LMFDB - Embedded newform 560.5.p.e.209.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [560,5,Mod(209,560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(560, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("560.209");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 560 = 2^{4} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	560.p (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$57.8871793270$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 6 $ x^2 + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 35)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			209.2
Root			$-2.44949i$ of defining polynomial
Character	$\chi$	$=$	560.209
Dual form			560.5.p.e.209.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.00000 q^{3} +(-5.00000 + 24.4949i) q^{5} +(35.0000 - 34.2929i) q^{7} -56.0000 q^{9} +O(q^{10})$	$q+5.00000 q^{3} +(-5.00000 + 24.4949i) q^{5} +(35.0000 - 34.2929i) q^{7} -56.0000 q^{9} -89.0000 q^{11} +5.00000 q^{13} +(-25.0000 + 122.474i) q^{15} +485.000 q^{17} -220.454i q^{19} +(175.000 - 171.464i) q^{21} +700.554i q^{23} +(-575.000 - 244.949i) q^{25} -685.000 q^{27} +191.000 q^{29} +1053.28i q^{31} -445.000 q^{33} +(665.000 + 1028.79i) q^{35} +1631.36i q^{37} +25.0000 q^{39} +2914.89i q^{41} -377.221i q^{43} +(280.000 - 1371.71i) q^{45} -2195.00 q^{47} +(49.0000 - 2400.50i) q^{49} +2425.00 q^{51} -1587.27i q^{53} +(445.000 - 2180.05i) q^{55} -1102.27i q^{57} +3625.24i q^{59} +1935.10i q^{61} +(-1960.00 + 1920.40i) q^{63} +(-25.0000 + 122.474i) q^{65} -2047.77i q^{67} +3502.77i q^{69} -4454.00 q^{71} -8650.00 q^{73} +(-2875.00 - 1224.74i) q^{75} +(-3115.00 + 3052.06i) q^{77} -5561.00 q^{79} +1111.00 q^{81} +1990.00 q^{83} +(-2425.00 + 11880.0i) q^{85} +955.000 q^{87} -808.332i q^{89} +(175.000 - 171.464i) q^{91} +5266.40i q^{93} +(5400.00 + 1102.27i) q^{95} -9235.00 q^{97} +4984.00 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 10 q^{3} - 10 q^{5} + 70 q^{7} - 112 q^{9}+O(q^{10}) $ 2 * q + 10 * q^3 - 10 * q^5 + 70 * q^7 - 112 * q^9	$ 2 q + 10 q^{3} - 10 q^{5} + 70 q^{7} - 112 q^{9} - 178 q^{11} + 10 q^{13} - 50 q^{15} + 970 q^{17} + 350 q^{21} - 1150 q^{25} - 1370 q^{27} + 382 q^{29} - 890 q^{33} + 1330 q^{35} + 50 q^{39} + 560 q^{45} - 4390 q^{47} + 98 q^{49} + 4850 q^{51} + 890 q^{55} - 3920 q^{63} - 50 q^{65} - 8908 q^{71} - 17300 q^{73} - 5750 q^{75} - 6230 q^{77} - 11122 q^{79} + 2222 q^{81} + 3980 q^{83} - 4850 q^{85} + 1910 q^{87} + 350 q^{91} + 10800 q^{95} - 18470 q^{97} + 9968 q^{99}+O(q^{100}) $ 2 * q + 10 * q^3 - 10 * q^5 + 70 * q^7 - 112 * q^9 - 178 * q^11 + 10 * q^13 - 50 * q^15 + 970 * q^17 + 350 * q^21 - 1150 * q^25 - 1370 * q^27 + 382 * q^29 - 890 * q^33 + 1330 * q^35 + 50 * q^39 + 560 * q^45 - 4390 * q^47 + 98 * q^49 + 4850 * q^51 + 890 * q^55 - 3920 * q^63 - 50 * q^65 - 8908 * q^71 - 17300 * q^73 - 5750 * q^75 - 6230 * q^77 - 11122 * q^79 + 2222 * q^81 + 3980 * q^83 - 4850 * q^85 + 1910 * q^87 + 350 * q^91 + 10800 * q^95 - 18470 * q^97 + 9968 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/560\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$351$	$421$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	5.00000			0.555556			0.277778	−	0.960645i	$-0.410402\pi$
$3$	5.00000			0.555556			0.277778	+	0.960645i	$0.410402\pi$
$4$		0			0
$5$	−5.00000	+	24.4949i	−0.200000	+	0.979796i
$5$	−5.00000	+	24.4949i	−0.200000	+	0.979796i
$6$		0			0
$7$	35.0000	−	34.2929i	0.714286	−	0.699854i
$7$	35.0000	−	34.2929i	0.714286	−	0.699854i
$8$		0			0
$9$	−56.0000			−0.691358
$10$		0			0
$11$	−89.0000			−0.735537			−0.367769	−	0.929917i	$-0.619878\pi$
$11$	−89.0000			−0.735537			−0.367769	+	0.929917i	$0.619878\pi$
$12$		0			0
$13$	5.00000			0.0295858			0.0147929	−	0.999891i	$-0.495291\pi$
$13$	5.00000			0.0295858			0.0147929	+	0.999891i	$0.495291\pi$
$14$		0			0
$15$	−25.0000	+	122.474i	−0.111111	+	0.544331i
$16$		0			0
$17$	485.000			1.67820			0.839100	−	0.543977i	$-0.183082\pi$
$17$	485.000			1.67820			0.839100	+	0.543977i	$0.183082\pi$
$18$		0			0
$19$		−	220.454i		−	0.610676i	−0.952244	−	0.305338i	$-0.901231\pi$
$19$		−	220.454i		−	0.610676i	0.952244	−	0.305338i	$-0.0987695\pi$
$20$		0			0
$21$	175.000	−	171.464i	0.396825	−	0.388808i
$22$		0			0
$23$			700.554i			1.32430i	0.749372	+	0.662149i	$0.230356\pi$
$23$			700.554i			1.32430i	−0.749372	+	0.662149i	$0.769644\pi$
$24$		0			0
$25$	−575.000	−	244.949i	−0.920000	−	0.391918i
$26$		0			0
$27$	−685.000			−0.939643
$28$		0			0
$29$	191.000			0.227111			0.113555	−	0.993532i	$-0.463776\pi$
$29$	191.000			0.227111			0.113555	+	0.993532i	$0.463776\pi$
$30$		0			0
$31$			1053.28i			1.09603i	0.836470	+	0.548013i	$0.184615\pi$
$31$			1053.28i			1.09603i	−0.836470	+	0.548013i	$0.815385\pi$
$32$		0			0
$33$	−445.000			−0.408632
$34$		0			0
$35$	665.000	+	1028.79i	0.542857	+	0.839825i
$36$		0			0
$37$			1631.36i			1.19164i	0.803117	+	0.595822i	$0.203174\pi$
$37$			1631.36i			1.19164i	−0.803117	+	0.595822i	$0.796826\pi$
$38$		0			0
$39$	25.0000			0.0164366
$40$		0			0
$41$			2914.89i			1.73402i	0.498288	+	0.867012i	$0.333962\pi$
$41$			2914.89i			1.73402i	−0.498288	+	0.867012i	$0.666038\pi$
$42$		0			0
$43$		−	377.221i		−	0.204014i	−0.994784	−	0.102007i	$-0.967474\pi$
$43$		−	377.221i		−	0.204014i	0.994784	−	0.102007i	$-0.0325264\pi$
$44$		0			0
$45$	280.000	−	1371.71i	0.138272	−	0.677390i
$46$		0			0
$47$	−2195.00			−0.993662			−0.496831	−	0.867847i	$-0.665503\pi$
$47$	−2195.00			−0.993662			−0.496831	+	0.867847i	$0.665503\pi$
$48$		0			0
$49$	49.0000	−	2400.50i	0.0204082	−	0.999792i
$50$		0			0
$51$	2425.00			0.932334
$52$		0			0
$53$		−	1587.27i		−	0.565066i	−0.959258	−	0.282533i	$-0.908825\pi$
$53$		−	1587.27i		−	0.565066i	0.959258	−	0.282533i	$-0.0911746\pi$
$54$		0			0
$55$	445.000	−	2180.05i	0.147107	−	0.720676i
$56$		0			0
$57$		−	1102.27i		−	0.339265i
$58$		0			0
$59$			3625.24i			1.04144i	0.853728	+	0.520719i	$0.174336\pi$
