LMFDB - Embedded newform 1120.3.c.g.209.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1120,3,Mod(209,1120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1120, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1120.209");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1120 = 2^{5} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1120.c (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:	$30.5177896084$
Analytic rank:	$0$
Dimension:	$80$
Twist minimal:	no (minimal twist has level 280)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			209.12
Character	$\chi$	$=$	1120.209
Dual form			1120.3.c.g.209.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.74983i q^{3} +(-4.72637 - 1.63138i) q^{5} +(1.41172 + 6.85617i) q^{7} +1.43843 q^{9} +O(q^{10})$	$q+2.74983i q^{3} +(-4.72637 - 1.63138i) q^{5} +(1.41172 + 6.85617i) q^{7} +1.43843 q^{9} -14.1984i q^{11} +5.89684i q^{13} +(4.48601 - 12.9967i) q^{15} +10.0079 q^{17} +18.9481 q^{19} +(-18.8533 + 3.88200i) q^{21} +11.5809i q^{23} +(19.6772 + 15.4210i) q^{25} +28.7039i q^{27} -31.7681i q^{29} +48.2661i q^{31} +39.0431 q^{33} +(4.51266 - 34.7079i) q^{35} +39.7143 q^{37} -16.2153 q^{39} +15.8176i q^{41} +30.0245 q^{43} +(-6.79856 - 2.34662i) q^{45} -83.3321 q^{47} +(-45.0141 + 19.3580i) q^{49} +27.5199i q^{51} -17.9248 q^{53} +(-23.1629 + 67.1069i) q^{55} +52.1041i q^{57} -109.697 q^{59} -35.4866 q^{61} +(2.03067 + 9.86212i) q^{63} +(9.61996 - 27.8707i) q^{65} +44.5578 q^{67} -31.8455 q^{69} +75.5330 q^{71} +81.9741 q^{73} +(-42.4051 + 54.1090i) q^{75} +(97.3465 - 20.0442i) q^{77} -89.6824 q^{79} -65.9850 q^{81} +107.645i q^{83} +(-47.3009 - 16.3266i) q^{85} +87.3568 q^{87} +145.006i q^{89} +(-40.4297 + 8.32471i) q^{91} -132.724 q^{93} +(-89.5558 - 30.9115i) q^{95} +13.4460 q^{97} -20.4234i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q - 224 q^{9}+O(q^{10}) $ 80 * q - 224 * q^9	$ 80 q - 224 q^{9} + 72 q^{15} - 104 q^{25} + 112 q^{39} + 192 q^{49} + 472 q^{65} - 800 q^{71} - 480 q^{79} - 896 q^{81} - 1176 q^{95}+O(q^{100}) $ 80 * q - 224 * q^9 + 72 * q^15 - 104 * q^25 + 112 * q^39 + 192 * q^49 + 472 * q^65 - 800 * q^71 - 480 * q^79 - 896 * q^81 - 1176 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1120\mathbb{Z}\right)^\times$.

$n$	$351$	$421$	$801$	$897$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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
$3$			2.74983i			0.916610i	0.888795	+	0.458305i	$0.151543\pi$
$3$			2.74983i			0.916610i	−0.888795	+	0.458305i	$0.848457\pi$
$4$		0			0
$5$	−4.72637	−	1.63138i	−0.945275	−	0.326275i
$5$	−4.72637	−	1.63138i	−0.945275	−	0.326275i
$6$		0			0
$7$	1.41172	+	6.85617i	0.201675	+	0.979453i
$7$	1.41172	+	6.85617i	0.201675	+	0.979453i
$8$		0			0
$9$	1.43843			0.159826
$10$		0			0
$11$		−	14.1984i		−	1.29076i	−0.763861	−	0.645381i	$-0.776699\pi$
$11$		−	14.1984i		−	1.29076i	0.763861	−	0.645381i	$-0.223301\pi$
$12$		0			0
$13$			5.89684i			0.453603i	0.973941	+	0.226801i	$0.0728268\pi$
$13$			5.89684i			0.453603i	−0.973941	+	0.226801i	$0.927173\pi$
$14$		0			0
$15$	4.48601	−	12.9967i	0.299067	−	0.866449i
$16$		0			0
$17$	10.0079			0.588698			0.294349	−	0.955698i	$-0.404897\pi$
$17$	10.0079			0.588698			0.294349	+	0.955698i	$0.404897\pi$
$18$		0			0
$19$	18.9481			0.997268			0.498634	−	0.866813i	$-0.333835\pi$
$19$	18.9481			0.997268			0.498634	+	0.866813i	$0.333835\pi$
$20$		0			0
$21$	−18.8533	+	3.88200i	−0.897776	+	0.184857i
$22$		0			0
$23$			11.5809i			0.503517i	0.967790	+	0.251759i	$0.0810089\pi$
$23$			11.5809i			0.503517i	−0.967790	+	0.251759i	$0.918991\pi$
$24$		0			0
$25$	19.6772	+	15.4210i	0.787089	+	0.616840i
$26$		0			0
$27$			28.7039i			1.06311i
$28$		0			0
$29$		−	31.7681i		−	1.09545i	−0.836658	−	0.547725i	$-0.815494\pi$
$29$		−	31.7681i		−	1.09545i	0.836658	−	0.547725i	$-0.184506\pi$
$30$		0			0
$31$			48.2661i			1.55697i	0.627662	+	0.778486i	$0.284012\pi$
$31$			48.2661i			1.55697i	−0.627662	+	0.778486i	$0.715988\pi$
$32$		0			0
$33$	39.0431			1.18313
$34$		0			0
$35$	4.51266	−	34.7079i	0.128933	−	0.991653i
$36$		0			0
$37$	39.7143			1.07336			0.536680	−	0.843786i	$-0.319678\pi$
$37$	39.7143			1.07336			0.536680	+	0.843786i	$0.319678\pi$
$38$		0			0
$39$	−16.2153			−0.415777
$40$		0			0
$41$			15.8176i			0.385794i	0.981219	+	0.192897i	$0.0617883\pi$
$41$			15.8176i			0.385794i	−0.981219	+	0.192897i	$0.938212\pi$
$42$		0			0
$43$	30.0245			0.698243			0.349122	−	0.937077i	$-0.386480\pi$
$43$	30.0245			0.698243			0.349122	+	0.937077i	$0.386480\pi$
$44$		0			0
$45$	−6.79856	−	2.34662i	−0.151079	−	0.0521472i
$46$		0			0
$47$	−83.3321			−1.77302			−0.886511	−	0.462707i	$-0.846878\pi$
$47$	−83.3321			−1.77302			−0.886511	+	0.462707i	$0.846878\pi$
$48$		0			0
$49$	−45.0141	+	19.3580i	−0.918654	+	0.395062i
$50$		0			0
$51$			27.5199i			0.539606i
$52$		0			0
$53$	−17.9248			−0.338203			−0.169102	−	0.985599i	$-0.554087\pi$
$53$	−17.9248			−0.338203			−0.169102	+	0.985599i	$0.554087\pi$
$54$		0			0
$55$	−23.1629	+	67.1069i	−0.421144	+	1.22012i
$56$		0			0
$57$			52.1041i			0.914106i
$58$		0			0
$59$	−109.697			−1.85927			−0.929634	−	0.368484i	$-0.879877\pi$
$59$	−109.697			−1.85927			−0.929634	+	0.368484i	$0.879877\pi$
$60$		0			0
$61$	−35.4866			−0.581748			−0.290874	−	0.956761i	$-0.593946\pi$
$61$	−35.4866			−0.581748			−0.290874	+	0.956761i	$0.593946\pi$
$62$		0			0
$63$	2.03067	+	9.86212i	0.0322328	+	0.156542i
$64$		0			0
$65$	9.61996	−	27.8707i	0.147999	−	0.428779i
$66$		0			0
$67$	44.5578			0.665042			0.332521	−	0.943096i	$-0.392101\pi$
$67$	44.5578			0.665042			0.332521	+	0.943096i	$0.392101\pi$
$68$		0			0
$69$	−31.8455			−0.461529
$70$		0			0
$71$	75.5330			1.06384			0.531922	−	0.846793i	$-0.321470\pi$
$71$	75.5330			1.06384			0.531922	+	0.846793i	$0.321470\pi$
