LMFDB - Embedded newform 560.4.g.c.449.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [560,4,Mod(449,560)]

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, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("560.449");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 560 = 2^{4} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	560.g (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:	$33.0410696032$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 70)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	560.449
Dual form			560.4.g.c.449.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.00000i q^{3} +(10.0000 + 5.00000i) q^{5} -7.00000i q^{7} -22.0000 q^{9} +O(q^{10})$	$q-7.00000i q^{3} +(10.0000 + 5.00000i) q^{5} -7.00000i q^{7} -22.0000 q^{9} +37.0000 q^{11} -51.0000i q^{13} +(35.0000 - 70.0000i) q^{15} +41.0000i q^{17} -108.000 q^{19} -49.0000 q^{21} -70.0000i q^{23} +(75.0000 + 100.000i) q^{25} -35.0000i q^{27} +249.000 q^{29} +134.000 q^{31} -259.000i q^{33} +(35.0000 - 70.0000i) q^{35} -334.000i q^{37} -357.000 q^{39} +206.000 q^{41} -376.000i q^{43} +(-220.000 - 110.000i) q^{45} +287.000i q^{47} -49.0000 q^{49} +287.000 q^{51} +6.00000i q^{53} +(370.000 + 185.000i) q^{55} +756.000i q^{57} -2.00000 q^{59} -940.000 q^{61} +154.000i q^{63} +(255.000 - 510.000i) q^{65} -106.000i q^{67} -490.000 q^{69} -456.000 q^{71} -650.000i q^{73} +(700.000 - 525.000i) q^{75} -259.000i q^{77} -1239.00 q^{79} -839.000 q^{81} +428.000i q^{83} +(-205.000 + 410.000i) q^{85} -1743.00i q^{87} +220.000 q^{89} -357.000 q^{91} -938.000i q^{93} +(-1080.00 - 540.000i) q^{95} -1055.00i q^{97} -814.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 20 q^{5} - 44 q^{9}+O(q^{10}) $ 2 * q + 20 * q^5 - 44 * q^9	$ 2 q + 20 q^{5} - 44 q^{9} + 74 q^{11} + 70 q^{15} - 216 q^{19} - 98 q^{21} + 150 q^{25} + 498 q^{29} + 268 q^{31} + 70 q^{35} - 714 q^{39} + 412 q^{41} - 440 q^{45} - 98 q^{49} + 574 q^{51} + 740 q^{55} - 4 q^{59} - 1880 q^{61} + 510 q^{65} - 980 q^{69} - 912 q^{71} + 1400 q^{75} - 2478 q^{79} - 1678 q^{81} - 410 q^{85} + 440 q^{89} - 714 q^{91} - 2160 q^{95} - 1628 q^{99}+O(q^{100}) $ 2 * q + 20 * q^5 - 44 * q^9 + 74 * q^11 + 70 * q^15 - 216 * q^19 - 98 * q^21 + 150 * q^25 + 498 * q^29 + 268 * q^31 + 70 * q^35 - 714 * q^39 + 412 * q^41 - 440 * q^45 - 98 * q^49 + 574 * q^51 + 740 * q^55 - 4 * q^59 - 1880 * q^61 + 510 * q^65 - 980 * q^69 - 912 * q^71 + 1400 * q^75 - 2478 * q^79 - 1678 * q^81 - 410 * q^85 + 440 * q^89 - 714 * q^91 - 2160 * q^95 - 1628 * 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
$3$		−	7.00000i		−	1.34715i	−0.739119	−	0.673575i	$-0.764758\pi$
$3$		−	7.00000i		−	1.34715i	0.739119	−	0.673575i	$-0.235242\pi$
$4$		0			0
$5$	10.0000	+	5.00000i	0.894427	+	0.447214i
$5$	10.0000	+	5.00000i	0.894427	+	0.447214i
$6$		0			0
$7$		−	7.00000i		−	0.377964i
$7$		−	7.00000i		−	0.377964i
$8$		0			0
$9$	−22.0000			−0.814815
$10$		0			0
$11$	37.0000			1.01417			0.507087	−	0.861895i	$-0.330722\pi$
$11$	37.0000			1.01417			0.507087	+	0.861895i	$0.330722\pi$
$12$		0			0
$13$		−	51.0000i		−	1.08807i	−0.839064	−	0.544033i	$-0.816897\pi$
$13$		−	51.0000i		−	1.08807i	0.839064	−	0.544033i	$-0.183103\pi$
$14$		0			0
$15$	35.0000	−	70.0000i	0.602464	−	1.20493i
$16$		0			0
$17$			41.0000i			0.584939i	0.956275	+	0.292469i	$0.0944770\pi$
$17$			41.0000i			0.584939i	−0.956275	+	0.292469i	$0.905523\pi$
$18$		0			0
$19$	−108.000			−1.30405			−0.652024	−	0.758199i	$-0.726080\pi$
$19$	−108.000			−1.30405			−0.652024	+	0.758199i	$0.726080\pi$
$20$		0			0
$21$	−49.0000			−0.509175
$22$		0			0
$23$		−	70.0000i		−	0.634609i	−0.948324	−	0.317305i	$-0.897222\pi$
$23$		−	70.0000i		−	0.634609i	0.948324	−	0.317305i	$-0.102778\pi$
$24$		0			0
$25$	75.0000	+	100.000i	0.600000	+	0.800000i
$26$		0			0
$27$		−	35.0000i		−	0.249472i
$28$		0			0
$29$	249.000			1.59442			0.797209	−	0.603703i	$-0.206309\pi$
$29$	249.000			1.59442			0.797209	+	0.603703i	$0.206309\pi$
$30$		0			0
$31$	134.000			0.776358			0.388179	−	0.921584i	$-0.373104\pi$
$31$	134.000			0.776358			0.388179	+	0.921584i	$0.373104\pi$
$32$		0			0
$33$		−	259.000i		−	1.36625i
$34$		0			0
$35$	35.0000	−	70.0000i	0.169031	−	0.338062i
$36$		0			0
$37$		−	334.000i		−	1.48403i	−0.670381	−	0.742017i	$-0.733869\pi$
$37$		−	334.000i		−	1.48403i	0.670381	−	0.742017i	$-0.266131\pi$
$38$		0			0
$39$	−357.000			−1.46579
$40$		0			0
$41$	206.000			0.784678			0.392339	−	0.919821i	$-0.371666\pi$
$41$	206.000			0.784678			0.392339	+	0.919821i	$0.371666\pi$
$42$		0			0
$43$		−	376.000i		−	1.33348i	−0.745292	−	0.666738i	$-0.767690\pi$
$43$		−	376.000i		−	1.33348i	0.745292	−	0.666738i	$-0.232310\pi$
$44$		0			0
$45$	−220.000	−	110.000i	−0.728793	−	0.364396i
$46$		0			0
$47$			287.000i			0.890708i	0.895355	+	0.445354i	$0.146922\pi$
$47$			287.000i			0.890708i	−0.895355	+	0.445354i	$0.853078\pi$
$48$		0			0
$49$	−49.0000			−0.142857
$50$		0			0
$51$	287.000			0.788001
$52$		0			0
$53$			6.00000i			0.0155503i	0.999970	+	0.00777513i	$0.00247492\pi$
$53$			6.00000i			0.0155503i	−0.999970	+	0.00777513i	$0.997525\pi$
$54$		0			0
$55$	370.000	+	185.000i	0.907105	+	0.453553i
$56$		0			0
$57$			756.000i			1.75675i
$58$		0			0
$59$	−2.00000			−0.00441318			−0.00220659	−	0.999998i	$-0.500702\pi$
$59$	−2.00000			−0.00441318			−0.00220659	+	0.999998i	$0.500702\pi$
$60$		0			0
$61$	−940.000			−1.97303			−0.986514	−	0.163679i	$-0.947664\pi$
$61$	−940.000			−1.97303			−0.986514	+	0.163679i	$0.947664\pi$
$62$		0			0
$63$			154.000i			0.307971i
$64$		0			0
$65$	255.000	−	510.000i	0.486598	−	0.973196i
$66$		0			0
$67$		−	106.000i		−	0.193283i	−0.995319	−	0.0966415i	$-0.969190\pi$
