LMFDB - Embedded newform 880.3.j.a.241.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [880,3,Mod(241,880)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(880, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("880.241"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 880 = 2^{4} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	880.j (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,-8,0,0,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$23.9782632637$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 30x^{6} + 280x^{4} + 890x^{2} + 895 $ x^8 + 30x^6 + 280x^4 + 890*x^2 + 895
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 55)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			241.8
Root			$3.15955i$ of defining polynomial
Character	$\chi$	$=$	880.241
Dual form			880.3.j.a.241.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.03112 q^{3} -2.23607 q^{5} +5.67155i q^{7} +0.187686 q^{9} +(-10.8573 - 1.76614i) q^{11} -17.2349i q^{13} -6.77779 q^{15} +3.43437i q^{17} +18.5738i q^{19} +17.1911i q^{21} -15.7918 q^{23} +5.00000 q^{25} -26.7112 q^{27} +52.6031i q^{29} -27.5484 q^{31} +(-32.9097 - 5.35337i) q^{33} -12.6820i q^{35} -40.1425 q^{37} -52.2411i q^{39} -74.8398i q^{41} +9.72983i q^{43} -0.419679 q^{45} -18.2212 q^{47} +16.8335 q^{49} +10.4100i q^{51} -75.2040 q^{53} +(24.2776 + 3.94920i) q^{55} +56.2994i q^{57} -1.75672 q^{59} -48.5900i q^{61} +1.06447i q^{63} +38.5385i q^{65} +35.4901 q^{67} -47.8668 q^{69} -110.499 q^{71} +50.7154i q^{73} +15.1556 q^{75} +(10.0167 - 61.5777i) q^{77} -12.7481i q^{79} -82.6540 q^{81} +112.062i q^{83} -7.67948i q^{85} +159.446i q^{87} +66.3600 q^{89} +97.7488 q^{91} -83.5024 q^{93} -41.5323i q^{95} +65.1302 q^{97} +(-2.03776 - 0.331479i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{3} - 4 q^{9} - 8 q^{11} - 20 q^{15} - 8 q^{23} + 40 q^{25} + 16 q^{27} - 36 q^{31} - 152 q^{33} - 88 q^{37} + 8 q^{47} + 172 q^{49} - 152 q^{53} + 20 q^{55} + 16 q^{59} + 88 q^{67} + 8 q^{69}+ \cdots + 184 q^{99}+O(q^{100}) $ 8 * q - 8 * q^3 - 4 * q^9 - 8 * q^11 - 20 * q^15 - 8 * q^23 + 40 * q^25 + 16 * q^27 - 36 * q^31 - 152 * q^33 - 88 * q^37 + 8 * q^47 + 172 * q^49 - 152 * q^53 + 20 * q^55 + 16 * q^59 + 88 * q^67 + 8 * q^69 - 276 * q^71 - 40 * q^75 + 160 * q^77 - 320 * q^81 + 444 * q^89 + 40 * q^91 - 104 * q^93 + 312 * q^97 + 184 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/880\mathbb{Z}\right)^\times$.

$n$	$111$	$177$	$321$	$661$
$\chi(n)$	$1$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	3.03112			1.01037			0.505187	−	0.863010i	$-0.331424\pi$
$3$	3.03112			1.01037			0.505187	+	0.863010i	$0.331424\pi$
$4$		0			0
$5$	−2.23607			−0.447214
$5$	−2.23607			−0.447214
$6$		0			0
$7$			5.67155i			0.810221i	0.914268	+	0.405111i	$0.132767\pi$
$7$			5.67155i			0.810221i	−0.914268	+	0.405111i	$0.867233\pi$
$8$		0			0
$9$	0.187686			0.0208540
$10$		0			0
$11$	−10.8573	−	1.76614i	−0.987026	−	0.160558i
$11$	−10.8573	−	1.76614i	−0.987026	−	0.160558i
$12$		0			0
$13$		−	17.2349i		−	1.32576i	−0.748724	−	0.662882i	$-0.769333\pi$
$13$		−	17.2349i		−	1.32576i	0.748724	−	0.662882i	$-0.230667\pi$
$14$		0			0
$15$	−6.77779			−0.451853
$16$		0			0
$17$			3.43437i			0.202022i	0.994885	+	0.101011i	$0.0322077\pi$
$17$			3.43437i			0.202022i	−0.994885	+	0.101011i	$0.967792\pi$
$18$		0			0
$19$			18.5738i			0.977569i	0.872405	+	0.488784i	$0.162559\pi$
$19$			18.5738i			0.977569i	−0.872405	+	0.488784i	$0.837441\pi$
$20$		0			0
$21$			17.1911i			0.818626i
$22$		0			0
$23$	−15.7918			−0.686599			−0.343300	−	0.939226i	$-0.611545\pi$
$23$	−15.7918			−0.686599			−0.343300	+	0.939226i	$0.611545\pi$
$24$		0			0
$25$	5.00000			0.200000
$26$		0			0
$27$	−26.7112			−0.989303
$28$		0			0
$29$			52.6031i			1.81390i	0.421238	+	0.906950i	$0.361596\pi$
$29$			52.6031i			1.81390i	−0.421238	+	0.906950i	$0.638404\pi$
$30$		0			0
$31$	−27.5484			−0.888657			−0.444328	−	0.895864i	$-0.646558\pi$
$31$	−27.5484			−0.888657			−0.444328	+	0.895864i	$0.646558\pi$
$32$		0			0
$33$	−32.9097	−	5.35337i	−0.997265	−	0.162223i
$34$		0			0
$35$		−	12.6820i		−	0.362342i
$36$		0			0
$37$	−40.1425			−1.08493			−0.542467	−	0.840077i	$-0.682510\pi$
$37$	−40.1425			−1.08493			−0.542467	+	0.840077i	$0.682510\pi$
$38$		0			0
$39$		−	52.2411i		−	1.33952i
$40$		0			0
$41$		−	74.8398i		−	1.82536i	−0.408675	−	0.912680i	$-0.634009\pi$
$41$		−	74.8398i		−	1.82536i	0.408675	−	0.912680i	$-0.365991\pi$
$42$		0			0
$43$			9.72983i			0.226275i	0.993579	+	0.113138i	$0.0360901\pi$
$43$			9.72983i			0.226275i	−0.993579	+	0.113138i	$0.963910\pi$
$44$		0			0
$45$	−0.419679			−0.00932620
$46$		0			0
$47$	−18.2212			−0.387686			−0.193843	−	0.981033i	$-0.562095\pi$
$47$	−18.2212			−0.387686			−0.193843	+	0.981033i	$0.562095\pi$
$48$		0			0
$49$	16.8335			0.343541
$50$		0			0
$51$			10.4100i			0.204117i
$52$		0			0
$53$	−75.2040			−1.41894			−0.709471	−	0.704734i	$-0.751066\pi$
$53$	−75.2040			−1.41894			−0.709471	+	0.704734i	$0.751066\pi$
$54$		0			0
$55$	24.2776	+	3.94920i	0.441412	+	0.0718036i
$56$		0			0
$57$			56.2994i			0.987709i
$58$		0			0
$59$	−1.75672			−0.0297750			−0.0148875	−	0.999889i	$-0.504739\pi$
$59$	−1.75672			−0.0297750			−0.0148875	+	0.999889i	$0.504739\pi$
$60$		0			0
$61$		−	48.5900i		−	0.796557i	−0.917265	−	0.398278i	$-0.869608\pi$
$61$		−	48.5900i		−	0.796557i	0.917265	−	0.398278i	$-0.130392\pi$
$62$		0			0
$63$			1.06447i			0.0168964i
