LMFDB - Embedded newform 1120.3.c.g.209.70

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			209.70
Character	$\chi$	$=$	1120.209
Dual form			1120.3.c.g.209.69

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.14158i q^{3} +(3.67215 - 3.39343i) q^{5} +(-3.38765 + 6.12567i) q^{7} -17.4359 q^{9} +O(q^{10})$	$q+5.14158i q^{3} +(3.67215 - 3.39343i) q^{5} +(-3.38765 + 6.12567i) q^{7} -17.4359 q^{9} -18.5106i q^{11} +12.0353i q^{13} +(17.4476 + 18.8806i) q^{15} -19.5965 q^{17} -6.87089 q^{19} +(-31.4956 - 17.4179i) q^{21} -20.3728i q^{23} +(1.96931 - 24.9223i) q^{25} -43.3737i q^{27} +2.31875i q^{29} -6.49424i q^{31} +95.1736 q^{33} +(8.34706 + 33.9901i) q^{35} -30.2885 q^{37} -61.8806 q^{39} -48.2859i q^{41} -70.0518 q^{43} +(-64.0270 + 59.1673i) q^{45} +68.9298 q^{47} +(-26.0476 - 41.5033i) q^{49} -100.757i q^{51} -14.6534 q^{53} +(-62.8143 - 67.9735i) q^{55} -35.3272i q^{57} -23.3458 q^{59} -65.4599 q^{61} +(59.0666 - 106.806i) q^{63} +(40.8410 + 44.1954i) q^{65} -38.4464 q^{67} +104.748 q^{69} -66.3125 q^{71} +49.3159 q^{73} +(128.140 + 10.1254i) q^{75} +(113.390 + 62.7074i) q^{77} +44.8588 q^{79} +66.0865 q^{81} -30.7499i q^{83} +(-71.9613 + 66.4994i) q^{85} -11.9221 q^{87} +135.358i q^{89} +(-73.7244 - 40.7715i) q^{91} +33.3907 q^{93} +(-25.2309 + 23.3158i) q^{95} -36.8568 q^{97} +322.748i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.681738\pi$
$11$		−	18.5106i		−	1.68278i	0.540429	−	0.841389i	$-0.318262\pi$
$12$		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	$-0.695531\pi$
$17$	−19.5965			−1.15274			−0.576369	+	0.817190i	$0.695531\pi$
$18$		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	$0.0127289\pi$
$29$			2.31875i			0.0799570i	−0.999201	+	0.0399785i	$0.987271\pi$
$30$		0			0
$31$		−	6.49424i		−	0.209492i	−0.994499	−	0.104746i	$-0.966597\pi$
$31$		−	6.49424i		−	0.209492i	0.994499	−	0.104746i	$-0.0334029\pi$
$32$		0			0
$33$	95.1736			2.88405
$34$		0			0
$35$	8.34706	+	33.9901i	0.238487	+	0.971146i
$36$		0			0
$37$	−30.2885			−0.818609			−0.409305	−	0.912398i	$-0.634229\pi$
$37$	−30.2885			−0.818609			−0.409305	+	0.912398i	$0.634229\pi$
$38$		0			0
$39$	−61.8806			−1.58668
$40$		0			0
$41$		−	48.2859i		−	1.17771i	−0.808240	−	0.588853i	$-0.799580\pi$
$41$		−	48.2859i		−	1.17771i	0.808240	−	0.588853i	$-0.200420\pi$
$42$		0			0
$43$	−70.0518			−1.62911			−0.814556	−	0.580085i	$-0.803019\pi$
$43$	−70.0518			−1.62911			−0.814556	+	0.580085i	$0.803019\pi$
$44$		0			0
$45$	−64.0270	+	59.1673i	−1.42282	+	1.31483i
$46$		0			0
$47$	68.9298			1.46659			0.733296	−	0.679909i	$-0.237981\pi$
$47$	68.9298			1.46659			0.733296	+	0.679909i	$0.237981\pi$
$48$		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	$-0.544144\pi$
$53$	−14.6534			−0.276480			−0.138240	+	0.990399i	$0.544144\pi$
$54$		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	$-0.592629\pi$
$67$	−38.4464			−0.573826			−0.286913	+	0.957957i	$0.592629\pi$
$68$		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	$-0.390324\pi$
$73$	49.3159			0.675560			0.337780	+	0.941225i	$0.390324\pi$
$74$		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	$0.275017\pi$
$89$			135.358i			1.52088i	−0.649408	+	0.760440i	$0.724983\pi$
$90$		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	$-0.560843\pi$
$97$	−36.8568			−0.379967			−0.189983	+	0.981787i	$0.560843\pi$
$98$		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	$-0.275082\pi$
