LMFDB - Embedded newform 320.6.c.l.129.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [320,6,Mod(129,320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(320, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("320.129");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	320.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$51.3228223402$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 6 x^{11} + 18 x^{10} + 348 x^{9} + 21226 x^{8} - 87824 x^{7} + 205428 x^{6} + 2113880 x^{5} + \cdots + 1072562500 $ x^12 - 6x^11 + 18x^10 + 348x^9 + 21226x^8 - 87824x^7 + 205428x^6 + 2113880x^5 + 27827225x^4 - 59798850x^3 + 89646050x^2 + 438522500*x + 1072562500
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{46}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 160)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.1
Root			$-8.04538 + 8.04538i$ of defining polynomial
Character	$\chi$	$=$	320.129
Dual form			320.6.c.l.129.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-21.9637i q^{3} +(-18.7585 - 52.6604i) q^{5} +29.6983i q^{7} -239.406 q^{9} +O(q^{10})$	$q-21.9637i q^{3} +(-18.7585 - 52.6604i) q^{5} +29.6983i q^{7} -239.406 q^{9} +227.001 q^{11} -1067.66i q^{13} +(-1156.62 + 412.007i) q^{15} +686.258i q^{17} -1309.11 q^{19} +652.286 q^{21} -4076.26i q^{23} +(-2421.24 + 1975.66i) q^{25} -78.9360i q^{27} -4535.21 q^{29} -7691.77 q^{31} -4985.79i q^{33} +(1563.92 - 557.096i) q^{35} +9329.38i q^{37} -23449.8 q^{39} +3753.20 q^{41} +2934.28i q^{43} +(4490.90 + 12607.2i) q^{45} -3690.55i q^{47} +15925.0 q^{49} +15072.8 q^{51} +4816.68i q^{53} +(-4258.20 - 11954.0i) q^{55} +28752.9i q^{57} +31168.2 q^{59} +33809.4 q^{61} -7109.96i q^{63} +(-56223.3 + 20027.7i) q^{65} +24844.0i q^{67} -89529.9 q^{69} -14153.6 q^{71} +73107.8i q^{73} +(43392.9 + 53179.4i) q^{75} +6741.55i q^{77} +73646.1 q^{79} -59909.4 q^{81} -67353.7i q^{83} +(36138.6 - 12873.2i) q^{85} +99610.3i q^{87} +60277.5 q^{89} +31707.6 q^{91} +168940. i q^{93} +(24556.9 + 68938.2i) q^{95} -15478.8i q^{97} -54345.4 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 60 q^{5} - 268 q^{9}+O(q^{10}) $ 12 * q + 60 * q^5 - 268 * q^9	$ 12 q + 60 q^{5} - 268 q^{9} - 12080 q^{21} - 16420 q^{25} + 7864 q^{29} + 65560 q^{41} + 60420 q^{45} + 17956 q^{49} + 118232 q^{61} + 11360 q^{65} - 204464 q^{69} - 346436 q^{81} - 96320 q^{85} + 354936 q^{89}+O(q^{100}) $ 12 * q + 60 * q^5 - 268 * q^9 - 12080 * q^21 - 16420 * q^25 + 7864 * q^29 + 65560 * q^41 + 60420 * q^45 + 17956 * q^49 + 118232 * q^61 + 11360 * q^65 - 204464 * q^69 - 346436 * q^81 - 96320 * q^85 + 354936 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$.

$n$	$191$	$257$	$261$
$\chi(n)$	$1$	$-1$	$1$

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
$3$		−	21.9637i		−	1.40897i	−0.709716	−	0.704487i	$-0.751177\pi$
$3$		−	21.9637i		−	1.40897i	0.709716	−	0.704487i	$-0.248823\pi$
$4$		0			0
$5$	−18.7585	−	52.6604i	−0.335563	−	0.942018i
$5$	−18.7585	−	52.6604i	−0.335563	−	0.942018i
$6$		0			0
$7$			29.6983i			0.229080i	0.993419	+	0.114540i	$0.0365394\pi$
$7$			29.6983i			0.229080i	−0.993419	+	0.114540i	$0.963461\pi$
$8$		0			0
$9$	−239.406			−0.985210
$10$		0			0
$11$	227.001			0.565648			0.282824	−	0.959172i	$-0.408729\pi$
$11$	227.001			0.565648			0.282824	+	0.959172i	$0.408729\pi$
$12$		0			0
$13$		−	1067.66i		−	1.75216i	−0.482165	−	0.876080i	$-0.660150\pi$
$13$		−	1067.66i		−	1.75216i	0.482165	−	0.876080i	$-0.339850\pi$
$14$		0			0
$15$	−1156.62	+	412.007i	−1.32728	+	0.472799i
$16$		0			0
$17$			686.258i			0.575924i	0.957642	+	0.287962i	$0.0929777\pi$
$17$			686.258i			0.575924i	−0.957642	+	0.287962i	$0.907022\pi$
$18$		0			0
$19$	−1309.11			−0.831940			−0.415970	−	0.909378i	$-0.636558\pi$
$19$	−1309.11			−0.831940			−0.415970	+	0.909378i	$0.636558\pi$
$20$		0			0
$21$	652.286			0.322768
$22$		0			0
$23$		−	4076.26i		−	1.60673i	−0.595489	−	0.803364i	$-0.703042\pi$
$23$		−	4076.26i		−	1.60673i	0.595489	−	0.803364i	$-0.296958\pi$
$24$		0			0
$25$	−2421.24	+	1975.66i	−0.774796	+	0.632212i
$26$		0			0
$27$		−	78.9360i		−	0.0208385i
$28$		0			0
$29$	−4535.21			−1.00139			−0.500694	−	0.865624i	$-0.666922\pi$
$29$	−4535.21			−1.00139			−0.500694	+	0.865624i	$0.666922\pi$
$30$		0			0
$31$	−7691.77			−1.43755			−0.718773	−	0.695245i	$-0.755296\pi$
$31$	−7691.77			−1.43755			−0.718773	+	0.695245i	$0.755296\pi$
$32$		0			0
$33$		−	4985.79i		−	0.796983i
$34$		0			0
$35$	1563.92	−	557.096i	0.215797	−	0.0768706i
$36$		0			0
$37$			9329.38i			1.12034i	0.828379	+	0.560168i	$0.189263\pi$
$37$			9329.38i			1.12034i	−0.828379	+	0.560168i	$0.810737\pi$
$38$		0			0
$39$	−23449.8			−2.46875
$40$		0			0
$41$	3753.20			0.348692			0.174346	−	0.984684i	$-0.444219\pi$
$41$	3753.20			0.348692			0.174346	+	0.984684i	$0.444219\pi$
$42$		0			0
$43$			2934.28i			0.242008i	0.992652	+	0.121004i	$0.0386114\pi$
$43$			2934.28i			0.242008i	−0.992652	+	0.121004i	$0.961389\pi$
$44$		0			0
$45$	4490.90	+	12607.2i	0.330600	+	0.928086i
$46$		0			0
$47$		−	3690.55i		−	0.243695i	−0.992549	−	0.121847i	$-0.961118\pi$
$47$		−	3690.55i		−	0.243695i	0.992549	−	0.121847i	$-0.0388819\pi$
$48$		0			0
$49$	15925.0			0.947522
$50$		0			0
$51$	15072.8			0.811463
$52$		0			0
$53$			4816.68i			0.235536i	0.993041	+	0.117768i	$0.0375740\pi$
$53$			4816.68i			0.235536i	−0.993041	+	0.117768i	$0.962426\pi$
$54$		0			0
$55$	−4258.20	−	11954.0i	−0.189810	−	0.532850i
$56$		0			0
$57$			28752.9i			1.17218i
$58$		0			0
$59$	31168.2			1.16569			0.582843	−	0.812585i	$-0.301940\pi$
$59$	31168.2			1.16569			0.582843	+	0.812585i	$0.301940\pi$
$60$		0			0
$61$	33809.4			1.16336			0.581678	−	0.813419i	$-0.302396\pi$
$61$	33809.4			1.16336			0.581678	+	0.813419i	$0.302396\pi$
$62$		0			0
$63$		−	7109.96i		−	0.225692i
$64$		0			0
