538118	−	0.842869i	$-0.680865\pi$
$113$		−	228816.i		−	1.68574i	0.538118	−	0.842869i	$-0.319135\pi$
$114$	124320.			0.895940
$115$	12460.0	−	68530.0i	0.0878564	−	0.483210i
$116$	−4320.00			−0.0298084
$117$			32148.0i			0.217115i
$118$			158960.i			1.05095i
$119$	−323584.			−2.09469
$120$	49280.0	+	8960.00i	0.312405	+	0.0568009i
$121$	−139147.			−0.863993
$122$		−	169192.i		−	1.02915i
$123$		−	33572.0i		−	0.200085i
$124$	32768.0			0.191380
$125$	149875.	+	89750.0i	0.857935	+	0.513759i
$126$	29704.0			0.166682
$127$		−	175238.i		−	0.964093i	−0.876146	−	0.482047i	$-0.839894\pi$
$127$		−	175238.i		−	0.964093i	0.876146	−	0.482047i	$-0.160106\pi$
$128$		−	16384.0i		−	0.0883883i
$129$	−32116.0			−0.169921
$130$	−150480.	−	27360.0i	−0.780945	−	0.141990i
$131$	299652.			1.52559			0.762797	−	0.646638i	$-0.223826\pi$
$131$	299652.			1.52559			0.762797	+	0.646638i	$0.223826\pi$
$132$			33152.0i			0.165606i
$133$			350760.i			1.71942i
$134$	128392.			0.617698
$135$	−40600.0	+	223300.i	−0.191731	+	1.05452i
$136$	−131072.			−0.607663
$137$		−	107928.i		−	0.491284i	−0.969361	−	0.245642i	$-0.921001\pi$
$137$		−	107928.i		−	0.491284i	0.969361	−	0.245642i	$-0.0789988\pi$
$138$			69776.0i			0.311895i
$139$	196460.			0.862456			0.431228	−	0.902243i	$-0.358080\pi$
$139$	196460.			0.862456			0.431228	+	0.902243i	$0.358080\pi$
$140$	−25280.0	+	139040.i	−0.109008	+	0.599542i
$141$	−149548.			−0.633480
$142$		−	16992.0i		−	0.0707170i
$143$		−	101232.i		−	0.413978i
$144$	12032.0			0.0483539
$145$	14850.0	+	2700.00i	0.0586552	+	0.0106646i
$146$	−120416.			−0.467522
$147$		−	114198.i		−	0.435878i
$148$		−	69952.0i		−	0.262510i
$149$	−138850.			−0.512366			−0.256183	−	0.966628i	$-0.582465\pi$
$149$	−138850.			−0.512366			−0.256183	+	0.966628i	$0.582465\pi$
$150$	−163800.	−	61600.0i	−0.594410	−	0.223539i
$151$	416152.			1.48528			0.742642	−	0.669688i	$-0.233572\pi$
$151$	416152.			1.48528			0.742642	+	0.669688i	$0.233572\pi$
$152$			142080.i			0.498797i
$153$		−	96256.0i		−	0.332429i
$154$	−93536.0			−0.317817
$155$	−112640.	−	20480.0i	−0.376585	−	0.0684701i
$156$	153216.			0.504072
$157$		−	433108.i		−	1.40232i	−0.713004	−	0.701160i	$-0.752666\pi$
$157$		−	433108.i		−	1.40232i	0.713004	−	0.701160i	$-0.247334\pi$
$158$		−	141120.i		−	0.449724i
$159$	−41496.0			−0.130171
$160$	−10240.0	+	56320.0i	−0.0316228	+	0.173925i
$161$	−196868.			−0.598564
$162$		−	181676.i		−	0.543889i
$163$			149134.i			0.439651i	0.975539	+	0.219825i	$0.0705487\pi$
$163$			149134.i			0.439651i	−0.975539	+	0.219825i	$0.929451\pi$
$164$	38368.0			0.111393
$165$	20720.0	−	113960.i	0.0592488	−	0.325869i
$166$	111304.			0.313502
$167$			559602.i			1.55270i	0.630301	+	0.776351i	$0.282932\pi$
$167$			559602.i			1.55270i	−0.630301	+	0.776351i	$0.717068\pi$
$168$		−	141568.i		−	0.386983i
$169$	−96563.0			−0.260072
$170$	450560.	+	81920.0i	1.19572	+	0.217404i
$171$	−104340.			−0.272873
$172$		−	36704.0i		−	0.0946002i
$173$			343804.i			0.873365i	0.899616	+	0.436682i	$0.143847\pi$
$173$			343804.i			0.873365i	−0.899616	+	0.436682i	$0.856153\pi$
$174$	−15120.0			−0.0378598
$175$	173800.	−	462150.i	0.428997	−	1.14074i
$176$	−37888.0			−0.0921977
$177$			556360.i			1.33482i
$178$			340840.i			0.806307i
$179$	−23980.0			−0.0559392			−0.0279696	−	0.999609i	$-0.508904\pi$
$179$	−23980.0			−0.0559392			−0.0279696	+	0.999609i	$0.508904\pi$
$180$	−41360.0	−	7520.00i	−0.0951479	−	0.0172996i
$181$	−651898.			−1.47905			−0.739526	−	0.673128i	$-0.764950\pi$
$181$	−651898.			−1.47905			−0.739526	+	0.673128i	$0.764950\pi$
$182$			432288.i			0.967375i
$183$		−	592172.i		−	1.30713i
$184$	−79744.0			−0.173641
$185$	−43720.0	+	240460.i	−0.0939184	+	0.516551i
$186$	114688.			0.243072
$187$			303104.i			0.633852i
$188$		−	170912.i		−	0.352678i
$189$	641480.			1.30626
$190$	88800.0	−	488400.i	0.178455	−	0.981503i
$191$	202752.			0.402144			0.201072	−	0.979576i	$-0.435557\pi$
$191$	202752.			0.402144			0.201072	+	0.979576i	$0.435557\pi$
$192$		−	57344.0i		−	0.112263i
$193$		−	452656.i		−	0.874732i	−0.899284	−	0.437366i	$-0.855911\pi$
$193$		−	452656.i		−	0.874732i	0.899284	−	0.437366i	$-0.144089\pi$
$194$	−388928.			−0.741933
$195$	−526680.	−	95760.0i	−0.991883	−	0.180342i
$196$	130512.			0.242667
$197$		−	337468.i		−	0.619537i	−0.950812	−	0.309768i	$-0.899748\pi$
$197$		−	337468.i		−	0.619537i	0.950812	−	0.309768i	$-0.100252\pi$
$198$		−	27824.0i		−	0.0504379i
$199$	561000.			1.00422			0.502112	−	0.864803i	$-0.332557\pi$
$199$	561000.			1.00422			0.502112	+	0.864803i	$0.332557\pi$
$200$	70400.0	−	187200.i	0.124451	−	0.330926i
$201$	449372.			0.784541