$103$	133.746			1.29850			0.649252	+	0.760573i	$0.275082\pi$
$104$		0			0
$105$	−174.763	+	42.9171i	−1.66441	+	0.408734i
$106$		0			0
$107$	74.3641			0.694991			0.347496	−	0.937682i	$-0.387032\pi$
$107$	74.3641			0.694991			0.347496	+	0.937682i	$0.387032\pi$
$108$		0			0
$109$		−	28.9108i		−	0.265236i	−0.991167	−	0.132618i	$-0.957662\pi$
$109$		−	28.9108i		−	0.265236i	0.991167	−	0.132618i	$-0.0423384\pi$
$110$		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	$-0.982152\pi$
$149$		−	16.7007i		−	0.112086i	0.998428	−	0.0560428i	$-0.0178483\pi$
$150$		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	$-0.759417\pi$
$163$	−237.235			−1.45543			−0.727713	+	0.685881i	$0.759417\pi$
$164$		0			0
$165$	349.491	−	322.965i	2.11813	−	1.95736i
$166$		0			0
$167$	−109.805			−0.657518			−0.328759	−	0.944414i	$-0.606630\pi$
$167$	−109.805			−0.657518			−0.328759	+	0.944414i	$0.606630\pi$
$168$		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
$175$	145.995	+	96.4915i	0.834254	+	0.551380i
$176$		0			0
$177$		−	120.034i		−	0.678159i
$178$		0			0
$179$		−	185.824i		−	1.03812i	−0.854737	−	0.519062i	$-0.826281\pi$
$179$		−	185.824i		−	1.03812i	0.854737	−	0.519062i	$-0.173719\pi$
$180$		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	$-0.759167\pi$
$197$	−286.507			−1.45435			−0.727175	+	0.686452i	$0.759167\pi$
$198$		0			0
$199$			102.976i			0.517467i	0.965949	+	0.258734i	$0.0833052\pi$
$199$			102.976i			0.517467i	−0.965949	+	0.258734i	$0.916695\pi$
$200$		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	$0.133501\pi$
$211$			171.846i			0.814437i	−0.913331	+	0.407218i	$0.866499\pi$
$212$		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	$-0.494691\pi$
$223$	7.43778			0.0333533			0.0166766	+	0.999861i	$0.494691\pi$
$224$		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	$-0.979208\pi$
$241$		−	31.4618i		−	0.130547i	0.997867	−	0.0652734i	$-0.0207919\pi$
$242$		0			0
$243$		−	50.5737i		−	0.208122i
$244$		0			0
$245$	−236.489	−	64.0153i	−0.965261	−	0.261287i
$246$		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	$-0.614598\pi$
$257$	−181.078			−0.704586			−0.352293	+	0.935890i	$0.614598\pi$
$258$		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	$-0.830476\pi$
$271$		−	275.203i		−	1.01551i	0.861502	−	0.507754i	$-0.169524\pi$
$272$		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	$-0.751308\pi$
$277$	−393.344			−1.42001			−0.710006	+	0.704195i	$0.751308\pi$
$278$		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	$-0.893274\pi$
$311$		−	204.664i		−	0.658085i	0.944315	−	0.329043i	$-0.106726\pi$
$312$		0			0
$313$	213.198			0.681143			0.340571	−	0.940219i	$-0.389379\pi$
$313$	213.198			0.681143			0.340571	+	0.940219i	$0.389379\pi$
$314$		0			0
$315$	−145.538	−	592.647i	−0.462026	−	1.88142i
$316$		0			0
$317$	169.509			0.534729			0.267364	−	0.963595i	$-0.413847\pi$
$317$	169.509			0.534729			0.267364	+	0.963595i	$0.413847\pi$
$318$		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	$-0.974609\pi$
$331$		−	52.7511i		−	0.159369i	0.996820	−	0.0796845i	$-0.0253913\pi$
$332$		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	$-0.661275\pi$
$347$	−336.770			−0.970518			−0.485259	+	0.874370i	$0.661275\pi$
$348$		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	$-0.766744\pi$
$353$	−524.774			−1.48661			−0.743307	+	0.668951i	$0.766744\pi$
