LMFDB - Embedded newform 1280.3.e.l.639.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,3,Mod(639,1280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1280, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1280.639");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1280 = 2^{8} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1280.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$34.8774738381$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 640)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			639.19
Character	$\chi$	$=$	1280.639
Dual form			1280.3.e.l.639.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.933412i q^{3} +(0.723593 - 4.94736i) q^{5} -6.91082 q^{7} +8.12874 q^{9} +O(q^{10})$	$q-0.933412i q^{3} +(0.723593 - 4.94736i) q^{5} -6.91082 q^{7} +8.12874 q^{9} +11.8128 q^{11} -13.4577 q^{13} +(-4.61793 - 0.675411i) q^{15} +28.7701i q^{17} +30.1635 q^{19} +6.45064i q^{21} +39.5266 q^{23} +(-23.9528 - 7.15976i) q^{25} -15.9882i q^{27} +18.9999i q^{29} -6.41602i q^{31} -11.0262i q^{33} +(-5.00062 + 34.1903i) q^{35} +51.5200 q^{37} +12.5616i q^{39} +17.0345 q^{41} -58.0207i q^{43} +(5.88190 - 40.2158i) q^{45} -48.1230 q^{47} -1.24057 q^{49} +26.8543 q^{51} -4.02173 q^{53} +(8.54765 - 58.4421i) q^{55} -28.1550i q^{57} +45.6972 q^{59} +37.8458i q^{61} -56.1763 q^{63} +(-9.73792 + 66.5802i) q^{65} -3.82746i q^{67} -36.8947i q^{69} -119.629i q^{71} -82.9311i q^{73} +(-6.68301 + 22.3579i) q^{75} -81.6360 q^{77} -83.2914i q^{79} +58.2351 q^{81} +115.831i q^{83} +(142.336 + 20.8178i) q^{85} +17.7348 q^{87} -54.7938 q^{89} +93.0039 q^{91} -5.98879 q^{93} +(21.8261 - 149.230i) q^{95} +51.8578i q^{97} +96.0230 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 72 q^{9}+O(q^{10}) $ 24 * q - 72 * q^9	$ 24 q - 72 q^{9} + 16 q^{11} - 48 q^{19} - 24 q^{25} + 112 q^{35} - 80 q^{41} + 168 q^{49} + 576 q^{51} - 496 q^{59} + 32 q^{65} - 224 q^{75} + 184 q^{81} - 144 q^{89} + 864 q^{91} - 368 q^{99}+O(q^{100}) $ 24 * q - 72 * q^9 + 16 * q^11 - 48 * q^19 - 24 * q^25 + 112 * q^35 - 80 * q^41 + 168 * q^49 + 576 * q^51 - 496 * q^59 + 32 * q^65 - 224 * q^75 + 184 * q^81 - 144 * q^89 + 864 * q^91 - 368 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times$.

$n$	$257$	$261$	$511$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$		−	0.933412i		−	0.311137i	−0.987825	−	0.155569i	$-0.950279\pi$
$3$		−	0.933412i		−	0.311137i	0.987825	−	0.155569i	$-0.0497210\pi$
$4$		0			0
$5$	0.723593	−	4.94736i	0.144719	−	0.989473i
$5$	0.723593	−	4.94736i	0.144719	−	0.989473i
$6$		0			0
$7$	−6.91082			−0.987260			−0.493630	−	0.869672i	$-0.664330\pi$
$7$	−6.91082			−0.987260			−0.493630	+	0.869672i	$0.664330\pi$
$8$		0			0
$9$	8.12874			0.903194
$10$		0			0
$11$	11.8128			1.07389			0.536944	−	0.843618i	$-0.319579\pi$
$11$	11.8128			1.07389			0.536944	+	0.843618i	$0.319579\pi$
$12$		0			0
$13$	−13.4577			−1.03521			−0.517605	−	0.855620i	$-0.673176\pi$
$13$	−13.4577			−1.03521			−0.517605	+	0.855620i	$0.673176\pi$
$14$		0			0
$15$	−4.61793	−	0.675411i	−0.307862	−	0.0450274i
$16$		0			0
$17$			28.7701i			1.69236i	0.532899	+	0.846179i	$0.321102\pi$
$17$			28.7701i			1.69236i	−0.532899	+	0.846179i	$0.678898\pi$
$18$		0			0
$19$	30.1635			1.58755			0.793777	−	0.608209i	$-0.208112\pi$
$19$	30.1635			1.58755			0.793777	+	0.608209i	$0.208112\pi$
$20$		0			0
$21$			6.45064i			0.307174i
$22$		0			0
$23$	39.5266			1.71855			0.859275	−	0.511514i	$-0.170915\pi$
$23$	39.5266			1.71855			0.859275	+	0.511514i	$0.170915\pi$
$24$		0			0
$25$	−23.9528	−	7.15976i	−0.958113	−	0.286390i
$26$		0			0
$27$		−	15.9882i		−	0.592155i
$28$		0			0
$29$			18.9999i			0.655170i	0.944822	+	0.327585i	$0.106235\pi$
$29$			18.9999i			0.655170i	−0.944822	+	0.327585i	$0.893765\pi$
$30$		0			0
$31$		−	6.41602i		−	0.206968i	−0.994631	−	0.103484i	$-0.967001\pi$
$31$		−	6.41602i		−	0.206968i	0.994631	−	0.103484i	$-0.0329991\pi$
$32$		0			0
$33$		−	11.0262i		−	0.334127i
$34$		0			0
$35$	−5.00062	+	34.1903i	−0.142875	+	0.976867i
$36$		0			0
$37$	51.5200			1.39243			0.696217	−	0.717832i	$-0.254865\pi$
$37$	51.5200			1.39243			0.696217	+	0.717832i	$0.254865\pi$
$38$		0			0
$39$			12.5616i			0.322092i
$40$		0			0
$41$	17.0345			0.415477			0.207738	−	0.978184i	$-0.433390\pi$
$41$	17.0345			0.415477			0.207738	+	0.978184i	$0.433390\pi$
$42$		0			0
$43$		−	58.0207i		−	1.34932i	−0.738129	−	0.674660i	$-0.764290\pi$
$43$		−	58.0207i		−	1.34932i	0.738129	−	0.674660i	$-0.235710\pi$
$44$		0			0
$45$	5.88190	−	40.2158i	0.130709	−	0.893685i
$46$		0			0
$47$	−48.1230			−1.02389			−0.511947	−	0.859017i	$-0.671076\pi$
$47$	−48.1230			−1.02389			−0.511947	+	0.859017i	$0.671076\pi$
$48$		0			0
$49$	−1.24057			−0.0253178
$50$		0			0
$51$	26.8543			0.526556
$52$		0			0
$53$	−4.02173			−0.0758817			−0.0379409	−	0.999280i	$-0.512080\pi$
$53$	−4.02173			−0.0758817			−0.0379409	+	0.999280i	$0.512080\pi$
$54$		0			0
$55$	8.54765	−	58.4421i	0.155412	−	1.06258i
$56$		0			0
$57$		−	28.1550i		−	0.493947i
$58$		0			0
$59$	45.6972			0.774529			0.387264	−	0.921969i	$-0.373420\pi$
$59$	45.6972			0.774529			0.387264	+	0.921969i	$0.373420\pi$
$60$		0			0
$61$			37.8458i			0.620422i	0.950668	+	0.310211i	$0.100400\pi$
$61$			37.8458i			0.620422i	−0.950668	+	0.310211i	$0.899600\pi$
$62$		0			0
$63$	−56.1763			−0.891687
$64$		0			0
$65$	−9.73792	+	66.5802i	−0.149814	+	1.02431i
$66$		0			0
$67$		−	3.82746i		−	0.0571263i	−0.999592	−	0.0285632i	$-0.990907\pi$
$67$		−	3.82746i		−	0.0571263i	0.999592	−	0.0285632i	$-0.00909317\pi$
$68$		0			0
$69$		−	36.8947i		−	0.534705i
$70$		0			0
$71$		−	119.629i		−	1.68492i	−0.538760	−	0.842459i	$-0.681107\pi$