$202$		−	17192.0i		−	0.0296448i
$203$		−	42660.0i		−	0.0726576i
$204$	−458752.			−0.771796
$205$	−131890.	−	23980.0i	−0.219193	−	0.0398533i
$206$	−496456.			−0.815103
$207$		−	58562.0i		−	0.0949927i
$208$			175104.i			0.280632i
$209$	328560.			0.520294
$210$	−88480.0	+	486640.i	−0.138451	+	0.761482i
$211$	−805548.			−1.24562			−0.622810	−	0.782373i	$-0.714009\pi$
$211$	−805548.			−1.24562			−0.622810	+	0.782373i	$0.714009\pi$
$212$		−	47424.0i		−	0.0724700i
$213$		−	59472.0i		−	0.0898180i
$214$	−169368.			−0.252812
$215$	−22940.0	+	126170.i	−0.0338452	+	0.186149i
$216$	259840.			0.378941
$217$			323584.i			0.466485i
$218$			143960.i			0.205164i
$219$	−421456.			−0.593802
$220$	130240.	+	23680.0i	0.181421	+	0.0329856i
$221$	1.40083e6			1.92932
$222$		−	244832.i		−	0.333415i
$223$			1.21855e6i			1.64090i	0.571717	+	0.820451i	$0.306278\pi$
$223$			1.21855e6i			1.64090i	−0.571717	+	0.820451i	$0.693722\pi$
$224$	161792.			0.215445
$225$	137475.	+	51700.0i	0.181037	+	0.0680823i
$226$	915264.			1.19200
$227$		−	564338.i		−	0.726900i	−0.931614	−	0.363450i	$-0.881599\pi$
$227$		−	564338.i		−	0.726900i	0.931614	−	0.363450i	$-0.118401\pi$
$228$			497280.i			0.633525i
$229$	−560330.			−0.706082			−0.353041	−	0.935608i	$-0.614852\pi$
$229$	−560330.			−0.706082			−0.353041	+	0.935608i	$0.614852\pi$
$230$	274120.	+	49840.0i	0.341681	+	0.0621239i
$231$	−327376.			−0.403661
$232$		−	17280.0i		−	0.0210777i
$233$		−	293576.i		−	0.354267i	−0.984187	−	0.177134i	$-0.943318\pi$
$233$		−	293576.i		−	0.354267i	0.984187	−	0.177134i	$-0.0566824\pi$
$234$	−128592.			−0.153523
$235$	−106820.	+	587510.i	−0.126178	+	0.693978i
$236$	−635840.			−0.743135
$237$		−	493920.i		−	0.571197i
$238$		−	1.29434e6i		−	1.48117i
$239$	−584240.			−0.661602			−0.330801	−	0.943701i	$-0.607319\pi$
$239$	−584240.			−0.661602			−0.330801	+	0.943701i	$0.607319\pi$
$240$	−35840.0	+	197120.i	−0.0401643	+	0.220903i
$241$	−563798.			−0.625289			−0.312645	−	0.949870i	$-0.601215\pi$
$241$	−563798.			−0.625289			−0.312645	+	0.949870i	$0.601215\pi$
$242$		−	556588.i		−	0.610936i
$243$			350714.i			0.381011i
$244$	676768.			0.727722
$245$	−448635.	−	81570.0i	−0.477505	−	0.0868191i
$246$	134288.			0.141481
$247$		−	1.51848e6i		−	1.58368i
$248$			131072.i			0.135326i
$249$	389564.			0.398181
$250$	−359000.	+	599500.i	−0.363282	+	0.606651i
$251$	−1.01975e6			−1.02167			−0.510833	−	0.859680i	$-0.670663\pi$
$251$	−1.01975e6			−1.02167			−0.510833	+	0.859680i	$0.670663\pi$
$252$			118816.i			0.117862i
$253$			184408.i			0.181125i
$254$	700952.			0.681717
$255$	1.57696e6	+	286720.i	1.51869	+	0.276126i
$256$	65536.0			0.0625000
$257$		−	657408.i		−	0.620872i	−0.950594	−	0.310436i	$-0.899525\pi$
$257$		−	657408.i		−	0.620872i	0.950594	−	0.310436i	$-0.100475\pi$
$258$		−	128464.i		−	0.120152i
$259$	690776.			0.639864
$260$	109440.	−	601920.i	0.100402	−	0.552211i
$261$	12690.0			0.0115308
$262$			1.19861e6i			1.07876i
$263$		−	562366.i		−	0.501337i	−0.968073	−	0.250668i	$-0.919350\pi$
$263$		−	562366.i		−	0.501337i	0.968073	−	0.250668i	$-0.0806504\pi$
$264$	−132608.			−0.117101
$265$	−29640.0	+	163020.i	−0.0259277	+	0.142602i
$266$	−1.40304e6			−1.21581
$267$			1.19294e6i			1.02410i
$268$			513568.i			0.436778i
$269$	−366570.			−0.308870			−0.154435	−	0.988003i	$-0.549356\pi$
$269$	−366570.			−0.308870			−0.154435	+	0.988003i	$0.549356\pi$
$270$	−893200.	−	162400.i	−0.745657	−	0.135574i
$271$	1.16075e6			0.960099			0.480050	−	0.877241i	$-0.340619\pi$
$271$	1.16075e6			0.960099			0.480050	+	0.877241i	$0.340619\pi$
$272$		−	524288.i		−	0.429682i
$273$			1.51301e6i			1.22867i
$274$	431712.			0.347390
$275$	−432900.	−	162800.i	−0.345188	−	0.129814i
$276$	−279104.			−0.220543
$277$			2.51501e6i			1.96943i	0.174172	+	0.984715i	$0.444275\pi$
$277$			2.51501e6i			1.96943i	−0.174172	+	0.984715i	$0.555725\pi$
$278$			785840.i			0.609849i
$279$	−96256.0			−0.0740316
$280$	−556160.	−	101120.i	−0.423940	−	0.0770800i
$281$	2.08600e6			1.57597			0.787987	−	0.615692i	$-0.211124\pi$
$281$	2.08600e6			1.57597			0.787987	+	0.615692i	$0.211124\pi$
$282$		−	598192.i		−	0.447938i
$283$		−	2.23803e6i		−	1.66111i	−0.556935	−	0.830556i	$-0.688023\pi$
$283$		−	2.23803e6i		−	1.66111i	0.556935	−	0.830556i	$-0.311977\pi$
$284$	67968.0			0.0500044
$285$	310800.	−	1.70940e6i	0.226657	−	1.24661i
$286$	404928.			0.292727
$287$			378884.i			0.271520i
$288$			48128.0i			0.0341914i
$289$	−2.77445e6			−1.95403
$290$	−10800.0	+	59400.0i	−0.00754100	+	0.0414755i
$291$	−1.36125e6			−0.942334
$292$		−	481664.i		−	0.330588i
$293$		−	975756.i		−	0.664006i	−0.943278	−	0.332003i	$-0.892276\pi$
