LMFDB - Embedded newform 1560.2.l.c.1249.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1560,2,Mod(1249,1560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1560, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1560.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1560.l (of order $2$, degree $1$, 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:	$12.4566627153$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.5161984.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 4x^{3} + 25x^{2} - 20x + 8 $ x^6 - 4x^3 + 25x^2 - 20*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.2
Root			$0.432320 + 0.432320i$ of defining polynomial
Character	$\chi$	$=$	1560.1249
Dual form			1560.2.l.c.1249.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +(0.432320 - 2.19388i) q^{5} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +(0.432320 - 2.19388i) q^{5} -1.00000 q^{9} -2.86464 q^{11} -1.00000i q^{13} +(-2.19388 - 0.432320i) q^{15} -5.52311i q^{17} -3.52311 q^{19} +7.52311i q^{23} +(-4.62620 - 1.89692i) q^{25} +1.00000i q^{27} -6.77551 q^{29} +5.72928 q^{31} +2.86464i q^{33} +3.72928i q^{37} -1.00000 q^{39} -10.1170 q^{41} -5.52311i q^{43} +(-0.432320 + 2.19388i) q^{45} -8.65847i q^{47} +7.00000 q^{49} -5.52311 q^{51} -6.77551i q^{53} +(-1.23844 + 6.28467i) q^{55} +3.52311i q^{57} -0.593923 q^{59} -5.25240 q^{61} +(-2.19388 - 0.432320i) q^{65} +10.5048i q^{67} +7.52311 q^{69} -2.38776 q^{71} +5.45856i q^{73} +(-1.89692 + 4.62620i) q^{75} -2.47689 q^{79} +1.00000 q^{81} +8.11704i q^{83} +(-12.1170 - 2.38776i) q^{85} +6.77551i q^{87} +14.1170 q^{89} -5.72928i q^{93} +(-1.52311 + 7.72928i) q^{95} -6.00000i q^{97} +2.86464 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^9	$ 6 q - 6 q^{9} - 12 q^{11} + 2 q^{15} + 4 q^{19} - 10 q^{25} + 20 q^{29} + 24 q^{31} - 6 q^{39} - 20 q^{41} + 42 q^{49} - 8 q^{51} - 20 q^{55} + 12 q^{59} + 4 q^{61} + 2 q^{65} + 20 q^{69} + 16 q^{71} - 4 q^{75} - 40 q^{79} + 6 q^{81} - 32 q^{85} + 44 q^{89} + 16 q^{95} + 12 q^{99}+O(q^{100}) $ 6 * q - 6 * q^9 - 12 * q^11 + 2 * q^15 + 4 * q^19 - 10 * q^25 + 20 * q^29 + 24 * q^31 - 6 * q^39 - 20 * q^41 + 42 * q^49 - 8 * q^51 - 20 * q^55 + 12 * q^59 + 4 * q^61 + 2 * q^65 + 20 * q^69 + 16 * q^71 - 4 * q^75 - 40 * q^79 + 6 * q^81 - 32 * q^85 + 44 * q^89 + 16 * q^95 + 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1560\mathbb{Z}\right)^\times$.

$n$	$391$	$521$	$781$	$937$	$1081$
$\chi(n)$	$1$	$1$	$1$	$-1$	$1$

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
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$	0.432320	−	2.19388i	0.193340	−	0.981132i
$5$	0.432320	−	2.19388i	0.193340	−	0.981132i
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−2.86464			−0.863722			−0.431861	−	0.901940i	$-0.642143\pi$
$11$	−2.86464			−0.863722			−0.431861	+	0.901940i	$0.642143\pi$
$12$		0			0
$13$		−	1.00000i		−	0.277350i
$13$		−	1.00000i		−	0.277350i
$14$		0			0
$15$	−2.19388	−	0.432320i	−0.566457	−	0.111625i
$16$		0			0
$17$		−	5.52311i		−	1.33955i	−0.742563	−	0.669776i	$-0.766390\pi$
$17$		−	5.52311i		−	1.33955i	0.742563	−	0.669776i	$-0.233610\pi$
$18$		0			0
$19$	−3.52311			−0.808258			−0.404129	−	0.914702i	$-0.632425\pi$
$19$	−3.52311			−0.808258			−0.404129	+	0.914702i	$0.632425\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			7.52311i			1.56868i	0.620333	+	0.784339i	$0.286998\pi$
$23$			7.52311i			1.56868i	−0.620333	+	0.784339i	$0.713002\pi$
$24$		0			0
$25$	−4.62620	−	1.89692i	−0.925240	−	0.379383i
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	−6.77551			−1.25818			−0.629090	−	0.777332i	$-0.716572\pi$
$29$	−6.77551			−1.25818			−0.629090	+	0.777332i	$0.716572\pi$
$30$		0			0
$31$	5.72928			1.02901			0.514505	−	0.857488i	$-0.327976\pi$
$31$	5.72928			1.02901			0.514505	+	0.857488i	$0.327976\pi$
$32$		0			0
$33$			2.86464i			0.498670i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			3.72928i			0.613090i	0.951856	+	0.306545i	$0.0991730\pi$
$37$			3.72928i			0.613090i	−0.951856	+	0.306545i	$0.900827\pi$
$38$		0			0
$39$	−1.00000			−0.160128
$40$		0			0
$41$	−10.1170			−1.58002			−0.790008	−	0.613097i	$-0.789924\pi$
$41$	−10.1170			−1.58002			−0.790008	+	0.613097i	$0.789924\pi$
$42$		0			0
$43$		−	5.52311i		−	0.842267i	−0.906999	−	0.421134i	$-0.861632\pi$
$43$		−	5.52311i		−	0.842267i	0.906999	−	0.421134i	$-0.138368\pi$
$44$		0			0
$45$	−0.432320	+	2.19388i	−0.0644465	+	0.327044i
$46$		0			0