$71$		−	119.629i		−	1.68492i	0.538760	−	0.842459i	$-0.318893\pi$
$72$		0			0
$73$		−	82.9311i		−	1.13604i	−0.823014	−	0.568021i	$-0.807709\pi$
$73$		−	82.9311i		−	1.13604i	0.823014	−	0.568021i	$-0.192291\pi$
$74$		0			0
$75$	−6.68301	+	22.3579i	−0.0891068	+	0.298105i
$76$		0			0
$77$	−81.6360			−1.06021
$78$		0			0
$79$		−	83.2914i		−	1.05432i	−0.849766	−	0.527161i	$-0.823257\pi$
$79$		−	83.2914i		−	1.05432i	0.849766	−	0.527161i	$-0.176743\pi$
$80$		0			0
$81$	58.2351			0.718952
$82$		0			0
$83$			115.831i			1.39555i	0.716317	+	0.697775i	$0.245826\pi$
$83$			115.831i			1.39555i	−0.716317	+	0.697775i	$0.754174\pi$
$84$		0			0
$85$	142.336	+	20.8178i	1.67454	+	0.244916i
$86$		0			0
$87$	17.7348			0.203848
$88$		0			0
$89$	−54.7938			−0.615661			−0.307830	−	0.951441i	$-0.599603\pi$
$89$	−54.7938			−0.615661			−0.307830	+	0.951441i	$0.599603\pi$
$90$		0			0
$91$	93.0039			1.02202
$92$		0			0
$93$	−5.98879			−0.0643956
$94$		0			0
$95$	21.8261	−	149.230i	0.229749	−	1.57084i
$96$		0			0
$97$			51.8578i			0.534617i	0.963611	+	0.267308i	$0.0861342\pi$
$97$			51.8578i			0.534617i	−0.963611	+	0.267308i	$0.913866\pi$
$98$		0			0
$99$	96.0230			0.969929
$100$		0			0
$101$		−	3.60567i		−	0.0356997i	−0.999841	−	0.0178498i	$-0.994318\pi$
$101$		−	3.60567i		−	0.0356997i	0.999841	−	0.0178498i	$-0.00568208\pi$
$102$		0			0
$103$	−8.24447			−0.0800434			−0.0400217	−	0.999199i	$-0.512743\pi$
$103$	−8.24447			−0.0800434			−0.0400217	+	0.999199i	$0.512743\pi$
$104$		0			0
$105$	31.9137	+	4.66764i	0.303940	+	0.0444537i
$106$		0			0
$107$		−	52.6034i		−	0.491620i	−0.969318	−	0.245810i	$-0.920946\pi$
$107$		−	52.6034i		−	0.491620i	0.969318	−	0.245810i	$-0.0790540\pi$
$108$		0			0
$109$		−	110.432i		−	1.01313i	−0.862201	−	0.506567i	$-0.830914\pi$
$109$		−	110.432i		−	1.01313i	0.862201	−	0.506567i	$-0.169086\pi$
$110$		0			0
$111$		−	48.0894i		−	0.433238i
$112$		0			0
$113$		−	88.3827i		−	0.782148i	−0.920359	−	0.391074i	$-0.872104\pi$
$113$		−	88.3827i		−	0.782148i	0.920359	−	0.391074i	$-0.127896\pi$
$114$		0			0
$115$	28.6012	−	195.553i	0.248706	−	1.70046i
$116$		0			0
$117$	−109.394			−0.934994
$118$		0			0
$119$		−	198.825i		−	1.67080i
$120$		0			0
$121$	18.5416			0.153237
$122$		0			0
$123$		−	15.9003i		−	0.129270i
$124$		0			0
$125$	−52.7541	+	113.323i	−0.422032	+	0.906581i
$126$		0			0
$127$	69.5834			0.547901			0.273950	−	0.961744i	$-0.411670\pi$
$127$	69.5834			0.547901			0.273950	+	0.961744i	$0.411670\pi$
$128$		0			0
$129$	−54.1573			−0.419824
$130$		0			0
$131$	154.902			1.18246			0.591230	−	0.806503i	$-0.298643\pi$
$131$	154.902			1.18246			0.591230	+	0.806503i	$0.298643\pi$
$132$		0			0
$133$	−208.455			−1.56733
$134$		0			0
$135$	−79.0993	−	11.5689i	−0.585921	−	0.0856959i
$136$		0			0
$137$			72.4528i			0.528852i	0.964406	+	0.264426i	$0.0851825\pi$
$137$			72.4528i			0.528852i	−0.964406	+	0.264426i	$0.914817\pi$
$138$		0			0
$139$	103.567			0.745087			0.372544	−	0.928015i	$-0.378486\pi$
$139$	103.567			0.745087			0.372544	+	0.928015i	$0.378486\pi$
$140$		0			0
$141$			44.9186i			0.318572i
$142$		0			0
$143$	−158.973			−1.11170
$144$		0			0
$145$	93.9995	+	13.7482i	0.648273	+	0.0948153i
$146$		0			0
$147$			1.15797i			0.00787731i
$148$		0			0
$149$			115.086i			0.772391i	0.922417	+	0.386195i	$0.126211\pi$
$149$			115.086i			0.772391i	−0.922417	+	0.386195i	$0.873789\pi$
$150$		0			0
$151$			176.561i			1.16928i	0.811293	+	0.584640i	$0.198764\pi$
$151$			176.561i			1.16928i	−0.811293	+	0.584640i	$0.801236\pi$
$152$		0			0
$153$			233.864i			1.52853i
$154$		0			0
$155$	−31.7424	−	4.64259i	−0.204790	−	0.0299522i
$156$		0			0
$157$	79.1290			0.504006			0.252003	−	0.967726i	$-0.418911\pi$
$157$	79.1290			0.504006			0.252003	+	0.967726i	$0.418911\pi$
$158$		0			0
$159$			3.75393i			0.0236096i
$160$		0			0
$161$	−273.161			−1.69666
$162$		0			0
$163$			10.6626i			0.0654147i	0.999465	+	0.0327074i	$0.0104129\pi$
$163$			10.6626i			0.0654147i	−0.999465	+	0.0327074i	$0.989587\pi$
$164$		0			0
$165$	−54.5506	−	7.97848i	−0.330610	−	0.0483544i
$166$		0			0
$167$	−19.5606			−0.117129			−0.0585647	−	0.998284i	$-0.518652\pi$
$167$	−19.5606			−0.117129			−0.0585647	+	0.998284i	$0.518652\pi$
$168$		0			0
$169$	12.1103			0.0716584
$170$		0			0
$171$	245.191			1.43387
$172$		0			0
$173$	202.219			1.16890			0.584448	−	0.811431i	$-0.301311\pi$
$173$	202.219			1.16890			0.584448	+	0.811431i	$0.301311\pi$
$174$		0			0
$175$	165.534	+	49.4798i	0.945907	+	0.282742i
$176$		0			0
$177$		−	42.6543i		−	0.240985i
$178$		0			0
$179$	−66.7608			−0.372965			−0.186483	−	0.982458i	$-0.559709\pi$
$179$	−66.7608			−0.372965			−0.186483	+	0.982458i	$0.559709\pi$
$180$		0			0
$181$		−	138.580i		−	0.765636i	−0.923824	−	0.382818i	$-0.874954\pi$
$181$		−	138.580i		−	0.765636i	0.923824	−	0.382818i	$-0.125046\pi$
$182$		0			0
$183$	35.3257			0.193037
$184$		0			0
$185$	37.2796	−	254.888i	0.201511	−	1.37778i
$186$		0			0
$187$			339.854i			1.81740i
$188$		0			0
$189$			110.491i			0.584611i
$190$		0			0
$191$			61.3227i			0.321061i	0.987031	+	0.160531i	$0.0513205\pi$
$191$			61.3227i			0.321061i	−0.987031	+	0.160531i	$0.948680\pi$
$192$		0			0
$193$		−	365.204i		−	1.89225i	−0.323800	−	0.946126i	$-0.604960\pi$
$193$		−	365.204i		−	1.89225i	0.323800	−	0.946126i	$-0.395040\pi$
$194$		0			0
$195$	62.1468	+	9.08949i	0.318702	+	0.0466128i
$196$		0			0
$197$	151.856			0.770844			0.385422	−	0.922740i	$-0.374056\pi$
$197$	151.856			0.770844			0.385422	+	0.922740i	$0.374056\pi$
$198$		0			0
$199$		−	29.1323i		−	0.146393i	−0.997318	−	0.0731967i	$-0.976680\pi$
$199$		−	29.1323i		−	0.146393i	0.997318	−	0.0731967i	$-0.0233201\pi$