$67$		−	106.000i		−	0.193283i	0.995319	−	0.0966415i	$-0.0308100\pi$
$68$		0			0
$69$	−490.000			−0.854914
$70$		0			0
$71$	−456.000			−0.762215			−0.381107	−	0.924531i	$-0.624457\pi$
$71$	−456.000			−0.762215			−0.381107	+	0.924531i	$0.624457\pi$
$72$		0			0
$73$		−	650.000i		−	1.04215i	−0.853512	−	0.521074i	$-0.825532\pi$
$73$		−	650.000i		−	1.04215i	0.853512	−	0.521074i	$-0.174468\pi$
$74$		0			0
$75$	700.000	−	525.000i	1.07772	−	0.808290i
$76$		0			0
$77$		−	259.000i		−	0.383322i
$78$		0			0
$79$	−1239.00			−1.76454			−0.882268	−	0.470747i	$-0.843984\pi$
$79$	−1239.00			−1.76454			−0.882268	+	0.470747i	$0.843984\pi$
$80$		0			0
$81$	−839.000			−1.15089
$82$		0			0
$83$			428.000i			0.566013i	0.959118	+	0.283007i	$0.0913319\pi$
$83$			428.000i			0.566013i	−0.959118	+	0.283007i	$0.908668\pi$
$84$		0			0
$85$	−205.000	+	410.000i	−0.261593	+	0.523185i
$86$		0			0
$87$		−	1743.00i		−	2.14792i
$88$		0			0
$89$	220.000			0.262022			0.131011	−	0.991381i	$-0.458178\pi$
$89$	220.000			0.262022			0.131011	+	0.991381i	$0.458178\pi$
$90$		0			0
$91$	−357.000			−0.411250
$92$		0			0
$93$		−	938.000i		−	1.04587i
$94$		0			0
$95$	−1080.00	−	540.000i	−1.16638	−	0.583188i
$96$		0			0
$97$		−	1055.00i		−	1.10432i	−0.833738	−	0.552160i	$-0.813804\pi$
$97$		−	1055.00i		−	1.10432i	0.833738	−	0.552160i	$-0.186196\pi$
$98$		0			0
$99$	−814.000			−0.826364
$100$		0			0
$101$	1960.00			1.93096			0.965482	−	0.260471i	$-0.0838779\pi$
$101$	1960.00			1.93096			0.965482	+	0.260471i	$0.0838779\pi$
$102$		0			0
$103$			1825.00i			1.74585i	0.487854	+	0.872925i	$0.337780\pi$
$103$			1825.00i			1.74585i	−0.487854	+	0.872925i	$0.662220\pi$
$104$		0			0
$105$	−490.000	−	245.000i	−0.455420	−	0.227710i
$106$		0			0
$107$			144.000i			0.130103i	0.997882	+	0.0650514i	$0.0207211\pi$
$107$			144.000i			0.130103i	−0.997882	+	0.0650514i	$0.979279\pi$
$108$		0			0
$109$	−1681.00			−1.47716			−0.738581	−	0.674165i	$-0.764504\pi$
$109$	−1681.00			−1.47716			−0.738581	+	0.674165i	$0.764504\pi$
$110$		0			0
$111$	−2338.00			−1.99922
$112$		0			0
$113$		−	798.000i		−	0.664332i	−0.943221	−	0.332166i	$-0.892221\pi$
$113$		−	798.000i		−	0.664332i	0.943221	−	0.332166i	$-0.107779\pi$
$114$		0			0
$115$	350.000	−	700.000i	0.283806	−	0.567612i
$116$		0			0
$117$			1122.00i			0.886572i
$118$		0			0
$119$	287.000			0.221086
$120$		0			0
$121$	38.0000			0.0285500
$122$		0			0
$123$		−	1442.00i		−	1.05708i
$124$		0			0
$125$	250.000	+	1375.00i	0.178885	+	0.983870i
$126$		0			0
$127$		−	434.000i		−	0.303238i	−0.988439	−	0.151619i	$-0.951551\pi$
$127$		−	434.000i		−	0.303238i	0.988439	−	0.151619i	$-0.0484487\pi$
$128$		0			0
$129$	−2632.00			−1.79639
$130$		0			0
$131$	1290.00			0.860365			0.430183	−	0.902742i	$-0.358449\pi$
$131$	1290.00			0.860365			0.430183	+	0.902742i	$0.358449\pi$
$132$		0			0
$133$			756.000i			0.492884i
$134$		0			0
$135$	175.000	−	350.000i	0.111567	−	0.223135i
$136$		0			0
$137$			192.000i			0.119735i	0.998206	+	0.0598674i	$0.0190678\pi$
$137$			192.000i			0.119735i	−0.998206	+	0.0598674i	$0.980932\pi$
$138$		0			0
$139$	1402.00			0.855511			0.427756	−	0.903894i	$-0.359304\pi$
$139$	1402.00			0.855511			0.427756	+	0.903894i	$0.359304\pi$
$140$		0			0
$141$	2009.00			1.19992
$142$		0			0
$143$		−	1887.00i		−	1.10349i
$144$		0			0
$145$	2490.00	+	1245.00i	1.42609	+	0.713046i
$146$		0			0
$147$			343.000i			0.192450i
$148$		0			0
$149$	302.000			0.166046			0.0830228	−	0.996548i	$-0.473543\pi$
$149$	302.000			0.166046			0.0830228	+	0.996548i	$0.473543\pi$
$150$		0			0
$151$	3167.00			1.70680			0.853400	−	0.521257i	$-0.174537\pi$
$151$	3167.00			1.70680			0.853400	+	0.521257i	$0.174537\pi$
$152$		0			0
$153$		−	902.000i		−	0.476617i
$154$		0			0
$155$	1340.00	+	670.000i	0.694396	+	0.347198i
$156$		0			0
$157$		−	470.000i		−	0.238918i	−0.992839	−	0.119459i	$-0.961884\pi$
$157$		−	470.000i		−	0.238918i	0.992839	−	0.119459i	$-0.0381160\pi$
$158$		0			0
$159$	42.0000			0.0209485
$160$		0			0
$161$	−490.000			−0.239860
$162$		0			0
$163$		−	2390.00i		−	1.14846i	−0.818693	−	0.574231i	$-0.805301\pi$
$163$		−	2390.00i		−	1.14846i	0.818693	−	0.574231i	$-0.194699\pi$
$164$		0			0
$165$	1295.00	−	2590.00i	0.611004	−	1.22201i
$166$		0			0
$167$			2631.00i			1.21912i	0.792740	+	0.609560i	$0.208654\pi$
$167$			2631.00i			1.21912i	−0.792740	+	0.609560i	$0.791346\pi$
$168$		0			0
$169$	−404.000			−0.183887
$170$		0			0
$171$	2376.00			1.06256
$172$		0			0
$173$			2243.00i			0.985735i	0.870104	+	0.492867i	$0.164051\pi$
$173$			2243.00i			0.985735i	−0.870104	+	0.492867i	$0.835949\pi$
$174$		0			0
$175$	700.000	−	525.000i	0.302372	−	0.226779i
$176$		0			0
$177$			14.0000i			0.00594522i
$178$		0			0
$179$	52.0000			0.0217132			0.0108566	−	0.999941i	$-0.496544\pi$
$179$	52.0000			0.0217132			0.0108566	+	0.999941i	$0.496544\pi$
$180$		0			0
$181$	2462.00			1.01104			0.505522	−	0.862814i	$-0.331300\pi$
$181$	2462.00			1.01104			0.505522	+	0.862814i	$0.331300\pi$
$182$		0			0
$183$			6580.00i			2.65797i
$184$		0			0
$185$	1670.00	−	3340.00i	0.663680	−	1.32736i
$186$		0			0
$187$			1517.00i			0.593230i
$188$		0			0
$189$	−245.000			−0.0942917
$190$		0			0
$191$	−3159.00			−1.19674			−0.598370	−	0.801220i	$-0.704185\pi$
$191$	−3159.00			−1.19674			−0.598370	+	0.801220i	$0.704185\pi$
$192$		0			0
$193$		−	2060.00i		−	0.768301i	−0.923271	−	0.384150i	$-0.874494\pi$
$193$		−	2060.00i		−	0.768301i	0.923271	−	0.384150i	$-0.125506\pi$
$194$		0			0
$195$	−3570.00	−	1785.00i	−1.31104	−	0.655521i
$196$		0			0
$197$			1738.00i			0.628565i	0.949329	+	0.314283i	$0.101764\pi$
$197$			1738.00i			0.628565i	−0.949329	+	0.314283i	$0.898236\pi$