$293$		−	975756.i		−	0.664006i	0.943278	−	0.332003i	$-0.107724\pi$
$294$	456792.			0.308212
$295$	2.18570e6	+	397400.i	1.46230	+	0.265872i
$296$	279808.			0.185623
$297$		−	600880.i		−	0.395273i
$298$		−	555400.i		−	0.362297i
$299$	852264.			0.551310
$300$	246400.	−	655200.i	0.158066	−	0.420311i
$301$	362452.			0.230587
$302$			1.66461e6i			1.05025i
$303$		−	60172.0i		−	0.0376520i
$304$	−568320.			−0.352703
$305$	−2.32639e6	−	422980.i	−1.43197	−	0.260358i
$306$	385024.			0.235063
$307$		−	87858.0i		−	0.0532029i	−0.999646	−	0.0266015i	$-0.991531\pi$
$307$		−	87858.0i		−	0.0532029i	0.999646	−	0.0266015i	$-0.00846850\pi$
$308$		−	374144.i		−	0.224730i
$309$	−1.73760e6			−1.03527
$310$	81920.0	−	450560.i	0.0484156	−	0.266286i
$311$	599352.			0.351383			0.175692	−	0.984445i	$-0.443784\pi$
$311$	599352.			0.351383			0.175692	+	0.984445i	$0.443784\pi$
$312$			612864.i			0.356433i
$313$		−	2.09342e6i		−	1.20780i	−0.797060	−	0.603900i	$-0.793613\pi$
$313$		−	2.09342e6i		−	1.20780i	0.797060	−	0.603900i	$-0.206387\pi$
$314$	1.73243e6			0.991590
$315$	74260.0	−	408430.i	0.0421676	−	0.231922i
$316$	564480.			0.318003
$317$			2.41625e6i			1.35050i	0.737590	+	0.675249i	$0.235964\pi$
$317$			2.41625e6i			1.35050i	−0.737590	+	0.675249i	$0.764036\pi$
$318$		−	165984.i		−	0.0920446i
$319$	−39960.0			−0.0219861
$320$	−225280.	−	40960.0i	−0.122984	−	0.0223607i
$321$	−592788.			−0.321097
$322$		−	787472.i		−	0.423249i
$323$			4.54656e6i			2.42480i
$324$	726704.			0.384587
$325$	−752400.	+	2.00070e6i	−0.395130	+	1.05069i
$326$	−596536.			−0.310880
$327$			503860.i			0.260580i
$328$			153472.i			0.0787670i
$329$	1.68776e6			0.859647
$330$	455840.	+	82880.0i	0.230424	+	0.0418953i
$331$	−1.64095e6			−0.823237			−0.411618	−	0.911356i	$-0.635036\pi$
$331$	−1.64095e6			−0.823237			−0.411618	+	0.911356i	$0.635036\pi$
$332$			445216.i			0.221680i
$333$			205484.i			0.101547i
$334$	−2.23841e6			−1.09793
$335$	320980.	−	1.76539e6i	0.156267	−	0.859466i
$336$	566272.			0.273638
$337$		−	2.18773e6i		−	1.04935i	−0.851304	−	0.524673i	$-0.824188\pi$
$337$		−	2.18773e6i		−	1.04935i	0.851304	−	0.524673i	$-0.175812\pi$
$338$		−	386252.i		−	0.183899i
$339$	3.20342e6			1.51396
$340$	−327680.	+	1.80224e6i	−0.153728	+	0.845503i
$341$	303104.			0.141158
$342$		−	417360.i		−	0.192950i
$343$		−	1.36670e6i		−	0.627246i
$344$	146816.			0.0668925
$345$	959420.	+	174440.i	0.433971	+	0.0789039i
$346$	−1.37522e6			−0.617562
$347$		−	2.74502e6i		−	1.22383i	−0.790923	−	0.611916i	$-0.790399\pi$
$347$		−	2.74502e6i		−	1.22383i	0.790923	−	0.611916i	$-0.209601\pi$
$348$		−	60480.0i		−	0.0267709i
$349$	2.65115e6			1.16512			0.582560	−	0.812788i	$-0.302051\pi$
$349$	2.65115e6			1.16512			0.582560	+	0.812788i	$0.302051\pi$
$350$	1.84860e6	+	695200.i	0.806627	+	0.303347i
$351$	−2.77704e6			−1.20313
$352$		−	151552.i		−	0.0651936i
$353$			3.05766e6i			1.30603i	0.757345	+	0.653015i	$0.226496\pi$
$353$			3.05766e6i			1.30603i	−0.757345	+	0.653015i	$0.773504\pi$
$354$	−2.22544e6			−0.943860
$355$	−233640.	−	42480.0i	−0.0983957	−	0.0178901i
$356$	−1.36336e6			−0.570145
$357$		−	4.53018e6i		−	1.88124i
$358$		−	95920.0i		−	0.0395550i
$359$	−3.79356e6			−1.55350			−0.776749	−	0.629810i	$-0.783133\pi$
$359$	−3.79356e6			−1.55350			−0.776749	+	0.629810i	$0.783133\pi$
$360$	30080.0	−	165440.i	0.0122327	−	0.0672797i
$361$	2.45230e6			0.990389
$362$		−	2.60759e6i		−	1.04585i
$363$		−	1.94806e6i		−	0.775953i
$364$	−1.72915e6			−0.684037
$365$	−301040.	+	1.65572e6i	−0.118275	+	0.650511i
$366$	2.36869e6			0.924283
$367$			3.11060e6i			1.20553i	0.797917	+	0.602767i	$0.205935\pi$
$367$			3.11060e6i			1.20553i	−0.797917	+	0.602767i	$0.794065\pi$
$368$		−	318976.i		−	0.122783i
$369$	−112706.			−0.0430905
$370$	−961840.	−	174880.i	−0.365257	−	0.0664104i
$371$	468312.			0.176645
$372$			458752.i			0.171878i
$373$		−	1.41520e6i		−	0.526677i	−0.964703	−	0.263339i	$-0.915176\pi$
$373$		−	1.41520e6i		−	0.526677i	0.964703	−	0.263339i	$-0.0848236\pi$
$374$	−1.21242e6			−0.448201
$375$	−1.25650e6	+	2.09825e6i	−0.461407	+	0.770511i
$376$	683648.			0.249381
$377$			184680.i			0.0669216i
$378$			2.56592e6i			0.923663i
$379$	3.90262e6			1.39559			0.697796	−	0.716297i	$-0.254164\pi$
$379$	3.90262e6			1.39559			0.697796	+	0.716297i	$0.254164\pi$
$380$	1.95360e6	+	355200.i	0.694027	+	0.126187i
$381$	2.45333e6			0.865852
$382$			811008.i			0.284359i
$383$			695674.i			0.242331i	0.992632	+	0.121165i	$0.0386632\pi$
$383$			695674.i			0.242331i	−0.992632	+	0.121165i	$0.961337\pi$
$384$	229376.			0.0793816
$385$	−233840.	+	1.28612e6i	−0.0804020	+	0.442211i