$200$		0			0
$201$	−3.57260			−0.0177741
$202$		0			0
$203$		−	131.305i		−	0.646823i
$204$		0			0
$205$	12.3261	−	84.2761i	0.0601272	−	0.411103i
$206$		0			0
$207$	321.302			1.55218
$208$		0			0
$209$	356.315			1.70486
$210$		0			0
$211$	227.463			1.07802			0.539012	−	0.842298i	$-0.318798\pi$
$211$	227.463			1.07802			0.539012	+	0.842298i	$0.318798\pi$
$212$		0			0
$213$	−111.663			−0.524241
$214$		0			0
$215$	−287.050	−	41.9834i	−1.33511	−	0.195272i
$216$		0			0
$217$			44.3399i			0.204332i
$218$		0			0
$219$	−77.4089			−0.353465
$220$		0			0
$221$		−	387.180i		−	1.75194i
$222$		0			0
$223$	−0.518789			−0.00232641			−0.00116320	−	0.999999i	$-0.500370\pi$
$223$	−0.518789			−0.00232641			−0.00116320	+	0.999999i	$0.500370\pi$
$224$		0			0
$225$	−194.706	−	58.1998i	−0.865361	−	0.258666i
$226$		0			0
$227$		−	375.370i		−	1.65361i	−0.562489	−	0.826805i	$-0.690156\pi$
$227$		−	375.370i		−	1.65361i	0.562489	−	0.826805i	$-0.309844\pi$
$228$		0			0
$229$			4.43677i			0.0193745i	0.999953	+	0.00968727i	$0.00308360\pi$
$229$			4.43677i			0.0193745i	−0.999953	+	0.00968727i	$0.996916\pi$
$230$		0			0
$231$			76.2000i			0.329870i
$232$		0			0
$233$			259.393i			1.11328i	0.830755	+	0.556638i	$0.187909\pi$
$233$			259.393i			1.11328i	−0.830755	+	0.556638i	$0.812091\pi$
$234$		0			0
$235$	−34.8215	+	238.082i	−0.148177	+	1.01312i
$236$		0			0
$237$	−77.7452			−0.328039
$238$		0			0
$239$		−	389.947i		−	1.63158i	−0.578351	−	0.815788i	$-0.696304\pi$
$239$		−	389.947i		−	1.63158i	0.578351	−	0.815788i	$-0.303696\pi$
$240$		0			0
$241$	216.169			0.896967			0.448483	−	0.893791i	$-0.351964\pi$
$241$	216.169			0.896967			0.448483	+	0.893791i	$0.351964\pi$
$242$		0			0
$243$		−	198.251i		−	0.815848i
$244$		0			0
$245$	−0.897670	+	6.13756i	−0.00366396	+	0.0250513i
$246$		0			0
$247$	−405.932			−1.64345
$248$		0			0
$249$	108.118			0.434208
$250$		0			0
$251$	181.548			0.723300			0.361650	−	0.932314i	$-0.382214\pi$
$251$	181.548			0.723300			0.361650	+	0.932314i	$0.382214\pi$
$252$		0			0
$253$	466.919			1.84553
$254$		0			0
$255$	19.4316	−	132.858i	0.0762024	−	0.521013i
$256$		0			0
$257$			256.554i			0.998266i	0.866525	+	0.499133i	$0.166348\pi$
$257$			256.554i			0.998266i	−0.866525	+	0.499133i	$0.833652\pi$
$258$		0			0
$259$	−356.046			−1.37469
$260$		0			0
$261$			154.445i			0.591745i
$262$		0			0
$263$	379.271			1.44209			0.721047	−	0.692886i	$-0.243661\pi$
$263$	379.271			1.44209			0.721047	+	0.692886i	$0.243661\pi$
$264$		0			0
$265$	−2.91010	+	19.8970i	−0.0109815	+	0.0750829i
$266$		0			0
$267$			51.1452i			0.191555i
$268$		0			0
$269$			203.365i			0.756005i	0.925804	+	0.378003i	$0.123389\pi$
$269$			203.365i			0.756005i	−0.925804	+	0.378003i	$0.876611\pi$
$270$		0			0
$271$			432.023i			1.59418i	0.603861	+	0.797090i	$0.293628\pi$
$271$			432.023i			1.59418i	−0.603861	+	0.797090i	$0.706372\pi$
$272$		0			0
$273$		−	86.8110i		−	0.317989i
$274$		0			0
$275$	−282.949	−	84.5766i	−1.02891	−	0.307551i
$276$		0			0
$277$	−199.461			−0.720077			−0.360039	−	0.932937i	$-0.617237\pi$
$277$	−199.461			−0.720077			−0.360039	+	0.932937i	$0.617237\pi$
$278$		0			0
$279$		−	52.1541i		−	0.186932i
$280$		0			0
$281$	−382.351			−1.36068			−0.680340	−	0.732897i	$-0.738168\pi$
$281$	−382.351			−1.36068			−0.680340	+	0.732897i	$0.738168\pi$
$282$		0			0
$283$		−	300.383i		−	1.06142i	−0.847552	−	0.530712i	$-0.821925\pi$
$283$		−	300.383i		−	1.06142i	0.847552	−	0.530712i	$-0.178075\pi$
$284$		0			0
$285$	−139.293	−	20.3728i	−0.488747	−	0.0714834i
$286$		0			0
$287$	−117.723			−0.410183
$288$		0			0
$289$	−538.717			−1.86407
$290$		0			0
$291$	48.4047			0.166339
$292$		0			0
$293$	13.5540			0.0462594			0.0231297	−	0.999732i	$-0.492637\pi$
$293$	13.5540			0.0462594			0.0231297	+	0.999732i	$0.492637\pi$
$294$		0			0
$295$	33.0662	−	226.081i	0.112089	−	0.766375i
$296$		0			0
$297$		−	188.865i		−	0.635908i
$298$		0			0
$299$	−531.939			−1.77906
$300$		0			0
$301$			400.971i			1.33213i
$302$		0			0
$303$	−3.36557			−0.0111075
$304$		0			0
$305$	187.237	+	27.3849i	0.613891	+	0.0897867i
$306$		0			0
$307$			305.607i			0.995463i	0.867331	+	0.497732i	$0.165834\pi$
$307$			305.607i			0.995463i	−0.867331	+	0.497732i	$0.834166\pi$
$308$		0			0
$309$			7.69549i			0.0249045i
$310$		0			0
$311$			266.374i			0.856509i	0.903658	+	0.428254i	$0.140871\pi$
$311$			266.374i			0.856509i	−0.903658	+	0.428254i	$0.859129\pi$
$312$		0			0
$313$			507.453i			1.62126i	0.585561	+	0.810628i	$0.300874\pi$
$313$			507.453i			1.62126i	−0.585561	+	0.810628i	$0.699126\pi$
$314$		0			0
$315$	−40.6488	+	277.924i	−0.129044	+	0.882300i
$316$		0			0
$317$	−425.070			−1.34092			−0.670458	−	0.741948i	$-0.733902\pi$
$317$	−425.070			−1.34092			−0.670458	+	0.741948i	$0.733902\pi$
$318$		0			0
$319$			224.442i			0.703579i
$320$		0			0
$321$	−49.1006			−0.152962
$322$		0			0
$323$			867.807i			2.68671i
$324$		0			0
$325$	322.350	+	96.3541i	0.991848	+	0.296474i
$326$		0			0
$327$	−103.078			−0.315224
$328$		0			0
$329$	332.570			1.01085
$330$		0			0
$331$	−245.761			−0.742479			−0.371240	−	0.928537i	$-0.621067\pi$
$331$	−245.761			−0.742479			−0.371240	+	0.928537i	$0.621067\pi$
$332$		0			0
$333$	418.793			1.25764
$334$		0			0
$335$	−18.9359	−	2.76953i	−0.0565249	−	0.00826725i
$336$		0			0
$337$		−	155.538i		−	0.461538i	−0.973009	−	0.230769i	$-0.925876\pi$
$337$		−	155.538i		−	0.461538i	0.973009	−	0.230769i	$-0.0741241\pi$
$338$		0			0
$339$	−82.4975			−0.243355
$340$		0			0
$341$		−	75.7910i		−	0.222261i
$342$		0			0
$343$	347.204			1.01226
$344$		0			0
$345$	−182.531	−	26.6967i	−0.529076	−	0.0773818i
$346$		0			0