$198$		0			0
$199$	−894.000			−0.318462			−0.159231	−	0.987241i	$-0.550901\pi$
$199$	−894.000			−0.318462			−0.159231	+	0.987241i	$0.550901\pi$
$200$		0			0
$201$	−742.000			−0.260381
$202$		0			0
$203$		−	1743.00i		−	0.602634i
$204$		0			0
$205$	2060.00	+	1030.00i	0.701837	+	0.350919i
$206$		0			0
$207$			1540.00i			0.517089i
$208$		0			0
$209$	−3996.00			−1.32253
$210$		0			0
$211$	4083.00			1.33216			0.666079	−	0.745881i	$-0.267971\pi$
$211$	4083.00			1.33216			0.666079	+	0.745881i	$0.267971\pi$
$212$		0			0
$213$			3192.00i			1.02682i
$214$		0			0
$215$	1880.00	−	3760.00i	0.596349	−	1.19270i
$216$		0			0
$217$		−	938.000i		−	0.293436i
$218$		0			0
$219$	−4550.00			−1.40393
$220$		0			0
$221$	2091.00			0.636452
$222$		0			0
$223$			377.000i			0.113210i	0.998397	+	0.0566049i	$0.0180275\pi$
$223$			377.000i			0.113210i	−0.998397	+	0.0566049i	$0.981972\pi$
$224$		0			0
$225$	−1650.00	−	2200.00i	−0.488889	−	0.651852i
$226$		0			0
$227$			2551.00i			0.745885i	0.927855	+	0.372942i	$0.121651\pi$
$227$			2551.00i			0.745885i	−0.927855	+	0.372942i	$0.878349\pi$
$228$		0			0
$229$	−74.0000			−0.0213540			−0.0106770	−	0.999943i	$-0.503399\pi$
$229$	−74.0000			−0.0213540			−0.0106770	+	0.999943i	$0.503399\pi$
$230$		0			0
$231$	−1813.00			−0.516392
$232$		0			0
$233$		−	1888.00i		−	0.530845i	−0.964132	−	0.265423i	$-0.914488\pi$
$233$		−	1888.00i		−	0.530845i	0.964132	−	0.265423i	$-0.0855115\pi$
$234$		0			0
$235$	−1435.00	+	2870.00i	−0.398337	+	0.796673i
$236$		0			0
$237$			8673.00i			2.37710i
$238$		0			0
$239$	4997.00			1.35242			0.676211	−	0.736708i	$-0.263621\pi$
$239$	4997.00			1.35242			0.676211	+	0.736708i	$0.263621\pi$
$240$		0			0
$241$	−3830.00			−1.02370			−0.511851	−	0.859074i	$-0.671040\pi$
$241$	−3830.00			−1.02370			−0.511851	+	0.859074i	$0.671040\pi$
$242$		0			0
$243$			4928.00i			1.30095i
$244$		0			0
$245$	−490.000	−	245.000i	−0.127775	−	0.0638877i
$246$		0			0
$247$			5508.00i			1.41889i
$248$		0			0
$249$	2996.00			0.762505
$250$		0			0
$251$	3390.00			0.852490			0.426245	−	0.904608i	$-0.359836\pi$
$251$	3390.00			0.852490			0.426245	+	0.904608i	$0.359836\pi$
$252$		0			0
$253$		−	2590.00i		−	0.643604i
$254$		0			0
$255$	2870.00	+	1435.00i	0.704809	+	0.352405i
$256$		0			0
$257$			7170.00i			1.74028i	0.492803	+	0.870141i	$0.335972\pi$
$257$			7170.00i			1.74028i	−0.492803	+	0.870141i	$0.664028\pi$
$258$		0			0
$259$	−2338.00			−0.560912
$260$		0			0
$261$	−5478.00			−1.29916
$262$		0			0
$263$		−	7672.00i		−	1.79877i	−0.437160	−	0.899384i	$-0.644016\pi$
$263$		−	7672.00i		−	1.79877i	0.437160	−	0.899384i	$-0.355984\pi$
$264$		0			0
$265$	−30.0000	+	60.0000i	−0.00695428	+	0.0139086i
$266$		0			0
$267$		−	1540.00i		−	0.352983i
$268$		0			0
$269$	54.0000			0.0122395			0.00611977	−	0.999981i	$-0.498052\pi$
$269$	54.0000			0.0122395			0.00611977	+	0.999981i	$0.498052\pi$
$270$		0			0
$271$	−2932.00			−0.657219			−0.328609	−	0.944466i	$-0.606580\pi$
$271$	−2932.00			−0.657219			−0.328609	+	0.944466i	$0.606580\pi$
$272$		0			0
$273$			2499.00i			0.554016i
$274$		0			0
$275$	2775.00	+	3700.00i	0.608505	+	0.811340i
$276$		0			0
$277$			3254.00i			0.705826i	0.935656	+	0.352913i	$0.114809\pi$
$277$			3254.00i			0.705826i	−0.935656	+	0.352913i	$0.885191\pi$
$278$		0			0
$279$	−2948.00			−0.632588
$280$		0			0
$281$	3327.00			0.706307			0.353153	−	0.935565i	$-0.385109\pi$
$281$	3327.00			0.706307			0.353153	+	0.935565i	$0.385109\pi$
$282$		0			0
$283$			4627.00i			0.971896i	0.873988	+	0.485948i	$0.161526\pi$
$283$			4627.00i			0.971896i	−0.873988	+	0.485948i	$0.838474\pi$
$284$		0			0
$285$	−3780.00	+	7560.00i	−0.785642	+	1.57128i
$286$		0			0
$287$		−	1442.00i		−	0.296580i
$288$		0			0
$289$	3232.00			0.657847
$290$		0			0
$291$	−7385.00			−1.48769
$292$		0			0
$293$			4083.00i			0.814100i	0.913406	+	0.407050i	$0.133443\pi$
$293$			4083.00i			0.814100i	−0.913406	+	0.407050i	$0.866557\pi$
$294$		0			0
$295$	−20.0000	−	10.0000i	−0.00394727	−	0.00197364i
$296$		0			0
$297$		−	1295.00i		−	0.253008i
$298$		0			0
$299$	−3570.00			−0.690496
$300$		0			0
$301$	−2632.00			−0.504007
$302$		0			0
$303$		−	13720.0i		−	2.60130i
$304$		0			0
$305$	−9400.00	−	4700.00i	−1.76473	−	0.882365i
$306$		0			0
$307$			4089.00i			0.760168i	0.924952	+	0.380084i	$0.124105\pi$
$307$			4089.00i			0.760168i	−0.924952	+	0.380084i	$0.875895\pi$
$308$		0			0
$309$	12775.0			2.35192
$310$		0			0
$311$	4008.00			0.730781			0.365390	−	0.930854i	$-0.380935\pi$
$311$	4008.00			0.730781			0.365390	+	0.930854i	$0.380935\pi$
$312$		0			0
$313$		−	7355.00i		−	1.32821i	−0.747640	−	0.664104i	$-0.768813\pi$
$313$		−	7355.00i		−	1.32821i	0.747640	−	0.664104i	$-0.231187\pi$
$314$		0			0
$315$	−770.000	+	1540.00i	−0.137729	+	0.275458i
$316$		0			0
$317$		−	1684.00i		−	0.298369i	−0.988809	−	0.149184i	$-0.952335\pi$
$317$		−	1684.00i		−	0.298369i	0.988809	−	0.149184i	$-0.0476648\pi$
$318$		0			0
$319$	9213.00			1.61702
$320$		0			0
$321$	1008.00			0.175268
$322$		0			0
$323$		−	4428.00i		−	0.762788i
$324$		0			0
$325$	5100.00	−	3825.00i	0.870453	−	0.652839i
$326$		0			0
$327$			11767.0i			1.98996i
$328$		0			0
$329$	2009.00			0.336656
$330$		0			0
$331$	1460.00			0.242444			0.121222	−	0.992625i	$-0.461319\pi$
$331$	1460.00			0.242444			0.121222	+	0.992625i	$0.461319\pi$
$332$		0			0
$333$			7348.00i			1.20921i
$334$		0			0
$335$	530.000	−	1060.00i	0.0864388	−	0.172878i
$336$		0			0
$337$			7514.00i			1.21458i	0.794480	+	0.607290i	$0.207744\pi$
$337$			7514.00i			1.21458i	−0.794480	+	0.607290i	$0.792256\pi$
$338$		0			0
$339$	−5586.00			−0.894955
$340$		0			0
$341$	4958.00			0.787363
$342$		0			0