$47$		−	8.65847i		−	1.26297i	−0.775389	−	0.631484i	$-0.782446\pi$
$47$		−	8.65847i		−	1.26297i	0.775389	−	0.631484i	$-0.217554\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$	−5.52311			−0.773391
$52$		0			0
$53$		−	6.77551i		−	0.930688i	−0.885130	−	0.465344i	$-0.845931\pi$
$53$		−	6.77551i		−	0.930688i	0.885130	−	0.465344i	$-0.154069\pi$
$54$		0			0
$55$	−1.23844	+	6.28467i	−0.166992	+	0.847425i
$56$		0			0
$57$			3.52311i			0.466648i
$58$		0			0
$59$	−0.593923			−0.0773221			−0.0386611	−	0.999252i	$-0.512309\pi$
$59$	−0.593923			−0.0773221			−0.0386611	+	0.999252i	$0.512309\pi$
$60$		0			0
$61$	−5.25240			−0.672500			−0.336250	−	0.941773i	$-0.609159\pi$
$61$	−5.25240			−0.672500			−0.336250	+	0.941773i	$0.609159\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.19388	−	0.432320i	−0.272117	−	0.0536228i
$66$		0			0
$67$			10.5048i			1.28336i	0.766971	+	0.641682i	$0.221763\pi$
$67$			10.5048i			1.28336i	−0.766971	+	0.641682i	$0.778237\pi$
$68$		0			0
$69$	7.52311			0.905677
$70$		0			0
$71$	−2.38776			−0.283374			−0.141687	−	0.989911i	$-0.545253\pi$
$71$	−2.38776			−0.283374			−0.141687	+	0.989911i	$0.545253\pi$
$72$		0			0
$73$			5.45856i			0.638877i	0.947607	+	0.319438i	$0.103494\pi$
$73$			5.45856i			0.638877i	−0.947607	+	0.319438i	$0.896506\pi$
$74$		0			0
$75$	−1.89692	+	4.62620i	−0.219037	+	0.534187i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−2.47689			−0.278671			−0.139336	−	0.990245i	$-0.544497\pi$
$79$	−2.47689			−0.278671			−0.139336	+	0.990245i	$0.544497\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			8.11704i			0.890961i	0.895292	+	0.445480i	$0.146967\pi$
$83$			8.11704i			0.890961i	−0.895292	+	0.445480i	$0.853033\pi$
$84$		0			0
$85$	−12.1170	−	2.38776i	−1.31428	−	0.258988i
$86$		0			0
$87$			6.77551i			0.726411i
$88$		0			0
$89$	14.1170			1.49640			0.748201	−	0.663472i	$-0.230918\pi$
$89$	14.1170			1.49640			0.748201	+	0.663472i	$0.230918\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		−	5.72928i		−	0.594099i
$94$		0			0
$95$	−1.52311	+	7.72928i	−0.156268	+	0.793008i
$96$		0			0
$97$		−	6.00000i		−	0.609208i	−0.952479	−	0.304604i	$-0.901476\pi$
$97$		−	6.00000i		−	0.609208i	0.952479	−	0.304604i	$-0.0985241\pi$
$98$		0			0
$99$	2.86464			0.287907
$100$		0			0
$101$	−11.7293			−1.16711			−0.583554	−	0.812075i	$-0.698338\pi$
$101$	−11.7293			−1.16711			−0.583554	+	0.812075i	$0.698338\pi$
$102$		0			0
$103$			14.7755i			1.45587i	0.685644	+	0.727937i	$0.259521\pi$
$103$			14.7755i			1.45587i	−0.685644	+	0.727937i	$0.740479\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	1.72928i		−	0.167176i	−0.996500	−	0.0835880i	$-0.973362\pi$
$107$		−	1.72928i		−	0.167176i	0.996500	−	0.0835880i	$-0.0266380\pi$
$108$		0			0
$109$	−12.0279			−1.15206			−0.576032	−	0.817427i	$-0.695400\pi$
$109$	−12.0279			−1.15206			−0.576032	+	0.817427i	$0.695400\pi$
$110$		0			0
$111$	3.72928			0.353968
$112$		0			0
$113$		−	7.25240i		−	0.682248i	−0.940018	−	0.341124i	$-0.889192\pi$
$113$		−	7.25240i		−	0.682248i	0.940018	−	0.341124i	$-0.110808\pi$
$114$		0			0
$115$	16.5048	+	3.25240i	1.53908	+	0.303288i
$116$		0			0
$117$			1.00000i			0.0924500i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−2.79383			−0.253985
$122$		0			0
$123$			10.1170i			0.912223i
$124$		0			0
$125$	−6.16160	+	9.32924i	−0.551110	+	0.834432i
$126$		0			0
$127$		−	17.2524i		−	1.53090i	−0.643494	−	0.765451i	$-0.722516\pi$
$127$		−	17.2524i		−	1.53090i	0.643494	−	0.765451i	$-0.277484\pi$
$128$		0			0
$129$	−5.52311			−0.486283
$130$		0			0
$131$	−8.77551			−0.766720			−0.383360	−	0.923599i	$-0.625233\pi$
$131$	−8.77551			−0.766720			−0.383360	+	0.923599i	$0.625233\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	2.19388	+	0.432320i	0.188819	+	0.0372082i
$136$		0			0
$137$		−	19.4340i		−	1.66036i	−0.557497	−	0.830179i	$-0.688238\pi$
$137$		−	19.4340i		−	1.66036i	0.557497	−	0.830179i	$-0.311762\pi$
$138$		0			0
$139$	−7.58767			−0.643577			−0.321789	−	0.946812i	$-0.604284\pi$
$139$	−7.58767			−0.643577			−0.321789	+	0.946812i	$0.604284\pi$
$140$		0			0
$141$	−8.65847			−0.729175
$142$		0			0
$143$			2.86464i			0.239553i
$144$		0			0
$145$	−2.92919	+	14.8646i	−0.243256	+	1.23444i
$146$		0			0
$147$		−	7.00000i		−	0.577350i
$148$		0			0
$149$	1.40608			0.115190			0.0575952	−	0.998340i	$-0.481657\pi$
$149$	1.40608			0.115190			0.0575952	+	0.998340i	$0.481657\pi$
$150$		0			0