$347$		−	143.795i		−	0.414395i	−0.978299	−	0.207198i	$-0.933566\pi$
$347$		−	143.795i		−	0.414395i	0.978299	−	0.207198i	$-0.0664343\pi$
$348$		0			0
$349$		−	384.926i		−	1.10294i	−0.834195	−	0.551470i	$-0.814067\pi$
$349$		−	384.926i		−	1.10294i	0.834195	−	0.551470i	$-0.185933\pi$
$350$		0			0
$351$			215.164i			0.613004i
$352$		0			0
$353$			463.015i			1.31166i	0.754910	+	0.655829i	$0.227681\pi$
$353$			463.015i			1.31166i	−0.754910	+	0.655829i	$0.772319\pi$
$354$		0			0
$355$	−591.849	−	86.5629i	−1.66718	−	0.243839i
$356$		0			0
$357$	−185.585			−0.519847
$358$		0			0
$359$			33.1612i			0.0923711i	0.998933	+	0.0461856i	$0.0147066\pi$
$359$			33.1612i			0.0923711i	−0.998933	+	0.0461856i	$0.985293\pi$
$360$		0			0
$361$	548.838			1.52033
$362$		0			0
$363$		−	17.3070i		−	0.0476777i
$364$		0			0
$365$	−410.291	−	60.0084i	−1.12408	−	0.164407i
$366$		0			0
$367$	−288.141			−0.785125			−0.392562	−	0.919725i	$-0.628411\pi$
$367$	−288.141			−0.785125			−0.392562	+	0.919725i	$0.628411\pi$
$368$		0			0
$369$	138.469			0.375256
$370$		0			0
$371$	27.7935			0.0749150
$372$		0			0
$373$	−139.091			−0.372898			−0.186449	−	0.982465i	$-0.559698\pi$
$373$	−139.091			−0.372898			−0.186449	+	0.982465i	$0.559698\pi$
$374$		0			0
$375$	105.777	+	49.2413i	0.282071	+	0.131310i
$376$		0			0
$377$		−	255.696i		−	0.678238i
$378$		0			0
$379$	−384.173			−1.01365			−0.506825	−	0.862049i	$-0.669181\pi$
$379$	−384.173			−1.01365			−0.506825	+	0.862049i	$0.669181\pi$
$380$		0			0
$381$		−	64.9500i		−	0.170472i
$382$		0			0
$383$	−357.786			−0.934167			−0.467084	−	0.884213i	$-0.654695\pi$
$383$	−357.786			−0.934167			−0.467084	+	0.884213i	$0.654695\pi$
$384$		0			0
$385$	−59.0712	+	403.883i	−0.153432	+	1.04905i
$386$		0			0
$387$		−	471.636i		−	1.21870i
$388$		0			0
$389$		−	511.950i		−	1.31607i	−0.752989	−	0.658033i	$-0.771389\pi$
$389$		−	511.950i		−	1.31607i	0.752989	−	0.658033i	$-0.228611\pi$
$390$		0			0
$391$			1137.18i			2.90840i
$392$		0			0
$393$		−	144.588i		−	0.367908i
$394$		0			0
$395$	−412.073	−	60.2691i	−1.04322	−	0.152580i
$396$		0			0
$397$	11.0817			0.0279137			0.0139568	−	0.999903i	$-0.495557\pi$
$397$	11.0817			0.0279137			0.0139568	+	0.999903i	$0.495557\pi$
$398$		0			0
$399$			194.574i			0.487654i
$400$		0			0
$401$	679.510			1.69454			0.847269	−	0.531164i	$-0.178245\pi$
$401$	679.510			1.69454			0.847269	+	0.531164i	$0.178245\pi$
$402$		0			0
$403$			86.3450i			0.214256i
$404$		0			0
$405$	42.1385	−	288.110i	0.104046	−	0.711384i
$406$		0			0
$407$	608.595			1.49532
$408$		0			0
$409$	−773.907			−1.89219			−0.946097	−	0.323884i	$-0.895011\pi$
$409$	−773.907			−1.89219			−0.946097	+	0.323884i	$0.895011\pi$
$410$		0			0
$411$	67.6283			0.164546
$412$		0			0
$413$	−315.805			−0.764661
$414$		0			0
$415$	573.056	+	83.8142i	1.38086	+	0.201962i
$416$		0			0
$417$		−	96.6709i		−	0.231825i
$418$		0			0
$419$	35.5511			0.0848474			0.0424237	−	0.999100i	$-0.486492\pi$
$419$	35.5511			0.0848474			0.0424237	+	0.999100i	$0.486492\pi$
$420$		0			0
$421$			549.424i			1.30504i	0.757770	+	0.652522i	$0.226289\pi$
$421$			549.424i			1.30504i	−0.757770	+	0.652522i	$0.773711\pi$
$422$		0			0
$423$	−391.180			−0.924775
$424$		0			0
$425$	205.987	−	689.125i	0.484675	−	1.62147i
$426$		0			0
$427$		−	261.545i		−	0.612518i
$428$		0			0
$429$			148.387i			0.345891i
$430$		0			0
$431$			707.347i			1.64118i	0.571520	+	0.820588i	$0.306354\pi$
$431$			707.347i			1.64118i	−0.571520	+	0.820588i	$0.693646\pi$
$432$		0			0
$433$		−	209.829i		−	0.484594i	−0.970202	−	0.242297i	$-0.922099\pi$
$433$		−	209.829i		−	0.484594i	0.970202	−	0.242297i	$-0.0779008\pi$
$434$		0			0
$435$	12.8328	−	87.7403i	0.0295006	−	0.201702i
$436$		0			0
$437$	1192.26			2.72829
$438$		0			0
$439$			551.120i			1.25540i	0.778456	+	0.627699i	$0.216003\pi$
$439$			551.120i			1.25540i	−0.778456	+	0.627699i	$0.783997\pi$
$440$		0			0
$441$	−10.0843			−0.0228669
$442$		0			0
$443$		−	374.038i		−	0.844330i	−0.906519	−	0.422165i	$-0.861270\pi$
$443$		−	374.038i		−	0.844330i	0.906519	−	0.422165i	$-0.138730\pi$
$444$		0			0
$445$	−39.6484	+	271.085i	−0.0890976	+	0.609180i
$446$		0			0
$447$	107.423			0.240320
$448$		0			0
$449$	−429.736			−0.957096			−0.478548	−	0.878061i	$-0.658837\pi$
$449$	−429.736			−0.957096			−0.478548	+	0.878061i	$0.658837\pi$
$450$		0			0
$451$	201.225			0.446176
$452$		0			0
$453$	164.804			0.363807
$454$		0			0
$455$	67.2970	−	460.124i	0.147905	−	1.01126i
$456$		0			0
$457$		−	25.8641i		−	0.0565953i	−0.999600	−	0.0282977i	$-0.990991\pi$
$457$		−	25.8641i		−	0.0565953i	0.999600	−	0.0282977i	$-0.00900863\pi$
$458$		0			0
$459$	459.981			1.00214
$460$		0			0
$461$			33.7766i			0.0732680i	0.999329	+	0.0366340i	$0.0116636\pi$
$461$			33.7766i			0.0732680i	−0.999329	+	0.0366340i	$0.988336\pi$
$462$		0			0
$463$	−598.561			−1.29279			−0.646394	−	0.763004i	$-0.723724\pi$
$463$	−598.561			−1.29279			−0.646394	+	0.763004i	$0.723724\pi$
$464$		0			0
$465$	−4.33345	+	29.6287i	−0.00931924	+	0.0637177i
$466$		0			0
$467$			620.732i			1.32919i	0.747204	+	0.664595i	$0.231396\pi$
$467$			620.732i			1.32919i	−0.747204	+	0.664595i	$0.768604\pi$
$468$		0			0
$469$			26.4509i			0.0563985i
$470$		0			0
$471$		−	73.8600i		−	0.156815i
$472$		0			0
$473$		−	685.386i		−	1.44902i
$474$		0			0
$475$	−722.502	−	215.964i	−1.52106	−	0.454660i
$476$		0			0
$477$	−32.6916			−0.0685359
$478$		0			0
$479$			183.309i			0.382690i	0.981523	+	0.191345i	$0.0612850\pi$
$479$			183.309i			0.382690i	−0.981523	+	0.191345i	$0.938715\pi$
$480$		0			0
$481$	−693.342			−1.44146
$482$		0			0
$483$			254.972i			0.527893i
$484$		0			0
$485$	256.559	+	37.5240i	0.528989	+	0.0773690i
