LMFDB - Embedded newform 1120.3.c.g.209.22

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			209.22
Character	$\chi$	$=$	1120.209
Dual form			1120.3.c.g.209.21

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.14158i q^{3} +(3.67215 - 3.39343i) q^{5} +(3.38765 + 6.12567i) q^{7} -17.4359 q^{9} +O(q^{10})$	$q+5.14158i q^{3} +(3.67215 - 3.39343i) q^{5} +(3.38765 + 6.12567i) q^{7} -17.4359 q^{9} +18.5106i q^{11} +12.0353i q^{13} +(17.4476 + 18.8806i) q^{15} +19.5965 q^{17} -6.87089 q^{19} +(-31.4956 + 17.4179i) q^{21} -20.3728i q^{23} +(1.96931 - 24.9223i) q^{25} -43.3737i q^{27} -2.31875i q^{29} +6.49424i q^{31} -95.1736 q^{33} +(33.2270 + 10.9986i) q^{35} +30.2885 q^{37} -61.8806 q^{39} +48.2859i q^{41} +70.0518 q^{43} +(-64.0270 + 59.1673i) q^{45} -68.9298 q^{47} +(-26.0476 + 41.5033i) q^{49} +100.757i q^{51} +14.6534 q^{53} +(62.8143 + 67.9735i) q^{55} -35.3272i q^{57} -23.3458 q^{59} -65.4599 q^{61} +(-59.0666 - 106.806i) q^{63} +(40.8410 + 44.1954i) q^{65} +38.4464 q^{67} +104.748 q^{69} -66.3125 q^{71} -49.3159 q^{73} +(128.140 + 10.1254i) q^{75} +(-113.390 + 62.7074i) q^{77} +44.8588 q^{79} +66.0865 q^{81} -30.7499i q^{83} +(71.9613 - 66.4994i) q^{85} +11.9221 q^{87} -135.358i q^{89} +(-73.7244 + 40.7715i) q^{91} -33.3907 q^{93} +(-25.2309 + 23.3158i) q^{95} +36.8568 q^{97} -322.748i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$			5.14158i			1.71386i	0.515432	+	0.856930i	$0.327631\pi$
$3$			5.14158i			1.71386i	−0.515432	+	0.856930i	$0.672369\pi$
$4$		0			0
$5$	3.67215	−	3.39343i	0.734429	−	0.678685i
$5$	3.67215	−	3.39343i	0.734429	−	0.678685i
$6$		0			0
$7$	3.38765	+	6.12567i	0.483950	+	0.875096i
$7$	3.38765	+	6.12567i	0.483950	+	0.875096i
$8$		0			0
$9$	−17.4359			−1.93732
$10$		0			0
$11$			18.5106i			1.68278i	0.540429	+	0.841389i	$0.318262\pi$
$11$			18.5106i			1.68278i	−0.540429	+	0.841389i	$0.681738\pi$
$12$		0			0
$13$			12.0353i			0.925794i	0.886412	+	0.462897i	$0.153190\pi$
$13$			12.0353i			0.925794i	−0.886412	+	0.462897i	$0.846810\pi$
$14$		0			0
$15$	17.4476	+	18.8806i	1.16317	+	1.25871i
$16$		0			0
$17$	19.5965			1.15274			0.576369	−	0.817190i	$-0.304469\pi$
$17$	19.5965			1.15274			0.576369	+	0.817190i	$0.304469\pi$
$18$		0			0
$19$	−6.87089			−0.361626			−0.180813	−	0.983518i	$-0.557873\pi$
$19$	−6.87089			−0.361626			−0.180813	+	0.983518i	$0.557873\pi$
$20$		0			0
$21$	−31.4956	+	17.4179i	−1.49979	+	0.829423i
$22$		0			0
$23$		−	20.3728i		−	0.885773i	−0.896578	−	0.442886i	$-0.853954\pi$
$23$		−	20.3728i		−	0.885773i	0.896578	−	0.442886i	$-0.146046\pi$
$24$		0			0
$25$	1.96931	−	24.9223i	0.0787723	−	0.996893i
$26$		0			0
$27$		−	43.3737i		−	1.60643i
$28$		0			0
$29$		−	2.31875i		−	0.0799570i	−0.999201	−	0.0399785i	$-0.987271\pi$
$29$		−	2.31875i		−	0.0799570i	0.999201	−	0.0399785i	$-0.0127289\pi$
$30$		0			0
$31$			6.49424i			0.209492i	0.994499	+	0.104746i	$0.0334029\pi$
$31$			6.49424i			0.209492i	−0.994499	+	0.104746i	$0.966597\pi$
$32$		0			0
$33$	−95.1736			−2.88405
$34$		0			0
$35$	33.2270	+	10.9986i	0.949342	+	0.314246i
$36$		0			0
$37$	30.2885			0.818609			0.409305	−	0.912398i	$-0.365771\pi$
$37$	30.2885			0.818609			0.409305	+	0.912398i	$0.365771\pi$
$38$		0			0
$39$	−61.8806			−1.58668
$40$		0			0
$41$			48.2859i			1.17771i	0.808240	+	0.588853i	$0.200420\pi$
$41$			48.2859i			1.17771i	−0.808240	+	0.588853i	$0.799580\pi$
$42$		0			0
$43$	70.0518			1.62911			0.814556	−	0.580085i	$-0.196981\pi$
$43$	70.0518			1.62911			0.814556	+	0.580085i	$0.196981\pi$
$44$		0			0
$45$	−64.0270	+	59.1673i	−1.42282	+	1.31483i
$46$		0			0
$47$	−68.9298			−1.46659			−0.733296	−	0.679909i	$-0.762019\pi$
$47$	−68.9298			−1.46659			−0.733296	+	0.679909i	$0.762019\pi$
$48$		0			0
$49$	−26.0476	+	41.5033i	−0.531584	+	0.847005i
$50$		0			0
$51$			100.757i			1.97563i
$52$		0			0
$53$	14.6534			0.276480			0.138240	−	0.990399i	$-0.455856\pi$
$53$	14.6534			0.276480			0.138240	+	0.990399i	$0.455856\pi$
$54$		0			0
$55$	62.8143	+	67.9735i	1.14208	+	1.23588i
$56$		0			0
$57$		−	35.3272i		−	0.619776i
$58$		0			0
$59$	−23.3458			−0.395691			−0.197845	−	0.980233i	$-0.563394\pi$
$59$	−23.3458			−0.395691			−0.197845	+	0.980233i	$0.563394\pi$
$60$		0			0
$61$	−65.4599			−1.07311			−0.536557	−	0.843864i	$-0.680275\pi$
$61$	−65.4599			−1.07311			−0.536557	+	0.843864i	$0.680275\pi$
$62$		0			0
$63$	−59.0666	−	106.806i	−0.937566	−	1.69534i
$64$		0			0
$65$	40.8410	+	44.1954i	0.628323	+	0.679930i
$66$		0			0
$67$	38.4464			0.573826			0.286913	−	0.957957i	$-0.407371\pi$
$67$	38.4464			0.573826			0.286913	+	0.957957i	$0.407371\pi$
$68$		0			0
$69$	104.748			1.51809
$70$		0			0
$71$	−66.3125			−0.933979			−0.466989	−	0.884263i	$-0.654661\pi$
$71$	−66.3125			−0.933979			−0.466989	+	0.884263i	$0.654661\pi$
$72$		0			0
$73$	−49.3159			−0.675560			−0.337780	−	0.941225i	$-0.609676\pi$
$73$	−49.3159			−0.675560			−0.337780	+	0.941225i	$0.609676\pi$
$74$		0			0
$75$	128.140	+	10.1254i	1.70853	+	0.135005i
$76$		0			0
$77$	−113.390	+	62.7074i	−1.47259	+	0.814381i
$78$		0			0
$79$	44.8588			0.567833			0.283917	−	0.958849i	$-0.408366\pi$
$79$	44.8588			0.567833			0.283917	+	0.958849i	$0.408366\pi$
$80$		0			0
$81$	66.0865			0.815883
$82$		0			0
$83$		−	30.7499i		−	0.370480i	−0.982693	−	0.185240i	$-0.940694\pi$
$83$		−	30.7499i		−	0.370480i	0.982693	−	0.185240i	$-0.0593063\pi$
$84$		0			0
$85$	71.9613	−	66.4994i	0.846604	−	0.782346i
$86$		0			0
$87$	11.9221			0.137035
$88$		0			0
$89$		−	135.358i		−	1.52088i	−0.649408	−	0.760440i	$-0.724983\pi$
$89$		−	135.358i		−	1.52088i	0.649408	−	0.760440i	$-0.275017\pi$