$59$			3625.24i			1.04144i	−0.853728	+	0.520719i	$0.825664\pi$
$60$		0			0
$61$			1935.10i			0.520048i	0.965602	+	0.260024i	$0.0837304\pi$
$61$			1935.10i			0.520048i	−0.965602	+	0.260024i	$0.916270\pi$
$62$		0			0
$63$	−1960.00	+	1920.40i	−0.493827	+	0.483850i
$64$		0			0
$65$	−25.0000	+	122.474i	−0.00591716	+	0.0289880i
$66$		0			0
$67$		−	2047.77i		−	0.456176i	−0.973641	−	0.228088i	$-0.926753\pi$
$67$		−	2047.77i		−	0.456176i	0.973641	−	0.228088i	$-0.0732474\pi$
$68$		0			0
$69$			3502.77i			0.735722i
$70$		0			0
$71$	−4454.00			−0.883555			−0.441777	−	0.897125i	$-0.645652\pi$
$71$	−4454.00			−0.883555			−0.441777	+	0.897125i	$0.645652\pi$
$72$		0			0
$73$	−8650.00			−1.62319			−0.811597	−	0.584218i	$-0.801401\pi$
$73$	−8650.00			−1.62319			−0.811597	+	0.584218i	$0.801401\pi$
$74$		0			0
$75$	−2875.00	−	1224.74i	−0.511111	−	0.217732i
$76$		0			0
$77$	−3115.00	+	3052.06i	−0.525384	+	0.514769i
$78$		0			0
$79$	−5561.00			−0.891043			−0.445522	−	0.895271i	$-0.646982\pi$
$79$	−5561.00			−0.891043			−0.445522	+	0.895271i	$0.646982\pi$
$80$		0			0
$81$	1111.00			0.169334
$82$		0			0
$83$	1990.00			0.288866			0.144433	−	0.989515i	$-0.453864\pi$
$83$	1990.00			0.288866			0.144433	+	0.989515i	$0.453864\pi$
$84$		0			0
$85$	−2425.00	+	11880.0i	−0.335640	+	1.64429i
$86$		0			0
$87$	955.000			0.126173
$88$		0			0
$89$		−	808.332i		−	0.102049i	−0.998697	−	0.0510246i	$-0.983751\pi$
$89$		−	808.332i		−	0.102049i	0.998697	−	0.0510246i	$-0.0162487\pi$
$90$		0			0
$91$	175.000	−	171.464i	0.0211327	−	0.0207057i
$92$		0			0
$93$			5266.40i			0.608903i
$94$		0			0
$95$	5400.00	+	1102.27i	0.598338	+	0.122135i
$96$		0			0
$97$	−9235.00			−0.981507			−0.490754	−	0.871298i	$-0.663278\pi$
$97$	−9235.00			−0.981507			−0.490754	+	0.871298i	$0.663278\pi$
$98$		0			0
$99$	4984.00			0.508520
$100$		0			0
$101$		−	4825.49i		−	0.473041i	−0.971626	−	0.236521i	$-0.923993\pi$
$101$		−	4825.49i		−	0.473041i	0.971626	−	0.236521i	$-0.0760071\pi$
$102$		0			0
$103$	−4715.00			−0.444434			−0.222217	−	0.974997i	$-0.571329\pi$
$103$	−4715.00			−0.444434			−0.222217	+	0.974997i	$0.571329\pi$
$104$		0			0
$105$	3325.00	+	5143.93i	0.301587	+	0.466569i
$106$		0			0
$107$			12668.8i			1.10654i	0.833002	+	0.553269i	$0.186620\pi$
$107$			12668.8i			1.10654i	−0.833002	+	0.553269i	$0.813380\pi$
$108$		0			0
$109$	12311.0			1.03619			0.518096	−	0.855322i	$-0.326641\pi$
$109$	12311.0			1.03619			0.518096	+	0.855322i	$0.326641\pi$
$110$		0			0
$111$			8156.80i			0.662024i
$112$		0			0
$113$			3120.65i			0.244393i	0.992506	+	0.122196i	$0.0389938\pi$
$113$			3120.65i			0.244393i	−0.992506	+	0.122196i	$0.961006\pi$
$114$		0			0
$115$	−17160.0	−	3502.77i	−1.29754	−	0.264860i
$116$		0			0
$117$	−280.000			−0.0204544
$118$		0			0
$119$	16975.0	−	16632.0i	1.19871	−	1.17450i
$120$		0			0
$121$	−6720.00			−0.458985
$122$		0			0
$123$			14574.5i			0.963346i
$124$		0			0
$125$	8875.00	−	12859.8i	0.568000	−	0.823029i
$126$		0			0
$127$			23867.8i			1.47981i	0.672712	+	0.739904i	$0.265129\pi$
$127$			23867.8i			1.47981i	−0.672712	+	0.739904i	$0.734871\pi$
$128$		0			0
$129$		−	1886.11i		−	0.113341i
$130$		0			0
$131$		−	6736.10i		−	0.392524i	−0.980552	−	0.196262i	$-0.937120\pi$
$131$		−	6736.10i		−	0.392524i	0.980552	−	0.196262i	$-0.0628802\pi$
$132$		0			0
$133$	−7560.00	−	7715.89i	−0.427384	−	0.436197i
$134$		0			0
$135$	3425.00	−	16779.0i	0.187929	−	0.920659i
$136$		0			0
$137$			23069.3i			1.22912i	0.788871	+	0.614558i	$0.210666\pi$
$137$			23069.3i			1.22912i	−0.788871	+	0.614558i	$0.789334\pi$
$138$		0			0
$139$			29491.9i			1.52641i	0.646154	+	0.763207i	$0.276376\pi$
$139$			29491.9i			1.52641i	−0.646154	+	0.763207i	$0.723624\pi$
$140$		0			0
$141$	−10975.0			−0.552035
$142$		0			0
$143$	−445.000			−0.0217615
$144$		0			0
$145$	−955.000	+	4678.53i	−0.0454221	+	0.222522i
$146$		0			0
$147$	245.000	−	12002.5i	0.0113379	−	0.555440i
$148$		0			0
$149$	28346.0			1.27679			0.638395	−	0.769709i	$-0.279599\pi$
$149$	28346.0			1.27679			0.638395	+	0.769709i	$0.279599\pi$
$150$		0			0
$151$	17551.0			0.769747			0.384873	−	0.922969i	$-0.374245\pi$
$151$	17551.0			0.769747			0.384873	+	0.922969i	$0.374245\pi$
$152$		0			0
$153$	−27160.0			−1.16024
$154$		0			0
$155$	−25800.0	−	5266.40i	−1.07388	−	0.219205i
$156$		0			0
$157$	25790.0			1.04629			0.523145	−	0.852244i	$-0.324759\pi$
$157$	25790.0			1.04629			0.523145	+	0.852244i	$0.324759\pi$
$158$		0			0
$159$		−	7936.35i		−	0.313925i
$160$		0			0
$161$	24024.0	+	24519.4i	0.926816	+	0.945928i
$162$		0			0
$163$			37153.9i			1.39839i	0.714930	+	0.699196i	$0.246458\pi$
$163$			37153.9i			1.39839i	−0.714930	+	0.699196i	$0.753542\pi$
$164$		0			0
$165$	2225.00	−	10900.2i	0.0817264	−	0.400376i
$166$		0			0
$167$	−20795.0			−0.745634			−0.372817	−	0.927905i	$-0.621608\pi$
$167$	−20795.0			−0.745634			−0.372817	+	0.927905i	$0.621608\pi$
$168$		0			0
$169$	−28536.0			−0.999125
$170$		0			0
$171$			12345.4i			0.422196i
$172$		0			0
$173$	−115.000			−0.00384243			−0.00192121	−	0.999998i	$-0.500612\pi$
$173$	−115.000			−0.00384243			−0.00192121	+	0.999998i	$0.500612\pi$
$174$		0			0
$175$	−28525.0	+	11145.2i	−0.931429	+	0.363924i
$176$		0			0
$177$			18126.2i			0.578577i
$178$		0			0
$179$	−5318.00			−0.165975			−0.0829874	−	0.996551i	$-0.526446\pi$
$179$	−5318.00			−0.165975			−0.0829874	+	0.996551i	$0.526446\pi$
$180$		0			0
$181$		−	12345.4i		−	0.376833i	−0.982089	−	0.188417i	$-0.939665\pi$
$181$		−	12345.4i		−	0.376833i	0.982089	−	0.188417i	$-0.0603355\pi$
$182$		0			0
$183$			9675.48i			0.288915i
$184$		0			0
$185$	−39960.0	−	8156.80i	−1.16757	−	0.238329i
$186$		0			0
$187$	−43165.0			−1.23438
$188$		0			0
$189$	−23975.0	+	23490.6i	−0.671174	+	0.657613i
$190$		0			0
$191$	14263.0			0.390971			0.195485	−	0.980707i	$-0.437372\pi$
$191$	14263.0			0.390971			0.195485	+	0.980707i	$0.437372\pi$
$192$		0			0
$193$		−	32005.0i		−	0.859219i	−0.903015	−	0.429609i	$-0.858651\pi$
