LMFDB - Embedded newform 1760.2.l.c.529.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1760,2,Mod(529,1760)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1760.529"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1760, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1760 = 2^{5} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1760.l (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [56,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$14.0536707557$
Analytic rank:	$0$
Dimension:	$56$
Twist minimal:	no (minimal twist has level 440)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			529.3
Character	$\chi$	$=$	1760.529
Dual form			1760.2.l.c.529.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.02021 q^{3} +(-0.177003 + 2.22905i) q^{5} -1.23027i q^{7} +6.12165 q^{9} +1.00000i q^{11} -3.80312 q^{13} +(0.534586 - 6.73220i) q^{15} -0.0304528i q^{17} +0.340188i q^{19} +3.71568i q^{21} +6.84884i q^{23} +(-4.93734 - 0.789098i) q^{25} -9.42804 q^{27} +5.57762i q^{29} +3.87751 q^{31} -3.02021i q^{33} +(2.74234 + 0.217762i) q^{35} +10.1323 q^{37} +11.4862 q^{39} -8.20176 q^{41} +5.88624 q^{43} +(-1.08355 + 13.6455i) q^{45} +1.91438i q^{47} +5.48643 q^{49} +0.0919738i q^{51} -11.1736 q^{53} +(-2.22905 - 0.177003i) q^{55} -1.02744i q^{57} -11.5516i q^{59} +2.50766i q^{61} -7.53130i q^{63} +(0.673164 - 8.47735i) q^{65} -12.1043 q^{67} -20.6849i q^{69} -11.3541 q^{71} -13.9669i q^{73} +(14.9118 + 2.38324i) q^{75} +1.23027 q^{77} -0.348432 q^{79} +10.1097 q^{81} -12.7615 q^{83} +(0.0678809 + 0.00539024i) q^{85} -16.8456i q^{87} -1.45203 q^{89} +4.67887i q^{91} -11.7109 q^{93} +(-0.758296 - 0.0602143i) q^{95} -7.37006i q^{97} +6.12165i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q + 56 q^{9} + 8 q^{15} + 16 q^{25} + 16 q^{39} - 32 q^{41} - 24 q^{49} + 8 q^{55} - 32 q^{65} - 48 q^{71} + 32 q^{79} + 88 q^{81} - 16 q^{89} - 24 q^{95}+O(q^{100}) $ 56 * q + 56 * q^9 + 8 * q^15 + 16 * q^25 + 16 * q^39 - 32 * q^41 - 24 * q^49 + 8 * q^55 - 32 * q^65 - 48 * q^71 + 32 * q^79 + 88 * q^81 - 16 * q^89 - 24 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1760\mathbb{Z}\right)^\times$.

$n$	$321$	$991$	$1057$	$1541$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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
$3$	−3.02021			−1.74372			−0.871859	−	0.489757i	$-0.837085\pi$
$3$	−3.02021			−1.74372			−0.871859	+	0.489757i	$0.837085\pi$
$4$		0			0
$5$	−0.177003	+	2.22905i	−0.0791582	+	0.996862i
$5$	−0.177003	+	2.22905i	−0.0791582	+	0.996862i
$6$		0			0
$7$		−	1.23027i		−	0.464999i	−0.972597	−	0.232500i	$-0.925310\pi$
$7$		−	1.23027i		−	0.464999i	0.972597	−	0.232500i	$-0.0746904\pi$
$8$		0			0
$9$	6.12165			2.04055
$10$		0			0
$11$			1.00000i			0.301511i
$11$			1.00000i			0.301511i
$12$		0			0
$13$	−3.80312			−1.05480			−0.527398	−	0.849619i	$-0.676832\pi$
$13$	−3.80312			−1.05480			−0.527398	+	0.849619i	$0.676832\pi$
$14$		0			0
$15$	0.534586	−	6.73220i	0.138030	−	1.73825i
$16$		0			0
$17$		−	0.0304528i		−	0.00738589i	−0.999993	−	0.00369295i	$-0.998824\pi$
$17$		−	0.0304528i		−	0.00738589i	0.999993	−	0.00369295i	$-0.00117550\pi$
$18$		0			0
$19$			0.340188i			0.0780444i	0.999238	+	0.0390222i	$0.0124243\pi$
$19$			0.340188i			0.0780444i	−0.999238	+	0.0390222i	$0.987576\pi$
$20$		0			0
$21$			3.71568i			0.810827i
$22$		0			0
$23$			6.84884i			1.42808i	0.700104	+	0.714040i	$0.253137\pi$
$23$			6.84884i			1.42808i	−0.700104	+	0.714040i	$0.746863\pi$
$24$		0			0
$25$	−4.93734	−	0.789098i	−0.987468	−	0.157820i
$26$		0			0
$27$	−9.42804			−1.81443
$28$		0			0
$29$			5.57762i			1.03574i	0.855460	+	0.517869i	$0.173275\pi$
$29$			5.57762i			1.03574i	−0.855460	+	0.517869i	$0.826725\pi$
$30$		0			0
$31$	3.87751			0.696420			0.348210	−	0.937416i	$-0.386790\pi$
$31$	3.87751			0.696420			0.348210	+	0.937416i	$0.386790\pi$
$32$		0			0
$33$		−	3.02021i		−	0.525751i
$34$		0			0
$35$	2.74234	+	0.217762i	0.463540	+	0.0368085i
$36$		0			0
$37$	10.1323			1.66575			0.832873	−	0.553464i	$-0.186694\pi$
$37$	10.1323			1.66575			0.832873	+	0.553464i	$0.186694\pi$
$38$		0			0
$39$	11.4862			1.83927
$40$		0			0
$41$	−8.20176			−1.28090			−0.640450	−	0.768000i	$-0.721252\pi$
$41$	−8.20176			−1.28090			−0.640450	+	0.768000i	$0.721252\pi$
$42$		0			0
$43$	5.88624			0.897643			0.448821	−	0.893621i	$-0.351844\pi$
$43$	5.88624			0.897643			0.448821	+	0.893621i	$0.351844\pi$
$44$		0			0
$45$	−1.08355	+	13.6455i	−0.161526	+	2.03415i
$46$		0			0
$47$			1.91438i			0.279242i	0.990205	+	0.139621i	$0.0445884\pi$
$47$			1.91438i			0.279242i	−0.990205	+	0.139621i	$0.955412\pi$
$48$		0			0
$49$	5.48643			0.783776
$50$		0			0
$51$			0.0919738i			0.0128789i
$52$		0			0
$53$	−11.1736			−1.53481			−0.767403	−	0.641165i	$-0.778451\pi$
$53$	−11.1736			−1.53481			−0.767403	+	0.641165i	$0.778451\pi$
$54$		0			0
$55$	−2.22905	−	0.177003i	−0.300565	−	0.0238671i
$56$		0			0
$57$		−	1.02744i		−	0.136087i
$58$		0			0
$59$		−	11.5516i		−	1.50390i	−0.659223	−	0.751948i	$-0.729114\pi$
$59$		−	11.5516i		−	1.50390i	0.659223	−	0.751948i	$-0.270886\pi$
$60$		0			0
$61$			2.50766i			0.321073i	0.987030	+	0.160537i	$0.0513225\pi$
$61$			2.50766i			0.321073i	−0.987030	+	0.160537i	$0.948678\pi$
$62$		0			0
$63$		−	7.53130i		−	0.948854i
$64$		0			0
$65$	0.673164	−	8.47735i	0.0834957	−	1.05149i
$66$		0			0
$67$	−12.1043			−1.47877			−0.739386	−	0.673282i	$-0.764884\pi$
$67$	−12.1043			−1.47877			−0.739386	+	0.673282i	$0.764884\pi$
$68$		0			0
$69$		−	20.6849i		−	2.49017i
$70$		0			0
$71$	−11.3541			−1.34749			−0.673744	−	0.738965i	$-0.735315\pi$
