LMFDB - Embedded newform 2240.2.e.g.2239.28

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(2239,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2240.2239");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.e (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:	$17.8864900528$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	no (minimal twist has level 1120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2239.28
Character	$\chi$	$=$	2240.2239
Dual form			2240.2.e.g.2239.27

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.21376i q^{3} +(1.81856 - 1.30110i) q^{5} +(-1.09155 + 2.41009i) q^{7} -1.90072 q^{9} +O(q^{10})$	\(q+2.21376i q^{3} +(1.81856 - 1.30110i) q^{5} +(-1.09155 + 2.41009i) q^{7} -1.90072 q^{9} +0.661829i q^{11} -0.235163 q^{13} +(2.88031 + 4.02584i) q^{15} -6.02258 q^{17} -7.98868 q^{19} +(-5.33535 - 2.41642i) q^{21} +8.48129 q^{23} +(1.61430 - 4.73223i) q^{25} +2.43354i q^{27} -7.86505 q^{29} -1.36954 q^{31} -1.46513 q^{33} +(1.15072 + 5.80309i) q^{35} +6.09846i q^{37} -0.520594i q^{39} +2.79815i q^{41} -9.54155 q^{43} +(-3.45657 + 2.47302i) q^{45} +7.72404i q^{47} +(-4.61705 - 5.26145i) q^{49} -13.3325i q^{51} -8.45461i q^{53} +(0.861103 + 1.20357i) q^{55} -17.6850i q^{57} +0.929088 q^{59} +2.28972i q^{61} +(2.07472 - 4.58090i) q^{63} +(-0.427658 + 0.305970i) q^{65} -3.19295 q^{67} +18.7755i q^{69} -0.619016i q^{71} +11.9176 q^{73} +(10.4760 + 3.57366i) q^{75} +(-1.59507 - 0.722417i) q^{77} +7.01658i q^{79} -11.0894 q^{81} +7.05444i q^{83} +(-10.9524 + 7.83596i) q^{85} -17.4113i q^{87} +9.97484i q^{89} +(0.256692 - 0.566764i) q^{91} -3.03183i q^{93} +(-14.5279 + 10.3940i) q^{95} -6.02258 q^{97} -1.25795i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 48 q^{9}+O(q^{10}) $ 48 * q - 48 * q^9	$ 48 q - 48 q^{9} - 8 q^{21} - 16 q^{25} - 16 q^{29} + 24 q^{49} - 16 q^{65} - 32 q^{81} - 32 q^{85}+O(q^{100}) $ 48 * q - 48 * q^9 - 8 * q^21 - 16 * q^25 - 16 * q^29 + 24 * q^49 - 16 * q^65 - 32 * q^81 - 32 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times$.

$n$	$897$	$1471$	$1541$	$1921$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$			2.21376i			1.27811i	0.769160	+	0.639057i	$0.220675\pi$
$3$			2.21376i			1.27811i	−0.769160	+	0.639057i	$0.779325\pi$
$4$		0			0
$5$	1.81856	−	1.30110i	0.813283	−	0.581868i
$5$	1.81856	−	1.30110i	0.813283	−	0.581868i
$6$		0			0
$7$	−1.09155	+	2.41009i	−0.412566	+	0.910928i
$7$	−1.09155	+	2.41009i	−0.412566	+	0.910928i
$8$		0			0
$9$	−1.90072			−0.633573
$10$		0			0
$11$			0.661829i			0.199549i	0.995010	+	0.0997745i	$0.0318121\pi$
$11$			0.661829i			0.199549i	−0.995010	+	0.0997745i	$0.968188\pi$
$12$		0			0
$13$	−0.235163			−0.0652226			−0.0326113	−	0.999468i	$-0.510382\pi$
$13$	−0.235163			−0.0652226			−0.0326113	+	0.999468i	$0.510382\pi$
$14$		0			0
$15$	2.88031	+	4.02584i	0.743693	+	1.03947i
$16$		0			0
$17$	−6.02258			−1.46069			−0.730345	−	0.683078i	$-0.760641\pi$
$17$	−6.02258			−1.46069			−0.730345	+	0.683078i	$0.760641\pi$
$18$		0			0
$19$	−7.98868			−1.83273			−0.916365	−	0.400344i	$-0.868891\pi$
$19$	−7.98868			−1.83273			−0.916365	+	0.400344i	$0.868891\pi$
$20$		0			0
$21$	−5.33535	−	2.41642i	−1.16427	−	0.527306i
$22$		0			0
$23$	8.48129			1.76847			0.884236	−	0.467041i	$-0.154680\pi$
$23$	8.48129			1.76847			0.884236	+	0.467041i	$0.154680\pi$
$24$		0			0
$25$	1.61430	−	4.73223i	0.322859	−	0.946447i
$26$		0			0
$27$			2.43354i			0.468335i
$28$		0			0
$29$	−7.86505			−1.46050			−0.730251	−	0.683179i	$-0.760597\pi$
$29$	−7.86505			−1.46050			−0.730251	+	0.683179i	$0.760597\pi$
$30$		0			0
$31$	−1.36954			−0.245977			−0.122988	−	0.992408i	$-0.539248\pi$
$31$	−1.36954			−0.245977			−0.122988	+	0.992408i	$0.539248\pi$
$32$		0			0
$33$	−1.46513			−0.255046
$34$		0			0
$35$	1.15072	+	5.80309i	0.194507	+	0.980901i
$36$		0			0
$37$			6.09846i			1.00258i	0.865279	+	0.501291i	$0.167141\pi$
$37$			6.09846i			1.00258i	−0.865279	+	0.501291i	$0.832859\pi$
$38$		0			0
$39$		−	0.520594i		−	0.0833618i
$40$		0			0
$41$			2.79815i			0.436998i	0.975837	+	0.218499i	$0.0701161\pi$
$41$			2.79815i			0.436998i	−0.975837	+	0.218499i	$0.929884\pi$
$42$		0			0
$43$	−9.54155			−1.45507			−0.727536	−	0.686069i	$-0.759335\pi$
$43$	−9.54155			−1.45507			−0.727536	+	0.686069i	$0.759335\pi$
$44$		0			0
$45$	−3.45657	+	2.47302i	−0.515274	+	0.368656i
$46$		0			0
$47$			7.72404i			1.12667i	0.826229	+	0.563334i	$0.190481\pi$
$47$			7.72404i			1.12667i	−0.826229	+	0.563334i	$0.809519\pi$
$48$		0			0
$49$	−4.61705	−	5.26145i	−0.659579	−	0.751635i
$50$		0			0
$51$		−	13.3325i		−	1.86693i
$52$		0			0
$53$		−	8.45461i		−	1.16133i	−0.814143	−	0.580665i	$-0.802793\pi$
$53$		−	8.45461i		−	1.16133i	0.814143	−	0.580665i	$-0.197207\pi$
$54$		0			0
$55$	0.861103	+	1.20357i	0.116111	+	0.162290i
$56$		0			0
$57$		−	17.6850i		−	2.34244i
$58$		0			0
$59$	0.929088			0.120957			0.0604785	−	0.998170i	$-0.480737\pi$
$59$	0.929088			0.120957			0.0604785	+	0.998170i	$0.480737\pi$
$60$		0			0
$61$			2.28972i			0.293168i	0.989198	+	0.146584i	$0.0468279\pi$
$61$			2.28972i			0.293168i	−0.989198	+	0.146584i	$0.953172\pi$
$62$		0			0
$63$	2.07472	−	4.58090i	0.261391	−	0.577139i
$64$		0			0
$65$	−0.427658	+	0.305970i	−0.0530444	+	0.0379509i
$66$		0			0
$67$	−3.19295			−0.390082			−0.195041	−	0.980795i	$-0.562484\pi$
$67$	−3.19295			−0.390082			−0.195041	+	0.980795i	$0.562484\pi$
$68$		0			0
$69$			18.7755i			2.26031i
$70$		0			0
$71$		−	0.619016i		−	0.0734637i	−0.999325	−	0.0367318i	$-0.988305\pi$
$71$		−	0.619016i		−	0.0734637i	0.999325	−	0.0367318i	$-0.0116947\pi$
$72$		0			0
$73$	11.9176			1.39485			0.697423	−	0.716660i	$-0.254330\pi$
$73$	11.9176			1.39485			0.697423	+	0.716660i	$0.254330\pi$
$74$		0			0