$151$	−1.31695			−0.107172			−0.0535858	−	0.998563i	$-0.517065\pi$
$151$	−1.31695			−0.107172			−0.0535858	+	0.998563i	$0.517065\pi$
$152$		0			0
$153$			5.52311i			0.446517i
$154$		0			0
$155$	2.47689	−	12.5693i	0.198948	−	1.00959i
$156$		0			0
$157$		−	20.7110i		−	1.65291i	−0.562999	−	0.826457i	$-0.690353\pi$
$157$		−	20.7110i		−	1.65291i	0.562999	−	0.826457i	$-0.309647\pi$
$158$		0			0
$159$	−6.77551			−0.537333
$160$		0			0
$161$		0			0
$162$		0			0
$163$			2.27072i			0.177856i	0.996038	+	0.0889282i	$0.0283442\pi$
$163$			2.27072i			0.177856i	−0.996038	+	0.0889282i	$0.971656\pi$
$164$		0			0
$165$	6.28467	+	1.23844i	0.489261	+	0.0964126i
$166$		0			0
$167$		−	7.34153i		−	0.568104i	−0.958809	−	0.284052i	$-0.908321\pi$
$167$		−	7.34153i		−	0.568104i	0.958809	−	0.284052i	$-0.0916789\pi$
$168$		0			0
$169$	−1.00000			−0.0769231
$170$		0			0
$171$	3.52311			0.269419
$172$		0			0
$173$		−	16.0925i		−	1.22349i	−0.791056	−	0.611743i	$-0.790468\pi$
$173$		−	16.0925i		−	1.22349i	0.791056	−	0.611743i	$-0.209532\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			0.593923i			0.0446420i
$178$		0			0
$179$	−15.0462			−1.12461			−0.562304	−	0.826931i	$-0.690085\pi$
$179$	−15.0462			−1.12461			−0.562304	+	0.826931i	$0.690085\pi$
$180$		0			0
$181$	5.58767			0.415328			0.207664	−	0.978200i	$-0.433414\pi$
$181$	5.58767			0.415328			0.207664	+	0.978200i	$0.433414\pi$
$182$		0			0
$183$			5.25240i			0.388268i
$184$		0			0
$185$	8.18159	+	1.61224i	0.601522	+	0.118535i
$186$		0			0
$187$			15.8217i			1.15700i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	17.5510			1.26995			0.634974	−	0.772534i	$-0.281011\pi$
$191$	17.5510			1.26995			0.634974	+	0.772534i	$0.281011\pi$
$192$		0			0
$193$			13.8217i			0.994911i	0.867490	+	0.497455i	$0.165732\pi$
$193$			13.8217i			0.994911i	−0.867490	+	0.497455i	$0.834268\pi$
$194$		0			0
$195$	−0.432320	+	2.19388i	−0.0309591	+	0.157107i
$196$		0			0
$197$		−	26.3511i		−	1.87744i	−0.344682	−	0.938719i	$-0.612013\pi$
$197$		−	26.3511i		−	1.87744i	0.344682	−	0.938719i	$-0.387987\pi$
$198$		0			0
$199$	23.0741			1.63568			0.817841	−	0.575444i	$-0.195171\pi$
$199$	23.0741			1.63568			0.817841	+	0.575444i	$0.195171\pi$
$200$		0			0
$201$	10.5048			0.740951
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−4.37380	+	22.1955i	−0.305480	+	1.55020i
$206$		0			0
$207$		−	7.52311i		−	0.522893i
$208$		0			0
$209$	10.0925			0.698110
$210$		0			0
$211$	1.52311			0.104856			0.0524278	−	0.998625i	$-0.483304\pi$
$211$	1.52311			0.104856			0.0524278	+	0.998625i	$0.483304\pi$
$212$		0			0
$213$			2.38776i			0.163606i
$214$		0			0
$215$	−12.1170	−	2.38776i	−0.826375	−	0.162844i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	5.45856			0.368856
$220$		0			0
$221$	−5.52311			−0.371525
$222$		0			0
$223$			13.3169i			0.891769i	0.895091	+	0.445884i	$0.147111\pi$
$223$			13.3169i			0.891769i	−0.895091	+	0.445884i	$0.852889\pi$
$224$		0			0
$225$	4.62620	+	1.89692i	0.308413	+	0.126461i
$226$		0			0
$227$			6.80009i			0.451338i	0.974204	+	0.225669i	$0.0724568\pi$
$227$			6.80009i			0.451338i	−0.974204	+	0.225669i	$0.927543\pi$
$228$		0			0
$229$	−1.52311			−0.100650			−0.0503251	−	0.998733i	$-0.516026\pi$
$229$	−1.52311			−0.100650			−0.0503251	+	0.998733i	$0.516026\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	10.7110i		−	0.701698i	−0.936432	−	0.350849i	$-0.885893\pi$
$233$		−	10.7110i		−	0.701698i	0.936432	−	0.350849i	$-0.114107\pi$
$234$		0			0
$235$	−18.9956	−	3.74324i	−1.23914	−	0.244182i
$236$		0			0
$237$			2.47689i			0.160891i
$238$		0			0
$239$	−4.65847			−0.301332			−0.150666	−	0.988585i	$-0.548142\pi$
$239$	−4.65847			−0.301332			−0.150666	+	0.988585i	$0.548142\pi$
$240$		0			0
$241$	5.22449			0.336539			0.168269	−	0.985741i	$-0.446182\pi$
$241$	5.22449			0.336539			0.168269	+	0.985741i	$0.446182\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	3.02624	−	15.3571i	0.193340	−	0.981132i
$246$		0			0
$247$			3.52311i			0.224170i
$248$		0			0
$249$	8.11704			0.514396
$250$		0			0
$251$	5.85838			0.369778			0.184889	−	0.982759i	$-0.440807\pi$
$251$	5.85838			0.369778			0.184889	+	0.982759i	$0.440807\pi$
$252$		0			0
$253$		−	21.5510i		−	1.35490i
$254$		0			0
$255$	−2.38776	+	12.1170i	−0.149527	+	0.758798i
$256$		0			0
$257$			2.29862i			0.143384i	0.997427	+	0.0716921i	$0.0228399\pi$
$257$			2.29862i			0.143384i	−0.997427	+	0.0716921i	$0.977160\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	6.77551			0.419394