$193$		−	32005.0i		−	0.859219i	0.903015	−	0.429609i	$-0.141349\pi$
$194$		0			0
$195$	−125.000	+	612.372i	−0.00328731	+	0.0161045i
$196$		0			0
$197$			3394.99i			0.0874795i	0.999043	+	0.0437398i	$0.0139272\pi$
$197$			3394.99i			0.0874795i	−0.999043	+	0.0437398i	$0.986073\pi$
$198$		0			0
$199$			19400.0i			0.489886i	0.969538	+	0.244943i	$0.0787692\pi$
$199$			19400.0i			0.489886i	−0.969538	+	0.244943i	$0.921231\pi$
$200$		0			0
$201$		−	10238.9i		−	0.253431i
$202$		0			0
$203$	6685.00	−	6549.94i	0.162222	−	0.158944i
$204$		0			0
$205$	−71400.0	−	14574.5i	−1.69899	−	0.346805i
$206$		0			0
$207$		−	39231.0i		−	0.915565i
$208$		0			0
$209$			19620.4i			0.449175i
$210$		0			0
$211$	−52817.0			−1.18634			−0.593170	−	0.805078i	$-0.702124\pi$
$211$	−52817.0			−1.18634			−0.593170	+	0.805078i	$0.702124\pi$
$212$		0			0
$213$	−22270.0			−0.490864
$214$		0			0
$215$	9240.00	+	1886.11i	0.199892	+	0.0408027i
$216$		0			0
$217$	36120.0	+	36864.8i	0.767058	+	0.782875i
$218$		0			0
$219$	−43250.0			−0.901774
$220$		0			0
$221$	2425.00			0.0496509
$222$		0			0
$223$	13645.0			0.274387			0.137194	−	0.990544i	$-0.456192\pi$
$223$	13645.0			0.274387			0.137194	+	0.990544i	$0.456192\pi$
$224$		0			0
$225$	32200.0	+	13717.1i	0.636049	+	0.270956i
$226$		0			0
$227$	74485.0			1.44550			0.722748	−	0.691111i	$-0.242879\pi$
$227$	74485.0			1.44550			0.722748	+	0.691111i	$0.242879\pi$
$228$		0			0
$229$			68463.2i			1.30553i	0.757561	+	0.652764i	$0.226391\pi$
$229$			68463.2i			1.30553i	−0.757561	+	0.652764i	$0.773609\pi$
$230$		0			0
$231$	−15575.0	+	15260.3i	−0.291880	+	0.285983i
$232$		0			0
$233$			60169.3i			1.10831i	0.832412	+	0.554157i	$0.186959\pi$
$233$			60169.3i			1.10831i	−0.832412	+	0.554157i	$0.813041\pi$
$234$		0			0
$235$	10975.0	−	53766.3i	0.198732	−	0.973586i
$236$		0			0
$237$	−27805.0			−0.495024
$238$		0			0
$239$	75367.0			1.31943			0.659714	−	0.751517i	$-0.270678\pi$
$239$	75367.0			1.31943			0.659714	+	0.751517i	$0.270678\pi$
$240$		0			0
$241$		−	77869.3i		−	1.34070i	−0.742044	−	0.670351i	$-0.766144\pi$
$241$		−	77869.3i		−	1.34070i	0.742044	−	0.670351i	$-0.233856\pi$
$242$		0			0
$243$	61040.0			1.03372
$244$		0			0
$245$	58555.0	+	13202.7i	0.975510	+	0.219954i
$246$		0			0
$247$		−	1102.27i		−	0.0180673i
$248$		0			0
$249$	9950.00			0.160481
$250$		0			0
$251$		−	50410.5i		−	0.800154i	−0.916482	−	0.400077i	$-0.868983\pi$
$251$		−	50410.5i		−	0.800154i	0.916482	−	0.400077i	$-0.131017\pi$
$252$		0			0
$253$		−	62349.3i		−	0.974071i
$254$		0			0
$255$	−12125.0	+	59400.1i	−0.186467	+	0.913497i
$256$		0			0
$257$	−3370.00			−0.0510227			−0.0255114	−	0.999675i	$-0.508121\pi$
$257$	−3370.00			−0.0510227			−0.0255114	+	0.999675i	$0.508121\pi$
$258$		0			0
$259$	55944.0	+	57097.6i	0.833977	+	0.851174i
$260$		0			0
$261$	−10696.0			−0.157015
$262$		0			0
$263$		−	59385.4i		−	0.858556i	−0.903173	−	0.429278i	$-0.858768\pi$
$263$		−	59385.4i		−	0.858556i	0.903173	−	0.429278i	$-0.141232\pi$
$264$		0			0
$265$	38880.0	+	7936.35i	0.553649	+	0.113013i
$266$		0			0
$267$		−	4041.66i		−	0.0566940i
$268$		0			0
$269$		−	77012.0i		−	1.06427i	−0.846658	−	0.532137i	$-0.821389\pi$
$269$		−	77012.0i		−	1.06427i	0.846658	−	0.532137i	$-0.178611\pi$
$270$		0			0
$271$		−	143222.i		−	1.95016i	−0.221855	−	0.975080i	$-0.571211\pi$
$271$		−	143222.i		−	1.95016i	0.221855	−	0.975080i	$-0.428789\pi$
$272$		0			0
$273$	875.000	−	857.321i	0.0117404	−	0.0115032i
$274$		0			0
$275$	51175.0	+	21800.5i	0.676694	+	0.288271i
$276$		0			0
$277$		−	101928.i		−	1.32842i	−0.747547	−	0.664209i	$-0.768769\pi$
$277$		−	101928.i		−	1.32842i	0.747547	−	0.664209i	$-0.231231\pi$
$278$		0			0
$279$		−	58983.7i		−	0.757746i
$280$		0			0
$281$	−47521.0			−0.601829			−0.300914	−	0.953651i	$-0.597292\pi$
$281$	−47521.0			−0.601829			−0.300914	+	0.953651i	$0.597292\pi$
$282$		0			0
$283$	−43115.0			−0.538339			−0.269169	−	0.963093i	$-0.586749\pi$
$283$	−43115.0			−0.538339			−0.269169	+	0.963093i	$0.586749\pi$
$284$		0			0
$285$	27000.0	+	5511.35i	0.332410	+	0.0678529i
$286$		0			0
$287$	99960.0	+	102021.i	1.21356	+	1.23859i
$288$		0			0
$289$	151704.			1.81636
$290$		0			0
$291$	−46175.0			−0.545282
$292$		0			0
$293$	162125.			1.88849			0.944245	−	0.329243i	$-0.106794\pi$
$293$	162125.			1.88849			0.944245	+	0.329243i	$0.106794\pi$
$294$		0			0
$295$	−88800.0	−	18126.2i	−1.02040	−	0.208288i
$296$		0			0
$297$	60965.0			0.691143
$298$		0			0
$299$			3502.77i			0.0391804i
$300$		0			0
$301$	−12936.0	−	13202.7i	−0.142780	−	0.145724i
$302$		0			0
$303$		−	24127.5i		−	0.262801i
$304$		0			0
$305$	−47400.0	−	9675.48i	−0.509540	−	0.104010i
$306$		0			0
$307$	33805.0			0.358678			0.179339	−	0.983787i	$-0.442604\pi$
$307$	33805.0			0.358678			0.179339	+	0.983787i	$0.442604\pi$
$308$		0			0
$309$	−23575.0			−0.246908
$310$		0			0
$311$			50435.0i			0.521448i	0.965413	+	0.260724i	$0.0839613\pi$
$311$			50435.0i			0.521448i	−0.965413	+	0.260724i	$0.916039\pi$
$312$		0			0
$313$	−68155.0			−0.695679			−0.347840	−	0.937554i	$-0.613085\pi$
$313$	−68155.0			−0.695679			−0.347840	+	0.937554i	$0.613085\pi$
$314$		0			0
$315$	−37240.0	−	57612.0i	−0.375309	−	0.580620i
$316$		0			0
$317$		−	42435.0i		−	0.422285i	−0.977455	−	0.211142i	$-0.932282\pi$
$317$		−	42435.0i		−	0.422285i	0.977455	−	0.211142i	$-0.0677184\pi$
$318$		0			0
$319$	−16999.0			−0.167048
$320$		0			0
$321$			63343.8i			0.614744i
$322$		0			0
$323$		−	106920.i		−	1.02484i
$324$		0			0
$325$	−2875.00	−	1224.74i	−0.0272189	−	0.0115952i
$326$		0			0
$327$	61555.0			0.575662
$328$		0			0
$329$	−76825.0	+	75272.8i	−0.709759	+	0.695419i
$330$		0			0
$331$	114706.			1.04696			0.523480	−	0.852038i	$-0.324633\pi$
$331$	114706.			1.04696			0.523480	+	0.852038i	$0.324633\pi$
$332$		0			0
$333$		−	91356.2i		−	0.823852i
$334$		0			0
$335$	50160.0	+	10238.9i	0.446959	+	0.0912352i
$336$		0			0
$337$			117007.i			1.03027i	0.857108	+	0.515137i	$0.172259\pi$
$337$			117007.i			1.03027i	−0.857108	+	0.515137i	$0.827741\pi$