$90$		0			0
$91$	−73.7244	+	40.7715i	−0.810158	+	0.448038i
$92$		0			0
$93$	−33.3907			−0.359039
$94$		0			0
$95$	−25.2309	+	23.3158i	−0.265588	+	0.245430i
$96$		0			0
$97$	36.8568			0.379967			0.189983	−	0.981787i	$-0.439157\pi$
$97$	36.8568			0.379967			0.189983	+	0.981787i	$0.439157\pi$
$98$		0			0
$99$		−	322.748i		−	3.26008i
$100$		0			0
$101$	117.202			1.16042			0.580208	−	0.814469i	$-0.302971\pi$
$101$	117.202			1.16042			0.580208	+	0.814469i	$0.302971\pi$
$102$		0			0
$103$	−133.746			−1.29850			−0.649252	−	0.760573i	$-0.724918\pi$
$103$	−133.746			−1.29850			−0.649252	+	0.760573i	$0.724918\pi$
$104$		0			0
$105$	−56.5502	+	170.839i	−0.538573	+	1.62704i
$106$		0			0
$107$	−74.3641			−0.694991			−0.347496	−	0.937682i	$-0.612968\pi$
$107$	−74.3641			−0.694991			−0.347496	+	0.937682i	$0.612968\pi$
$108$		0			0
$109$			28.9108i			0.265236i	0.991167	+	0.132618i	$0.0423384\pi$
$109$			28.9108i			0.265236i	−0.991167	+	0.132618i	$0.957662\pi$
$110$		0			0
$111$			155.731i			1.40298i
$112$		0			0
$113$		−	142.601i		−	1.26195i	−0.775801	−	0.630977i	$-0.782654\pi$
$113$		−	142.601i		−	1.26195i	0.775801	−	0.630977i	$-0.217346\pi$
$114$		0			0
$115$	−69.1335	−	74.8118i	−0.601161	−	0.650537i
$116$		0			0
$117$		−	209.846i		−	1.79356i
$118$		0			0
$119$	66.3862	+	120.042i	0.557867	+	1.00876i
$120$		0			0
$121$	−221.641			−1.83174
$122$		0			0
$123$	−248.266			−2.01842
$124$		0			0
$125$	−77.3405	−	98.2011i	−0.618724	−	0.785609i
$126$		0			0
$127$			164.684i			1.29673i	0.761331	+	0.648363i	$0.224546\pi$
$127$			164.684i			1.29673i	−0.761331	+	0.648363i	$0.775454\pi$
$128$		0			0
$129$			360.177i			2.79207i
$130$		0			0
$131$	−0.684303			−0.00522369			−0.00261184	−	0.999997i	$-0.500831\pi$
$131$	−0.684303			−0.00522369			−0.00261184	+	0.999997i	$0.500831\pi$
$132$		0			0
$133$	−23.2762	−	42.0888i	−0.175009	−	0.316457i
$134$		0			0
$135$	−147.185	−	159.274i	−1.09026	−	1.17981i
$136$		0			0
$137$		−	158.086i		−	1.15392i	−0.816774	−	0.576958i	$-0.804240\pi$
$137$		−	158.086i		−	1.15392i	0.816774	−	0.576958i	$-0.195760\pi$
$138$		0			0
$139$	13.1028			0.0942645			0.0471322	−	0.998889i	$-0.484992\pi$
$139$	13.1028			0.0942645			0.0471322	+	0.998889i	$0.484992\pi$
$140$		0			0
$141$		−	354.408i		−	2.51354i
$142$		0			0
$143$	−222.781			−1.55791
$144$		0			0
$145$	−7.86852	−	8.51480i	−0.0542657	−	0.0587228i
$146$		0			0
$147$	−213.392	−	133.926i	−1.45165	−	0.911061i
$148$		0			0
$149$			16.7007i			0.112086i	0.998428	+	0.0560428i	$0.0178483\pi$
$149$			16.7007i			0.112086i	−0.998428	+	0.0560428i	$0.982152\pi$
$150$		0			0
$151$	−108.405			−0.717914			−0.358957	−	0.933354i	$-0.616868\pi$
$151$	−108.405			−0.717914			−0.358957	+	0.933354i	$0.616868\pi$
$152$		0			0
$153$	−341.682			−2.23322
$154$		0			0
$155$	22.0377	+	23.8478i	0.142179	+	0.153857i
$156$		0			0
$157$			60.9427i			0.388170i	0.980985	+	0.194085i	$0.0621738\pi$
$157$			60.9427i			0.388170i	−0.980985	+	0.194085i	$0.937826\pi$
$158$		0			0
$159$			75.3418i			0.473847i
$160$		0			0
$161$	124.797	−	69.0159i	0.775136	−	0.428670i
$162$		0			0
$163$	237.235			1.45543			0.727713	−	0.685881i	$-0.240583\pi$
$163$	237.235			1.45543			0.727713	+	0.685881i	$0.240583\pi$
$164$		0			0
$165$	−349.491	+	322.965i	−2.11813	+	1.95736i
$166$		0			0
$167$	109.805			0.657518			0.328759	−	0.944414i	$-0.393370\pi$
$167$	109.805			0.657518			0.328759	+	0.944414i	$0.393370\pi$
$168$		0			0
$169$	24.1512			0.142906
$170$		0			0
$171$	119.800			0.700584
$172$		0			0
$173$			25.0358i			0.144715i	0.997379	+	0.0723577i	$0.0230523\pi$
$173$			25.0358i			0.144715i	−0.997379	+	0.0723577i	$0.976948\pi$
$174$		0			0
$175$	159.337	−	72.3648i	0.910498	−	0.413513i
$176$		0			0
$177$		−	120.034i		−	0.678159i
$178$		0			0
$179$			185.824i			1.03812i	0.854737	+	0.519062i	$0.173719\pi$
$179$			185.824i			1.03812i	−0.854737	+	0.519062i	$0.826281\pi$
$180$		0			0
$181$	112.689			0.622590			0.311295	−	0.950313i	$-0.399237\pi$
$181$	112.689			0.622590			0.311295	+	0.950313i	$0.399237\pi$
$182$		0			0
$183$		−	336.568i		−	1.83917i
$184$		0			0
$185$	111.224	−	102.782i	0.601210	−	0.555578i
$186$		0			0
$187$			362.743i			1.93980i
$188$		0			0
$189$	265.693	−	146.935i	1.40578	−	0.777433i
$190$		0			0
$191$	293.237			1.53527			0.767636	−	0.640886i	$-0.221433\pi$
$191$	293.237			1.53527			0.767636	+	0.640886i	$0.221433\pi$
$192$		0			0
$193$		−	98.3546i		−	0.509609i	−0.966993	−	0.254805i	$-0.917989\pi$
$193$		−	98.3546i		−	0.509609i	0.966993	−	0.254805i	$-0.0820111\pi$
$194$		0			0
$195$	−227.234	+	209.987i	−1.16530	+	1.07686i
$196$		0			0
$197$	286.507			1.45435			0.727175	−	0.686452i	$-0.240833\pi$
$197$	286.507			1.45435			0.727175	+	0.686452i	$0.240833\pi$
$198$		0			0
$199$		−	102.976i		−	0.517467i	−0.965949	−	0.258734i	$-0.916695\pi$
$199$		−	102.976i		−	0.517467i	0.965949	−	0.258734i	$-0.0833052\pi$
$200$		0			0
$201$			197.675i			0.983458i
$202$		0			0
$203$	14.2039	−	7.85513i	0.0699700	−	0.0386952i
$204$		0			0
$205$	163.855	+	177.313i	0.799291	+	0.864941i
$206$		0			0
$207$			355.217i			1.71602i
$208$		0			0
$209$		−	127.184i		−	0.608536i
$210$		0			0
$211$		−	171.846i		−	0.814437i	−0.913331	−	0.407218i	$-0.866499\pi$
$211$		−	171.846i		−	0.814437i	0.913331	−	0.407218i	$-0.133501\pi$
$212$		0			0
$213$		−	340.951i		−	1.60071i
$214$		0			0
$215$	257.240	−	237.716i	1.19647	−	1.10565i
$216$		0			0
$217$	−39.7815	+	22.0002i	−0.183325	+	0.101383i
$218$		0			0
$219$		−	253.562i		−	1.15782i
$220$		0			0
$221$			235.850i			1.06720i
$222$		0			0
$223$	−7.43778			−0.0333533			−0.0166766	−	0.999861i	$-0.505309\pi$