$75$	10.4760	+	3.57366i	1.20967	+	0.412651i
$76$		0			0
$77$	−1.59507	−	0.722417i	−0.181775	−	0.0823271i
$78$		0			0
$79$			7.01658i			0.789427i	0.918804	+	0.394713i	$0.129156\pi$
$79$			7.01658i			0.789427i	−0.918804	+	0.394713i	$0.870844\pi$
$80$		0			0
$81$	−11.0894			−1.23216
$82$		0			0
$83$			7.05444i			0.774326i	0.922011	+	0.387163i	$0.126545\pi$
$83$			7.05444i			0.774326i	−0.922011	+	0.387163i	$0.873455\pi$
$84$		0			0
$85$	−10.9524	+	7.83596i	−1.18796	+	0.849929i
$86$		0			0
$87$		−	17.4113i		−	1.86669i
$88$		0			0
$89$			9.97484i			1.05733i	0.848830	+	0.528665i	$0.177307\pi$
$89$			9.97484i			1.05733i	−0.848830	+	0.528665i	$0.822693\pi$
$90$		0			0
$91$	0.256692	−	0.566764i	0.0269086	−	0.0594130i
$92$		0			0
$93$		−	3.03183i		−	0.314386i
$94$		0			0
$95$	−14.5279	+	10.3940i	−1.49053	+	1.06641i
$96$		0			0
$97$	−6.02258			−0.611500			−0.305750	−	0.952112i	$-0.598907\pi$
$97$	−6.02258			−0.611500			−0.305750	+	0.952112i	$0.598907\pi$
$98$		0			0
$99$		−	1.25795i		−	0.126429i
$100$		0			0
$101$			10.4833i			1.04312i	0.853213	+	0.521562i	$0.174651\pi$
$101$			10.4833i			1.04312i	−0.853213	+	0.521562i	$0.825349\pi$
$102$		0			0
$103$		−	4.79144i		−	0.472115i	−0.971739	−	0.236058i	$-0.924145\pi$
$103$		−	4.79144i		−	0.472115i	0.971739	−	0.236058i	$-0.0758554\pi$
$104$		0			0
$105$	−12.8466	+	2.54741i	−1.25370	+	0.248602i
$106$		0			0
$107$	3.19295			0.308675			0.154337	−	0.988018i	$-0.450676\pi$
$107$	3.19295			0.308675			0.154337	+	0.988018i	$0.450676\pi$
$108$		0			0
$109$	3.36906			0.322698			0.161349	−	0.986897i	$-0.448416\pi$
$109$	3.36906			0.322698			0.161349	+	0.986897i	$0.448416\pi$
$110$		0			0
$111$	−13.5005			−1.28141
$112$		0			0
$113$		−	10.7455i		−	1.01085i	−0.862871	−	0.505424i	$-0.831336\pi$
$113$		−	10.7455i		−	1.01085i	0.862871	−	0.505424i	$-0.168664\pi$
$114$		0			0
$115$	15.4237	−	11.0350i	1.43827	−	1.02902i
$116$		0			0
$117$	0.446979			0.0413233
$118$		0			0
$119$	6.57392	−	14.5150i	0.602631	−	1.33058i
$120$		0			0
$121$	10.5620			0.960180
$122$		0			0
$123$	−6.19443			−0.558533
$124$		0			0
$125$	−3.22140	−	10.7062i	−0.288131	−	0.957591i
$126$		0			0
$127$	7.94371			0.704891			0.352445	−	0.935832i	$-0.385350\pi$
$127$	7.94371			0.704891			0.352445	+	0.935832i	$0.385350\pi$
$128$		0			0
$129$		−	21.1227i		−	1.85975i
$130$		0			0
$131$	−14.8215			−1.29496			−0.647481	−	0.762081i	$-0.724178\pi$
$131$	−14.8215			−1.29496			−0.647481	+	0.762081i	$0.724178\pi$
$132$		0			0
$133$	8.72002	−	19.2534i	0.756121	−	1.66948i
$134$		0			0
$135$	3.16627	+	4.42553i	0.272509	+	0.380889i
$136$		0			0
$137$			18.9726i			1.62094i	0.585780	+	0.810470i	$0.300788\pi$
$137$			18.9726i			1.62094i	−0.585780	+	0.810470i	$0.699212\pi$
$138$		0			0
$139$	−9.47968			−0.804056			−0.402028	−	0.915627i	$-0.631695\pi$
$139$	−9.47968			−0.804056			−0.402028	+	0.915627i	$0.631695\pi$
$140$		0			0
$141$	−17.0992			−1.44001
$142$		0			0
$143$		−	0.155638i		−	0.0130151i
$144$		0			0
$145$	−14.3030	+	10.2332i	−1.18780	+	0.849820i
$146$		0			0
$147$	11.6476	−	10.2210i	0.960675	−	0.843017i
$148$		0			0
$149$	13.0440			1.06860			0.534302	−	0.845294i	$-0.320574\pi$
$149$	13.0440			1.06860			0.534302	+	0.845294i	$0.320574\pi$
$150$		0			0
$151$		−	15.0683i		−	1.22624i	−0.789991	−	0.613119i	$-0.789915\pi$
$151$		−	15.0683i		−	1.22624i	0.789991	−	0.613119i	$-0.210085\pi$
$152$		0			0
$153$	11.4472			0.925454
$154$		0			0
$155$	−2.49058	+	1.78190i	−0.200049	+	0.143126i
$156$		0			0
$157$	−12.5780			−1.00384			−0.501918	−	0.864915i	$-0.667372\pi$
$157$	−12.5780			−1.00384			−0.501918	+	0.864915i	$0.667372\pi$
$158$		0			0
$159$	18.7164			1.48431
$160$		0			0
$161$	−9.25772	+	20.4407i	−0.729611	+	1.61095i
$162$		0			0
$163$	−1.22660			−0.0960748			−0.0480374	−	0.998846i	$-0.515297\pi$
$163$	−1.22660			−0.0960748			−0.0480374	+	0.998846i	$0.515297\pi$
$164$		0			0
$165$	−2.66442	+	1.90627i	−0.207425	+	0.148403i
$166$		0			0
$167$			4.41726i			0.341818i	0.985287	+	0.170909i	$0.0546704\pi$
$167$			4.41726i			0.341818i	−0.985287	+	0.170909i	$0.945330\pi$
$168$		0			0
$169$	−12.9447			−0.995746
$170$		0			0
$171$	15.1842			1.16117
$172$		0			0
$173$	7.47071			0.567988			0.283994	−	0.958826i	$-0.408340\pi$
$173$	7.47071			0.567988			0.283994	+	0.958826i	$0.408340\pi$
$174$		0			0
$175$	9.64303	+	9.05605i	0.728944	+	0.684573i
$176$		0			0
$177$			2.05677i			0.154597i
$178$		0			0
$179$			12.5662i			0.939242i	0.882868	+	0.469621i	$0.155609\pi$
$179$			12.5662i			0.939242i	−0.882868	+	0.469621i	$0.844391\pi$
$180$		0			0
$181$		−	18.7214i		−	1.39155i	−0.718260	−	0.695775i	$-0.755061\pi$
$181$		−	18.7214i		−	1.39155i	0.718260	−	0.695775i	$-0.244939\pi$
$182$		0			0
$183$	−5.06888			−0.374702
$184$		0			0
$185$	7.93469	+	11.0904i	0.583370	+	0.815383i
$186$		0			0
$187$		−	3.98592i		−	0.291479i
$188$		0			0
$189$	−5.86505	−	2.65632i	−0.426619	−	0.193219i
$190$		0			0
$191$		−	16.1501i		−	1.16858i	−0.811545	−	0.584290i	$-0.801373\pi$
$191$		−	16.1501i		−	1.16858i	0.811545	−	0.584290i	$-0.198627\pi$
$192$		0			0
$193$			2.01309i			0.144905i	0.997372	+	0.0724526i	$0.0230826\pi$
$193$			2.01309i			0.144905i	−0.997372	+	0.0724526i	$0.976917\pi$
$194$		0			0
$195$	−0.677343	−	0.946730i	−0.0485056	−	0.0677968i
$196$		0			0
$197$		−	24.3517i		−	1.73499i	−0.497448	−	0.867494i	$-0.665729\pi$
$197$		−	24.3517i		−	1.73499i	0.497448	−	0.867494i	$-0.334271\pi$
$198$		0			0
$199$	8.55060			0.606136			0.303068	−	0.952969i	$-0.401989\pi$
$199$	8.55060			0.606136			0.303068	+	0.952969i	$0.401989\pi$
$200$		0			0
$201$		−	7.06843i		−	0.498568i
$202$		0			0
$203$	8.58506	−	18.9555i	0.602553	−	1.33041i
$204$		0			0
$205$	3.64067	+	5.08860i	0.254275	+	0.355403i
$206$		0			0
$207$	−16.1206			−1.12046