$338$		0			0
$339$			15603.2i			0.135774i
$340$		0			0
$341$		−	93742.0i		−	0.806168i
$342$		0			0
$343$	−80605.0	−	85697.8i	−0.685131	−	0.728420i
$344$		0			0
$345$	−85800.0	−	17513.9i	−0.720857	−	0.147144i
$346$		0			0
$347$			70481.6i			0.585352i	0.956212	+	0.292676i	$0.0945457\pi$
$347$			70481.6i			0.585352i	−0.956212	+	0.292676i	$0.905454\pi$
$348$		0			0
$349$		−	91390.5i		−	0.750326i	−0.926959	−	0.375163i	$-0.877587\pi$
$349$		−	91390.5i		−	0.750326i	0.926959	−	0.375163i	$-0.122413\pi$
$350$		0			0
$351$	−3425.00			−0.0278001
$352$		0			0
$353$	11405.0			0.0915263			0.0457631	−	0.998952i	$-0.485428\pi$
$353$	11405.0			0.0915263			0.0457631	+	0.998952i	$0.485428\pi$
$354$		0			0
$355$	22270.0	−	109100.i	0.176711	−	0.865703i
$356$		0			0
$357$	84875.0	−	83160.2i	0.665953	−	0.652498i
$358$		0			0
$359$	−230366.			−1.78743			−0.893716	−	0.448633i	$-0.851911\pi$
$359$	−230366.			−1.78743			−0.893716	+	0.448633i	$0.851911\pi$
$360$		0			0
$361$	81721.0			0.627075
$362$		0			0
$363$	−33600.0			−0.254992
$364$		0			0
$365$	43250.0	−	211881.i	0.324639	−	1.59040i
$366$		0			0
$367$	74485.0			0.553015			0.276507	−	0.961012i	$-0.410823\pi$
$367$	74485.0			0.553015			0.276507	+	0.961012i	$0.410823\pi$
$368$		0			0
$369$		−	163234.i		−	1.19883i
$370$		0			0
$371$	−54432.0	−	55554.4i	−0.395464	−	0.403618i
$372$		0			0
$373$		−	102550.i		−	0.737088i	−0.929610	−	0.368544i	$-0.879856\pi$
$373$		−	102550.i		−	0.737088i	0.929610	−	0.368544i	$-0.120144\pi$
$374$		0			0
$375$	44375.0	−	64299.1i	0.315556	−	0.457238i
$376$		0			0
$377$	955.000			0.00671925
$378$		0			0
$379$	172954.			1.20407			0.602036	−	0.798469i	$-0.294357\pi$
$379$	172954.			1.20407			0.602036	+	0.798469i	$0.294357\pi$
$380$		0			0
$381$			119339.i			0.822116i
$382$		0			0
$383$	−144050.			−0.982010			−0.491005	−	0.871157i	$-0.663370\pi$
$383$	−144050.			−0.982010			−0.491005	+	0.871157i	$0.663370\pi$
$384$		0			0
$385$	−59185.0	−	91561.9i	−0.399292	−	0.617723i
$386$		0			0
$387$			21124.4i			0.141047i
$388$		0			0
$389$	−148969.			−0.984457			−0.492228	−	0.870466i	$-0.663818\pi$
$389$	−148969.			−0.984457			−0.492228	+	0.870466i	$0.663818\pi$
$390$		0			0
$391$			339769.i			2.22244i
$392$		0			0
$393$		−	33680.5i		−	0.218069i
$394$		0			0
$395$	27805.0	−	136216.i	0.178209	−	0.873040i
$396$		0			0
$397$	256925.			1.63014			0.815071	−	0.579361i	$-0.196698\pi$
$397$	256925.			1.63014			0.815071	+	0.579361i	$0.196698\pi$
$398$		0			0
$399$	−37800.0	−	38579.5i	−0.237436	−	0.242332i
$400$		0			0
$401$	−35641.0			−0.221647			−0.110823	−	0.993840i	$-0.535349\pi$
$401$	−35641.0			−0.221647			−0.110823	+	0.993840i	$0.535349\pi$
$402$		0			0
$403$			5266.40i			0.0324268i
$404$		0			0
$405$	−5555.00	+	27213.8i	−0.0338668	+	0.165913i
$406$		0			0
$407$		−	145191.i		−	0.876498i
$408$		0			0
$409$			151721.i			0.906985i	0.891260	+	0.453493i	$0.149822\pi$
$409$			151721.i			0.906985i	−0.891260	+	0.453493i	$0.850178\pi$
$410$		0			0
$411$			115346.i			0.682843i
$412$		0			0
$413$	124320.	+	126884.i	0.728855	+	0.743884i
$414$		0			0
$415$	−9950.00	+	48744.8i	−0.0577733	+	0.283030i
$416$		0			0
$417$			147459.i			0.848008i
$418$		0			0
$419$			150889.i			0.859465i	0.902956	+	0.429733i	$0.141392\pi$
$419$			150889.i			0.859465i	−0.902956	+	0.429733i	$0.858608\pi$
$420$		0			0
$421$	−4681.00			−0.0264104			−0.0132052	−	0.999913i	$-0.504203\pi$
$421$	−4681.00			−0.0264104			−0.0132052	+	0.999913i	$0.504203\pi$
$422$		0			0
$423$	122920.			0.686976
$424$		0			0
$425$	−278875.	−	118800.i	−1.54394	−	0.657718i
$426$		0			0
$427$	66360.0	+	67728.4i	0.363957	+	0.371463i
$428$		0			0
$429$	−2225.00			−0.0120897
$430$		0			0
$431$	87103.0			0.468898			0.234449	−	0.972128i	$-0.424671\pi$
$431$	87103.0			0.468898			0.234449	+	0.972128i	$0.424671\pi$
$432$		0			0
$433$	−231730.			−1.23597			−0.617983	−	0.786192i	$-0.712050\pi$
$433$	−231730.			−1.23597			−0.617983	+	0.786192i	$0.712050\pi$
$434$		0			0
$435$	−4775.00	+	23392.6i	−0.0252345	+	0.123623i
$436$		0			0
$437$	154440.			0.808718
$438$		0			0
$439$		−	246272.i		−	1.27787i	−0.769262	−	0.638933i	$-0.779376\pi$
$439$		−	246272.i		−	1.27787i	0.769262	−	0.638933i	$-0.220624\pi$
$440$		0			0
$441$	−2744.00	+	134428.i	−0.0141093	+	0.691214i
$442$		0			0
$443$		−	349841.i		−	1.78264i	−0.453376	−	0.891319i	$-0.649781\pi$
$443$		−	349841.i		−	1.78264i	0.453376	−	0.891319i	$-0.350219\pi$
$444$		0			0
$445$	19800.0	+	4041.66i	0.0999874	+	0.0204098i
$446$		0			0
$447$	141730.			0.709327
$448$		0			0
$449$	254711.			1.26344			0.631721	−	0.775196i	$-0.282349\pi$
$449$	254711.			1.26344			0.631721	+	0.775196i	$0.282349\pi$
$450$		0			0
$451$		−	259425.i		−	1.27544i
$452$		0			0
$453$	87755.0			0.427637
$454$		0			0
$455$	3325.00	+	5143.93i	0.0160609	+	0.0248469i
$456$		0			0
$457$			263629.i			1.26229i	0.775663	+	0.631147i	$0.217415\pi$
$457$			263629.i			1.26229i	−0.775663	+	0.631147i	$0.782585\pi$
$458$		0			0
$459$	−332225.			−1.57691
$460$		0			0
$461$			128255.i			0.603495i	0.953388	+	0.301747i	$0.0975699\pi$
$461$			128255.i			0.603495i	−0.953388	+	0.301747i	$0.902430\pi$
$462$		0			0
$463$			81857.0i			0.381851i	0.981605	+	0.190926i	$0.0611489\pi$
$463$			81857.0i			0.381851i	−0.981605	+	0.190926i	$0.938851\pi$
$464$		0			0
$465$	−129000.	−	26332.0i	−0.596601	−	0.121781i
$466$		0			0
$467$	204445.			0.937438			0.468719	−	0.883347i	$-0.344716\pi$
$467$	204445.			0.937438			0.468719	+	0.883347i	$0.344716\pi$
$468$		0			0
$469$	−70224.0	−	71672.1i	−0.319257	−	0.325840i
$470$		0			0
$471$	128950.			0.581272
$472$		0			0
$473$			33572.7i			0.150060i
$474$		0			0
$475$	−54000.0	+	126761.i	−0.239335	+	0.561822i
$476$		0			0
$477$			88887.1i			0.390663i
$478$		0			0
$479$		−	79534.9i		−	0.346647i	−0.984865	−	0.173323i	$-0.944549\pi$
$479$		−	79534.9i		−	0.346647i	0.984865	−	0.173323i	$-0.0554505\pi$
$480$		0			0
$481$			8156.80i			0.0352557i
$482$		0			0