$223$	−7.43778			−0.0333533			−0.0166766	+	0.999861i	$0.505309\pi$
$224$		0			0
$225$	−34.3366	+	434.542i	−0.152607	+	1.93130i
$226$		0			0
$227$		−	75.9403i		−	0.334539i	−0.985911	−	0.167269i	$-0.946505\pi$
$227$		−	75.9403i		−	0.334539i	0.985911	−	0.167269i	$-0.0534949\pi$
$228$		0			0
$229$	−219.553			−0.958749			−0.479374	−	0.877611i	$-0.659136\pi$
$229$	−219.553			−0.958749			−0.479374	+	0.877611i	$0.659136\pi$
$230$		0			0
$231$	−322.415	−	583.002i	−1.39574	−	2.52382i
$232$		0			0
$233$			352.875i			1.51448i	0.653134	+	0.757242i	$0.273454\pi$
$233$			352.875i			1.51448i	−0.653134	+	0.757242i	$0.726546\pi$
$234$		0			0
$235$	−253.120	+	233.908i	−1.07711	+	0.995355i
$236$		0			0
$237$			230.645i			0.973187i
$238$		0			0
$239$	23.9344			0.100144			0.0500721	−	0.998746i	$-0.484055\pi$
$239$	23.9344			0.100144			0.0500721	+	0.998746i	$0.484055\pi$
$240$		0			0
$241$			31.4618i			0.130547i	0.997867	+	0.0652734i	$0.0207919\pi$
$241$			31.4618i			0.130547i	−0.997867	+	0.0652734i	$0.979208\pi$
$242$		0			0
$243$		−	50.5737i		−	0.208122i
$244$		0			0
$245$	45.1876	+	240.797i	0.184439	+	0.982844i
$246$		0			0
$247$		−	82.6933i		−	0.334791i
$248$		0			0
$249$	158.103			0.634952
$250$		0			0
$251$	−429.837			−1.71250			−0.856249	−	0.516564i	$-0.827211\pi$
$251$	−429.837			−1.71250			−0.856249	+	0.516564i	$0.827211\pi$
$252$		0			0
$253$	377.112			1.49056
$254$		0			0
$255$	341.912	+	369.995i	1.34083	+	1.45096i
$256$		0			0
$257$	181.078			0.704586			0.352293	−	0.935890i	$-0.385402\pi$
$257$	181.078			0.704586			0.352293	+	0.935890i	$0.385402\pi$
$258$		0			0
$259$	102.607	+	185.538i	0.396166	+	0.716361i
$260$		0			0
$261$			40.4295i			0.154902i
$262$		0			0
$263$		−	247.354i		−	0.940508i	−0.882531	−	0.470254i	$-0.844162\pi$
$263$		−	247.354i		−	0.940508i	0.882531	−	0.470254i	$-0.155838\pi$
$264$		0			0
$265$	53.8095	−	49.7253i	0.203055	−	0.187643i
$266$		0			0
$267$	695.956			2.60658
$268$		0			0
$269$	−314.890			−1.17059			−0.585297	−	0.810819i	$-0.699022\pi$
$269$	−314.890			−1.17059			−0.585297	+	0.810819i	$0.699022\pi$
$270$		0			0
$271$			275.203i			1.01551i	0.861502	+	0.507754i	$0.169524\pi$
$271$			275.203i			1.01551i	−0.861502	+	0.507754i	$0.830476\pi$
$272$		0			0
$273$	−209.630	−	379.060i	−0.767875	−	1.38850i
$274$		0			0
$275$	461.326	+	36.4530i	1.67755	+	0.132556i
$276$		0			0
$277$	393.344			1.42001			0.710006	−	0.704195i	$-0.248692\pi$
$277$	393.344			1.42001			0.710006	+	0.704195i	$0.248692\pi$
$278$		0			0
$279$		−	113.233i		−	0.405852i
$280$		0			0
$281$	165.416			0.588667			0.294334	−	0.955703i	$-0.404902\pi$
$281$	165.416			0.588667			0.294334	+	0.955703i	$0.404902\pi$
$282$		0			0
$283$			501.665i			1.77267i	0.463048	+	0.886333i	$0.346756\pi$
$283$			501.665i			1.77267i	−0.463048	+	0.886333i	$0.653244\pi$
$284$		0			0
$285$	−119.880	−	129.727i	−0.420633	−	0.455181i
$286$		0			0
$287$	−295.784	+	163.576i	−1.03060	+	0.569951i
$288$		0			0
$289$	95.0241			0.328803
$290$		0			0
$291$			189.502i			0.651210i
$292$		0			0
$293$			551.653i			1.88277i	0.337327	+	0.941387i	$0.390477\pi$
$293$			551.653i			1.88277i	−0.337327	+	0.941387i	$0.609523\pi$
$294$		0			0
$295$	−85.7290	+	79.2221i	−0.290607	+	0.268550i
$296$		0			0
$297$	802.871			2.70327
$298$		0			0
$299$	245.193			0.820043
$300$		0			0
$301$	237.311	+	429.114i	0.788409	+	1.42563i
$302$		0			0
$303$			602.604i			1.98879i
$304$		0			0
$305$	−240.378	+	222.134i	−0.788126	+	0.728307i
$306$		0			0
$307$			143.310i			0.466809i	0.972380	+	0.233405i	$0.0749867\pi$
$307$			143.310i			0.466809i	−0.972380	+	0.233405i	$0.925013\pi$
$308$		0			0
$309$		−	687.666i		−	2.22546i
$310$		0			0
$311$			204.664i			0.658085i	0.944315	+	0.329043i	$0.106726\pi$
$311$			204.664i			0.658085i	−0.944315	+	0.329043i	$0.893274\pi$
$312$		0			0
$313$	−213.198			−0.681143			−0.340571	−	0.940219i	$-0.610621\pi$
$313$	−213.198			−0.681143			−0.340571	+	0.940219i	$0.610621\pi$
$314$		0			0
$315$	−579.341	−	191.770i	−1.83918	−	0.608794i
$316$		0			0
$317$	−169.509			−0.534729			−0.267364	−	0.963595i	$-0.586153\pi$
$317$	−169.509			−0.534729			−0.267364	+	0.963595i	$0.586153\pi$
$318$		0			0
$319$	42.9214			0.134550
$320$		0			0
$321$		−	382.349i		−	1.19112i
$322$		0			0
$323$	−134.646			−0.416859
$324$		0			0
$325$	299.948	+	23.7012i	0.922917	+	0.0729269i
$326$		0			0
$327$	−148.647			−0.454578
$328$		0			0
$329$	−233.510	−	422.241i	−0.709758	−	1.28341i
$330$		0			0
$331$			52.7511i			0.159369i	0.996820	+	0.0796845i	$0.0253913\pi$
$331$			52.7511i			0.159369i	−0.996820	+	0.0796845i	$0.974609\pi$
$332$		0			0
$333$	−528.107			−1.58591
$334$		0			0
$335$	141.181	−	130.465i	0.421435	−	0.389447i
$336$		0			0
$337$			103.564i			0.307312i	0.988124	+	0.153656i	$0.0491048\pi$
$337$			103.564i			0.307312i	−0.988124	+	0.153656i	$0.950895\pi$
$338$		0			0
$339$	733.194			2.16281
$340$		0			0
$341$	−120.212			−0.352528
$342$		0			0
$343$	−342.476	−	18.9605i	−0.998471	−	0.0552786i
$344$		0			0
$345$	384.651	−	355.456i	1.11493	−	1.03031i
$346$		0			0
$347$	336.770			0.970518			0.485259	−	0.874370i	$-0.338725\pi$
$347$	336.770			0.970518			0.485259	+	0.874370i	$0.338725\pi$
$348$		0			0
$349$	294.617			0.844175			0.422088	−	0.906555i	$-0.361297\pi$
$349$	294.617			0.844175			0.422088	+	0.906555i	$0.361297\pi$
$350$		0			0
$351$	522.016			1.48722
$352$		0			0
$353$	524.774			1.48661			0.743307	−	0.668951i	$-0.233256\pi$
$353$	524.774			1.48661			0.743307	+	0.668951i	$0.233256\pi$
$354$		0			0
$355$	−243.509	+	225.027i	−0.685941	+	0.633878i
$356$		0			0