$64$		0			0
$65$			38.5385i			0.592900i
$66$		0			0
$67$	35.4901			0.529702			0.264851	−	0.964289i	$-0.414677\pi$
$67$	35.4901			0.529702			0.264851	+	0.964289i	$0.414677\pi$
$68$		0			0
$69$	−47.8668			−0.693721
$70$		0			0
$71$	−110.499			−1.55632			−0.778159	−	0.628067i	$-0.783846\pi$
$71$	−110.499			−1.55632			−0.778159	+	0.628067i	$0.783846\pi$
$72$		0			0
$73$			50.7154i			0.694732i	0.937730	+	0.347366i	$0.112924\pi$
$73$			50.7154i			0.694732i	−0.937730	+	0.347366i	$0.887076\pi$
$74$		0			0
$75$	15.1556			0.202075
$76$		0			0
$77$	10.0167	−	61.5777i	0.130087	−	0.799710i
$78$		0			0
$79$		−	12.7481i		−	0.161368i	−0.996740	−	0.0806842i	$-0.974289\pi$
$79$		−	12.7481i		−	0.161368i	0.996740	−	0.0806842i	$-0.0257105\pi$
$80$		0			0
$81$	−82.6540			−1.02042
$82$		0			0
$83$			112.062i			1.35015i	0.737749	+	0.675075i	$0.235889\pi$
$83$			112.062i			1.35015i	−0.737749	+	0.675075i	$0.764111\pi$
$84$		0			0
$85$		−	7.67948i		−	0.0903468i
$86$		0			0
$87$			159.446i			1.83272i
$88$		0			0
$89$	66.3600			0.745618			0.372809	−	0.927908i	$-0.378395\pi$
$89$	66.3600			0.745618			0.372809	+	0.927908i	$0.378395\pi$
$90$		0			0
$91$	97.7488			1.07416
$92$		0			0
$93$	−83.5024			−0.897875
$94$		0			0
$95$		−	41.5323i		−	0.437182i
$96$		0			0
$97$	65.1302			0.671445			0.335723	−	0.941961i	$-0.391020\pi$
$97$	65.1302			0.671445			0.335723	+	0.941961i	$0.391020\pi$
$98$		0			0
$99$	−2.03776	−	0.331479i	−0.0205835	−	0.00334828i
$100$		0			0
$101$			172.019i			1.70316i	0.524229	+	0.851578i	$0.324354\pi$
$101$			172.019i			1.70316i	−0.524229	+	0.851578i	$0.675646\pi$
$102$		0			0
$103$	−28.3278			−0.275027			−0.137514	−	0.990500i	$-0.543911\pi$
$103$	−28.3278			−0.275027			−0.137514	+	0.990500i	$0.543911\pi$
$104$		0			0
$105$		−	38.4406i		−	0.366101i
$106$		0			0
$107$		−	63.2464i		−	0.591088i	−0.955329	−	0.295544i	$-0.904499\pi$
$107$		−	63.2464i		−	0.591088i	0.955329	−	0.295544i	$-0.0955009\pi$
$108$		0			0
$109$		−	128.641i		−	1.18020i	−0.807331	−	0.590098i	$-0.799089\pi$
$109$		−	128.641i		−	1.18020i	0.807331	−	0.590098i	$-0.200911\pi$
$110$		0			0
$111$	−121.677			−1.09619
$112$		0			0
$113$	−35.4694			−0.313889			−0.156944	−	0.987607i	$-0.550164\pi$
$113$	−35.4694			−0.313889			−0.156944	+	0.987607i	$0.550164\pi$
$114$		0			0
$115$	35.3115			0.307056
$116$		0			0
$117$		−	3.23476i		−	0.0276475i
$118$		0			0
$119$	−19.4782			−0.163682
$120$		0			0
$121$	114.762	+	38.3509i	0.948442	+	0.316949i
$122$		0			0
$123$		−	226.848i		−	1.84430i
$124$		0			0
$125$	−11.1803			−0.0894427
$126$		0			0
$127$		−	210.686i		−	1.65895i	−0.558547	−	0.829473i	$-0.688641\pi$
$127$		−	210.686i		−	1.65895i	0.558547	−	0.829473i	$-0.311359\pi$
$128$		0			0
$129$			29.4923i			0.228622i
$130$		0			0
$131$		−	131.724i		−	1.00553i	−0.864424	−	0.502764i	$-0.832316\pi$
$131$		−	131.724i		−	1.00553i	0.864424	−	0.502764i	$-0.167684\pi$
$132$		0			0
$133$	−105.342			−0.792047
$134$		0			0
$135$	59.7280			0.442430
$136$		0			0
$137$	167.934			1.22580			0.612898	−	0.790162i	$-0.290004\pi$
$137$	167.934			1.22580			0.612898	+	0.790162i	$0.290004\pi$
$138$		0			0
$139$			90.6142i			0.651901i	0.945387	+	0.325950i	$0.105684\pi$
$139$			90.6142i			0.651901i	−0.945387	+	0.325950i	$0.894316\pi$
$140$		0			0
$141$	−55.2308			−0.391708
$142$		0			0
$143$	−30.4392	+	187.125i	−0.212862	+	1.30856i
$144$		0			0
$145$		−	117.624i		−	0.811201i
$146$		0			0
$147$	51.0244			0.347105
$148$		0			0
$149$			72.8977i			0.489246i	0.969618	+	0.244623i	$0.0786642\pi$
$149$			72.8977i			0.489246i	−0.969618	+	0.244623i	$0.921336\pi$
$150$		0			0
$151$			237.180i			1.57073i	0.619035	+	0.785364i	$0.287524\pi$
$151$			237.180i			1.57073i	−0.619035	+	0.785364i	$0.712476\pi$
$152$		0			0
$153$			0.644584i			0.00421296i
$154$		0			0
$155$	61.6000			0.397419
$156$		0			0
$157$	105.915			0.674620			0.337310	−	0.941394i	$-0.390483\pi$
$157$	105.915			0.674620			0.337310	+	0.941394i	$0.390483\pi$
$158$		0			0
$159$	−227.952			−1.43366
$160$		0			0
$161$		−	89.5639i		−	0.556297i
$162$		0			0
$163$	−143.233			−0.878730			−0.439365	−	0.898309i	$-0.644797\pi$
$163$	−143.233			−0.878730			−0.439365	+	0.898309i	$0.644797\pi$
$164$		0			0
$165$	73.5884	+	11.9705i	0.445991	+	0.0725484i
$166$		0			0
$167$			35.5974i			0.213158i	0.994304	+	0.106579i	$0.0339897\pi$
$167$			35.5974i			0.213158i	−0.994304	+	0.106579i	$0.966010\pi$
$168$		0			0
$169$	−128.043			−0.757650
$170$		0			0
$171$			3.48605i			0.0203862i
$172$		0			0
$173$			196.047i			1.13322i	0.823987	+	0.566609i	$0.191745\pi$
$173$			196.047i			1.13322i	−0.823987	+	0.566609i	$0.808255\pi$
$174$		0			0
$175$			28.3578i			0.162044i
$176$		0			0
$177$	−5.32484			−0.0300838
$178$		0			0
$179$	65.1824			0.364148			0.182074	−	0.983285i	$-0.441719\pi$
$179$	65.1824			0.364148			0.182074	+	0.983285i	$0.441719\pi$
$180$		0			0
$181$	−157.016			−0.867493			−0.433746	−	0.901035i	$-0.642809\pi$
$181$	−157.016			−0.867493			−0.433746	+	0.901035i	$0.642809\pi$
$182$		0			0
$183$		−	147.282i		−	0.804820i
$184$		0			0
$185$	89.7615			0.485197
$186$		0			0
$187$	6.06556	−	37.2879i	0.0324361	−	0.199401i
$188$		0			0
$189$		−	151.494i		−	0.801554i
$190$		0			0
$191$	−161.967			−0.847995			−0.423997	−	0.905663i	$-0.639373\pi$
$191$	−161.967			−0.847995			−0.423997	+	0.905663i	$0.639373\pi$
$192$		0			0
$193$			0.742915i			0.00384930i	0.999998	+	0.00192465i	$0.000612635\pi$
$193$			0.742915i			0.00384930i	−0.999998	+	0.00192465i	$0.999387\pi$