$343$			343.000i			0.0539949i
$344$		0			0
$345$	−4900.00	−	2450.00i	−0.764658	−	0.382329i
$346$		0			0
$347$			2862.00i			0.442767i	0.975187	+	0.221384i	$0.0710573\pi$
$347$			2862.00i			0.442767i	−0.975187	+	0.221384i	$0.928943\pi$
$348$		0			0
$349$	6368.00			0.976708			0.488354	−	0.872646i	$-0.337597\pi$
$349$	6368.00			0.976708			0.488354	+	0.872646i	$0.337597\pi$
$350$		0			0
$351$	−1785.00			−0.271442
$352$		0			0
$353$			3635.00i			0.548078i	0.961719	+	0.274039i	$0.0883597\pi$
$353$			3635.00i			0.548078i	−0.961719	+	0.274039i	$0.911640\pi$
$354$		0			0
$355$	−4560.00	−	2280.00i	−0.681746	−	0.340873i
$356$		0			0
$357$		−	2009.00i		−	0.297836i
$358$		0			0
$359$	7116.00			1.04615			0.523075	−	0.852286i	$-0.324785\pi$
$359$	7116.00			1.04615			0.523075	+	0.852286i	$0.324785\pi$
$360$		0			0
$361$	4805.00			0.700539
$362$		0			0
$363$		−	266.000i		−	0.0384611i
$364$		0			0
$365$	3250.00	−	6500.00i	0.466062	−	0.932125i
$366$		0			0
$367$			319.000i			0.0453724i	0.999743	+	0.0226862i	$0.00722186\pi$
$367$			319.000i			0.0453724i	−0.999743	+	0.0226862i	$0.992778\pi$
$368$		0			0
$369$	−4532.00			−0.639367
$370$		0			0
$371$	42.0000			0.00587744
$372$		0			0
$373$			11652.0i			1.61747i	0.588171	+	0.808737i	$0.299848\pi$
$373$			11652.0i			1.61747i	−0.588171	+	0.808737i	$0.700152\pi$
$374$		0			0
$375$	9625.00	−	1750.00i	1.32542	−	0.240986i
$376$		0			0
$377$		−	12699.0i		−	1.73483i
$378$		0			0
$379$	7748.00			1.05010			0.525050	−	0.851071i	$-0.324047\pi$
$379$	7748.00			1.05010			0.525050	+	0.851071i	$0.324047\pi$
$380$		0			0
$381$	−3038.00			−0.408508
$382$		0			0
$383$			8680.00i			1.15803i	0.815315	+	0.579017i	$0.196564\pi$
$383$			8680.00i			1.15803i	−0.815315	+	0.579017i	$0.803436\pi$
$384$		0			0
$385$	1295.00	−	2590.00i	0.171427	−	0.342854i
$386$		0			0
$387$			8272.00i			1.08654i
$388$		0			0
$389$	1711.00			0.223011			0.111505	−	0.993764i	$-0.464433\pi$
$389$	1711.00			0.223011			0.111505	+	0.993764i	$0.464433\pi$
$390$		0			0
$391$	2870.00			0.371208
$392$		0			0
$393$		−	9030.00i		−	1.15904i
$394$		0			0
$395$	−12390.0	−	6195.00i	−1.57825	−	0.789125i
$396$		0			0
$397$			1589.00i			0.200881i	0.994943	+	0.100440i	$0.0320252\pi$
$397$			1589.00i			0.200881i	−0.994943	+	0.100440i	$0.967975\pi$
$398$		0			0
$399$	5292.00			0.663988
$400$		0			0
$401$	−5147.00			−0.640970			−0.320485	−	0.947254i	$-0.603846\pi$
$401$	−5147.00			−0.640970			−0.320485	+	0.947254i	$0.603846\pi$
$402$		0			0
$403$		−	6834.00i		−	0.844729i
$404$		0			0
$405$	−8390.00	−	4195.00i	−1.02939	−	0.514694i
$406$		0			0
$407$		−	12358.0i		−	1.50507i
$408$		0			0
$409$	9100.00			1.10016			0.550081	−	0.835111i	$-0.314597\pi$
$409$	9100.00			1.10016			0.550081	+	0.835111i	$0.314597\pi$
$410$		0			0
$411$	1344.00			0.161301
$412$		0			0
$413$			14.0000i			0.00166803i
$414$		0			0
$415$	−2140.00	+	4280.00i	−0.253129	+	0.506258i
$416$		0			0
$417$		−	9814.00i		−	1.15250i
$418$		0			0
$419$	2618.00			0.305245			0.152623	−	0.988285i	$-0.451228\pi$
$419$	2618.00			0.305245			0.152623	+	0.988285i	$0.451228\pi$
$420$		0			0
$421$	−3695.00			−0.427751			−0.213876	−	0.976861i	$-0.568609\pi$
$421$	−3695.00			−0.427751			−0.213876	+	0.976861i	$0.568609\pi$
$422$		0			0
$423$		−	6314.00i		−	0.725762i
$424$		0			0
$425$	−4100.00	+	3075.00i	−0.467951	+	0.350963i
$426$		0			0
$427$			6580.00i			0.745734i
$428$		0			0
$429$	−13209.0			−1.48657
$430$		0			0
$431$	−15779.0			−1.76345			−0.881726	−	0.471762i	$-0.843618\pi$
$431$	−15779.0			−1.76345			−0.881726	+	0.471762i	$0.843618\pi$
$432$		0			0
$433$		−	7238.00i		−	0.803317i	−0.915790	−	0.401658i	$-0.868434\pi$
$433$		−	7238.00i		−	0.803317i	0.915790	−	0.401658i	$-0.131566\pi$
$434$		0			0
$435$	8715.00	−	17430.0i	0.960580	−	1.92116i
$436$		0			0
$437$			7560.00i			0.827560i
$438$		0			0
$439$	−2646.00			−0.287669			−0.143834	−	0.989602i	$-0.545943\pi$
$439$	−2646.00			−0.287669			−0.143834	+	0.989602i	$0.545943\pi$
$440$		0			0
$441$	1078.00			0.116402
$442$		0			0
$443$			5688.00i			0.610034i	0.952347	+	0.305017i	$0.0986621\pi$
$443$			5688.00i			0.610034i	−0.952347	+	0.305017i	$0.901338\pi$
$444$		0			0
$445$	2200.00	+	1100.00i	0.234360	+	0.117180i
$446$		0			0
$447$		−	2114.00i		−	0.223689i
$448$		0			0
$449$	3285.00			0.345276			0.172638	−	0.984985i	$-0.444771\pi$
$449$	3285.00			0.345276			0.172638	+	0.984985i	$0.444771\pi$
$450$		0			0
$451$	7622.00			0.795800
$452$		0			0
$453$		−	22169.0i		−	2.29932i
$454$		0			0
$455$	−3570.00	−	1785.00i	−0.367833	−	0.183917i
$456$		0			0
$457$			14834.0i			1.51839i	0.650862	+	0.759196i	$0.274408\pi$
$457$			14834.0i			1.51839i	−0.650862	+	0.759196i	$0.725592\pi$
$458$		0			0
$459$	1435.00			0.145926
$460$		0			0
$461$	−9972.00			−1.00747			−0.503734	−	0.863859i	$-0.668041\pi$
$461$	−9972.00			−1.00747			−0.503734	+	0.863859i	$0.668041\pi$
$462$		0			0
$463$		−	9096.00i		−	0.913017i	−0.889719	−	0.456509i	$-0.849100\pi$
$463$		−	9096.00i		−	0.913017i	0.889719	−	0.456509i	$-0.150900\pi$
$464$		0			0
$465$	4690.00	−	9380.00i	0.467728	−	0.935456i
$466$		0			0
$467$		−	15867.0i		−	1.57224i	−0.618072	−	0.786121i	$-0.712086\pi$
$467$		−	15867.0i		−	1.57224i	0.618072	−	0.786121i	$-0.287914\pi$
$468$		0			0
$469$	−742.000			−0.0730541
$470$		0			0
$471$	−3290.00			−0.321858
$472$		0			0
$473$		−	13912.0i		−	1.35238i
$474$		0			0
$475$	−8100.00	−	10800.0i	−0.782428	−	1.04324i
$476$		0			0
$477$		−	132.000i		−	0.0126706i
$478$		0			0
$479$	242.000			0.0230841			0.0115420	−	0.999933i	$-0.496326\pi$
$479$	242.000			0.0230841			0.0115420	+	0.999933i	$0.496326\pi$
$480$		0			0
$481$	−17034.0			−1.61473
$482$		0			0
$483$			3430.00i			0.323127i