$357$	−617.205	+	341.330i	−1.72887	+	0.956107i
$358$		0			0
$359$	−267.798			−0.745954			−0.372977	−	0.927840i	$-0.621663\pi$
$359$	−267.798			−0.745954			−0.372977	+	0.927840i	$0.621663\pi$
$360$		0			0
$361$	−313.791			−0.869227
$362$		0			0
$363$		−	1139.59i		−	3.13936i
$364$		0			0
$365$	−181.095	+	167.350i	−0.496151	+	0.458493i
$366$		0			0
$367$	358.839			0.977763			0.488882	−	0.872350i	$-0.337405\pi$
$367$	358.839			0.977763			0.488882	+	0.872350i	$0.337405\pi$
$368$		0			0
$369$		−	841.907i		−	2.28159i
$370$		0			0
$371$	49.6407	+	89.7620i	0.133802	+	0.241946i
$372$		0			0
$373$	76.3933			0.204808			0.102404	−	0.994743i	$-0.467347\pi$
$373$	76.3933			0.204808			0.102404	+	0.994743i	$0.467347\pi$
$374$		0			0
$375$	504.909	−	397.652i	1.34642	−	1.06041i
$376$		0			0
$377$	27.9069			0.0740237
$378$		0			0
$379$		−	435.652i		−	1.14948i	−0.818337	−	0.574739i	$-0.805104\pi$
$379$		−	435.652i		−	1.14948i	0.818337	−	0.574739i	$-0.194896\pi$
$380$		0			0
$381$	−846.738			−2.22241
$382$		0			0
$383$	196.615			0.513356			0.256678	−	0.966497i	$-0.417372\pi$
$383$	196.615			0.513356			0.256678	+	0.966497i	$0.417372\pi$
$384$		0			0
$385$	−203.590	+	615.050i	−0.528806	+	1.59753i
$386$		0			0
$387$	−1221.41			−3.15611
$388$		0			0
$389$		−	473.370i		−	1.21689i	−0.793596	−	0.608445i	$-0.791794\pi$
$389$		−	473.370i		−	1.21689i	0.793596	−	0.608445i	$-0.208206\pi$
$390$		0			0
$391$		−	399.236i		−	1.02106i
$392$		0			0
$393$		−	3.51840i		−	0.00895267i
$394$		0			0
$395$	164.728	−	152.225i	0.417033	−	0.385380i
$396$		0			0
$397$			102.950i			0.259320i	0.991559	+	0.129660i	$0.0413886\pi$
$397$			102.950i			0.259320i	−0.991559	+	0.129660i	$0.958611\pi$
$398$		0			0
$399$	216.403	−	119.676i	0.542363	−	0.299941i
$400$		0			0
$401$	274.940			0.685636			0.342818	−	0.939402i	$-0.388619\pi$
$401$	274.940			0.685636			0.342818	+	0.939402i	$0.388619\pi$
$402$		0			0
$403$	−78.1602			−0.193946
$404$		0			0
$405$	242.679	−	224.260i	0.599208	−	0.553728i
$406$		0			0
$407$			560.658i			1.37754i
$408$		0			0
$409$			319.228i			0.780508i	0.920707	+	0.390254i	$0.127613\pi$
$409$			319.228i			0.780508i	−0.920707	+	0.390254i	$0.872387\pi$
$410$		0			0
$411$	812.814			1.97765
$412$		0			0
$413$	−79.0873	−	143.008i	−0.191495	−	0.346267i
$414$		0			0
$415$	−104.347	−	112.918i	−0.251440	−	0.272092i
$416$		0			0
$417$			67.3689i			0.161556i
$418$		0			0
$419$	509.481			1.21594			0.607972	−	0.793958i	$-0.291983\pi$
$419$	509.481			1.21594			0.607972	+	0.793958i	$0.291983\pi$
$420$		0			0
$421$			635.328i			1.50909i	0.656247	+	0.754546i	$0.272143\pi$
$421$			635.328i			1.50909i	−0.656247	+	0.754546i	$0.727857\pi$
$422$		0			0
$423$	1201.85			2.84126
$424$		0			0
$425$	38.5916	−	488.391i	0.0908038	−	1.14916i
$426$		0			0
$427$	−221.756	−	400.986i	−0.519334	−	0.939077i
$428$		0			0
$429$		−	1145.44i		−	2.67003i
$430$		0			0
$431$	271.590			0.630138			0.315069	−	0.949069i	$-0.397972\pi$
$431$	271.590			0.630138			0.315069	+	0.949069i	$0.397972\pi$
$432$		0			0
$433$	679.871			1.57014			0.785071	−	0.619406i	$-0.212627\pi$
$433$	679.871			1.57014			0.785071	+	0.619406i	$0.212627\pi$
$434$		0			0
$435$	43.7795	−	40.4566i	0.100643	−	0.0930038i
$436$		0			0
$437$			139.979i			0.320318i
$438$		0			0
$439$		−	258.502i		−	0.588843i	−0.955676	−	0.294421i	$-0.904873\pi$
$439$		−	258.502i		−	0.588843i	0.955676	−	0.294421i	$-0.0951269\pi$
$440$		0			0
$441$	454.163	−	723.645i	1.02985	−	1.64092i
$442$		0			0
$443$	139.976			0.315974			0.157987	−	0.987441i	$-0.449500\pi$
$443$	139.976			0.315974			0.157987	+	0.987441i	$0.449500\pi$
$444$		0			0
$445$	−459.329	−	497.056i	−1.03220	−	1.11698i
$446$		0			0
$447$	−85.8682			−0.192099
$448$		0			0
$449$	246.858			0.549794			0.274897	−	0.961474i	$-0.411356\pi$
$449$	246.858			0.549794			0.274897	+	0.961474i	$0.411356\pi$
$450$		0			0
$451$	−893.800			−1.98182
$452$		0			0
$453$		−	557.373i		−	1.23040i
$454$		0			0
$455$	−132.372	+	399.897i	−0.290927	+	0.878894i
$456$		0			0
$457$		−	445.711i		−	0.975297i	−0.873040	−	0.487648i	$-0.837855\pi$
$457$		−	445.711i		−	0.975297i	0.873040	−	0.487648i	$-0.162145\pi$
$458$		0			0
$459$		−	849.974i		−	1.85179i
$460$		0			0
$461$	−194.924			−0.422828			−0.211414	−	0.977397i	$-0.567807\pi$
$461$	−194.924			−0.422828			−0.211414	+	0.977397i	$0.567807\pi$
$462$		0			0
$463$		−	279.703i		−	0.604109i	−0.953291	−	0.302055i	$-0.902327\pi$
$463$		−	279.703i		−	0.604109i	0.953291	−	0.302055i	$-0.0976725\pi$
$464$		0			0
$465$	−122.615	+	113.309i	−0.263689	+	0.243675i
$466$		0			0
$467$		−	590.130i		−	1.26366i	−0.775106	−	0.631831i	$-0.782304\pi$
$467$		−	590.130i		−	1.26366i	0.775106	−	0.631831i	$-0.217696\pi$
$468$		0			0
$469$	130.243	+	235.510i	0.277703	+	0.502153i
$470$		0			0
$471$	−313.342			−0.665270
$472$		0			0
$473$			1296.70i			2.74143i
$474$		0			0
$475$	−13.5309	+	171.238i	−0.0284861	+	0.360502i
$476$		0			0
$477$	−255.495			−0.535629
$478$		0			0
$479$		−	259.677i		−	0.542122i	−0.962562	−	0.271061i	$-0.912625\pi$
$479$		−	259.677i		−	0.542122i	0.962562	−	0.271061i	$-0.0873746\pi$
$480$		0			0
$481$			364.532i			0.757863i
$482$		0			0
$483$	354.851	+	641.653i	0.734681	+	1.32847i
$484$		0			0
$485$	135.343	−	125.071i	0.279059	−	0.257878i
$486$		0			0
$487$		−	743.395i		−	1.52648i	−0.646116	−	0.763239i	$-0.723608\pi$
$487$		−	743.395i		−	1.52648i	0.646116	−	0.763239i	$-0.276392\pi$
$488$		0			0
$489$			1219.76i			2.49440i
$490$		0			0
$491$			489.837i			0.997631i	0.866708	+	0.498816i	$0.166232\pi$
$491$			489.837i			0.997631i	−0.866708	+	0.498816i	$0.833768\pi$