$262$		0			0
$263$		−	31.5789i		−	1.94724i	−0.228175	−	0.973620i	$-0.573276\pi$
$263$		−	31.5789i		−	1.94724i	0.228175	−	0.973620i	$-0.426724\pi$
$264$		0			0
$265$	−14.8646	−	2.92919i	−0.913128	−	0.179939i
$266$		0			0
$267$		−	14.1170i		−	0.863949i
$268$		0			0
$269$	30.3632			1.85128			0.925638	−	0.378411i	$-0.123529\pi$
$269$	30.3632			1.85128			0.925638	+	0.378411i	$0.123529\pi$
$270$		0			0
$271$	28.5972			1.73716			0.868580	−	0.495550i	$-0.165033\pi$
$271$	28.5972			1.73716			0.868580	+	0.495550i	$0.165033\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	13.2524	+	5.43398i	0.799150	+	0.327682i
$276$		0			0
$277$		−	1.79383i		−	0.107781i	−0.998547	−	0.0538905i	$-0.982838\pi$
$277$		−	1.79383i		−	0.107781i	0.998547	−	0.0538905i	$-0.0171622\pi$
$278$		0			0
$279$	−5.72928			−0.343003
$280$		0			0
$281$	32.2095			1.92146			0.960729	−	0.277489i	$-0.0895024\pi$
$281$	32.2095			1.92146			0.960729	+	0.277489i	$0.0895024\pi$
$282$		0			0
$283$			18.0925i			1.07548i	0.843109	+	0.537742i	$0.180723\pi$
$283$			18.0925i			1.07548i	−0.843109	+	0.537742i	$0.819277\pi$
$284$		0			0
$285$	7.72928	+	1.52311i	0.457843	+	0.0902215i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−13.5048			−0.794400
$290$		0			0
$291$	−6.00000			−0.351726
$292$		0			0
$293$		−	12.6218i		−	0.737375i	−0.929553	−	0.368688i	$-0.879807\pi$
$293$		−	12.6218i		−	0.737375i	0.929553	−	0.368688i	$-0.120193\pi$
$294$		0			0
$295$	−0.256765	+	1.30299i	−0.0149494	+	0.0758632i
$296$		0			0
$297$		−	2.86464i		−	0.166223i
$298$		0			0
$299$	7.52311			0.435073
$300$		0			0
$301$		0			0
$302$		0			0
$303$			11.7293i			0.673830i
$304$		0			0
$305$	−2.27072	+	11.5231i	−0.130021	+	0.659812i
$306$		0			0
$307$		−	20.0000i		−	1.14146i	−0.821138	−	0.570730i	$-0.806660\pi$
$307$		−	20.0000i		−	1.14146i	0.821138	−	0.570730i	$-0.193340\pi$
$308$		0			0
$309$	14.7755			0.840549
$310$		0			0
$311$	−31.8217			−1.80445			−0.902223	−	0.431271i	$-0.858065\pi$
$311$	−31.8217			−1.80445			−0.902223	+	0.431271i	$0.858065\pi$
$312$		0			0
$313$			11.7938i			0.666627i	0.942816	+	0.333313i	$0.108167\pi$
$313$			11.7938i			0.666627i	−0.942816	+	0.333313i	$0.891833\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	11.4340i		−	0.642197i	−0.947046	−	0.321098i	$-0.895948\pi$
$317$		−	11.4340i		−	0.642197i	0.947046	−	0.321098i	$-0.104052\pi$
$318$		0			0
$319$	19.4094			1.08672
$320$		0			0
$321$	−1.72928			−0.0965191
$322$		0			0
$323$			19.4586i			1.08270i
$324$		0			0
$325$	−1.89692	+	4.62620i	−0.105222	+	0.256615i
$326$		0			0
$327$			12.0279i			0.665145i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	2.20617			0.121262			0.0606310	−	0.998160i	$-0.480689\pi$
$331$	2.20617			0.121262			0.0606310	+	0.998160i	$0.480689\pi$
$332$		0			0
$333$		−	3.72928i		−	0.204363i
$334$		0			0
$335$	23.0462	+	4.54144i	1.25915	+	0.248125i
$336$		0			0
$337$			16.8034i			0.915340i	0.889122	+	0.457670i	$0.151316\pi$
$337$			16.8034i			0.915340i	−0.889122	+	0.457670i	$0.848684\pi$
$338$		0			0
$339$	−7.25240			−0.393896
$340$		0			0
$341$	−16.4123			−0.888778
$342$		0			0
$343$		0			0
$344$		0			0
$345$	3.25240	−	16.5048i	0.175103	−	0.888588i
$346$		0			0
$347$		−	9.96336i		−	0.534861i	−0.963577	−	0.267430i	$-0.913825\pi$
$347$		−	9.96336i		−	0.534861i	0.963577	−	0.267430i	$-0.0861746\pi$
$348$		0			0
$349$	21.3449			1.14256			0.571282	−	0.820754i	$-0.306446\pi$
$349$	21.3449			1.14256			0.571282	+	0.820754i	$0.306446\pi$
$350$		0			0
$351$	1.00000			0.0533761
$352$		0			0
$353$			33.3973i			1.77756i	0.458333	+	0.888781i	$0.348447\pi$
$353$			33.3973i			1.77756i	−0.458333	+	0.888781i	$0.651553\pi$
$354$		0			0
$355$	−1.03228	+	5.23844i	−0.0547875	+	0.278028i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−21.9754			−1.15982			−0.579909	−	0.814681i	$-0.696912\pi$
$359$	−21.9754			−1.15982			−0.579909	+	0.814681i	$0.696912\pi$
$360$		0			0
$361$	−6.58767			−0.346719
$362$		0			0
$363$			2.79383i			0.146638i
$364$		0			0
$365$	11.9754	+	2.35985i	0.626822	+	0.123520i
$366$		0			0
$367$		−	29.2803i		−	1.52842i	−0.644968	−	0.764210i	$-0.723129\pi$
$367$		−	29.2803i		−	1.52842i	0.644968	−	0.764210i	$-0.276871\pi$
$368$		0			0
$369$	10.1170			0.526672
$370$		0			0
$371$		0			0
$372$		0			0
$373$			2.41233i			0.124906i	0.998048	+	0.0624530i	$0.0198923\pi$
$373$			2.41233i			0.124906i	−0.998048	+	0.0624530i	$0.980108\pi$
$374$		0			0