$354$		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	$-0.662595\pi$
$367$	−358.839			−0.977763			−0.488882	+	0.872350i	$0.662595\pi$
$368$		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	$-0.532653\pi$
$373$	−76.3933			−0.204808			−0.102404	+	0.994743i	$0.532653\pi$
$374$		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	$0.194896\pi$
$379$			435.652i			1.14948i	−0.818337	+	0.574739i	$0.805104\pi$
$380$		0			0
$381$	−846.738			−2.22241
$382$		0			0
$383$	−196.615			−0.513356			−0.256678	−	0.966497i	$-0.582628\pi$
$383$	−196.615			−0.513356			−0.256678	+	0.966497i	$0.582628\pi$
$384$		0			0
$385$	629.176	−	154.509i	1.63422	−	0.401321i
$386$		0			0
$387$	1221.41			3.15611
$388$		0			0
$389$			473.370i			1.21689i	0.793596	+	0.608445i	$0.208206\pi$
$389$			473.370i			1.21689i	−0.793596	+	0.608445i	$0.791794\pi$
$390$		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	$-0.872387\pi$
$409$		−	319.228i		−	0.780508i	0.920707	−	0.390254i	$-0.127613\pi$
$410$		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	$-0.727857\pi$
$421$		−	635.328i		−	1.50909i	0.656247	−	0.754546i	$-0.272143\pi$
$422$		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	$-0.787373\pi$
$433$	−679.871			−1.57014			−0.785071	+	0.619406i	$0.787373\pi$
$434$		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	$0.0951269\pi$
$439$			258.502i			0.588843i	−0.955676	+	0.294421i	$0.904873\pi$
$440$		0			0
$441$	454.163	+	723.645i	1.02985	+	1.64092i
$442$		0			0
$443$	−139.976			−0.315974			−0.157987	−	0.987441i	$-0.550500\pi$
$443$	−139.976			−0.315974			−0.157987	+	0.987441i	$0.550500\pi$
$444$		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
$455$	−409.082	+	100.459i	−0.899080	+	0.220790i
$456$		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	$0.0873746\pi$
$479$			259.677i			0.542122i	−0.962562	+	0.271061i	$0.912625\pi$
$480$		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	$-0.833768\pi$
$491$		−	489.837i		−	0.997631i	0.866708	−	0.498816i	$-0.166232\pi$
$492$		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	$0.146726\pi$
$499$			443.911i			0.889602i	−0.895630	+	0.444801i	$0.853274\pi$
$500$		0			0
$501$		−	564.574i		−	1.12689i
$502$		0			0
$503$	−909.757			−1.80866			−0.904331	−	0.426833i	$-0.859629\pi$
$503$	−909.757			−1.80866			−0.904331	+	0.426833i	$0.859629\pi$
$504$		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	$-0.963881\pi$
$521$		−	117.982i		−	0.226453i	0.993569	−	0.113227i	$-0.0361186\pi$
$522$		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
$525$	−496.119	+	750.643i	−0.944988	+	1.42980i
$526$		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	$-0.990764\pi$
$541$		−	31.3915i		−	0.0580249i	0.999579	−	0.0290125i	$-0.00923625\pi$
$542$		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	$-0.555474\pi$
$547$	−189.695			−0.346791			−0.173396	+	0.984852i	$0.555474\pi$
$548$		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	$-0.740871\pi$
$557$	−764.805			−1.37308			−0.686539	+	0.727093i	$0.740871\pi$
$558$		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	$-0.960768\pi$
$571$		−	140.396i		−	0.245878i	0.992414	−	0.122939i	$-0.0392319\pi$
$572$		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	$-0.212947\pi$
$577$	905.252			1.56889			0.784447	+	0.620195i	$0.212947\pi$
$578$		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	$-0.601939\pi$
$593$	−373.359			−0.629611			−0.314805	+	0.949156i	$0.601939\pi$