$194$		0			0
$195$			116.815i			0.599050i
$196$		0			0
$197$		−	43.0945i		−	0.218754i	−0.994000	−	0.109377i	$-0.965114\pi$
$197$		−	43.0945i		−	0.218754i	0.994000	−	0.109377i	$-0.0348855\pi$
$198$		0			0
$199$	201.933			1.01474			0.507370	−	0.861729i	$-0.330618\pi$
$199$	201.933			1.01474			0.507370	+	0.861729i	$0.330618\pi$
$200$		0			0
$201$	107.575			0.535197
$202$		0			0
$203$	−298.341			−1.46966
$204$		0			0
$205$			167.347i			0.816326i
$206$		0			0
$207$	−2.96390			−0.0143184
$208$		0			0
$209$	32.8038	−	201.661i	0.156956	−	0.964886i
$210$		0			0
$211$			53.4705i			0.253415i	0.991940	+	0.126707i	$0.0404409\pi$
$211$			53.4705i			0.253415i	−0.991940	+	0.126707i	$0.959559\pi$
$212$		0			0
$213$	−334.934			−1.57246
$214$		0			0
$215$		−	21.7566i		−	0.101193i
$216$		0			0
$217$		−	156.242i		−	0.720009i
$218$		0			0
$219$			153.725i			0.701939i
$220$		0			0
$221$	59.1911			0.267833
$222$		0			0
$223$	282.600			1.26726			0.633632	−	0.773635i	$-0.281563\pi$
$223$	282.600			1.26726			0.633632	+	0.773635i	$0.281563\pi$
$224$		0			0
$225$	0.938431			0.00417081
$226$		0			0
$227$			111.251i			0.490092i	0.969511	+	0.245046i	$0.0788031\pi$
$227$			111.251i			0.490092i	−0.969511	+	0.245046i	$0.921197\pi$
$228$		0			0
$229$	−53.3814			−0.233107			−0.116553	−	0.993184i	$-0.537185\pi$
$229$	−53.3814			−0.233107			−0.116553	+	0.993184i	$0.537185\pi$
$230$		0			0
$231$	30.3619	−	186.649i	0.131437	−	0.808006i
$232$		0			0
$233$		−	42.7082i		−	0.183297i	−0.995791	−	0.0916485i	$-0.970786\pi$
$233$		−	42.7082i		−	0.183297i	0.995791	−	0.0916485i	$-0.0292136\pi$
$234$		0			0
$235$	40.7439			0.173378
$236$		0			0
$237$		−	38.6411i		−	0.163042i
$238$		0			0
$239$			216.974i			0.907840i	0.891042	+	0.453920i	$0.149975\pi$
$239$			216.974i			0.907840i	−0.891042	+	0.453920i	$0.850025\pi$
$240$		0			0
$241$			155.081i			0.643488i	0.946827	+	0.321744i	$0.104269\pi$
$241$			155.081i			0.643488i	−0.946827	+	0.321744i	$0.895731\pi$
$242$		0			0
$243$	−10.1334			−0.0417013
$244$		0			0
$245$	−37.6409			−0.153636
$246$		0			0
$247$	320.118			1.29603
$248$		0			0
$249$			339.675i			1.36416i
$250$		0			0
$251$	116.539			0.464300			0.232150	−	0.972680i	$-0.425424\pi$
$251$	116.539			0.464300			0.232150	+	0.972680i	$0.425424\pi$
$252$		0			0
$253$	171.456	+	27.8904i	0.677692	+	0.110239i
$254$		0			0
$255$		−	23.2774i		−	0.0912840i
$256$		0			0
$257$	−85.7024			−0.333472			−0.166736	−	0.986002i	$-0.553323\pi$
$257$	−85.7024			−0.333472			−0.166736	+	0.986002i	$0.553323\pi$
$258$		0			0
$259$		−	227.670i		−	0.879037i
$260$		0			0
$261$			9.87288i			0.0378271i
$262$		0			0
$263$		−	398.864i		−	1.51659i	−0.651911	−	0.758296i	$-0.726032\pi$
$263$		−	398.864i		−	1.51659i	0.651911	−	0.758296i	$-0.273968\pi$
$264$		0			0
$265$	168.161			0.634570
$266$		0			0
$267$	201.145			0.753353
$268$		0			0
$269$	−237.209			−0.881819			−0.440910	−	0.897552i	$-0.645344\pi$
$269$	−237.209			−0.881819			−0.440910	+	0.897552i	$0.645344\pi$
$270$		0			0
$271$		−	34.0378i		−	0.125601i	−0.998026	−	0.0628003i	$-0.979997\pi$
$271$		−	34.0378i		−	0.125601i	0.998026	−	0.0628003i	$-0.0200031\pi$
$272$		0			0
$273$	296.288			1.08530
$274$		0			0
$275$	−54.2865	−	8.83068i	−0.197405	−	0.0321116i
$276$		0			0
$277$			202.654i			0.731604i	0.930693	+	0.365802i	$0.119205\pi$
$277$			202.654i			0.731604i	−0.930693	+	0.365802i	$0.880795\pi$
$278$		0			0
$279$	−5.17045			−0.0185321
$280$		0			0
$281$			100.966i			0.359311i	0.983730	+	0.179655i	$0.0574982\pi$
$281$			100.966i			0.359311i	−0.983730	+	0.179655i	$0.942502\pi$
$282$		0			0
$283$		−	162.113i		−	0.572836i	−0.958105	−	0.286418i	$-0.907535\pi$
$283$		−	162.113i		−	0.572836i	0.958105	−	0.286418i	$-0.0924646\pi$
$284$		0			0
$285$		−	125.889i		−	0.441717i
$286$		0			0
$287$	424.458			1.47895
$288$		0			0
$289$	277.205			0.959187
$290$		0			0
$291$	197.417			0.678410
$292$		0			0
$293$		−	372.772i		−	1.27226i	−0.771582	−	0.636130i	$-0.780534\pi$
$293$		−	372.772i		−	1.27226i	0.771582	−	0.636130i	$-0.219466\pi$
$294$		0			0
$295$	3.92815			0.0133158
$296$		0			0
$297$	290.011	+	47.1756i	0.976468	+	0.158840i
$298$		0			0
$299$			272.170i			0.910268i
$300$		0			0
$301$	−55.1832			−0.183333
$302$		0			0
$303$			521.409i			1.72082i
$304$		0			0
$305$			108.650i			0.356231i
$306$		0			0
$307$		−	445.994i		−	1.45275i	−0.687300	−	0.726374i	$-0.741204\pi$
$307$		−	445.994i		−	1.45275i	0.687300	−	0.726374i	$-0.258796\pi$
$308$		0			0
$309$	−85.8650			−0.277880
$310$		0			0
$311$	249.222			0.801357			0.400679	−	0.916219i	$-0.368774\pi$
$311$	249.222			0.801357			0.400679	+	0.916219i	$0.368774\pi$
$312$		0			0
$313$	463.222			1.47994			0.739971	−	0.672638i	$-0.234839\pi$
$313$	463.222			1.47994			0.739971	+	0.672638i	$0.234839\pi$
$314$		0			0
$315$		−	2.38023i		−	0.00755629i
$316$		0			0
$317$	167.548			0.528542			0.264271	−	0.964448i	$-0.414869\pi$
$317$	167.548			0.528542			0.264271	+	0.964448i	$0.414869\pi$
$318$		0			0
$319$	92.9042	−	571.127i	0.291236	−	1.79037i
$320$		0			0
$321$		−	191.707i		−	0.597220i
$322$		0			0
$323$	−63.7893			−0.197490
$324$		0			0
$325$		−	86.1747i		−	0.265153i
$326$		0			0
$327$		−	389.928i		−	1.19244i
$328$		0			0
$329$		−	103.343i		−	0.314112i
$330$		0			0
$331$	−8.98005			−0.0271301			−0.0135650	−	0.999908i	$-0.504318\pi$
$331$	−8.98005			−0.0271301			−0.0135650	+	0.999908i	$0.504318\pi$
$332$		0			0
$333$	−7.53420			−0.0226252