$72$		0			0
$73$	81.9741			1.12293			0.561466	−	0.827500i	$-0.310237\pi$
$73$	81.9741			1.12293			0.561466	+	0.827500i	$0.310237\pi$
$74$		0			0
$75$	−42.4051	+	54.1090i	−0.565402	+	0.721454i
$76$		0			0
$77$	97.3465	−	20.0442i	1.26424	−	0.260314i
$78$		0			0
$79$	−89.6824			−1.13522			−0.567610	−	0.823297i	$-0.692132\pi$
$79$	−89.6824			−1.13522			−0.567610	+	0.823297i	$0.692132\pi$
$80$		0			0
$81$	−65.9850			−0.814630
$82$		0			0
$83$			107.645i			1.29693i	0.761246	+	0.648463i	$0.224588\pi$
$83$			107.645i			1.29693i	−0.761246	+	0.648463i	$0.775412\pi$
$84$		0			0
$85$	−47.3009	−	16.3266i	−0.556481	−	0.192078i
$86$		0			0
$87$	87.3568			1.00410
$88$		0			0
$89$			145.006i			1.62928i	0.579965	+	0.814641i	$0.303066\pi$
$89$			145.006i			1.62928i	−0.579965	+	0.814641i	$0.696934\pi$
$90$		0			0
$91$	−40.4297	+	8.32471i	−0.444282	+	0.0914803i
$92$		0			0
$93$	−132.724			−1.42714
$94$		0			0
$95$	−89.5558	−	30.9115i	−0.942693	−	0.325384i
$96$		0			0
$97$	13.4460			0.138619			0.0693093	−	0.997595i	$-0.477920\pi$
$97$	13.4460			0.138619			0.0693093	+	0.997595i	$0.477920\pi$
$98$		0			0
$99$		−	20.4234i		−	0.206297i
$100$		0			0
$101$	45.8794			0.454251			0.227126	−	0.973865i	$-0.427067\pi$
$101$	45.8794			0.454251			0.227126	+	0.973865i	$0.427067\pi$
$102$		0			0
$103$	103.275			1.00267			0.501334	−	0.865254i	$-0.332843\pi$
$103$	103.275			1.00267			0.501334	+	0.865254i	$0.332843\pi$
$104$		0			0
$105$	95.4408	+	12.4090i	0.908960	+	0.118181i
$106$		0			0
$107$	−183.182			−1.71198			−0.855992	−	0.516990i	$-0.827052\pi$
$107$	−183.182			−1.71198			−0.855992	+	0.516990i	$0.827052\pi$
$108$		0			0
$109$			183.478i			1.68328i	0.540035	+	0.841642i	$0.318411\pi$
$109$			183.478i			1.68328i	−0.540035	+	0.841642i	$0.681589\pi$
$110$		0			0
$111$			109.208i			0.983853i
$112$		0			0
$113$		−	50.0317i		−	0.442759i	−0.975188	−	0.221379i	$-0.928944\pi$
$113$		−	50.0317i		−	0.442759i	0.975188	−	0.221379i	$-0.0710559\pi$
$114$		0			0
$115$	18.8928	−	54.7356i	0.164285	−	0.475962i
$116$		0			0
$117$			8.48219i			0.0724974i
$118$		0			0
$119$	14.1283	+	68.6156i	0.118726	+	0.576602i
$120$		0			0
$121$	−80.5940			−0.666066
$122$		0			0
$123$	−43.4956			−0.353623
$124$		0			0
$125$	−67.8445	−	104.986i	−0.542756	−	0.839891i
$126$		0			0
$127$			82.7028i			0.651203i	0.945507	+	0.325602i	$0.105567\pi$
$127$			82.7028i			0.651203i	−0.945507	+	0.325602i	$0.894433\pi$
$128$		0			0
$129$			82.5622i			0.640017i
$130$		0			0
$131$	−87.5299			−0.668167			−0.334084	−	0.942543i	$-0.608427\pi$
$131$	−87.5299			−0.668167			−0.334084	+	0.942543i	$0.608427\pi$
$132$		0			0
$133$	26.7495	+	129.911i	0.201124	+	0.976777i
$134$		0			0
$135$	46.8269	−	135.665i	0.346866	−	1.00493i
$136$		0			0
$137$		−	61.5426i		−	0.449216i	−0.974449	−	0.224608i	$-0.927890\pi$
$137$		−	61.5426i		−	0.449216i	0.974449	−	0.224608i	$-0.0721101\pi$
$138$		0			0
$139$	−15.9192			−0.114527			−0.0572634	−	0.998359i	$-0.518237\pi$
$139$	−15.9192			−0.114527			−0.0572634	+	0.998359i	$0.518237\pi$
$140$		0			0
$141$		−	229.149i		−	1.62517i
$142$		0			0
$143$	83.7255			0.585493
$144$		0			0
$145$	−51.8257	+	150.148i	−0.357419	+	1.03550i
$146$		0			0
$147$	−53.2313	−	123.781i	−0.362118	−	0.842048i
$148$		0			0
$149$			56.3566i			0.378232i	0.981955	+	0.189116i	$0.0605623\pi$
$149$			56.3566i			0.378232i	−0.981955	+	0.189116i	$0.939438\pi$
$150$		0			0
$151$	50.9865			0.337659			0.168829	−	0.985645i	$-0.446001\pi$
$151$	50.9865			0.337659			0.168829	+	0.985645i	$0.446001\pi$
$152$		0			0
$153$	14.3956			0.0940890
$154$		0			0
$155$	78.7402	−	228.124i	0.508001	−	1.47177i
$156$		0			0
$157$			222.491i			1.41714i	0.705641	+	0.708570i	$0.250659\pi$
$157$			222.491i			1.41714i	−0.705641	+	0.708570i	$0.749341\pi$
$158$		0			0
$159$		−	49.2901i		−	0.310001i
$160$		0			0
$161$	−79.4005	+	16.3490i	−0.493171	+	0.101547i
$162$		0			0
$163$	−69.7727			−0.428053			−0.214027	−	0.976828i	$-0.568658\pi$
$163$	−69.7727			−0.428053			−0.214027	+	0.976828i	$0.568658\pi$
$164$		0			0
$165$	−184.532	−	63.6941i	−1.11838	−	0.386025i
$166$		0			0
$167$	36.9782			0.221426			0.110713	−	0.993852i	$-0.464687\pi$
$167$	36.9782			0.221426			0.110713	+	0.993852i	$0.464687\pi$
$168$		0			0
$169$	134.227			0.794244
$170$		0			0
$171$	27.2555			0.159389
$172$		0			0
$173$		−	111.750i		−	0.645954i	−0.946407	−	0.322977i	$-0.895316\pi$
$173$		−	111.750i		−	0.645954i	0.946407	−	0.322977i	$-0.104684\pi$
$174$		0			0
$175$	−77.9501	+	156.681i	−0.445429	+	0.895317i
$176$		0			0
$177$		−	301.648i		−	1.70422i
$178$		0			0
$179$			157.855i			0.881874i	0.897538	+	0.440937i	$0.145354\pi$
$179$			157.855i			0.881874i	−0.897538	+	0.440937i	$0.854646\pi$
$180$		0			0
$181$	−89.1362			−0.492465			−0.246232	−	0.969211i	$-0.579193\pi$
$181$	−89.1362			−0.492465			−0.246232	+	0.969211i	$0.579193\pi$
$182$		0			0
$183$		−	97.5822i		−	0.533236i
$184$		0			0
$185$	−187.705	−	64.7890i	−1.01462	−	0.350211i
$186$		0			0
$187$		−	142.095i		−	0.759869i
$188$		0			0
$189$	−196.799	+	40.5220i	−1.04126	+	0.214402i
$190$		0			0
$191$	102.321			0.535711			0.267855	−	0.963459i	$-0.413685\pi$
$191$	102.321			0.535711			0.267855	+	0.963459i	$0.413685\pi$
$192$		0			0
$193$			141.825i			0.734844i	0.930054	+	0.367422i	$0.119760\pi$
$193$			141.825i			0.734844i	−0.930054	+	0.367422i	$0.880240\pi$
$194$		0			0
$195$	76.6396	+	26.4533i	0.393024	+	0.135658i
$196$		0			0
$197$	51.0345			0.259058			0.129529	−	0.991576i	$-0.458653\pi$
$197$	51.0345			0.259058			0.129529	+	0.991576i	$0.458653\pi$
$198$		0			0
$199$		−	75.4770i		−	0.379281i	−0.981854	−	0.189641i	$-0.939268\pi$
$199$		−	75.4770i		−	0.379281i	0.981854	−	0.189641i	$-0.0607323\pi$
$200$		0			0
$201$			122.526i			0.609584i
$202$		0			0