$65$	−56223.3	+	20027.7i	−1.65057	+	0.587959i
$66$		0			0
$67$			24844.0i			0.676136i	0.941122	+	0.338068i	$0.109773\pi$
$67$			24844.0i			0.676136i	−0.941122	+	0.338068i	$0.890227\pi$
$68$		0			0
$69$	−89529.9			−2.26384
$70$		0			0
$71$	−14153.6			−0.333212			−0.166606	−	0.986024i	$-0.553281\pi$
$71$	−14153.6			−0.333212			−0.166606	+	0.986024i	$0.553281\pi$
$72$		0			0
$73$			73107.8i			1.60567i	0.596200	+	0.802836i	$0.296677\pi$
$73$			73107.8i			1.60567i	−0.596200	+	0.802836i	$0.703323\pi$
$74$		0			0
$75$	43392.9	+	53179.4i	0.890771	+	1.09167i
$76$		0			0
$77$			6741.55i			0.129578i
$78$		0			0
$79$	73646.1			1.32764			0.663822	−	0.747890i	$-0.268933\pi$
$79$	73646.1			1.32764			0.663822	+	0.747890i	$0.268933\pi$
$80$		0			0
$81$	−59909.4			−1.01457
$82$		0			0
$83$		−	67353.7i		−	1.07316i	−0.843848	−	0.536582i	$-0.819715\pi$
$83$		−	67353.7i		−	1.07316i	0.843848	−	0.536582i	$-0.180285\pi$
$84$		0			0
$85$	36138.6	−	12873.2i	0.542531	−	0.193259i
$86$		0			0
$87$			99610.3i			1.41093i
$88$		0			0
$89$	60277.5			0.806641			0.403320	−	0.915059i	$-0.367856\pi$
$89$	60277.5			0.806641			0.403320	+	0.915059i	$0.367856\pi$
$90$		0			0
$91$	31707.6			0.401384
$92$		0			0
$93$			168940.i			2.02547i
$94$		0			0
$95$	24556.9	+	68938.2i	0.279168	+	0.783702i
$96$		0			0
$97$		−	15478.8i		−	0.167035i	−0.996506	−	0.0835175i	$-0.973385\pi$
$97$		−	15478.8i		−	0.167035i	0.996506	−	0.0835175i	$-0.0266155\pi$
$98$		0			0
$99$	−54345.4			−0.557282
$100$		0			0
$101$	−37535.9			−0.366137			−0.183069	−	0.983100i	$-0.558603\pi$
$101$	−37535.9			−0.366137			−0.183069	+	0.983100i	$0.558603\pi$
$102$		0			0
$103$		−	183229.i		−	1.70177i	−0.525354	−	0.850884i	$-0.676067\pi$
$103$		−	183229.i		−	1.70177i	0.525354	−	0.850884i	$-0.323933\pi$
$104$		0			0
$105$	−12235.9	−	34349.6i	−0.108309	−	0.304053i
$106$		0			0
$107$		−	72550.4i		−	0.612605i	−0.951934	−	0.306302i	$-0.900908\pi$
$107$		−	72550.4i		−	0.612605i	0.951934	−	0.306302i	$-0.0990919\pi$
$108$		0			0
$109$	−159664.			−1.28719			−0.643593	−	0.765368i	$-0.722557\pi$
$109$	−159664.			−1.28719			−0.643593	+	0.765368i	$0.722557\pi$
$110$		0			0
$111$	204908.			1.57853
$112$		0			0
$113$			130294.i			0.959905i	0.877294	+	0.479953i	$0.159346\pi$
$113$			130294.i			0.959905i	−0.877294	+	0.479953i	$0.840654\pi$
$114$		0			0
$115$	−214657.	+	76464.6i	−1.51357	+	0.539158i
$116$		0			0
$117$			255604.i			1.72625i
$118$		0			0
$119$	−20380.7			−0.131933
$120$		0			0
$121$	−109522.			−0.680043
$122$		0			0
$123$		−	82434.4i		−	0.491299i
$124$		0			0
$125$	149458.	+	90442.8i	0.855547	+	0.517725i
$126$		0			0
$127$			240270.i			1.32187i	0.750442	+	0.660936i	$0.229841\pi$
$127$			240270.i			1.32187i	−0.750442	+	0.660936i	$0.770159\pi$
$128$		0			0
$129$	64447.7			0.340984
$130$		0			0
$131$	−389775.			−1.98443			−0.992214	−	0.124546i	$-0.960253\pi$
$131$	−389775.			−1.98443			−0.992214	+	0.124546i	$0.960253\pi$
$132$		0			0
$133$		−	38878.3i		−	0.190581i
$134$		0			0
$135$	−4156.80	+	1480.72i	−0.0196302	+	0.00699261i
$136$		0			0
$137$			434535.i			1.97799i	0.147962	+	0.988993i	$0.452729\pi$
$137$			434535.i			1.97799i	−0.147962	+	0.988993i	$0.547271\pi$
$138$		0			0
$139$	−114647.			−0.503298			−0.251649	−	0.967819i	$-0.580973\pi$
$139$	−114647.			−0.503298			−0.251649	+	0.967819i	$0.580973\pi$
$140$		0			0
$141$	−81058.3			−0.343360
$142$		0			0
$143$		−	242359.i		−	0.991105i
$144$		0			0
$145$	85073.9	+	238826.i	0.336029	+	0.943326i
$146$		0			0
$147$		−	349773.i		−	1.33504i
$148$		0			0
$149$	294922.			1.08828			0.544141	−	0.838994i	$-0.316856\pi$
$149$	294922.			1.08828			0.544141	+	0.838994i	$0.316856\pi$
$150$		0			0
$151$	−404524.			−1.44378			−0.721892	−	0.692006i	$-0.756727\pi$
$151$	−404524.			−1.44378			−0.721892	+	0.692006i	$0.756727\pi$
$152$		0			0
$153$		−	164294.i		−	0.567406i
$154$		0			0
$155$	144286.	+	405051.i	0.482387	+	1.35419i
$156$		0			0
$157$		−	377457.i		−	1.22213i	−0.791579	−	0.611067i	$-0.790741\pi$
$157$		−	377457.i		−	1.22213i	0.791579	−	0.611067i	$-0.209259\pi$
$158$		0			0
$159$	105792.			0.331865
$160$		0			0
$161$	121058.			0.368069
$162$		0			0
$163$		−	422416.i		−	1.24529i	−0.782503	−	0.622647i	$-0.786057\pi$
$163$		−	422416.i		−	1.24529i	0.782503	−	0.622647i	$-0.213943\pi$
$164$		0			0
$165$	−262554.	+	93526.1i	−0.750773	+	0.267438i
$166$		0			0
$167$			360875.i			1.00130i	0.865649	+	0.500652i	$0.166906\pi$
$167$			360875.i			1.00130i	−0.865649	+	0.500652i	$0.833094\pi$
$168$		0			0
$169$	−768601.			−2.07007
$170$		0			0
$171$	313409.			0.819636
$172$		0			0
$173$		−	405196.i		−	1.02932i	−0.857395	−	0.514659i	$-0.827919\pi$
$173$		−	405196.i		−	1.02932i	0.857395	−	0.514659i	$-0.172081\pi$
$174$		0			0
$175$	−58673.8	−	71906.6i	−0.144827	−	0.177490i
$176$		0			0
$177$		−	684570.i		−	1.64242i
$178$		0			0
$179$	−198995.			−0.464205			−0.232102	−	0.972691i	$-0.574560\pi$
$179$	−198995.			−0.464205			−0.232102	+	0.972691i	$0.574560\pi$
$180$		0			0
$181$	−243402.			−0.552240			−0.276120	−	0.961123i	$-0.589049\pi$
$181$	−243402.			−0.552240			−0.276120	+	0.961123i	$0.589049\pi$
$182$		0			0
$183$		−	742581.i		−	1.63914i
$184$		0			0
$185$	491289.	−	175005.i	1.05538	−	0.375943i
$186$		0			0
$187$			155781.i			0.325770i
$188$		0			0
$189$	2344.27			0.00477367
$190$		0			0
$191$	239201.			0.474438			0.237219	−	0.971456i	$-0.423764\pi$
$191$	239201.			0.474438			0.237219	+	0.971456i	$0.423764\pi$
$192$		0			0
$193$		−	506878.i		−	0.979513i	−0.871859	−	0.489757i	$-0.837086\pi$
$193$		−	506878.i		−	0.979513i	0.871859	−	0.489757i	$-0.162914\pi$
$194$		0			0
$195$	439883.	+	1.23487e6i	0.828420	+	2.32561i
$196$		0			0
$197$			493617.i			0.906201i	0.891460	+	0.453100i	$0.149682\pi$
$197$			493617.i			0.906201i	−0.891460	+	0.453100i	$0.850318\pi$