$375$	9.32924	+	6.16160i	0.481760	+	0.318184i
$376$		0			0
$377$			6.77551i			0.348957i
$378$		0			0
$379$	−19.5231			−1.00284			−0.501418	−	0.865205i	$-0.667188\pi$
$379$	−19.5231			−1.00284			−0.501418	+	0.865205i	$0.667188\pi$
$380$		0			0
$381$	−17.2524			−0.883867
$382$		0			0
$383$		−	23.7047i		−	1.21125i	−0.795749	−	0.605627i	$-0.792922\pi$
$383$		−	23.7047i		−	1.21125i	0.795749	−	0.605627i	$-0.207078\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$			5.52311i			0.280756i
$388$		0			0
$389$	14.0000			0.709828			0.354914	−	0.934899i	$-0.384510\pi$
$389$	14.0000			0.709828			0.354914	+	0.934899i	$0.384510\pi$
$390$		0			0
$391$	41.5510			2.10133
$392$		0			0
$393$			8.77551i			0.442666i
$394$		0			0
$395$	−1.07081	+	5.43398i	−0.0538782	+	0.273413i
$396$		0			0
$397$		−	28.8680i		−	1.44884i	−0.689358	−	0.724421i	$-0.742107\pi$
$397$		−	28.8680i		−	1.44884i	0.689358	−	0.724421i	$-0.257893\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	20.0804			1.00277			0.501383	−	0.865225i	$-0.332825\pi$
$401$	20.0804			1.00277			0.501383	+	0.865225i	$0.332825\pi$
$402$		0			0
$403$		−	5.72928i		−	0.285396i
$404$		0			0
$405$	0.432320	−	2.19388i	0.0214822	−	0.109015i
$406$		0			0
$407$		−	10.6831i		−	0.529539i
$408$		0			0
$409$	13.6926			0.677057			0.338529	−	0.940956i	$-0.390071\pi$
$409$	13.6926			0.677057			0.338529	+	0.940956i	$0.390071\pi$
$410$		0			0
$411$	−19.4340			−0.958608
$412$		0			0
$413$		0			0
$414$		0			0
$415$	17.8078	+	3.50916i	0.874150	+	0.172258i
$416$		0			0
$417$			7.58767i			0.371570i
$418$		0			0
$419$	16.3632			0.799393			0.399697	−	0.916647i	$-0.369115\pi$
$419$	16.3632			0.799393			0.399697	+	0.916647i	$0.369115\pi$
$420$		0			0
$421$	24.9817			1.21753			0.608766	−	0.793350i	$-0.291665\pi$
$421$	24.9817			1.21753			0.608766	+	0.793350i	$0.291665\pi$
$422$		0			0
$423$			8.65847i			0.420989i
$424$		0			0
$425$	−10.4769	+	25.5510i	−0.508204	+	1.23941i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	2.86464			0.138306
$430$		0			0
$431$	−27.9388			−1.34576			−0.672882	−	0.739750i	$-0.734944\pi$
$431$	−27.9388			−1.34576			−0.672882	+	0.739750i	$0.734944\pi$
$432$		0			0
$433$		−	26.7110i		−	1.28365i	−0.766852	−	0.641823i	$-0.778178\pi$
$433$		−	26.7110i		−	1.28365i	0.766852	−	0.641823i	$-0.221822\pi$
$434$		0			0
$435$	14.8646	+	2.92919i	0.712705	+	0.140444i
$436$		0			0
$437$		−	26.5048i		−	1.26790i
$438$		0			0
$439$	5.00958			0.239094			0.119547	−	0.992829i	$-0.461856\pi$
$439$	5.00958			0.239094			0.119547	+	0.992829i	$0.461856\pi$
$440$		0			0
$441$	−7.00000			−0.333333
$442$		0			0
$443$		−	19.6926i		−	0.935625i	−0.883828	−	0.467813i	$-0.845042\pi$
$443$		−	19.6926i		−	0.935625i	0.883828	−	0.467813i	$-0.154958\pi$
$444$		0			0
$445$	6.10308	−	30.9711i	0.289314	−	1.46817i
$446$		0			0
$447$		−	1.40608i		−	0.0665052i
$448$		0			0
$449$	−38.7143			−1.82704			−0.913520	−	0.406794i	$-0.866647\pi$
$449$	−38.7143			−1.82704			−0.913520	+	0.406794i	$0.866647\pi$
$450$		0			0
$451$	28.9817			1.36469
$452$		0			0
$453$			1.31695i			0.0618756i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	10.9538i		−	0.512396i	−0.966624	−	0.256198i	$-0.917530\pi$
$457$		−	10.9538i		−	0.512396i	0.966624	−	0.256198i	$-0.0824699\pi$
$458$		0			0
$459$	5.52311			0.257797
$460$		0			0
$461$	−0.864641			−0.0402703			−0.0201352	−	0.999797i	$-0.506410\pi$
$461$	−0.864641			−0.0402703			−0.0201352	+	0.999797i	$0.506410\pi$
$462$		0			0
$463$			16.0000i			0.743583i	0.928316	+	0.371792i	$0.121256\pi$
$463$			16.0000i			0.743583i	−0.928316	+	0.371792i	$0.878744\pi$
$464$		0			0
$465$	−12.5693	−	2.47689i	−0.582889	−	0.114863i
$466$		0			0
$467$		−	20.7755i		−	0.961376i	−0.876892	−	0.480688i	$-0.840387\pi$
$467$		−	20.7755i		−	0.961376i	0.876892	−	0.480688i	$-0.159613\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−20.7110			−0.954311
$472$		0			0
$473$			15.8217i			0.727484i
$474$		0			0
$475$	16.2986	+	6.68305i	0.747832	+	0.306640i
$476$		0			0
$477$			6.77551i			0.310229i
$478$		0			0
$479$	−20.6585			−0.943910			−0.471955	−	0.881623i	$-0.656451\pi$
$479$	−20.6585			−0.943910			−0.471955	+	0.881623i	$0.656451\pi$
$480$		0			0
$481$	3.72928			0.170041
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−13.1633	−	2.59392i	−0.597713	−	0.117784i
$486$		0			0
$487$			16.3632i			0.741486i	0.928735	+	0.370743i	$0.120897\pi$
$487$			16.3632i			0.741486i	−0.928735	+	0.370743i	$0.879103\pi$
$488$		0			0