$386$	1.81062e6			0.618529
$387$			107818.i			0.0365943i
$388$		−	1.55571e6i		−	0.524626i
$389$	−498290.			−0.166958			−0.0834792	−	0.996510i	$-0.526603\pi$
$389$	−498290.			−0.166958			−0.0834792	+	0.996510i	$0.526603\pi$
$390$	383040.	−	2.10672e6i	0.127521	−	0.701367i
$391$	−2.55181e6			−0.844124
$392$			522048.i			0.171591i
$393$			4.19513e6i			1.37014i
$394$	1.34987e6			0.438079
$395$	−1.94040e6	−	352800.i	−0.625747	−	0.113772i
$396$	111296.			0.0356649
$397$		−	1.09567e6i		−	0.348901i	−0.984666	−	0.174451i	$-0.944185\pi$
$397$		−	1.09567e6i		−	0.348901i	0.984666	−	0.174451i	$-0.0558150\pi$
$398$			2.24400e6i			0.710093i
$399$	−4.91064e6			−1.54421
$400$	748800.	+	281600.i	0.234000	+	0.0880000i
$401$	−2.49160e6			−0.773779			−0.386890	−	0.922126i	$-0.626451\pi$
$401$	−2.49160e6			−0.773779			−0.386890	+	0.922126i	$0.626451\pi$
$402$			1.79749e6i			0.554755i
$403$		−	1.40083e6i		−	0.429659i
$404$	68768.0			0.0209620
$405$	−2.49804e6	−	454190.i	−0.756768	−	0.137594i
$406$	170640.			0.0513766
$407$		−	647056.i		−	0.193623i
$408$		−	1.83501e6i		−	0.545742i
$409$	3.63349e6			1.07403			0.537014	−	0.843573i	$-0.319552\pi$
$409$	3.63349e6			1.07403			0.537014	+	0.843573i	$0.319552\pi$
$410$	95920.0	−	527560.i	0.0281806	−	0.154993i
$411$	1.51099e6			0.441222
$412$		−	1.98582e6i		−	0.576365i
$413$		−	6.27892e6i		−	1.81138i
$414$	234248.			0.0671700
$415$	278260.	−	1.53043e6i	0.0793105	−	0.436208i
$416$	−700416.			−0.198437
$417$			2.75044e6i			0.774572i
$418$			1.31424e6i			0.367904i
$419$	3.64378e6			1.01395			0.506976	−	0.861960i	$-0.330763\pi$
$419$	3.64378e6			1.01395			0.506976	+	0.861960i	$0.330763\pi$
$420$	−1.94656e6	−	353920.i	−0.538449	−	0.0978998i
$421$	−1.82530e6			−0.501913			−0.250957	−	0.967998i	$-0.580745\pi$
$421$	−1.82530e6			−0.501913			−0.250957	+	0.967998i	$0.580745\pi$
$422$		−	3.22219e6i		−	0.880786i
$423$			502054.i			0.136427i
$424$	189696.			0.0512441
$425$	2.25280e6	−	5.99040e6i	0.604993	−	1.60873i
$426$	237888.			0.0635109
$427$			6.68308e6i			1.77381i
$428$		−	677472.i		−	0.178765i
$429$	1.41725e6			0.371794
$430$	−504680.	−	91760.0i	−0.131627	−	0.0239322i
$431$	2.85435e6			0.740141			0.370070	−	0.929004i	$-0.379334\pi$
$431$	2.85435e6			0.740141			0.370070	+	0.929004i	$0.379334\pi$
$432$			1.03936e6i			0.267952i
$433$		−	587776.i		−	0.150658i	−0.997159	−	0.0753290i	$-0.975999\pi$
$433$		−	587776.i		−	0.150658i	0.997159	−	0.0753290i	$-0.0240007\pi$
$434$	−1.29434e6			−0.329855
$435$	−37800.0	+	207900.i	−0.00957786	+	0.0526783i
$436$	−575840.			−0.145073
$437$			2.76612e6i			0.692895i
$438$		−	1.68582e6i		−	0.419882i
$439$	−6.11604e6			−1.51464			−0.757319	−	0.653045i	$-0.773491\pi$
$439$	−6.11604e6			−1.51464			−0.757319	+	0.653045i	$0.773491\pi$
$440$	−94720.0	+	520960.i	−0.0233244	+	0.128284i
$441$	−383379.			−0.0938711
$442$			5.60333e6i			1.36424i
$443$		−	2.35771e6i		−	0.570795i	−0.958409	−	0.285398i	$-0.907874\pi$
$443$		−	2.35771e6i		−	0.570795i	0.958409	−	0.285398i	$-0.0921257\pi$
$444$	979328.			0.235760
$445$	4.68655e6	+	852100.i	1.12190	+	0.203981i
$446$	−4.87422e6			−1.16029
$447$		−	1.94390e6i		−	0.460156i
$448$			647168.i			0.152343i
$449$	−5.49735e6			−1.28688			−0.643439	−	0.765497i	$-0.722493\pi$
$449$	−5.49735e6			−1.28688			−0.643439	+	0.765497i	$0.722493\pi$
$450$	−206800.	+	549900.i	−0.0481415	+	0.128013i
$451$	354904.			0.0821617
$452$			3.66106e6i			0.842869i
$453$			5.82613e6i			1.33393i
$454$	2.25735e6			0.513996
$455$	5.94396e6	+	1.08072e6i	1.34601	+	0.244729i
$456$	−1.98912e6			−0.447970
$457$			1.16039e6i			0.259905i	0.991520	+	0.129952i	$0.0414824\pi$
$457$			1.16039e6i			0.259905i	−0.991520	+	0.129952i	$0.958518\pi$
$458$		−	2.24132e6i		−	0.499275i
$459$	8.31488e6			1.84215
$460$	−199360.	+	1.09648e6i	−0.0439282	+	0.241605i
$461$	−2.30330e6			−0.504775			−0.252387	−	0.967626i	$-0.581216\pi$
$461$	−2.30330e6			−0.504775			−0.252387	+	0.967626i	$0.581216\pi$
$462$		−	1.30950e6i		−	0.285431i
$463$			2.71343e6i			0.588257i	0.955766	+	0.294128i	$0.0950293\pi$
$463$			2.71343e6i			0.588257i	−0.955766	+	0.294128i	$0.904971\pi$
$464$	69120.0			0.0149042
$465$	286720.	−	1.57696e6i	0.0614930	−	0.338211i
$466$	1.17430e6			0.250505
$467$		−	4.05050e6i		−	0.859441i	−0.902962	−	0.429721i	$-0.858612\pi$
$467$		−	4.05050e6i		−	0.859441i	0.902962	−	0.429721i	$-0.141388\pi$
$468$		−	514368.i		−	0.108557i
$469$	−5.07148e6			−1.06464
$470$	−2.35004e6	−	427280.i	−0.490716	−	0.0892212i
$471$	6.06351e6			1.25942
$472$		−	2.54336e6i		−	0.525476i
$473$		−	339512.i		−	0.0697754i
$474$	1.97568e6			0.403897