$483$	120120.	+	122597.i	0.514898	+	0.525515i
$484$		0			0
$485$	46175.0	−	226210.i	0.196301	−	0.961677i
$486$		0			0
$487$			164958.i			0.695531i	0.937582	+	0.347766i	$0.113059\pi$
$487$			164958.i			0.695531i	−0.937582	+	0.347766i	$0.886941\pi$
$488$		0			0
$489$			185769.i			0.776884i
$490$		0			0
$491$	74191.0			0.307743			0.153872	−	0.988091i	$-0.450826\pi$
$491$	74191.0			0.307743			0.153872	+	0.988091i	$0.450826\pi$
$492$		0			0
$493$	92635.0			0.381137
$494$		0			0
$495$	−24920.0	+	122083.i	−0.101704	+	0.498245i
$496$		0			0
$497$	−155890.	+	152740.i	−0.631111	+	0.618360i
$498$		0			0
$499$	278527.			1.11858			0.559289	−	0.828973i	$-0.311074\pi$
$499$	278527.			1.11858			0.559289	+	0.828973i	$0.311074\pi$
$500$		0			0
$501$	−103975.			−0.414241
$502$		0			0
$503$	−60275.0			−0.238233			−0.119116	−	0.992880i	$-0.538006\pi$
$503$	−60275.0			−0.238233			−0.119116	+	0.992880i	$0.538006\pi$
$504$		0			0
$505$	118200.	+	24127.5i	0.463484	+	0.0946083i
$506$		0			0
$507$	−142680.			−0.555069
$508$		0			0
$509$		−	348709.i		−	1.34595i	−0.739666	−	0.672974i	$-0.765017\pi$
$509$		−	348709.i		−	1.34595i	0.739666	−	0.672974i	$-0.234983\pi$
$510$		0			0
$511$	−302750.	+	296633.i	−1.15942	+	1.13600i
$512$		0			0
$513$			151011.i			0.573818i
$514$		0			0
$515$	23575.0	−	115493.i	0.0888868	−	0.435455i
$516$		0			0
$517$	195355.			0.730876
$518$		0			0
$519$	−575.000			−0.00213468
$520$		0			0
$521$		−	84727.9i		−	0.312141i	−0.987746	−	0.156070i	$-0.950117\pi$
$521$		−	84727.9i		−	0.312141i	0.987746	−	0.156070i	$-0.0498827\pi$
$522$		0			0
$523$	−235610.			−0.861371			−0.430686	−	0.902502i	$-0.641728\pi$
$523$	−235610.			−0.861371			−0.430686	+	0.902502i	$0.641728\pi$
$524$		0			0
$525$	−142625.	+	55725.9i	−0.517460	+	0.202180i
$526$		0			0
$527$			510841.i			1.83935i
$528$		0			0
$529$	−210935.			−0.753767
$530$		0			0
$531$		−	203014.i		−	0.720006i
$532$		0			0
$533$			14574.5i			0.0513025i
$534$		0			0
$535$	−310320.	−	63343.8i	−1.08418	−	0.221308i
$536$		0			0
$537$	−26590.0			−0.0922082
$538$		0			0
$539$	−4361.00	+	213644.i	−0.0150110	+	0.735384i
$540$		0			0
$541$	259199.			0.885602			0.442801	−	0.896620i	$-0.353985\pi$
$541$	259199.			0.885602			0.442801	+	0.896620i	$0.353985\pi$
$542$		0			0
$543$		−	61727.1i		−	0.209352i
$544$		0			0
$545$	−61555.0	+	301557.i	−0.207238	+	1.01526i
$546$		0			0
$547$			423894.i			1.41672i	0.705854	+	0.708358i	$0.250564\pi$
$547$			423894.i			1.41672i	−0.705854	+	0.708358i	$0.749436\pi$
$548$		0			0
$549$		−	108365.i		−	0.359539i
$550$		0			0
$551$		−	42106.7i		−	0.138691i
$552$		0			0
$553$	−194635.	+	190703.i	−0.636459	+	0.623600i
$554$		0			0
$555$	−199800.	−	40784.0i	−0.648649	−	0.132405i
$556$		0			0
$557$			133389.i			0.429943i	0.976620	+	0.214972i	$0.0689659\pi$
$557$			133389.i			0.429943i	−0.976620	+	0.214972i	$0.931034\pi$
$558$		0			0
$559$		−	1886.11i		−	0.00603591i
$560$		0			0
$561$	−215825.			−0.685766
$562$		0			0
$563$	−369770.			−1.16658			−0.583290	−	0.812264i	$-0.698235\pi$
$563$	−369770.			−1.16658			−0.583290	+	0.812264i	$0.698235\pi$
$564$		0			0
$565$	−76440.0	−	15603.2i	−0.239455	−	0.0488785i
$566$		0			0
$567$	38885.0	−	38099.4i	0.120953	−	0.118509i
$568$		0			0
$569$	−326134.			−1.00733			−0.503665	−	0.863899i	$-0.668015\pi$
$569$	−326134.			−1.00733			−0.503665	+	0.863899i	$0.668015\pi$
$570$		0			0
$571$	−306374.			−0.939679			−0.469840	−	0.882752i	$-0.655688\pi$
$571$	−306374.			−0.939679			−0.469840	+	0.882752i	$0.655688\pi$
$572$		0			0
$573$	71315.0			0.217206
$574$		0			0
$575$	171600.	−	402819.i	0.519017	−	1.21835i
$576$		0			0
$577$	−387595.			−1.16420			−0.582099	−	0.813118i	$-0.697768\pi$
$577$	−387595.			−1.16420			−0.582099	+	0.813118i	$0.697768\pi$
$578$		0			0
$579$		−	160025.i		−	0.477344i
$580$		0			0
$581$	69650.0	−	68242.8i	0.206333	−	0.202164i
$582$		0			0
$583$			141267.i			0.415627i
$584$		0			0
$585$	1400.00	−	6858.57i	0.00409088	−	0.0200411i
$586$		0			0
$587$	467350.			1.35633			0.678166	−	0.734909i	$-0.262775\pi$
$587$	467350.			1.35633			0.678166	+	0.734909i	$0.262775\pi$
$588$		0			0
$589$	232200.			0.669317
$590$		0			0
$591$			16975.0i			0.0485997i
$592$		0			0
$593$	−214315.			−0.609457			−0.304729	−	0.952439i	$-0.598566\pi$
$593$	−214315.			−0.609457			−0.304729	+	0.952439i	$0.598566\pi$
$594$		0			0
$595$	322525.	+	498961.i	0.911023	+	1.40939i
$596$		0			0
$597$			96999.8i			0.272159i
$598$		0			0
$599$	283999.			0.791522			0.395761	−	0.918353i	$-0.370481\pi$
$599$	283999.			0.791522			0.395761	+	0.918353i	$0.370481\pi$
$600$		0			0
$601$			91219.0i			0.252544i	0.991996	+	0.126272i	$0.0403011\pi$
$601$			91219.0i			0.252544i	−0.991996	+	0.126272i	$0.959699\pi$
$602$		0			0
$603$			114675.i			0.315381i
$604$		0			0
$605$	33600.0	−	164606.i	0.0917970	−	0.449712i
$606$		0			0
$607$	−193715.			−0.525758			−0.262879	−	0.964829i	$-0.584672\pi$
$607$	−193715.			−0.525758			−0.262879	+	0.964829i	$0.584672\pi$
$608$		0			0
$609$	33425.0	−	32749.7i	0.0901232	−	0.0883024i
$610$		0			0
$611$	−10975.0			−0.0293983
$612$		0			0
$613$		−	296231.i		−	0.788334i	−0.919039	−	0.394167i	$-0.871033\pi$
$613$		−	296231.i		−	0.788334i	0.919039	−	0.394167i	$-0.128967\pi$
$614$		0			0
$615$	−357000.	−	72872.3i	−0.943883	−	0.192669i
$616$		0			0
$617$		−	424751.i		−	1.11574i	−0.829927	−	0.557872i	$-0.811618\pi$
$617$		−	424751.i		−	1.11574i	0.829927	−	0.557872i	$-0.188382\pi$
$618$		0			0
$619$		−	375825.i		−	0.980855i	−0.871482	−	0.490427i	$-0.836841\pi$
$619$		−	375825.i		−	0.980855i	0.871482	−	0.490427i	$-0.163159\pi$
$620$		0			0
$621$		−	479880.i		−	1.24437i
$622$		0			0
$623$	−27720.0	−	28291.6i	−0.0714196	−	0.0728923i
$624$		0			0
$625$	270625.	+	281691.i	0.692800	+	0.721130i
$626$		0			0
$627$			98102.1i			0.249542i
$628$		0			0
$629$			791210.i			1.99982i
$630$		0			0
$631$	471511.			1.18422			0.592111	−	0.805856i	$-0.298295\pi$
$631$	471511.			1.18422			0.592111	+	0.805856i	$0.298295\pi$