$208$		0			0
$209$		−	5.28714i		−	0.365719i
$210$		0			0
$211$		−	19.0444i		−	1.31107i	−0.755164	−	0.655536i	$-0.772443\pi$
$211$		−	19.0444i		−	1.31107i	0.755164	−	0.655536i	$-0.227557\pi$
$212$		0			0
$213$	1.37035			0.0938949
$214$		0			0
$215$	−17.3519	+	12.4145i	−1.18339	+	0.846660i
$216$		0			0
$217$	1.49492	−	3.30071i	0.101481	−	0.224067i
$218$		0			0
$219$			26.3826i			1.78277i
$220$		0			0
$221$	1.41629			0.0952700
$222$		0			0
$223$			14.9124i			0.998607i	0.866427	+	0.499304i	$0.166411\pi$
$223$			14.9124i			0.998607i	−0.866427	+	0.499304i	$0.833589\pi$
$224$		0			0
$225$	−3.06833	+	8.99465i	−0.204555	+	0.599643i
$226$		0			0
$227$			0.365162i			0.0242366i	0.999927	+	0.0121183i	$0.00385748\pi$
$227$			0.365162i			0.0242366i	−0.999927	+	0.0121183i	$0.996143\pi$
$228$		0			0
$229$		−	3.77634i		−	0.249548i	−0.992185	−	0.124774i	$-0.960179\pi$
$229$		−	3.77634i		−	0.249548i	0.992185	−	0.124774i	$-0.0398206\pi$
$230$		0			0
$231$	1.59926	−	3.53109i	0.105223	−	0.232329i
$232$		0			0
$233$			3.46456i			0.226971i	0.993540	+	0.113485i	$0.0362015\pi$
$233$			3.46456i			0.226971i	−0.993540	+	0.113485i	$0.963798\pi$
$234$		0			0
$235$	10.0497	+	14.0466i	0.655572	+	0.916300i
$236$		0			0
$237$	−15.5330			−1.00898
$238$		0			0
$239$			3.89143i			0.251716i	0.992048	+	0.125858i	$0.0401683\pi$
$239$			3.89143i			0.251716i	−0.992048	+	0.125858i	$0.959832\pi$
$240$		0			0
$241$		−	4.65633i		−	0.299941i	−0.988691	−	0.149970i	$-0.952082\pi$
$241$		−	4.65633i		−	0.299941i	0.988691	−	0.149970i	$-0.0479178\pi$
$242$		0			0
$243$		−	17.2487i		−	1.10650i
$244$		0			0
$245$	−15.2420	−	3.56101i	−0.973777	−	0.227504i
$246$		0			0
$247$	1.87864			0.119535
$248$		0			0
$249$	−15.6168			−0.989676
$250$		0			0
$251$	8.73327			0.551239			0.275620	−	0.961267i	$-0.411117\pi$
$251$	8.73327			0.551239			0.275620	+	0.961267i	$0.411117\pi$
$252$		0			0
$253$			5.61317i			0.352897i
$254$		0			0
$255$	−17.3469	−	24.2460i	−1.08631	−	1.51834i
$256$		0			0
$257$	5.70661			0.355968			0.177984	−	0.984033i	$-0.443042\pi$
$257$	5.70661			0.355968			0.177984	+	0.984033i	$0.443042\pi$
$258$		0			0
$259$	−14.6978	−	6.65675i	−0.913279	−	0.413631i
$260$		0			0
$261$	14.9492			0.925335
$262$		0			0
$263$	−9.79832			−0.604190			−0.302095	−	0.953278i	$-0.597686\pi$
$263$	−9.79832			−0.604190			−0.302095	+	0.953278i	$0.597686\pi$
$264$		0			0
$265$	−11.0003	−	15.3752i	−0.675740	−	0.944490i
$266$		0			0
$267$	−22.0819			−1.35139
$268$		0			0
$269$			20.8330i			1.27021i	0.772426	+	0.635104i	$0.219043\pi$
$269$			20.8330i			1.27021i	−0.772426	+	0.635104i	$0.780957\pi$
$270$		0			0
$271$	13.4035			0.814202			0.407101	−	0.913383i	$-0.366540\pi$
$271$	13.4035			0.814202			0.407101	+	0.913383i	$0.366540\pi$
$272$		0			0
$273$	1.25468	+	0.568253i	0.0759366	+	0.0343922i
$274$		0			0
$275$	3.13193	+	1.06839i	0.188863	+	0.0644263i
$276$		0			0
$277$			7.77892i			0.467390i	0.972310	+	0.233695i	$0.0750818\pi$
$277$			7.77892i			0.467390i	−0.972310	+	0.233695i	$0.924918\pi$
$278$		0			0
$279$	2.60311			0.155844
$280$		0			0
$281$	−18.0580			−1.07725			−0.538624	−	0.842546i	$-0.681056\pi$
$281$	−18.0580			−1.07725			−0.538624	+	0.842546i	$0.681056\pi$
$282$		0			0
$283$			17.6208i			1.04745i	0.851888	+	0.523725i	$0.175458\pi$
$283$			17.6208i			1.04745i	−0.851888	+	0.523725i	$0.824542\pi$
$284$		0			0
$285$	−23.0099	−	32.1612i	−1.36299	−	1.90506i
$286$		0			0
$287$	−6.74380	−	3.05431i	−0.398074	−	0.180290i
$288$		0			0
$289$	19.2715			1.13362
$290$		0			0
$291$		−	13.3325i		−	0.781567i
$292$		0			0
$293$	19.9862			1.16761			0.583803	−	0.811895i	$-0.301564\pi$
$293$	19.9862			1.16761			0.583803	+	0.811895i	$0.301564\pi$
$294$		0			0
$295$	1.68960	−	1.20883i	0.0983722	−	0.0703810i
$296$		0			0
$297$	−1.61059			−0.0934558
$298$		0			0
$299$	−1.99449			−0.115344
$300$		0			0
$301$	10.4150	−	22.9960i	0.600313	−	1.32547i
$302$		0			0
$303$	−23.2074			−1.33323
$304$		0			0
$305$	2.97914	+	4.16398i	0.170585	+	0.238429i
$306$		0			0
$307$			21.1613i			1.20774i	0.797083	+	0.603870i	$0.206375\pi$
$307$			21.1613i			1.20774i	−0.797083	+	0.603870i	$0.793625\pi$
$308$		0			0
$309$	10.6071			0.603416
$310$		0			0
$311$	−24.7737			−1.40479			−0.702393	−	0.711789i	$-0.747885\pi$
$311$	−24.7737			−1.40479			−0.702393	+	0.711789i	$0.747885\pi$
$312$		0			0
$313$	4.74418			0.268157			0.134078	−	0.990971i	$-0.457193\pi$
$313$	4.74418			0.268157			0.134078	+	0.990971i	$0.457193\pi$
$314$		0			0
$315$	−2.18719	−	11.0300i	−0.123234	−	0.621473i
$316$		0			0
$317$		−	0.397922i		−	0.0223495i	−0.999938	−	0.0111748i	$-0.996443\pi$
$317$		−	0.397922i		−	0.0223495i	0.999938	−	0.0111748i	$-0.00355711\pi$
$318$		0			0
$319$		−	5.20532i		−	0.291442i
$320$		0			0
$321$			7.06843i			0.394521i
$322$		0			0
$323$	48.1125			2.67705
$324$		0			0
$325$	−0.379623	+	1.11285i	−0.0210577	+	0.0617297i
$326$		0			0
$327$			7.45829i			0.412444i
$328$		0			0
$329$	−18.6156	−	8.43115i	−1.02631	−	0.464824i
$330$		0			0
$331$		−	4.18036i		−	0.229773i	−0.993379	−	0.114887i	$-0.963350\pi$
$331$		−	4.18036i		−	0.229773i	0.993379	−	0.114887i	$-0.0366505\pi$
$332$		0			0
$333$		−	11.5915i		−	0.635209i
$334$		0			0
$335$	−5.80657	+	4.15434i	−0.317247	+	0.226976i
$336$		0			0
$337$			26.3737i			1.43667i	0.695699	+	0.718333i	$0.255095\pi$
$337$			26.3737i			1.43667i	−0.695699	+	0.718333i	$0.744905\pi$
$338$		0			0
$339$	23.7878			1.29198
$340$		0			0
$341$		−	0.906401i		−	0.0490844i
$342$		0			0
$343$	17.7203	−	5.38440i	0.956805	−	0.290730i
$344$		0			0
$345$	24.4288	+	34.1443i	1.31520	+	1.83827i
$346$		0			0
$347$	−2.38649			−0.128114			−0.0640568	−	0.997946i	$-0.520404\pi$
$347$	−2.38649			−0.128114			−0.0640568	+	0.997946i	$0.520404\pi$
$348$		0			0