$475$	−6.49350e6	−	2.44200e6i	−1.32052	−	0.496606i
$476$	5.17734e6			1.04734
$477$			139308.i			0.0280337i
$478$		−	2.33696e6i		−	0.467823i
$479$	−5.60528e6			−1.11624			−0.558121	−	0.829759i	$-0.688478\pi$
$479$	−5.60528e6			−1.11624			−0.558121	+	0.829759i	$0.688478\pi$
$480$	−788480.	−	143360.i	−0.156202	−	0.0284004i
$481$	−2.99045e6			−0.589350
$482$		−	2.25519e6i		−	0.442146i
$483$		−	2.75615e6i		−	0.537570i
$484$	2.22635e6			0.431997
$485$	−972320.	+	5.34776e6i	−0.187696	+	1.03233i
$486$	−1.40286e6			−0.269415
$487$		−	7.13168e6i		−	1.36260i	−0.732003	−	0.681301i	$-0.761414\pi$
$487$		−	7.13168e6i		−	1.36260i	0.732003	−	0.681301i	$-0.238586\pi$
$488$			2.70707e6i			0.514577i
$489$	−2.08788e6			−0.394850
$490$	326280.	−	1.79454e6i	0.0613904	−	0.337647i
$491$	5.88145e6			1.10098			0.550492	−	0.834841i	$-0.314440\pi$
$491$	5.88145e6			1.10098			0.550492	+	0.834841i	$0.314440\pi$
$492$			537152.i			0.100042i
$493$		−	552960.i		−	0.102465i
$494$	6.07392e6			1.11983
$495$	−382580.	−	69560.0i	−0.0701793	−	0.0127599i
$496$	−524288.			−0.0956898
$497$			671184.i			0.121885i
$498$			1.55826e6i			0.281556i
$499$	−1.75710e6			−0.315897			−0.157948	−	0.987447i	$-0.550488\pi$
$499$	−1.75710e6			−0.315897			−0.157948	+	0.987447i	$0.550488\pi$
$500$	−2.39800e6	−	1.43600e6i	−0.428967	−	0.256879i
$501$	−7.83443e6			−1.39448
$502$		−	4.07899e6i		−	0.722426i
$503$			4.91411e6i			0.866015i	0.901390	+	0.433007i	$0.142548\pi$
$503$			4.91411e6i			0.866015i	−0.901390	+	0.433007i	$0.857452\pi$
$504$	−475264.			−0.0833410
$505$	−236390.	−	42980.0i	−0.0412478	−	0.00749960i
$506$	−737632.			−0.128075
$507$		−	1.35188e6i		−	0.233571i
$508$			2.80381e6i			0.482047i
$509$	5.75499e6			0.984578			0.492289	−	0.870432i	$-0.336160\pi$
$509$	5.75499e6			0.984578			0.492289	+	0.870432i	$0.336160\pi$
$510$	−1.14688e6	+	6.30784e6i	−0.195251	+	1.07388i
$511$	4.75643e6			0.805803
$512$			262144.i			0.0441942i
$513$		−	9.01320e6i		−	1.51212i
$514$	2.62963e6			0.439023
$515$	−1.24114e6	+	6.82627e6i	−0.206207	+	1.13414i
$516$	513856.			0.0849605
$517$		−	1.58094e6i		−	0.260128i
$518$			2.76310e6i			0.452452i
$519$	−4.81326e6			−0.784369
$520$	2.40768e6	+	437760.i	0.390472	+	0.0709950i
$521$	−1.61980e6			−0.261437			−0.130718	−	0.991420i	$-0.541728\pi$
$521$	−1.61980e6			−0.261437			−0.130718	+	0.991420i	$0.541728\pi$
$522$			50760.0i			0.00815352i
$523$			1.19117e7i			1.90422i	0.305751	+	0.952112i	$0.401093\pi$
$523$			1.19117e7i			1.90422i	−0.305751	+	0.952112i	$0.598907\pi$
$524$	−4.79443e6			−0.762797
$525$	6.47010e6	+	2.43320e6i	1.02450	+	0.385283i
$526$	2.24946e6			0.354499
$527$			4.19430e6i			0.657860i
$528$		−	530432.i		−	0.0828028i
$529$	4.88383e6			0.758789
$530$	−652080.	−	118560.i	−0.100835	−	0.0183336i
$531$	1.86778e6			0.287468
$532$		−	5.61216e6i		−	0.859708i
$533$		−	1.64023e6i		−	0.250085i
$534$	−4.77176e6			−0.724145
$535$	−423420.	+	2.32881e6i	−0.0639568	+	0.351763i
$536$	−2.05427e6			−0.308849
$537$		−	335720.i		−	0.0502391i
$538$		−	1.46628e6i		−	0.218404i
$539$	1.20724e6			0.178986
$540$	649600.	−	3.57280e6i	0.0958653	−	0.527259i
$541$	4.07630e6			0.598788			0.299394	−	0.954130i	$-0.403215\pi$
$541$	4.07630e6			0.598788			0.299394	+	0.954130i	$0.403215\pi$
$542$			4.64301e6i			0.678893i
$543$		−	9.12657e6i		−	1.32834i
$544$	2.09715e6			0.303831
$545$	1.97945e6	+	359900.i	0.285465	+	0.0519028i
$546$	−6.05203e6			−0.868800
$547$		−	1.23680e7i		−	1.76739i	−0.468065	−	0.883694i	$-0.655049\pi$
$547$		−	1.23680e7i		−	1.76739i	0.468065	−	0.883694i	$-0.344951\pi$
$548$			1.72685e6i			0.245642i
$549$	−1.98801e6			−0.281505
$550$	651200.	−	1.73160e6i	0.0917926	−	0.244085i
$551$	−599400.			−0.0841081
$552$		−	1.11642e6i		−	0.155947i
$553$			5.57424e6i			0.775127i
$554$	−1.00600e7			−1.39260
$555$	−3.36644e6	−	612080.i	−0.463915	−	0.0843482i
$556$	−3.14336e6			−0.431228
$557$		−	130308.i		−	0.0177964i	−0.999960	−	0.00889822i	$-0.997168\pi$
$557$		−	130308.i		−	0.0177964i	0.999960	−	0.00889822i	$-0.00283243\pi$
$558$		−	385024.i		−	0.0523483i
$559$	−1.56910e6			−0.212383
$560$	404480.	−	2.22464e6i	0.0545038	−	0.299771i
$561$	−4.24346e6			−0.569262
$562$			8.34401e6i			1.11438i
$563$		−	5.91687e6i		−	0.786721i	−0.919384	−	0.393361i	$-0.871312\pi$
$563$		−	5.91687e6i		−	0.786721i	0.919384	−	0.393361i	$-0.128688\pi$
$564$	2.39277e6			0.316740
$565$	2.28816e6	−	1.25849e7i	0.301554	−	1.65855i
$566$	8.95210e6			1.17458
$567$			7.17620e6i			0.937426i
$568$			271872.i			0.0353585i
$569$	9.03013e6			1.16927			0.584633	−	0.811298i	$-0.301239\pi$
$569$	9.03013e6			1.16927			0.584633	+	0.811298i	$0.301239\pi$