$203$	217.807	−	44.8478i	1.07294	−	0.220925i
$204$		0			0
$205$	25.8044	−	74.7597i	0.125875	−	0.364681i
$206$		0			0
$207$			16.6583i			0.0804749i
$208$		0			0
$209$		−	269.032i		−	1.28724i
$210$		0			0
$211$		−	187.728i		−	0.889706i	−0.895604	−	0.444853i	$-0.853256\pi$
$211$		−	187.728i		−	0.889706i	0.895604	−	0.444853i	$-0.146744\pi$
$212$		0			0
$213$			207.703i			0.975131i
$214$		0			0
$215$	−141.907	−	48.9812i	−0.660032	−	0.227820i
$216$		0			0
$217$	−330.921	+	68.1384i	−1.52498	+	0.314002i
$218$		0			0
$219$			225.415i			1.02929i
$220$		0			0
$221$			59.0147i			0.267035i
$222$		0			0
$223$	315.595			1.41523			0.707613	−	0.706600i	$-0.249772\pi$
$223$	315.595			1.41523			0.707613	+	0.706600i	$0.249772\pi$
$224$		0			0
$225$	28.3043	+	22.1820i	0.125797	+	0.0985868i
$226$		0			0
$227$			7.37331i			0.0324815i	0.999868	+	0.0162408i	$0.00516983\pi$
$227$			7.37331i			0.0324815i	−0.999868	+	0.0162408i	$0.994830\pi$
$228$		0			0
$229$	−60.8674			−0.265797			−0.132898	−	0.991130i	$-0.542428\pi$
$229$	−60.8674			−0.265797			−0.132898	+	0.991130i	$0.542428\pi$
$230$		0			0
$231$	55.1182	+	267.686i	0.238607	+	1.15882i
$232$		0			0
$233$			336.839i			1.44566i	0.691024	+	0.722831i	$0.257160\pi$
$233$			336.839i			1.44566i	−0.691024	+	0.722831i	$0.742840\pi$
$234$		0			0
$235$	393.859	+	135.946i	1.67599	+	0.578494i
$236$		0			0
$237$		−	246.611i		−	1.04055i
$238$		0			0
$239$	−135.916			−0.568688			−0.284344	−	0.958722i	$-0.591776\pi$
$239$	−135.916			−0.568688			−0.284344	+	0.958722i	$0.591776\pi$
$240$		0			0
$241$			67.8933i			0.281715i	0.990030	+	0.140857i	$0.0449859\pi$
$241$			67.8933i			0.281715i	−0.990030	+	0.140857i	$0.955014\pi$
$242$		0			0
$243$			76.8876i			0.316410i
$244$		0			0
$245$	244.334	−	18.0584i	0.997280	−	0.0737078i
$246$		0			0
$247$			111.734i			0.452364i
$248$		0			0
$249$	−296.005			−1.18878
$250$		0			0
$251$	224.184			0.893164			0.446582	−	0.894743i	$-0.352641\pi$
$251$	224.184			0.893164			0.446582	+	0.894743i	$0.352641\pi$
$252$		0			0
$253$	164.430			0.649921
$254$		0			0
$255$	44.8954	−	130.069i	0.176060	−	0.510076i
$256$		0			0
$257$	−55.9674			−0.217772			−0.108886	−	0.994054i	$-0.534728\pi$
$257$	−55.9674			−0.217772			−0.108886	+	0.994054i	$0.534728\pi$
$258$		0			0
$259$	56.0657	+	272.288i	0.216470	+	1.05131i
$260$		0			0
$261$		−	45.6962i		−	0.175081i
$262$		0			0
$263$			74.4566i			0.283105i	0.989931	+	0.141552i	$0.0452094\pi$
$263$			74.4566i			0.283105i	−0.989931	+	0.141552i	$0.954791\pi$
$264$		0			0
$265$	84.7192	+	29.2421i	0.319695	+	0.110347i
$266$		0			0
$267$	−398.742			−1.49342
$268$		0			0
$269$	377.527			1.40345			0.701724	−	0.712449i	$-0.252414\pi$
$269$	377.527			1.40345			0.701724	+	0.712449i	$0.252414\pi$
$270$		0			0
$271$		−	463.315i		−	1.70965i	−0.518915	−	0.854826i	$-0.673664\pi$
$271$		−	463.315i		−	1.70965i	0.518915	−	0.854826i	$-0.326336\pi$
$272$		0			0
$273$	−22.8915	−	111.175i	−0.0838518	−	0.407234i
$274$		0			0
$275$	218.953	−	279.385i	0.796193	−	1.01594i
$276$		0			0
$277$	319.935			1.15500			0.577500	−	0.816391i	$-0.304028\pi$
$277$	319.935			1.15500			0.577500	+	0.816391i	$0.304028\pi$
$278$		0			0
$279$			69.4275i			0.248844i
$280$		0			0
$281$	398.504			1.41816			0.709081	−	0.705127i	$-0.249110\pi$
$281$	398.504			1.41816			0.709081	+	0.705127i	$0.249110\pi$
$282$		0			0
$283$			211.268i			0.746529i	0.927725	+	0.373264i	$0.121762\pi$
$283$			211.268i			0.746529i	−0.927725	+	0.373264i	$0.878238\pi$
$284$		0			0
$285$	85.0014	−	246.263i	0.298250	−	0.864082i
$286$		0			0
$287$	−108.448	+	22.3300i	−0.377867	+	0.0778050i
$288$		0			0
$289$	−188.843			−0.653435
$290$		0			0
$291$			36.9742i			0.127059i
$292$		0			0
$293$			68.4081i			0.233475i	0.993163	+	0.116737i	$0.0372436\pi$
$293$			68.4081i			0.233475i	−0.993163	+	0.116737i	$0.962756\pi$
$294$		0			0
$295$	518.468	+	178.957i	1.75752	+	0.606633i
$296$		0			0
$297$	407.549			1.37222
$298$		0			0
$299$	−68.2906			−0.228397
$300$		0			0
$301$	42.3863	+	205.853i	0.140818	+	0.683896i
$302$		0			0
$303$			126.160i			0.416371i
$304$		0			0
$305$	167.723	+	57.8920i	0.549912	+	0.189810i
$306$		0			0
$307$			127.999i			0.416936i	0.978029	+	0.208468i	$0.0668477\pi$
$307$			127.999i			0.416936i	−0.978029	+	0.208468i	$0.933152\pi$
$308$		0			0
$309$			283.988i			0.919056i
$310$		0			0
$311$		−	460.410i		−	1.48042i	−0.672376	−	0.740210i	$-0.734726\pi$
$311$		−	460.410i		−	1.48042i	0.672376	−	0.740210i	$-0.265274\pi$
$312$		0			0
$313$	−158.669			−0.506930			−0.253465	−	0.967345i	$-0.581570\pi$
$313$	−158.669			−0.506930			−0.253465	+	0.967345i	$0.581570\pi$
$314$		0			0
$315$	6.49114	−	49.9249i	0.0206068	−	0.158492i
$316$		0			0
$317$	298.341			0.941138			0.470569	−	0.882363i	$-0.344049\pi$
$317$	298.341			0.941138			0.470569	+	0.882363i	$0.344049\pi$
$318$		0			0
$319$	−451.055			−1.41397
$320$		0			0
$321$		−	503.720i		−	1.56922i
$322$		0			0
$323$	189.630			0.587090
$324$		0			0
$325$	−90.9351	+	116.033i	−0.279800	+	0.357026i
$326$		0			0
$327$	−504.533			−1.54292
$328$		0			0
$329$	−117.642	−	571.339i	−0.357574	−	1.73659i
$330$		0			0
$331$		−	255.386i		−	0.771558i	−0.922591	−	0.385779i	$-0.873933\pi$
$331$		−	255.386i		−	0.771558i	0.922591	−	0.385779i	$-0.126067\pi$
$332$		0			0
$333$	57.1263			0.171551
$334$		0			0
$335$	−210.597	−	72.6906i	−0.628647	−	0.216987i
$336$		0			0
$337$		−	98.5741i		−	0.292505i	−0.989247	−	0.146252i	$-0.953279\pi$
$337$		−	98.5741i		−	0.292505i	0.989247	−	0.146252i	$-0.0467212\pi$
$338$		0			0
$339$	137.579			0.405837
$340$		0			0
$341$	685.301			2.00968
$342$		0			0
$343$	−196.269	−	281.296i	−0.572214	−	0.820104i
$344$		0			0
$345$	150.514	+	51.9520i	0.436272	+	0.150586i
$346$		0			0
$347$	−367.417			−1.05884			−0.529420	−	0.848360i	$-0.677590\pi$