$489$	2.27072			0.102685
$490$		0			0
$491$	26.7389			1.20671			0.603354	−	0.797473i	$-0.293831\pi$
$491$	26.7389			1.20671			0.603354	+	0.797473i	$0.293831\pi$
$492$		0			0
$493$			37.4219i			1.68540i
$494$		0			0
$495$	1.23844	−	6.28467i	0.0556639	−	0.282475i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−4.24281			−0.189934			−0.0949672	−	0.995480i	$-0.530275\pi$
$499$	−4.24281			−0.189934			−0.0949672	+	0.995480i	$0.530275\pi$
$500$		0			0
$501$	−7.34153			−0.327995
$502$		0			0
$503$		−	0.889220i		−	0.0396484i	−0.999803	−	0.0198242i	$-0.993689\pi$
$503$		−	0.889220i		−	0.0396484i	0.999803	−	0.0198242i	$-0.00631065\pi$
$504$		0			0
$505$	−5.07081	+	25.7326i	−0.225648	+	1.14509i
$506$		0			0
$507$			1.00000i			0.0444116i
$508$		0			0
$509$	11.3694			0.503941			0.251971	−	0.967735i	$-0.418921\pi$
$509$	11.3694			0.503941			0.251971	+	0.967735i	$0.418921\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		−	3.52311i		−	0.155549i
$514$		0			0
$515$	32.4157	+	6.38776i	1.42840	+	0.281478i
$516$		0			0
$517$			24.8034i			1.09085i
$518$		0			0
$519$	−16.0925			−0.706380
$520$		0			0
$521$	3.18785			0.139662			0.0698310	−	0.997559i	$-0.477754\pi$
$521$	3.18785			0.139662			0.0698310	+	0.997559i	$0.477754\pi$
$522$		0			0
$523$		−	37.1666i		−	1.62518i	−0.582835	−	0.812591i	$-0.698056\pi$
$523$		−	37.1666i		−	1.62518i	0.582835	−	0.812591i	$-0.301944\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	31.6435i		−	1.37841i
$528$		0			0
$529$	−33.5972			−1.46075
$530$		0			0
$531$	0.593923			0.0257740
$532$		0			0
$533$			10.1170i			0.438218i
$534$		0			0
$535$	−3.79383	−	0.747604i	−0.164022	−	0.0323217i
$536$		0			0
$537$			15.0462i			0.649293i
$538$		0			0
$539$	−20.0525			−0.863722
$540$		0			0
$541$	−8.02791			−0.345147			−0.172573	−	0.984997i	$-0.555208\pi$
$541$	−8.02791			−0.345147			−0.172573	+	0.984997i	$0.555208\pi$
$542$		0			0
$543$		−	5.58767i		−	0.239790i
$544$		0			0
$545$	−5.19991	+	26.3878i	−0.222740	+	1.13033i
$546$		0			0
$547$		−	31.2032i		−	1.33415i	−0.744989	−	0.667077i	$-0.767545\pi$
$547$		−	31.2032i		−	1.33415i	0.744989	−	0.667077i	$-0.232455\pi$
$548$		0			0
$549$	5.25240			0.224167
$550$		0			0
$551$	23.8709			1.01693
$552$		0			0
$553$		0			0
$554$		0			0
$555$	1.61224	−	8.18159i	0.0684360	−	0.347289i
$556$		0			0
$557$			29.2191i			1.23805i	0.785370	+	0.619026i	$0.212472\pi$
$557$			29.2191i			1.23805i	−0.785370	+	0.619026i	$0.787528\pi$
$558$		0			0
$559$	−5.52311			−0.233603
$560$		0			0
$561$	15.8217			0.667994
$562$		0			0
$563$			30.5048i			1.28562i	0.766024	+	0.642812i	$0.222232\pi$
$563$			30.5048i			1.28562i	−0.766024	+	0.642812i	$0.777768\pi$
$564$		0			0
$565$	−15.9109	−	3.13536i	−0.669375	−	0.131906i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	12.1416			0.509003			0.254502	−	0.967072i	$-0.418089\pi$
$569$	12.1416			0.509003			0.254502	+	0.967072i	$0.418089\pi$
$570$		0			0
$571$	−8.85258			−0.370469			−0.185234	−	0.982694i	$-0.559304\pi$
$571$	−8.85258			−0.370469			−0.185234	+	0.982694i	$0.559304\pi$
$572$		0			0
$573$		−	17.5510i		−	0.733204i
$574$		0			0
$575$	14.2707	−	34.8034i	0.595130	−	1.45140i
$576$		0			0
$577$			30.2341i			1.25866i	0.777138	+	0.629330i	$0.216671\pi$
$577$			30.2341i			1.25866i	−0.777138	+	0.629330i	$0.783329\pi$
$578$		0			0
$579$	13.8217			0.574412
$580$		0			0
$581$		0			0
$582$		0			0
$583$			19.4094i			0.803855i
$584$		0			0
$585$	2.19388	+	0.432320i	0.0907057	+	0.0178743i
$586$		0			0
$587$			23.0342i			0.950722i	0.879791	+	0.475361i	$0.157682\pi$
$587$			23.0342i			0.950722i	−0.879791	+	0.475361i	$0.842318\pi$
$588$		0			0
$589$	−20.1849			−0.831705
$590$		0			0
$591$	−26.3511			−1.08394
$592$		0			0
$593$			27.9754i			1.14881i	0.818570	+	0.574406i	$0.194767\pi$
$593$			27.9754i			1.14881i	−0.818570	+	0.574406i	$0.805233\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	23.0741i		−	0.944361i
$598$		0			0
$599$	−22.7389			−0.929085			−0.464542	−	0.885551i	$-0.653781\pi$
$599$	−22.7389			−0.929085			−0.464542	+	0.885551i	$0.653781\pi$
$600$		0			0
$601$	−11.2158			−0.457500			−0.228750	−	0.973485i	$-0.573464\pi$
$601$	−11.2158			−0.457500			−0.228750	+	0.973485i	$0.573464\pi$
$602$		0			0
$603$		−	10.5048i		−	0.427788i
$604$		0			0
$605$	−1.20783	+	6.12933i	−0.0491053	+	0.249193i
$606$		0			0
$607$		−	23.7293i		−	0.963142i	−0.876407	−	0.481571i	$-0.840066\pi$