$570$	6.83760e6	+	1.24320e6i	0.881488	+	0.160271i
$571$	−1.07093e7			−1.37459			−0.687294	−	0.726379i	$-0.741202\pi$
$571$	−1.07093e7			−1.37459			−0.687294	+	0.726379i	$0.741202\pi$
$572$			1.61971e6i			0.206989i
$573$			2.83853e6i			0.361166i
$574$	−1.51554e6			−0.191994
$575$	1.37060e6	−	3.64455e6i	0.172879	−	0.459700i
$576$	−192512.			−0.0241770
$577$			1.22051e6i			0.152617i	0.997084	+	0.0763084i	$0.0243134\pi$
$577$			1.22051e6i			0.152617i	−0.997084	+	0.0763084i	$0.975687\pi$
$578$		−	1.10978e7i		−	1.38171i
$579$	6.33718e6			0.785597
$580$	−237600.	−	43200.0i	−0.0293276	−	0.00533229i
$581$	−4.39651e6			−0.540341
$582$		−	5.44499e6i		−	0.666331i
$583$		−	438672.i		−	0.0534526i
$584$	1.92666e6			0.233761
$585$	−321480.	+	1.76814e6i	−0.0388387	+	0.213613i
$586$	3.90302e6			0.469523
$587$			1.47104e7i			1.76210i	0.473026	+	0.881049i	$0.343162\pi$
$587$			1.47104e7i			1.76210i	−0.473026	+	0.881049i	$0.656838\pi$
$588$			1.82717e6i			0.217939i
$589$	4.54656e6			0.540001
$590$	−1.58960e6	+	8.74280e6i	−0.188000	+	1.03400i
$591$	4.72455e6			0.556406
$592$			1.11923e6i			0.131255i
$593$			8.52014e6i			0.994970i	0.867472	+	0.497485i	$0.165743\pi$
$593$			8.52014e6i			0.994970i	−0.867472	+	0.497485i	$0.834257\pi$
$594$	2.40352e6			0.279500
$595$	−1.77971e7	−	3.23584e6i	−2.06090	−	0.374709i
$596$	2.22160e6			0.256183
$597$			7.85400e6i			0.901893i
$598$			3.40906e6i			0.389835i
$599$	−2.90100e6			−0.330355			−0.165177	−	0.986264i	$-0.552820\pi$
$599$	−2.90100e6			−0.330355			−0.165177	+	0.986264i	$0.552820\pi$
$600$	2.62080e6	+	985600.i	0.297205	+	0.111769i
$601$	5.72760e6			0.646825			0.323412	−	0.946258i	$-0.395170\pi$
$601$	5.72760e6			0.646825			0.323412	+	0.946258i	$0.395170\pi$
$602$			1.44981e6i			0.163049i
$603$		−	1.50861e6i		−	0.168959i
$604$	−6.65843e6			−0.742642
$605$	−7.65308e6	−	1.39147e6i	−0.850057	−	0.154556i
$606$	240688.			0.0266240
$607$			8.79924e6i			0.969334i	0.874699	+	0.484667i	$0.161059\pi$
$607$			8.79924e6i			0.969334i	−0.874699	+	0.484667i	$0.838941\pi$
$608$		−	2.27328e6i		−	0.249399i
$609$	597240.			0.0652538
$610$	1.69192e6	−	9.30556e6i	0.184101	−	1.01255i
$611$	−7.30649e6			−0.791782
$612$			1.54010e6i			0.166215i
$613$			1.03408e6i			0.111149i	0.998455	+	0.0555744i	$0.0176990\pi$
$613$			1.03408e6i			0.111149i	−0.998455	+	0.0555744i	$0.982301\pi$
$614$	351432.			0.0376201
$615$	335720.	−	1.84646e6i	0.0357923	−	0.196858i
$616$	1.49658e6			0.158908
$617$		−	1.29854e7i		−	1.37323i	−0.727020	−	0.686616i	$-0.759095\pi$
$617$		−	1.29854e7i		−	1.37323i	0.727020	−	0.686616i	$-0.240905\pi$
$618$		−	6.95038e6i		−	0.732045i
$619$	−7.92002e6			−0.830806			−0.415403	−	0.909637i	$-0.636359\pi$
$619$	−7.92002e6			−0.830806			−0.415403	+	0.909637i	$0.636359\pi$
$620$	1.80224e6	+	327680.i	0.188293	+	0.0342350i
$621$	5.05876e6			0.526399
$622$			2.39741e6i			0.248465i
$623$		−	1.34632e7i		−	1.38972i
$624$	−2.45146e6			−0.252036
$625$	7.34562e6	+	6.43500e6i	0.752192	+	0.658944i
$626$	8.37366e6			0.854043
$627$			4.59984e6i			0.467276i
$628$			6.92973e6i			0.701160i
$629$	8.95386e6			0.902368
$630$	1.63372e6	+	297040.i	0.163993	+	0.0298170i
$631$	1.68218e7			1.68189			0.840945	−	0.541120i	$-0.181999\pi$
$631$	1.68218e7			1.68189			0.840945	+	0.541120i	$0.181999\pi$
$632$			2.25792e6i			0.224862i
$633$		−	1.12777e7i		−	1.11869i
$634$	−9.66501e6			−0.954947
$635$	1.75238e6	−	9.63809e6i	0.172462	−	0.948542i
$636$	663936.			0.0650854
$637$		−	5.57939e6i		−	0.544801i
$638$		−	159840.i		−	0.0155465i
$639$	−199656.			−0.0193433
$640$	163840.	−	901120.i	0.0158114	−	0.0869626i
$641$	−1.55154e7			−1.49148			−0.745741	−	0.666236i	$-0.767904\pi$
$641$	−1.55154e7			−1.49148			−0.745741	+	0.666236i	$0.767904\pi$
$642$		−	2.37115e6i		−	0.227050i
$643$			1.05801e7i			1.00916i	0.863364	+	0.504582i	$0.168354\pi$
$643$			1.05801e7i			1.00916i	−0.863364	+	0.504582i	$0.831646\pi$
$644$	3.14989e6			0.299282
$645$	−1.76638e6	−	321160.i	−0.167180	−	0.0303964i
$646$	−1.81862e7			−1.71460
$647$		−	1.37883e7i		−	1.29494i	−0.762090	−	0.647471i	$-0.775827\pi$
$647$		−	1.37883e7i		−	1.29494i	0.762090	−	0.647471i	$-0.224173\pi$
$648$			2.90682e6i			0.271944i
$649$	−5.88152e6			−0.548123
$650$	−8.00280e6	−	3.00960e6i	−0.742948	−	0.279399i
$651$	−4.53018e6			−0.418950
$652$		−	2.38614e6i		−	0.219825i
$653$		−	1.58924e6i		−	0.145850i	−0.997337	−	0.0729248i	$-0.976767\pi$
$653$		−	1.58924e6i		−	0.145850i	0.997337	−	0.0729248i	$-0.0232333\pi$
$654$	−2.01544e6			−0.184258
$655$	1.64809e7	+	2.99652e6i	1.50099	+	0.272907i
$656$	−613888.			−0.0556967
$657$			1.41489e6i			0.127882i