$198$		0			0
$199$	−489526.			−0.876281			−0.438140	−	0.898907i	$-0.644363\pi$
$199$	−489526.			−0.876281			−0.438140	+	0.898907i	$0.644363\pi$
$200$		0			0
$201$	545666.			0.952658
$202$		0			0
$203$		−	134688.i		−	0.229398i
$204$		0			0
$205$	−70404.5	−	197645.i	−0.117008	−	0.328474i
$206$		0			0
$207$			975881.i			1.58296i
$208$		0			0
$209$	−297169.			−0.470585
$210$		0			0
$211$	1.19087e6			1.84145			0.920723	−	0.390217i	$-0.127600\pi$
$211$	1.19087e6			1.84145			0.920723	+	0.390217i	$0.127600\pi$
$212$		0			0
$213$			310866.i			0.469487i
$214$		0			0
$215$	154520.	−	55042.7i	0.227976	−	0.0812089i
$216$		0			0
$217$		−	228432.i		−	0.329313i
$218$		0			0
$219$	1.60572e6			2.26235
$220$		0			0
$221$	732689.			1.00911
$222$		0			0
$223$		−	876410.i		−	1.18017i	−0.807341	−	0.590086i	$-0.799094\pi$
$223$		−	876410.i		−	1.18017i	0.807341	−	0.590086i	$-0.200906\pi$
$224$		0			0
$225$	579659.	−	472986.i	0.763336	−	0.622862i
$226$		0			0
$227$		−	1.06583e6i		−	1.37285i	−0.727200	−	0.686425i	$-0.759179\pi$
$227$		−	1.06583e6i		−	1.37285i	0.727200	−	0.686425i	$-0.240821\pi$
$228$		0			0
$229$	657279.			0.828250			0.414125	−	0.910220i	$-0.364088\pi$
$229$	657279.			0.828250			0.414125	+	0.910220i	$0.364088\pi$
$230$		0			0
$231$	148070.			0.182573
$232$		0			0
$233$			549169.i			0.662699i	0.943508	+	0.331349i	$0.107504\pi$
$233$			549169.i			0.662699i	−0.943508	+	0.331349i	$0.892496\pi$
$234$		0			0
$235$	−194346.	+	69229.3i	−0.229565	+	0.0817749i
$236$		0			0
$237$		−	1.61754e6i		−	1.87062i
$238$		0			0
$239$	−453609.			−0.513673			−0.256836	−	0.966455i	$-0.582680\pi$
$239$	−453609.			−0.513673			−0.256836	+	0.966455i	$0.582680\pi$
$240$		0			0
$241$	1.08405e6			1.20229			0.601143	−	0.799142i	$-0.294712\pi$
$241$	1.08405e6			1.20229			0.601143	+	0.799142i	$0.294712\pi$
$242$		0			0
$243$			1.29665e6i			1.40867i
$244$		0			0
$245$	−298730.	−	838617.i	−0.317953	−	0.892583i
$246$		0			0
$247$			1.39768e6i			1.45769i
$248$		0			0
$249$	−1.47934e6			−1.51206
$250$		0			0
$251$	−1.10334e6			−1.10541			−0.552705	−	0.833377i	$-0.686405\pi$
$251$	−1.10334e6			−1.10541			−0.552705	+	0.833377i	$0.686405\pi$
$252$		0			0
$253$		−	925315.i		−	0.908842i
$254$		0			0
$255$	−282743.	−	793740.i	−0.272297	−	0.764412i
$256$		0			0
$257$		−	360858.i		−	0.340803i	−0.985375	−	0.170402i	$-0.945493\pi$
$257$		−	360858.i		−	0.340803i	0.985375	−	0.170402i	$-0.0545065\pi$
$258$		0			0
$259$	−277067.			−0.256646
$260$		0			0
$261$	1.08576e6			0.986578
$262$		0			0
$263$		−	492090.i		−	0.438687i	−0.975648	−	0.219344i	$-0.929608\pi$
$263$		−	492090.i		−	0.438687i	0.975648	−	0.219344i	$-0.0703916\pi$
$264$		0			0
$265$	253648.	−	90353.8i	0.221880	−	0.0790372i
$266$		0			0
$267$		−	1.32392e6i		−	1.13654i
$268$		0			0
$269$	−809822.			−0.682352			−0.341176	−	0.939999i	$-0.610825\pi$
$269$	−809822.			−0.682352			−0.341176	+	0.939999i	$0.610825\pi$
$270$		0			0
$271$	−1.07473e6			−0.888949			−0.444475	−	0.895791i	$-0.646610\pi$
$271$	−1.07473e6			−0.888949			−0.444475	+	0.895791i	$0.646610\pi$
$272$		0			0
$273$		−	696419.i		−	0.565541i
$274$		0			0
$275$	−549623.	+	448477.i	−0.438261	+	0.357609i
$276$		0			0
$277$		−	625727.i		−	0.489988i	−0.969525	−	0.244994i	$-0.921214\pi$
$277$		−	625727.i		−	0.489988i	0.969525	−	0.244994i	$-0.0787859\pi$
$278$		0			0
$279$	1.84146e6			1.41629
$280$		0			0
$281$	239071.			0.180618			0.0903091	−	0.995914i	$-0.471215\pi$
$281$	239071.			0.180618			0.0903091	+	0.995914i	$0.471215\pi$
$282$		0			0
$283$			96643.3i			0.0717308i	0.999357	+	0.0358654i	$0.0114188\pi$
$283$			96643.3i			0.0717308i	−0.999357	+	0.0358654i	$0.988581\pi$
$284$		0			0
$285$	1.51414e6	−	539363.i	1.10422	−	0.393340i
$286$		0			0
$287$			111464.i			0.0798783i
$288$		0			0
$289$	948906.			0.668311
$290$		0			0
$291$	−339972.			−0.235348
$292$		0			0
$293$			695903.i			0.473565i	0.971563	+	0.236783i	$0.0760929\pi$
$293$			695903.i			0.473565i	−0.971563	+	0.236783i	$0.923907\pi$
$294$		0			0
$295$	−584669.	−	1.64133e6i	−0.391160	−	1.09810i
$296$		0			0
$297$		−	17918.5i		−	0.0117872i
$298$		0			0
$299$	−4.35205e6			−2.81524
$300$		0			0
$301$	−87143.1			−0.0554392
$302$		0			0
$303$			824430.i			0.515878i
$304$		0			0
$305$	−634214.	−	1.78042e6i	−0.390379	−	1.09590i
$306$		0			0
$307$		−	1.89838e6i		−	1.14958i	−0.818302	−	0.574788i	$-0.805084\pi$
$307$		−	1.89838e6i		−	1.14958i	0.818302	−	0.574788i	$-0.194916\pi$
$308$		0			0
$309$	−4.02439e6			−2.39775
$310$		0			0
$311$	1.32482e6			0.776706			0.388353	−	0.921511i	$-0.373044\pi$
$311$	1.32482e6			0.776706			0.388353	+	0.921511i	$0.373044\pi$
$312$		0			0
$313$		−	671307.i		−	0.387311i	−0.981070	−	0.193656i	$-0.937966\pi$
$313$		−	671307.i		−	0.387311i	0.981070	−	0.193656i	$-0.0620345\pi$
$314$		0			0
$315$	−374413.	+	133372.i	−0.212606	+	0.0757337i
$316$		0			0
$317$			2.05383e6i			1.14793i	0.818879	+	0.573967i	$0.194596\pi$
$317$			2.05383e6i			1.14793i	−0.818879	+	0.573967i	$0.805404\pi$
$318$		0			0
$319$	−1.02950e6			−0.566433
$320$		0			0
$321$	−1.59348e6			−0.863145
$322$		0			0
$323$		−	898387.i		−	0.479134i
$324$		0			0
$325$	2.10933e6	+	2.58505e6i	1.10774	+	1.35757i
$326$		0			0
$327$			3.50683e6i			1.81361i
$328$		0			0
$329$	109603.			0.0558256
$330$		0			0
$331$	−2.74106e6			−1.37515			−0.687573	−	0.726115i	$-0.741324\pi$
$331$	−2.74106e6			−1.37515			−0.687573	+	0.726115i	$0.741324\pi$
$332$		0			0
$333$		−	2.23351e6i		−	1.10377i
$334$		0			0
$335$	1.30829e6	−	466036.i	0.636932	−	0.226886i
$336$		0			0
$337$		−	1.46812e6i		−	0.704183i	−0.935966	−	0.352092i	$-0.885471\pi$
$337$		−	1.46812e6i		−	0.704183i	0.935966	−	0.352092i	$-0.114529\pi$
$338$		0			0
$339$	2.86175e6			1.35248
$340$		0			0
$341$	−1.74604e6			−0.813145
$342$		0			0
$343$			972085.i			0.446138i