$632$		0			0
$633$	−264085.			−0.659077
$634$		0			0
$635$	−584640.	−	119339.i	−1.44991	−	0.295962i
$636$		0			0
$637$	245.000	−	12002.5i	0.000603792	−	0.0295796i
$638$		0			0
$639$	249424.			0.610853
$640$		0			0
$641$	558794.			1.35999			0.679995	−	0.733217i	$-0.261982\pi$
$641$	558794.			1.35999			0.679995	+	0.733217i	$0.261982\pi$
$642$		0			0
$643$	−454235.			−1.09865			−0.549324	−	0.835609i	$-0.685115\pi$
$643$	−454235.			−1.09865			−0.549324	+	0.835609i	$0.685115\pi$
$644$		0			0
$645$	46200.0	+	9430.54i	0.111051	+	0.0226682i
$646$		0			0
$647$	−129530.			−0.309430			−0.154715	−	0.987959i	$-0.549446\pi$
$647$	−129530.			−0.309430			−0.154715	+	0.987959i	$0.549446\pi$
$648$		0			0
$649$		−	322647.i		−	0.766016i
$650$		0			0
$651$	180600.	+	184324.i	0.426143	+	0.434931i
$652$		0			0
$653$		−	343903.i		−	0.806511i	−0.915087	−	0.403255i	$-0.867879\pi$
$653$		−	343903.i		−	0.806511i	0.915087	−	0.403255i	$-0.132121\pi$
$654$		0			0
$655$	165000.	+	33680.5i	0.384593	+	0.0785047i
$656$		0			0
$657$	484400.			1.12221
$658$		0			0
$659$	−526913.			−1.21330			−0.606650	−	0.794969i	$-0.707487\pi$
$659$	−526913.			−1.21330			−0.606650	+	0.794969i	$0.707487\pi$
$660$		0			0
$661$			191844.i			0.439082i	0.975603	+	0.219541i	$0.0704559\pi$
$661$			191844.i			0.439082i	−0.975603	+	0.219541i	$0.929544\pi$
$662$		0			0
$663$	12125.0			0.0275838
$664$		0			0
$665$	226800.	−	146602.i	0.512861	−	0.331510i
$666$		0			0
$667$			133806.i			0.300762i
$668$		0			0
$669$	68225.0			0.152437
$670$		0			0
$671$		−	172224.i		−	0.382514i
$672$		0			0
$673$			429792.i			0.948918i	0.880278	+	0.474459i	$0.157356\pi$
$673$			429792.i			0.948918i	−0.880278	+	0.474459i	$0.842644\pi$
$674$		0			0
$675$	393875.	+	167790.i	0.864472	+	0.368263i
$676$		0			0
$677$	79685.0			0.173860			0.0869299	−	0.996214i	$-0.472294\pi$
$677$	79685.0			0.173860			0.0869299	+	0.996214i	$0.472294\pi$
$678$		0			0
$679$	−323225.	+	316695.i	−0.701076	+	0.686912i
$680$		0			0
$681$	372425.			0.803054
$682$		0			0
$683$			719577.i			1.54254i	0.636510	+	0.771269i	$0.280378\pi$
$683$			719577.i			1.54254i	−0.636510	+	0.771269i	$0.719622\pi$
$684$		0			0
$685$	−565080.	−	115346.i	−1.20428	−	0.245823i
$686$		0			0
$687$			342316.i			0.725294i
$688$		0			0
$689$		−	7936.35i		−	0.0167179i
$690$		0			0
$691$		−	398532.i		−	0.834655i	−0.908756	−	0.417328i	$-0.862967\pi$
$691$		−	398532.i		−	0.834655i	0.908756	−	0.417328i	$-0.137033\pi$
$692$		0			0
$693$	174440.	−	170916.i	0.363228	−	0.355890i
$694$		0			0
$695$	−722400.	−	147459.i	−1.49557	−	0.305283i
$696$		0			0
$697$			1.41372e6i			2.91004i
$698$		0			0
$699$			300846.i			0.615730i
$700$		0			0
$701$	−658873.			−1.34081			−0.670403	−	0.741998i	$-0.733879\pi$
$701$	−658873.			−1.34081			−0.670403	+	0.741998i	$0.733879\pi$
$702$		0			0
$703$	359640.			0.727708
$704$		0			0
$705$	54875.0	−	268831.i	0.110407	−	0.540881i
$706$		0			0
$707$	−165480.	−	168892.i	−0.331060	−	0.337887i
$708$		0			0
$709$	−593737.			−1.18114			−0.590570	−	0.806986i	$-0.701097\pi$
$709$	−593737.			−1.18114			−0.590570	+	0.806986i	$0.701097\pi$
$710$		0			0
$711$	311416.			0.616030
$712$		0			0
$713$	−737880.			−1.45147
$714$		0			0
$715$	2225.00	−	10900.2i	0.00435229	−	0.0213218i
$716$		0			0
$717$	376835.			0.733015
$718$		0			0
$719$		−	75321.8i		−	0.145701i	−0.997343	−	0.0728506i	$-0.976790\pi$
$719$		−	75321.8i		−	0.145701i	0.997343	−	0.0728506i	$-0.0232096\pi$
$720$		0			0
$721$	−165025.	+	161691.i	−0.317453	+	0.311039i
$722$		0			0
$723$		−	389346.i		−	0.744834i
$724$		0			0
$725$	−109825.	−	46785.3i	−0.208942	−	0.0890088i
$726$		0			0
$727$	342190.			0.647438			0.323719	−	0.946153i	$-0.395067\pi$
$727$	342190.			0.647438			0.323719	+	0.946153i	$0.395067\pi$
$728$		0			0
$729$	215209.			0.404954
$730$		0			0
$731$		−	182952.i		−	0.342376i
$732$		0			0
$733$	−958555.			−1.78406			−0.892029	−	0.451978i	$-0.850719\pi$
$733$	−958555.			−1.78406			−0.892029	+	0.451978i	$0.850719\pi$
$734$		0			0
$735$	292775.	+	66013.7i	0.541950	+	0.122197i
$736$		0			0
$737$			182252.i			0.335534i
$738$		0			0
$739$	229399.			0.420052			0.210026	−	0.977696i	$-0.432645\pi$
$739$	229399.			0.420052			0.210026	+	0.977696i	$0.432645\pi$
$740$		0			0
$741$		−	5511.35i		−	0.0100374i
$742$		0			0
$743$			97803.2i			0.177164i	0.996069	+	0.0885820i	$0.0282335\pi$
$743$			97803.2i			0.177164i	−0.996069	+	0.0885820i	$0.971766\pi$
$744$		0			0
$745$	−141730.	+	694332.i	−0.255358	+	1.25099i
$746$		0			0
$747$	−111440.			−0.199710
$748$		0			0
$749$	434448.	+	443407.i	0.774416	+	0.790385i
$750$		0			0
$751$	−246977.			−0.437902			−0.218951	−	0.975736i	$-0.570263\pi$
$751$	−246977.			−0.437902			−0.218951	+	0.975736i	$0.570263\pi$
$752$		0			0
$753$		−	252052.i		−	0.444530i
$754$		0			0
$755$	−87755.0	+	429910.i	−0.153949	+	0.754195i
$756$		0			0
$757$			1.05636e6i			1.84340i	0.387906	+	0.921699i	$0.373198\pi$
$757$			1.05636e6i			1.84340i	−0.387906	+	0.921699i	$0.626802\pi$
$758$		0			0
$759$		−	311747.i		−	0.541151i
$760$		0			0
$761$		−	406836.i		−	0.702506i	−0.936281	−	0.351253i	$-0.885756\pi$
$761$		−	406836.i		−	0.702506i	0.936281	−	0.351253i	$-0.114244\pi$
$762$		0			0
$763$	430885.	−	422179.i	0.740137	−	0.725184i
$764$		0			0
$765$	135800.	−	665281.i	0.232048	−	1.13680i
$766$		0			0
$767$			18126.2i			0.0308118i
$768$		0			0
$769$		−	592238.i		−	1.00148i	−0.865597	−	0.500741i	$-0.833061\pi$
$769$		−	592238.i		−	1.00148i	0.865597	−	0.500741i	$-0.166939\pi$
$770$		0			0
$771$	−16850.0			−0.0283460
$772$		0			0
$773$	401165.			0.671373			0.335687	−	0.941974i	$-0.391032\pi$
$773$	401165.			0.671373			0.335687	+	0.941974i	$0.391032\pi$
$774$		0			0
$775$	258000.	−	605636.i	0.429553	−	1.00834i
$776$		0			0
$777$	279720.	+	285488.i	0.463320	+	0.472874i
$778$		0			0
$779$	642600.			1.05893
$780$		0			0
$781$	396406.			0.649887
$782$		0			0
$783$	−130835.			−0.213403
$784$		0			0
$785$	−128950.	+	631723.i	−0.209258	+	1.02515i