$349$			23.1032i			1.23669i	0.785908	+	0.618344i	$0.212196\pi$
$349$			23.1032i			1.23669i	−0.785908	+	0.618344i	$0.787804\pi$
$350$		0			0
$351$		−	0.572279i		−	0.0305460i
$352$		0			0
$353$	24.6590			1.31246			0.656232	−	0.754559i	$-0.272149\pi$
$353$	24.6590			1.31246			0.656232	+	0.754559i	$0.272149\pi$
$354$		0			0
$355$	−0.805399	−	1.12572i	−0.0427461	−	0.0597468i
$356$		0			0
$357$	32.1326	+	14.5531i	1.70064	+	0.770230i
$358$		0			0
$359$		−	7.34704i		−	0.387762i	−0.981025	−	0.193881i	$-0.937892\pi$
$359$		−	7.34704i		−	0.387762i	0.981025	−	0.193881i	$-0.0621076\pi$
$360$		0			0
$361$	44.8191			2.35890
$362$		0			0
$363$			23.3817i			1.22722i
$364$		0			0
$365$	21.6728	−	15.5059i	1.13440	−	0.811616i
$366$		0			0
$367$			1.57622i			0.0822782i	0.999153	+	0.0411391i	$0.0130987\pi$
$367$			1.57622i			0.0822782i	−0.999153	+	0.0411391i	$0.986901\pi$
$368$		0			0
$369$		−	5.31851i		−	0.276870i
$370$		0			0
$371$	20.3764	+	9.22859i	1.05789	+	0.479125i
$372$		0			0
$373$			32.5286i			1.68427i	0.539271	+	0.842133i	$0.318700\pi$
$373$			32.5286i			1.68427i	−0.539271	+	0.842133i	$0.681300\pi$
$374$		0			0
$375$	23.7009	−	7.13140i	1.22391	−	0.368264i
$376$		0			0
$377$	1.84957			0.0952577
$378$		0			0
$379$			36.0655i			1.85256i	0.376839	+	0.926279i	$0.377011\pi$
$379$			36.0655i			1.85256i	−0.376839	+	0.926279i	$0.622989\pi$
$380$		0			0
$381$			17.5855i			0.900930i
$382$		0			0
$383$			18.6692i			0.953951i	0.878917	+	0.476976i	$0.158267\pi$
$383$			18.6692i			0.953951i	−0.878917	+	0.476976i	$0.841733\pi$
$384$		0			0
$385$	−3.84065	+	0.761579i	−0.195738	+	0.0388137i
$386$		0			0
$387$	18.1358			0.921895
$388$		0			0
$389$	20.9929			1.06438			0.532192	−	0.846624i	$-0.321368\pi$
$389$	20.9929			1.06438			0.532192	+	0.846624i	$0.321368\pi$
$390$		0			0
$391$	−51.0793			−2.58319
$392$		0			0
$393$		−	32.8113i		−	1.65511i
$394$		0			0
$395$	9.12924	+	12.7600i	0.459342	+	0.642027i
$396$		0			0
$397$	36.1967			1.81666			0.908329	−	0.418256i	$-0.137359\pi$
$397$	36.1967			1.81666			0.908329	+	0.418256i	$0.137359\pi$
$398$		0			0
$399$	42.6224	+	19.3040i	2.13379	+	0.966409i
$400$		0			0
$401$	9.76155			0.487469			0.243734	−	0.969842i	$-0.421628\pi$
$401$	9.76155			0.487469			0.243734	+	0.969842i	$0.421628\pi$
$402$		0			0
$403$	0.322065			0.0160432
$404$		0			0
$405$	−20.1667	+	14.4284i	−1.00209	+	0.716953i
$406$		0			0
$407$	−4.03614			−0.200064
$408$		0			0
$409$		−	21.8238i		−	1.07912i	−0.841949	−	0.539558i	$-0.818591\pi$
$409$		−	21.8238i		−	1.07912i	0.841949	−	0.539558i	$-0.181409\pi$
$410$		0			0
$411$	−42.0008			−2.07175
$412$		0			0
$413$	−1.01414	+	2.23918i	−0.0499027	+	0.110183i
$414$		0			0
$415$	9.17851	+	12.8289i	0.450555	+	0.629746i
$416$		0			0
$417$		−	20.9857i		−	1.02767i
$418$		0			0
$419$	−32.2899			−1.57746			−0.788732	−	0.614737i	$-0.789262\pi$
$419$	−32.2899			−1.57746			−0.788732	+	0.614737i	$0.789262\pi$
$420$		0			0
$421$	8.26869			0.402991			0.201496	−	0.979489i	$-0.435420\pi$
$421$	8.26869			0.402991			0.201496	+	0.979489i	$0.435420\pi$
$422$		0			0
$423$		−	14.6812i		−	0.713826i
$424$		0			0
$425$	−9.72223	+	28.5003i	−0.471598	+	1.38247i
$426$		0			0
$427$	−5.51842	−	2.49933i	−0.267055	−	0.120951i
$428$		0			0
$429$	0.344545			0.0166348
$430$		0			0
$431$			36.0982i			1.73879i	0.494120	+	0.869394i	$0.335490\pi$
$431$			36.0982i			1.73879i	−0.494120	+	0.869394i	$0.664510\pi$
$432$		0			0
$433$	30.9851			1.48905			0.744524	−	0.667595i	$-0.232676\pi$
$433$	30.9851			1.48905			0.744524	+	0.667595i	$0.232676\pi$
$434$		0			0
$435$	−22.6538	−	31.6634i	−1.08617	−	1.51815i
$436$		0			0
$437$	−67.7544			−3.24113
$438$		0			0
$439$	−12.9148			−0.616391			−0.308195	−	0.951323i	$-0.599725\pi$
$439$	−12.9148			−0.616391			−0.308195	+	0.951323i	$0.599725\pi$
$440$		0			0
$441$	8.77573	+	10.0005i	0.417892	+	0.476216i
$442$		0			0
$443$	11.7872			0.560028			0.280014	−	0.959996i	$-0.409661\pi$
$443$	11.7872			0.560028			0.280014	+	0.959996i	$0.409661\pi$
$444$		0			0
$445$	12.9782	+	18.1398i	0.615227	+	0.859909i
$446$		0			0
$447$			28.8762i			1.36580i
$448$		0			0
$449$	−0.434095			−0.0204862			−0.0102431	−	0.999948i	$-0.503261\pi$
$449$	−0.434095			−0.0204862			−0.0102431	+	0.999948i	$0.503261\pi$
$450$		0			0
$451$	−1.85190			−0.0872026
$452$		0			0
$453$	33.3575			1.56727
$454$		0			0
$455$	−0.270607	−	1.36467i	−0.0126862	−	0.0639769i
$456$		0			0
$457$			17.4775i			0.817561i	0.912633	+	0.408781i	$0.134046\pi$
$457$			17.4775i			0.817561i	−0.912633	+	0.408781i	$0.865954\pi$
$458$		0			0
$459$		−	14.6562i		−	0.684092i
$460$		0			0
$461$		−	6.32441i		−	0.294557i	−0.989095	−	0.147279i	$-0.952949\pi$
$461$		−	6.32441i		−	0.294557i	0.989095	−	0.147279i	$-0.0470514\pi$
$462$		0			0
$463$	−26.5240			−1.23268			−0.616338	−	0.787482i	$-0.711385\pi$
$463$	−26.5240			−1.23268			−0.616338	+	0.787482i	$0.711385\pi$
$464$		0			0
$465$	−3.94470	−	5.51355i	−0.182931	−	0.255685i
$466$		0			0
$467$			18.6029i			0.860839i	0.902629	+	0.430419i	$0.141634\pi$
$467$			18.6029i			0.860839i	−0.902629	+	0.430419i	$0.858366\pi$
$468$		0			0
$469$	3.48526	−	7.69530i	0.160934	−	0.355336i
$470$		0			0
$471$		−	27.8447i		−	1.28302i
$472$		0			0
$473$		−	6.31488i		−	0.290358i
$474$		0			0
$475$	−12.8961	+	37.8043i	−0.591714	+	1.73458i
$476$		0			0
$477$			16.0698i			0.735787i
$478$		0			0
$479$	−8.79631			−0.401914			−0.200957	−	0.979600i	$-0.564405\pi$
$479$	−8.79631			−0.401914			−0.200957	+	0.979600i	$0.564405\pi$
$480$		0			0
$481$		−	1.43413i		−	0.0653909i
$482$		0			0
$483$	−45.2507	−	20.4943i	−2.05898	−	0.932525i
$484$		0			0
$485$	−10.9524	+	7.83596i	−0.497323	+	0.355813i
$486$		0			0
$487$	35.2570			1.59765			0.798824	−	0.601564i	$-0.205456\pi$