$344$		0			0
$345$	1.67945e6	+	4.71468e6i	0.759659	+	2.13258i
$346$		0			0
$347$		−	1.12356e6i		−	0.500926i	−0.968126	−	0.250463i	$-0.919417\pi$
$347$		−	1.12356e6i		−	0.500926i	0.968126	−	0.250463i	$-0.0805828\pi$
$348$		0			0
$349$	−1.71043e6			−0.751693			−0.375846	−	0.926682i	$-0.622648\pi$
$349$	−1.71043e6			−0.751693			−0.375846	+	0.926682i	$0.622648\pi$
$350$		0			0
$351$	−84276.6			−0.0365123
$352$		0			0
$353$		−	2.43615e6i		−	1.04056i	−0.853996	−	0.520280i	$-0.825827\pi$
$353$		−	2.43615e6i		−	1.04056i	0.853996	−	0.520280i	$-0.174173\pi$
$354$		0			0
$355$	265500.	+	745334.i	0.111813	+	0.313892i
$356$		0			0
$357$			447637.i			0.185890i
$358$		0			0
$359$	2.00932e6			0.822837			0.411418	−	0.911447i	$-0.365034\pi$
$359$	2.00932e6			0.822837			0.411418	+	0.911447i	$0.365034\pi$
$360$		0			0
$361$	−762332.			−0.307876
$362$		0			0
$363$			2.40550e6i			0.958163i
$364$		0			0
$365$	3.84989e6	−	1.37139e6i	1.51257	−	0.538803i
$366$		0			0
$367$		−	3.79537e6i		−	1.47092i	−0.677567	−	0.735461i	$-0.736966\pi$
$367$		−	3.79537e6i		−	1.47092i	0.677567	−	0.735461i	$-0.263034\pi$
$368$		0			0
$369$	−898539.			−0.343535
$370$		0			0
$371$	−143047.			−0.0539566
$372$		0			0
$373$			2.29348e6i			0.853537i	0.904361	+	0.426769i	$0.140348\pi$
$373$			2.29348e6i			0.853537i	−0.904361	+	0.426769i	$0.859652\pi$
$374$		0			0
$375$	1.98646e6	−	3.28266e6i	0.729461	−	1.20544i
$376$		0			0
$377$			4.84206e6i			1.75459i
$378$		0			0
$379$	2.13249e6			0.762588			0.381294	−	0.924454i	$-0.375479\pi$
$379$	2.13249e6			0.762588			0.381294	+	0.924454i	$0.375479\pi$
$380$		0			0
$381$	5.27722e6			1.86248
$382$		0			0
$383$			4.10113e6i			1.42859i	0.699847	+	0.714293i	$0.253251\pi$
$383$			4.10113e6i			1.42859i	−0.699847	+	0.714293i	$0.746749\pi$
$384$		0			0
$385$	355012.	−	126461.i	0.122065	−	0.0434817i
$386$		0			0
$387$		−	702484.i		−	0.238429i
$388$		0			0
$389$	−4.88764e6			−1.63766			−0.818832	−	0.574033i	$-0.805378\pi$
$389$	−4.88764e6			−1.63766			−0.818832	+	0.574033i	$0.805378\pi$
$390$		0			0
$391$	2.79737e6			0.925353
$392$		0			0
$393$			8.56091e6i			2.79601i
$394$		0			0
$395$	−1.38149e6	−	3.87823e6i	−0.445508	−	1.25067i
$396$		0			0
$397$		−	674221.i		−	0.214697i	−0.994221	−	0.107348i	$-0.965764\pi$
$397$		−	674221.i		−	0.214697i	0.994221	−	0.107348i	$-0.0342360\pi$
$398$		0			0
$399$	−853914.			−0.268523
$400$		0			0
$401$	2.13743e6			0.663789			0.331894	−	0.943317i	$-0.392312\pi$
$401$	2.13743e6			0.663789			0.331894	+	0.943317i	$0.392312\pi$
$402$		0			0
$403$			8.21218e6i			2.51881i
$404$		0			0
$405$	1.12381e6	+	3.15485e6i	0.340452	+	0.955744i
$406$		0			0
$407$			2.11778e6i			0.633716i
$408$		0			0
$409$	−1.64322e6			−0.485722			−0.242861	−	0.970061i	$-0.578086\pi$
$409$	−1.64322e6			−0.485722			−0.242861	+	0.970061i	$0.578086\pi$
$410$		0			0
$411$	9.54401e6			2.78693
$412$		0			0
$413$			925642.i			0.267035i
$414$		0			0
$415$	−3.54687e6	+	1.26346e6i	−1.01094	+	0.360114i
$416$		0			0
$417$			2.51807e6i			0.709134i
$418$		0			0
$419$	−1.29320e6			−0.359859			−0.179929	−	0.983680i	$-0.557587\pi$
$419$	−1.29320e6			−0.359859			−0.179929	+	0.983680i	$0.557587\pi$
$420$		0			0
$421$	−131852.			−0.0362560			−0.0181280	−	0.999836i	$-0.505771\pi$
$421$	−131852.			−0.0362560			−0.0181280	+	0.999836i	$0.505771\pi$
$422$		0			0
$423$			883540.i			0.240091i
$424$		0			0
$425$	−1.35581e6	−	1.66159e6i	−0.364106	−	0.446224i
$426$		0			0
$427$			1.00408e6i			0.266501i
$428$		0			0
$429$	−5.32312e6			−1.39644
$430$		0			0
$431$	−1.09123e6			−0.282960			−0.141480	−	0.989941i	$-0.545186\pi$
$431$	−1.09123e6			−0.282960			−0.141480	+	0.989941i	$0.545186\pi$
$432$		0			0
$433$		−	1.10549e6i		−	0.283357i	−0.989913	−	0.141679i	$-0.954750\pi$
$433$		−	1.10549e6i		−	0.283357i	0.989913	−	0.141679i	$-0.0452499\pi$
$434$		0			0
$435$	5.24552e6	−	1.86854e6i	1.32912	−	0.473456i
$436$		0			0
$437$			5.33627e6i			1.33670i
$438$		0			0
$439$	−5.82681e6			−1.44301			−0.721505	−	0.692409i	$-0.756549\pi$
$439$	−5.82681e6			−1.44301			−0.721505	+	0.692409i	$0.756549\pi$
$440$		0			0
$441$	−3.81254e6			−0.933509
$442$		0			0
$443$			1.88621e6i			0.456647i	0.973585	+	0.228324i	$0.0733245\pi$
$443$			1.88621e6i			0.456647i	−0.973585	+	0.228324i	$0.926676\pi$
$444$		0			0
$445$	−1.13072e6	−	3.17424e6i	−0.270678	−	0.759870i
$446$		0			0
$447$		−	6.47760e6i		−	1.53336i
$448$		0			0
$449$	−8.06463e6			−1.88786			−0.943928	−	0.330152i	$-0.892900\pi$
$449$	−8.06463e6			−1.88786			−0.943928	+	0.330152i	$0.892900\pi$
$450$		0			0
$451$	851980.			0.197237
$452$		0			0
$453$			8.88487e6i			2.03426i
$454$		0			0
$455$	−594788.	−	1.66974e6i	−0.134690	−	0.378111i
$456$		0			0
$457$		−	3.33016e6i		−	0.745889i	−0.927854	−	0.372945i	$-0.878348\pi$
$457$		−	3.33016e6i		−	0.745889i	0.927854	−	0.372945i	$-0.121652\pi$
$458$		0			0
$459$	54170.5			0.0120014
$460$		0			0
$461$	−4.99208e6			−1.09403			−0.547015	−	0.837123i	$-0.684236\pi$
$461$	−4.99208e6			−1.09403			−0.547015	+	0.837123i	$0.684236\pi$
$462$		0			0
$463$			1.89423e6i			0.410659i	0.978693	+	0.205329i	$0.0658265\pi$
$463$			1.89423e6i			0.410659i	−0.978693	+	0.205329i	$0.934173\pi$
$464$		0			0
$465$	8.89645e6	−	3.16906e6i	1.90803	−	0.679671i
$466$		0			0
$467$		−	3.60284e6i		−	0.764455i	−0.924068	−	0.382228i	$-0.875157\pi$
$467$		−	3.60284e6i		−	0.764455i	0.924068	−	0.382228i	$-0.124843\pi$
$468$		0			0
$469$	−737824.			−0.154889
$470$		0			0
$471$	−8.29037e6			−1.72196
$472$		0			0
$473$			666084.i			0.136891i
$474$		0			0
$475$	3.16966e6	−	2.58636e6i	0.644583	−	0.525962i
$476$		0			0
$477$		−	1.15314e6i		−	0.232053i
$478$		0			0
$479$	5.65624e6			1.12639			0.563196	−	0.826324i	$-0.309572\pi$
$479$	5.65624e6			1.12639			0.563196	+	0.826324i	$0.309572\pi$
$480$		0			0