$71$	−11.3541			−1.34749			−0.673744	+	0.738965i	$0.735315\pi$
$72$		0			0
$73$		−	13.9669i		−	1.63471i	−0.576136	−	0.817354i	$-0.695440\pi$
$73$		−	13.9669i		−	1.63471i	0.576136	−	0.817354i	$-0.304560\pi$
$74$		0			0
$75$	14.9118	+	2.38324i	1.72187	+	0.275193i
$76$		0			0
$77$	1.23027			0.140203
$78$		0			0
$79$	−0.348432			−0.0392016			−0.0196008	−	0.999808i	$-0.506240\pi$
$79$	−0.348432			−0.0392016			−0.0196008	+	0.999808i	$0.506240\pi$
$80$		0			0
$81$	10.1097			1.12330
$82$		0			0
$83$	−12.7615			−1.40076			−0.700379	−	0.713772i	$-0.746985\pi$
$83$	−12.7615			−1.40076			−0.700379	+	0.713772i	$0.746985\pi$
$84$		0			0
$85$	0.0678809	+	0.00539024i	0.00736272	+	0.000584654i
$86$		0			0
$87$		−	16.8456i		−	1.80604i
$88$		0			0
$89$	−1.45203			−0.153914			−0.0769572	−	0.997034i	$-0.524520\pi$
$89$	−1.45203			−0.153914			−0.0769572	+	0.997034i	$0.524520\pi$
$90$		0			0
$91$			4.67887i			0.490479i
$92$		0			0
$93$	−11.7109			−1.21436
$94$		0			0
$95$	−0.758296	−	0.0602143i	−0.0777995	−	0.00617785i
$96$		0			0
$97$		−	7.37006i		−	0.748316i	−0.927365	−	0.374158i	$-0.877932\pi$
$97$		−	7.37006i		−	0.748316i	0.927365	−	0.374158i	$-0.122068\pi$
$98$		0			0
$99$			6.12165i			0.615249i
$100$		0			0
$101$			6.14874i			0.611823i	0.952060	+	0.305911i	$0.0989611\pi$
$101$			6.14874i			0.611823i	−0.952060	+	0.305911i	$0.901039\pi$
$102$		0			0
$103$		−	0.0287339i		−	0.00283124i	−0.999999	−	0.00141562i	$-0.999549\pi$
$103$		−	0.0287339i		−	0.00283124i	0.999999	−	0.00141562i	$-0.000450605\pi$
$104$		0			0
$105$	−8.28243	−	0.657686i	−0.808283	−	0.0641836i
$106$		0			0
$107$	0.0702483			0.00679116			0.00339558	−	0.999994i	$-0.498919\pi$
$107$	0.0702483			0.00679116			0.00339558	+	0.999994i	$0.498919\pi$
$108$		0			0
$109$		−	5.22488i		−	0.500453i	−0.968187	−	0.250226i	$-0.919495\pi$
$109$		−	5.22488i		−	0.500453i	0.968187	−	0.250226i	$-0.0805051\pi$
$110$		0			0
$111$	−30.6018			−2.90459
$112$		0			0
$113$			8.06929i			0.759095i	0.925172	+	0.379547i	$0.123920\pi$
$113$			8.06929i			0.759095i	−0.925172	+	0.379547i	$0.876080\pi$
$114$		0			0
$115$	−15.2664	−	1.21227i	−1.42360	−	0.113044i
$116$		0			0
$117$	−23.2814			−2.15236
$118$		0			0
$119$	−0.0374652			−0.00343443
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$	24.7710			2.23353
$124$		0			0
$125$	2.63286	−	10.8659i	0.235491	−	0.971877i
$126$		0			0
$127$			4.90985i			0.435679i	0.975985	+	0.217839i	$0.0699009\pi$
$127$			4.90985i			0.435679i	−0.975985	+	0.217839i	$0.930099\pi$
$128$		0			0
$129$	−17.7777			−1.56524
$130$		0			0
$131$		−	19.0414i		−	1.66366i	−0.555033	−	0.831828i	$-0.687294\pi$
$131$		−	19.0414i		−	1.66366i	0.555033	−	0.831828i	$-0.312706\pi$
$132$		0			0
$133$	0.418523			0.0362906
$134$		0			0
$135$	1.66879	−	21.0156i	0.143627	−	1.80873i
$136$		0			0
$137$			9.81605i			0.838642i	0.907838	+	0.419321i	$0.137732\pi$
$137$			9.81605i			0.838642i	−0.907838	+	0.419321i	$0.862268\pi$
$138$		0			0
$139$		−	21.1605i		−	1.79481i	−0.441208	−	0.897405i	$-0.645450\pi$
$139$		−	21.1605i		−	1.79481i	0.441208	−	0.897405i	$-0.354550\pi$
$140$		0			0
$141$		−	5.78184i		−	0.486919i
$142$		0			0
$143$		−	3.80312i		−	0.318033i
$144$		0			0
$145$	−12.4328	−	0.987256i	−1.03249	−	0.0819872i
$146$		0			0
$147$	−16.5702			−1.36668
$148$		0			0
$149$		−	11.8241i		−	0.968670i	−0.874883	−	0.484335i	$-0.839062\pi$
$149$		−	11.8241i		−	0.968670i	0.874883	−	0.484335i	$-0.160938\pi$
$150$		0			0
$151$	−10.9659			−0.892393			−0.446196	−	0.894935i	$-0.647222\pi$
$151$	−10.9659			−0.892393			−0.446196	+	0.894935i	$0.647222\pi$
$152$		0			0
$153$		−	0.186422i		−	0.0150713i
$154$		0			0
$155$	−0.686330	+	8.64316i	−0.0551274	+	0.694235i
$156$		0			0
$157$	−6.37368			−0.508675			−0.254337	−	0.967116i	$-0.581857\pi$
$157$	−6.37368			−0.508675			−0.254337	+	0.967116i	$0.581857\pi$
$158$		0			0
$159$	33.7465			2.67627
$160$		0			0
$161$	8.42593			0.664056
$162$		0			0
$163$	−1.22308			−0.0957991			−0.0478995	−	0.998852i	$-0.515253\pi$
$163$	−1.22308			−0.0957991			−0.0478995	+	0.998852i	$0.515253\pi$
$164$		0			0
$165$	6.73220	+	0.534586i	0.524101	+	0.0416175i
$166$		0			0
$167$			0.0712766i			0.00551555i	0.999996	+	0.00275778i	$0.000877829\pi$
$167$			0.0712766i			0.00551555i	−0.999996	+	0.00275778i	$0.999122\pi$
$168$		0			0
$169$	1.46371			0.112593
$170$		0			0
$171$			2.08251i			0.159254i
$172$		0			0
$173$	17.4588			1.32737			0.663686	−	0.748012i	$-0.268991\pi$
$173$	17.4588			1.32737			0.663686	+	0.748012i	$0.268991\pi$
$174$		0			0
$175$	−0.970805	+	6.07427i	−0.0733860	+	0.459172i
$176$		0			0
$177$			34.8884i			2.62237i
$178$		0			0
$179$		−	4.21685i		−	0.315182i	−0.987504	−	0.157591i	$-0.949627\pi$
$179$		−	4.21685i		−	0.315182i	0.987504	−	0.157591i	$-0.0503727\pi$
$180$		0			0
$181$		−	14.8896i		−	1.10673i	−0.832938	−	0.553367i	$-0.813343\pi$
$181$		−	14.8896i		−	1.10673i	0.832938	−	0.553367i	$-0.186657\pi$
$182$		0			0
$183$		−	7.57366i		−	0.559861i
$184$		0			0
$185$	−1.79346	+	22.5855i	−0.131857	+	1.66052i
$186$		0			0
$187$	0.0304528			0.00222693
$188$		0			0
$189$			11.5990i			0.843707i
$190$		0			0
$191$	15.8442			1.14645			0.573225	−	0.819398i	$-0.305692\pi$
$191$	15.8442			1.14645			0.573225	+	0.819398i	$0.305692\pi$
$192$		0			0
$193$		−	22.2998i		−	1.60518i	−0.596533	−	0.802589i	$-0.703456\pi$
$193$		−	22.2998i		−	1.60518i	0.596533	−	0.802589i	$-0.296544\pi$
$194$		0			0
$195$	−2.03309	+	25.6033i	−0.145593	+	1.83349i
$196$		0			0
$197$	−14.4317			−1.02821			−0.514107	−	0.857726i	$-0.671876\pi$
$197$	−14.4317			−1.02821			−0.514107	+	0.857726i	$0.671876\pi$