$347$	−367.417			−1.05884			−0.529420	+	0.848360i	$0.677590\pi$
$348$		0			0
$349$	−170.008			−0.487128			−0.243564	−	0.969885i	$-0.578317\pi$
$349$	−170.008			−0.487128			−0.243564	+	0.969885i	$0.578317\pi$
$350$		0			0
$351$	−169.262			−0.482229
$352$		0			0
$353$	573.815			1.62554			0.812769	−	0.582587i	$-0.197959\pi$
$353$	573.815			1.62554			0.812769	+	0.582587i	$0.197959\pi$
$354$		0			0
$355$	−356.997	−	123.223i	−1.00563	−	0.347106i
$356$		0			0
$357$	−188.681	+	38.8506i	−0.528519	+	0.108825i
$358$		0			0
$359$	130.637			0.363890			0.181945	−	0.983309i	$-0.441761\pi$
$359$	130.637			0.363890			0.181945	+	0.983309i	$0.441761\pi$
$360$		0			0
$361$	−1.96954			−0.00545579
$362$		0			0
$363$		−	221.620i		−	0.610523i
$364$		0			0
$365$	−387.440	−	133.731i	−1.06148	−	0.366385i
$366$		0			0
$367$	338.502			0.922349			0.461175	−	0.887309i	$-0.347428\pi$
$367$	338.502			0.922349			0.461175	+	0.887309i	$0.347428\pi$
$368$		0			0
$369$			22.7525i			0.0616598i
$370$		0			0
$371$	−25.3049	−	122.895i	−0.0682071	−	0.331254i
$372$		0			0
$373$	−433.913			−1.16330			−0.581652	−	0.813438i	$-0.697594\pi$
$373$	−433.913			−1.16330			−0.581652	+	0.813438i	$0.697594\pi$
$374$		0			0
$375$	288.695	−	186.561i	0.769852	−	0.497495i
$376$		0			0
$377$	187.331			0.496900
$378$		0			0
$379$			2.34294i			0.00618191i	0.999995	+	0.00309096i	$0.000983883\pi$
$379$			2.34294i			0.00618191i	−0.999995	+	0.00309096i	$0.999016\pi$
$380$		0			0
$381$	−227.419			−0.596900
$382$		0			0
$383$	−752.048			−1.96357			−0.981786	−	0.189992i	$-0.939154\pi$
$383$	−752.048			−1.96357			−0.981786	+	0.189992i	$0.939154\pi$
$384$		0			0
$385$	−492.795	−	64.0724i	−1.27999	−	0.166422i
$386$		0			0
$387$	43.1881			0.111597
$388$		0			0
$389$			365.187i			0.938784i	0.882990	+	0.469392i	$0.155527\pi$
$389$			365.187i			0.938784i	−0.882990	+	0.469392i	$0.844473\pi$
$390$		0			0
$391$			115.900i			0.296419i
$392$		0			0
$393$		−	240.692i		−	0.612449i
$394$		0			0
$395$	423.873	+	146.306i	1.07310	+	0.370394i
$396$		0			0
$397$		−	675.156i		−	1.70064i	−0.526262	−	0.850322i	$-0.676407\pi$
$397$		−	675.156i		−	1.70064i	0.526262	−	0.850322i	$-0.323593\pi$
$398$		0			0
$399$	−357.234	+	73.5566i	−0.895324	+	0.184352i
$400$		0			0
$401$	719.884			1.79522			0.897611	−	0.440789i	$-0.145301\pi$
$401$	719.884			1.79522			0.897611	+	0.440789i	$0.145301\pi$
$402$		0			0
$403$	−284.617			−0.706247
$404$		0			0
$405$	311.870	+	107.646i	0.770049	+	0.265794i
$406$		0			0
$407$		−	563.879i		−	1.38545i
$408$		0			0
$409$		−	352.382i		−	0.861570i	−0.902455	−	0.430785i	$-0.858237\pi$
$409$		−	352.382i		−	0.861570i	0.902455	−	0.430785i	$-0.141763\pi$
$410$		0			0
$411$	169.232			0.411756
$412$		0			0
$413$	−154.862	−	752.100i	−0.374968	−	1.82107i
$414$		0			0
$415$	175.609	−	508.770i	0.423155	−	1.22595i
$416$		0			0
$417$		−	43.7752i		−	0.104976i
$418$		0			0
$419$	207.738			0.495795			0.247898	−	0.968786i	$-0.420260\pi$
$419$	207.738			0.495795			0.247898	+	0.968786i	$0.420260\pi$
$420$		0			0
$421$		−	480.271i		−	1.14079i	−0.821372	−	0.570393i	$-0.806791\pi$
$421$		−	480.271i		−	1.14079i	0.821372	−	0.570393i	$-0.193209\pi$
$422$		0			0
$423$	−119.867			−0.283375
$424$		0			0
$425$	196.927	+	154.331i	0.463357	+	0.363132i
$426$		0			0
$427$	−50.0973	−	243.302i	−0.117324	−	0.569794i
$428$		0			0
$429$			230.231i			0.536669i
$430$		0			0
$431$	362.214			0.840405			0.420202	−	0.907430i	$-0.361959\pi$
$431$	362.214			0.840405			0.420202	+	0.907430i	$0.361959\pi$
$432$		0			0
$433$	−603.870			−1.39462			−0.697309	−	0.716771i	$-0.745619\pi$
$433$	−603.870			−1.39462			−0.697309	+	0.716771i	$0.745619\pi$
$434$		0			0
$435$	−412.881	−	142.512i	−0.949152	−	0.327614i
$436$		0			0
$437$			219.436i			0.502142i
$438$		0			0
$439$		−	173.388i		−	0.394961i	−0.980307	−	0.197480i	$-0.936724\pi$
$439$		−	173.388i		−	0.394961i	0.980307	−	0.197480i	$-0.0632759\pi$
$440$		0			0
$441$	−64.7496	+	27.8452i	−0.146825	+	0.0631410i
$442$		0			0
$443$	350.334			0.790821			0.395410	−	0.918505i	$-0.370602\pi$
$443$	350.334			0.790821			0.395410	+	0.918505i	$0.370602\pi$
$444$		0			0
$445$	236.560	−	685.353i	0.531595	−	1.54012i
$446$		0			0
$447$	−154.971			−0.346692
$448$		0			0
$449$	−768.172			−1.71085			−0.855426	−	0.517926i	$-0.826704\pi$
$449$	−768.172			−1.71085			−0.855426	+	0.517926i	$0.826704\pi$
$450$		0			0
$451$	224.584			0.497968
$452$		0			0
$453$			140.204i			0.309502i
$454$		0			0
$455$	204.667	+	26.6104i	0.449817	+	0.0584844i
$456$		0			0
$457$		−	375.067i		−	0.820716i	−0.911925	−	0.410358i	$-0.865404\pi$
$457$		−	375.067i		−	0.820716i	0.911925	−	0.410358i	$-0.134596\pi$
$458$		0			0
$459$			287.265i			0.625849i
$460$		0			0
$461$	39.7150			0.0861497			0.0430748	−	0.999072i	$-0.486285\pi$
$461$	39.7150			0.0861497			0.0430748	+	0.999072i	$0.486285\pi$
$462$		0			0
$463$			22.9529i			0.0495743i	0.999693	+	0.0247872i	$0.00789081\pi$
$463$			22.9529i			0.0495743i	−0.999693	+	0.0247872i	$0.992109\pi$
$464$		0			0
$465$	627.302	+	216.522i	1.34904	+	0.465639i
$466$		0			0
$467$		−	41.0354i		−	0.0878702i	−0.999034	−	0.0439351i	$-0.986011\pi$
$467$		−	41.0354i		−	0.0878702i	0.999034	−	0.0439351i	$-0.0139895\pi$
$468$		0			0
$469$	62.9033	+	305.496i	0.134122	+	0.651377i
$470$		0			0
$471$	−611.812			−1.29896
$472$		0			0
$473$		−	426.299i		−	0.901266i
$474$		0			0
$475$	372.846	+	292.198i	0.784939	+	0.615155i
$476$		0			0
$477$	−25.7836			−0.0540536
$478$		0			0
$479$		−	740.675i		−	1.54629i	−0.634227	−	0.773147i	$-0.718681\pi$
$479$		−	740.675i		−	1.54629i	0.634227	−	0.773147i	$-0.281319\pi$
$480$		0			0
$481$			234.189i			0.486879i
$482$		0			0
$483$	−44.9571	−	218.338i	−0.0930788	−	0.452046i
$484$		0			0
$485$	−63.5509	−	21.9355i	−0.131033	−	0.0452278i
$486$		0			0