$481$	9.96059e6			1.96301
$482$		0			0
$483$		−	2.65889e6i		−	0.518600i
$484$		0			0
$485$	−815119.	+	290359.i	−0.157350	+	0.0560507i
$486$		0			0
$487$		−	5.12375e6i		−	0.978961i	−0.872014	−	0.489480i	$-0.837187\pi$
$487$		−	5.12375e6i		−	0.978961i	0.872014	−	0.489480i	$-0.162813\pi$
$488$		0			0
$489$	−9.27784e6			−1.75459
$490$		0			0
$491$	−8.93216e6			−1.67206			−0.836032	−	0.548681i	$-0.815130\pi$
$491$	−8.93216e6			−1.67206			−0.836032	+	0.548681i	$0.815130\pi$
$492$		0			0
$493$		−	3.11233e6i		−	0.576724i
$494$		0			0
$495$	1.01944e6	+	2.86185e6i	0.187003	+	0.524970i
$496$		0			0
$497$		−	420338.i		−	0.0763321i
$498$		0			0
$499$	1.78948e6			0.321719			0.160859	−	0.986977i	$-0.448573\pi$
$499$	1.78948e6			0.321719			0.160859	+	0.986977i	$0.448573\pi$
$500$		0			0
$501$	7.92617e6			1.41081
$502$		0			0
$503$		−	296978.i		−	0.0523365i	−0.999658	−	0.0261683i	$-0.991669\pi$
$503$		−	296978.i		−	0.0523365i	0.999658	−	0.0261683i	$-0.00833057\pi$
$504$		0			0
$505$	704118.	+	1.97666e6i	0.122862	+	0.344908i
$506$		0			0
$507$			1.68814e7i			2.91667i
$508$		0			0
$509$	3.91977e6			0.670603			0.335302	−	0.942111i	$-0.391162\pi$
$509$	3.91977e6			0.670603			0.335302	+	0.942111i	$0.391162\pi$
$510$		0			0
$511$	−2.17118e6			−0.367827
$512$		0			0
$513$			103336.i			0.0173363i
$514$		0			0
$515$	−9.64889e6	+	3.43710e6i	−1.60310	+	0.571049i
$516$		0			0
$517$		−	837759.i		−	0.137845i
$518$		0			0
$519$	−8.89962e6			−1.45028
$520$		0			0
$521$	4.74987e6			0.766633			0.383317	−	0.923617i	$-0.374782\pi$
$521$	4.74987e6			0.766633			0.383317	+	0.923617i	$0.374782\pi$
$522$		0			0
$523$		−	8.15275e6i		−	1.30332i	−0.758513	−	0.651658i	$-0.774074\pi$
$523$		−	8.15275e6i		−	1.30332i	0.758513	−	0.651658i	$-0.225926\pi$
$524$		0			0
$525$	−1.57934e6	+	1.28870e6i	−0.250079	+	0.204057i
$526$		0			0
$527$		−	5.27854e6i		−	0.827918i
$528$		0			0
$529$	−1.01795e7			−1.58157
$530$		0			0
$531$	−7.46185e6			−1.14844
$532$		0			0
$533$		−	4.00714e6i		−	0.610965i
$534$		0			0
$535$	−3.82053e6	+	1.36094e6i	−0.577085	+	0.205567i
$536$		0			0
$537$			4.37068e6i			0.654053i
$538$		0			0
$539$	3.61499e6			0.535964
$540$		0			0
$541$	835785.			0.122773			0.0613863	−	0.998114i	$-0.480448\pi$
$541$	835785.			0.122773			0.0613863	+	0.998114i	$0.480448\pi$
$542$		0			0
$543$			5.34602e6i			0.778092i
$544$		0			0
$545$	2.99507e6	+	8.40799e6i	0.431932	+	1.21255i
$546$		0			0
$547$		−	7.88859e6i		−	1.12728i	−0.826021	−	0.563639i	$-0.809401\pi$
$547$		−	7.88859e6i		−	1.12728i	0.826021	−	0.563639i	$-0.190599\pi$
$548$		0			0
$549$	−8.09418e6			−1.14615
$550$		0			0
$551$	5.93709e6			0.833095
$552$		0			0
$553$			2.18716e6i			0.304137i
$554$		0			0
$555$	−3.84377e6	−	1.07905e7i	−0.529694	−	1.48700i
$556$		0			0
$557$		−	7.42914e6i		−	1.01461i	−0.861765	−	0.507307i	$-0.830641\pi$
$557$		−	7.42914e6i		−	1.01461i	0.861765	−	0.507307i	$-0.169359\pi$
$558$		0			0
$559$	3.13281e6			0.424037
$560$		0			0
$561$	3.42154e6			0.459002
$562$		0			0
$563$		−	5.23381e6i		−	0.695901i	−0.937513	−	0.347950i	$-0.886878\pi$
$563$		−	5.23381e6i		−	0.695901i	0.937513	−	0.347950i	$-0.113122\pi$
$564$		0			0
$565$	6.86134e6	−	2.44412e6i	0.904248	−	0.322108i
$566$		0			0
$567$		−	1.77921e6i		−	0.232418i
$568$		0			0
$569$	−7.64257e6			−0.989598			−0.494799	−	0.869008i	$-0.664758\pi$
$569$	−7.64257e6			−0.989598			−0.494799	+	0.869008i	$0.664758\pi$
$570$		0			0
$571$	1.06264e7			1.36394			0.681971	−	0.731379i	$-0.261123\pi$
$571$	1.06264e7			1.36394			0.681971	+	0.731379i	$0.261123\pi$
$572$		0			0
$573$		−	5.25375e6i		−	0.668471i
$574$		0			0
$575$	8.05331e6	+	9.86959e6i	1.01579	+	1.24489i
$576$		0			0
$577$		−	7.92313e6i		−	0.990735i	−0.868684	−	0.495367i	$-0.835033\pi$
$577$		−	7.92313e6i		−	0.990735i	0.868684	−	0.495367i	$-0.164967\pi$
$578$		0			0
$579$	−1.11329e7			−1.38011
$580$		0			0
$581$	2.00029e6			0.245840
$582$		0			0
$583$			1.09339e6i			0.133231i
$584$		0			0
$585$	1.34602e7	−	4.79475e6i	1.62615	−	0.579264i
$586$		0			0
$587$			1.41402e7i			1.69380i	0.531755	+	0.846898i	$0.321533\pi$
$587$			1.41402e7i			1.69380i	−0.531755	+	0.846898i	$0.678467\pi$
$588$		0			0
$589$	1.00694e7			1.19595
$590$		0			0
$591$	1.08417e7			1.27681
$592$		0			0
$593$		−	6.60665e6i		−	0.771516i	−0.922600	−	0.385758i	$-0.873940\pi$
$593$		−	6.60665e6i		−	0.771516i	0.922600	−	0.385758i	$-0.126060\pi$
$594$		0			0
$595$	382312.	+	1.07326e6i	0.0442716	+	0.124283i
$596$		0			0
$597$			1.07518e7i			1.23466i
$598$		0			0
$599$	1.73390e7			1.97450			0.987252	−	0.159166i	$-0.0508805\pi$
$599$	1.73390e7			1.97450			0.987252	+	0.159166i	$0.0508805\pi$
$600$		0			0
$601$	6.83953e6			0.772397			0.386198	−	0.922416i	$-0.373788\pi$
$601$	6.83953e6			0.772397			0.386198	+	0.922416i	$0.373788\pi$
$602$		0			0
$603$		−	5.94780e6i		−	0.666136i
$604$		0			0
$605$	2.05446e6	+	5.76745e6i	0.228197	+	0.640612i
$606$		0			0
$607$		−	5.78383e6i		−	0.637153i	−0.947897	−	0.318576i	$-0.896795\pi$
$607$		−	5.78383e6i		−	0.637153i	0.947897	−	0.318576i	$-0.103205\pi$
$608$		0			0
$609$	−2.95826e6			−0.323216
$610$		0			0
$611$	−3.94025e6			−0.426993
$612$		0			0
$613$		−	1.29204e7i		−	1.38875i	−0.719614	−	0.694374i	$-0.755681\pi$
$613$		−	1.29204e7i		−	1.38875i	0.719614	−	0.694374i	$-0.244319\pi$
$614$		0			0
$615$	−4.34103e6	+	1.54635e6i	−0.462812	+	0.164861i
$616$		0			0
$617$			3.21433e6i			0.339920i	0.985451	+	0.169960i	$0.0543639\pi$
$617$			3.21433e6i			0.339920i	−0.985451	+	0.169960i	$0.945636\pi$
$618$		0			0
$619$	−3.18358e6			−0.333956			−0.166978	−	0.985961i	$-0.553401\pi$
$619$	−3.18358e6			−0.333956			−0.166978	+	0.985961i	$0.553401\pi$
$620$		0			0
$621$	−321764.			−0.0334817
$622$		0			0
$623$			1.79014e6i			0.184785i
$624$		0			0
$625$	1.95914e6	−	9.56709e6i	0.200616	−	0.979670i