$484$		0			0
$485$	5275.00	−	10550.0i	0.493867	−	0.987734i
$486$		0			0
$487$			3558.00i			0.331064i	0.986204	+	0.165532i	$0.0529342\pi$
$487$			3558.00i			0.331064i	−0.986204	+	0.165532i	$0.947066\pi$
$488$		0			0
$489$	−16730.0			−1.54715
$490$		0			0
$491$	−1473.00			−0.135388			−0.0676941	−	0.997706i	$-0.521564\pi$
$491$	−1473.00			−0.135388			−0.0676941	+	0.997706i	$0.521564\pi$
$492$		0			0
$493$			10209.0i			0.932637i
$494$		0			0
$495$	−8140.00	−	4070.00i	−0.739123	−	0.369561i
$496$		0			0
$497$			3192.00i			0.288090i
$498$		0			0
$499$	603.000			0.0540962			0.0270481	−	0.999634i	$-0.491389\pi$
$499$	603.000			0.0540962			0.0270481	+	0.999634i	$0.491389\pi$
$500$		0			0
$501$	18417.0			1.64234
$502$		0			0
$503$			18387.0i			1.62989i	0.579537	+	0.814946i	$0.303234\pi$
$503$			18387.0i			1.62989i	−0.579537	+	0.814946i	$0.696766\pi$
$504$		0			0
$505$	19600.0	+	9800.00i	1.72711	+	0.863553i
$506$		0			0
$507$			2828.00i			0.247724i
$508$		0			0
$509$	−9018.00			−0.785296			−0.392648	−	0.919689i	$-0.628441\pi$
$509$	−9018.00			−0.785296			−0.392648	+	0.919689i	$0.628441\pi$
$510$		0			0
$511$	−4550.00			−0.393895
$512$		0			0
$513$			3780.00i			0.325324i
$514$		0			0
$515$	−9125.00	+	18250.0i	−0.780768	+	1.56154i
$516$		0			0
$517$			10619.0i			0.903333i
$518$		0			0
$519$	15701.0			1.32793
$520$		0			0
$521$	4624.00			0.388831			0.194416	−	0.980919i	$-0.437719\pi$
$521$	4624.00			0.388831			0.194416	+	0.980919i	$0.437719\pi$
$522$		0			0
$523$		−	5876.00i		−	0.491280i	−0.969361	−	0.245640i	$-0.921002\pi$
$523$		−	5876.00i		−	0.491280i	0.969361	−	0.245640i	$-0.0789981\pi$
$524$		0			0
$525$	−3675.00	−	4900.00i	−0.305505	−	0.407340i
$526$		0			0
$527$			5494.00i			0.454122i
$528$		0			0
$529$	7267.00			0.597271
$530$		0			0
$531$	44.0000			0.00359593
$532$		0			0
$533$		−	10506.0i		−	0.853781i
$534$		0			0
$535$	−720.000	+	1440.00i	−0.0581838	+	0.116368i
$536$		0			0
$537$		−	364.000i		−	0.0292509i
$538$		0			0
$539$	−1813.00			−0.144882
$540$		0			0
$541$	−8537.00			−0.678437			−0.339218	−	0.940708i	$-0.610163\pi$
$541$	−8537.00			−0.678437			−0.339218	+	0.940708i	$0.610163\pi$
$542$		0			0
$543$		−	17234.0i		−	1.36203i
$544$		0			0
$545$	−16810.0	−	8405.00i	−1.32121	−	0.660607i
$546$		0			0
$547$			13060.0i			1.02085i	0.859922	+	0.510425i	$0.170512\pi$
$547$			13060.0i			1.02085i	−0.859922	+	0.510425i	$0.829488\pi$
$548$		0			0
$549$	20680.0			1.60765
$550$		0			0
$551$	−26892.0			−2.07920
$552$		0			0
$553$			8673.00i			0.666932i
$554$		0			0
$555$	−23380.0	−	11690.0i	−1.78815	−	0.894077i
$556$		0			0
$557$			21372.0i			1.62578i	0.582416	+	0.812891i	$0.302108\pi$
$557$			21372.0i			1.62578i	−0.582416	+	0.812891i	$0.697892\pi$
$558$		0			0
$559$	−19176.0			−1.45091
$560$		0			0
$561$	10619.0			0.799170
$562$		0			0
$563$		−	12704.0i		−	0.950994i	−0.879717	−	0.475497i	$-0.842268\pi$
$563$		−	12704.0i		−	0.950994i	0.879717	−	0.475497i	$-0.157732\pi$
$564$		0			0
$565$	3990.00	−	7980.00i	0.297098	−	0.594197i
$566$		0			0
$567$			5873.00i			0.434996i
$568$		0			0
$569$	−8762.00			−0.645557			−0.322779	−	0.946474i	$-0.604617\pi$
$569$	−8762.00			−0.645557			−0.322779	+	0.946474i	$0.604617\pi$
$570$		0			0
$571$	24764.0			1.81496			0.907479	−	0.420097i	$-0.138004\pi$
$571$	24764.0			1.81496			0.907479	+	0.420097i	$0.138004\pi$
$572$		0			0
$573$			22113.0i			1.61219i
$574$		0			0
$575$	7000.00	−	5250.00i	0.507687	−	0.380765i
$576$		0			0
$577$			1811.00i			0.130664i	0.997864	+	0.0653318i	$0.0208106\pi$
$577$			1811.00i			0.130664i	−0.997864	+	0.0653318i	$0.979189\pi$
$578$		0			0
$579$	−14420.0			−1.03502
$580$		0			0
$581$	2996.00			0.213933
$582$		0			0
$583$			222.000i			0.0157707i
$584$		0			0
$585$	−5610.00	+	11220.0i	−0.396487	+	0.792974i
$586$		0			0
$587$		−	10548.0i		−	0.741674i	−0.928698	−	0.370837i	$-0.879071\pi$
$587$		−	10548.0i		−	0.741674i	0.928698	−	0.370837i	$-0.120929\pi$
$588$		0			0
$589$	−14472.0			−1.01241
$590$		0			0
$591$	12166.0			0.846772
$592$		0			0
$593$		−	17439.0i		−	1.20765i	−0.797119	−	0.603823i	$-0.793643\pi$
$593$		−	17439.0i		−	1.20765i	0.797119	−	0.603823i	$-0.206357\pi$
$594$		0			0
$595$	2870.00	+	1435.00i	0.197745	+	0.0988727i
$596$		0			0
$597$			6258.00i			0.429017i
$598$		0			0
$599$	2451.00			0.167187			0.0835936	−	0.996500i	$-0.473360\pi$
$599$	2451.00			0.167187			0.0835936	+	0.996500i	$0.473360\pi$
$600$		0			0
$601$	−7792.00			−0.528856			−0.264428	−	0.964405i	$-0.585183\pi$
$601$	−7792.00			−0.528856			−0.264428	+	0.964405i	$0.585183\pi$
$602$		0			0
$603$			2332.00i			0.157490i
$604$		0			0
$605$	380.000	+	190.000i	0.0255359	+	0.0127679i
$606$		0			0
$607$			1937.00i			0.129523i	0.997901	+	0.0647615i	$0.0206286\pi$
$607$			1937.00i			0.129523i	−0.997901	+	0.0647615i	$0.979371\pi$
$608$		0			0
$609$	−12201.0			−0.811838
$610$		0			0
$611$	14637.0			0.969148
$612$		0			0
$613$		−	5036.00i		−	0.331814i	−0.986141	−	0.165907i	$-0.946945\pi$
$613$		−	5036.00i		−	0.331814i	0.986141	−	0.165907i	$-0.0530552\pi$
$614$		0			0
$615$	7210.00	−	14420.0i	0.472740	−	0.945481i
$616$		0			0
$617$		−	27286.0i		−	1.78038i	−0.455592	−	0.890189i	$-0.650572\pi$
$617$		−	27286.0i		−	1.78038i	0.455592	−	0.890189i	$-0.349428\pi$
$618$		0			0
$619$	28538.0			1.85305			0.926526	−	0.376231i	$-0.122780\pi$
$619$	28538.0			1.85305			0.926526	+	0.376231i	$0.122780\pi$
$620$		0			0
$621$	−2450.00			−0.158317
$622$		0			0
$623$		−	1540.00i		−	0.0990350i
$624$		0			0
$625$	−4375.00	+	15000.0i	−0.280000	+	0.960000i
$626$		0			0
$627$			27972.0i			1.78165i
$628$		0			0
$629$	13694.0			0.868069
$630$		0			0
$631$	−25007.0			−1.57768			−0.788838	−	0.614602i	$-0.789317\pi$