$594$		0			0
$595$	−163.573	−	666.088i	−0.274913	−	1.11948i
$596$		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	$-0.952355\pi$
$601$		−	179.246i		−	0.298246i	0.988819	−	0.149123i	$-0.0476451\pi$
$602$		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	$-0.336137\pi$
$607$	597.716			0.984705			0.492353	+	0.870396i	$0.336137\pi$
$608$		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	$-0.635261\pi$
$613$	−505.433			−0.824523			−0.412261	+	0.911066i	$0.635261\pi$
$614$		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	$-0.532181\pi$
$647$	−130.602			−0.201858			−0.100929	+	0.994894i	$0.532181\pi$
$648$		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	$-0.428337\pi$
$653$	291.551			0.446479			0.223240	+	0.974764i	$0.428337\pi$
$654$		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	$-0.654383\pi$
$659$		−	1166.00i		−	1.76934i	0.466217	−	0.884670i	$-0.345617\pi$
$660$		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
$665$	−57.3517	−	233.542i	−0.0862431	−	0.351191i
$666$		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	$-0.678976\pi$
$683$	−728.224			−1.06621			−0.533107	+	0.846048i	$0.678976\pi$
$684$		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	$-0.910063\pi$
$701$		−	390.880i		−	0.557603i	0.960349	−	0.278801i	$-0.0899371\pi$
$702$		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	$-0.811875\pi$
$709$		−	790.112i		−	1.11440i	0.830377	−	0.557202i	$-0.188125\pi$
$710$		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	$0.387977\pi$
$719$			1349.86i			1.87742i	−0.344710	+	0.938709i	$0.612023\pi$
$720$		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	$-0.378085\pi$
$727$	543.377			0.747423			0.373712	+	0.927545i	$0.378085\pi$
$728$		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
$735$	329.140	−	1215.93i	0.447809	−	1.65432i
$736$		0			0
$737$			711.664i			0.965623i
$738$		0			0
$739$			1333.98i			1.80511i	0.430570	+	0.902557i	$0.358313\pi$
$739$			1333.98i			1.80511i	−0.430570	+	0.902557i	$0.641687\pi$
$740$		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	$-0.279640\pi$
$757$	966.377			1.27659			0.638294	+	0.769792i	$0.279640\pi$
$758$		0			0
$759$		−	1938.95i		−	2.55461i
$760$		0			0
$761$		−	892.095i		−	1.17227i	−0.810215	−	0.586133i	$-0.800649\pi$
$761$		−	892.095i		−	1.17227i	0.810215	−	0.586133i	$-0.199351\pi$
$762$		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	$-0.875568\pi$
$769$		−	586.031i		−	0.762069i	0.924561	−	0.381035i	$-0.124432\pi$
$770$		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
$805$	692.473	−	170.053i	0.860214	−	0.211246i
$806$		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	$-0.991492\pi$
$821$		−	43.8853i		−	0.0534535i	0.999643	−	0.0267268i	$-0.00850841\pi$
$822$		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	$-0.352518\pi$
$827$	739.219			0.893856			0.446928	+	0.894570i	$0.352518\pi$
$828$		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	$-0.982989\pi$
$839$		−	89.6345i		−	0.106835i	0.998572	−	0.0534174i	$-0.0170114\pi$
$840$		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	$-0.525069\pi$
$857$	−134.851			−0.157352			−0.0786759	+	0.996900i	$0.525069\pi$
$858$		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
$875$	863.550	−	141.091i	0.986914	−	0.161247i
$876$		0			0
$877$	−552.461			−0.629944			−0.314972	−	0.949101i	$-0.601995\pi$