$492$		0			0
$493$		−	45.4395i		−	0.0921694i
$494$		0			0
$495$	−1095.22	−	1185.18i	−2.21257	−	2.39430i
$496$		0			0
$497$	−224.644	−	406.208i	−0.451999	−	0.817321i
$498$		0			0
$499$		−	443.911i		−	0.889602i	−0.895630	−	0.444801i	$-0.853274\pi$
$499$		−	443.911i		−	0.889602i	0.895630	−	0.444801i	$-0.146726\pi$
$500$		0			0
$501$			564.574i			1.12689i
$502$		0			0
$503$	909.757			1.80866			0.904331	−	0.426833i	$-0.140371\pi$
$503$	909.757			1.80866			0.904331	+	0.426833i	$0.140371\pi$
$504$		0			0
$505$	430.383	−	397.716i	0.852243	−	0.787557i
$506$		0			0
$507$			124.175i			0.244921i
$508$		0			0
$509$	343.817			0.675475			0.337738	−	0.941240i	$-0.390338\pi$
$509$	343.817			0.675475			0.337738	+	0.941240i	$0.390338\pi$
$510$		0			0
$511$	−167.065	−	302.093i	−0.326938	−	0.591180i
$512$		0			0
$513$			298.016i			0.580927i
$514$		0			0
$515$	−491.135	+	453.857i	−0.953660	+	0.881276i
$516$		0			0
$517$		−	1275.93i		−	2.46795i
$518$		0			0
$519$	−128.723			−0.248022
$520$		0			0
$521$			117.982i			0.226453i	0.993569	+	0.113227i	$0.0361186\pi$
$521$			117.982i			0.226453i	−0.993569	+	0.113227i	$0.963881\pi$
$522$		0			0
$523$			590.248i			1.12858i	0.825576	+	0.564290i	$0.190850\pi$
$523$			590.248i			1.12858i	−0.825576	+	0.564290i	$0.809150\pi$
$524$		0			0
$525$	372.070	+	819.245i	0.708704	+	1.56047i
$526$		0			0
$527$			127.265i			0.241489i
$528$		0			0
$529$	113.950			0.215407
$530$		0			0
$531$	407.053			0.766579
$532$		0			0
$533$	−581.136			−1.09031
$534$		0			0
$535$	−273.076	+	252.349i	−0.510422	+	0.471680i
$536$		0			0
$537$	−955.430			−1.77920
$538$		0			0
$539$	−768.249	−	482.156i	−1.42532	−	0.894539i
$540$		0			0
$541$			31.3915i			0.0580249i	0.999579	+	0.0290125i	$0.00923625\pi$
$541$			31.3915i			0.0580249i	−0.999579	+	0.0290125i	$0.990764\pi$
$542$		0			0
$543$			579.399i			1.06703i
$544$		0			0
$545$	98.1066	+	106.165i	0.180012	+	0.194797i
$546$		0			0
$547$	189.695			0.346791			0.173396	−	0.984852i	$-0.444526\pi$
$547$	189.695			0.346791			0.173396	+	0.984852i	$0.444526\pi$
$548$		0			0
$549$	1141.35			2.07896
$550$		0			0
$551$			15.9319i			0.0289145i
$552$		0			0
$553$	151.966	+	274.790i	0.274803	+	0.496908i
$554$		0			0
$555$	528.462	+	571.867i	0.952183	+	1.03039i
$556$		0			0
$557$	764.805			1.37308			0.686539	−	0.727093i	$-0.259129\pi$
$557$	764.805			1.37308			0.686539	+	0.727093i	$0.259129\pi$
$558$		0			0
$559$			843.095i			1.50822i
$560$		0			0
$561$	−1865.07			−3.32455
$562$		0			0
$563$			340.073i			0.604037i	0.953302	+	0.302018i	$0.0976604\pi$
$563$			340.073i			0.604037i	−0.953302	+	0.302018i	$0.902340\pi$
$564$		0			0
$565$	−483.906	−	523.651i	−0.856470	−	0.926816i
$566$		0			0
$567$	223.878	+	404.824i	0.394847	+	0.713976i
$568$		0			0
$569$	86.0796			0.151282			0.0756411	−	0.997135i	$-0.475900\pi$
$569$	86.0796			0.151282			0.0756411	+	0.997135i	$0.475900\pi$
$570$		0			0
$571$			140.396i			0.245878i	0.992414	+	0.122939i	$0.0392319\pi$
$571$			140.396i			0.245878i	−0.992414	+	0.122939i	$0.960768\pi$
$572$		0			0
$573$			1507.70i			2.63124i
$574$		0			0
$575$	−507.737	−	40.1203i	−0.883020	−	0.0697744i
$576$		0			0
$577$	−905.252			−1.56889			−0.784447	−	0.620195i	$-0.787053\pi$
$577$	−905.252			−1.56889			−0.784447	+	0.620195i	$0.787053\pi$
$578$		0			0
$579$	505.698			0.873399
$580$		0			0
$581$	188.363	−	104.170i	0.324206	−	0.179294i
$582$		0			0
$583$			271.243i			0.465254i
$584$		0			0
$585$	−712.097	−	770.586i	−1.21726	−	1.31724i
$586$		0			0
$587$			263.910i			0.449592i	0.974406	+	0.224796i	$0.0721715\pi$
$587$			263.910i			0.449592i	−0.974406	+	0.224796i	$0.927828\pi$
$588$		0			0
$589$		−	44.6212i		−	0.0757575i
$590$		0			0
$591$			1473.10i			2.49255i
$592$		0			0
$593$	373.359			0.629611			0.314805	−	0.949156i	$-0.398061\pi$
$593$	373.359			0.629611			0.314805	+	0.949156i	$0.398061\pi$
$594$		0			0
$595$	651.133	+	215.534i	1.09434	+	0.362243i
$596$		0			0
$597$	529.459			0.886866
$598$		0			0
$599$	148.536			0.247974			0.123987	−	0.992284i	$-0.460432\pi$
$599$	148.536			0.247974			0.123987	+	0.992284i	$0.460432\pi$
$600$		0			0
$601$			179.246i			0.298246i	0.988819	+	0.149123i	$0.0476451\pi$
$601$			179.246i			0.298246i	−0.988819	+	0.149123i	$0.952355\pi$
$602$		0			0
$603$	−670.345			−1.11168
$604$		0			0
$605$	−813.898	+	752.123i	−1.34529	+	1.24318i
$606$		0			0
$607$	−597.716			−0.984705			−0.492353	−	0.870396i	$-0.663863\pi$
$607$	−597.716			−0.984705			−0.492353	+	0.870396i	$0.663863\pi$
$608$		0			0
$609$	40.3878	+	73.0306i	0.0663182	+	0.119919i
$610$		0			0
$611$		−	829.593i		−	1.35776i
$612$		0			0
$613$	505.433			0.824523			0.412261	−	0.911066i	$-0.364739\pi$
$613$	505.433			0.824523			0.412261	+	0.911066i	$0.364739\pi$
$614$		0			0
$615$	−911.669	+	842.473i	−1.48239	+	1.36987i
$616$		0			0
$617$		−	579.639i		−	0.939448i	−0.882813	−	0.469724i	$-0.844353\pi$
$617$		−	579.639i		−	0.939448i	0.882813	−	0.469724i	$-0.155647\pi$
$618$		0			0
$619$	−638.836			−1.03205			−0.516023	−	0.856575i	$-0.672588\pi$
$619$	−638.836			−1.03205			−0.516023	+	0.856575i	$0.672588\pi$
$620$		0			0
$621$	−883.642			−1.42293
$622$		0			0
$623$	829.161	−	458.547i	1.33092	−	0.736031i
$624$		0			0
$625$	−617.244	−	98.1594i	−0.987590	−	0.157055i
$626$		0			0
$627$	653.927			1.04295
$628$		0			0
$629$	593.550			0.943641
$630$		0			0
$631$	427.036			0.676761			0.338381	−	0.941009i	$-0.390121\pi$
$631$	427.036			0.676761			0.338381	+	0.941009i	$0.390121\pi$
$632$		0			0
$633$	883.561			1.39583
$634$		0			0
$635$	558.844	+	604.745i	0.880069	+	0.952354i
$636$		0			0