$607$		−	23.7293i		−	0.963142i	0.876407	−	0.481571i	$-0.159934\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−8.65847			−0.350284
$612$		0			0
$613$		−	18.2341i		−	0.736467i	−0.929733	−	0.368234i	$-0.879963\pi$
$613$		−	18.2341i		−	0.736467i	0.929733	−	0.368234i	$-0.120037\pi$
$614$		0			0
$615$	22.1955	+	4.37380i	0.895011	+	0.176369i
$616$		0			0
$617$		−	0.0245796i		−	0.000989536i	−1.00000		0.000494768i	$-0.999843\pi$
$617$		−	0.0245796i		−	0.000989536i	1.00000		0.000494768i	$-0.000157490\pi$
$618$		0			0
$619$	−31.2158			−1.25467			−0.627334	−	0.778751i	$-0.715854\pi$
$619$	−31.2158			−1.25467			−0.627334	+	0.778751i	$0.715854\pi$
$620$		0			0
$621$	−7.52311			−0.301892
$622$		0			0
$623$		0			0
$624$		0			0
$625$	17.8034	+	17.5510i	0.712137	+	0.702041i
$626$		0			0
$627$		−	10.0925i		−	0.403054i
$628$		0			0
$629$	20.5972			0.821266
$630$		0			0
$631$	13.4460			0.535279			0.267639	−	0.963519i	$-0.413756\pi$
$631$	13.4460			0.535279			0.267639	+	0.963519i	$0.413756\pi$
$632$		0			0
$633$		−	1.52311i		−	0.0605384i
$634$		0			0
$635$	−37.8496	−	7.45856i	−1.50202	−	0.295984i
$636$		0			0
$637$		−	7.00000i		−	0.277350i
$638$		0			0
$639$	2.38776			0.0944581
$640$		0			0
$641$	2.54144			0.100381			0.0501904	−	0.998740i	$-0.484017\pi$
$641$	2.54144			0.100381			0.0501904	+	0.998740i	$0.484017\pi$
$642$		0			0
$643$			13.9634i			0.550661i	0.961350	+	0.275330i	$0.0887873\pi$
$643$			13.9634i			0.550661i	−0.961350	+	0.275330i	$0.911213\pi$
$644$		0			0
$645$	−2.38776	+	12.1170i	−0.0940178	+	0.477108i
$646$		0			0
$647$			30.8034i			1.21101i	0.795843	+	0.605504i	$0.207028\pi$
$647$			30.8034i			1.21101i	−0.795843	+	0.605504i	$0.792972\pi$
$648$		0			0
$649$	1.70138			0.0667848
$650$		0			0
$651$		0			0
$652$		0			0
$653$			2.00000i			0.0782660i	0.999234	+	0.0391330i	$0.0124596\pi$
$653$			2.00000i			0.0782660i	−0.999234	+	0.0391330i	$0.987540\pi$
$654$		0			0
$655$	−3.79383	+	19.2524i	−0.148237	+	0.752253i
$656$		0			0
$657$		−	5.45856i		−	0.212959i
$658$		0			0
$659$	4.05581			0.157992			0.0789960	−	0.996875i	$-0.474829\pi$
$659$	4.05581			0.157992			0.0789960	+	0.996875i	$0.474829\pi$
$660$		0			0
$661$	−32.9325			−1.28093			−0.640463	−	0.767989i	$-0.721258\pi$
$661$	−32.9325			−1.28093			−0.640463	+	0.767989i	$0.721258\pi$
$662$		0			0
$663$			5.52311i			0.214500i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	50.9729i		−	1.97368i
$668$		0			0
$669$	13.3169			0.514863
$670$		0			0
$671$	15.0462			0.580853
$672$		0			0
$673$			9.42192i			0.363188i	0.983374	+	0.181594i	$0.0581257\pi$
$673$			9.42192i			0.363188i	−0.983374	+	0.181594i	$0.941874\pi$
$674$		0			0
$675$	1.89692	−	4.62620i	0.0730123	−	0.178062i
$676$		0			0
$677$		−	42.2341i		−	1.62319i	−0.584222	−	0.811594i	$-0.698600\pi$
$677$		−	42.2341i		−	1.62319i	0.584222	−	0.811594i	$-0.301400\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	6.80009			0.260580
$682$		0			0
$683$		−	18.7509i		−	0.717484i	−0.933437	−	0.358742i	$-0.883206\pi$
$683$		−	18.7509i		−	0.717484i	0.933437	−	0.358742i	$-0.116794\pi$
$684$		0			0
$685$	−42.6358	−	8.40171i	−1.62903	−	0.321013i
$686$		0			0
$687$			1.52311i			0.0581104i
$688$		0			0
$689$	−6.77551			−0.258126
$690$		0			0
$691$	45.2524			1.72148			0.860741	−	0.509043i	$-0.170001\pi$
$691$	45.2524			1.72148			0.860741	+	0.509043i	$0.170001\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−3.28030	+	16.6464i	−0.124429	+	0.631434i
$696$		0			0
$697$			55.8776i			2.11651i
$698$		0			0
$699$	−10.7110			−0.405126
$700$		0			0
$701$	−37.4586			−1.41479			−0.707395	−	0.706818i	$-0.750130\pi$
$701$	−37.4586			−1.41479			−0.707395	+	0.706818i	$0.750130\pi$
$702$		0			0
$703$		−	13.1387i		−	0.495535i
$704$		0			0
$705$	−3.74324	+	18.9956i	−0.140978	+	0.715417i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−11.8988			−0.446869			−0.223435	−	0.974719i	$-0.571727\pi$
$709$	−11.8988			−0.446869			−0.223435	+	0.974719i	$0.571727\pi$
$710$		0			0
$711$	2.47689			0.0928905
$712$		0			0
$713$			43.1020i			1.61418i
$714$		0			0
$715$	6.28467	+	1.23844i	0.235033	+	0.0463151i
$716$		0			0
$717$			4.65847i			0.173974i
$718$		0			0
$719$	−25.7851			−0.961622			−0.480811	−	0.876824i	$-0.659658\pi$
$719$	−25.7851			−0.961622			−0.480811	+	0.876824i	$0.659658\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		−	5.22449i		−	0.194301i
$724$		0			0
$725$	31.3449	+	12.8526i	1.16412	+	0.477333i
$726$		0			0
$727$		−	34.8034i		−	1.29079i	−0.763850	−	0.645394i	$-0.776693\pi$