$877$	−552.461			−0.629944			−0.314972	+	0.949101i	$0.601995\pi$
$878$		0			0
$879$	−2836.37			−3.22681
$880$		0			0
$881$			1608.54i			1.82582i	0.408166	+	0.912908i	$0.366168\pi$
$881$			1608.54i			1.82582i	−0.408166	+	0.912908i	$0.633832\pi$
$882$		0			0
$883$	1245.56			1.41060			0.705298	−	0.708911i	$-0.250813\pi$
$883$	1245.56			1.41060			0.705298	+	0.708911i	$0.250813\pi$
$884$		0			0
$885$	−407.327	−	440.783i	−0.460257	−	0.498060i
$886$		0			0
$887$	795.632			0.896992			0.448496	−	0.893785i	$-0.351960\pi$
$887$	795.632			0.896992			0.448496	+	0.893785i	$0.351960\pi$
$888$		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	$-0.315732\pi$
$907$	992.438			1.09420			0.547099	+	0.837068i	$0.315732\pi$
$908$		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	$-0.923242\pi$
$929$		−	443.715i		−	0.477627i	0.971065	−	0.238813i	$-0.0767584\pi$
$930$		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	$-0.324718\pi$
$937$	980.575			1.04650			0.523252	+	0.852178i	$0.324718\pi$
$938$		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
$945$	1474.28	−	362.043i	1.56008	−	0.383114i
$946$		0			0
$947$	−39.6674			−0.0418874			−0.0209437	−	0.999781i	$-0.506667\pi$
$947$	−39.6674			−0.0418874			−0.0209437	+	0.999781i	$0.506667\pi$
$948$		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	$-0.798576\pi$
$983$	−1585.34			−1.61276			−0.806379	+	0.591399i	$0.798576\pi$
$984$		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.209.70		80
4.3	odd	2		280.3.c.g.69.79	yes	80
5.4	even	2	inner	1120.3.c.g.209.21		80
7.6	odd	2	inner	1120.3.c.g.209.57		80
8.3	odd	2		280.3.c.g.69.4	yes	80
8.5	even	2	inner	1120.3.c.g.209.55		80
20.19	odd	2		280.3.c.g.69.2	yes	80
28.27	even	2		280.3.c.g.69.80	yes	80
35.34	odd	2	inner	1120.3.c.g.209.56		80
40.19	odd	2		280.3.c.g.69.77	yes	80
40.29	even	2	inner	1120.3.c.g.209.58		80
56.13	odd	2	inner	1120.3.c.g.209.22		80
56.27	even	2		280.3.c.g.69.3	yes	80
140.139	even	2		280.3.c.g.69.1	✓	80
280.69	odd	2	inner	1120.3.c.g.209.69		80
280.139	even	2		280.3.c.g.69.78	yes	80

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+5.14158i q^{3} +(3.67215 - 3.39343i) q^{5} +(-3.38765 + 6.12567i) q^{7} -17.4359 q^{9} +O(q^{10})\)	\(q+5.14158i q^{3} +(3.67215 - 3.39343i) q^{5} +(-3.38765 + 6.12567i) q^{7} -17.4359 q^{9} -18.5106i q^{11} +12.0353i q^{13} +(17.4476 + 18.8806i) q^{15} -19.5965 q^{17} -6.87089 q^{19} +(-31.4956 - 17.4179i) q^{21} -20.3728i q^{23} +(1.96931 - 24.9223i) q^{25} -43.3737i q^{27} +2.31875i q^{29} -6.49424i q^{31} +95.1736 q^{33} +(8.34706 + 33.9901i) q^{35} -30.2885 q^{37} -61.8806 q^{39} -48.2859i q^{41} -70.0518 q^{43} +(-64.0270 + 59.1673i) q^{45} +68.9298 q^{47} +(-26.0476 - 41.5033i) q^{49} -100.757i q^{51} -14.6534 q^{53} +(-62.8143 - 67.9735i) q^{55} -35.3272i q^{57} -23.3458 q^{59} -65.4599 q^{61} +(59.0666 - 106.806i) q^{63} +(40.8410 + 44.1954i) q^{65} -38.4464 q^{67} +104.748 q^{69} -66.3125 q^{71} +49.3159 q^{73} +(128.140 + 10.1254i) q^{75} +(113.390 + 62.7074i) q^{77} +44.8588 q^{79} +66.0865 q^{81} -30.7499i q^{83} +(-71.9613 + 66.4994i) q^{85} -11.9221 q^{87} +135.358i q^{89} +(-73.7244 - 40.7715i) q^{91} +33.3907 q^{93} +(-25.2309 + 23.3158i) q^{95} -36.8568 q^{97} +322.748i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 224 q^{9}+O(q^{10}) \) 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

Related objects

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