$486$		0			0
$487$	311.586			0.639807			0.319903	−	0.947450i	$-0.396350\pi$
$487$	311.586			0.639807			0.319903	+	0.947450i	$0.396350\pi$
$488$		0			0
$489$	9.95260			0.0203530
$490$		0			0
$491$	707.884			1.44172			0.720859	−	0.693081i	$-0.243747\pi$
$491$	707.884			1.44172			0.720859	+	0.693081i	$0.243747\pi$
$492$		0			0
$493$	−546.629			−1.10878
$494$		0			0
$495$	69.4816	−	475.061i	0.140367	−	0.959719i
$496$		0			0
$497$			826.736i			1.66345i
$498$		0			0
$499$	−583.958			−1.17026			−0.585128	−	0.810941i	$-0.698956\pi$
$499$	−583.958			−1.17026			−0.585128	+	0.810941i	$0.698956\pi$
$500$		0			0
$501$			18.2581i			0.0364433i
$502$		0			0
$503$	767.659			1.52616			0.763080	−	0.646304i	$-0.223686\pi$
$503$	767.659			1.52616			0.763080	+	0.646304i	$0.223686\pi$
$504$		0			0
$505$	−17.8385	−	2.60904i	−0.0353238	−	0.00516641i
$506$		0			0
$507$		−	11.3039i		−	0.0222956i
$508$		0			0
$509$			70.8826i			0.139259i	0.997573	+	0.0696293i	$0.0221816\pi$
$509$			70.8826i			0.139259i	−0.997573	+	0.0696293i	$0.977818\pi$
$510$		0			0
$511$			573.122i			1.12157i
$512$		0			0
$513$		−	482.260i		−	0.940077i
$514$		0			0
$515$	−5.96565	+	40.7884i	−0.0115838	+	0.0792008i
$516$		0			0
$517$	−568.467			−1.09955
$518$		0			0
$519$		−	188.754i		−	0.363688i
$520$		0			0
$521$	−167.378			−0.321263			−0.160632	−	0.987014i	$-0.551353\pi$
$521$	−167.378			−0.321263			−0.160632	+	0.987014i	$0.551353\pi$
$522$		0			0
$523$			342.892i			0.655624i	0.944743	+	0.327812i	$0.106311\pi$
$523$			342.892i			0.655624i	−0.944743	+	0.327812i	$0.893689\pi$
$524$		0			0
$525$	46.1851	−	154.511i	0.0879716	−	0.294307i
$526$		0			0
$527$	184.589			0.350264
$528$		0			0
$529$	1033.36			1.95341
$530$		0			0
$531$	371.461			0.699549
$532$		0			0
$533$	−229.246			−0.430105
$534$		0			0
$535$	−260.248	−	38.0635i	−0.486445	−	0.0711467i
$536$		0			0
$537$			62.3154i			0.116044i
$538$		0			0
$539$	−14.6546			−0.0271885
$540$		0			0
$541$			176.843i			0.326881i	0.986553	+	0.163441i	$0.0522592\pi$
$541$			176.843i			0.326881i	−0.986553	+	0.163441i	$0.947741\pi$
$542$		0			0
$543$	−129.352			−0.238218
$544$		0			0
$545$	−546.345	−	79.9075i	−1.00247	−	0.146619i
$546$		0			0
$547$			278.331i			0.508832i	0.967095	+	0.254416i	$0.0818832\pi$
$547$			278.331i			0.508832i	−0.967095	+	0.254416i	$0.918117\pi$
$548$		0			0
$549$			307.638i			0.560361i
$550$		0			0
$551$			573.105i			1.04012i
$552$		0			0
$553$			575.612i			1.04089i
$554$		0			0
$555$	−237.916	−	34.7972i	−0.428677	−	0.0626977i
$556$		0			0
$557$	445.245			0.799363			0.399682	−	0.916654i	$-0.369121\pi$
$557$	445.245			0.799363			0.399682	+	0.916654i	$0.369121\pi$
$558$		0			0
$559$			780.827i			1.39683i
$560$		0			0
$561$	317.224			0.565462
$562$		0			0
$563$			191.846i			0.340758i	0.985379	+	0.170379i	$0.0544991\pi$
$563$			191.846i			0.340758i	−0.985379	+	0.170379i	$0.945501\pi$
$564$		0			0
$565$	−437.261	−	63.9532i	−0.773914	−	0.113191i
$566$		0			0
$567$	−402.452			−0.709793
$568$		0			0
$569$	198.465			0.348796			0.174398	−	0.984675i	$-0.444202\pi$
$569$	198.465			0.348796			0.174398	+	0.984675i	$0.444202\pi$
$570$		0			0
$571$	−198.296			−0.347279			−0.173639	−	0.984809i	$-0.555553\pi$
$571$	−198.296			−0.347279			−0.173639	+	0.984809i	$0.555553\pi$
$572$		0			0
$573$	57.2393			0.0998941
$574$		0			0
$575$	−946.775	−	283.001i	−1.64656	−	0.492176i
$576$		0			0
$577$		−	146.589i		−	0.254054i	−0.991899	−	0.127027i	$-0.959456\pi$
$577$		−	146.589i		−	0.254054i	0.991899	−	0.127027i	$-0.0405435\pi$
$578$		0			0
$579$	−340.886			−0.588750
$580$		0			0
$581$		−	800.484i		−	1.37777i
$582$		0			0
$583$	−47.5078			−0.0814885
$584$		0			0
$585$	−79.1570	+	541.214i	−0.135311	+	0.925152i
$586$		0			0
$587$			167.717i			0.285718i	0.989743	+	0.142859i	$0.0456296\pi$
$587$			167.717i			0.285718i	−0.989743	+	0.142859i	$0.954370\pi$
$588$		0			0
$589$		−	193.530i		−	0.328573i
$590$		0			0
$591$		−	141.745i		−	0.239839i
$592$		0			0
$593$		−	443.963i		−	0.748674i	−0.927293	−	0.374337i	$-0.877870\pi$
$593$		−	443.963i		−	0.748674i	0.927293	−	0.374337i	$-0.122130\pi$
$594$		0			0
$595$	−983.659	−	143.868i	−1.65321	−	0.241795i
$596$		0			0
$597$	−27.1924			−0.0455485
$598$		0			0
$599$		−	426.294i		−	0.711676i	−0.934548	−	0.355838i	$-0.884195\pi$
$599$		−	426.294i		−	0.711676i	0.934548	−	0.355838i	$-0.115805\pi$
$600$		0			0
$601$	−54.1068			−0.0900279			−0.0450140	−	0.998986i	$-0.514333\pi$
$601$	−54.1068			−0.0900279			−0.0450140	+	0.998986i	$0.514333\pi$
$602$		0			0
$603$		−	31.1125i		−	0.0515961i
$604$		0			0
$605$	13.4166	−	91.7323i	0.0221762	−	0.151624i
$606$		0			0
$607$	408.990			0.673790			0.336895	−	0.941542i	$-0.390623\pi$
$607$	408.990			0.673790			0.336895	+	0.941542i	$0.390623\pi$
$608$		0			0
$609$	−122.562			−0.201251
$610$		0			0
$611$	647.626			1.05994
$612$		0			0
$613$	−185.150			−0.302039			−0.151020	−	0.988531i	$-0.548256\pi$
$613$	−185.150			−0.302039			−0.151020	+	0.988531i	$0.548256\pi$
$614$		0			0
$615$	−78.6643	−	11.5053i	−0.127909	−	0.0187078i
$616$		0			0
$617$		−	415.587i		−	0.673561i	−0.941583	−	0.336781i	$-0.890662\pi$
$617$		−	415.587i		−	0.673561i	0.941583	−	0.336781i	$-0.109338\pi$
$618$		0			0
$619$	−935.377			−1.51111			−0.755555	−	0.655086i	$-0.772633\pi$
$619$	−935.377			−1.51111			−0.755555	+	0.655086i	$0.772633\pi$
$620$		0			0
$621$		−	631.959i		−	1.01765i
$622$		0			0
$623$	378.670			0.607817
$624$		0			0
$625$	522.476	+	342.993i	0.835961	+	0.548789i
$626$		0			0
$627$		−	332.589i		−	0.530444i
$628$		0			0
$629$			1482.24i			2.35649i
$630$		0			0
$631$			366.911i			0.581475i	0.956803	+	0.290738i	$0.0939007\pi$