$487$			701.882i			1.44124i	0.693332	+	0.720618i	$0.256142\pi$
$487$			701.882i			1.44124i	−0.693332	+	0.720618i	$0.743858\pi$
$488$		0			0
$489$		−	191.863i		−	0.392358i
$490$		0			0
$491$			725.694i			1.47799i	0.673709	+	0.738996i	$0.264700\pi$
$491$			725.694i			1.47799i	−0.673709	+	0.738996i	$0.735300\pi$
$492$		0			0
$493$		−	317.931i		−	0.644890i
$494$		0			0
$495$	−33.3182	+	96.5286i	−0.0673096	+	0.195007i
$496$		0			0
$497$	106.632	+	517.867i	0.214551	+	1.04199i
$498$		0			0
$499$			624.305i			1.25111i	0.780179	+	0.625556i	$0.215128\pi$
$499$			624.305i			1.25111i	−0.780179	+	0.625556i	$0.784872\pi$
$500$		0			0
$501$			101.684i			0.202962i
$502$		0			0
$503$	419.530			0.834057			0.417028	−	0.908894i	$-0.363072\pi$
$503$	419.530			0.834057			0.417028	+	0.908894i	$0.363072\pi$
$504$		0			0
$505$	−216.843	−	74.8465i	−0.429392	−	0.148211i
$506$		0			0
$507$			369.102i			0.728013i
$508$		0			0
$509$	−312.970			−0.614872			−0.307436	−	0.951569i	$-0.599471\pi$
$509$	−312.970			−0.614872			−0.307436	+	0.951569i	$0.599471\pi$
$510$		0			0
$511$	115.725	+	562.028i	0.226467	+	1.09986i
$512$		0			0
$513$			543.885i			1.06020i
$514$		0			0
$515$	−488.115	−	168.480i	−0.947797	−	0.327146i
$516$		0			0
$517$			1183.18i			2.28855i
$518$		0			0
$519$	307.294			0.592088
$520$		0			0
$521$		−	519.109i		−	0.996370i	−0.867071	−	0.498185i	$-0.834000\pi$
$521$		−	519.109i		−	0.996370i	0.867071	−	0.498185i	$-0.166000\pi$
$522$		0			0
$523$		−	422.665i		−	0.808155i	−0.914725	−	0.404078i	$-0.867593\pi$
$523$		−	422.665i		−	0.808155i	0.914725	−	0.404078i	$-0.132407\pi$
$524$		0			0
$525$	−430.845	−	214.350i	−0.820657	−	0.408285i
$526$		0			0
$527$			483.041i			0.916586i
$528$		0			0
$529$	394.883			0.746471
$530$		0			0
$531$	−157.791			−0.297159
$532$		0			0
$533$	−93.2735			−0.174997
$534$		0			0
$535$	865.787	+	298.839i	1.61829	+	0.558578i
$536$		0			0
$537$	−434.076			−0.808335
$538$		0			0
$539$	274.853	+	639.127i	0.509931	+	1.18576i
$540$		0			0
$541$		−	423.091i		−	0.782053i	−0.920379	−	0.391026i	$-0.872120\pi$
$541$		−	423.091i		−	0.782053i	0.920379	−	0.391026i	$-0.127880\pi$
$542$		0			0
$543$		−	245.109i		−	0.451398i
$544$		0			0
$545$	299.322	−	867.186i	0.549214	−	1.59117i
$546$		0			0
$547$	451.054			0.824597			0.412298	−	0.911049i	$-0.364726\pi$
$547$	451.054			0.824597			0.412298	+	0.911049i	$0.364726\pi$
$548$		0			0
$549$	−51.0450			−0.0929782
$550$		0			0
$551$		−	601.945i		−	1.09246i
$552$		0			0
$553$	−126.607	−	614.878i	−0.228945	−	1.11189i
$554$		0			0
$555$	178.159	−	516.156i	0.321007	−	0.930012i
$556$		0			0
$557$	263.706			0.473439			0.236720	−	0.971578i	$-0.423928\pi$
$557$	263.706			0.473439			0.236720	+	0.971578i	$0.423928\pi$
$558$		0			0
$559$			177.049i			0.316725i
$560$		0			0
$561$	390.738			0.696503
$562$		0			0
$563$			529.494i			0.940487i	0.882537	+	0.470244i	$0.155834\pi$
$563$			529.494i			0.940487i	−0.882537	+	0.470244i	$0.844166\pi$
$564$		0			0
$565$	−81.6206	+	236.469i	−0.144461	+	0.418529i
$566$		0			0
$567$	−93.1527	−	452.404i	−0.164290	−	0.797892i
$568$		0			0
$569$	−605.747			−1.06458			−0.532291	−	0.846562i	$-0.678669\pi$
$569$	−605.747			−1.06458			−0.532291	+	0.846562i	$0.678669\pi$
$570$		0			0
$571$		−	618.848i		−	1.08380i	−0.840444	−	0.541898i	$-0.817706\pi$
$571$		−	618.848i		−	1.08380i	0.840444	−	0.541898i	$-0.182294\pi$
$572$		0			0
$573$			281.365i			0.491038i
$574$		0			0
$575$	−178.589	+	227.880i	−0.310589	+	0.396313i
$576$		0			0
$577$	−98.2803			−0.170330			−0.0851649	−	0.996367i	$-0.527142\pi$
$577$	−98.2803			−0.170330			−0.0851649	+	0.996367i	$0.527142\pi$
$578$		0			0
$579$	−389.995			−0.673566
$580$		0			0
$581$	−738.031	+	151.965i	−1.27028	+	0.261557i
$582$		0			0
$583$			254.503i			0.436540i
$584$		0			0
$585$	13.8377	−	40.0900i	0.0236541	−	0.0685299i
$586$		0			0
$587$		−	191.410i		−	0.326081i	−0.986619	−	0.163041i	$-0.947870\pi$
$587$		−	191.410i		−	0.326081i	0.986619	−	0.163041i	$-0.0521301\pi$
$588$		0			0
$589$			914.551i			1.55272i
$590$		0			0
$591$			140.336i			0.237456i
$592$		0			0
$593$	−926.369			−1.56217			−0.781087	−	0.624423i	$-0.785334\pi$
$593$	−926.369			−1.56217			−0.781087	+	0.624423i	$0.785334\pi$
$594$		0			0
$595$	45.1620	−	347.352i	0.0759026	−	0.583784i
$596$		0			0
$597$	207.549			0.347653
$598$		0			0
$599$	−378.246			−0.631462			−0.315731	−	0.948849i	$-0.602250\pi$
$599$	−378.246			−0.631462			−0.315731	+	0.948849i	$0.602250\pi$
$600$		0			0
$601$			893.893i			1.48734i	0.668545	+	0.743671i	$0.266917\pi$
$601$			893.893i			1.48734i	−0.668545	+	0.743671i	$0.733083\pi$
$602$		0			0
$603$	64.0933			0.106291
$604$		0			0
$605$	380.917	+	131.479i	0.629616	+	0.217321i
$606$		0			0
$607$	433.820			0.714696			0.357348	−	0.933971i	$-0.383681\pi$
$607$	433.820			0.714696			0.357348	+	0.933971i	$0.383681\pi$
$608$		0			0
$609$	123.324	+	598.933i	0.202502	+	0.983470i
$610$		0			0
$611$		−	491.396i		−	0.804248i
$612$		0			0
$613$	−202.703			−0.330674			−0.165337	−	0.986237i	$-0.552871\pi$
$613$	−202.703			−0.330674			−0.165337	+	0.986237i	$0.552871\pi$
$614$		0			0
$615$	205.576	+	70.9577i	0.334271	+	0.115378i
$616$		0			0
$617$			952.378i			1.54356i	0.635888	+	0.771781i	$0.280634\pi$
$617$			952.378i			1.54356i	−0.635888	+	0.771781i	$0.719366\pi$
$618$		0			0
$619$	−742.737			−1.19990			−0.599949	−	0.800038i	$-0.704813\pi$
$619$	−742.737			−1.19990			−0.599949	+	0.800038i	$0.704813\pi$
$620$		0			0
$621$	−332.417			−0.535293
$622$		0			0
$623$	−994.187	+	204.709i	−1.59581	+	0.328585i
$624$		0			0
$625$	149.386	+	606.885i	0.239018	+	0.971015i
$626$		0			0
$627$	739.793			1.17989
$628$		0			0
$629$	397.456			0.631885
$630$		0			0
$631$	−707.912			−1.12189			−0.560944	−	0.827854i	$-0.689562\pi$
$631$	−707.912			−1.12189			−0.560944	+	0.827854i	$0.689562\pi$