$198$		0			0
$199$	−7.01349			−0.497173			−0.248587	−	0.968610i	$-0.579966\pi$
$199$	−7.01349			−0.497173			−0.248587	+	0.968610i	$0.579966\pi$
$200$		0			0
$201$	36.5574			2.57856
$202$		0			0
$203$	6.86199			0.481617
$204$		0			0
$205$	1.45174	−	18.2821i	0.101394	−	1.27688i
$206$		0			0
$207$			41.9262i			2.91407i
$208$		0			0
$209$	−0.340188			−0.0235313
$210$		0			0
$211$			23.4999i			1.61780i	0.587946	+	0.808900i	$0.299937\pi$
$211$			23.4999i			1.61780i	−0.587946	+	0.808900i	$0.700063\pi$
$212$		0			0
$213$	34.2919			2.34964
$214$		0			0
$215$	−1.04188	+	13.1207i	−0.0710558	+	0.894826i
$216$		0			0
$217$		−	4.77039i		−	0.323835i
$218$		0			0
$219$			42.1831i			2.85047i
$220$		0			0
$221$			0.115816i			0.00779061i
$222$		0			0
$223$		−	0.819775i		−	0.0548962i	−0.999623	−	0.0274481i	$-0.991262\pi$
$223$		−	0.819775i		−	0.0548962i	0.999623	−	0.0274481i	$-0.00873810\pi$
$224$		0			0
$225$	−30.2247	−	4.83058i	−2.01498	−	0.322039i
$226$		0			0
$227$	−0.715564			−0.0474937			−0.0237468	−	0.999718i	$-0.507560\pi$
$227$	−0.715564			−0.0474937			−0.0237468	+	0.999718i	$0.507560\pi$
$228$		0			0
$229$		−	25.1232i		−	1.66018i	−0.557626	−	0.830092i	$-0.688288\pi$
$229$		−	25.1232i		−	1.66018i	0.557626	−	0.830092i	$-0.311712\pi$
$230$		0			0
$231$	−3.71568			−0.244474
$232$		0			0
$233$		−	13.6856i		−	0.896574i	−0.893890	−	0.448287i	$-0.852034\pi$
$233$		−	13.6856i		−	0.896574i	0.893890	−	0.448287i	$-0.147966\pi$
$234$		0			0
$235$	−4.26726	−	0.338852i	−0.278365	−	0.0221043i
$236$		0			0
$237$	1.05234			0.0683566
$238$		0			0
$239$	−18.5024			−1.19682			−0.598412	−	0.801189i	$-0.704201\pi$
$239$	−18.5024			−1.19682			−0.598412	+	0.801189i	$0.704201\pi$
$240$		0			0
$241$	−28.7403			−1.85133			−0.925664	−	0.378346i	$-0.876493\pi$
$241$	−28.7403			−1.85133			−0.925664	+	0.378346i	$0.876493\pi$
$242$		0			0
$243$	−2.24918			−0.144285
$244$		0			0
$245$	−0.971115	+	12.2295i	−0.0620423	+	0.781316i
$246$		0			0
$247$		−	1.29377i		−	0.0823209i
$248$		0			0
$249$	38.5424			2.44252
$250$		0			0
$251$		−	10.0246i		−	0.632745i	−0.948635	−	0.316372i	$-0.897535\pi$
$251$		−	10.0246i		−	0.632745i	0.948635	−	0.316372i	$-0.102465\pi$
$252$		0			0
$253$	−6.84884			−0.430583
$254$		0			0
$255$	−0.205014	−	0.0162797i	−0.0128385	−	0.00101947i
$256$		0			0
$257$			4.52077i			0.281998i	0.990010	+	0.140999i	$0.0450314\pi$
$257$			4.52077i			0.281998i	−0.990010	+	0.140999i	$0.954969\pi$
$258$		0			0
$259$		−	12.4655i		−	0.774570i
$260$		0			0
$261$			34.1443i			2.11348i
$262$		0			0
$263$			22.0727i			1.36106i	0.732721	+	0.680529i	$0.238250\pi$
$263$			22.0727i			1.36106i	−0.732721	+	0.680529i	$0.761750\pi$
$264$		0			0
$265$	1.97775	−	24.9064i	0.121492	−	1.52999i
$266$		0			0
$267$	4.38542			0.268383
$268$		0			0
$269$			29.1556i			1.77765i	0.458248	+	0.888824i	$0.348477\pi$
$269$			29.1556i			1.77765i	−0.458248	+	0.888824i	$0.651523\pi$
$270$		0			0
$271$	−13.9674			−0.848456			−0.424228	−	0.905555i	$-0.639455\pi$
$271$	−13.9674			−0.848456			−0.424228	+	0.905555i	$0.639455\pi$
$272$		0			0
$273$		−	14.1312i		−	0.855257i
$274$		0			0
$275$	0.789098	−	4.93734i	0.0475844	−	0.297733i
$276$		0			0
$277$	−4.58067			−0.275226			−0.137613	−	0.990486i	$-0.543943\pi$
$277$	−4.58067			−0.275226			−0.137613	+	0.990486i	$0.543943\pi$
$278$		0			0
$279$	23.7367			1.42108
$280$		0			0
$281$	5.31591			0.317121			0.158560	−	0.987349i	$-0.449315\pi$
$281$	5.31591			0.317121			0.158560	+	0.987349i	$0.449315\pi$
$282$		0			0
$283$	9.48374			0.563750			0.281875	−	0.959451i	$-0.409044\pi$
$283$	9.48374			0.563750			0.281875	+	0.959451i	$0.409044\pi$
$284$		0			0
$285$	2.29021	+	0.181860i	0.135660	+	0.0107724i
$286$		0			0
$287$			10.0904i			0.595617i
$288$		0			0
$289$	16.9991			0.999945
$290$		0			0
$291$			22.2591i			1.30485i
$292$		0			0
$293$	−12.5808			−0.734980			−0.367490	−	0.930027i	$-0.619783\pi$
$293$	−12.5808			−0.734980			−0.367490	+	0.930027i	$0.619783\pi$
$294$		0			0
$295$	25.7492	+	2.04468i	1.49918	+	0.119046i
$296$		0			0
$297$		−	9.42804i		−	0.547070i
$298$		0			0
$299$		−	26.0469i		−	1.50633i
$300$		0			0
$301$		−	7.24167i		−	0.417403i
$302$		0			0
$303$		−	18.5705i		−	1.06685i
$304$		0			0
$305$	−5.58971	−	0.443864i	−0.320066	−	0.0254156i
$306$		0			0
$307$	−15.0059			−0.856433			−0.428217	−	0.903676i	$-0.640858\pi$
$307$	−15.0059			−0.856433			−0.428217	+	0.903676i	$0.640858\pi$
$308$		0			0
$309$			0.0867823i			0.00493687i
$310$		0			0
$311$	−0.515721			−0.0292439			−0.0146219	−	0.999893i	$-0.504654\pi$
$311$	−0.515721			−0.0292439			−0.0146219	+	0.999893i	$0.504654\pi$
$312$		0			0
$313$			16.7399i			0.946195i	0.881010	+	0.473097i	$0.156864\pi$
$313$			16.7399i			0.946195i	−0.881010	+	0.473097i	$0.843136\pi$
$314$		0			0
$315$	16.7876	+	1.33306i	0.945877	+	0.0751096i
$316$		0			0
$317$	−11.3238			−0.636010			−0.318005	−	0.948089i	$-0.603013\pi$
$317$	−11.3238			−0.636010			−0.318005	+	0.948089i	$0.603013\pi$
$318$		0			0
$319$	−5.57762			−0.312287
$320$		0			0
$321$	−0.212164			−0.0118419
$322$		0			0
$323$	0.0103597			0.000576427
$324$		0			0
$325$	18.7773	+	3.00103i	1.04158	+	0.166467i
$326$		0			0
$327$			15.7802i			0.872649i
$328$		0			0
$329$	2.35521			0.129847
$330$		0			0
$331$			26.8748i			1.47717i	0.674160	+	0.738586i	$0.264506\pi$
$331$			26.8748i			1.47717i	−0.674160	+	0.738586i	$0.735494\pi$
$332$		0			0
$333$	62.0266			3.39904
$334$		0			0
$335$	2.14249	−	26.9810i	0.117057	−	1.47413i
$336$		0			0
$337$			2.32186i			0.126480i	0.997998	+	0.0632400i	$0.0201433\pi$