$631$	−25007.0			−1.57768			−0.788838	+	0.614602i	$0.789317\pi$
$632$		0			0
$633$		−	28581.0i		−	1.79462i
$634$		0			0
$635$	2170.00	−	4340.00i	0.135612	−	0.271225i
$636$		0			0
$637$			2499.00i			0.155438i
$638$		0			0
$639$	10032.0			0.621064
$640$		0			0
$641$	−12130.0			−0.747436			−0.373718	−	0.927542i	$-0.621917\pi$
$641$	−12130.0			−0.747436			−0.373718	+	0.927542i	$0.621917\pi$
$642$		0			0
$643$			14385.0i			0.882254i	0.897445	+	0.441127i	$0.145421\pi$
$643$			14385.0i			0.882254i	−0.897445	+	0.441127i	$0.854579\pi$
$644$		0			0
$645$	−26320.0	−	13160.0i	−1.60674	−	0.803371i
$646$		0			0
$647$			2208.00i			0.134166i	0.997747	+	0.0670830i	$0.0213692\pi$
$647$			2208.00i			0.134166i	−0.997747	+	0.0670830i	$0.978631\pi$
$648$		0			0
$649$	−74.0000			−0.00447574
$650$		0			0
$651$	−6566.00			−0.395302
$652$		0			0
$653$		−	22448.0i		−	1.34527i	−0.739977	−	0.672633i	$-0.765163\pi$
$653$		−	22448.0i		−	1.34527i	0.739977	−	0.672633i	$-0.234837\pi$
$654$		0			0
$655$	12900.0	+	6450.00i	0.769534	+	0.384767i
$656$		0			0
$657$			14300.0i			0.849157i
$658$		0			0
$659$	8791.00			0.519649			0.259825	−	0.965656i	$-0.416335\pi$
$659$	8791.00			0.519649			0.259825	+	0.965656i	$0.416335\pi$
$660$		0			0
$661$	−13180.0			−0.775556			−0.387778	−	0.921753i	$-0.626757\pi$
$661$	−13180.0			−0.775556			−0.387778	+	0.921753i	$0.626757\pi$
$662$		0			0
$663$		−	14637.0i		−	0.857397i
$664$		0			0
$665$	−3780.00	+	7560.00i	−0.220424	+	0.440848i
$666$		0			0
$667$		−	17430.0i		−	1.01183i
$668$		0			0
$669$	2639.00			0.152511
$670$		0			0
$671$	−34780.0			−2.00099
$672$		0			0
$673$			7164.00i			0.410330i	0.978727	+	0.205165i	$0.0657731\pi$
$673$			7164.00i			0.410330i	−0.978727	+	0.205165i	$0.934227\pi$
$674$		0			0
$675$	3500.00	−	2625.00i	0.199578	−	0.149683i
$676$		0			0
$677$			12335.0i			0.700255i	0.936702	+	0.350127i	$0.113862\pi$
$677$			12335.0i			0.700255i	−0.936702	+	0.350127i	$0.886138\pi$
$678$		0			0
$679$	−7385.00			−0.417394
$680$		0			0
$681$	17857.0			1.00482
$682$		0			0
$683$		−	15436.0i		−	0.864776i	−0.901688	−	0.432388i	$-0.857671\pi$
$683$		−	15436.0i		−	0.864776i	0.901688	−	0.432388i	$-0.142329\pi$
$684$		0			0
$685$	−960.000	+	1920.00i	−0.0535470	+	0.107094i
$686$		0			0
$687$			518.000i			0.0287670i
$688$		0			0
$689$	306.000			0.0169197
$690$		0			0
$691$	19184.0			1.05614			0.528071	−	0.849200i	$-0.322916\pi$
$691$	19184.0			1.05614			0.528071	+	0.849200i	$0.322916\pi$
$692$		0			0
$693$			5698.00i			0.312336i
$694$		0			0
$695$	14020.0	+	7010.00i	0.765193	+	0.382596i
$696$		0			0
$697$			8446.00i			0.458989i
$698$		0			0
$699$	−13216.0			−0.715129
$700$		0			0
$701$	32975.0			1.77667			0.888337	−	0.459192i	$-0.151861\pi$
$701$	32975.0			1.77667			0.888337	+	0.459192i	$0.151861\pi$
$702$		0			0
$703$			36072.0i			1.93525i
$704$		0			0
$705$	20090.0	+	10045.0i	1.07324	+	0.536619i
$706$		0			0
$707$		−	13720.0i		−	0.729836i
$708$		0			0
$709$	31497.0			1.66840			0.834199	−	0.551463i	$-0.185930\pi$
$709$	31497.0			1.66840			0.834199	+	0.551463i	$0.185930\pi$
$710$		0			0
$711$	27258.0			1.43777
$712$		0			0
$713$		−	9380.00i		−	0.492684i
$714$		0			0
$715$	9435.00	−	18870.0i	0.493495	−	0.986990i
$716$		0			0
$717$		−	34979.0i		−	1.82192i
$718$		0			0
$719$	−18610.0			−0.965279			−0.482640	−	0.875819i	$-0.660322\pi$
$719$	−18610.0			−0.965279			−0.482640	+	0.875819i	$0.660322\pi$
$720$		0			0
$721$	12775.0			0.659869
$722$		0			0
$723$			26810.0i			1.37908i
$724$		0			0
$725$	18675.0	+	24900.0i	0.956651	+	1.27553i
$726$		0			0
$727$			17508.0i			0.893172i	0.894741	+	0.446586i	$0.147360\pi$
$727$			17508.0i			0.893172i	−0.894741	+	0.446586i	$0.852640\pi$
$728$		0			0
$729$	11843.0			0.601687
$730$		0			0
$731$	15416.0			0.780002
$732$		0			0
$733$			4685.00i			0.236077i	0.993009	+	0.118038i	$0.0376606\pi$
$733$			4685.00i			0.236077i	−0.993009	+	0.118038i	$0.962339\pi$
$734$		0			0
$735$	−1715.00	+	3430.00i	−0.0860663	+	0.172133i
$736$		0			0
$737$		−	3922.00i		−	0.196023i
$738$		0			0
$739$	−25925.0			−1.29048			−0.645241	−	0.763979i	$-0.723243\pi$
$739$	−25925.0			−1.29048			−0.645241	+	0.763979i	$0.723243\pi$
$740$		0			0
$741$	38556.0			1.91146
$742$		0			0
$743$			25578.0i			1.26294i	0.775400	+	0.631471i	$0.217548\pi$
$743$			25578.0i			1.26294i	−0.775400	+	0.631471i	$0.782452\pi$
$744$		0			0
$745$	3020.00	+	1510.00i	0.148516	+	0.0742579i
$746$		0			0
$747$		−	9416.00i		−	0.461196i
$748$		0			0
$749$	1008.00			0.0491743
$750$		0			0
$751$	4291.00			0.208496			0.104248	−	0.994551i	$-0.466756\pi$
$751$	4291.00			0.208496			0.104248	+	0.994551i	$0.466756\pi$
$752$		0			0
$753$		−	23730.0i		−	1.14843i
$754$		0			0
$755$	31670.0	+	15835.0i	1.52661	+	0.763304i
$756$		0			0
$757$		−	31528.0i		−	1.51374i	−0.653563	−	0.756872i	$-0.726726\pi$
$757$		−	31528.0i		−	1.51374i	0.653563	−	0.756872i	$-0.273274\pi$
$758$		0			0
$759$	−18130.0			−0.867032
$760$		0			0
$761$	−23154.0			−1.10293			−0.551466	−	0.834197i	$-0.685932\pi$
$761$	−23154.0			−1.10293			−0.551466	+	0.834197i	$0.685932\pi$
$762$		0			0
$763$			11767.0i			0.558315i
$764$		0			0
$765$	4510.00	−	9020.00i	0.213150	−	0.426299i
$766$		0			0
$767$			102.000i			0.00480183i
$768$		0			0
$769$	−13992.0			−0.656131			−0.328065	−	0.944655i	$-0.606397\pi$
$769$	−13992.0			−0.656131			−0.328065	+	0.944655i	$0.606397\pi$
$770$		0			0
$771$	50190.0			2.34442
$772$		0			0
$773$			21681.0i			1.00881i	0.863467	+	0.504406i	$0.168288\pi$
$773$			21681.0i			1.00881i	−0.863467	+	0.504406i	$0.831712\pi$
$774$		0			0
$775$	10050.0	+	13400.0i	0.465815	+	0.621087i
$776$		0			0
$777$			16366.0i			0.755633i
$778$		0			0