$334$		0			0
$335$	−79.3582			−0.236890
$336$		0			0
$337$		−	263.901i		−	0.783090i	−0.920159	−	0.391545i	$-0.871941\pi$
$337$		−	263.901i		−	0.783090i	0.920159	−	0.391545i	$-0.128059\pi$
$338$		0			0
$339$	−107.512			−0.317145
$340$		0			0
$341$	299.101	+	48.6541i	0.877128	+	0.142681i
$342$		0			0
$343$			373.378i			1.08857i
$344$		0			0
$345$	107.033			0.310242
$346$		0			0
$347$			611.749i			1.76297i	0.472216	+	0.881483i	$0.343454\pi$
$347$			611.749i			1.76297i	−0.472216	+	0.881483i	$0.656546\pi$
$348$		0			0
$349$			282.223i			0.808662i	0.914613	+	0.404331i	$0.132496\pi$
$349$			282.223i			0.808662i	−0.914613	+	0.404331i	$0.867504\pi$
$350$		0			0
$351$			460.365i			1.31158i
$352$		0			0
$353$	−173.629			−0.491868			−0.245934	−	0.969287i	$-0.579095\pi$
$353$	−173.629			−0.491868			−0.245934	+	0.969287i	$0.579095\pi$
$354$		0			0
$355$	247.082			0.696007
$356$		0			0
$357$	−59.0407			−0.165380
$358$		0			0
$359$		−	285.384i		−	0.794941i	−0.917615	−	0.397471i	$-0.869888\pi$
$359$		−	285.384i		−	0.794941i	0.917615	−	0.397471i	$-0.130112\pi$
$360$		0			0
$361$	16.0139			0.0443597
$362$		0			0
$363$	347.856	+	116.246i	0.958281	+	0.320237i
$364$		0			0
$365$		−	113.403i		−	0.310694i
$366$		0			0
$367$	324.275			0.883582			0.441791	−	0.897118i	$-0.354343\pi$
$367$	324.275			0.883582			0.441791	+	0.897118i	$0.354343\pi$
$368$		0			0
$369$		−	14.0464i		−	0.0380661i
$370$		0			0
$371$		−	426.523i		−	1.14966i
$372$		0			0
$373$			128.367i			0.344148i	0.985084	+	0.172074i	$0.0550469\pi$
$373$			128.367i			0.344148i	−0.985084	+	0.172074i	$0.944953\pi$
$374$		0			0
$375$	−33.8889			−0.0903705
$376$		0			0
$377$	906.611			2.40480
$378$		0			0
$379$	−684.510			−1.80610			−0.903048	−	0.429540i	$-0.858676\pi$
$379$	−684.510			−1.80610			−0.903048	+	0.429540i	$0.858676\pi$
$380$		0			0
$381$		−	638.615i		−	1.67615i
$382$		0			0
$383$	−338.766			−0.884506			−0.442253	−	0.896890i	$-0.645821\pi$
$383$	−338.766			−0.884506			−0.442253	+	0.896890i	$0.645821\pi$
$384$		0			0
$385$	−22.3981	+	137.692i	−0.0581768	+	0.357641i
$386$		0			0
$387$			1.82616i			0.00471875i
$388$		0			0
$389$	−552.950			−1.42146			−0.710732	−	0.703463i	$-0.751636\pi$
$389$	−552.950			−1.42146			−0.710732	+	0.703463i	$0.751636\pi$
$390$		0			0
$391$		−	54.2348i		−	0.138708i
$392$		0			0
$393$		−	399.272i		−	1.01596i
$394$		0			0
$395$			28.5056i			0.0721662i
$396$		0			0
$397$	111.881			0.281815			0.140908	−	0.990023i	$-0.454998\pi$
$397$	111.881			0.281815			0.140908	+	0.990023i	$0.454998\pi$
$398$		0			0
$399$	−319.305			−0.800263
$400$		0			0
$401$	−284.972			−0.710654			−0.355327	−	0.934742i	$-0.615630\pi$
$401$	−284.972			−0.710654			−0.355327	+	0.934742i	$0.615630\pi$
$402$		0			0
$403$			474.794i			1.17815i
$404$		0			0
$405$	184.820			0.456345
$406$		0			0
$407$	435.839	+	70.8972i	1.07086	+	0.174195i
$408$		0			0
$409$		−	122.119i		−	0.298580i	−0.988793	−	0.149290i	$-0.952301\pi$
$409$		−	122.119i		−	0.298580i	0.988793	−	0.149290i	$-0.0476989\pi$
$410$		0			0
$411$	509.028			1.23851
$412$		0			0
$413$		−	9.96334i		−	0.0241243i
$414$		0			0
$415$		−	250.579i		−	0.603806i
$416$		0			0
$417$			274.663i			0.658663i
$418$		0			0
$419$	−407.625			−0.972852			−0.486426	−	0.873722i	$-0.661700\pi$
$419$	−407.625			−0.972852			−0.486426	+	0.873722i	$0.661700\pi$
$420$		0			0
$421$	−348.212			−0.827107			−0.413554	−	0.910480i	$-0.635713\pi$
$421$	−348.212			−0.827107			−0.413554	+	0.910480i	$0.635713\pi$
$422$		0			0
$423$	−3.41988			−0.00808481
$424$		0			0
$425$			17.1718i			0.0404043i
$426$		0			0
$427$	275.580			0.645387
$428$		0			0
$429$	−92.2649	+	567.197i	−0.215070	+	1.32214i
$430$		0			0
$431$		−	453.606i		−	1.05245i	−0.850346	−	0.526225i	$-0.823607\pi$
$431$		−	453.606i		−	1.05245i	0.850346	−	0.526225i	$-0.176393\pi$
$432$		0			0
$433$	760.149			1.75554			0.877771	−	0.479081i	$-0.159030\pi$
$433$	760.149			1.75554			0.877771	+	0.479081i	$0.159030\pi$
$434$		0			0
$435$		−	356.533i		−	0.819616i
$436$		0			0
$437$		−	293.313i		−	0.671198i
$438$		0			0
$439$			343.308i			0.782023i	0.920386	+	0.391011i	$0.127875\pi$
$439$			343.308i			0.782023i	−0.920386	+	0.391011i	$0.872125\pi$
$440$		0			0
$441$	3.15942			0.00716422
$442$		0			0
$443$	−47.4703			−0.107156			−0.0535782	−	0.998564i	$-0.517063\pi$
$443$	−47.4703			−0.107156			−0.0535782	+	0.998564i	$0.517063\pi$
$444$		0			0
$445$	−148.385			−0.333451
$446$		0			0
$447$			220.962i			0.494321i
$448$		0			0
$449$	−339.087			−0.755205			−0.377602	−	0.925968i	$-0.623251\pi$
$449$	−339.087			−0.755205			−0.377602	+	0.925968i	$0.623251\pi$
$450$		0			0
$451$	−132.177	+	812.557i	−0.293076	+	1.80168i
$452$		0			0
$453$			718.921i			1.58702i
$454$		0			0
$455$	−218.573			−0.480380
$456$		0			0
$457$			276.791i			0.605669i	0.953043	+	0.302835i	$0.0979330\pi$
$457$			276.791i			0.605669i	−0.953043	+	0.302835i	$0.902067\pi$
$458$		0			0
$459$		−	91.7360i		−	0.199861i
$460$		0			0
$461$			156.898i			0.340342i	0.985415	+	0.170171i	$0.0544321\pi$
$461$			156.898i			0.340342i	−0.985415	+	0.170171i	$0.945568\pi$
$462$		0			0
$463$	−407.669			−0.880494			−0.440247	−	0.897877i	$-0.645109\pi$
$463$	−407.669			−0.880494			−0.440247	+	0.897877i	$0.645109\pi$
$464$		0			0
$465$	186.717			0.401542
$466$		0			0
$467$	−749.076			−1.60402			−0.802008	−	0.597313i	$-0.796235\pi$
$467$	−749.076			−1.60402			−0.802008	+	0.597313i	$0.796235\pi$
$468$		0			0