$637$	−499.505	−	313.491i	−0.784152	−	0.492137i
$638$		0			0
$639$	1156.22			1.80941
$640$		0			0
$641$	934.945			1.45857			0.729286	−	0.684209i	$-0.239852\pi$
$641$	934.945			1.45857			0.729286	+	0.684209i	$0.239852\pi$
$642$		0			0
$643$		−	386.015i		−	0.600334i	−0.953887	−	0.300167i	$-0.902958\pi$
$643$		−	386.015i		−	0.600334i	0.953887	−	0.300167i	$-0.0970425\pi$
$644$		0			0
$645$	1222.23	+	1322.62i	1.89494	+	2.05058i
$646$		0			0
$647$	130.602			0.201858			0.100929	−	0.994894i	$-0.467819\pi$
$647$	130.602			0.201858			0.100929	+	0.994894i	$0.467819\pi$
$648$		0			0
$649$		−	432.143i		−	0.665860i
$650$		0			0
$651$	−113.116	−	204.540i	−0.173757	−	0.314194i
$652$		0			0
$653$	−291.551			−0.446479			−0.223240	−	0.974764i	$-0.571663\pi$
$653$	−291.551			−0.446479			−0.223240	+	0.974764i	$0.571663\pi$
$654$		0			0
$655$	−2.51286	+	2.32213i	−0.00383643	+	0.00354524i
$656$		0			0
$657$	859.865			1.30878
$658$		0			0
$659$			1166.00i			1.76934i	0.466217	+	0.884670i	$0.345617\pi$
$659$			1166.00i			1.76934i	−0.466217	+	0.884670i	$0.654383\pi$
$660$		0			0
$661$	640.576			0.969101			0.484551	−	0.874763i	$-0.338983\pi$
$661$	640.576			0.969101			0.484551	+	0.874763i	$0.338983\pi$
$662$		0			0
$663$	−1212.64			−1.82903
$664$		0			0
$665$	−228.299	−	75.5701i	−0.343306	−	0.113639i
$666$		0			0
$667$	−47.2394			−0.0708238
$668$		0			0
$669$		−	38.2420i		−	0.0571629i
$670$		0			0
$671$		−	1211.70i		−	1.80581i
$672$		0			0
$673$			188.966i			0.280782i	0.990096	+	0.140391i	$0.0448360\pi$
$673$			188.966i			0.280782i	−0.990096	+	0.140391i	$0.955164\pi$
$674$		0			0
$675$	−1080.97	−	85.4161i	−1.60144	−	0.126542i
$676$		0			0
$677$		−	330.026i		−	0.487483i	−0.969840	−	0.243741i	$-0.921625\pi$
$677$		−	330.026i		−	0.487483i	0.969840	−	0.243741i	$-0.0783748\pi$
$678$		0			0
$679$	124.858	+	225.772i	0.183885	+	0.332507i
$680$		0			0
$681$	390.453			0.573353
$682$		0			0
$683$	728.224			1.06621			0.533107	−	0.846048i	$-0.321024\pi$
$683$	728.224			1.06621			0.533107	+	0.846048i	$0.321024\pi$
$684$		0			0
$685$	−536.455	−	580.516i	−0.783145	−	0.847469i
$686$		0			0
$687$		−	1128.85i		−	1.64316i
$688$		0			0
$689$			176.359i			0.255963i
$690$		0			0
$691$	−1206.78			−1.74643			−0.873216	−	0.487334i	$-0.837970\pi$
$691$	−1206.78			−1.74643			−0.873216	+	0.487334i	$0.837970\pi$
$692$		0			0
$693$	1977.05	−	1093.36i	2.85288	−	1.57772i
$694$		0			0
$695$	48.1153	−	44.4633i	0.0692306	−	0.0639759i
$696$		0			0
$697$			946.237i			1.35758i
$698$		0			0
$699$	−1814.33			−2.59561
$700$		0			0
$701$			390.880i			0.557603i	0.960349	+	0.278801i	$0.0899371\pi$
$701$			390.880i			0.557603i	−0.960349	+	0.278801i	$0.910063\pi$
$702$		0			0
$703$	−208.109			−0.296030
$704$		0			0
$705$	−1202.66	−	1301.44i	−1.70590	−	1.84601i
$706$		0			0
$707$	397.039	+	717.940i	0.561583	+	1.01547i
$708$		0			0
$709$			790.112i			1.11440i	0.830377	+	0.557202i	$0.188125\pi$
$709$			790.112i			1.11440i	−0.830377	+	0.557202i	$0.811875\pi$
$710$		0			0
$711$	−782.153			−1.10007
$712$		0			0
$713$	132.306			0.185562
$714$		0			0
$715$	−818.083	+	755.989i	−1.14417	+	1.05733i
$716$		0			0
$717$			123.061i			0.171633i
$718$		0			0
$719$		−	1349.86i		−	1.87742i	−0.344710	−	0.938709i	$-0.612023\pi$
$719$		−	1349.86i		−	1.87742i	0.344710	−	0.938709i	$-0.387977\pi$
$720$		0			0
$721$	−453.085	−	819.283i	−0.628412	−	1.13632i
$722$		0			0
$723$	−161.763			−0.223739
$724$		0			0
$725$	−57.7887	−	4.56634i	−0.0797086	−	0.00629840i
$726$		0			0
$727$	−543.377			−0.747423			−0.373712	−	0.927545i	$-0.621915\pi$
$727$	−543.377			−0.747423			−0.373712	+	0.927545i	$0.621915\pi$
$728$		0			0
$729$	854.808			1.17258
$730$		0			0
$731$	1372.77			1.87794
$732$		0			0
$733$			772.693i			1.05415i	0.849818	+	0.527076i	$0.176712\pi$
$733$			772.693i			1.05415i	−0.849818	+	0.527076i	$0.823288\pi$
$734$		0			0
$735$	−1238.08	+	232.336i	−1.68446	+	0.316103i
$736$		0			0
$737$			711.664i			0.965623i
$738$		0			0
$739$		−	1333.98i		−	1.80511i	−0.430570	−	0.902557i	$-0.641687\pi$
$739$		−	1333.98i		−	1.80511i	0.430570	−	0.902557i	$-0.358313\pi$
$740$		0			0
$741$	425.174			0.573784
$742$		0			0
$743$			1224.39i			1.64790i	0.566666	+	0.823948i	$0.308233\pi$
$743$			1224.39i			1.64790i	−0.566666	+	0.823948i	$0.691767\pi$
$744$		0			0
$745$	56.6727	+	61.3276i	0.0760708	+	0.0823189i
$746$		0			0
$747$			536.150i			0.717738i
$748$		0			0
$749$	−251.920	−	455.530i	−0.336341	−	0.608184i
$750$		0			0
$751$	−1436.98			−1.91342			−0.956711	−	0.291039i	$-0.905999\pi$
$751$	−1436.98			−1.91342			−0.956711	+	0.291039i	$0.905999\pi$
$752$		0			0
$753$		−	2210.04i		−	2.93498i
$754$		0			0
$755$	−398.079	+	367.865i	−0.527257	+	0.487238i
$756$		0			0
$757$	−966.377			−1.27659			−0.638294	−	0.769792i	$-0.720360\pi$
$757$	−966.377			−1.27659			−0.638294	+	0.769792i	$0.720360\pi$
$758$		0			0
$759$			1938.95i			2.55461i
$760$		0			0
$761$			892.095i			1.17227i	0.810215	+	0.586133i	$0.199351\pi$
$761$			892.095i			1.17227i	−0.810215	+	0.586133i	$0.800649\pi$
$762$		0			0
$763$	−177.098	+	97.9396i	−0.232107	+	0.128361i
$764$		0			0
$765$	−1254.71	+	1159.47i	−1.64014	+	1.51565i
$766$		0			0
$767$		−	280.974i		−	0.366328i
$768$		0			0
$769$			586.031i			0.762069i	0.924561	+	0.381035i	$0.124432\pi$
$769$			586.031i			0.762069i	−0.924561	+	0.381035i	$0.875568\pi$
$770$		0			0
$771$			931.030i			1.20756i
$772$		0			0
$773$		−	1458.07i		−	1.88625i	−0.332434	−	0.943127i	$-0.607870\pi$
$773$		−	1458.07i		−	1.88625i	0.332434	−	0.943127i	$-0.392130\pi$
$774$		0			0
$775$	161.851	+	12.7892i	0.208841	+	0.0165021i