$337$			2.32186i			0.126480i	−0.997998	+	0.0632400i	$0.979857\pi$
$338$		0			0
$339$		−	24.3709i		−	1.32365i
$340$		0			0
$341$			3.87751i			0.209979i
$342$		0			0
$343$		−	15.3617i		−	0.829454i
$344$		0			0
$345$	46.1077	+	3.66129i	2.48236	+	0.197117i
$346$		0			0
$347$	27.5505			1.47899			0.739495	−	0.673162i	$-0.235064\pi$
$347$	27.5505			1.47899			0.739495	+	0.673162i	$0.235064\pi$
$348$		0			0
$349$			21.2033i			1.13499i	0.823378	+	0.567493i	$0.192087\pi$
$349$			21.2033i			1.13499i	−0.823378	+	0.567493i	$0.807913\pi$
$350$		0			0
$351$	35.8559			1.91385
$352$		0			0
$353$			11.3136i			0.602165i	0.953598	+	0.301082i	$0.0973479\pi$
$353$			11.3136i			0.602165i	−0.953598	+	0.301082i	$0.902652\pi$
$354$		0			0
$355$	2.00972	−	25.3090i	0.106665	−	1.34326i
$356$		0			0
$357$	0.113153			0.00598868
$358$		0			0
$359$	6.31172			0.333120			0.166560	−	0.986031i	$-0.446734\pi$
$359$	6.31172			0.333120			0.166560	+	0.986031i	$0.446734\pi$
$360$		0			0
$361$	18.8843			0.993909
$362$		0			0
$363$	3.02021			0.158520
$364$		0			0
$365$	31.1330	+	2.47219i	1.62958	+	0.129400i
$366$		0			0
$367$		−	0.106810i		−	0.00557546i	−0.999996	−	0.00278773i	$-0.999113\pi$
$367$		−	0.106810i		−	0.00557546i	0.999996	−	0.00278773i	$-0.000887363\pi$
$368$		0			0
$369$	−50.2083			−2.61374
$370$		0			0
$371$			13.7465i			0.713683i
$372$		0			0
$373$	−21.0446			−1.08965			−0.544824	−	0.838551i	$-0.683403\pi$
$373$	−21.0446			−1.08965			−0.544824	+	0.838551i	$0.683403\pi$
$374$		0			0
$375$	−7.95180	+	32.8173i	−0.410629	+	1.69468i
$376$		0			0
$377$		−	21.2124i		−	1.09249i
$378$		0			0
$379$			14.8971i			0.765211i	0.923912	+	0.382606i	$0.124973\pi$
$379$			14.8971i			0.765211i	−0.923912	+	0.382606i	$0.875027\pi$
$380$		0			0
$381$		−	14.8288i		−	0.759701i
$382$		0			0
$383$		−	12.7457i		−	0.651276i	−0.945494	−	0.325638i	$-0.894421\pi$
$383$		−	12.7457i		−	0.651276i	0.945494	−	0.325638i	$-0.105579\pi$
$384$		0			0
$385$	−0.217762	+	2.74234i	−0.0110982	+	0.139763i
$386$		0			0
$387$	36.0335			1.83169
$388$		0			0
$389$		−	21.5588i		−	1.09307i	−0.837435	−	0.546536i	$-0.815946\pi$
$389$		−	21.5588i		−	1.09307i	0.837435	−	0.546536i	$-0.184054\pi$
$390$		0			0
$391$	0.208566			0.0105477
$392$		0			0
$393$			57.5090i			2.90095i
$394$		0			0
$395$	0.0616735	−	0.776672i	0.00310313	−	0.0390786i
$396$		0			0
$397$	−8.35200			−0.419175			−0.209588	−	0.977790i	$-0.567212\pi$
$397$	−8.35200			−0.419175			−0.209588	+	0.977790i	$0.567212\pi$
$398$		0			0
$399$	−1.26403			−0.0632805
$400$		0			0
$401$	−4.96500			−0.247940			−0.123970	−	0.992286i	$-0.539563\pi$
$401$	−4.96500			−0.247940			−0.123970	+	0.992286i	$0.539563\pi$
$402$		0			0
$403$	−14.7466			−0.734581
$404$		0			0
$405$	−1.78944	+	22.5350i	−0.0889181	+	1.11977i
$406$		0			0
$407$			10.1323i			0.502241i
$408$		0			0
$409$	5.94435			0.293929			0.146965	−	0.989142i	$-0.453050\pi$
$409$	5.94435			0.293929			0.146965	+	0.989142i	$0.453050\pi$
$410$		0			0
$411$		−	29.6465i		−	1.46235i
$412$		0			0
$413$	−14.2117			−0.699310
$414$		0			0
$415$	2.25883	−	28.4461i	0.110881	−	1.39636i
$416$		0			0
$417$			63.9091i			3.12964i
$418$		0			0
$419$			4.16751i			0.203596i	0.994805	+	0.101798i	$0.0324596\pi$
$419$			4.16751i			0.203596i	−0.994805	+	0.101798i	$0.967540\pi$
$420$		0			0
$421$			2.30672i			0.112423i	0.998419	+	0.0562113i	$0.0179020\pi$
$421$			2.30672i			0.112423i	−0.998419	+	0.0562113i	$0.982098\pi$
$422$		0			0
$423$			11.7192i			0.569807i
$424$		0			0
$425$	−0.0240303	+	0.150356i	−0.00116564	+	0.00729333i
$426$		0			0
$427$	3.08511			0.149299
$428$		0			0
$429$			11.4862i			0.554559i
$430$		0			0
$431$	−23.5127			−1.13257			−0.566284	−	0.824210i	$-0.691619\pi$
$431$	−23.5127			−1.13257			−0.566284	+	0.824210i	$0.691619\pi$
$432$		0			0
$433$			17.0454i			0.819148i	0.912277	+	0.409574i	$0.134323\pi$
$433$			17.0454i			0.819148i	−0.912277	+	0.409574i	$0.865677\pi$
$434$		0			0
$435$	37.5496	+	2.98172i	1.80037	+	0.142962i
$436$		0			0
$437$	−2.32989			−0.111454
$438$		0			0
$439$	−10.5695			−0.504457			−0.252228	−	0.967668i	$-0.581163\pi$
$439$	−10.5695			−0.504457			−0.252228	+	0.967668i	$0.581163\pi$
$440$		0			0
$441$	33.5860			1.59933
$442$		0			0
$443$	31.8419			1.51285			0.756427	−	0.654078i	$-0.226943\pi$
$443$	31.8419			1.51285			0.756427	+	0.654078i	$0.226943\pi$
$444$		0			0
$445$	0.257013	−	3.23664i	0.0121836	−	0.153431i
$446$		0			0
$447$			35.7113i			1.68909i
$448$		0			0
$449$	−24.8757			−1.17396			−0.586979	−	0.809602i	$-0.699683\pi$
$449$	−24.8757			−1.17396			−0.586979	+	0.809602i	$0.699683\pi$
$450$		0			0
$451$		−	8.20176i		−	0.386206i
$452$		0			0
$453$	33.1193			1.55608
$454$		0			0
$455$	−10.4294	−	0.828175i	−0.488940	−	0.0388254i
$456$		0			0
$457$			19.9744i			0.934361i	0.884162	+	0.467180i	$0.154730\pi$
$457$			19.9744i			0.934361i	−0.884162	+	0.467180i	$0.845270\pi$
$458$		0			0
$459$			0.287110i			0.0134012i
$460$		0			0
$461$			5.43282i			0.253032i	0.991965	+	0.126516i	$0.0403794\pi$
$461$			5.43282i			0.253032i	−0.991965	+	0.126516i	$0.959621\pi$
$462$		0			0
$463$		−	7.01967i		−	0.326232i	−0.986607	−	0.163116i	$-0.947846\pi$
$463$		−	7.01967i		−	0.326232i	0.986607	−	0.163116i	$-0.0521545\pi$
$464$		0			0
$465$	2.07286	−	26.1041i	0.0961266	−	1.21055i
$466$		0			0
$467$	−6.97230			−0.322640			−0.161320	−	0.986902i	$-0.551575\pi$
$467$	−6.97230			−0.322640			−0.161320	+	0.986902i	$0.551575\pi$
$468$		0			0
$469$			14.8915i			0.687627i
$470$		0			0
$471$	19.2498			0.886985
$472$		0			0
$473$			5.88624i			0.270650i
$474$		0			0
$475$	0.268441	−	1.67962i	0.0123169	−	0.0770663i