$487$	35.2570			1.59765			0.798824	+	0.601564i	$0.205456\pi$
$488$		0			0
$489$		−	2.71540i		−	0.122794i
$490$		0			0
$491$			38.4114i			1.73348i	0.498759	+	0.866741i	$0.333789\pi$
$491$			38.4114i			1.73348i	−0.498759	+	0.866741i	$0.666211\pi$
$492$		0			0
$493$	47.3679			2.13334
$494$		0			0
$495$	−1.63672	−	2.28766i	−0.0735649	−	0.102823i
$496$		0			0
$497$	1.49188	+	0.675684i	0.0669201	+	0.0303086i
$498$		0			0
$499$		−	20.1510i		−	0.902084i	−0.892503	−	0.451042i	$-0.851052\pi$
$499$		−	20.1510i		−	0.902084i	0.892503	−	0.451042i	$-0.148948\pi$
$500$		0			0
$501$	−9.77875			−0.436882
$502$		0			0
$503$			25.4467i			1.13461i	0.823507	+	0.567306i	$0.192014\pi$
$503$			25.4467i			1.13461i	−0.823507	+	0.567306i	$0.807986\pi$
$504$		0			0
$505$	13.6397	+	19.0644i	0.606961	+	0.848356i
$506$		0			0
$507$		−	28.6564i		−	1.27268i
$508$		0			0
$509$		−	2.87999i		−	0.127654i	−0.997961	−	0.0638268i	$-0.979669\pi$
$509$		−	2.87999i		−	0.127654i	0.997961	−	0.0638268i	$-0.0203305\pi$
$510$		0			0
$511$	−13.0086	+	28.7224i	−0.575465	+	1.27060i
$512$		0			0
$513$		−	19.4408i		−	0.858331i
$514$		0			0
$515$	−6.23413	−	8.71351i	−0.274709	−	0.383963i
$516$		0			0
$517$	−5.11200			−0.224825
$518$		0			0
$519$			16.5383i			0.725952i
$520$		0			0
$521$		−	33.1526i		−	1.45244i	−0.687462	−	0.726220i	$-0.741275\pi$
$521$		−	33.1526i		−	1.45244i	0.687462	−	0.726220i	$-0.258725\pi$
$522$		0			0
$523$			3.73506i			0.163323i	0.996660	+	0.0816615i	$0.0260226\pi$
$523$			3.73506i			0.163323i	−0.996660	+	0.0816615i	$0.973977\pi$
$524$		0			0
$525$	−20.0479	+	21.3473i	−0.874962	+	0.931673i
$526$		0			0
$527$	8.24816			0.359296
$528$		0			0
$529$	48.9323			2.12749
$530$		0			0
$531$	−1.76594			−0.0766351
$532$		0			0
$533$		−	0.658023i		−	0.0285021i
$534$		0			0
$535$	5.80657	−	4.15434i	0.251040	−	0.179608i
$536$		0			0
$537$	−27.8185			−1.20046
$538$		0			0
$539$	3.48218	−	3.05570i	0.149988	−	0.131618i
$540$		0			0
$541$	11.0946			0.476996			0.238498	−	0.971143i	$-0.423345\pi$
$541$	11.0946			0.476996			0.238498	+	0.971143i	$0.423345\pi$
$542$		0			0
$543$	41.4446			1.77856
$544$		0			0
$545$	6.12683	−	4.38347i	0.262445	−	0.187767i
$546$		0			0
$547$	23.4626			1.00319			0.501595	−	0.865103i	$-0.332747\pi$
$547$	23.4626			1.00319			0.501595	+	0.865103i	$0.332747\pi$
$548$		0			0
$549$		−	4.35211i		−	0.185744i
$550$		0			0
$551$	62.8314			2.67671
$552$		0			0
$553$	−16.9106	−	7.65892i	−0.719111	−	0.325690i
$554$		0			0
$555$	−24.5515	+	17.5655i	−1.04215	+	0.745613i
$556$		0			0
$557$			13.7120i			0.580996i	0.956876	+	0.290498i	$0.0938209\pi$
$557$			13.7120i			0.580996i	−0.956876	+	0.290498i	$0.906179\pi$
$558$		0			0
$559$	2.24382			0.0949036
$560$		0			0
$561$	8.82386			0.372544
$562$		0			0
$563$		−	19.4840i		−	0.821155i	−0.911826	−	0.410577i	$-0.865327\pi$
$563$		−	19.4840i		−	0.821155i	0.911826	−	0.410577i	$-0.134673\pi$
$564$		0			0
$565$	−13.9809	−	19.5412i	−0.588180	−	0.822105i
$566$		0			0
$567$	12.1046	−	26.7265i	0.508346	−	1.12241i
$568$		0			0
$569$	−1.64067			−0.0687804			−0.0343902	−	0.999408i	$-0.510949\pi$
$569$	−1.64067			−0.0687804			−0.0343902	+	0.999408i	$0.510949\pi$
$570$		0			0
$571$		−	11.3282i		−	0.474069i	−0.971501	−	0.237034i	$-0.923825\pi$
$571$		−	11.3282i		−	0.474069i	0.971501	−	0.237034i	$-0.0761754\pi$
$572$		0			0
$573$	35.7524			1.49358
$574$		0			0
$575$	13.6913	−	40.1355i	0.570968	−	1.67376i
$576$		0			0
$577$	−23.6001			−0.982483			−0.491242	−	0.871023i	$-0.663457\pi$
$577$	−23.6001			−0.982483			−0.491242	+	0.871023i	$0.663457\pi$
$578$		0			0
$579$	−4.45648			−0.185205
$580$		0			0
$581$	−17.0018	−	7.70025i	−0.705355	−	0.319460i
$582$		0			0
$583$	5.59551			0.231742
$584$		0			0
$585$	0.812857	−	0.581563i	0.0336075	−	0.0240447i
$586$		0			0
$587$			4.16287i			0.171820i	0.996303	+	0.0859099i	$0.0273797\pi$
$587$			4.16287i			0.171820i	−0.996303	+	0.0859099i	$0.972620\pi$
$588$		0			0
$589$	10.9408			0.450808
$590$		0			0
$591$	53.9088			2.21751
$592$		0			0
$593$	11.5914			0.476003			0.238002	−	0.971265i	$-0.423508\pi$
$593$	11.5914			0.476003			0.238002	+	0.971265i	$0.423508\pi$
$594$		0			0
$595$	−6.93030	−	34.9496i	−0.284115	−	1.43279i
$596$		0			0
$597$			18.9289i			0.774710i
$598$		0			0
$599$		−	27.8564i		−	1.13818i	−0.822275	−	0.569090i	$-0.807296\pi$
$599$		−	27.8564i		−	1.13818i	0.822275	−	0.569090i	$-0.192704\pi$
$600$		0			0
$601$			11.8809i			0.484632i	0.970197	+	0.242316i	$0.0779070\pi$
$601$			11.8809i			0.484632i	−0.970197	+	0.242316i	$0.922093\pi$
$602$		0			0
$603$	6.06891			0.247145
$604$		0			0
$605$	19.2076	−	13.7422i	0.780898	−	0.558698i
$606$		0			0
$607$		−	4.44568i		−	0.180445i	−0.995922	−	0.0902224i	$-0.971242\pi$
$607$		−	4.44568i		−	0.180445i	0.995922	−	0.0902224i	$-0.0287578\pi$
$608$		0			0
$609$	41.9628	+	19.0052i	1.70042	+	0.770131i
$610$		0			0
$611$		−	1.81641i		−	0.0734841i
$612$		0			0
$613$			24.4542i			0.987697i	0.869548	+	0.493848i	$0.164410\pi$
$613$			24.4542i			0.987697i	−0.869548	+	0.493848i	$0.835590\pi$
$614$		0			0
$615$	−11.2649	+	8.05955i	−0.454246	+	0.324993i
$616$		0			0
$617$			45.9516i			1.84994i	0.380037	+	0.924971i	$0.375911\pi$
$617$			45.9516i			1.84994i	−0.380037	+	0.924971i	$0.624089\pi$
$618$		0			0
$619$	8.63803			0.347192			0.173596	−	0.984817i	$-0.444461\pi$
$619$	8.63803			0.347192			0.173596	+	0.984817i	$0.444461\pi$
$620$		0			0
$621$			20.6396i			0.828237i
$622$		0			0
$623$	−24.0402	−	10.8880i	−0.963152	−	0.436218i
$624$		0			0
$625$	−19.7881	−	15.2785i	−0.791524	−	0.611138i
$626$		0			0
$627$	11.7045			0.467431
$628$		0			0
$629$		−	36.7285i		−	1.46446i
$630$		0			0
$631$			23.5730i			0.938426i	0.883085	+	0.469213i	$0.155462\pi$
$631$			23.5730i			0.938426i	−0.883085	+	0.469213i	$0.844538\pi$