$776$		0			0
$777$	−953.956	+	527.562i	−1.22774	+	0.678973i
$778$		0			0
$779$		−	331.767i		−	0.425888i
$780$		0			0
$781$		−	1227.48i		−	1.57168i
$782$		0			0
$783$	−100.573			−0.128446
$784$		0			0
$785$	206.805	+	223.791i	0.263446	+	0.285084i
$786$		0			0
$787$		−	966.035i		−	1.22749i	−0.789504	−	0.613746i	$-0.789662\pi$
$787$		−	966.035i		−	1.22749i	0.789504	−	0.613746i	$-0.210338\pi$
$788$		0			0
$789$	1271.79			1.61190
$790$		0			0
$791$	873.526	−	483.082i	1.10433	−	0.610723i
$792$		0			0
$793$		−	787.831i		−	0.993482i
$794$		0			0
$795$	255.667	+	276.666i	0.321593	+	0.348007i
$796$		0			0
$797$			1314.28i			1.64903i	0.565840	+	0.824515i	$0.308552\pi$
$797$			1314.28i			1.64903i	−0.565840	+	0.824515i	$0.691448\pi$
$798$		0			0
$799$	−1350.79			−1.69060
$800$		0			0
$801$			2360.09i			2.94643i
$802$		0			0
$803$		−	912.866i		−	1.13682i
$804$		0			0
$805$	224.072	−	676.925i	0.278350	−	0.840901i
$806$		0			0
$807$		−	1619.03i		−	2.00623i
$808$		0			0
$809$	203.836			0.251960			0.125980	−	0.992033i	$-0.459792\pi$
$809$	203.836			0.251960			0.125980	+	0.992033i	$0.459792\pi$
$810$		0			0
$811$	221.257			0.272820			0.136410	−	0.990652i	$-0.456444\pi$
$811$	221.257			0.272820			0.136410	+	0.990652i	$0.456444\pi$
$812$		0			0
$813$	−1414.98			−1.74044
$814$		0			0
$815$	871.160	−	805.038i	1.06891	−	0.987777i
$816$		0			0
$817$	−481.318			−0.589128
$818$		0			0
$819$	1285.45	−	710.886i	1.56953	−	0.867992i
$820$		0			0
$821$			43.8853i			0.0534535i	0.999643	+	0.0267268i	$0.00850841\pi$
$821$			43.8853i			0.0534535i	−0.999643	+	0.0267268i	$0.991492\pi$
$822$		0			0
$823$			628.726i			0.763944i	0.924174	+	0.381972i	$0.124755\pi$
$823$			628.726i			0.763944i	−0.924174	+	0.381972i	$0.875245\pi$
$824$		0			0
$825$	−187.426	+	2371.95i	−0.227183	+	2.87509i
$826$		0			0
$827$	−739.219			−0.893856			−0.446928	−	0.894570i	$-0.647482\pi$
$827$	−739.219			−0.893856			−0.446928	+	0.894570i	$0.647482\pi$
$828$		0			0
$829$	−269.289			−0.324836			−0.162418	−	0.986722i	$-0.551929\pi$
$829$	−269.289			−0.324836			−0.162418	+	0.986722i	$0.551929\pi$
$830$		0			0
$831$			2022.41i			2.43370i
$832$		0			0
$833$	−510.443	+	813.320i	−0.612777	+	0.976375i
$834$		0			0
$835$	403.222	−	372.617i	0.482900	−	0.446248i
$836$		0			0
$837$	281.679			0.336534
$838$		0			0
$839$			89.6345i			0.106835i	0.998572	+	0.0534174i	$0.0170114\pi$
$839$			89.6345i			0.106835i	−0.998572	+	0.0534174i	$0.982989\pi$
$840$		0			0
$841$	835.623			0.993607
$842$		0			0
$843$			850.498i			1.00889i
$844$		0			0
$845$	88.6866	−	81.9552i	0.104955	−	0.0969884i
$846$		0			0
$847$	−750.843	−	1357.70i	−0.886473	−	1.60295i
$848$		0			0
$849$	−2579.35			−3.03810
$850$		0			0
$851$		−	617.062i		−	0.725102i
$852$		0			0
$853$			601.901i			0.705628i	0.935693	+	0.352814i	$0.114775\pi$
$853$			601.901i			0.705628i	−0.935693	+	0.352814i	$0.885225\pi$
$854$		0			0
$855$	439.922	−	406.532i	0.514529	−	0.475476i
$856$		0			0
$857$	134.851			0.157352			0.0786759	−	0.996900i	$-0.474931\pi$
$857$	134.851			0.157352			0.0786759	+	0.996900i	$0.474931\pi$
$858$		0			0
$859$	849.790			0.989278			0.494639	−	0.869099i	$-0.335300\pi$
$859$	849.790			0.989278			0.494639	+	0.869099i	$0.335300\pi$
$860$		0			0
$861$	−841.039	−	1520.80i	−0.976816	−	1.76631i
$862$		0			0
$863$		−	612.428i		−	0.709649i	−0.934933	−	0.354825i	$-0.884541\pi$
$863$		−	612.428i		−	0.709649i	0.934933	−	0.354825i	$-0.115459\pi$
$864$		0			0
$865$	84.9571	+	91.9350i	0.0982163	+	0.106283i
$866$		0			0
$867$			488.574i			0.563522i
$868$		0			0
$869$			830.363i			0.955538i
$870$		0			0
$871$			462.714i			0.531245i
$872$		0			0
$873$	−642.629			−0.736116
$874$		0			0
$875$	339.545	−	806.433i	0.388051	−	0.921638i
$876$		0			0
$877$	552.461			0.629944			0.314972	−	0.949101i	$-0.398005\pi$
$877$	552.461			0.629944			0.314972	+	0.949101i	$0.398005\pi$
$878$		0			0
$879$	−2836.37			−3.22681
$880$		0			0
$881$		−	1608.54i		−	1.82582i	−0.408166	−	0.912908i	$-0.633832\pi$
$881$		−	1608.54i		−	1.82582i	0.408166	−	0.912908i	$-0.366168\pi$
$882$		0			0
$883$	−1245.56			−1.41060			−0.705298	−	0.708911i	$-0.749187\pi$
$883$	−1245.56			−1.41060			−0.705298	+	0.708911i	$0.749187\pi$
$884$		0			0
$885$	−407.327	−	440.783i	−0.460257	−	0.498060i
$886$		0			0
$887$	−795.632			−0.896992			−0.448496	−	0.893785i	$-0.648040\pi$
$887$	−795.632			−0.896992			−0.448496	+	0.893785i	$0.648040\pi$
$888$		0			0
$889$	−1008.80	+	557.893i	−1.13476	+	0.627551i
$890$		0			0
$891$			1223.30i			1.37295i
$892$		0			0
$893$	473.609			0.530357
$894$		0			0
$895$	630.580	+	682.373i	0.704559	+	0.762428i
$896$		0			0
$897$			1260.68i			1.40544i
$898$		0			0
$899$	15.0585			0.0167503
$900$		0			0
$901$	287.156			0.318708
$902$		0			0
$903$	−2206.32	+	1220.15i	−2.44333	+	1.35122i
$904$		0			0
$905$	413.810	−	382.401i	0.457248	−	0.422543i
$906$		0			0
$907$	−992.438			−1.09420			−0.547099	−	0.837068i	$-0.684268\pi$
$907$	−992.438			−1.09420			−0.547099	+	0.837068i	$0.684268\pi$
$908$		0			0
$909$	−2043.52			−2.24809
$910$		0			0
$911$	97.6148			0.107151			0.0535756	−	0.998564i	$-0.482938\pi$
$911$	97.6148			0.107151			0.0535756	+	0.998564i	$0.482938\pi$
$912$		0			0
$913$	569.197			0.623436
$914$		0			0
$915$	−1142.12	−	1235.93i	−1.24822	−	1.35074i
$916$		0			0
$917$	−2.31818	−	4.19181i	−0.00252800	−	0.00457122i
$918$		0			0
$919$	1321.08			1.43751			0.718757	−	0.695261i	$-0.244711\pi$
$919$	1321.08			1.43751			0.718757	+	0.695261i	$0.244711\pi$
$920$		0			0
$921$	−736.842			−0.800046
$922$		0			0
$923$		−	798.092i		−	0.864672i
$924$		0			0