$476$		0			0
$477$	−68.4006			−3.13185
$478$		0			0
$479$	28.1631			1.28680			0.643402	−	0.765529i	$-0.277523\pi$
$479$	28.1631			1.28680			0.643402	+	0.765529i	$0.277523\pi$
$480$		0			0
$481$	−38.5345			−1.75702
$482$		0			0
$483$	−25.4481			−1.15793
$484$		0			0
$485$	16.4282	+	1.30452i	0.745968	+	0.0592354i
$486$		0			0
$487$		−	13.8662i		−	0.628336i	−0.949367	−	0.314168i	$-0.898275\pi$
$487$		−	13.8662i		−	0.628336i	0.949367	−	0.314168i	$-0.101725\pi$
$488$		0			0
$489$	3.69396			0.167047
$490$		0			0
$491$		−	32.5993i		−	1.47119i	−0.677423	−	0.735594i	$-0.736903\pi$
$491$		−	32.5993i		−	1.47119i	0.677423	−	0.735594i	$-0.263097\pi$
$492$		0			0
$493$	0.169854			0.00764985
$494$		0			0
$495$	−13.6455	−	1.08355i	−0.613319	−	0.0487020i
$496$		0			0
$497$			13.9687i			0.626581i
$498$		0			0
$499$			1.61926i			0.0724880i	0.999343	+	0.0362440i	$0.0115394\pi$
$499$			1.61926i			0.0724880i	−0.999343	+	0.0362440i	$0.988461\pi$
$500$		0			0
$501$		−	0.215270i		−	0.00961757i
$502$		0			0
$503$			9.41640i			0.419856i	0.977717	+	0.209928i	$0.0673230\pi$
$503$			9.41640i			0.419856i	−0.977717	+	0.209928i	$0.932677\pi$
$504$		0			0
$505$	−13.7059	−	1.08835i	−0.609903	−	0.0484308i
$506$		0			0
$507$	−4.42072			−0.196331
$508$		0			0
$509$			0.257770i			0.0114254i	0.999984	+	0.00571272i	$0.00181842\pi$
$509$			0.257770i			0.0114254i	−0.999984	+	0.00571272i	$0.998182\pi$
$510$		0			0
$511$	−17.1831			−0.760137
$512$		0			0
$513$		−	3.20730i		−	0.141606i
$514$		0			0
$515$	0.0640493	+	0.00508599i	0.00282235	+	0.000224116i
$516$		0			0
$517$	−1.91438			−0.0841945
$518$		0			0
$519$	−52.7293			−2.31456
$520$		0			0
$521$	−3.74585			−0.164108			−0.0820542	−	0.996628i	$-0.526148\pi$
$521$	−3.74585			−0.164108			−0.0820542	+	0.996628i	$0.526148\pi$
$522$		0			0
$523$	2.51018			0.109762			0.0548812	−	0.998493i	$-0.482522\pi$
$523$	2.51018			0.109762			0.0548812	+	0.998493i	$0.482522\pi$
$524$		0			0
$525$	2.93203	−	18.3456i	0.127964	−	0.800666i
$526$		0			0
$527$		−	0.118081i		−	0.00514369i
$528$		0			0
$529$	−23.9065			−1.03941
$530$		0			0
$531$		−	70.7151i		−	3.06878i
$532$		0			0
$533$	31.1923			1.35109
$534$		0			0
$535$	−0.0124342	+	0.156587i	−0.000537576	+	0.00676985i
$536$		0			0
$537$			12.7358i			0.549588i
$538$		0			0
$539$			5.48643i			0.236317i
$540$		0			0
$541$		−	30.4550i		−	1.30936i	−0.755906	−	0.654680i	$-0.772803\pi$
$541$		−	30.4550i		−	1.30936i	0.755906	−	0.654680i	$-0.227197\pi$
$542$		0			0
$543$			44.9696i			1.92983i
$544$		0			0
$545$	11.6465	+	0.924820i	0.498883	+	0.0396150i
$546$		0			0
$547$	−42.7674			−1.82860			−0.914301	−	0.405036i	$-0.867259\pi$
$547$	−42.7674			−1.82860			−0.914301	+	0.405036i	$0.867259\pi$
$548$		0			0
$549$			15.3510i			0.655166i
$550$		0			0
$551$	−1.89744			−0.0808336
$552$		0			0
$553$			0.428666i			0.0182287i
$554$		0			0
$555$	5.41661	−	68.2129i	0.229922	−	2.89548i
$556$		0			0
$557$	43.3665			1.83750			0.918749	−	0.394843i	$-0.129201\pi$
$557$	43.3665			1.83750			0.918749	+	0.394843i	$0.129201\pi$
$558$		0			0
$559$	−22.3861			−0.946830
$560$		0			0
$561$	−0.0919738			−0.00388314
$562$		0			0
$563$	−21.7140			−0.915138			−0.457569	−	0.889174i	$-0.651280\pi$
$563$	−21.7140			−0.915138			−0.457569	+	0.889174i	$0.651280\pi$
$564$		0			0
$565$	−17.9869	−	1.42829i	−0.756713	−	0.0600886i
$566$		0			0
$567$		−	12.4376i		−	0.522332i
$568$		0			0
$569$	−31.6955			−1.32875			−0.664373	−	0.747401i	$-0.731301\pi$
$569$	−31.6955			−1.32875			−0.664373	+	0.747401i	$0.731301\pi$
$570$		0			0
$571$			3.41171i			0.142776i	0.997449	+	0.0713878i	$0.0227428\pi$
$571$			3.41171i			0.142776i	−0.997449	+	0.0713878i	$0.977257\pi$
$572$		0			0
$573$	−47.8529			−1.99908
$574$		0			0
$575$	5.40440	−	33.8150i	0.225379	−	1.41018i
$576$		0			0
$577$		−	23.5647i		−	0.981013i	−0.871438	−	0.490506i	$-0.836812\pi$
$577$		−	23.5647i		−	0.981013i	0.871438	−	0.490506i	$-0.163188\pi$
$578$		0			0
$579$			67.3501i			2.79898i
$580$		0			0
$581$			15.7001i			0.651351i
$582$		0			0
$583$		−	11.1736i		−	0.462761i
$584$		0			0
$585$	4.12088	−	51.8954i	0.170377	−	2.14561i
$586$		0			0
$587$	14.4902			0.598075			0.299038	−	0.954241i	$-0.403334\pi$
$587$	14.4902			0.598075			0.299038	+	0.954241i	$0.403334\pi$
$588$		0			0
$589$			1.31908i			0.0543517i
$590$		0			0
$591$	43.5866			1.79291
$592$		0			0
$593$			17.5719i			0.721592i	0.932645	+	0.360796i	$0.117495\pi$
$593$			17.5719i			0.721592i	−0.932645	+	0.360796i	$0.882505\pi$
$594$		0			0
$595$	0.00663147	−	0.0835120i	0.000271864	−	0.00342366i
$596$		0			0
$597$	21.1822			0.866930
$598$		0			0
$599$	14.3427			0.586026			0.293013	−	0.956108i	$-0.405342\pi$
$599$	14.3427			0.586026			0.293013	+	0.956108i	$0.405342\pi$
$600$		0			0
$601$	4.23245			0.172645			0.0863226	−	0.996267i	$-0.472488\pi$
$601$	4.23245			0.172645			0.0863226	+	0.996267i	$0.472488\pi$
$602$		0			0
$603$	−74.0981			−3.01751
$604$		0			0
$605$	0.177003	−	2.22905i	0.00719620	−	0.0906238i
$606$		0			0
$607$		−	30.7964i		−	1.24999i	−0.780629	−	0.624994i	$-0.785101\pi$
$607$		−	30.7964i		−	1.24999i	0.780629	−	0.624994i	$-0.214899\pi$
$608$		0			0
$609$	−20.7246			−0.839805
$610$		0			0
$611$		−	7.28063i		−	0.294543i
$612$		0			0
$613$	−22.6121			−0.913296			−0.456648	−	0.889648i	$-0.650950\pi$
$613$	−22.6121			−0.913296			−0.456648	+	0.889648i	$0.650950\pi$
$614$		0			0
$615$	−4.38455	+	55.2159i	−0.176802	+	2.22652i
$616$		0			0
$617$			22.0095i			0.886071i	0.896504	+	0.443036i	$0.146098\pi$
$617$			22.0095i			0.886071i	−0.896504	+	0.443036i	$0.853902\pi$
$618$		0			0