$779$	−22248.0			−1.02326
$780$		0			0
$781$	−16872.0			−0.773019
$782$		0			0
$783$		−	8715.00i		−	0.397763i
$784$		0			0
$785$	2350.00	−	4700.00i	0.106847	−	0.213695i
$786$		0			0
$787$			16903.0i			0.765600i	0.923831	+	0.382800i	$0.125040\pi$
$787$			16903.0i			0.765600i	−0.923831	+	0.382800i	$0.874960\pi$
$788$		0			0
$789$	−53704.0			−2.42321
$790$		0			0
$791$	−5586.00			−0.251094
$792$		0			0
$793$			47940.0i			2.14678i
$794$		0			0
$795$	420.000	+	210.000i	0.0187369	+	0.00936847i
$796$		0			0
$797$			18905.0i			0.840213i	0.907475	+	0.420106i	$0.138007\pi$
$797$			18905.0i			0.840213i	−0.907475	+	0.420106i	$0.861993\pi$
$798$		0			0
$799$	−11767.0			−0.521009
$800$		0			0
$801$	−4840.00			−0.213499
$802$		0			0
$803$		−	24050.0i		−	1.05692i
$804$		0			0
$805$	−4900.00	−	2450.00i	−0.214537	−	0.107269i
$806$		0			0
$807$		−	378.000i		−	0.0164885i
$808$		0			0
$809$	5571.00			0.242109			0.121054	−	0.992646i	$-0.461372\pi$
$809$	5571.00			0.242109			0.121054	+	0.992646i	$0.461372\pi$
$810$		0			0
$811$	−10894.0			−0.471689			−0.235845	−	0.971791i	$-0.575786\pi$
$811$	−10894.0			−0.471689			−0.235845	+	0.971791i	$0.575786\pi$
$812$		0			0
$813$			20524.0i			0.885373i
$814$		0			0
$815$	11950.0	−	23900.0i	0.513608	−	1.02722i
$816$		0			0
$817$			40608.0i			1.73892i
$818$		0			0
$819$	7854.00			0.335093
$820$		0			0
$821$	−30731.0			−1.30636			−0.653179	−	0.757204i	$-0.726565\pi$
$821$	−30731.0			−1.30636			−0.653179	+	0.757204i	$0.726565\pi$
$822$		0			0
$823$			1038.00i			0.0439640i	0.999758	+	0.0219820i	$0.00699766\pi$
$823$			1038.00i			0.0439640i	−0.999758	+	0.0219820i	$0.993002\pi$
$824$		0			0
$825$	25900.0	−	19425.0i	1.09300	−	0.819748i
$826$		0			0
$827$		−	7958.00i		−	0.334615i	−0.985905	−	0.167308i	$-0.946493\pi$
$827$		−	7958.00i		−	0.334615i	0.985905	−	0.167308i	$-0.0535073\pi$
$828$		0			0
$829$	30666.0			1.28477			0.642385	−	0.766382i	$-0.277945\pi$
$829$	30666.0			1.28477			0.642385	+	0.766382i	$0.277945\pi$
$830$		0			0
$831$	22778.0			0.950854
$832$		0			0
$833$		−	2009.00i		−	0.0835627i
$834$		0			0
$835$	−13155.0	+	26310.0i	−0.545207	+	1.09041i
$836$		0			0
$837$		−	4690.00i		−	0.193680i
$838$		0			0
$839$	−5354.00			−0.220311			−0.110155	−	0.993914i	$-0.535135\pi$
$839$	−5354.00			−0.220311			−0.110155	+	0.993914i	$0.535135\pi$
$840$		0			0
$841$	37612.0			1.54217
$842$		0			0
$843$		−	23289.0i		−	0.951502i
$844$		0			0
$845$	−4040.00	−	2020.00i	−0.164474	−	0.0822368i
$846$		0			0
$847$		−	266.000i		−	0.0107909i
$848$		0			0
$849$	32389.0			1.30929
$850$		0			0
$851$	−23380.0			−0.941782
$852$		0			0
$853$			42890.0i			1.72160i	0.508943	+	0.860800i	$0.330037\pi$
$853$			42890.0i			1.72160i	−0.508943	+	0.860800i	$0.669963\pi$
$854$		0			0
$855$	23760.0	+	11880.0i	0.950380	+	0.475190i
$856$		0			0
$857$			22950.0i			0.914769i	0.889269	+	0.457385i	$0.151214\pi$
$857$			22950.0i			0.914769i	−0.889269	+	0.457385i	$0.848786\pi$
$858$		0			0
$859$	−2824.00			−0.112170			−0.0560848	−	0.998426i	$-0.517862\pi$
$859$	−2824.00			−0.112170			−0.0560848	+	0.998426i	$0.517862\pi$
$860$		0			0
$861$	−10094.0			−0.399538
$862$		0			0
$863$		−	4866.00i		−	0.191936i	−0.995384	−	0.0959679i	$-0.969405\pi$
$863$		−	4866.00i		−	0.191936i	0.995384	−	0.0959679i	$-0.0305946\pi$
$864$		0			0
$865$	−11215.0	+	22430.0i	−0.440834	+	0.881668i
$866$		0			0
$867$		−	22624.0i		−	0.886218i
$868$		0			0
$869$	−45843.0			−1.78955
$870$		0			0
$871$	−5406.00			−0.210305
$872$		0			0
$873$			23210.0i			0.899816i
$874$		0			0
$875$	9625.00	−	1750.00i	0.371868	−	0.0676123i
$876$		0			0
$877$		−	10676.0i		−	0.411064i	−0.978650	−	0.205532i	$-0.934108\pi$
$877$		−	10676.0i		−	0.411064i	0.978650	−	0.205532i	$-0.0658924\pi$
$878$		0			0
$879$	28581.0			1.09672
$880$		0			0
$881$	−29856.0			−1.14174			−0.570871	−	0.821040i	$-0.693394\pi$
$881$	−29856.0			−1.14174			−0.570871	+	0.821040i	$0.693394\pi$
$882$		0			0
$883$		−	1944.00i		−	0.0740893i	−0.999314	−	0.0370446i	$-0.988206\pi$
$883$		−	1944.00i		−	0.0740893i	0.999314	−	0.0370446i	$-0.0117944\pi$
$884$		0			0
$885$	−70.0000	+	140.000i	−0.00265878	+	0.00531757i
$886$		0			0
$887$			14628.0i			0.553732i	0.960909	+	0.276866i	$0.0892958\pi$
$887$			14628.0i			0.553732i	−0.960909	+	0.276866i	$0.910704\pi$
$888$		0			0
$889$	−3038.00			−0.114613
$890$		0			0
$891$	−31043.0			−1.16720
$892$		0			0
$893$		−	30996.0i		−	1.16152i
$894$		0			0
$895$	520.000	+	260.000i	0.0194209	+	0.00971044i
$896$		0			0
$897$			24990.0i			0.930203i
$898$		0			0
$899$	33366.0			1.23784
$900$		0			0
$901$	−246.000			−0.00909595
$902$		0			0
$903$			18424.0i			0.678973i
$904$		0			0
$905$	24620.0	+	12310.0i	0.904306	+	0.452153i
$906$		0			0
$907$			12858.0i			0.470720i	0.971908	+	0.235360i	$0.0756269\pi$
$907$			12858.0i			0.470720i	−0.971908	+	0.235360i	$0.924373\pi$
$908$		0			0
$909$	−43120.0			−1.57338
$910$		0			0
$911$	18324.0			0.666412			0.333206	−	0.942854i	$-0.391870\pi$
$911$	18324.0			0.666412			0.333206	+	0.942854i	$0.391870\pi$
$912$		0			0
$913$			15836.0i			0.574036i
$914$		0			0
$915$	−32900.0	+	65800.0i	−1.18868	+	2.37736i
$916$		0			0
$917$		−	9030.00i		−	0.325187i
$918$		0			0
$919$	−14751.0			−0.529478			−0.264739	−	0.964320i	$-0.585286\pi$
$919$	−14751.0			−0.529478			−0.264739	+	0.964320i	$0.585286\pi$
$920$		0			0
$921$	28623.0			1.02406
$922$		0			0
$923$			23256.0i			0.829340i
$924$		0			0
$925$	33400.0	−	25050.0i	1.18723	−	0.890420i
$926$		0			0
$927$		−	40150.0i		−	1.42254i
$928$		0			0
$929$	47922.0			1.69243			0.846216	−	0.532840i	$-0.178875\pi$
$929$	47922.0			1.69243			0.846216	+	0.532840i	$0.178875\pi$
$930$		0			0
$931$	5292.00			0.186292