$632$		0			0
$633$	516.220			0.815513
$634$		0			0
$635$	134.919	−	390.884i	0.212472	−	0.615566i
$636$		0			0
$637$	−114.151	−	265.441i	−0.179201	−	0.416704i
$638$		0			0
$639$	108.649			0.170030
$640$		0			0
$641$	−256.060			−0.399470			−0.199735	−	0.979850i	$-0.564008\pi$
$641$	−256.060			−0.399470			−0.199735	+	0.979850i	$0.564008\pi$
$642$		0			0
$643$		−	743.551i		−	1.15638i	−0.815903	−	0.578189i	$-0.803760\pi$
$643$		−	743.551i		−	1.15638i	0.815903	−	0.578189i	$-0.196240\pi$
$644$		0			0
$645$	134.690	−	390.220i	0.208822	−	0.604992i
$646$		0			0
$647$	780.148			1.20579			0.602896	−	0.797819i	$-0.294013\pi$
$647$	780.148			1.20579			0.602896	+	0.797819i	$0.294013\pi$
$648$		0			0
$649$			1557.52i			2.39987i
$650$		0			0
$651$	−187.369	−	909.976i	−0.287817	−	1.39781i
$652$		0			0
$653$	800.379			1.22570			0.612848	−	0.790201i	$-0.290024\pi$
$653$	800.379			1.22570			0.612848	+	0.790201i	$0.290024\pi$
$654$		0			0
$655$	413.699	+	142.794i	0.631602	+	0.218006i
$656$		0			0
$657$	117.914			0.179473
$658$		0			0
$659$			824.020i			1.25041i	0.780461	+	0.625205i	$0.214985\pi$
$659$			824.020i			1.25041i	−0.780461	+	0.625205i	$0.785015\pi$
$660$		0			0
$661$	1004.18			1.51918			0.759589	−	0.650403i	$-0.225400\pi$
$661$	1004.18			1.51918			0.759589	+	0.650403i	$0.225400\pi$
$662$		0			0
$663$	−162.281			−0.244767
$664$		0			0
$665$	85.5062	−	657.648i	0.128581	−	0.988944i
$666$		0			0
$667$	367.903			0.551578
$668$		0			0
$669$			867.834i			1.29721i
$670$		0			0
$671$			503.853i			0.750898i
$672$		0			0
$673$		−	209.582i		−	0.311414i	−0.987803	−	0.155707i	$-0.950234\pi$
$673$		−	209.582i		−	0.311414i	0.987803	−	0.155707i	$-0.0497655\pi$
$674$		0			0
$675$	−442.643	+	564.813i	−0.655767	+	0.836760i
$676$		0			0
$677$			910.828i			1.34539i	0.739920	+	0.672695i	$0.234863\pi$
$677$			910.828i			1.34539i	−0.739920	+	0.672695i	$0.765137\pi$
$678$		0			0
$679$	18.9821	+	92.1881i	0.0279559	+	0.135770i
$680$		0			0
$681$	−20.2754			−0.0297729
$682$		0			0
$683$	313.801			0.459445			0.229722	−	0.973256i	$-0.426218\pi$
$683$	313.801			0.459445			0.229722	+	0.973256i	$0.426218\pi$
$684$		0			0
$685$	−100.399	+	290.873i	−0.146568	+	0.424632i
$686$		0			0
$687$		−	167.375i		−	0.243632i
$688$		0			0
$689$		−	105.700i		−	0.153410i
$690$		0			0
$691$	−2.99680			−0.00433690			−0.00216845	−	0.999998i	$-0.500690\pi$
$691$	−2.99680			−0.00433690			−0.00216845	+	0.999998i	$0.500690\pi$
$692$		0			0
$693$	140.026	−	28.8322i	0.202058	−	0.0416049i
$694$		0			0
$695$	75.2402	+	25.9703i	0.108259	+	0.0373673i
$696$		0			0
$697$			158.300i			0.227116i
$698$		0			0
$699$	−926.251			−1.32511
$700$		0			0
$701$			276.084i			0.393843i	0.980419	+	0.196921i	$0.0630944\pi$
$701$			276.084i			0.393843i	−0.980419	+	0.196921i	$0.936906\pi$
$702$		0			0
$703$	752.511			1.07043
$704$		0			0
$705$	−373.829	+	1083.04i	−0.530253	+	1.53623i
$706$		0			0
$707$	64.7690	+	314.557i	0.0916110	+	0.444917i
$708$		0			0
$709$		−	364.044i		−	0.513461i	−0.966483	−	0.256731i	$-0.917355\pi$
$709$		−	364.044i		−	0.513461i	0.966483	−	0.256731i	$-0.0826453\pi$
$710$		0			0
$711$	−129.002			−0.181437
$712$		0			0
$713$	−558.965			−0.783962
$714$		0			0
$715$	−395.718	−	136.588i	−0.553452	−	0.191032i
$716$		0			0
$717$		−	373.747i		−	0.521265i
$718$		0			0
$719$			200.157i			0.278383i	0.990265	+	0.139191i	$0.0444503\pi$
$719$			200.157i			0.278383i	−0.990265	+	0.139191i	$0.955550\pi$
$720$		0			0
$721$	145.796	+	708.069i	0.202213	+	0.982065i
$722$		0			0
$723$	−186.695			−0.258223
$724$		0			0
$725$	489.895	−	625.107i	0.675718	−	0.862217i
$726$		0			0
$727$	417.422			0.574170			0.287085	−	0.957905i	$-0.407314\pi$
$727$	417.422			0.574170			0.287085	+	0.957905i	$0.407314\pi$
$728$		0			0
$729$	−805.293			−1.10465
$730$		0			0
$731$	300.481			0.411054
$732$		0			0
$733$		−	1102.88i		−	1.50461i	−0.658814	−	0.752306i	$-0.728942\pi$
$733$		−	1102.88i		−	1.50461i	0.658814	−	0.752306i	$-0.271058\pi$
$734$		0			0
$735$	49.6576	+	671.876i	0.0675613	+	0.914117i
$736$		0			0
$737$		−	632.649i		−	0.858411i
$738$		0			0
$739$		−	721.307i		−	0.976059i	−0.872827	−	0.488029i	$-0.837716\pi$
$739$		−	721.307i		−	0.976059i	0.872827	−	0.488029i	$-0.162284\pi$
$740$		0			0
$741$	−307.249			−0.414641
$742$		0			0
$743$			588.343i			0.791847i	0.918283	+	0.395924i	$0.129576\pi$
$743$			588.343i			0.791847i	−0.918283	+	0.395924i	$0.870424\pi$
$744$		0			0
$745$	91.9388	−	266.362i	0.123408	−	0.357533i
$746$		0			0
$747$			154.840i			0.207282i
$748$		0			0
$749$	−258.603	−	1255.93i	−0.345264	−	1.67681i
$750$		0			0
$751$	1459.31			1.94315			0.971576	−	0.236730i	$-0.0760756\pi$
$751$	1459.31			1.94315			0.971576	+	0.236730i	$0.0760756\pi$
$752$		0			0
$753$			616.468i			0.818683i
$754$		0			0
$755$	−240.981	−	83.1782i	−0.319180	−	0.110170i
$756$		0			0
$757$	−982.060			−1.29730			−0.648652	−	0.761085i	$-0.724667\pi$
$757$	−982.060			−1.29730			−0.648652	+	0.761085i	$0.724667\pi$
$758$		0			0
$759$			452.154i			0.595724i
$760$		0			0
$761$		−	900.760i		−	1.18365i	−0.806065	−	0.591826i	$-0.798407\pi$
$761$		−	900.760i		−	1.18365i	0.806065	−	0.591826i	$-0.201593\pi$
$762$		0			0
$763$	−1257.96	+	259.020i	−1.64870	+	0.339476i
$764$		0			0
$765$	−68.0391	−	23.4847i	−0.0889400	−	0.0306989i
$766$		0			0
$767$		−	646.864i		−	0.843369i
$768$		0			0
$769$			282.871i			0.367843i	0.982941	+	0.183921i	$0.0588792\pi$
$769$			282.871i			0.367843i	−0.982941	+	0.183921i	$0.941121\pi$
$770$		0			0
$771$		−	153.901i		−	0.199612i
$772$		0			0
$773$			849.275i			1.09867i	0.835601	+	0.549337i	$0.185120\pi$
$773$			849.275i			1.09867i	−0.835601	+	0.549337i	$0.814880\pi$
$774$		0			0
$775$	−744.311	+	949.743i	−0.960402	+	1.22547i
$776$		0			0
$777$	−748.746	+	154.171i	−0.963637	+	0.198418i