$632$		0			0
$633$	42.1597			1.67570
$634$		0			0
$635$	14.4461	−	10.3355i	0.573276	−	0.410153i
$636$		0			0
$637$	1.08576	+	1.23730i	0.0430194	+	0.0490236i
$638$		0			0
$639$			1.17658i			0.0465446i
$640$		0			0
$641$	−27.3036			−1.07843			−0.539214	−	0.842169i	$-0.681279\pi$
$641$	−27.3036			−1.07843			−0.539214	+	0.842169i	$0.681279\pi$
$642$		0			0
$643$			17.5703i			0.692904i	0.938068	+	0.346452i	$0.112614\pi$
$643$			17.5703i			0.692904i	−0.938068	+	0.346452i	$0.887386\pi$
$644$		0			0
$645$	−27.4826	−	38.4128i	−1.08213	−	1.51250i
$646$		0			0
$647$		−	36.3691i		−	1.42982i	−0.699218	−	0.714908i	$-0.746468\pi$
$647$		−	36.3691i		−	1.42982i	0.699218	−	0.714908i	$-0.253532\pi$
$648$		0			0
$649$			0.614897i			0.0241368i
$650$		0			0
$651$	7.30697	+	3.30938i	0.286383	+	0.129705i
$652$		0			0
$653$		−	31.1711i		−	1.21982i	−0.792471	−	0.609910i	$-0.791206\pi$
$653$		−	31.1711i		−	1.21982i	0.792471	−	0.609910i	$-0.208794\pi$
$654$		0			0
$655$	−26.9538	+	19.2842i	−1.05317	+	0.753497i
$656$		0			0
$657$	−22.6519			−0.883737
$658$		0			0
$659$		−	28.3559i		−	1.10459i	−0.833649	−	0.552295i	$-0.813752\pi$
$659$		−	28.3559i		−	1.10459i	0.833649	−	0.552295i	$-0.186248\pi$
$660$		0			0
$661$		−	22.4537i		−	0.873346i	−0.899620	−	0.436673i	$-0.856157\pi$
$661$		−	22.4537i		−	0.873346i	0.899620	−	0.436673i	$-0.143843\pi$
$662$		0			0
$663$			3.13532i			0.121766i
$664$		0			0
$665$	−9.19273	−	46.3590i	−0.356479	−	1.79773i
$666$		0			0
$667$	−66.7058			−2.58286
$668$		0			0
$669$	−33.0124			−1.27633
$670$		0			0
$671$	−1.51540			−0.0585015
$672$		0			0
$673$		−	9.02934i		−	0.348056i	−0.984741	−	0.174028i	$-0.944322\pi$
$673$		−	9.02934i		−	0.348056i	0.984741	−	0.174028i	$-0.0556783\pi$
$674$		0			0
$675$	11.5161	+	3.92845i	0.443254	+	0.151206i
$676$		0			0
$677$	−2.14690			−0.0825120			−0.0412560	−	0.999149i	$-0.513136\pi$
$677$	−2.14690			−0.0825120			−0.0412560	+	0.999149i	$0.513136\pi$
$678$		0			0
$679$	6.57392	−	14.5150i	0.252284	−	0.557033i
$680$		0			0
$681$	−0.808380			−0.0309772
$682$		0			0
$683$	−35.6440			−1.36388			−0.681939	−	0.731409i	$-0.738863\pi$
$683$	−35.6440			−1.36388			−0.681939	+	0.731409i	$0.738863\pi$
$684$		0			0
$685$	24.6852	+	34.5028i	0.943174	+	1.31828i
$686$		0			0
$687$	8.35991			0.318950
$688$		0			0
$689$			1.98821i			0.0757449i
$690$		0			0
$691$	4.87253			0.185360			0.0926800	−	0.995696i	$-0.470457\pi$
$691$	4.87253			0.185360			0.0926800	+	0.995696i	$0.470457\pi$
$692$		0			0
$693$	3.03178	+	1.37311i	0.115168	+	0.0521602i
$694$		0			0
$695$	−17.2393	+	12.3340i	−0.653925	+	0.467855i
$696$		0			0
$697$		−	16.8521i		−	0.638319i
$698$		0			0
$699$	−7.66969			−0.290094
$700$		0			0
$701$	29.9296			1.13043			0.565213	−	0.824945i	$-0.308794\pi$
$701$	29.9296			1.13043			0.565213	+	0.824945i	$0.308794\pi$
$702$		0			0
$703$		−	48.7187i		−	1.83746i
$704$		0			0
$705$	−31.0958	+	22.2476i	−1.17113	+	0.837895i
$706$		0			0
$707$	−25.2656	−	11.4430i	−0.950211	−	0.430357i
$708$		0			0
$709$	26.7899			1.00611			0.503057	−	0.864253i	$-0.332208\pi$
$709$	26.7899			1.00611			0.503057	+	0.864253i	$0.332208\pi$
$710$		0			0
$711$		−	13.3365i		−	0.500160i
$712$		0			0
$713$	−11.6155			−0.435003
$714$		0			0
$715$	−0.202500	−	0.283036i	−0.00757307	−	0.0105850i
$716$		0			0
$717$	−8.61468			−0.321721
$718$		0			0
$719$	−39.7507			−1.48245			−0.741225	−	0.671256i	$-0.765755\pi$
$719$	−39.7507			−1.48245			−0.741225	+	0.671256i	$0.765755\pi$
$720$		0			0
$721$	11.5478	+	5.23008i	0.430063	+	0.194778i
$722$		0			0
$723$	10.3080			0.383358
$724$		0			0
$725$	−12.6965	+	37.2192i	−0.471537	+	1.38229i
$726$		0			0
$727$		−	16.8141i		−	0.623599i	−0.950148	−	0.311799i	$-0.899068\pi$
$727$		−	16.8141i		−	0.623599i	0.950148	−	0.311799i	$-0.100932\pi$
$728$		0			0
$729$	4.91609			0.182077
$730$		0			0
$731$	57.4648			2.12541
$732$		0			0
$733$	−13.5587			−0.500803			−0.250401	−	0.968142i	$-0.580563\pi$
$733$	−13.5587			−0.500803			−0.250401	+	0.968142i	$0.580563\pi$
$734$		0			0
$735$	7.88320	−	33.7421i	0.290776	−	1.24460i
$736$		0			0
$737$		−	2.11319i		−	0.0778404i
$738$		0			0
$739$		−	29.5016i		−	1.08523i	−0.839980	−	0.542617i	$-0.817433\pi$
$739$		−	29.5016i		−	1.08523i	0.839980	−	0.542617i	$-0.182567\pi$
$740$		0			0
$741$			4.15886i			0.152780i
$742$		0			0
$743$	−31.3403			−1.14976			−0.574882	−	0.818237i	$-0.694952\pi$
$743$	−31.3403			−1.14976			−0.574882	+	0.818237i	$0.694952\pi$
$744$		0			0
$745$	23.7212	−	16.9715i	0.869078	−	0.621787i
$746$		0			0
$747$		−	13.4085i		−	0.490592i
$748$		0			0
$749$	−3.48526	+	7.69530i	−0.127349	+	0.281180i
$750$		0			0
$751$			49.3249i			1.79989i	0.436002	+	0.899946i	$0.356394\pi$
$751$			49.3249i			1.79989i	−0.436002	+	0.899946i	$0.643606\pi$
$752$		0			0
$753$			19.3333i			0.704546i
$754$		0			0
$755$	−19.6053	−	27.4025i	−0.713508	−	0.997279i
$756$		0			0
$757$			25.4706i			0.925743i	0.886425	+	0.462872i	$0.153181\pi$
$757$			25.4706i			0.925743i	−0.886425	+	0.462872i	$0.846819\pi$
$758$		0			0
$759$	−12.4262			−0.451042
$760$		0			0
$761$		−	39.7349i		−	1.44039i	−0.693772	−	0.720194i	$-0.744053\pi$
$761$		−	39.7349i		−	1.44039i	0.693772	−	0.720194i	$-0.255947\pi$
$762$		0			0
$763$	−3.67749	+	8.11974i	−0.133134	+	0.293954i
$764$		0			0
$765$	20.8175	−	14.8940i	0.752657	−	0.538492i
$766$		0			0
$767$	−0.218487			−0.00788912
$768$		0			0
$769$		−	48.0580i		−	1.73302i	−0.499163	−	0.866508i	$-0.666359\pi$
$769$		−	48.0580i		−	1.73302i	0.499163	−	0.866508i	$-0.333641\pi$
$770$		0			0
$771$			12.6330i			0.454968i
$772$		0			0
$773$	2.33155			0.0838599			0.0419300	−	0.999121i	$-0.486649\pi$
$773$	2.33155			0.0838599			0.0419300	+	0.999121i	$0.486649\pi$
$774$		0			0
$775$	−2.21084	+	6.48098i	−0.0794158	+	0.232804i