$727$		−	34.8034i		−	1.29079i	0.763850	−	0.645394i	$-0.223307\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−30.5048			−1.12826
$732$		0			0
$733$			19.5510i			0.722133i	0.932540	+	0.361067i	$0.117587\pi$
$733$			19.5510i			0.722133i	−0.932540	+	0.361067i	$0.882413\pi$
$734$		0			0
$735$	−15.3571	−	3.02624i	−0.566457	−	0.111625i
$736$		0			0
$737$		−	30.0925i		−	1.10847i
$738$		0			0
$739$	−12.2986			−0.452412			−0.226206	−	0.974079i	$-0.572632\pi$
$739$	−12.2986			−0.452412			−0.226206	+	0.974079i	$0.572632\pi$
$740$		0			0
$741$	3.52311			0.129425
$742$		0			0
$743$			40.9975i			1.50405i	0.659133	+	0.752027i	$0.270924\pi$
$743$			40.9975i			1.50405i	−0.659133	+	0.752027i	$0.729076\pi$
$744$		0			0
$745$	0.607876	−	3.08476i	0.0222709	−	0.113017i
$746$		0			0
$747$		−	8.11704i		−	0.296987i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	33.5510			1.22429			0.612147	−	0.790744i	$-0.290306\pi$
$751$	33.5510			1.22429			0.612147	+	0.790744i	$0.290306\pi$
$752$		0			0
$753$		−	5.85838i		−	0.213491i
$754$		0			0
$755$	−0.569343	+	2.88922i	−0.0207205	+	0.105149i
$756$		0			0
$757$		−	25.1020i		−	0.912349i	−0.889890	−	0.456175i	$-0.849219\pi$
$757$		−	25.1020i		−	0.912349i	0.889890	−	0.456175i	$-0.150781\pi$
$758$		0			0
$759$	−21.5510			−0.782253
$760$		0			0
$761$	−39.9021			−1.44645			−0.723226	−	0.690612i	$-0.757341\pi$
$761$	−39.9021			−1.44645			−0.723226	+	0.690612i	$0.757341\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	12.1170	+	2.38776i	0.438092	+	0.0863295i
$766$		0			0
$767$			0.593923i			0.0214453i
$768$		0			0
$769$	15.5510			0.560784			0.280392	−	0.959886i	$-0.409536\pi$
$769$	15.5510			0.560784			0.280392	+	0.959886i	$0.409536\pi$
$770$		0			0
$771$	2.29862			0.0827830
$772$		0			0
$773$			27.7972i			0.999794i	0.866085	+	0.499897i	$0.166629\pi$
$773$			27.7972i			0.999794i	−0.866085	+	0.499897i	$0.833371\pi$
$774$		0			0
$775$	−26.5048	−	10.8680i	−0.952080	−	0.390389i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	35.6435			1.27706
$780$		0			0
$781$	6.84006			0.244757
$782$		0			0
$783$		−	6.77551i		−	0.242137i
$784$		0			0
$785$	−45.4373	−	8.95377i	−1.62173	−	0.319574i
$786$		0			0
$787$			30.6339i			1.09198i	0.837792	+	0.545990i	$0.183846\pi$
$787$			30.6339i			1.09198i	−0.837792	+	0.545990i	$0.816154\pi$
$788$		0			0
$789$	−31.5789			−1.12424
$790$		0			0
$791$		0			0
$792$		0			0
$793$			5.25240i			0.186518i
$794$		0			0
$795$	−2.92919	+	14.8646i	−0.103888	+	0.527194i
$796$		0			0
$797$			18.6464i			0.660490i	0.943895	+	0.330245i	$0.107131\pi$
$797$			18.6464i			0.660490i	−0.943895	+	0.330245i	$0.892869\pi$

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:	\( N \)	\(=\)	\( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1560.l (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +(0.432320 - 2.19388i) q^{5} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +(0.432320 - 2.19388i) q^{5} -1.00000 q^{9} -2.86464 q^{11} -1.00000i q^{13} +(-2.19388 - 0.432320i) q^{15} -5.52311i q^{17} -3.52311 q^{19} +7.52311i q^{23} +(-4.62620 - 1.89692i) q^{25} +1.00000i q^{27} -6.77551 q^{29} +5.72928 q^{31} +2.86464i q^{33} +3.72928i q^{37} -1.00000 q^{39} -10.1170 q^{41} -5.52311i q^{43} +(-0.432320 + 2.19388i) q^{45} -8.65847i q^{47} +7.00000 q^{49} -5.52311 q^{51} -6.77551i q^{53} +(-1.23844 + 6.28467i) q^{55} +3.52311i q^{57} -0.593923 q^{59} -5.25240 q^{61} +(-2.19388 - 0.432320i) q^{65} +10.5048i q^{67} +7.52311 q^{69} -2.38776 q^{71} +5.45856i q^{73} +(-1.89692 + 4.62620i) q^{75} -2.47689 q^{79} +1.00000 q^{81} +8.11704i q^{83} +(-12.1170 - 2.38776i) q^{85} +6.77551i q^{87} +14.1170 q^{89} -5.72928i q^{93} +(-1.52311 + 7.72928i) q^{95} -6.00000i q^{97} +2.86464 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^9	\( 6 q - 6 q^{9} - 12 q^{11} + 2 q^{15} + 4 q^{19} - 10 q^{25} + 20 q^{29} + 24 q^{31} - 6 q^{39} - 20 q^{41} + 42 q^{49} - 8 q^{51} - 20 q^{55} + 12 q^{59} + 4 q^{61} + 2 q^{65} + 20 q^{69} + 16 q^{71} - 4 q^{75} - 40 q^{79} + 6 q^{81} - 32 q^{85} + 44 q^{89} + 16 q^{95} + 12 q^{99}+O(q^{100}) \) 6 * q - 6 * q^9 - 12 * q^11 + 2 * q^15 + 4 * q^19 - 10 * q^25 + 20 * q^29 + 24 * q^31 - 6 * q^39 - 20 * q^41 + 42 * q^49 - 8 * q^51 - 20 * q^55 + 12 * q^59 + 4 * q^61 + 2 * q^65 + 20 * q^69 + 16 * q^71 - 4 * q^75 - 40 * q^79 + 6 * q^81 - 32 * q^85 + 44 * q^89 + 16 * q^95 + 12 * q^99

Introduction

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