$619$			24.0876i			0.968162i	0.875023	+	0.484081i	$0.160846\pi$
$619$			24.0876i			0.968162i	−0.875023	+	0.484081i	$0.839154\pi$
$620$		0			0
$621$		−	64.5711i		−	2.59115i
$622$		0			0
$623$			1.78639i			0.0715701i
$624$		0			0
$625$	23.7546	+	7.79209i	0.950186	+	0.311684i
$626$		0			0
$627$	1.02744			0.0410319
$628$		0			0
$629$		−	0.308558i		−	0.0123030i
$630$		0			0
$631$	−9.81595			−0.390767			−0.195383	−	0.980727i	$-0.562595\pi$
$631$	−9.81595			−0.390767			−0.195383	+	0.980727i	$0.562595\pi$
$632$		0			0
$633$		−	70.9746i		−	2.82099i
$634$		0			0
$635$	−10.9443	−	0.869059i	−0.434312	−	0.0344875i
$636$		0			0
$637$	−20.8655			−0.826723
$638$		0			0
$639$	−69.5061			−2.74962
$640$		0			0
$641$	−26.6019			−1.05071			−0.525357	−	0.850882i	$-0.676068\pi$
$641$	−26.6019			−1.05071			−0.525357	+	0.850882i	$0.676068\pi$
$642$		0			0
$643$	17.1963			0.678157			0.339078	−	0.940758i	$-0.389885\pi$
$643$	17.1963			0.678157			0.339078	+	0.940758i	$0.389885\pi$
$644$		0			0
$645$	3.14670	−	39.6273i	0.123901	−	1.56032i
$646$		0			0
$647$			21.0574i			0.827850i	0.910311	+	0.413925i	$0.135842\pi$
$647$			21.0574i			0.827850i	−0.910311	+	0.413925i	$0.864158\pi$
$648$		0			0
$649$	11.5516			0.453442
$650$		0			0
$651$			14.4076i			0.564677i
$652$		0			0
$653$	−19.3219			−0.756124			−0.378062	−	0.925780i	$-0.623409\pi$
$653$	−19.3219			−0.756124			−0.378062	+	0.925780i	$0.623409\pi$
$654$		0			0
$655$	42.4443	+	3.37039i	1.65844	+	0.131692i
$656$		0			0
$657$		−	85.5008i		−	3.33570i
$658$		0			0
$659$		−	40.3740i		−	1.57275i	−0.617750	−	0.786374i	$-0.711956\pi$
$659$		−	40.3740i		−	1.57275i	0.617750	−	0.786374i	$-0.288044\pi$
$660$		0			0
$661$			41.4368i			1.61170i	0.592117	+	0.805852i	$0.298292\pi$
$661$			41.4368i			1.61170i	−0.592117	+	0.805852i	$0.701708\pi$
$662$		0			0
$663$		−	0.349787i		−	0.0135846i
$664$		0			0
$665$	−0.0740799	+	0.932910i	−0.00287270	+	0.0361767i
$666$		0			0
$667$	−38.2002			−1.47912
$668$		0			0
$669$			2.47589i			0.0957235i
$670$		0			0
$671$	−2.50766			−0.0968072
$672$		0			0
$673$		−	21.4755i		−	0.827821i	−0.910318	−	0.413910i	$-0.864163\pi$
$673$		−	21.4755i		−	0.827821i	0.910318	−	0.413910i	$-0.135837\pi$
$674$		0			0
$675$	46.5494	+	7.43965i	1.79169	+	0.286352i
$676$		0			0
$677$	32.9052			1.26465			0.632325	−	0.774703i	$-0.282101\pi$
$677$	32.9052			1.26465			0.632325	+	0.774703i	$0.282101\pi$
$678$		0			0
$679$	−9.06718			−0.347967
$680$		0			0
$681$	2.16115			0.0828156
$682$		0			0
$683$	0.0734613			0.00281092			0.00140546	−	0.999999i	$-0.499553\pi$
$683$	0.0734613			0.00281092			0.00140546	+	0.999999i	$0.499553\pi$
$684$		0			0
$685$	−21.8805	−	1.73747i	−0.836010	−	0.0663854i
$686$		0			0
$687$			75.8771i			2.89489i
$688$		0			0
$689$	42.4944			1.61891
$690$		0			0
$691$			17.0275i			0.647758i	0.946099	+	0.323879i	$0.104987\pi$
$691$			17.0275i			0.647758i	−0.946099	+	0.323879i	$0.895013\pi$
$692$		0			0
$693$	7.53130			0.286090
$694$		0			0
$695$	47.1678	+	3.74547i	1.78918	+	0.142074i
$696$		0			0
$697$			0.249767i			0.00946059i
$698$		0			0
$699$			41.3334i			1.56337i
$700$		0			0
$701$			18.5032i			0.698855i	0.936963	+	0.349428i	$0.113624\pi$
$701$			18.5032i			0.698855i	−0.936963	+	0.349428i	$0.886376\pi$
$702$		0			0
$703$			3.44690i			0.130002i
$704$		0			0
$705$	12.8880	+	1.02340i	0.485391	+	0.0385436i
$706$		0			0
$707$	7.56463			0.284497
$708$		0			0
$709$			37.1678i			1.39587i	0.716162	+	0.697934i	$0.245897\pi$
$709$			37.1678i			1.39587i	−0.716162	+	0.697934i	$0.754103\pi$
$710$		0			0
$711$	−2.13298			−0.0799929
$712$		0			0
$713$			26.5564i			0.994545i
$714$		0			0
$715$	8.47735	+	0.673164i	0.317035	+	0.0251749i
$716$		0			0
$717$	55.8812			2.08692
$718$		0			0
$719$	−31.5730			−1.17748			−0.588738	−	0.808324i	$-0.700375\pi$
$719$	−31.5730			−1.17748			−0.588738	+	0.808324i	$0.700375\pi$
$720$		0			0
$721$	−0.0353505			−0.00131652
$722$		0			0
$723$	86.8018			3.22819
$724$		0			0
$725$	4.40129	−	27.5386i	0.163460	−	1.02276i
$726$		0			0
$727$		−	9.05786i		−	0.335938i	−0.985792	−	0.167969i	$-0.946279\pi$
$727$		−	9.05786i		−	0.335938i	0.985792	−	0.167969i	$-0.0537208\pi$
$728$		0			0
$729$	−23.5360			−0.871704
$730$		0			0
$731$		−	0.179253i		−	0.00662989i
$732$		0			0
$733$	20.8006			0.768287			0.384143	−	0.923273i	$-0.374497\pi$
$733$	20.8006			0.768287			0.384143	+	0.923273i	$0.374497\pi$
$734$		0			0
$735$	2.93297	−	36.9357i	0.108184	−	1.36240i
$736$		0			0
$737$		−	12.1043i		−	0.445866i
$738$		0			0
$739$			33.3544i			1.22696i	0.789710	+	0.613481i	$0.210231\pi$
$739$			33.3544i			1.22696i	−0.789710	+	0.613481i	$0.789769\pi$
$740$		0			0
$741$			3.90747i			0.143544i
$742$		0			0
$743$		−	42.4956i		−	1.55901i	−0.626394	−	0.779507i	$-0.715470\pi$
$743$		−	42.4956i		−	1.55901i	0.626394	−	0.779507i	$-0.284530\pi$
$744$		0			0
$745$	26.3566	+	2.09291i	0.965630	+	0.0766782i
$746$		0			0
$747$	−78.1215			−2.85832
$748$		0			0
$749$		−	0.0864245i		−	0.00315788i
$750$		0			0
$751$	−7.99715			−0.291820			−0.145910	−	0.989298i	$-0.546611\pi$
$751$	−7.99715			−0.291820			−0.145910	+	0.989298i	$0.546611\pi$
$752$		0			0
$753$			30.2763i			1.10333i
$754$		0			0
$755$	1.94100	−	24.4436i	0.0706402	−	0.889593i
$756$		0			0
$757$	−9.94886			−0.361597			−0.180799	−	0.983520i	$-0.557868\pi$
$757$	−9.94886			−0.361597			−0.180799	+	0.983520i	$0.557868\pi$
$758$		0			0
$759$	20.6849			0.750814
$760$		0			0
$761$	−20.2801			−0.735151			−0.367576	−	0.929994i	$-0.619812\pi$
$761$	−20.2801			−0.735151			−0.367576	+	0.929994i	$0.619812\pi$
$762$		0			0
$763$	−6.42803			−0.232710
$764$		0			0