$469$			201.284i			0.429176i
$470$		0			0
$471$	321.042			0.681618
$472$		0			0
$473$	17.1842	−	105.640i	0.0363302	−	0.223340i
$474$		0			0
$475$			92.8690i			0.195514i
$476$		0			0
$477$	−14.1147			−0.0295907
$478$		0			0
$479$			11.2596i			0.0235064i	0.999931	+	0.0117532i	$0.00374124\pi$
$479$			11.2596i			0.0235064i	−0.999931	+	0.0117532i	$0.996259\pi$
$480$		0			0
$481$			691.854i			1.43837i
$482$		0			0
$483$		−	271.479i		−	0.562068i
$484$		0			0
$485$	−145.636			−0.300279
$486$		0			0
$487$	734.846			1.50892			0.754462	−	0.656344i	$-0.227898\pi$
$487$	734.846			1.50892			0.754462	+	0.656344i	$0.227898\pi$
$488$		0			0
$489$	−434.156			−0.887845
$490$		0			0
$491$		−	347.360i		−	0.707455i	−0.935349	−	0.353727i	$-0.884914\pi$
$491$		−	347.360i		−	0.707455i	0.935349	−	0.353727i	$-0.115086\pi$
$492$		0			0
$493$	−180.658			−0.366447
$494$		0			0
$495$	4.55658	+	0.741210i	0.00920521	+	0.00149739i
$496$		0			0
$497$		−	626.698i		−	1.26096i
$498$		0			0
$499$	−875.439			−1.75439			−0.877193	−	0.480138i	$-0.840587\pi$
$499$	−875.439			−1.75439			−0.877193	+	0.480138i	$0.840587\pi$
$500$		0			0
$501$			107.900i			0.215369i
$502$		0			0
$503$		−	505.554i		−	1.00508i	−0.864554	−	0.502539i	$-0.832399\pi$
$503$		−	505.554i		−	1.00508i	0.864554	−	0.502539i	$-0.167601\pi$
$504$		0			0
$505$		−	384.645i		−	0.761674i
$506$		0			0
$507$	−388.113			−0.765509
$508$		0			0
$509$	−778.269			−1.52902			−0.764508	−	0.644614i	$-0.777018\pi$
$509$	−778.269			−1.52902			−0.764508	+	0.644614i	$0.777018\pi$
$510$		0			0
$511$	−287.635			−0.562887
$512$		0			0
$513$		−	496.128i		−	0.967111i
$514$		0			0
$515$	63.3429			0.122996
$516$		0			0
$517$	197.833	+	32.1812i	0.382656	+	0.0622460i
$518$		0			0
$519$			594.241i			1.14497i
$520$		0			0
$521$	−273.723			−0.525380			−0.262690	−	0.964880i	$-0.584610\pi$
$521$	−273.723			−0.525380			−0.262690	+	0.964880i	$0.584610\pi$
$522$		0			0
$523$		−	628.316i		−	1.20137i	−0.799486	−	0.600685i	$-0.794895\pi$
$523$		−	628.316i		−	1.20137i	0.799486	−	0.600685i	$-0.205105\pi$
$524$		0			0
$525$			85.9557i			0.163725i
$526$		0			0
$527$		−	94.6112i		−	0.179528i
$528$		0			0
$529$	−279.620			−0.528582
$530$		0			0
$531$	−0.329713			−0.000620928
$532$		0			0
$533$	−1289.86			−2.42000
$534$		0			0
$535$			141.423i			0.264343i
$536$		0			0
$537$	197.576			0.367925
$538$		0			0
$539$	−182.766	−	29.7303i	−0.339084	−	0.0551582i
$540$		0			0
$541$		−	543.648i		−	1.00490i	−0.864608	−	0.502448i	$-0.832433\pi$
$541$		−	543.648i		−	1.00490i	0.864608	−	0.502448i	$-0.167567\pi$
$542$		0			0
$543$	−475.935			−0.876492
$544$		0			0
$545$			287.651i			0.527800i
$546$		0			0
$547$			421.038i			0.769722i	0.922975	+	0.384861i	$0.125751\pi$
$547$			421.038i			0.769722i	−0.922975	+	0.384861i	$0.874249\pi$
$548$		0			0
$549$		−	9.11967i		−	0.0166114i
$550$		0			0
$551$	−977.040			−1.77321
$552$		0			0
$553$	72.3016			0.130744
$554$		0			0
$555$	272.078			0.490230
$556$		0			0
$557$			940.508i			1.68852i	0.535930	+	0.844262i	$0.319961\pi$
$557$			940.508i			1.68852i	−0.535930	+	0.844262i	$0.680039\pi$
$558$		0			0
$559$	167.693			0.299987
$560$		0			0
$561$	18.3854	−	113.024i	0.0327726	−	0.201469i
$562$		0			0
$563$			285.806i			0.507649i	0.967250	+	0.253824i	$0.0816885\pi$
$563$			285.806i			0.507649i	−0.967250	+	0.253824i	$0.918312\pi$
$564$		0			0
$565$	79.3120			0.140375
$566$		0			0
$567$		−	468.776i		−	0.826765i
$568$		0			0
$569$		−	307.137i		−	0.539783i	−0.962891	−	0.269892i	$-0.913012\pi$
$569$		−	307.137i		−	0.539783i	0.962891	−	0.269892i	$-0.0869879\pi$
$570$		0			0
$571$		−	390.713i		−	0.684261i	−0.939652	−	0.342130i	$-0.888852\pi$
$571$		−	390.713i		−	0.684261i	0.939652	−	0.342130i	$-0.111148\pi$
$572$		0			0
$573$	−490.941			−0.856791
$574$		0			0
$575$	−78.9589			−0.137320
$576$		0			0
$577$	−119.491			−0.207090			−0.103545	−	0.994625i	$-0.533019\pi$
$577$	−119.491			−0.207090			−0.103545	+	0.994625i	$0.533019\pi$
$578$		0			0
$579$			2.25186i			0.00388923i
$580$		0			0
$581$	−635.568			−1.09392
$582$		0			0
$583$	816.511	+	132.820i	1.40053	+	0.227822i
$584$		0			0
$585$			7.23314i			0.0123643i
$586$		0			0
$587$	340.259			0.579658			0.289829	−	0.957078i	$-0.406402\pi$
$587$	340.259			0.579658			0.289829	+	0.957078i	$0.406402\pi$
$588$		0			0
$589$		−	511.678i		−	0.868723i
$590$		0			0
$591$		−	130.625i		−	0.221023i
$592$		0			0
$593$			752.004i			1.26813i	0.773278	+	0.634067i	$0.218616\pi$
$593$			752.004i			1.26813i	−0.773278	+	0.634067i	$0.781384\pi$
$594$		0			0
$595$	43.5546			0.0732009
$596$		0			0
$597$	612.083			1.02527
$598$		0			0
$599$	−274.841			−0.458833			−0.229416	−	0.973328i	$-0.573682\pi$
$599$	−274.841			−0.458833			−0.229416	+	0.973328i	$0.573682\pi$
$600$		0			0
$601$		−	397.025i		−	0.660608i	−0.943875	−	0.330304i	$-0.892849\pi$
$601$		−	397.025i		−	0.660608i	0.943875	−	0.330304i	$-0.107151\pi$
$602$		0			0
$603$	6.66100			0.0110464
$604$		0			0
$605$	−256.615	−	85.7552i	−0.424156	−	0.141744i
$606$		0			0
$607$			351.597i			0.579238i	0.957142	+	0.289619i	$0.0935286\pi$
$607$			351.597i			0.579238i	−0.957142	+	0.289619i	$0.906471\pi$
$608$		0			0
$609$	−904.308			−1.48491
$610$		0			0
$611$			314.042i			0.513980i
$612$		0			0
$613$		−	547.965i		−	0.893907i	−0.894557	−	0.446953i	$-0.852509\pi$
$613$		−	547.965i		−	0.893907i	0.894557	−	0.446953i	$-0.147491\pi$
$614$		0			0
$615$			507.248i			0.824794i
$616$		0			0
$617$	−1124.68			−1.82282			−0.911408	−	0.411505i	$-0.865003\pi$