$786$		0			0
$787$	357565.			0.577305			0.288653	−	0.957434i	$-0.406793\pi$
$787$	357565.			0.577305			0.288653	+	0.957434i	$0.406793\pi$
$788$		0			0
$789$		−	296927.i		−	0.476975i
$790$		0			0
$791$	107016.	+	109223.i	0.171039	+	0.174566i
$792$		0			0
$793$			9675.48i			0.0153860i
$794$		0			0
$795$	194400.	+	39681.7i	0.307583	+	0.0627851i
$796$		0			0
$797$	1.18188e6			1.86062			0.930312	−	0.366769i	$-0.119536\pi$
$797$	1.18188e6			1.86062			0.930312	+	0.366769i	$0.119536\pi$
$798$		0			0
$799$	−1.06458e6			−1.66756
$800$		0			0
$801$			45266.6i			0.0705525i
$802$		0			0
$803$	769850.			1.19392
$804$		0			0
$805$	−720720.	+	465868.i	−1.11218	+	0.718905i
$806$		0			0
$807$		−	385060.i		−	0.591264i
$808$		0			0
$809$	−272449.			−0.416283			−0.208141	−	0.978099i	$-0.566741\pi$
$809$	−272449.			−0.416283			−0.208141	+	0.978099i	$0.566741\pi$
$810$		0			0
$811$			1.22117e6i			1.85667i	0.371749	+	0.928333i	$0.378758\pi$
$811$			1.22117e6i			1.85667i	−0.371749	+	0.928333i	$0.621242\pi$
$812$		0			0
$813$		−	716108.i		−	1.08342i
$814$		0			0
$815$	−910080.	−	185769.i	−1.37014	−	0.279678i
$816$		0			0
$817$	−83160.0			−0.124586
$818$		0			0
$819$	−9800.00	+	9602.00i	−0.0146103	+	0.0143151i
$820$		0			0
$821$	−621793.			−0.922485			−0.461243	−	0.887274i	$-0.652596\pi$
$821$	−621793.			−0.922485			−0.461243	+	0.887274i	$0.652596\pi$
$822$		0			0
$823$			117394.i			0.173319i	0.996238	+	0.0866597i	$0.0276193\pi$
$823$			117394.i			0.173319i	−0.996238	+	0.0866597i	$0.972381\pi$
$824$		0			0
$825$	255875.	+	109002.i	0.375941	+	0.160150i
$826$		0			0
$827$			199790.i			0.292121i	0.989276	+	0.146061i	$0.0466595\pi$
$827$			199790.i			0.292121i	−0.989276	+	0.146061i	$0.953341\pi$
$828$		0			0
$829$		−	197894.i		−	0.287955i	−0.989581	−	0.143977i	$-0.954011\pi$
$829$		−	197894.i		−	0.287955i	0.989581	−	0.143977i	$-0.0459892\pi$
$830$		0			0
$831$		−	509641.i		−	0.738010i
$832$		0			0
$833$	23765.0	−	1.16424e6i	0.0342490	−	1.67785i
$834$		0			0
$835$	103975.	−	509371.i	0.149127	−	0.730570i
$836$		0			0
$837$		−	721497.i		−	1.02987i
$838$		0			0
$839$			826972.i			1.17481i	0.809294	+	0.587404i	$0.199850\pi$
$839$			826972.i			1.17481i	−0.809294	+	0.587404i	$0.800150\pi$
$840$		0			0
$841$	−670800.			−0.948421
$842$		0			0
$843$	−237605.			−0.334349
$844$		0			0
$845$	142680.	−	698986.i	0.199825	−	0.978938i
$846$		0			0
$847$	−235200.	+	230448.i	−0.327846	+	0.321223i
$848$		0			0
$849$	−215575.			−0.299077
$850$		0			0
$851$	−1.14286e6			−1.57809
$852$		0			0
$853$	−385450.			−0.529749			−0.264874	−	0.964283i	$-0.585330\pi$
$853$	−385450.			−0.529749			−0.264874	+	0.964283i	$0.585330\pi$
$854$		0			0
$855$	−302400.	−	61727.1i	−0.413666	−	0.0844392i
$856$		0			0
$857$	356750.			0.485738			0.242869	−	0.970059i	$-0.421911\pi$
$857$	356750.			0.485738			0.242869	+	0.970059i	$0.421911\pi$
$858$		0			0
$859$		−	578227.i		−	0.783631i	−0.920044	−	0.391816i	$-0.871847\pi$
$859$		−	578227.i		−	0.783631i	0.920044	−	0.391816i	$-0.128153\pi$
$860$		0			0
$861$	499800.	+	510106.i	0.674202	+	0.688104i
$862$		0			0
$863$		−	151217.i		−	0.203039i	−0.994834	−	0.101519i	$-0.967630\pi$
$863$		−	151217.i		−	0.203039i	0.994834	−	0.101519i	$-0.0323704\pi$
$864$		0			0
$865$	575.000	−	2816.91i	0.000768485	−	0.00376479i
$866$		0			0
$867$	758520.			1.00909
$868$		0			0
$869$	494929.			0.655395
$870$		0			0
$871$		−	10238.9i		−	0.0134963i
$872$		0			0
$873$	517160.			0.678573
$874$		0			0
$875$	−130375.	−	754443.i	−0.170286	−	0.985395i
$876$		0			0
$877$		−	553981.i		−	0.720271i	−0.932900	−	0.360136i	$-0.882730\pi$
$877$		−	553981.i		−	0.720271i	0.932900	−	0.360136i	$-0.117270\pi$
$878$		0			0
$879$	810625.			1.04916
$880$		0			0
$881$			531882.i			0.685273i	0.939468	+	0.342637i	$0.111320\pi$
$881$			531882.i			0.685273i	−0.939468	+	0.342637i	$0.888680\pi$
$882$		0			0
$883$		−	606675.i		−	0.778099i	−0.921217	−	0.389049i	$-0.872804\pi$
$883$		−	606675.i		−	0.778099i	0.921217	−	0.389049i	$-0.127196\pi$
$884$		0			0
$885$	−444000.	−	90631.1i	−0.566887	−	0.115715i
$886$		0			0
$887$	1.35703e6			1.72481			0.862407	−	0.506216i	$-0.168956\pi$
$887$	1.35703e6			1.72481			0.862407	+	0.506216i	$0.168956\pi$
$888$		0			0
$889$	818496.	+	835374.i	1.03565	+	1.05701i
$890$		0			0
$891$	−98879.0			−0.124551
$892$		0			0
$893$			483897.i			0.606806i
$894$		0			0
$895$	26590.0	−	130264.i	0.0331950	−	0.162621i
$896$		0			0
$897$			17513.9i			0.0217669i
$898$		0			0
$899$			201177.i			0.248919i
$900$		0			0
$901$		−	769826.i		−	0.948294i
$902$		0			0
$903$	−64680.0	−	66013.7i	−0.0793222	−	0.0809578i
$904$		0			0
$905$	302400.	+	61727.1i	0.369219	+	0.0753666i
$906$		0			0
$907$			452230.i			0.549724i	0.961484	+	0.274862i	$0.0886321\pi$
$907$			452230.i			0.549724i	−0.961484	+	0.274862i	$0.911368\pi$
$908$		0			0
$909$			270228.i			0.327041i
$910$		0			0
$911$	948106.			1.14241			0.571203	−	0.820809i	$-0.306477\pi$
$911$	948106.			1.14241			0.571203	+	0.820809i	$0.306477\pi$
$912$		0			0
$913$	−177110.			−0.212472
$914$		0			0
$915$	−237000.	−	48377.4i	−0.283078	−	0.0577831i
$916$		0			0
$917$	−231000.	−	235763.i	−0.274709	−	0.280374i
$918$		0			0
$919$	−394481.			−0.467084			−0.233542	−	0.972347i	$-0.575032\pi$
$919$	−394481.			−0.467084			−0.233542	+	0.972347i	$0.575032\pi$
$920$		0			0
$921$	169025.			0.199265
$922$		0			0
$923$	−22270.0			−0.0261407
$924$		0			0
$925$	399600.	−	938032.i	0.467027	−	1.09631i
$926$		0			0
$927$	264040.			0.307263
$928$		0			0
$929$		−	139523.i		−	0.161664i	−0.996728	−	0.0808322i	$-0.974242\pi$
$929$		−	139523.i		−	0.161664i	0.996728	−	0.0808322i	$-0.0257578\pi$
$930$		0			0
$931$	−529200.	−	10802.2i	−0.610549	−	0.0124628i
$932$		0			0
$933$			252175.i			0.289693i
$934$		0			0
$935$	215825.	−	1.05732e6i	0.246876	−	1.20944i
$936$		0			0
$937$	434285.			0.494647			0.247324	−	0.968933i	$-0.420449\pi$
$937$	434285.			0.494647			0.247324	+	0.968933i	$0.420449\pi$
$938$		0			0