$776$		0			0
$777$	14.7364	−	32.5374i	0.528667	−	1.16727i
$778$		0			0
$779$		−	22.3536i		−	0.800900i
$780$		0			0
$781$	0.409683			0.0146596
$782$		0			0
$783$		−	19.1399i		−	0.684004i
$784$		0			0
$785$	−22.8739	+	16.3652i	−0.816403	+	0.584100i
$786$		0			0
$787$			17.9954i			0.641467i	0.947170	+	0.320733i	$0.103929\pi$
$787$			17.9954i			0.641467i	−0.947170	+	0.320733i	$0.896071\pi$
$788$		0			0
$789$		−	21.6911i		−	0.772224i
$790$		0			0
$791$	25.8975	+	11.7292i	0.920809	+	0.417041i
$792$		0			0
$793$		−	0.538458i		−	0.0191212i
$794$		0			0
$795$	34.0369	−	24.3519i	1.20716	−	0.863673i
$796$		0			0
$797$	−25.2255			−0.893533			−0.446767	−	0.894650i	$-0.647425\pi$
$797$	−25.2255			−0.893533			−0.446767	+	0.894650i	$0.647425\pi$
$798$		0			0
$799$		−	46.5187i		−	1.64571i
$800$		0			0
$801$		−	18.9594i		−	0.669896i
$802$		0			0
$803$			7.88739i			0.278340i
$804$		0			0
$805$	9.75958	+	49.2177i	0.343980	+	1.73470i
$806$		0			0
$807$	−46.1191			−1.62347
$808$		0			0
$809$	−32.3648			−1.13789			−0.568943	−	0.822377i	$-0.692648\pi$
$809$	−32.3648			−1.13789			−0.568943	+	0.822377i	$0.692648\pi$
$810$		0			0
$811$	22.1516			0.777847			0.388924	−	0.921270i	$-0.372847\pi$
$811$	22.1516			0.777847			0.388924	+	0.921270i	$0.372847\pi$
$812$		0			0
$813$			29.6720i			1.04064i
$814$		0			0
$815$	−2.23064	+	1.59593i	−0.0781360	+	0.0559028i
$816$		0			0
$817$	76.2244			2.66676
$818$		0			0
$819$	−0.487899	+	1.07726i	−0.0170486	+	0.0376425i
$820$		0			0
$821$	−32.6307			−1.13882			−0.569409	−	0.822054i	$-0.692828\pi$
$821$	−32.6307			−1.13882			−0.569409	+	0.822054i	$0.692828\pi$
$822$		0			0
$823$	45.3224			1.57984			0.789920	−	0.613210i	$-0.210122\pi$
$823$	45.3224			1.57984			0.789920	+	0.613210i	$0.210122\pi$
$824$		0			0
$825$	−2.36515	+	6.93333i	−0.0823441	+	0.241388i
$826$		0			0
$827$	5.55080			0.193020			0.0965102	−	0.995332i	$-0.469232\pi$
$827$	5.55080			0.193020			0.0965102	+	0.995332i	$0.469232\pi$
$828$		0			0
$829$			28.0542i			0.974363i	0.873301	+	0.487182i	$0.161975\pi$
$829$			28.0542i			0.974363i	−0.873301	+	0.487182i	$0.838025\pi$
$830$		0			0
$831$	−17.2206			−0.597378
$832$		0			0
$833$	27.8066	+	31.6875i	0.963441	+	1.09791i
$834$		0			0
$835$	5.74728	+	8.03304i	0.198893	+	0.277995i
$836$		0			0
$837$		−	3.33283i		−	0.115199i
$838$		0			0
$839$	−20.1929			−0.697138			−0.348569	−	0.937283i	$-0.613332\pi$
$839$	−20.1929			−0.697138			−0.348569	+	0.937283i	$0.613332\pi$
$840$		0			0
$841$	32.8589			1.13307
$842$		0			0
$843$		−	39.9759i		−	1.37684i
$844$		0			0
$845$	−23.5407	+	16.8423i	−0.809824	+	0.579393i
$846$		0			0
$847$	−11.5289	+	25.4553i	−0.396137	+	0.874655i
$848$		0			0
$849$	−39.0082			−1.33876
$850$		0			0
$851$			51.7229i			1.77304i
$852$		0			0
$853$	−41.4211			−1.41823			−0.709116	−	0.705092i	$-0.750906\pi$
$853$	−41.4211			−1.41823			−0.709116	+	0.705092i	$0.750906\pi$
$854$		0			0
$855$	27.6134	−	19.7562i	0.944359	−	0.675647i
$856$		0			0
$857$	−51.7709			−1.76846			−0.884229	−	0.467053i	$-0.845316\pi$
$857$	−51.7709			−1.76846			−0.884229	+	0.467053i	$0.845316\pi$
$858$		0			0
$859$	15.6976			0.535596			0.267798	−	0.963475i	$-0.413704\pi$
$859$	15.6976			0.535596			0.267798	+	0.963475i	$0.413704\pi$
$860$		0			0
$861$	6.76151	−	14.9291i	0.230432	−	0.508783i
$862$		0			0
$863$	−5.91090			−0.201210			−0.100605	−	0.994926i	$-0.532078\pi$
$863$	−5.91090			−0.201210			−0.100605	+	0.994926i	$0.532078\pi$
$864$		0			0
$865$	13.5859	−	9.72011i	0.461935	−	0.330494i
$866$		0			0
$867$			42.6624i			1.44889i
$868$		0			0
$869$	−4.64378			−0.157529
$870$		0			0
$871$	0.750866			0.0254421
$872$		0			0
$873$	11.4472			0.387430
$874$		0			0
$875$	29.3192	+	3.92244i	0.991169	+	0.132603i
$876$		0			0
$877$		−	37.5650i		−	1.26848i	−0.773136	−	0.634240i	$-0.781313\pi$
$877$		−	37.5650i		−	1.26848i	0.773136	−	0.634240i	$-0.218687\pi$
$878$		0			0
$879$			44.2446i			1.49233i
$880$		0			0
$881$			41.3815i			1.39418i	0.716984	+	0.697089i	$0.245522\pi$
$881$			41.3815i			1.39418i	−0.716984	+	0.697089i	$0.754478\pi$
$882$		0			0
$883$	13.8550			0.466258			0.233129	−	0.972446i	$-0.425103\pi$
$883$	13.8550			0.466258			0.233129	+	0.972446i	$0.425103\pi$
$884$		0			0
$885$	2.67606	+	3.74036i	0.0899548	+	0.125731i
$886$		0			0
$887$			1.16763i			0.0392053i	0.999808	+	0.0196027i	$0.00624012\pi$
$887$			1.16763i			0.0392053i	−0.999808	+	0.0196027i	$0.993760\pi$
$888$		0			0
$889$	−8.67093	+	19.1451i	−0.290814	+	0.642105i
$890$		0			0
$891$		−	7.33930i		−	0.245876i
$892$		0			0
$893$		−	61.7049i		−	2.06488i
$894$		0			0
$895$	16.3498	+	22.8523i	0.546515	+	0.763870i
$896$		0			0
$897$		−	4.41531i		−	0.147423i
$898$		0			0
$899$	10.7715			0.359249
$900$		0			0
$901$			50.9186i			1.69634i
$902$		0			0
$903$	50.9075	+	23.0564i	1.69410	+	0.767268i
$904$		0			0
$905$	−24.3583	−	34.0459i	−0.809698	−	1.13172i
$906$		0			0
$907$	39.8376			1.32278			0.661392	−	0.750040i	$-0.269966\pi$
$907$	39.8376			1.32278			0.661392	+	0.750040i	$0.269966\pi$
$908$		0			0
$909$		−	19.9258i		−	0.660896i
$910$		0			0
$911$		−	5.38823i		−	0.178520i	−0.996008	−	0.0892600i	$-0.971550\pi$
$911$		−	5.38823i		−	0.178520i	0.996008	−	0.0892600i	$-0.0284502\pi$
$912$		0			0
$913$	−4.66884			−0.154516
$914$		0			0
$915$	−9.21804	+	6.59510i	−0.304739	+	0.218027i
$916$		0			0
$917$	16.1784	−	35.7212i	0.534257	−	1.17962i
$918$		0			0
$919$			12.9544i			0.427326i	0.976907	+	0.213663i	$0.0685395\pi$
$919$			12.9544i			0.427326i	−0.976907	+	0.213663i	$0.931461\pi$
$920$		0			0
$921$	−46.8460			−1.54363
$922$		0			0
$923$			0.145570i			0.00479149i
$924$		0			0
$925$	28.8594	+	9.84473i	0.948890	+	0.323693i
$926$		0			0
$927$			9.10719i			0.299119i
$928$		0			0