$617$	−1124.68			−1.82282			−0.911408	+	0.411505i	$0.865003\pi$
$618$		0			0
$619$	−82.7193			−0.133634			−0.0668169	−	0.997765i	$-0.521284\pi$
$619$	−82.7193			−0.133634			−0.0668169	+	0.997765i	$0.521284\pi$
$620$		0			0
$621$	421.817			0.679254
$622$		0			0
$623$			376.364i			0.604116i
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	99.4324	−	611.259i	0.158584	−	0.974895i
$628$		0			0
$629$		−	137.864i		−	0.219180i
$630$		0			0
$631$	−317.879			−0.503771			−0.251885	−	0.967757i	$-0.581051\pi$
$631$	−317.879			−0.503771			−0.251885	+	0.967757i	$0.581051\pi$
$632$		0			0
$633$			162.076i			0.256043i
$634$		0			0
$635$			471.108i			0.741903i
$636$		0			0
$637$		−	290.125i		−	0.455454i
$638$		0			0
$639$	−20.7391			−0.0324555
$640$		0			0
$641$	432.182			0.674230			0.337115	−	0.941463i	$-0.390549\pi$
$641$	432.182			0.674230			0.337115	+	0.941463i	$0.390549\pi$
$642$		0			0
$643$	−229.511			−0.356937			−0.178469	−	0.983946i	$-0.557114\pi$
$643$	−229.511			−0.356937			−0.178469	+	0.983946i	$0.557114\pi$
$644$		0			0
$645$		−	65.9467i		−	0.102243i
$646$		0			0
$647$	768.342			1.18755			0.593773	−	0.804633i	$-0.297638\pi$
$647$	768.342			1.18755			0.593773	+	0.804633i	$0.297638\pi$
$648$		0			0
$649$	19.0733	+	3.10261i	0.0293887	+	0.00478060i
$650$		0			0
$651$		−	473.588i		−	0.727478i
$652$		0			0
$653$	624.977			0.957085			0.478543	−	0.878064i	$-0.341165\pi$
$653$	624.977			0.957085			0.478543	+	0.878064i	$0.341165\pi$
$654$		0			0
$655$			294.544i			0.449686i
$656$		0			0
$657$			9.51859i			0.0144880i
$658$		0			0
$659$			1104.04i			1.67533i	0.546185	+	0.837665i	$0.316080\pi$
$659$			1104.04i			1.67533i	−0.546185	+	0.837665i	$0.683920\pi$
$660$		0			0
$661$	246.985			0.373653			0.186827	−	0.982393i	$-0.440180\pi$
$661$	246.985			0.373653			0.186827	+	0.982393i	$0.440180\pi$
$662$		0			0
$663$	179.415			0.270611
$664$		0			0
$665$	235.552			0.354214
$666$		0			0
$667$		−	830.697i		−	1.24542i
$668$		0			0
$669$	856.594			1.28041
$670$		0			0
$671$	−85.8165	+	527.555i	−0.127893	+	0.786223i
$672$		0			0
$673$			715.709i			1.06346i	0.846914	+	0.531730i	$0.178458\pi$
$673$			715.709i			1.06346i	−0.846914	+	0.531730i	$0.821542\pi$
$674$		0			0
$675$	−133.556			−0.197861
$676$		0			0
$677$		−	946.798i		−	1.39852i	−0.714867	−	0.699260i	$-0.753513\pi$
$677$		−	946.798i		−	1.39852i	0.714867	−	0.699260i	$-0.246487\pi$
$678$		0			0
$679$			369.389i			0.544019i
$680$		0			0
$681$			337.215i			0.495176i
$682$		0			0
$683$	−67.4542			−0.0987616			−0.0493808	−	0.998780i	$-0.515725\pi$
$683$	−67.4542			−0.0987616			−0.0493808	+	0.998780i	$0.515725\pi$
$684$		0			0
$685$	−375.512			−0.548193
$686$		0			0
$687$	−161.805			−0.235525
$688$		0			0
$689$			1296.13i			1.88118i
$690$		0			0
$691$	396.379			0.573632			0.286816	−	0.957986i	$-0.407403\pi$
$691$	396.379			0.573632			0.286816	+	0.957986i	$0.407403\pi$
$692$		0			0
$693$	1.88000	−	11.5573i	0.00271284	−	0.0166772i
$694$		0			0
$695$		−	202.620i		−	0.291539i
$696$		0			0
$697$	257.027			0.368762
$698$		0			0
$699$		−	129.454i		−	0.185198i
$700$		0			0
$701$			884.194i			1.26133i	0.776054	+	0.630666i	$0.217218\pi$
$701$			884.194i			1.26133i	−0.776054	+	0.630666i	$0.782782\pi$
$702$		0			0
$703$		−	745.600i		−	1.06060i
$704$		0			0
$705$	123.500			0.175177
$706$		0			0
$707$	−975.613			−1.37993
$708$		0			0
$709$	−902.302			−1.27264			−0.636320	−	0.771425i	$-0.719544\pi$
$709$	−902.302			−1.27264			−0.636320	+	0.771425i	$0.719544\pi$
$710$		0			0
$711$		−	2.39265i		−	0.00336518i
$712$		0			0
$713$	435.038			0.610151
$714$		0			0
$715$	68.0642	−	418.423i	0.0951946	−	0.585208i
$716$		0			0
$717$			657.673i			0.917257i
$718$		0			0
$719$	−1021.01			−1.42004			−0.710021	−	0.704180i	$-0.751315\pi$
$719$	−1021.01			−1.42004			−0.710021	+	0.704180i	$0.751315\pi$
$720$		0			0
$721$		−	160.663i		−	0.222833i
$722$		0			0
$723$			470.068i			0.650163i
$724$		0			0
$725$			263.016i			0.362780i
$726$		0			0
$727$	−637.169			−0.876436			−0.438218	−	0.898869i	$-0.644390\pi$
$727$	−637.169			−0.876436			−0.438218	+	0.898869i	$0.644390\pi$
$728$		0			0
$729$	713.170			0.978285
$730$		0			0
$731$	−33.4158			−0.0457125
$732$		0			0
$733$		−	261.937i		−	0.357349i	−0.983908	−	0.178674i	$-0.942819\pi$
$733$		−	261.937i		−	0.357349i	0.983908	−	0.178674i	$-0.0571809\pi$
$734$		0			0
$735$	−114.094			−0.155230
$736$		0			0
$737$	−385.326	−	62.6803i	−0.522830	−	0.0850478i
$738$		0			0
$739$		−	734.865i		−	0.994404i	−0.867635	−	0.497202i	$-0.834361\pi$
$739$		−	734.865i		−	0.994404i	0.867635	−	0.497202i	$-0.165639\pi$
$740$		0			0
$741$	970.317			1.30947
$742$		0			0
$743$			161.161i			0.216905i	0.994102	+	0.108453i	$0.0345896\pi$
$743$			161.161i			0.216905i	−0.994102	+	0.108453i	$0.965410\pi$
$744$		0			0
$745$		−	163.004i		−	0.218797i
$746$		0			0
$747$			21.0326i			0.0281561i
$748$		0			0
$749$	358.705			0.478912
$750$		0			0
$751$	−1141.77			−1.52033			−0.760167	−	0.649728i	$-0.774883\pi$
$751$	−1141.77			−1.52033			−0.760167	+	0.649728i	$0.774883\pi$
$752$		0			0
$753$	353.244			0.469116
$754$		0			0
$755$		−	530.350i		−	0.702451i
$756$		0			0
$757$	119.677			0.158094			0.0790469	−	0.996871i	$-0.474812\pi$
$757$	119.677			0.158094			0.0790469	+	0.996871i	$0.474812\pi$
$758$		0			0
$759$	519.704	+	84.5392i	0.684721	+	0.111382i
$760$		0			0
$761$			157.280i			0.206676i	0.994646	+	0.103338i	$0.0329523\pi$