$939$	−340775.			−0.386488
$940$		0			0
$941$		−	539108.i		−	0.608831i	−0.952539	−	0.304415i	$-0.901539\pi$
$941$		−	539108.i		−	0.608831i	0.952539	−	0.304415i	$-0.0984611\pi$
$942$		0			0
$943$	−2.04204e6			−2.29636
$944$		0			0
$945$	−455525.	−	704718.i	−0.510092	−	0.789136i
$946$		0			0
$947$		−	167540.i		−	0.186818i	−0.995628	−	0.0934091i	$-0.970224\pi$
$947$		−	167540.i		−	0.186818i	0.995628	−	0.0934091i	$-0.0297764\pi$
$948$		0			0
$949$	−43250.0			−0.0480235
$950$		0			0
$951$		−	212175.i		−	0.234603i
$952$		0			0
$953$		−	1.31304e6i		−	1.44575i	−0.690981	−	0.722873i	$-0.742821\pi$
$953$		−	1.31304e6i		−	1.44575i	0.690981	−	0.722873i	$-0.257179\pi$
$954$		0			0
$955$	−71315.0	+	349371.i	−0.0781941	+	0.383071i
$956$		0			0
$957$	−84995.0			−0.0928046
$958$		0			0
$959$	791112.	+	807425.i	0.860203	+	0.877941i
$960$		0			0
$961$	−185879.			−0.201272
$962$		0			0
$963$		−	709451.i		−	0.765014i
$964$		0			0
$965$	783960.	+	160025.i	0.841859	+	0.171844i
$966$		0			0
$967$			338745.i			0.362259i	0.983459	+	0.181130i	$0.0579754\pi$
$967$			338745.i			0.362259i	−0.983459	+	0.181130i	$0.942025\pi$
$968$		0			0
$969$		−	534601.i		−	0.569354i
$970$		0			0
$971$		−	1.60718e6i		−	1.70462i	−0.523039	−	0.852309i	$-0.675202\pi$
$971$		−	1.60718e6i		−	1.70462i	0.523039	−	0.852309i	$-0.324798\pi$
$972$		0			0
$973$	1.01136e6	+	1.03221e6i	1.06827	+	1.09030i
$974$		0			0
$975$	−14375.0	−	6123.72i	−0.0151216	−	0.00644179i
$976$		0			0
$977$		−	1.40070e6i		−	1.46743i	−0.679459	−	0.733713i	$-0.737785\pi$
$977$		−	1.40070e6i		−	1.46743i	0.679459	−	0.733713i	$-0.262215\pi$
$978$		0			0
$979$			71941.5i			0.0750610i
$980$		0			0
$981$	−689416.			−0.716380
$982$		0			0
$983$	1.73768e6			1.79831			0.899154	−	0.437633i	$-0.144183\pi$
$983$	1.73768e6			1.79831			0.899154	+	0.437633i	$0.144183\pi$
$984$		0			0
$985$	−83160.0	−	16975.0i	−0.0857121	−	0.0174959i
$986$		0			0
$987$	−384125.	+	376364.i	−0.394310	+	0.386344i
$988$		0			0
$989$	264264.			0.270175
$990$		0			0
$991$	423586.			0.431315			0.215657	−	0.976469i	$-0.430811\pi$
$991$	423586.			0.431315			0.215657	+	0.976469i	$0.430811\pi$
$992$		0			0
$993$	573530.			0.581645
$994$		0			0
$995$	−475200.	−	96999.8i	−0.479988	−	0.0979771i
$996$		0			0
$997$	−548035.			−0.551338			−0.275669	−	0.961253i	$-0.588899\pi$
$997$	−548035.			−0.551338			−0.275669	+	0.961253i	$0.588899\pi$
$998$		0			0
$999$		−	1.11748e6i		−	1.11972i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.5.p.e.209.2		2
4.3	odd	2		35.5.c.c.34.2	yes	2
5.4	even	2		560.5.p.d.209.2		2
7.6	odd	2		560.5.p.d.209.1		2
12.11	even	2		315.5.e.d.244.1		2
20.3	even	4		175.5.d.e.76.3		4
20.7	even	4		175.5.d.e.76.2		4
20.19	odd	2		35.5.c.d.34.1	yes	2
28.27	even	2		35.5.c.d.34.2	yes	2
35.34	odd	2	inner	560.5.p.e.209.1		2
60.59	even	2		315.5.e.c.244.2		2
84.83	odd	2		315.5.e.c.244.1		2
140.27	odd	4		175.5.d.e.76.1		4
140.83	odd	4		175.5.d.e.76.4		4
140.139	even	2		35.5.c.c.34.1	✓	2
420.419	odd	2		315.5.e.d.244.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.5.c.c.34.1	✓	2	140.139	even	2
35.5.c.c.34.2	yes	2	4.3	odd	2
35.5.c.d.34.1	yes	2	20.19	odd	2
35.5.c.d.34.2	yes	2	28.27	even	2
175.5.d.e.76.1		4	140.27	odd	4
175.5.d.e.76.2		4	20.7	even	4
175.5.d.e.76.3		4	20.3	even	4
175.5.d.e.76.4		4	140.83	odd	4
315.5.e.c.244.1		2	84.83	odd	2
315.5.e.c.244.2		2	60.59	even	2
315.5.e.d.244.1		2	12.11	even	2
315.5.e.d.244.2		2	420.419	odd	2
560.5.p.d.209.1		2	7.6	odd	2
560.5.p.d.209.2		2	5.4	even	2
560.5.p.e.209.1		2	35.34	odd	2	inner
560.5.p.e.209.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	560.p (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+5.00000 q^{3} +(-5.00000 + 24.4949i) q^{5} +(35.0000 - 34.2929i) q^{7} -56.0000 q^{9} +O(q^{10})\)	\(q+5.00000 q^{3} +(-5.00000 + 24.4949i) q^{5} +(35.0000 - 34.2929i) q^{7} -56.0000 q^{9} -89.0000 q^{11} +5.00000 q^{13} +(-25.0000 + 122.474i) q^{15} +485.000 q^{17} -220.454i q^{19} +(175.000 - 171.464i) q^{21} +700.554i q^{23} +(-575.000 - 244.949i) q^{25} -685.000 q^{27} +191.000 q^{29} +1053.28i q^{31} -445.000 q^{33} +(665.000 + 1028.79i) q^{35} +1631.36i q^{37} +25.0000 q^{39} +2914.89i q^{41} -377.221i q^{43} +(280.000 - 1371.71i) q^{45} -2195.00 q^{47} +(49.0000 - 2400.50i) q^{49} +2425.00 q^{51} -1587.27i q^{53} +(445.000 - 2180.05i) q^{55} -1102.27i q^{57} +3625.24i q^{59} +1935.10i q^{61} +(-1960.00 + 1920.40i) q^{63} +(-25.0000 + 122.474i) q^{65} -2047.77i q^{67} +3502.77i q^{69} -4454.00 q^{71} -8650.00 q^{73} +(-2875.00 - 1224.74i) q^{75} +(-3115.00 + 3052.06i) q^{77} -5561.00 q^{79} +1111.00 q^{81} +1990.00 q^{83} +(-2425.00 + 11880.0i) q^{85} +955.000 q^{87} -808.332i q^{89} +(175.000 - 171.464i) q^{91} +5266.40i q^{93} +(5400.00 + 1102.27i) q^{95} -9235.00 q^{97} +4984.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 10 q^{3} - 10 q^{5} + 70 q^{7} - 112 q^{9}+O(q^{10}) \) 2 * q + 10 * q^3 - 10 * q^5 + 70 * q^7 - 112 * q^9	\( 2 q + 10 q^{3} - 10 q^{5} + 70 q^{7} - 112 q^{9} - 178 q^{11} + 10 q^{13} - 50 q^{15} + 970 q^{17} + 350 q^{21} - 1150 q^{25} - 1370 q^{27} + 382 q^{29} - 890 q^{33} + 1330 q^{35} + 50 q^{39} + 560 q^{45} - 4390 q^{47} + 98 q^{49} + 4850 q^{51} + 890 q^{55} - 3920 q^{63} - 50 q^{65} - 8908 q^{71} - 17300 q^{73} - 5750 q^{75} - 6230 q^{77} - 11122 q^{79} + 2222 q^{81} + 3980 q^{83} - 4850 q^{85} + 1910 q^{87} + 350 q^{91} + 10800 q^{95} - 18470 q^{97} + 9968 q^{99}+O(q^{100}) \) 2 * q + 10 * q^3 - 10 * q^5 + 70 * q^7 - 112 * q^9 - 178 * q^11 + 10 * q^13 - 50 * q^15 + 970 * q^17 + 350 * q^21 - 1150 * q^25 - 1370 * q^27 + 382 * q^29 - 890 * q^33 + 1330 * q^35 + 50 * q^39 + 560 * q^45 - 4390 * q^47 + 98 * q^49 + 4850 * q^51 + 890 * q^55 - 3920 * q^63 - 50 * q^65 - 8908 * q^71 - 17300 * q^73 - 5750 * q^75 - 6230 * q^77 - 11122 * q^79 + 2222 * q^81 + 3980 * q^83 - 4850 * q^85 + 1910 * q^87 + 350 * q^91 + 10800 * q^95 - 18470 * q^97 + 9968 * q^99

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