$765$	0.415543	+	0.0329972i	0.0150240	+	0.00119302i
$766$		0			0
$767$			43.9323i			1.58630i
$768$		0			0
$769$	23.3928			0.843565			0.421782	−	0.906697i	$-0.361405\pi$
$769$	23.3928			0.843565			0.421782	+	0.906697i	$0.361405\pi$
$770$		0			0
$771$		−	13.6537i		−	0.491725i
$772$		0			0
$773$	−1.69226			−0.0608665			−0.0304332	−	0.999537i	$-0.509689\pi$
$773$	−1.69226			−0.0608665			−0.0304332	+	0.999537i	$0.509689\pi$
$774$		0			0
$775$	−19.1446	−	3.05973i	−0.687693	−	0.109909i
$776$		0			0
$777$			37.6485i			1.35063i
$778$		0			0
$779$		−	2.79014i		−	0.0999670i
$780$		0			0
$781$		−	11.3541i		−	0.406283i
$782$		0			0
$783$		−	52.5860i		−	1.87927i
$784$		0			0
$785$	1.12816	−	14.2073i	0.0402658	−	0.507079i
$786$		0			0
$787$	17.3403			0.618116			0.309058	−	0.951043i	$-0.399986\pi$
$787$	17.3403			0.618116			0.309058	+	0.951043i	$0.399986\pi$
$788$		0			0
$789$		−	66.6640i		−	2.37330i
$790$		0			0
$791$	9.92742			0.352978
$792$		0			0
$793$		−	9.53694i		−	0.338667i
$794$		0			0
$795$	−5.97323	+	75.2226i	−0.211849	+	2.66787i
$796$		0			0
$797$	−10.7430			−0.380538			−0.190269	−	0.981732i	$-0.560936\pi$
$797$	−10.7430			−0.380538			−0.190269	+	0.981732i	$0.560936\pi$
$798$		0			0
$799$	0.0582984			0.00206245
$800$		0			0
$801$	−8.88880			−0.314070
$802$		0			0
$803$	13.9669			0.492883
$804$		0			0
$805$	−1.49142	+	18.7818i	−0.0525655	+	0.661973i
$806$		0			0
$807$		−	88.0560i		−	3.09972i
$808$		0			0
$809$	18.1558			0.638325			0.319162	−	0.947700i	$-0.396598\pi$
$809$	18.1558			0.638325			0.319162	+	0.947700i	$0.396598\pi$
$810$		0			0
$811$		−	15.5321i		−	0.545405i	−0.962098	−	0.272702i	$-0.912083\pi$
$811$		−	15.5321i		−	0.545405i	0.962098	−	0.272702i	$-0.0879174\pi$
$812$		0			0
$813$	42.1843			1.47947
$814$		0			0
$815$	0.216489	−	2.72631i	0.00758328	−	0.0954985i
$816$		0			0
$817$			2.00243i			0.0700560i
$818$		0			0
$819$			28.6424i			1.00085i
$820$		0			0
$821$			32.7264i			1.14216i	0.820895	+	0.571079i	$0.193475\pi$
$821$			32.7264i			1.14216i	−0.820895	+	0.571079i	$0.806525\pi$
$822$		0			0
$823$		−	23.7463i		−	0.827744i	−0.910335	−	0.413872i	$-0.864176\pi$
$823$		−	23.7463i		−	0.827744i	0.910335	−	0.413872i	$-0.135824\pi$
$824$		0			0
$825$	−2.38324	+	14.9118i	−0.0829738	+	0.519162i
$826$		0			0
$827$	−19.8970			−0.691887			−0.345943	−	0.938255i	$-0.612441\pi$
$827$	−19.8970			−0.691887			−0.345943	+	0.938255i	$0.612441\pi$
$828$		0			0
$829$		−	15.9190i		−	0.552888i	−0.961030	−	0.276444i	$-0.910844\pi$
$829$		−	15.9190i		−	0.552888i	0.961030	−	0.276444i	$-0.0891560\pi$
$830$		0			0
$831$	13.8346			0.479916
$832$		0			0
$833$		−	0.167077i		−	0.00578888i
$834$		0			0
$835$	−0.158879	−	0.0126162i	−0.00549825	−	0.000436601i
$836$		0			0
$837$	−36.5573			−1.26360
$838$		0			0
$839$	−21.0800			−0.727763			−0.363881	−	0.931445i	$-0.618549\pi$
$839$	−21.0800			−0.727763			−0.363881	+	0.931445i	$0.618549\pi$
$840$		0			0
$841$	−2.10986			−0.0727538
$842$		0			0
$843$	−16.0552			−0.552969
$844$		0			0
$845$	−0.259082	+	3.26269i	−0.00891269	+	0.112240i
$846$		0			0
$847$			1.23027i			0.0422726i
$848$		0			0
$849$	−28.6429			−0.983021
$850$		0			0
$851$			69.3947i			2.37882i
$852$		0			0
$853$	−38.3248			−1.31222			−0.656109	−	0.754666i	$-0.727799\pi$
$853$	−38.3248			−1.31222			−0.656109	+	0.754666i	$0.727799\pi$
$854$		0			0
$855$	−4.64202	−	0.368611i	−0.158754	−	0.0126062i
$856$		0			0
$857$		−	39.9567i		−	1.36490i	−0.730934	−	0.682448i	$-0.760915\pi$
$857$		−	39.9567i		−	1.36490i	0.730934	−	0.682448i	$-0.239085\pi$
$858$		0			0
$859$		−	27.8648i		−	0.950736i	−0.879787	−	0.475368i	$-0.842315\pi$
$859$		−	27.8648i		−	0.950736i	0.879787	−	0.475368i	$-0.157685\pi$
$860$		0			0
$861$		−	30.4751i		−	1.03859i
$862$		0			0
$863$		−	3.95489i		−	0.134626i	−0.997732	−	0.0673130i	$-0.978557\pi$
$863$		−	3.95489i		−	0.134626i	0.997732	−	0.0673130i	$-0.0214426\pi$
$864$		0			0
$865$	−3.09027	+	38.9167i	−0.105072	+	1.32321i
$866$		0			0
$867$	−51.3407			−1.74362
$868$		0			0
$869$		−	0.348432i		−	0.0118197i
$870$		0			0
$871$	46.0340			1.55980
$872$		0			0
$873$		−	45.1170i		−	1.52698i
$874$		0			0
$875$	−13.3680	−	3.23914i	−0.451922	−	0.109503i
$876$		0			0
$877$	−11.8619			−0.400548			−0.200274	−	0.979740i	$-0.564183\pi$
$877$	−11.8619			−0.400548			−0.200274	+	0.979740i	$0.564183\pi$
$878$		0			0
$879$	37.9967			1.28160
$880$		0			0
$881$	35.7527			1.20454			0.602270	−	0.798292i	$-0.294263\pi$
$881$	35.7527			1.20454			0.602270	+	0.798292i	$0.294263\pi$
$882$		0			0
$883$	42.2030			1.42024			0.710122	−	0.704079i	$-0.248640\pi$
$883$	42.2030			1.42024			0.710122	+	0.704079i	$0.248640\pi$
$884$		0			0
$885$	−77.7679	−	6.17535i	−2.61414	−	0.207582i
$886$		0			0
$887$		−	41.6257i		−	1.39766i	−0.715290	−	0.698828i	$-0.753705\pi$
$887$		−	41.6257i		−	1.39766i	0.715290	−	0.698828i	$-0.246295\pi$
$888$		0			0
$889$	6.04045			0.202590
$890$		0			0
$891$			10.1097i			0.338687i
$892$		0			0
$893$	−0.651250			−0.0217932
$894$		0			0
$895$	9.39957	+	0.746395i	0.314193	+	0.0249492i
$896$		0			0
$897$			78.6671i			2.62662i
$898$		0			0
$899$			21.6273i			0.721309i
$900$		0			0
$901$			0.340266i			0.0113359i
$902$		0			0
$903$			21.8714i			0.727833i
$904$		0			0
$905$	33.1896	+	2.63550i	1.10326	+	0.0876071i
$906$		0			0
$907$	−9.58417			−0.318237			−0.159119	−	0.987259i	$-0.550865\pi$
$907$	−9.58417			−0.318237			−0.159119	+	0.987259i	$0.550865\pi$
$908$		0			0
$909$			37.6405i			1.24846i
$910$		0			0
$911$	36.7150			1.21642			0.608211	−	0.793775i	$-0.291887\pi$
$911$	36.7150			1.21642			0.608211	+	0.793775i	$0.291887\pi$