$626$		0			0
$627$			6.52695e6i			0.663042i
$628$		0			0
$629$	−6.40236e6			−0.645229
$630$		0			0
$631$	2.08225e6			0.208190			0.104095	−	0.994567i	$-0.466805\pi$
$631$	2.08225e6			0.208190			0.104095	+	0.994567i	$0.466805\pi$
$632$		0			0
$633$		−	2.61560e7i		−	2.59455i
$634$		0			0
$635$	1.26527e7	−	4.50710e6i	1.24523	−	0.443571i
$636$		0			0
$637$		−	1.70025e7i		−	1.66021i
$638$		0			0
$639$	3.38845e6			0.328284
$640$		0			0
$641$	−1.96753e6			−0.189137			−0.0945687	−	0.995518i	$-0.530147\pi$
$641$	−1.96753e6			−0.189137			−0.0945687	+	0.995518i	$0.530147\pi$
$642$		0			0
$643$			6.15416e6i			0.587004i	0.955958	+	0.293502i	$0.0948208\pi$
$643$			6.15416e6i			0.587004i	−0.955958	+	0.293502i	$0.905179\pi$
$644$		0			0
$645$	−1.20894e6	−	3.39384e6i	−0.114421	−	0.321213i
$646$		0			0
$647$			8.21959e6i			0.771951i	0.922509	+	0.385975i	$0.126135\pi$
$647$			8.21959e6i			0.771951i	−0.922509	+	0.385975i	$0.873865\pi$
$648$		0			0
$649$	7.07520e6			0.659367
$650$		0			0
$651$	−5.01723e6			−0.463993
$652$		0			0
$653$			7.81613e6i			0.717314i	0.933470	+	0.358657i	$0.116765\pi$
$653$			7.81613e6i			0.717314i	−0.933470	+	0.358657i	$0.883235\pi$
$654$		0			0
$655$	7.31159e6	+	2.05257e7i	0.665900	+	1.86937i
$656$		0			0
$657$		−	1.75025e7i		−	1.58192i
$658$		0			0
$659$	−100743.			−0.00903649			−0.00451824	−	0.999990i	$-0.501438\pi$
$659$	−100743.			−0.00903649			−0.00451824	+	0.999990i	$0.501438\pi$
$660$		0			0
$661$	−5.45043e6			−0.485207			−0.242603	−	0.970126i	$-0.578001\pi$
$661$	−5.45043e6			−0.485207			−0.242603	+	0.970126i	$0.578001\pi$
$662$		0			0
$663$		−	1.60926e7i		−	1.42181i
$664$		0			0
$665$	−2.04735e6	+	729300.i	−0.179530	+	0.0639517i
$666$		0			0
$667$			1.84867e7i			1.60896i
$668$		0			0
$669$	−1.92492e7			−1.66283
$670$		0			0
$671$	7.67477e6			0.658050
$672$		0			0
$673$		−	3.25735e6i		−	0.277221i	−0.990347	−	0.138611i	$-0.955736\pi$
$673$		−	3.25735e6i		−	0.277221i	0.990347	−	0.138611i	$-0.0442637\pi$
$674$		0			0
$675$	155951.	+	191123.i	0.0131743	+	0.0161455i
$676$		0			0
$677$		−	5.82826e6i		−	0.488728i	−0.969684	−	0.244364i	$-0.921421\pi$
$677$		−	5.82826e6i		−	0.488728i	0.969684	−	0.244364i	$-0.0785792\pi$
$678$		0			0
$679$	459694.			0.0382643
$680$		0			0
$681$	−2.34096e7			−1.93431
$682$		0			0
$683$		−	2.13611e7i		−	1.75215i	−0.482174	−	0.876076i	$-0.660153\pi$
$683$		−	2.13611e7i		−	1.75215i	0.482174	−	0.876076i	$-0.339847\pi$
$684$		0			0
$685$	2.28828e7	−	8.15123e6i	1.86330	−	0.663738i
$686$		0			0
$687$		−	1.44363e7i		−	1.16698i
$688$		0			0
$689$	5.14257e6			0.412698
$690$		0			0
$691$	1.58254e7			1.26083			0.630417	−	0.776256i	$-0.282884\pi$
$691$	1.58254e7			1.26083			0.630417	+	0.776256i	$0.282884\pi$
$692$		0			0
$693$		−	1.61397e6i		−	0.127662i
$694$		0			0
$695$	2.15060e6	+	6.03735e6i	0.168888	+	0.474115i
$696$		0			0
$697$			2.57567e6i			0.200820i
$698$		0			0
$699$	1.20618e7			0.933726
$700$		0			0
$701$	2.75175e6			0.211502			0.105751	−	0.994393i	$-0.466275\pi$
$701$	2.75175e6			0.211502			0.105751	+	0.994393i	$0.466275\pi$
$702$		0			0
$703$		−	1.22132e7i		−	0.932052i
$704$		0			0
$705$	1.52053e6	+	4.26856e6i	0.115219	+	0.323451i
$706$		0			0
$707$		−	1.11475e6i		−	0.0838746i
$708$		0			0
$709$	−1.10623e7			−0.826474			−0.413237	−	0.910624i	$-0.635602\pi$
$709$	−1.10623e7			−0.826474			−0.413237	+	0.910624i	$0.635602\pi$
$710$		0			0
$711$	−1.76313e7			−1.30801
$712$		0			0
$713$			3.13536e7i			2.30974i
$714$		0			0
$715$	−1.27627e7	+	4.54630e6i	−0.933639	+	0.332578i
$716$		0			0
$717$			9.96294e6i			0.723752i
$718$		0			0
$719$	−2.57064e7			−1.85447			−0.927234	−	0.374483i	$-0.877820\pi$
$719$	−2.57064e7			−1.85447			−0.927234	+	0.374483i	$0.877820\pi$
$720$		0			0
$721$	5.44158e6			0.389840
$722$		0			0
$723$		−	2.38099e7i		−	1.69399i
$724$		0			0
$725$	1.09808e7	−	8.96005e6i	0.775872	−	0.633090i
$726$		0			0
$727$			1.44657e7i			1.01509i	0.861626	+	0.507544i	$0.169447\pi$
$727$			1.44657e7i			1.01509i	−0.861626	+	0.507544i	$0.830553\pi$
$728$		0			0
$729$	1.39214e7			0.970205
$730$		0			0
$731$	−2.01367e6			−0.139378
$732$		0			0
$733$		−	6.59256e6i		−	0.453204i	−0.973987	−	0.226602i	$-0.927238\pi$
$733$		−	6.59256e6i		−	0.453204i	0.973987	−	0.226602i	$-0.0727617\pi$
$734$		0			0
$735$	−1.84192e7	+	6.56122e6i	−1.25763	+	0.447988i
$736$		0			0
$737$			5.63960e6i			0.382455i
$738$		0			0
$739$	8.77086e6			0.590788			0.295394	−	0.955376i	$-0.404549\pi$
$739$	8.77086e6			0.590788			0.295394	+	0.955376i	$0.404549\pi$
$740$		0			0
$741$	3.06983e7			2.05385
$742$		0			0
$743$		−	5.10837e6i		−	0.339477i	−0.985489	−	0.169738i	$-0.945708\pi$
$743$		−	5.10837e6i		−	0.339477i	0.985489	−	0.169738i	$-0.0542923\pi$
$744$		0			0
$745$	−5.53230e6	−	1.55307e7i	−0.365187	−	1.02518i
$746$		0			0
$747$			1.61249e7i			1.05729i
$748$		0			0
$749$	2.15462e6			0.140335
$750$		0			0
$751$	−1.65673e7			−1.07190			−0.535948	−	0.844251i	$-0.680046\pi$
$751$	−1.65673e7			−1.07190			−0.535948	+	0.844251i	$0.680046\pi$
$752$		0			0
$753$			2.42334e7i			1.55750i
$754$		0			0
$755$	7.58828e6	+	2.13024e7i	0.484480	+	1.36007i
$756$		0			0
$757$		−	1.57190e7i		−	0.996978i	−0.866896	−	0.498489i	$-0.833888\pi$
$757$		−	1.57190e7i		−	0.996978i	0.866896	−	0.498489i	$-0.166112\pi$
$758$		0			0
$759$	−2.03234e7			−1.28054
$760$		0			0
$761$	2.04524e7			1.28022			0.640108	−	0.768285i	$-0.278890\pi$
$761$	2.04524e7			1.28022			0.640108	+	0.768285i	$0.278890\pi$
$762$		0			0
$763$		−	4.74176e6i		−	0.294868i
$764$		0			0
$765$	−8.65181e6	+	3.08192e6i	−0.534507	+	0.190400i
$766$		0			0
$767$		−	3.32769e7i		−	2.04247i
$768$		0			0
$769$	−3.29593e6			−0.200984			−0.100492	−	0.994938i	$-0.532042\pi$
$769$	−3.29593e6			−0.200984			−0.100492	+	0.994938i	$0.532042\pi$
$770$		0			0
$771$	−7.92580e6			−0.480183