$631$			366.911i			0.581475i	−0.956803	+	0.290738i	$0.906099\pi$
$632$		0			0
$633$		−	212.317i		−	0.335414i
$634$		0			0
$635$	50.3501	−	344.254i	0.0792915	−	0.542133i
$636$		0			0
$637$	16.6953			0.0262092
$638$		0			0
$639$		−	972.435i		−	1.52181i
$640$		0			0
$641$	−627.281			−0.978597			−0.489299	−	0.872116i	$-0.662747\pi$
$641$	−627.281			−0.978597			−0.489299	+	0.872116i	$0.662747\pi$
$642$		0			0
$643$		−	1231.97i		−	1.91598i	−0.286808	−	0.957988i	$-0.592594\pi$
$643$		−	1231.97i		−	1.91598i	0.286808	−	0.957988i	$-0.407406\pi$
$644$		0			0
$645$	−39.1878	+	267.936i	−0.0607563	+	0.415404i
$646$		0			0
$647$	−341.377			−0.527630			−0.263815	−	0.964573i	$-0.584981\pi$
$647$	−341.377			−0.527630			−0.263815	+	0.964573i	$0.584981\pi$
$648$		0			0
$649$	539.811			0.831758
$650$		0			0
$651$	41.3874			0.0635752
$652$		0			0
$653$	−404.426			−0.619336			−0.309668	−	0.950845i	$-0.600218\pi$
$653$	−404.426			−0.619336			−0.309668	+	0.950845i	$0.600218\pi$
$654$		0			0
$655$	112.086	−	766.358i	0.171124	−	1.17001i
$656$		0			0
$657$		−	674.126i		−	1.02607i
$658$		0			0
$659$	161.287			0.244745			0.122373	−	0.992484i	$-0.460950\pi$
$659$	161.287			0.244745			0.122373	+	0.992484i	$0.460950\pi$
$660$		0			0
$661$		−	630.605i		−	0.954017i	−0.878899	−	0.477009i	$-0.841721\pi$
$661$		−	630.605i		−	0.954017i	0.878899	−	0.477009i	$-0.158279\pi$
$662$		0			0
$663$	−361.398			−0.545095
$664$		0			0
$665$	−150.836	+	1031.30i	−0.226822	+	1.55083i
$666$		0			0
$667$			751.003i			1.12594i
$668$		0			0
$669$			0.484244i			0.000723833i
$670$		0			0
$671$			447.064i			0.666265i
$672$		0			0
$673$			799.682i			1.18823i	0.804378	+	0.594117i	$0.202499\pi$
$673$			799.682i			1.18823i	−0.804378	+	0.594117i	$0.797501\pi$
$674$		0			0
$675$	−114.472	+	382.962i	−0.169587	+	0.567351i
$676$		0			0
$677$	473.351			0.699189			0.349595	−	0.936901i	$-0.386319\pi$
$677$	473.351			0.699189			0.349595	+	0.936901i	$0.386319\pi$
$678$		0			0
$679$		−	358.380i		−	0.527805i
$680$		0			0
$681$	−350.375			−0.514500
$682$		0			0
$683$		−	421.137i		−	0.616598i	−0.951289	−	0.308299i	$-0.900240\pi$
$683$		−	421.137i		−	0.616598i	0.951289	−	0.308299i	$-0.0997597\pi$
$684$		0			0
$685$	358.450	+	52.4263i	0.523285	+	0.0765348i
$686$		0			0
$687$	4.14134			0.00602814
$688$		0			0
$689$	54.1233			0.0785535
$690$		0			0
$691$	−103.427			−0.149677			−0.0748387	−	0.997196i	$-0.523844\pi$
$691$	−103.427			−0.149677			−0.0748387	+	0.997196i	$0.523844\pi$
$692$		0			0
$693$	−663.598			−0.957572
$694$		0			0
$695$	74.9405	−	512.384i	0.107828	−	0.737244i
$696$		0			0
$697$			490.085i			0.703135i
$698$		0			0
$699$	242.121			0.346382
$700$		0			0
$701$			1025.80i			1.46333i	0.681664	+	0.731666i	$0.261257\pi$
$701$			1025.80i			1.46333i	−0.681664	+	0.731666i	$0.738743\pi$
$702$		0			0
$703$	1554.03			2.21056
$704$		0			0
$705$	222.229	+	32.5028i	0.315218	+	0.0461033i
$706$		0			0
$707$			24.9181i			0.0352448i
$708$		0			0
$709$		−	1216.11i		−	1.71525i	−0.514279	−	0.857623i	$-0.671941\pi$
$709$		−	1216.11i		−	1.71525i	0.514279	−	0.857623i	$-0.328059\pi$
$710$		0			0
$711$		−	677.054i		−	0.952257i
$712$		0			0
$713$		−	253.604i		−	0.355685i
$714$		0			0
$715$	−115.032	+	786.497i	−0.160884	+	1.10000i
$716$		0			0
$717$	−363.981			−0.507644
$718$		0			0
$719$		−	1009.14i		−	1.40353i	−0.712407	−	0.701767i	$-0.752395\pi$
$719$		−	1009.14i		−	1.40353i	0.712407	−	0.701767i	$-0.247605\pi$
$720$		0			0
$721$	56.9761			0.0790237
$722$		0			0
$723$		−	201.775i		−	0.279080i
$724$		0			0
$725$	136.035	−	455.102i	0.187634	−	0.627727i
$726$		0			0
$727$	−94.2492			−0.129641			−0.0648206	−	0.997897i	$-0.520648\pi$
$727$	−94.2492			−0.129641			−0.0648206	+	0.997897i	$0.520648\pi$
$728$		0			0
$729$	339.066			0.465111
$730$		0			0
$731$	1669.26			2.28353
$732$		0			0
$733$	504.116			0.687743			0.343872	−	0.939017i	$-0.388262\pi$
$733$	504.116			0.687743			0.343872	+	0.939017i	$0.388262\pi$
$734$		0			0
$735$	5.72888	+	0.837896i	0.00779439	+	0.00113999i
$736$		0			0
$737$		−	45.2130i		−	0.0613473i
$738$		0			0
$739$	117.256			0.158669			0.0793343	−	0.996848i	$-0.474721\pi$
$739$	117.256			0.158669			0.0793343	+	0.996848i	$0.474721\pi$
$740$		0			0
$741$			378.902i			0.511339i
$742$		0			0
$743$	−790.636			−1.06411			−0.532056	−	0.846709i	$-0.678581\pi$
$743$	−790.636			−1.06411			−0.532056	+	0.846709i	$0.678581\pi$
$744$		0			0
$745$	569.374	+	83.2757i	0.764260	+	0.111779i
$746$		0			0
$747$			941.557i			1.26045i
$748$		0			0
$749$			363.533i			0.485357i
$750$		0			0
$751$			310.766i			0.413803i	0.978362	+	0.206901i	$0.0663380\pi$
$751$			310.766i			0.413803i	−0.978362	+	0.206901i	$0.933662\pi$
$752$		0			0
$753$		−	169.459i		−	0.225046i
$754$		0			0
$755$	873.513	+	127.759i	1.15697	+	0.169217i
$756$		0			0
$757$	−540.769			−0.714358			−0.357179	−	0.934036i	$-0.616261\pi$
$757$	−540.769			−0.714358			−0.357179	+	0.934036i	$0.616261\pi$
$758$		0			0
$759$		−	435.828i		−	0.574214i
$760$		0			0
$761$	28.6841			0.0376926			0.0188463	−	0.999822i	$-0.494001\pi$
$761$	28.6841			0.0376926			0.0188463	+	0.999822i	$0.494001\pi$
$762$		0			0
$763$			763.172i			1.00023i
$764$		0			0
$765$	1157.01	+	169.223i	1.51244	+	0.221206i
$766$		0			0
$767$	−614.980			−0.801799
$768$		0			0
$769$	−704.747			−0.916447			−0.458223	−	0.888837i	$-0.651514\pi$
$769$	−704.747			−0.916447			−0.458223	+	0.888837i	$0.651514\pi$
$770$		0			0
$771$	239.471			0.310598
$772$		0			0
$773$	−74.4042			−0.0962538			−0.0481269	−	0.998841i	$-0.515325\pi$
$773$	−74.4042			−0.0962538			−0.0481269	+	0.998841i	$0.515325\pi$
$774$		0			0
$775$	−45.9371	+	153.682i	−0.0592737	+	0.198299i
$776$		0			0