$932$		0			0
$933$		−	28056.0i		−	0.984472i
$934$		0			0
$935$	−7585.00	+	15170.0i	−0.265301	+	0.530601i
$936$		0			0
$937$		−	44987.0i		−	1.56848i	−0.620461	−	0.784238i	$-0.713054\pi$
$937$		−	44987.0i		−	1.56848i	0.620461	−	0.784238i	$-0.286946\pi$
$938$		0			0
$939$	−51485.0			−1.78930
$940$		0			0
$941$	−20356.0			−0.705193			−0.352597	−	0.935775i	$-0.614701\pi$
$941$	−20356.0			−0.705193			−0.352597	+	0.935775i	$0.614701\pi$
$942$		0			0
$943$		−	14420.0i		−	0.497964i
$944$		0			0
$945$	−2450.00	−	1225.00i	−0.0843370	−	0.0421685i
$946$		0			0
$947$			27786.0i			0.953457i	0.879051	+	0.476728i	$0.158177\pi$
$947$			27786.0i			0.953457i	−0.879051	+	0.476728i	$0.841823\pi$
$948$		0			0
$949$	−33150.0			−1.13392
$950$		0			0
$951$	−11788.0			−0.401948
$952$		0			0
$953$			48674.0i			1.65447i	0.561859	+	0.827233i	$0.310086\pi$
$953$			48674.0i			1.65447i	−0.561859	+	0.827233i	$0.689914\pi$
$954$		0			0
$955$	−31590.0	−	15795.0i	−1.07040	−	0.535198i
$956$		0			0
$957$		−	64491.0i		−	2.17837i
$958$		0			0
$959$	1344.00			0.0452555
$960$		0			0
$961$	−11835.0			−0.397268
$962$		0			0
$963$		−	3168.00i		−	0.106010i
$964$		0			0
$965$	10300.0	−	20600.0i	0.343595	−	0.687189i
$966$		0			0
$967$			11168.0i			0.371395i	0.982607	+	0.185697i	$0.0594544\pi$
$967$			11168.0i			0.371395i	−0.982607	+	0.185697i	$0.940546\pi$
$968$		0			0
$969$	−30996.0			−1.02759
$970$		0			0
$971$	−20094.0			−0.664106			−0.332053	−	0.943261i	$-0.607741\pi$
$971$	−20094.0			−0.664106			−0.332053	+	0.943261i	$0.607741\pi$
$972$		0			0
$973$		−	9814.00i		−	0.323353i
$974$		0			0
$975$	−26775.0	−	35700.0i	−0.879473	−	1.17263i
$976$		0			0
$977$			49104.0i			1.60796i	0.594657	+	0.803980i	$0.297288\pi$
$977$			49104.0i			1.60796i	−0.594657	+	0.803980i	$0.702712\pi$
$978$		0			0
$979$	8140.00			0.265736
$980$		0			0
$981$	36982.0			1.20361
$982$		0			0
$983$			27751.0i			0.900427i	0.892921	+	0.450213i	$0.148652\pi$
$983$			27751.0i			0.900427i	−0.892921	+	0.450213i	$0.851348\pi$
$984$		0			0
$985$	−8690.00	+	17380.0i	−0.281103	+	0.562206i
$986$		0			0
$987$		−	14063.0i		−	0.453526i
$988$		0			0
$989$	−26320.0			−0.846236
$990$		0			0
$991$	−37600.0			−1.20525			−0.602625	−	0.798024i	$-0.705879\pi$
$991$	−37600.0			−1.20525			−0.602625	+	0.798024i	$0.705879\pi$
$992$		0			0
$993$		−	10220.0i		−	0.326608i
$994$		0			0
$995$	−8940.00	−	4470.00i	−0.284841	−	0.142421i
$996$		0			0
$997$		−	10911.0i		−	0.346595i	−0.984870	−	0.173297i	$-0.944558\pi$
$997$		−	10911.0i		−	0.346595i	0.984870	−	0.173297i	$-0.0554422\pi$
$998$		0			0
$999$	−11690.0			−0.370225

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.4.g.c.449.1		2
4.3	odd	2		70.4.c.a.29.2	yes	2
5.4	even	2	inner	560.4.g.c.449.2		2
12.11	even	2		630.4.g.a.379.1		2
20.3	even	4		350.4.a.m.1.1		1
20.7	even	4		350.4.a.i.1.1		1
20.19	odd	2		70.4.c.a.29.1	✓	2
28.27	even	2		490.4.c.a.99.2		2
60.59	even	2		630.4.g.a.379.2		2
140.27	odd	4		2450.4.a.c.1.1		1
140.83	odd	4		2450.4.a.bn.1.1		1
140.139	even	2		490.4.c.a.99.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.4.c.a.29.1	✓	2	20.19	odd	2
70.4.c.a.29.2	yes	2	4.3	odd	2
350.4.a.i.1.1		1	20.7	even	4
350.4.a.m.1.1		1	20.3	even	4
490.4.c.a.99.1		2	140.139	even	2
490.4.c.a.99.2		2	28.27	even	2
560.4.g.c.449.1		2	1.1	even	1	trivial
560.4.g.c.449.2		2	5.4	even	2	inner
630.4.g.a.379.1		2	12.11	even	2
630.4.g.a.379.2		2	60.59	even	2
2450.4.a.c.1.1		1	140.27	odd	4
2450.4.a.bn.1.1		1	140.83	odd	4

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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	560.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-7.00000i q^{3} +(10.0000 + 5.00000i) q^{5} -7.00000i q^{7} -22.0000 q^{9} +O(q^{10})\)	\(q-7.00000i q^{3} +(10.0000 + 5.00000i) q^{5} -7.00000i q^{7} -22.0000 q^{9} +37.0000 q^{11} -51.0000i q^{13} +(35.0000 - 70.0000i) q^{15} +41.0000i q^{17} -108.000 q^{19} -49.0000 q^{21} -70.0000i q^{23} +(75.0000 + 100.000i) q^{25} -35.0000i q^{27} +249.000 q^{29} +134.000 q^{31} -259.000i q^{33} +(35.0000 - 70.0000i) q^{35} -334.000i q^{37} -357.000 q^{39} +206.000 q^{41} -376.000i q^{43} +(-220.000 - 110.000i) q^{45} +287.000i q^{47} -49.0000 q^{49} +287.000 q^{51} +6.00000i q^{53} +(370.000 + 185.000i) q^{55} +756.000i q^{57} -2.00000 q^{59} -940.000 q^{61} +154.000i q^{63} +(255.000 - 510.000i) q^{65} -106.000i q^{67} -490.000 q^{69} -456.000 q^{71} -650.000i q^{73} +(700.000 - 525.000i) q^{75} -259.000i q^{77} -1239.00 q^{79} -839.000 q^{81} +428.000i q^{83} +(-205.000 + 410.000i) q^{85} -1743.00i q^{87} +220.000 q^{89} -357.000 q^{91} -938.000i q^{93} +(-1080.00 - 540.000i) q^{95} -1055.00i q^{97} -814.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 20 q^{5} - 44 q^{9}+O(q^{10}) \) 2 * q + 20 * q^5 - 44 * q^9	\( 2 q + 20 q^{5} - 44 q^{9} + 74 q^{11} + 70 q^{15} - 216 q^{19} - 98 q^{21} + 150 q^{25} + 498 q^{29} + 268 q^{31} + 70 q^{35} - 714 q^{39} + 412 q^{41} - 440 q^{45} - 98 q^{49} + 574 q^{51} + 740 q^{55} - 4 q^{59} - 1880 q^{61} + 510 q^{65} - 980 q^{69} - 912 q^{71} + 1400 q^{75} - 2478 q^{79} - 1678 q^{81} - 410 q^{85} + 440 q^{89} - 714 q^{91} - 2160 q^{95} - 1628 q^{99}+O(q^{100}) \) 2 * q + 20 * q^5 - 44 * q^9 + 74 * q^11 + 70 * q^15 - 216 * q^19 - 98 * q^21 + 150 * q^25 + 498 * q^29 + 268 * q^31 + 70 * q^35 - 714 * q^39 + 412 * q^41 - 440 * q^45 - 98 * q^49 + 574 * q^51 + 740 * q^55 - 4 * q^59 - 1880 * q^61 + 510 * q^65 - 980 * q^69 - 912 * q^71 + 1400 * q^75 - 2478 * q^79 - 1678 * q^81 - 410 * q^85 + 440 * q^89 - 714 * q^91 - 2160 * q^95 - 1628 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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