$772$		0			0
$773$			1.32963e7i			0.800354i	0.916438	+	0.400177i	$0.131051\pi$
$773$			1.32963e7i			0.800354i	−0.916438	+	0.400177i	$0.868949\pi$
$774$		0			0
$775$	1.86236e7	−	1.51963e7i	1.11380	−	0.908834i
$776$		0			0
$777$			6.08542e6i			0.361608i
$778$		0			0
$779$	−4.91335e6			−0.290091
$780$		0			0
$781$	−3.21288e6			−0.188481
$782$		0			0
$783$			357991.i			0.0208674i
$784$		0			0
$785$	−1.98770e7	+	7.08054e6i	−1.15127	+	0.410102i
$786$		0			0
$787$			1.98876e6i			0.114458i	0.998361	+	0.0572290i	$0.0182265\pi$
$787$			1.98876e6i			0.114458i	−0.998361	+	0.0572290i	$0.981773\pi$
$788$		0			0
$789$	−1.08081e7			−0.618099
$790$		0			0
$791$	−3.86951e6			−0.219895
$792$		0			0
$793$		−	3.60969e7i		−	2.03839i
$794$		0			0
$795$	−1.98451e6	−	5.57107e6i	−0.111361	−	0.312623i
$796$		0			0
$797$			2.20524e7i			1.22973i	0.788632	+	0.614866i	$0.210790\pi$
$797$			2.20524e7i			1.22973i	−0.788632	+	0.614866i	$0.789210\pi$
$798$		0			0
$799$	2.53267e6			0.140350
$800$		0			0
$801$	−1.44308e7			−0.794710
$802$		0			0
$803$			1.65956e7i			0.908245i
$804$		0			0
$805$	−2.27087e6	−	6.37496e6i	−0.123510	−	0.346727i
$806$		0			0
$807$			1.77867e7i			0.961417i
$808$		0			0
$809$	5.20415e6			0.279562			0.139781	−	0.990182i	$-0.455360\pi$
$809$	5.20415e6			0.279562			0.139781	+	0.990182i	$0.455360\pi$
$810$		0			0
$811$	−1.00506e7			−0.536585			−0.268293	−	0.963337i	$-0.586459\pi$
$811$	−1.00506e7			−0.536585			−0.268293	+	0.963337i	$0.586459\pi$
$812$		0			0
$813$			2.36051e7i			1.25251i
$814$		0			0
$815$	−2.22446e7	+	7.92390e6i	−1.17309	+	0.417874i
$816$		0			0
$817$		−	3.84129e6i		−	0.201336i
$818$		0			0
$819$	−7.59100e6			−0.395448
$820$		0			0
$821$	−1.48945e7			−0.771204			−0.385602	−	0.922665i	$-0.626006\pi$
$821$	−1.48945e7			−0.771204			−0.385602	+	0.922665i	$0.626006\pi$
$822$		0			0
$823$		−	4.97038e6i		−	0.255794i	−0.991787	−	0.127897i	$-0.959177\pi$
$823$		−	4.97038e6i		−	0.255794i	0.991787	−	0.127897i	$-0.0408227\pi$
$824$		0			0
$825$	9.85024e6	+	1.20718e7i	0.503862	+	0.617499i
$826$		0			0
$827$		−	1.50182e7i		−	0.763579i	−0.924249	−	0.381790i	$-0.875308\pi$
$827$		−	1.50182e7i		−	0.763579i	0.924249	−	0.381790i	$-0.124692\pi$
$828$		0			0
$829$	−2.69399e7			−1.36148			−0.680738	−	0.732527i	$-0.738341\pi$
$829$	−2.69399e7			−1.36148			−0.680738	+	0.732527i	$0.738341\pi$
$830$		0			0
$831$	−1.37433e7			−0.690380
$832$		0			0
$833$			1.09287e7i			0.545701i
$834$		0			0
$835$	1.90038e7	−	6.76948e6i	0.943246	−	0.336000i
$836$		0			0
$837$			607157.i			0.0299562i
$838$		0			0
$839$	1.35679e7			0.665441			0.332720	−	0.943026i	$-0.392034\pi$
$839$	1.35679e7			0.665441			0.332720	+	0.943026i	$0.392034\pi$
$840$		0			0
$841$	57008.6			0.00277939
$842$		0			0
$843$		−	5.25090e6i		−	0.254487i
$844$		0			0
$845$	1.44178e7	+	4.04748e7i	0.694637	+	1.95004i
$846$		0			0
$847$		−	3.25261e6i		−	0.155784i
$848$		0			0
$849$	2.12265e6			0.101067
$850$		0			0
$851$	3.80290e7			1.80007
$852$		0			0
$853$		−	8.75046e6i		−	0.411773i	−0.978576	−	0.205887i	$-0.933992\pi$
$853$		−	8.75046e6i		−	0.411773i	0.978576	−	0.205887i	$-0.0660078\pi$
$854$		0			0
$855$	−5.87908e6	−	1.65042e7i	−0.275039	−	0.772111i
$856$		0			0
$857$		−	2.85540e7i		−	1.32805i	−0.747710	−	0.664025i	$-0.768847\pi$
$857$		−	2.85540e7i		−	1.32805i	0.747710	−	0.664025i	$-0.231153\pi$
$858$		0			0
$859$	−1.50591e7			−0.696333			−0.348167	−	0.937433i	$-0.613196\pi$
$859$	−1.50591e7			−0.696333			−0.348167	+	0.937433i	$0.613196\pi$
$860$		0			0
$861$	2.44816e6			0.112547
$862$		0			0
$863$			1.17155e7i			0.535467i	0.963493	+	0.267734i	$0.0862747\pi$
$863$			1.17155e7i			0.535467i	−0.963493	+	0.267734i	$0.913725\pi$
$864$		0			0
$865$	−2.13378e7	+	7.60087e6i	−0.969636	+	0.345401i
$866$		0			0
$867$		−	2.08415e7i		−	0.941634i
$868$		0			0
$869$	1.67177e7			0.750979
$870$		0			0
$871$	2.65249e7			1.18470
$872$		0			0
$873$			3.70572e6i			0.164565i
$874$		0			0
$875$	−2.68600e6	+	4.43865e6i	−0.118600	+	0.195989i
$876$		0			0
$877$			2.27293e7i			0.997901i	0.866630	+	0.498951i	$0.166281\pi$
$877$			2.27293e7i			0.997901i	−0.866630	+	0.498951i	$0.833719\pi$
$878$		0			0
$879$	1.52846e7			0.667241
$880$		0			0
$881$	1.47515e7			0.640317			0.320159	−	0.947364i	$-0.396264\pi$
$881$	1.47515e7			0.640317			0.320159	+	0.947364i	$0.396264\pi$
$882$		0			0
$883$			2.41244e7i			1.04125i	0.853786	+	0.520625i	$0.174301\pi$
$883$			2.41244e7i			1.04125i	−0.853786	+	0.520625i	$0.825699\pi$
$884$		0			0
$885$	−3.60497e7	+	1.28415e7i	−1.54719	+	0.551135i
$886$		0			0
$887$			1.83011e7i			0.781031i	0.920596	+	0.390515i	$0.127703\pi$
$887$			1.83011e7i			0.781031i	−0.920596	+	0.390515i	$0.872297\pi$
$888$		0			0
$889$	−7.13560e6			−0.302814
$890$		0			0
$891$	−1.35995e7			−0.573890
$892$		0			0
$893$			4.83133e6i			0.202740i
$894$		0			0
$895$	3.73285e6	+	1.04792e7i	0.155770	+	0.437289i
$896$		0			0
$897$			9.55874e7i			3.96661i
$898$		0			0
$899$	3.48838e7			1.43954
$900$		0			0
$901$	−3.30549e6			−0.135651
$902$		0			0
$903$			1.91399e6i			0.0781124i
$904$		0			0
$905$	4.56586e6	+	1.28176e7i	0.185311	+	0.520220i
$906$		0			0
$907$		−	1.52809e7i		−	0.616782i	−0.951260	−	0.308391i	$-0.900210\pi$
$907$		−	1.52809e7i		−	0.616782i	0.951260	−	0.308391i	$-0.0997905\pi$
$908$		0			0
$909$	8.98633e6			0.360722
$910$		0			0
$911$	1.77783e7			0.709732			0.354866	−	0.934917i	$-0.384527\pi$
$911$	1.77783e7			0.709732			0.354866	+	0.934917i	$0.384527\pi$
$912$		0			0
$913$		−	1.52894e7i		−	0.607033i
$914$		0			0
$915$	−3.91046e7	+	1.39297e7i	−1.54410	+	0.550034i
$916$		0			0
$917$		−	1.15756e7i		−	0.454592i
$918$		0			0
$919$	4.10179e7			1.60208			0.801040	−	0.598611i	$-0.204280\pi$
$919$	4.10179e7			1.60208			0.801040	+	0.598611i	$0.204280\pi$
$920$		0			0