$777$			332.337i			0.427719i
$778$		0			0
$779$	513.822			0.659592
$780$		0			0
$781$		−	1413.15i		−	1.80941i
$782$		0			0
$783$	303.774			0.387962
$784$		0			0
$785$	57.2572	−	391.480i	0.0729391	−	0.498700i
$786$		0			0
$787$		−	1039.46i		−	1.32079i	−0.750918	−	0.660396i	$-0.770388\pi$
$787$		−	1039.46i		−	1.32079i	0.750918	−	0.660396i	$-0.229612\pi$
$788$		0			0
$789$		−	354.016i		−	0.448689i
$790$		0			0
$791$			610.797i			0.772183i
$792$		0			0
$793$		−	509.318i		−	0.642267i
$794$		0			0
$795$	18.5721	+	2.71632i	0.0233611	+	0.00341676i
$796$		0			0
$797$	1215.20			1.52472			0.762360	−	0.647153i	$-0.224040\pi$
$797$	1215.20			1.52472			0.762360	+	0.647153i	$0.224040\pi$
$798$		0			0
$799$		−	1384.50i		−	1.73280i
$800$		0			0
$801$	−445.405			−0.556061
$802$		0			0
$803$		−	979.647i		−	1.21998i
$804$		0			0
$805$	−197.658	+	1351.43i	−0.245538	+	1.67879i
$806$		0			0
$807$	189.824			0.235222
$808$		0			0
$809$	−888.501			−1.09827			−0.549135	−	0.835733i	$-0.685043\pi$
$809$	−888.501			−1.09827			−0.549135	+	0.835733i	$0.685043\pi$
$810$		0			0
$811$	1255.39			1.54796			0.773978	−	0.633212i	$-0.218264\pi$
$811$	1255.39			1.54796			0.773978	+	0.633212i	$0.218264\pi$
$812$		0			0
$813$	403.255			0.496009
$814$		0			0
$815$	52.7517	+	7.71539i	0.0647261	+	0.00946673i
$816$		0			0
$817$		−	1750.11i		−	2.14212i
$818$		0			0
$819$	756.005			0.923082
$820$		0			0
$821$			952.864i			1.16061i	0.814398	+	0.580307i	$0.197067\pi$
$821$			952.864i			1.16061i	−0.814398	+	0.580307i	$0.802933\pi$
$822$		0			0
$823$	−1353.86			−1.64503			−0.822514	−	0.568745i	$-0.807429\pi$
$823$	−1353.86			−1.64503			−0.822514	+	0.568745i	$0.807429\pi$
$824$		0			0
$825$	−78.9449	+	264.108i	−0.0956908	+	0.320131i
$826$		0			0
$827$		−	1590.93i		−	1.92373i	−0.273521	−	0.961866i	$-0.588188\pi$
$827$		−	1590.93i		−	1.92373i	0.273521	−	0.961866i	$-0.411812\pi$
$828$		0			0
$829$			266.156i			0.321057i	0.987031	+	0.160528i	$0.0513198\pi$
$829$			266.156i			0.321057i	−0.987031	+	0.160528i	$0.948680\pi$
$830$		0			0
$831$			186.180i			0.224043i
$832$		0			0
$833$		−	35.6914i		−	0.0428468i
$834$		0			0
$835$	−14.1539	+	96.7734i	−0.0169508	+	0.115896i
$836$		0			0
$837$	−102.580			−0.122557
$838$		0			0
$839$		−	531.574i		−	0.633581i	−0.948496	−	0.316790i	$-0.897395\pi$
$839$		−	531.574i		−	0.633581i	0.948496	−	0.316790i	$-0.102605\pi$
$840$		0			0
$841$	480.003			0.570753
$842$		0			0
$843$			356.891i			0.423358i
$844$		0			0
$845$	8.76291	−	59.9139i	0.0103703	−	0.0709040i
$846$		0			0
$847$	−128.138			−0.151284
$848$		0			0
$849$	−280.381			−0.330248
$850$		0			0
$851$	2036.41			2.39297
$852$		0			0
$853$	1363.88			1.59892			0.799459	−	0.600720i	$-0.205119\pi$
$853$	1363.88			1.59892			0.799459	+	0.600720i	$0.205119\pi$
$854$		0			0
$855$	177.419	−	1213.05i	0.207508	−	1.41877i
$856$		0			0
$857$			649.944i			0.758395i	0.925316	+	0.379197i	$0.123800\pi$
$857$			649.944i			0.758395i	−0.925316	+	0.379197i	$0.876200\pi$
$858$		0			0
$859$	−551.416			−0.641927			−0.320964	−	0.947092i	$-0.604007\pi$
$859$	−551.416			−0.641927			−0.320964	+	0.947092i	$0.604007\pi$
$860$		0			0
$861$			109.884i			0.127623i
$862$		0			0
$863$	−443.453			−0.513851			−0.256926	−	0.966431i	$-0.582710\pi$
$863$	−443.453			−0.513851			−0.256926	+	0.966431i	$0.582710\pi$
$864$		0			0
$865$	146.324	−	1000.45i	0.169161	−	1.15659i
$866$		0			0
$867$			502.845i			0.579983i
$868$		0			0
$869$		−	983.903i		−	1.13222i
$870$		0			0
$871$			51.5089i			0.0591377i
$872$		0			0
$873$			421.539i			0.482862i
$874$		0			0
$875$	364.574	−	783.152i	0.416656	−	0.895031i
$876$		0			0
$877$	1131.01			1.28964			0.644819	−	0.764336i	$-0.276933\pi$
$877$	1131.01			1.28964			0.644819	+	0.764336i	$0.276933\pi$
$878$		0			0
$879$		−	12.6515i		−	0.0143930i
$880$		0			0
$881$	180.319			0.204675			0.102338	−	0.994750i	$-0.467368\pi$
$881$	180.319			0.204675			0.102338	+	0.994750i	$0.467368\pi$
$882$		0			0
$883$			561.730i			0.636161i	0.948064	+	0.318080i	$0.103038\pi$
$883$			561.730i			0.636161i	−0.948064	+	0.318080i	$0.896962\pi$
$884$		0			0
$885$	−211.026	−	30.8644i	−0.238448	−	0.0348750i
$886$		0			0
$887$	−602.367			−0.679106			−0.339553	−	0.940587i	$-0.610276\pi$
$887$	−602.367			−0.679106			−0.339553	+	0.940587i	$0.610276\pi$
$888$		0			0
$889$	−480.878			−0.540921
$890$		0			0
$891$	687.918			0.772074
$892$		0			0
$893$	−1451.56			−1.62549
$894$		0			0
$895$	−48.3077	+	330.290i	−0.0539751	+	0.369039i
$896$		0			0
$897$			496.518i			0.553532i
$898$		0			0
$899$	121.904			0.135599
$900$		0			0
$901$		−	115.706i		−	0.128419i
$902$		0			0
$903$	374.271			0.414475
$904$		0			0
$905$	−685.606	−	100.276i	−0.757576	−	0.110802i
$906$		0			0
$907$		−	1581.18i		−	1.74330i	−0.490127	−	0.871651i	$-0.663049\pi$
$907$		−	1581.18i		−	1.74330i	0.490127	−	0.871651i	$-0.336951\pi$
$908$		0			0
$909$		−	29.3095i		−	0.0322437i
$910$		0			0
$911$		−	407.039i		−	0.446805i	−0.974726	−	0.223402i	$-0.928284\pi$
$911$		−	407.039i		−	0.446805i	0.974726	−	0.223402i	$-0.0717164\pi$
$912$		0			0
$913$			1368.28i			1.49866i
$914$		0			0
$915$	25.5614	−	174.769i	0.0279360	−	0.191004i
$916$		0			0
$917$	−1070.50			−1.16740
$918$		0			0
$919$			330.930i			0.360098i	0.983658	+	0.180049i	$0.0576256\pi$
$919$			330.930i			0.360098i	−0.983658	+	0.180049i	$0.942374\pi$
$920$		0			0
$921$	285.257			0.309726
$922$		0			0
$923$			1609.94i			1.74424i
$924$		0			0
$925$	−1234.05	−	368.871i	−1.33411	−	0.398780i
$926$		0			0
$927$	−67.0172			−0.0722947
$928$		0			0
$929$	713.506			0.768036			0.384018	−	0.923326i	$-0.374540\pi$
$929$	713.506			0.768036			0.384018	+	0.923326i	$0.374540\pi$