$912$		0			0
$913$		−	12.7615i		−	0.422344i
$914$		0			0
$915$	16.8821	+	1.34056i	0.558104	+	0.0443176i
$916$		0			0
$917$	−23.4261			−0.773599
$918$		0			0
$919$	18.2182			0.600965			0.300482	−	0.953787i	$-0.402852\pi$
$919$	18.2182			0.600965			0.300482	+	0.953787i	$0.402852\pi$
$920$		0			0
$921$	45.3210			1.49338
$922$		0			0
$923$	43.1811			1.42132
$924$		0			0
$925$	−50.0268	−	7.99541i	−1.64487	−	0.262887i
$926$		0			0
$927$		−	0.175899i		−	0.00577728i
$928$		0			0
$929$	−9.03564			−0.296450			−0.148225	−	0.988954i	$-0.547356\pi$
$929$	−9.03564			−0.296450			−0.148225	+	0.988954i	$0.547356\pi$
$930$		0			0
$931$			1.86642i			0.0611693i
$932$		0			0
$933$	1.55758			0.0509930
$934$		0			0
$935$	−0.00539024	+	0.0678809i	−0.000176280	+	0.00221994i
$936$		0			0
$937$		−	17.5696i		−	0.573974i	−0.957935	−	0.286987i	$-0.907346\pi$
$937$		−	17.5696i		−	0.573974i	0.957935	−	0.286987i	$-0.0926536\pi$
$938$		0			0
$939$		−	50.5579i		−	1.64990i
$940$		0			0
$941$		−	19.7859i		−	0.645001i	−0.946569	−	0.322501i	$-0.895477\pi$
$941$		−	19.7859i		−	0.645001i	0.946569	−	0.322501i	$-0.104523\pi$
$942$		0			0
$943$		−	56.1725i		−	1.82923i
$944$		0			0
$945$	−25.8549	−	2.05307i	−0.841059	−	0.0667863i
$946$		0			0
$947$	−19.5398			−0.634957			−0.317478	−	0.948265i	$-0.602836\pi$
$947$	−19.5398			−0.634957			−0.317478	+	0.948265i	$0.602836\pi$
$948$		0			0
$949$			53.1180i			1.72428i
$950$		0			0
$951$	34.2003			1.10902
$952$		0			0
$953$		−	0.995459i		−	0.0322461i	−0.999870	−	0.0161230i	$-0.994868\pi$
$953$		−	0.995459i		−	0.0322461i	0.999870	−	0.0161230i	$-0.00513235\pi$
$954$		0			0
$955$	−2.80448	+	35.3176i	−0.0907509	+	1.14285i
$956$		0			0
$957$	16.8456			0.544540
$958$		0			0
$959$	12.0764			0.389968
$960$		0			0
$961$	−15.9650			−0.514999
$962$		0			0
$963$	0.430035			0.0138577
$964$		0			0
$965$	49.7075	+	3.94714i	1.60014	+	0.127063i
$966$		0			0
$967$		−	53.1391i		−	1.70884i	−0.519583	−	0.854420i	$-0.673913\pi$
$967$		−	53.1391i		−	1.70884i	0.519583	−	0.854420i	$-0.326087\pi$
$968$		0			0
$969$	−0.0312884			−0.00100513
$970$		0			0
$971$			53.2228i			1.70800i	0.520271	+	0.854001i	$0.325831\pi$
$971$			53.2228i			1.70800i	−0.520271	+	0.854001i	$0.674169\pi$
$972$		0			0
$973$	−26.0332			−0.834585
$974$		0			0
$975$	−56.7113	−	9.06374i	−1.81622	−	0.290272i
$976$		0			0
$977$		−	50.2443i		−	1.60746i	−0.594997	−	0.803728i	$-0.702847\pi$
$977$		−	50.2443i		−	1.60746i	0.594997	−	0.803728i	$-0.297153\pi$
$978$		0			0
$979$		−	1.45203i		−	0.0464069i
$980$		0			0
$981$		−	31.9849i		−	1.02120i
$982$		0			0
$983$			27.0658i			0.863266i	0.902049	+	0.431633i	$0.142062\pi$
$983$			27.0658i			0.863266i	−0.902049	+	0.431633i	$0.857938\pi$
$984$		0			0
$985$	2.55445	−	32.1689i	0.0813915	−	1.02499i
$986$		0			0
$987$	−7.11323			−0.226417
$988$		0			0
$989$			40.3139i			1.28191i
$990$		0			0
$991$	29.4884			0.936729			0.468365	−	0.883535i	$-0.344843\pi$
$991$	29.4884			0.936729			0.468365	+	0.883535i	$0.344843\pi$
$992$		0			0
$993$		−	81.1674i		−	2.57577i
$994$		0			0
$995$	1.24141	−	15.6334i	0.0393554	−	0.495613i
$996$		0			0
$997$	1.74831			0.0553695			0.0276848	−	0.999617i	$-0.491187\pi$
$997$	1.74831			0.0553695			0.0276848	+	0.999617i	$0.491187\pi$
$998$		0			0
$999$	−95.5280			−3.02237

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1760.2.l.c.529.3		56
4.3	odd	2		440.2.l.c.309.41	yes	56
5.4	even	2	inner	1760.2.l.c.529.54		56
8.3	odd	2		440.2.l.c.309.15	✓	56
8.5	even	2	inner	1760.2.l.c.529.53		56
20.19	odd	2		440.2.l.c.309.16	yes	56
40.19	odd	2		440.2.l.c.309.42	yes	56
40.29	even	2	inner	1760.2.l.c.529.4		56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.l.c.309.15	✓	56	8.3	odd	2
440.2.l.c.309.16	yes	56	20.19	odd	2
440.2.l.c.309.41	yes	56	4.3	odd	2
440.2.l.c.309.42	yes	56	40.19	odd	2
1760.2.l.c.529.3		56	1.1	even	1	trivial
1760.2.l.c.529.4		56	40.29	even	2	inner
1760.2.l.c.529.53		56	8.5	even	2	inner
1760.2.l.c.529.54		56	5.4	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1760 = 2^{5} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1760.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.02021 q^{3} +(-0.177003 + 2.22905i) q^{5} -1.23027i q^{7} +6.12165 q^{9} +1.00000i q^{11} -3.80312 q^{13} +(0.534586 - 6.73220i) q^{15} -0.0304528i q^{17} +0.340188i q^{19} +3.71568i q^{21} +6.84884i q^{23} +(-4.93734 - 0.789098i) q^{25} -9.42804 q^{27} +5.57762i q^{29} +3.87751 q^{31} -3.02021i q^{33} +(2.74234 + 0.217762i) q^{35} +10.1323 q^{37} +11.4862 q^{39} -8.20176 q^{41} +5.88624 q^{43} +(-1.08355 + 13.6455i) q^{45} +1.91438i q^{47} +5.48643 q^{49} +0.0919738i q^{51} -11.1736 q^{53} +(-2.22905 - 0.177003i) q^{55} -1.02744i q^{57} -11.5516i q^{59} +2.50766i q^{61} -7.53130i q^{63} +(0.673164 - 8.47735i) q^{65} -12.1043 q^{67} -20.6849i q^{69} -11.3541 q^{71} -13.9669i q^{73} +(14.9118 + 2.38324i) q^{75} +1.23027 q^{77} -0.348432 q^{79} +10.1097 q^{81} -12.7615 q^{83} +(0.0678809 + 0.00539024i) q^{85} -16.8456i q^{87} -1.45203 q^{89} +4.67887i q^{91} -11.7109 q^{93} +(-0.758296 - 0.0602143i) q^{95} -7.37006i q^{97} +6.12165i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q + 56 q^{9} + 8 q^{15} + 16 q^{25} + 16 q^{39} - 32 q^{41} - 24 q^{49} + 8 q^{55} - 32 q^{65} - 48 q^{71} + 32 q^{79} + 88 q^{81} - 16 q^{89} - 24 q^{95}+O(q^{100}) \) 56 * q + 56 * q^9 + 8 * q^15 + 16 * q^25 + 16 * q^39 - 32 * q^41 - 24 * q^49 + 8 * q^55 - 32 * q^65 - 48 * q^71 + 32 * q^79 + 88 * q^81 - 16 * q^89 - 24 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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