$921$	−4.16956e7			−1.61972
$922$		0			0
$923$			1.51112e7i			0.583841i
$924$		0			0
$925$	−1.84317e7	−	2.25886e7i	−0.708290	−	0.868031i
$926$		0			0
$927$			4.38660e7i			1.67660i
$928$		0			0
$929$	−2.93427e7			−1.11548			−0.557739	−	0.830016i	$-0.688331\pi$
$929$	−2.93427e7			−1.11548			−0.557739	+	0.830016i	$0.688331\pi$
$930$		0			0
$931$	−2.08476e7			−0.788282
$932$		0			0
$933$		−	2.90981e7i		−	1.09436i
$934$		0			0
$935$	8.20351e6	−	2.92223e6i	0.306881	−	0.109316i
$936$		0			0
$937$		−	2.04953e7i		−	0.762614i	−0.924448	−	0.381307i	$-0.875474\pi$
$937$		−	2.04953e7i		−	0.762614i	0.924448	−	0.381307i	$-0.124526\pi$
$938$		0			0
$939$	−1.47444e7			−0.545712
$940$		0			0
$941$	−4.61324e7			−1.69837			−0.849184	−	0.528097i	$-0.822906\pi$
$941$	−4.61324e7			−1.69837			−0.849184	+	0.528097i	$0.822906\pi$
$942$		0			0
$943$		−	1.52990e7i		−	0.560253i
$944$		0			0
$945$	−43974.9	−	123450.i	−0.00160186	−	0.00449688i
$946$		0			0
$947$		−	2.69893e7i		−	0.977949i	−0.872298	−	0.488974i	$-0.837371\pi$
$947$		−	2.69893e7i		−	0.977949i	0.872298	−	0.488974i	$-0.162629\pi$
$948$		0			0
$949$	7.80542e7			2.81339
$950$		0			0
$951$	4.51098e7			1.61741
$952$		0			0
$953$			2.07300e7i			0.739377i	0.929156	+	0.369689i	$0.120536\pi$
$953$			2.07300e7i			0.739377i	−0.929156	+	0.369689i	$0.879464\pi$
$954$		0			0
$955$	−4.48705e6	−	1.25964e7i	−0.159204	−	0.446929i
$956$		0			0
$957$			2.26116e7i			0.798090i
$958$		0			0
$959$	−1.29050e7			−0.453116
$960$		0			0
$961$	3.05341e7			1.06654
$962$		0			0
$963$			1.73690e7i			0.603544i
$964$		0			0
$965$	−2.66924e7	+	9.50828e6i	−0.922719	+	0.328688i
$966$		0			0
$967$			1.26875e7i			0.436324i	0.975913	+	0.218162i	$0.0700062\pi$
$967$			1.26875e7i			0.436324i	−0.975913	+	0.218162i	$0.929994\pi$
$968$		0			0
$969$	−1.97319e7			−0.675088
$970$		0			0
$971$	1.10734e7			0.376905			0.188452	−	0.982082i	$-0.439653\pi$
$971$	1.10734e7			0.376905			0.188452	+	0.982082i	$0.439653\pi$
$972$		0			0
$973$		−	3.40482e6i		−	0.115295i
$974$		0			0
$975$	5.67774e7	−	4.63288e7i	1.91278	−	1.56077i
$976$		0			0
$977$			1.14221e7i			0.382833i	0.981509	+	0.191417i	$0.0613082\pi$
$977$			1.14221e7i			0.382833i	−0.981509	+	0.191417i	$0.938692\pi$
$978$		0			0
$979$	1.36830e7			0.456274
$980$		0			0
$981$	3.82246e7			1.26815
$982$		0			0
$983$			8.53757e6i			0.281806i	0.990023	+	0.140903i	$0.0450006\pi$
$983$			8.53757e6i			0.281806i	−0.990023	+	0.140903i	$0.954999\pi$
$984$		0			0
$985$	2.59941e7	−	9.25952e6i	0.853657	−	0.304087i
$986$		0			0
$987$		−	2.40730e6i		−	0.0786568i
$988$		0			0
$989$	1.19609e7			0.388841
$990$		0			0
$991$	−8.29735e6			−0.268383			−0.134192	−	0.990955i	$-0.542844\pi$
$991$	−8.29735e6			−0.268383			−0.134192	+	0.990955i	$0.542844\pi$
$992$		0			0
$993$			6.02040e7i			1.93755i
$994$		0			0
$995$	9.18278e6	+	2.57786e7i	0.294047	+	0.825472i
$996$		0			0
$997$			1.92324e6i			0.0612767i	0.999531	+	0.0306384i	$0.00975402\pi$
$997$			1.92324e6i			0.0612767i	−0.999531	+	0.0306384i	$0.990246\pi$
$998$		0			0
$999$	736424.			0.0233461

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.6.c.l.129.1		12
4.3	odd	2	inner	320.6.c.l.129.11		12
5.4	even	2	inner	320.6.c.l.129.12		12
8.3	odd	2		160.6.c.d.129.2	yes	12
8.5	even	2		160.6.c.d.129.12	yes	12
20.19	odd	2	inner	320.6.c.l.129.2		12
40.3	even	4		800.6.a.bb.1.6		6
40.13	odd	4		800.6.a.w.1.1		6
40.19	odd	2		160.6.c.d.129.11	yes	12
40.27	even	4		800.6.a.w.1.2		6
40.29	even	2		160.6.c.d.129.1	✓	12
40.37	odd	4		800.6.a.bb.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.6.c.d.129.1	✓	12	40.29	even	2
160.6.c.d.129.2	yes	12	8.3	odd	2
160.6.c.d.129.11	yes	12	40.19	odd	2
160.6.c.d.129.12	yes	12	8.5	even	2
320.6.c.l.129.1		12	1.1	even	1	trivial
320.6.c.l.129.2		12	20.19	odd	2	inner
320.6.c.l.129.11		12	4.3	odd	2	inner
320.6.c.l.129.12		12	5.4	even	2	inner
800.6.a.w.1.1		6	40.13	odd	4
800.6.a.w.1.2		6	40.27	even	4
800.6.a.bb.1.5		6	40.37	odd	4
800.6.a.bb.1.6		6	40.3	even	4

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:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	320.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-21.9637i q^{3} +(-18.7585 - 52.6604i) q^{5} +29.6983i q^{7} -239.406 q^{9} +O(q^{10})\)	\(q-21.9637i q^{3} +(-18.7585 - 52.6604i) q^{5} +29.6983i q^{7} -239.406 q^{9} +227.001 q^{11} -1067.66i q^{13} +(-1156.62 + 412.007i) q^{15} +686.258i q^{17} -1309.11 q^{19} +652.286 q^{21} -4076.26i q^{23} +(-2421.24 + 1975.66i) q^{25} -78.9360i q^{27} -4535.21 q^{29} -7691.77 q^{31} -4985.79i q^{33} +(1563.92 - 557.096i) q^{35} +9329.38i q^{37} -23449.8 q^{39} +3753.20 q^{41} +2934.28i q^{43} +(4490.90 + 12607.2i) q^{45} -3690.55i q^{47} +15925.0 q^{49} +15072.8 q^{51} +4816.68i q^{53} +(-4258.20 - 11954.0i) q^{55} +28752.9i q^{57} +31168.2 q^{59} +33809.4 q^{61} -7109.96i q^{63} +(-56223.3 + 20027.7i) q^{65} +24844.0i q^{67} -89529.9 q^{69} -14153.6 q^{71} +73107.8i q^{73} +(43392.9 + 53179.4i) q^{75} +6741.55i q^{77} +73646.1 q^{79} -59909.4 q^{81} -67353.7i q^{83} +(36138.6 - 12873.2i) q^{85} +99610.3i q^{87} +60277.5 q^{89} +31707.6 q^{91} +168940. i q^{93} +(24556.9 + 68938.2i) q^{95} -15478.8i q^{97} -54345.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 60 q^{5} - 268 q^{9}+O(q^{10}) \) 12 * q + 60 * q^5 - 268 * q^9	\( 12 q + 60 q^{5} - 268 q^{9} - 12080 q^{21} - 16420 q^{25} + 7864 q^{29} + 65560 q^{41} + 60420 q^{45} + 17956 q^{49} + 118232 q^{61} + 11360 q^{65} - 204464 q^{69} - 346436 q^{81} - 96320 q^{85} + 354936 q^{89}+O(q^{100}) \) 12 * q + 60 * q^5 - 268 * q^9 - 12080 * q^21 - 16420 * q^25 + 7864 * q^29 + 65560 * q^41 + 60420 * q^45 + 17956 * q^49 + 118232 * q^61 + 11360 * q^65 - 204464 * q^69 - 346436 * q^81 - 96320 * q^85 + 354936 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more