$658$			6.75102e6i			0.607862i
$659$	9.12434e6			0.818442			0.409221	−	0.912435i	$-0.365801\pi$
$659$	9.12434e6			0.818442			0.409221	+	0.912435i	$0.365801\pi$
$660$	−331520.	+	1.82336e6i	−0.0296244	+	0.162934i
$661$	6.50310e6			0.578918			0.289459	−	0.957190i	$-0.406525\pi$
$661$	6.50310e6			0.578918			0.289459	+	0.957190i	$0.406525\pi$
$662$		−	6.56379e6i		−	0.582116i
$663$			1.96116e7i			1.73273i
$664$	−1.78086e6			−0.156751
$665$	−3.50760e6	+	1.92918e7i	−0.307578	+	1.69168i
$666$	−821936.			−0.0718046
$667$		−	336420.i		−	0.0292797i
$668$		−	8.95363e6i		−	0.776351i
$669$	−1.70598e7			−1.47369
$670$	7.06156e6	+	1.28392e6i	0.607734	+	0.110497i
$671$	6.26010e6			0.536754
$672$			2.26509e6i			0.193492i
$673$		−	2.17810e6i		−	0.185370i	−0.995695	−	0.0926850i	$-0.970455\pi$
$673$		−	2.17810e6i		−	0.185370i	0.995695	−	0.0926850i	$-0.0295449\pi$
$674$	8.75091e6			0.741999
$675$	−4.46600e6	+	1.18755e7i	−0.377276	+	1.00321i
$676$	1.54501e6			0.130036
$677$		−	3.98419e6i		−	0.334094i	−0.985949	−	0.167047i	$-0.946577\pi$
$677$		−	3.98419e6i		−	0.334094i	0.985949	−	0.167047i	$-0.0534231\pi$
$678$			1.28137e7i			1.07053i
$679$	1.53627e7			1.27877
$680$	−7.20896e6	−	1.31072e6i	−0.597861	−	0.108702i
$681$	7.90073e6			0.652829
$682$			1.21242e6i			0.0998138i
$683$		−	5.91563e6i		−	0.485231i	−0.970122	−	0.242616i	$-0.921995\pi$
$683$		−	5.91563e6i		−	0.485231i	0.970122	−	0.242616i	$-0.0780054\pi$
$684$	1.66944e6			0.136436
$685$	1.07928e6	−	5.93604e6i	0.0878836	−	0.483360i
$686$	5.46680e6			0.443530
$687$		−	7.84462e6i		−	0.634133i
$688$			587264.i			0.0473001i
$689$	−2.02738e6			−0.162700
$690$	−697760.	+	3.83768e6i	−0.0557935	+	0.306864i
$691$	−1.55471e7			−1.23867			−0.619335	−	0.785127i	$-0.712598\pi$
$691$	−1.55471e7			−1.23867			−0.619335	+	0.785127i	$0.712598\pi$
$692$		−	5.50086e6i		−	0.436682i
$693$			1.09905e6i			0.0869328i
$694$	1.09801e7			0.865379
$695$	1.08053e7	+	1.96460e6i	0.848545	+	0.154281i
$696$	241920.			0.0189299
$697$			4.91110e6i			0.382910i
$698$			1.06046e7i			0.823864i
$699$	4.11006e6			0.318167
$700$	−2.78080e6	+	7.39440e6i	−0.214499	+	0.570372i
$701$	−2.27103e7			−1.74553			−0.872766	−	0.488139i	$-0.837676\pi$
$701$	−2.27103e7			−1.74553			−0.872766	+	0.488139i	$0.837676\pi$
$702$		−	1.11082e7i		−	0.850745i
$703$		−	9.70584e6i		−	0.740704i
$704$	606208.			0.0460988
$705$	−8.22514e6	−	1.49548e6i	−0.623262	−	0.113320i
$706$	−1.22307e7			−0.923502
$707$			679084.i			0.0510946i
$708$		−	8.90176e6i		−	0.667410i
$709$	−6.29841e6			−0.470560			−0.235280	−	0.971928i	$-0.575601\pi$
$709$	−6.29841e6			−0.470560			−0.235280	+	0.971928i	$0.575601\pi$
$710$	169920.	−	934560.i	0.0126502	−	0.0695763i
$711$	−1.65816e6			−0.123013
$712$		−	5.45344e6i		−	0.403154i
$713$			2.55181e6i			0.187985i
$714$	1.81207e7			1.33024
$715$	1.01232e6	−	5.56776e6i	0.0740547	−	0.407301i
$716$	383680.			0.0279696
$717$		−	8.17936e6i		−	0.594185i
$718$		−	1.51742e7i		−	1.09849i
$719$	−2.11911e7			−1.52873			−0.764367	−	0.644782i	$-0.776948\pi$
$719$	−2.11911e7			−1.52873			−0.764367	+	0.644782i	$0.776948\pi$
$720$	661760.	+	120320.i	0.0475740	+	0.00864981i
$721$	1.96100e7			1.40488
$722$			9.80920e6i			0.700311i
$723$		−	7.89317e6i		−	0.561572i
$724$	1.04304e7			0.739526
$725$	789750.	+	297000.i	0.0558013	+	0.0209851i
$726$	7.79223e6			0.548682
$727$		−	1.35610e7i		−	0.951605i	−0.879552	−	0.475803i	$-0.842158\pi$
$727$		−	1.35610e7i		−	0.951605i	0.879552	−	0.475803i	$-0.157842\pi$
$728$		−	6.91661e6i		−	0.483687i
$729$	−1.59468e7			−1.11136
$730$	−6.62288e6	−	1.20416e6i	−0.459981	−	0.0836329i
$731$	4.69811e6			0.325185
$732$			9.47475e6i			0.653567i
$733$			2.69413e7i			1.85208i	0.377429	+	0.926038i	$0.376808\pi$
$733$			2.69413e7i			1.85208i	−0.377429	+	0.926038i	$0.623192\pi$
$734$	−1.24424e7			−0.852441
$735$	1.14198e6	−	6.28089e6i	0.0779723	−	0.428847i
$736$	1.27590e6			0.0868207
$737$			4.75050e6i			0.322160i
$738$		−	450824.i		−	0.0304696i
$739$	−2.77414e6			−0.186860			−0.0934302	−	0.995626i	$-0.529783\pi$
$739$	−2.77414e6			−0.186860			−0.0934302	+	0.995626i	$0.529783\pi$
$740$	699520.	−	3.84736e6i	0.0469592	−	0.258276i
$741$	2.12587e7			1.42230
$742$			1.87325e6i			0.124907i
$743$			1.85538e7i			1.23299i	0.787358	+	0.616497i	$0.211449\pi$
$743$			1.85538e7i			1.23299i	−0.787358	+	0.616497i	$0.788551\pi$
$744$	−1.83501e6			−0.121536
$745$	−7.63675e6	−	1.38850e6i	−0.504101	−	0.0916548i
$746$	5.66078e6			0.372417
$747$		−	1.30782e6i		−	0.0857526i
$748$		−	4.84966e6i		−	0.316926i
$749$	6.69004e6			0.435736
$750$	−8.39300e6	−	5.02600e6i	−0.544834	−	0.326264i