$761$			157.280i			0.206676i	−0.994646	+	0.103338i	$0.967048\pi$
$762$		0			0
$763$	729.596			0.956221
$764$		0			0
$765$		−	1.44133i		−	0.00188409i
$766$		0			0
$767$			30.2770i			0.0394746i
$768$		0			0
$769$		−	1498.58i		−	1.94874i	−0.224947	−	0.974371i	$-0.572221\pi$
$769$		−	1498.58i		−	1.94874i	0.224947	−	0.974371i	$-0.427779\pi$
$770$		0			0
$771$	−259.774			−0.336931
$772$		0			0
$773$	−1528.87			−1.97784			−0.988920	−	0.148450i	$-0.952572\pi$
$773$	−1528.87			−1.97784			−0.988920	+	0.148450i	$0.952572\pi$
$774$		0			0
$775$	−137.742			−0.177731
$776$		0			0
$777$		−	690.096i		−	0.888155i
$778$		0			0
$779$	1390.06			1.78441
$780$		0			0
$781$	1199.72	+	195.155i	1.53613	+	0.249879i
$782$		0			0
$783$		−	1405.09i		−	1.79450i
$784$		0			0
$785$	−236.834			−0.301699
$786$		0			0
$787$			930.613i			1.18248i	0.806495	+	0.591241i	$0.201362\pi$
$787$			930.613i			1.18248i	−0.806495	+	0.591241i	$0.798638\pi$
$788$		0			0
$789$		−	1209.00i		−	1.53232i
$790$		0			0
$791$		−	201.167i		−	0.254319i
$792$		0			0
$793$	−837.445			−1.05605
$794$		0			0
$795$	509.717			0.641153
$796$		0			0
$797$	−332.628			−0.417350			−0.208675	−	0.977985i	$-0.566915\pi$
$797$	−332.628			−0.417350			−0.208675	+	0.977985i	$0.566915\pi$
$798$		0			0
$799$		−	62.5784i		−	0.0783210i
$800$		0			0
$801$	12.4549			0.0155491
$802$		0			0
$803$	89.5703	−	550.632i	0.111545	−	0.685719i
$804$		0			0
$805$			200.271i			0.248784i
$806$		0			0
$807$	−719.010			−0.890966
$808$		0			0
$809$			870.764i			1.07635i	0.842834	+	0.538173i	$0.180885\pi$
$809$			870.764i			1.07635i	−0.842834	+	0.538173i	$0.819115\pi$
$810$		0			0
$811$			1369.14i			1.68821i	0.536180	+	0.844103i	$0.319867\pi$
$811$			1369.14i			1.68821i	−0.536180	+	0.844103i	$0.680133\pi$
$812$		0			0
$813$		−	103.173i		−	0.126904i
$814$		0			0
$815$	320.279			0.392980
$816$		0			0
$817$	−180.720			−0.221199
$818$		0			0
$819$	18.3461			0.0224006
$820$		0			0
$821$			759.486i			0.925075i	0.886600	+	0.462537i	$0.153061\pi$
$821$			759.486i			0.925075i	−0.886600	+	0.462537i	$0.846939\pi$
$822$		0			0
$823$	−252.673			−0.307015			−0.153507	−	0.988148i	$-0.549057\pi$
$823$	−252.673			−0.307015			−0.153507	+	0.988148i	$0.549057\pi$
$824$		0			0
$825$	−164.549	−	26.7668i	−0.199453	−	0.0324447i
$826$		0			0
$827$		−	172.995i		−	0.209184i	−0.994515	−	0.104592i	$-0.966646\pi$
$827$		−	172.995i		−	0.209184i	0.994515	−	0.104592i	$-0.0333536\pi$
$828$		0			0
$829$	1145.40			1.38167			0.690834	−	0.723013i	$-0.257244\pi$
$829$	1145.40			1.38167			0.690834	+	0.723013i	$0.257244\pi$
$830$		0			0
$831$			614.270i			0.739193i
$832$		0			0
$833$			57.8125i			0.0694027i
$834$		0			0
$835$		−	79.5983i		−	0.0953273i
$836$		0			0
$837$	735.849			0.879151
$838$		0			0
$839$	41.7126			0.0497170			0.0248585	−	0.999691i	$-0.492086\pi$
$839$	41.7126			0.0497170			0.0248585	+	0.999691i	$0.492086\pi$
$840$		0			0
$841$	−1926.09			−2.29023
$842$		0			0
$843$			306.041i			0.363038i
$844$		0			0
$845$	286.312			0.338831
$846$		0			0
$847$	−217.509	+	650.876i	−0.256799	+	0.768448i
$848$		0			0
$849$		−	491.383i		−	0.578778i
$850$		0			0
$851$	633.922			0.744915
$852$		0			0
$853$			725.094i			0.850052i	0.905181	+	0.425026i	$0.139735\pi$
$853$			725.094i			0.850052i	−0.905181	+	0.425026i	$0.860265\pi$
$854$		0			0
$855$		−	7.79504i		−	0.00911700i
$856$		0			0
$857$		−	302.221i		−	0.352650i	−0.984332	−	0.176325i	$-0.943579\pi$
$857$		−	302.221i		−	0.352650i	0.984332	−	0.176325i	$-0.0564210\pi$
$858$		0			0
$859$	207.101			0.241096			0.120548	−	0.992708i	$-0.461535\pi$
$859$	207.101			0.241096			0.120548	+	0.992708i	$0.461535\pi$
$860$		0			0
$861$	1286.58			1.49429
$862$		0			0
$863$	−482.444			−0.559032			−0.279516	−	0.960141i	$-0.590174\pi$
$863$	−482.444			−0.559032			−0.279516	+	0.960141i	$0.590174\pi$
$864$		0			0
$865$		−	438.374i		−	0.506791i
$866$		0			0
$867$	840.242			0.969137
$868$		0			0
$869$	−22.5149	+	138.410i	−0.0259090	+	0.159275i
$870$		0			0
$871$		−	611.669i		−	0.702260i
$872$		0			0
$873$	12.2240			0.0140023
$874$		0			0
$875$		−	63.4099i		−	0.0724684i
$876$		0			0
$877$			377.864i			0.430860i	0.976519	+	0.215430i	$0.0691153\pi$
$877$			377.864i			0.430860i	−0.976519	+	0.215430i	$0.930885\pi$
$878$		0			0
$879$		−	1129.92i		−	1.28546i
$880$		0			0
$881$	−1459.75			−1.65693			−0.828463	−	0.560045i	$-0.810784\pi$
$881$	−1459.75			−1.65693			−0.828463	+	0.560045i	$0.810784\pi$
$882$		0			0
$883$	353.252			0.400059			0.200029	−	0.979790i	$-0.435896\pi$
$883$	353.252			0.400059			0.200029	+	0.979790i	$0.435896\pi$
$884$		0			0
$885$	11.9067			0.0134539
$886$		0			0
$887$			174.064i			0.196239i	0.995175	+	0.0981194i	$0.0312827\pi$
$887$			174.064i			0.196239i	−0.995175	+	0.0981194i	$0.968717\pi$
$888$		0			0
$889$	1194.92			1.34411
$890$		0			0
$891$	897.398	+	145.978i	1.00718	+	0.163836i
$892$		0			0
$893$		−	338.438i		−	0.378990i
$894$		0			0
$895$	−145.752			−0.162852
$896$		0			0
$897$			824.981i			0.919711i
$898$		0			0
$899$		−	1449.13i		−	1.61193i
$900$		0			0
$901$		−	258.278i		−	0.286657i
$902$		0			0
$903$	−167.267			−0.185235
$904$		0			0
$905$	351.099			0.387955
$906$		0			0
$907$	1341.78			1.47936			0.739679	−	0.672960i	$-0.234977\pi$
$907$	1341.78			1.47936			0.739679	+	0.672960i	$0.234977\pi$
$908$		0			0
$909$			32.2855i			0.0355176i
$910$		0			0
$911$	−81.8460			−0.0898419			−0.0449210	−	0.998991i	$-0.514304\pi$
$911$	−81.8460			−0.0898419			−0.0449210	+	0.998991i	$0.514304\pi$