$930$		0			0
$931$	−37.4200			−0.0401934
$932$		0			0
$933$	248.637			0.266492
$934$		0			0
$935$	1681.38	+	245.916i	1.79827	+	0.263012i
$936$		0			0
$937$			309.292i			0.330087i	0.986286	+	0.165044i	$0.0527765\pi$
$937$			309.292i			0.330087i	−0.986286	+	0.165044i	$0.947223\pi$
$938$		0			0
$939$	473.663			0.504434
$940$		0			0
$941$			439.051i			0.466579i	0.972407	+	0.233290i	$0.0749490\pi$
$941$			439.051i			0.466579i	−0.972407	+	0.233290i	$0.925051\pi$
$942$		0			0
$943$	673.318			0.714017
$944$		0			0
$945$	546.641	+	79.9509i	0.578456	+	0.0846041i
$946$		0			0
$947$		−	52.2559i		−	0.0551804i	−0.999619	−	0.0275902i	$-0.991217\pi$
$947$		−	52.2559i		−	0.0551804i	0.999619	−	0.0275902i	$-0.00878335\pi$
$948$		0			0
$949$			1116.06i			1.17604i
$950$		0			0
$951$			396.766i			0.417209i
$952$		0			0
$953$		−	138.100i		−	0.144911i	−0.997372	−	0.0724553i	$-0.976917\pi$
$953$		−	138.100i		−	0.144911i	0.997372	−	0.0724553i	$-0.0230835\pi$
$954$		0			0
$955$	303.385	+	44.3727i	0.317681	+	0.0464635i
$956$		0			0
$957$	209.497			0.218910
$958$		0			0
$959$		−	500.708i		−	0.522115i
$960$		0			0
$961$	919.835			0.957164
$962$		0			0
$963$		−	427.599i		−	0.444028i
$964$		0			0
$965$	−1806.80	−	264.260i	−1.87233	−	0.273844i
$966$		0			0
$967$	1231.77			1.27380			0.636902	−	0.770945i	$-0.280216\pi$
$967$	1231.77			1.27380			0.636902	+	0.770945i	$0.280216\pi$
$968$		0			0
$969$	810.021			0.835935
$970$		0			0
$971$	198.353			0.204277			0.102138	−	0.994770i	$-0.467432\pi$
$971$	198.353			0.204277			0.102138	+	0.994770i	$0.467432\pi$
$972$		0			0
$973$	−715.734			−0.735595
$974$		0			0
$975$	89.9381	−	300.886i	0.0922442	−	0.308601i
$976$		0			0
$977$			1521.90i			1.55773i	0.627194	+	0.778863i	$0.284203\pi$
$977$			1521.90i			1.55773i	−0.627194	+	0.778863i	$0.715797\pi$
$978$		0			0
$979$	−647.267			−0.661151
$980$		0			0
$981$		−	897.669i		−	0.915055i
$982$		0			0
$983$	313.432			0.318852			0.159426	−	0.987210i	$-0.449036\pi$
$983$	313.432			0.318852			0.159426	+	0.987210i	$0.449036\pi$
$984$		0			0
$985$	109.882	−	751.289i	0.111556	−	0.762730i
$986$		0			0
$987$		−	310.425i		−	0.314513i
$988$		0			0
$989$		−	2293.36i		−	2.31887i
$990$		0			0
$991$			751.646i			0.758472i	0.925300	+	0.379236i	$0.123813\pi$
$991$			751.646i			0.758472i	−0.925300	+	0.379236i	$0.876187\pi$
$992$		0			0
$993$			229.396i			0.231013i
$994$		0			0
$995$	−144.128	−	21.0799i	−0.144852	−	0.0211859i
$996$		0			0
$997$	589.360			0.591134			0.295567	−	0.955322i	$-0.404491\pi$
$997$	589.360			0.591134			0.295567	+	0.955322i	$0.404491\pi$
$998$		0			0
$999$		−	823.711i		−	0.824536i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1280.3.e.l.639.19		24
4.3	odd	2		1280.3.e.k.639.6		24
5.4	even	2	inner	1280.3.e.l.639.6		24
8.3	odd	2	inner	1280.3.e.l.639.5		24
8.5	even	2		1280.3.e.k.639.20		24
16.3	odd	4		640.3.h.a.639.12	yes	24
16.5	even	4		640.3.h.b.639.12	yes	24
16.11	odd	4		640.3.h.b.639.13	yes	24
16.13	even	4		640.3.h.a.639.13	yes	24
20.19	odd	2		1280.3.e.k.639.19		24
40.19	odd	2	inner	1280.3.e.l.639.20		24
40.29	even	2		1280.3.e.k.639.5		24
80.19	odd	4		640.3.h.a.639.14	yes	24
80.29	even	4		640.3.h.a.639.11	✓	24
80.59	odd	4		640.3.h.b.639.11	yes	24
80.69	even	4		640.3.h.b.639.14	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
640.3.h.a.639.11	✓	24	80.29	even	4
640.3.h.a.639.12	yes	24	16.3	odd	4
640.3.h.a.639.13	yes	24	16.13	even	4
640.3.h.a.639.14	yes	24	80.19	odd	4
640.3.h.b.639.11	yes	24	80.59	odd	4
640.3.h.b.639.12	yes	24	16.5	even	4
640.3.h.b.639.13	yes	24	16.11	odd	4
640.3.h.b.639.14	yes	24	80.69	even	4
1280.3.e.k.639.5		24	40.29	even	2
1280.3.e.k.639.6		24	4.3	odd	2
1280.3.e.k.639.19		24	20.19	odd	2
1280.3.e.k.639.20		24	8.5	even	2
1280.3.e.l.639.5		24	8.3	odd	2	inner
1280.3.e.l.639.6		24	5.4	even	2	inner
1280.3.e.l.639.19		24	1.1	even	1	trivial
1280.3.e.l.639.20		24	40.19	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1280 = 2^{8} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1280.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.933412i q^{3} +(0.723593 - 4.94736i) q^{5} -6.91082 q^{7} +8.12874 q^{9} +O(q^{10})\)	\(q-0.933412i q^{3} +(0.723593 - 4.94736i) q^{5} -6.91082 q^{7} +8.12874 q^{9} +11.8128 q^{11} -13.4577 q^{13} +(-4.61793 - 0.675411i) q^{15} +28.7701i q^{17} +30.1635 q^{19} +6.45064i q^{21} +39.5266 q^{23} +(-23.9528 - 7.15976i) q^{25} -15.9882i q^{27} +18.9999i q^{29} -6.41602i q^{31} -11.0262i q^{33} +(-5.00062 + 34.1903i) q^{35} +51.5200 q^{37} +12.5616i q^{39} +17.0345 q^{41} -58.0207i q^{43} +(5.88190 - 40.2158i) q^{45} -48.1230 q^{47} -1.24057 q^{49} +26.8543 q^{51} -4.02173 q^{53} +(8.54765 - 58.4421i) q^{55} -28.1550i q^{57} +45.6972 q^{59} +37.8458i q^{61} -56.1763 q^{63} +(-9.73792 + 66.5802i) q^{65} -3.82746i q^{67} -36.8947i q^{69} -119.629i q^{71} -82.9311i q^{73} +(-6.68301 + 22.3579i) q^{75} -81.6360 q^{77} -83.2914i q^{79} +58.2351 q^{81} +115.831i q^{83} +(142.336 + 20.8178i) q^{85} +17.7348 q^{87} -54.7938 q^{89} +93.0039 q^{91} -5.98879 q^{93} +(21.8261 - 149.230i) q^{95} +51.8578i q^{97} +96.0230 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 72 q^{9}+O(q^{10}) \) 24 * q - 72 * q^9	\( 24 q - 72 q^{9} + 16 q^{11} - 48 q^{19} - 24 q^{25} + 112 q^{35} - 80 q^{41} + 168 q^{49} + 576 q^{51} - 496 q^{59} + 32 q^{65} - 224 q^{75} + 184 q^{81} - 144 q^{89} + 864 q^{91} - 368 q^{99}+O(q^{100}) \) 24 * q - 72 * q^9 + 16 * q^11 - 48 * q^19 - 24 * q^25 + 112 * q^35 - 80 * q^41 + 168 * q^49 + 576 * q^51 - 496 * q^59 + 32 * q^65 - 224 * q^75 + 184 * q^81 - 144 * q^89 + 864 * q^91 - 368 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more