$751$	−2.19285e6			−0.141876			−0.0709380	−	0.997481i	$-0.522599\pi$
$751$	−2.19285e6			−0.141876			−0.0709380	+	0.997481i	$0.522599\pi$
$752$			2.73459e6i			0.176339i
$753$		−	1.42765e7i		−	0.917558i
$754$	−738720.			−0.0473207
$755$	2.28884e7	+	4.16152e6i	1.46133	+	0.265696i
$756$	−1.02637e7			−0.653128
$757$			9.48749e6i			0.601744i	0.953665	+	0.300872i	$0.0972777\pi$
$757$			9.48749e6i			0.601744i	−0.953665	+	0.300872i	$0.902722\pi$
$758$			1.56105e7i			0.986832i
$759$	−2.58171e6			−0.162668
$760$	−1.42080e6	+	7.81440e6i	−0.0892275	+	0.490752i
$761$	9.69580e6			0.606907			0.303453	−	0.952846i	$-0.401860\pi$
$761$	9.69580e6			0.606907			0.303453	+	0.952846i	$0.401860\pi$
$762$			9.81333e6i			0.612250i
$763$		−	5.68642e6i		−	0.353612i
$764$	−3.24403e6			−0.201072
$765$	962560.	−	5.29408e6i	0.0594668	−	0.327067i
$766$	−2.78270e6			−0.171354
$767$			2.71822e7i			1.66838i
$768$			917504.i			0.0561313i
$769$	−9.32787e6			−0.568809			−0.284405	−	0.958704i	$-0.591796\pi$
$769$	−9.32787e6			−0.568809			−0.284405	+	0.958704i	$0.591796\pi$
$770$	−5.14448e6	−	935360.i	−0.312690	−	0.0568528i
$771$	9.20371e6			0.557606
$772$			7.24250e6i			0.437366i
$773$		−	9.68080e6i		−	0.582723i	−0.956613	−	0.291362i	$-0.905892\pi$
$773$		−	9.68080e6i		−	0.582723i	0.956613	−	0.291362i	$-0.0941083\pi$
$774$	−431272.			−0.0258761
$775$	−5.99040e6

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:	\( N \)	\(=\)	\( 10 = 2 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	10.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.00000i q^{2} +14.0000i q^{3} -16.0000 q^{4} +(55.0000 + 10.0000i) q^{5} -56.0000 q^{6} -158.000i q^{7} -64.0000i q^{8} +47.0000 q^{9} +(-40.0000 + 220.000i) q^{10} -148.000 q^{11} -224.000i q^{12} +684.000i q^{13} +632.000 q^{14} +(-140.000 + 770.000i) q^{15} +256.000 q^{16} -2048.00i q^{17} +188.000i q^{18} -2220.00 q^{19} +(-880.000 - 160.000i) q^{20} +2212.00 q^{21} -592.000i q^{22} -1246.00i q^{23} +896.000 q^{24} +(2925.00 + 1100.00i) q^{25} -2736.00 q^{26} +4060.00i q^{27} +2528.00i q^{28} +270.000 q^{29} +(-3080.00 - 560.000i) q^{30} -2048.00 q^{31} +1024.00i q^{32} -2072.00i q^{33} +8192.00 q^{34} +(1580.00 - 8690.00i) q^{35} -752.000 q^{36} +4372.00i q^{37} -8880.00i q^{38} -9576.00 q^{39} +(640.000 - 3520.00i) q^{40} -2398.00 q^{41} +8848.00i q^{42} +2294.00i q^{43} +2368.00 q^{44} +(2585.00 + 470.000i) q^{45} +4984.00 q^{46} +10682.0i q^{47} +3584.00i q^{48} -8157.00 q^{49} +(-4400.00 + 11700.0i) q^{50} +28672.0 q^{51} -10944.0i q^{52} +2964.00i q^{53} -16240.0 q^{54} +(-8140.00 - 1480.00i) q^{55} -10112.0 q^{56} -31080.0i q^{57} +1080.00i q^{58} +39740.0 q^{59} +(2240.00 - 12320.0i) q^{60} -42298.0 q^{61} -8192.00i q^{62} -7426.00i q^{63} -4096.00 q^{64} +(-6840.00 + 37620.0i) q^{65} +8288.00 q^{66} -32098.0i q^{67} +32768.0i q^{68} +17444.0 q^{69} +(34760.0 + 6320.00i) q^{70} -4248.00 q^{71} -3008.00i q^{72} +30104.0i q^{73} -17488.0 q^{74} +(-15400.0 + 40950.0i) q^{75} +35520.0 q^{76} +23384.0i q^{77} -38304.0i q^{78} -35280.0 q^{79} +(14080.0 + 2560.00i) q^{80} -45419.0 q^{81} -9592.00i q^{82} -27826.0i q^{83} -35392.0 q^{84} +(20480.0 - 112640. i) q^{85} -9176.00 q^{86} +3780.00i q^{87} +9472.00i q^{88} +85210.0 q^{89} +(-1880.00 + 10340.0i) q^{90} +108072. q^{91} +19936.0i q^{92} -28672.0i q^{93} -42728.0 q^{94} +(-122100. - 22200.0i) q^{95} -14336.0 q^{96} +97232.0i q^{97} -32628.0i q^{98} -6956.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 32 q^{4} + 110 q^{5} - 112 q^{6} + 94 q^{9} - 80 q^{10} - 296 q^{11} + 1264 q^{14} - 280 q^{15} + 512 q^{16} - 4440 q^{19} - 1760 q^{20} + 4424 q^{21} + 1792 q^{24} + 5850 q^{25} - 5472 q^{26} + 540 q^{29}+ \cdots - 13912 q^{99}+O(q^{100}) \) 2 * q - 32 * q^4 + 110 * q^5 - 112 * q^6 + 94 * q^9 - 80 * q^10 - 296 * q^11 + 1264 * q^14 - 280 * q^15 + 512 * q^16 - 4440 * q^19 - 1760 * q^20 + 4424 * q^21 + 1792 * q^24 + 5850 * q^25 - 5472 * q^26 + 540 * q^29 - 6160 * q^30 - 4096 * q^31 + 16384 * q^34 + 3160 * q^35 - 1504 * q^36 - 19152 * q^39 + 1280 * q^40 - 4796 * q^41 + 4736 * q^44 + 5170 * q^45 + 9968 * q^46 - 16314 * q^49 - 8800 * q^50 + 57344 * q^51 - 32480 * q^54 - 16280 * q^55 - 20224 * q^56 + 79480 * q^59 + 4480 * q^60 - 84596 * q^61 - 8192 * q^64 - 13680 * q^65 + 16576 * q^66 + 34888 * q^69 + 69520 * q^70 - 8496 * q^71 - 34976 * q^74 - 30800 * q^75 + 71040 * q^76 - 70560 * q^79 + 28160 * q^80 - 90838 * q^81 - 70784 * q^84 + 40960 * q^85 - 18352 * q^86 + 170420 * q^89 - 3760 * q^90 + 216144 * q^91 - 85456 * q^94 - 244200 * q^95 - 28672 * q^96 - 13912 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more