$925$	59.6475	−	754.860i	0.0644837	−	0.816065i
$926$		0			0
$927$	2331.98			2.51562
$928$		0			0
$929$			443.715i			0.477627i	0.971065	+	0.238813i	$0.0767584\pi$
$929$			443.715i			0.477627i	−0.971065	+	0.238813i	$0.923242\pi$
$930$		0			0
$931$	178.970	−	285.164i	0.192234	−	0.306299i
$932$		0			0
$933$	−1052.30			−1.12787
$934$		0			0
$935$	1230.94	+	1332.04i	1.31652	+	1.42465i
$936$		0			0
$937$	−980.575			−1.04650			−0.523252	−	0.852178i	$-0.675282\pi$
$937$	−980.575			−1.04650			−0.523252	+	0.852178i	$0.675282\pi$
$938$		0			0
$939$		−	1096.17i		−	1.16738i
$940$		0			0
$941$	1308.56			1.39061			0.695305	−	0.718715i	$-0.255269\pi$
$941$	1308.56			1.39061			0.695305	+	0.718715i	$0.255269\pi$
$942$		0			0
$943$	983.718			1.04318
$944$		0			0
$945$	477.050	−	1441.18i	0.504814	−	1.52505i
$946$		0			0
$947$	39.6674			0.0418874			0.0209437	−	0.999781i	$-0.493333\pi$
$947$	39.6674			0.0418874			0.0209437	+	0.999781i	$0.493333\pi$
$948$		0			0
$949$		−	593.533i		−	0.625429i
$950$		0			0
$951$		−	871.545i		−	0.916451i
$952$		0			0
$953$		−	844.003i		−	0.885627i	−0.896614	−	0.442813i	$-0.853980\pi$
$953$		−	844.003i		−	0.885627i	0.896614	−	0.442813i	$-0.146020\pi$
$954$		0			0
$955$	1076.81	−	995.078i	1.12755	−	1.04197i
$956$		0			0
$957$			220.684i			0.230600i
$958$		0			0
$959$	968.385	−	535.542i	1.00979	−	0.558438i
$960$		0			0
$961$	918.825			0.956113
$962$		0			0
$963$	1296.60			1.34642
$964$		0			0
$965$	−333.759	−	361.172i	−0.345864	−	0.374272i
$966$		0			0
$967$		−	45.5145i		−	0.0470678i	−0.999723	−	0.0235339i	$-0.992508\pi$
$967$		−	45.5145i		−	0.0470678i	0.999723	−	0.0235339i	$-0.00749176\pi$
$968$		0			0
$969$		−	692.291i		−	0.714439i
$970$		0			0
$971$	−1334.58			−1.37444			−0.687219	−	0.726450i	$-0.741169\pi$
$971$	−1334.58			−1.37444			−0.687219	+	0.726450i	$0.741169\pi$
$972$		0			0
$973$	44.3876	+	80.2632i	0.0456193	+	0.0824904i
$974$		0			0
$975$	−121.862	+	1542.21i	−0.124987	+	1.58175i
$976$		0			0
$977$		−	1759.33i		−	1.80075i	−0.435120	−	0.900373i	$-0.643294\pi$
$977$		−	1759.33i		−	1.80075i	0.435120	−	0.900373i	$-0.356706\pi$
$978$		0			0
$979$	2505.56			2.55931
$980$		0			0
$981$		−	504.084i		−	0.513847i
$982$		0			0
$983$	1585.34			1.61276			0.806379	−	0.591399i	$-0.201424\pi$
$983$	1585.34			1.61276			0.806379	+	0.591399i	$0.201424\pi$
$984$		0			0
$985$	1052.09	−	972.240i	1.06812	−	0.987046i
$986$		0			0
$987$	2170.99	−	1200.61i	2.19958	−	1.21643i
$988$		0			0
$989$		−	1427.15i		−	1.44302i
$990$		0			0
$991$	891.378			0.899473			0.449737	−	0.893161i	$-0.351518\pi$
$991$	891.378			0.899473			0.449737	+	0.893161i	$0.351518\pi$
$992$		0			0
$993$	−271.224			−0.273136
$994$		0			0
$995$	−349.441	−	378.143i	−0.351197	−	0.380043i
$996$		0			0
$997$			1014.33i			1.01738i	0.860950	+	0.508690i	$0.169870\pi$
$997$			1014.33i			1.01738i	−0.860950	+	0.508690i	$0.830130\pi$
$998$		0			0
$999$		−	1313.73i		−	1.31504i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1120.3.c.g.209.22		80
4.3	odd	2		280.3.c.g.69.3	yes	80
5.4	even	2	inner	1120.3.c.g.209.69		80
7.6	odd	2	inner	1120.3.c.g.209.55		80
8.3	odd	2		280.3.c.g.69.80	yes	80
8.5	even	2	inner	1120.3.c.g.209.57		80
20.19	odd	2		280.3.c.g.69.78	yes	80
28.27	even	2		280.3.c.g.69.4	yes	80
35.34	odd	2	inner	1120.3.c.g.209.58		80
40.19	odd	2		280.3.c.g.69.1	✓	80
40.29	even	2	inner	1120.3.c.g.209.56		80
56.13	odd	2	inner	1120.3.c.g.209.70		80
56.27	even	2		280.3.c.g.69.79	yes	80
140.139	even	2		280.3.c.g.69.77	yes	80
280.69	odd	2	inner	1120.3.c.g.209.21		80
280.139	even	2		280.3.c.g.69.2	yes	80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.3.c.g.69.1	✓	80	40.19	odd	2
280.3.c.g.69.2	yes	80	280.139	even	2
280.3.c.g.69.3	yes	80	4.3	odd	2
280.3.c.g.69.4	yes	80	28.27	even	2
280.3.c.g.69.77	yes	80	140.139	even	2
280.3.c.g.69.78	yes	80	20.19	odd	2
280.3.c.g.69.79	yes	80	56.27	even	2
280.3.c.g.69.80	yes	80	8.3	odd	2
1120.3.c.g.209.21		80	280.69	odd	2	inner
1120.3.c.g.209.22		80	1.1	even	1	trivial
1120.3.c.g.209.55		80	7.6	odd	2	inner
1120.3.c.g.209.56		80	40.29	even	2	inner
1120.3.c.g.209.57		80	8.5	even	2	inner
1120.3.c.g.209.58		80	35.34	odd	2	inner
1120.3.c.g.209.69		80	5.4	even	2	inner
1120.3.c.g.209.70		80	56.13	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)

\(f(q)\)	\(=\)	\(q+5.14158i q^{3} +(3.67215 - 3.39343i) q^{5} +(3.38765 + 6.12567i) q^{7} -17.4359 q^{9} +O(q^{10})\)	\(q+5.14158i q^{3} +(3.67215 - 3.39343i) q^{5} +(3.38765 + 6.12567i) q^{7} -17.4359 q^{9} +18.5106i q^{11} +12.0353i q^{13} +(17.4476 + 18.8806i) q^{15} +19.5965 q^{17} -6.87089 q^{19} +(-31.4956 + 17.4179i) q^{21} -20.3728i q^{23} +(1.96931 - 24.9223i) q^{25} -43.3737i q^{27} -2.31875i q^{29} +6.49424i q^{31} -95.1736 q^{33} +(33.2270 + 10.9986i) q^{35} +30.2885 q^{37} -61.8806 q^{39} +48.2859i q^{41} +70.0518 q^{43} +(-64.0270 + 59.1673i) q^{45} -68.9298 q^{47} +(-26.0476 + 41.5033i) q^{49} +100.757i q^{51} +14.6534 q^{53} +(62.8143 + 67.9735i) q^{55} -35.3272i q^{57} -23.3458 q^{59} -65.4599 q^{61} +(-59.0666 - 106.806i) q^{63} +(40.8410 + 44.1954i) q^{65} +38.4464 q^{67} +104.748 q^{69} -66.3125 q^{71} -49.3159 q^{73} +(128.140 + 10.1254i) q^{75} +(-113.390 + 62.7074i) q^{77} +44.8588 q^{79} +66.0865 q^{81} -30.7499i q^{83} +(71.9613 - 66.4994i) q^{85} +11.9221 q^{87} -135.358i q^{89} +(-73.7244 + 40.7715i) q^{91} -33.3907 q^{93} +(-25.2309 + 23.3158i) q^{95} +36.8568 q^{97} -322.748i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 224 q^{9}+O(q^{10}) \) 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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