$929$			40.1582i			1.31755i	0.752341	+	0.658774i	$0.228924\pi$
$929$			40.1582i			1.31755i	−0.752341	+	0.658774i	$0.771076\pi$
$930$		0			0
$931$	36.8842	+	42.0320i	1.20883	+	1.37754i
$932$		0			0
$933$		−	54.8429i		−	1.79548i
$934$		0			0
$935$	−5.18607	−	7.24862i	−0.169602	−	0.237055i
$936$		0			0
$937$	6.49291			0.212114			0.106057	−	0.994360i	$-0.466177\pi$
$937$	6.49291			0.212114			0.106057	+	0.994360i	$0.466177\pi$
$938$		0			0
$939$			10.5025i			0.342735i
$940$		0			0
$941$			37.3828i			1.21864i	0.792923	+	0.609322i	$0.208558\pi$
$941$			37.3828i			1.21864i	−0.792923	+	0.609322i	$0.791442\pi$
$942$		0			0
$943$			23.7320i			0.772819i
$944$		0			0
$945$	−14.1220	+	2.80032i	−0.459390	+	0.0910944i
$946$		0			0
$947$	14.7989			0.480901			0.240450	−	0.970661i	$-0.422705\pi$
$947$	14.7989			0.480901			0.240450	+	0.970661i	$0.422705\pi$
$948$		0			0
$949$	−2.80257			−0.0909754
$950$		0			0
$951$	0.880903			0.0285653
$952$		0			0
$953$		−	18.4435i		−	0.597443i	−0.954340	−	0.298722i	$-0.903440\pi$
$953$		−	18.4435i		−	0.597443i	0.954340	−	0.298722i	$-0.0965602\pi$
$954$		0			0
$955$	−21.0128	−	29.3699i	−0.679960	−	0.950387i
$956$		0			0
$957$	11.5233			0.372496
$958$		0			0
$959$	−45.7257	−	20.7095i	−1.47656	−	0.668745i
$960$		0			0
$961$	−29.1244			−0.939496
$962$		0			0
$963$	−6.06891			−0.195568
$964$		0			0
$965$	2.61922	+	3.66091i	0.0843157	+	0.117849i
$966$		0			0
$967$	−16.7295			−0.537983			−0.268991	−	0.963143i	$-0.586690\pi$
$967$	−16.7295			−0.537983			−0.268991	+	0.963143i	$0.586690\pi$
$968$		0			0
$969$			106.509i			3.42157i
$970$		0			0
$971$	−42.8284			−1.37443			−0.687214	−	0.726455i	$-0.741166\pi$
$971$	−42.8284			−1.37443			−0.687214	+	0.726455i	$0.741166\pi$
$972$		0			0
$973$	10.3475	−	22.8469i	0.331726	−	0.732437i
$974$		0			0
$975$	−2.46357	−	0.840394i	−0.0788975	−	0.0269141i
$976$		0			0
$977$		−	11.9457i		−	0.382177i	−0.981573	−	0.191088i	$-0.938798\pi$
$977$		−	11.9457i		−	0.382177i	0.981573	−	0.191088i	$-0.0612017\pi$
$978$		0			0
$979$	−6.60164			−0.210989
$980$		0			0
$981$	−6.40364			−0.204453
$982$		0			0
$983$		−	19.4014i		−	0.618809i	−0.950930	−	0.309405i	$-0.899870\pi$
$983$		−	19.4014i		−	0.618809i	0.950930	−	0.309405i	$-0.100130\pi$
$984$		0			0
$985$	−31.6839	−	44.2850i	−1.00953	−	1.41104i
$986$		0			0
$987$	18.6645	−	41.2105i	0.594098	−	1.31174i
$988$		0			0
$989$	−80.9247			−2.57326
$990$		0			0
$991$		−	26.6231i		−	0.845712i	−0.906197	−	0.422856i	$-0.861028\pi$
$991$		−	26.6231i		−	0.845712i	0.906197	−	0.422856i	$-0.138972\pi$
$992$		0			0
$993$	9.25430			0.293676
$994$		0			0
$995$	15.5497	−	11.1251i	0.492960	−	0.352691i
$996$		0			0
$997$	−22.6223			−0.716454			−0.358227	−	0.933635i	$-0.616619\pi$
$997$	−22.6223			−0.716454			−0.358227	+	0.933635i	$0.616619\pi$
$998$		0			0
$999$	−14.8409			−0.469544

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.e.g.2239.28		48
4.3	odd	2	inner	2240.2.e.g.2239.45		48
5.4	even	2	inner	2240.2.e.g.2239.1		48
7.6	odd	2	inner	2240.2.e.g.2239.25		48
8.3	odd	2		1120.2.e.a.1119.26	yes	48
8.5	even	2		1120.2.e.a.1119.1	✓	48
20.19	odd	2	inner	2240.2.e.g.2239.26		48
28.27	even	2	inner	2240.2.e.g.2239.2		48
35.34	odd	2	inner	2240.2.e.g.2239.46		48
40.19	odd	2		1120.2.e.a.1119.45	yes	48
40.29	even	2		1120.2.e.a.1119.28	yes	48
56.13	odd	2		1120.2.e.a.1119.46	yes	48
56.27	even	2		1120.2.e.a.1119.27	yes	48
140.139	even	2	inner	2240.2.e.g.2239.27		48
280.69	odd	2		1120.2.e.a.1119.25	yes	48
280.139	even	2		1120.2.e.a.1119.2	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1120.2.e.a.1119.1	✓	48	8.5	even	2
1120.2.e.a.1119.2	yes	48	280.139	even	2
1120.2.e.a.1119.25	yes	48	280.69	odd	2
1120.2.e.a.1119.26	yes	48	8.3	odd	2
1120.2.e.a.1119.27	yes	48	56.27	even	2
1120.2.e.a.1119.28	yes	48	40.29	even	2
1120.2.e.a.1119.45	yes	48	40.19	odd	2
1120.2.e.a.1119.46	yes	48	56.13	odd	2
2240.2.e.g.2239.1		48	5.4	even	2	inner
2240.2.e.g.2239.2		48	28.27	even	2	inner
2240.2.e.g.2239.25		48	7.6	odd	2	inner
2240.2.e.g.2239.26		48	20.19	odd	2	inner
2240.2.e.g.2239.27		48	140.139	even	2	inner
2240.2.e.g.2239.28		48	1.1	even	1	trivial
2240.2.e.g.2239.45		48	4.3	odd	2	inner
2240.2.e.g.2239.46		48	35.34	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.21376i q^{3} +(1.81856 - 1.30110i) q^{5} +(-1.09155 + 2.41009i) q^{7} -1.90072 q^{9} +O(q^{10})\)	\(q+2.21376i q^{3} +(1.81856 - 1.30110i) q^{5} +(-1.09155 + 2.41009i) q^{7} -1.90072 q^{9} +0.661829i q^{11} -0.235163 q^{13} +(2.88031 + 4.02584i) q^{15} -6.02258 q^{17} -7.98868 q^{19} +(-5.33535 - 2.41642i) q^{21} +8.48129 q^{23} +(1.61430 - 4.73223i) q^{25} +2.43354i q^{27} -7.86505 q^{29} -1.36954 q^{31} -1.46513 q^{33} +(1.15072 + 5.80309i) q^{35} +6.09846i q^{37} -0.520594i q^{39} +2.79815i q^{41} -9.54155 q^{43} +(-3.45657 + 2.47302i) q^{45} +7.72404i q^{47} +(-4.61705 - 5.26145i) q^{49} -13.3325i q^{51} -8.45461i q^{53} +(0.861103 + 1.20357i) q^{55} -17.6850i q^{57} +0.929088 q^{59} +2.28972i q^{61} +(2.07472 - 4.58090i) q^{63} +(-0.427658 + 0.305970i) q^{65} -3.19295 q^{67} +18.7755i q^{69} -0.619016i q^{71} +11.9176 q^{73} +(10.4760 + 3.57366i) q^{75} +(-1.59507 - 0.722417i) q^{77} +7.01658i q^{79} -11.0894 q^{81} +7.05444i q^{83} +(-10.9524 + 7.83596i) q^{85} -17.4113i q^{87} +9.97484i q^{89} +(0.256692 - 0.566764i) q^{91} -3.03183i q^{93} +(-14.5279 + 10.3940i) q^{95} -6.02258 q^{97} -1.25795i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 48 q^{9}+O(q^{10}) \) 48 * q - 48 * q^9	\( 48 q - 48 q^{9} - 8 q^{21} - 16 q^{25} - 16 q^{29} + 24 q^{49} - 16 q^{65} - 32 q^{81} - 32 q^{85}+O(q^{100}) \) 48 * q - 48 * q^9 - 8 * q^21 - 16 * q^25 - 16 * q^29 + 24 * q^49 - 16 * q^65 - 32 * q^81 - 32 * q^85

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