$778$		0			0
$779$			299.713i			0.384740i
$780$		0			0
$781$		−	1072.45i		−	1.37317i
$782$		0			0
$783$	911.868			1.16458
$784$		0			0
$785$	362.966	−	1051.57i	0.462378	−	1.33959i
$786$		0			0
$787$			1087.66i			1.38204i	0.722838	+	0.691018i	$0.242837\pi$
$787$			1087.66i			1.38204i	−0.722838	+	0.691018i	$0.757163\pi$
$788$		0			0
$789$	−204.743			−0.259497
$790$		0			0
$791$	343.026	−	70.6310i	0.433661	−	0.0892933i
$792$		0			0
$793$		−	209.259i		−	0.263882i
$794$		0			0
$795$	−80.4108	+	232.964i	−0.101146	+	0.293036i
$796$		0			0
$797$		−	1145.86i		−	1.43772i	−0.695155	−	0.718860i	$-0.744664\pi$
$797$		−	1145.86i		−	1.43772i	0.695155	−	0.718860i	$-0.255336\pi$
$798$		0			0
$799$	−833.976			−1.04377
$800$		0			0
$801$			208.581i			0.260401i
$802$		0			0
$803$		−	1163.90i		−	1.44944i
$804$		0			0
$805$	401.948	+	52.2606i	0.499314	+	0.0649200i
$806$		0			0
$807$			1038.14i			1.28641i
$808$		0			0
$809$	614.722			0.759854			0.379927	−	0.925017i	$-0.375949\pi$
$809$	614.722			0.759854			0.379927	+	0.925017i	$0.375949\pi$
$810$		0			0
$811$	−979.064			−1.20723			−0.603616	−	0.797276i	$-0.706274\pi$
$811$	−979.064			−1.20723			−0.603616	+	0.797276i	$0.706274\pi$
$812$		0			0
$813$	1274.04			1.56708
$814$		0			0
$815$	329.772	+	113.825i	0.404628	+	0.139663i
$816$		0			0
$817$	568.906			0.696336
$818$		0			0
$819$	−58.1553	+	11.9745i	−0.0710077	+	0.0146209i
$820$		0			0
$821$			289.488i			0.352605i	0.984336	+	0.176302i	$0.0564136\pi$
$821$			289.488i			0.352605i	−0.984336	+	0.176302i	$0.943586\pi$
$822$		0			0
$823$		−	520.650i		−	0.632624i	−0.948655	−	0.316312i	$-0.897555\pi$
$823$		−	520.650i		−	0.632624i	0.948655	−	0.316312i	$-0.102445\pi$
$824$		0			0
$825$	768.261	+	602.084i	0.931225	+	0.729799i
$826$		0			0
$827$	468.146			0.566077			0.283039	−	0.959109i	$-0.408658\pi$
$827$	468.146			0.566077			0.283039	+	0.959109i	$0.408658\pi$
$828$		0			0
$829$	−952.556			−1.14904			−0.574521	−	0.818490i	$-0.694812\pi$
$829$	−952.556			−1.14904			−0.574521	+	0.818490i	$0.694812\pi$
$830$		0			0
$831$			879.767i			1.05868i
$832$		0			0
$833$	−450.495	+	193.733i	−0.540810	+	0.232572i
$834$		0			0
$835$	−174.773	−	60.3253i	−0.209309	−	0.0722459i
$836$		0			0
$837$	−1385.43			−1.65523
$838$		0			0
$839$		−	619.564i		−	0.738455i	−0.929339	−	0.369228i	$-0.879622\pi$
$839$		−	619.564i		−	0.738455i	0.929339	−	0.369228i	$-0.120378\pi$
$840$		0			0
$841$	−168.211			−0.200013
$842$		0			0
$843$			1095.82i			1.29990i
$844$		0			0
$845$	−634.408	−	218.975i	−0.750779	−	0.259142i
$846$		0			0
$847$	−113.777	−	552.566i	−0.134329	−	0.652380i
$848$		0			0
$849$	−580.950			−0.684276
$850$		0			0
$851$			459.927i			0.540455i
$852$		0			0
$853$			1263.00i			1.48065i	0.672247	+	0.740327i	$0.265329\pi$
$853$			1263.00i			1.48065i	−0.672247	+	0.740327i	$0.734671\pi$
$854$		0			0
$855$	−128.820	−	44.4640i	−0.150666	−	0.0520047i
$856$		0			0
$857$	1025.13			1.19619			0.598093	−	0.801427i	$-0.295925\pi$
$857$	1025.13			1.19619			0.598093	+	0.801427i	$0.295925\pi$
$858$		0			0
$859$	−1243.94			−1.44812			−0.724062	−	0.689735i	$-0.757727\pi$
$859$	−1243.94			−1.44812			−0.724062	+	0.689735i	$0.757727\pi$
$860$		0			0
$861$	−61.4038	−	298.213i	−0.0713168	−	0.346357i
$862$		0			0
$863$		−	1443.37i		−	1.67250i	−0.548350	−	0.836249i	$-0.684744\pi$
$863$		−	1443.37i		−	1.67250i	0.548350	−	0.836249i	$-0.315256\pi$
$864$		0			0
$865$	−182.307	+	528.173i	−0.210759	+	0.610604i
$866$		0			0
$867$		−	519.285i		−	0.598945i
$868$		0			0
$869$			1273.35i			1.46530i
$870$		0			0
$871$			262.750i			0.301665i
$872$		0			0
$873$	19.3412			0.0221548
$874$		0			0
$875$	624.026	−	613.365i	0.713173	−	0.700988i
$876$		0			0
$877$	−1233.98			−1.40704			−0.703522	−	0.710674i	$-0.748390\pi$
$877$	−1233.98			−1.40704			−0.703522	+	0.710674i	$0.748390\pi$
$878$		0			0
$879$	−188.111			−0.214005
$880$		0			0
$881$			718.191i			0.815199i	0.913161	+	0.407600i	$0.133634\pi$
$881$			718.191i			0.815199i	−0.913161	+	0.407600i	$0.866366\pi$
$882$		0			0
$883$	484.335			0.548511			0.274255	−	0.961657i	$-0.411569\pi$
$883$	484.335			0.548511			0.274255	+	0.961657i	$0.411569\pi$
$884$		0			0
$885$	−492.101	+	1425.70i	−0.556046	+	1.61096i
$886$		0			0
$887$	−653.308			−0.736537			−0.368268	−	0.929720i	$-0.620049\pi$
$887$	−653.308			−0.736537			−0.368268	+	0.929720i	$0.620049\pi$
$888$		0			0
$889$	−567.024	+	116.754i	−0.637823	+	0.131331i
$890$		0			0
$891$			936.881i			1.05149i
$892$		0			0
$893$	−1578.98			−1.76818
$894$		0			0
$895$	257.522	−	746.084i	0.287734	−	0.833613i
$896$		0			0
$897$		−	187.788i		−	0.209351i
$898$		0			0
$899$	1533.32			1.70559
$900$		0			0
$901$	−179.389			−0.199100
$902$		0			0
$903$	−566.060	+	116.555i	−0.626866	+	0.129075i
$904$		0			0
$905$	421.291	+	145.415i	0.465515	+	0.160679i
$906$		0			0
$907$	3.93877			0.00434263			0.00217131	−	0.999998i	$-0.499309\pi$
$907$	3.93877			0.00434263			0.00217131	+	0.999998i	$0.499309\pi$
$908$		0			0
$909$	65.9943			0.0726010
$910$		0			0
$911$	238.571			0.261878			0.130939	−	0.991390i	$-0.458201\pi$
$911$	238.571			0.261878			0.130939	+	0.991390i	$0.458201\pi$
$912$		0			0
$913$	1528.38			1.67402
$914$		0			0
$915$	−159.193	+	461.210i	−0.173982	+	0.504055i
$916$		0			0
$917$	−123.568	−	600.120i	−0.134753	−	0.654438i
$918$		0			0
$919$	−1711.75			−1.86262			−0.931311	−	0.364226i	$-0.881334\pi$
$919$	−1711.75			−1.86262			−0.931311	+	0.364226i	$0.881334\pi$
$920$		0			0
$921$	−351.977			−0.382168
$922$		0			0
$923$			445.406i			0.482563i
$924$		0			0
$925$	781.468	+	612.434i	0.844830	+	0.662091i
$926$		0			0
$927$	148.554			0.160252
$928$		0			0
$929$		−	1572.39i		−	1.69257i	−0.532734	−	0.846283i	$-0.678835\pi$
$929$		−	1572.39i		−	1.69257i	0.532734	−	0.846283i	$-0.321165\pi$
$930$		0			0
$931$	−852.931	+	366.798i	−0.916145	+	0.393983i
$932$		0			0
$933$	1266.05			1.35697
$934$		0			0
$935$	−231.811	+	671.596i	−0.247926	+	0.718285i
$936$		0			0
$937$	945.134			1.00868			0.504341	−	0.863505i	$-0.331736\pi$
$937$	945.134			1.00868			0.504341	+	0.863505i	$0.331736\pi$
$938$		0			0
$939$		−	436.313i		−	0.464658i
$940$		0			0
$941$	−352.792			−0.374912			−0.187456	−	0.982273i	$-0.560024\pi$
$941$	−352.792			−0.374912			−0.187456	+	0.982273i	$0.560024\pi$
$942$		0			0
$943$	−183.181			−0.194254
$944$		0			0
$945$	996.252	+	129.531i	1.05423	+	0.137070i
$946$		0			0
$947$	−996.916			−1.05271			−0.526355	−	0.850265i	$-0.676442\pi$
$947$	−996.916			−1.05271			−0.526355	+	0.850265i	$0.676442\pi$
$948$		0			0
$949$			483.388i			0.509365i
$950$		0			0
$951$			820.387i			0.862657i
$952$		0			0
$953$			1694.90i			1.77849i	0.457429	+	0.889246i	$0.348770\pi$
$953$			1694.90i			1.77849i	−0.457429	+	0.889246i	$0.651230\pi$
$954$		0			0
$955$	−483.606	−	166.924i	−0.506394	−	0.174789i
$956$		0			0
$957$		−	1240.33i		−	1.29606i
$958$		0			0
$959$	421.946	−	86.8811i	0.439985	−	0.0905955i
$960$		0			0
$961$	−1368.62			−1.42416
$962$		0			0
$963$	−263.495			−0.273619
$964$		0			0
$965$	231.370	−	670.318i	0.239762	−	0.694630i
$966$		0			0
$967$		−	1018.54i		−	1.05330i	−0.850082	−	0.526651i	$-0.823447\pi$
$967$		−	1018.54i		−	1.05330i	0.850082	−	0.526651i	$-0.176553\pi$
$968$		0			0
$969$			521.450i			0.538132i
$970$		0			0
$971$	1295.00			1.33368			0.666840	−	0.745201i	$-0.267646\pi$
$971$	1295.00			1.33368			0.666840	+	0.745201i	$0.267646\pi$
$972$		0			0
$973$	−22.4736	−	109.145i	−0.0230972	−	0.112174i
$974$		0			0
$975$	−319.072	−	250.056i	−0.327253	−	0.256468i
$976$		0			0
$977$		−	608.808i		−	0.623140i	−0.950223	−	0.311570i	$-0.899145\pi$
$977$		−	608.808i		−	0.623140i	0.950223	−	0.311570i	$-0.100855\pi$
$978$		0			0
$979$	2058.85			2.10302
$980$		0			0
$981$			263.920i			0.269032i
$982$		0			0
$983$	−364.756			−0.371064			−0.185532	−	0.982638i	$-0.559401\pi$
$983$	−364.756			−0.371064			−0.185532	+	0.982638i	$0.559401\pi$
$984$		0			0
$985$	−241.208	−	83.2565i	−0.244881	−	0.0845244i
$986$		0			0
$987$	1571.08	−	323.495i	1.59178	−	0.327756i
$988$		0			0
$989$			347.710i			0.351577i
$990$		0			0
$991$	1618.63			1.63333			0.816667	−	0.577110i	$-0.195819\pi$
$991$	1618.63			1.63333			0.816667	+	0.577110i	$0.195819\pi$
$992$		0			0
$993$	702.268			0.707218
$994$		0			0
$995$	−123.131	+	356.732i	−0.123750	+	0.358525i
$996$		0			0
$997$			1049.40i			1.05256i	0.850313	+	0.526278i	$0.176413\pi$
$997$			1049.40i			1.05256i	−0.850313	+	0.526278i	$0.823587\pi$
$998$		0			0
$999$			1139.96i			1.14110i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1120.3.c.g.209.12		80
4.3	odd	2		280.3.c.g.69.65	yes	80
5.4	even	2	inner	1120.3.c.g.209.25		80
7.6	odd	2	inner	1120.3.c.g.209.15		80
8.3	odd	2		280.3.c.g.69.14	yes	80
8.5	even	2	inner	1120.3.c.g.209.43		80
20.19	odd	2		280.3.c.g.69.16	yes	80
28.27	even	2		280.3.c.g.69.66	yes	80
35.34	odd	2	inner	1120.3.c.g.209.44		80
40.19	odd	2		280.3.c.g.69.67	yes	80
40.29	even	2	inner	1120.3.c.g.209.16		80
56.13	odd	2	inner	1120.3.c.g.209.26		80
56.27	even	2		280.3.c.g.69.13	✓	80
140.139	even	2		280.3.c.g.69.15	yes	80
280.69	odd	2	inner	1120.3.c.g.209.11		80
280.139	even	2		280.3.c.g.69.68	yes	80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.3.c.g.69.13	✓	80	56.27	even	2
280.3.c.g.69.14	yes	80	8.3	odd	2
280.3.c.g.69.15	yes	80	140.139	even	2
280.3.c.g.69.16	yes	80	20.19	odd	2
280.3.c.g.69.65	yes	80	4.3	odd	2
280.3.c.g.69.66	yes	80	28.27	even	2
280.3.c.g.69.67	yes	80	40.19	odd	2
280.3.c.g.69.68	yes	80	280.139	even	2
1120.3.c.g.209.11		80	280.69	odd	2	inner
1120.3.c.g.209.12		80	1.1	even	1	trivial
1120.3.c.g.209.15		80	7.6	odd	2	inner
1120.3.c.g.209.16		80	40.29	even	2	inner
1120.3.c.g.209.25		80	5.4	even	2	inner
1120.3.c.g.209.26		80	56.13	odd	2	inner
1120.3.c.g.209.43		80	8.5	even	2	inner
1120.3.c.g.209.44		80	35.34	odd	2	inner

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:	\( N \)	\(=\)	\( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1120.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.74983i q^{3} +(-4.72637 - 1.63138i) q^{5} +(1.41172 + 6.85617i) q^{7} +1.43843 q^{9} +O(q^{10})\)	\(q+2.74983i q^{3} +(-4.72637 - 1.63138i) q^{5} +(1.41172 + 6.85617i) q^{7} +1.43843 q^{9} -14.1984i q^{11} +5.89684i q^{13} +(4.48601 - 12.9967i) q^{15} +10.0079 q^{17} +18.9481 q^{19} +(-18.8533 + 3.88200i) q^{21} +11.5809i q^{23} +(19.6772 + 15.4210i) q^{25} +28.7039i q^{27} -31.7681i q^{29} +48.2661i q^{31} +39.0431 q^{33} +(4.51266 - 34.7079i) q^{35} +39.7143 q^{37} -16.2153 q^{39} +15.8176i q^{41} +30.0245 q^{43} +(-6.79856 - 2.34662i) q^{45} -83.3321 q^{47} +(-45.0141 + 19.3580i) q^{49} +27.5199i q^{51} -17.9248 q^{53} +(-23.1629 + 67.1069i) q^{55} +52.1041i q^{57} -109.697 q^{59} -35.4866 q^{61} +(2.03067 + 9.86212i) q^{63} +(9.61996 - 27.8707i) q^{65} +44.5578 q^{67} -31.8455 q^{69} +75.5330 q^{71} +81.9741 q^{73} +(-42.4051 + 54.1090i) q^{75} +(97.3465 - 20.0442i) q^{77} -89.6824 q^{79} -65.9850 q^{81} +107.645i q^{83} +(-47.3009 - 16.3266i) q^{85} +87.3568 q^{87} +145.006i q^{89} +(-40.4297 + 8.32471i) q^{91} -132.724 q^{93} +(-89.5558 - 30.9115i) q^{95} +13.4460 q^{97} -20.4234i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 224 q^{9}+O(q^{10}) \) 80 * q - 224 * q^9	\( 80 q - 224 q^{9} + 72 q^{15} - 104 q^{25} + 112 q^{39} + 192 q^{49} + 472 q^{65} - 800 q^{71} - 480 q^{79} - 896 q^{81} - 1176 q^{95}+O(q^{100}) \) 80 * q - 224 * q^9 + 72 * q^15 - 104 * q^25 + 112 * q^39 + 192 * q^49 + 472 * q^65 - 800 * q^71 - 480 * q^79 - 896 * q^81 - 1176 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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