$912$		0			0
$913$	197.918	−	1216.70i	0.216777	−	1.33263i
$914$		0			0
$915$			329.333i			0.359926i
$916$		0			0
$917$	747.080			0.814700
$918$		0			0
$919$		−	146.259i		−	0.159150i	−0.996829	−	0.0795752i	$-0.974644\pi$
$919$		−	146.259i		−	0.159150i	0.996829	−	0.0795752i	$-0.0253564\pi$
$920$		0			0
$921$		−	1351.86i		−	1.46782i
$922$		0			0
$923$			1904.44i			2.06331i
$924$		0			0
$925$	−200.713			−0.216987
$926$		0			0
$927$	−5.31674			−0.00573543
$928$		0			0
$929$	−624.735			−0.672481			−0.336240	−	0.941776i	$-0.609155\pi$
$929$	−624.735			−0.672481			−0.336240	+	0.941776i	$0.609155\pi$
$930$		0			0
$931$			312.662i			0.335835i
$932$		0			0
$933$	755.422			0.809670
$934$		0			0
$935$	−13.5630	+	83.3783i	−0.0145059	+	0.0891747i
$936$		0			0
$937$			1737.04i			1.85383i	0.375273	+	0.926915i	$0.377549\pi$
$937$			1737.04i			1.85383i	−0.375273	+	0.926915i	$0.622451\pi$
$938$		0			0
$939$	1404.08			1.49529
$940$		0			0
$941$		−	1214.68i		−	1.29084i	−0.763830	−	0.645418i	$-0.776683\pi$
$941$		−	1214.68i		−	1.29084i	0.763830	−	0.645418i	$-0.223317\pi$
$942$		0			0
$943$			1181.85i			1.25329i
$944$		0			0
$945$			338.750i			0.358466i
$946$		0			0
$947$	−1465.25			−1.54725			−0.773627	−	0.633641i	$-0.781560\pi$
$947$	−1465.25			−1.54725			−0.773627	+	0.633641i	$0.781560\pi$
$948$		0			0
$949$	874.077			0.921051
$950$		0			0
$951$	507.857			0.534025
$952$		0			0
$953$			464.526i			0.487435i	0.969846	+	0.243717i	$0.0783670\pi$
$953$			464.526i			0.487435i	−0.969846	+	0.243717i	$0.921633\pi$
$954$		0			0
$955$	362.169			0.379235
$956$		0			0
$957$	281.604	−	1731.15i	0.294257	−	1.80894i
$958$		0			0
$959$			952.446i			0.993166i
$960$		0			0
$961$	−202.088			−0.210289
$962$		0			0
$963$		−	11.8705i		−	0.0123266i
$964$		0			0
$965$		−	1.66121i		−	0.00172146i
$966$		0			0
$967$		−	667.349i		−	0.690123i	−0.938580	−	0.345062i	$-0.887858\pi$
$967$		−	667.349i		−	0.690123i	0.938580	−	0.345062i	$-0.112142\pi$
$968$		0			0
$969$	−193.353			−0.199539
$970$		0			0
$971$	−980.665			−1.00995			−0.504977	−	0.863133i	$-0.668499\pi$
$971$	−980.665			−1.00995			−0.504977	+	0.863133i	$0.668499\pi$
$972$		0			0
$973$	−513.923			−0.528184
$974$		0			0
$975$		−	261.206i		−	0.267903i
$976$		0			0
$977$	342.678			0.350745			0.175372	−	0.984502i	$-0.443887\pi$
$977$	342.678			0.350745			0.175372	+	0.984502i	$0.443887\pi$
$978$		0			0
$979$	−720.490	−	117.201i	−0.735945	−	0.119715i
$980$		0			0
$981$		−	24.1442i		−	0.0246118i
$982$		0			0
$983$	447.215			0.454949			0.227475	−	0.973784i	$-0.426953\pi$
$983$	447.215			0.454949			0.227475	+	0.973784i	$0.426953\pi$
$984$		0			0
$985$			96.3622i			0.0978297i
$986$		0			0
$987$		−	313.244i		−	0.317370i
$988$		0			0
$989$		−	153.651i		−	0.155360i
$990$		0			0
$991$	−135.932			−0.137166			−0.0685831	−	0.997645i	$-0.521848\pi$
$991$	−135.932			−0.137166			−0.0685831	+	0.997645i	$0.521848\pi$
$992$		0			0
$993$	−27.2196			−0.0274115
$994$		0			0
$995$	−451.536			−0.453805
$996$		0			0
$997$		−	303.075i		−	0.303987i	−0.988381	−	0.151993i	$-0.951431\pi$
$997$		−	303.075i		−	0.303987i	0.988381	−	0.151993i	$-0.0485693\pi$
$998$		0			0
$999$	1072.25			1.07333

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	880.3.j.a.241.8		8
4.3	odd	2		55.3.c.a.21.2	✓	8
11.10	odd	2	inner	880.3.j.a.241.7		8
12.11	even	2		495.3.b.a.406.7		8
20.3	even	4		275.3.d.c.274.3		16
20.7	even	4		275.3.d.c.274.14		16
20.19	odd	2		275.3.c.f.76.7		8
44.43	even	2		55.3.c.a.21.7	yes	8
132.131	odd	2		495.3.b.a.406.2		8
220.43	odd	4		275.3.d.c.274.13		16
220.87	odd	4		275.3.d.c.274.4		16
220.219	even	2		275.3.c.f.76.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.3.c.a.21.2	✓	8	4.3	odd	2
55.3.c.a.21.7	yes	8	44.43	even	2
275.3.c.f.76.2		8	220.219	even	2
275.3.c.f.76.7		8	20.19	odd	2
275.3.d.c.274.3		16	20.3	even	4
275.3.d.c.274.4		16	220.87	odd	4
275.3.d.c.274.13		16	220.43	odd	4
275.3.d.c.274.14		16	20.7	even	4
495.3.b.a.406.2		8	132.131	odd	2
495.3.b.a.406.7		8	12.11	even	2
880.3.j.a.241.7		8	11.10	odd	2	inner
880.3.j.a.241.8		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	880.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.03112 q^{3} -2.23607 q^{5} +5.67155i q^{7} +0.187686 q^{9} +(-10.8573 - 1.76614i) q^{11} -17.2349i q^{13} -6.77779 q^{15} +3.43437i q^{17} +18.5738i q^{19} +17.1911i q^{21} -15.7918 q^{23} +5.00000 q^{25} -26.7112 q^{27} +52.6031i q^{29} -27.5484 q^{31} +(-32.9097 - 5.35337i) q^{33} -12.6820i q^{35} -40.1425 q^{37} -52.2411i q^{39} -74.8398i q^{41} +9.72983i q^{43} -0.419679 q^{45} -18.2212 q^{47} +16.8335 q^{49} +10.4100i q^{51} -75.2040 q^{53} +(24.2776 + 3.94920i) q^{55} +56.2994i q^{57} -1.75672 q^{59} -48.5900i q^{61} +1.06447i q^{63} +38.5385i q^{65} +35.4901 q^{67} -47.8668 q^{69} -110.499 q^{71} +50.7154i q^{73} +15.1556 q^{75} +(10.0167 - 61.5777i) q^{77} -12.7481i q^{79} -82.6540 q^{81} +112.062i q^{83} -7.67948i q^{85} +159.446i q^{87} +66.3600 q^{89} +97.7488 q^{91} -83.5024 q^{93} -41.5323i q^{95} +65.1302 q^{97} +(-2.03776 - 0.331479i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{3} - 4 q^{9} - 8 q^{11} - 20 q^{15} - 8 q^{23} + 40 q^{25} + 16 q^{27} - 36 q^{31} - 152 q^{33} - 88 q^{37} + 8 q^{47} + 172 q^{49} - 152 q^{53} + 20 q^{55} + 16 q^{59} + 88 q^{67} + 8 q^{69}+ \cdots + 184 q^{99}+O(q^{100}) \) 8 * q - 8 * q^3 - 4 * q^9 - 8 * q^11 - 20 * q^15 - 8 * q^23 + 40 * q^25 + 16 * q^27 - 36 * q^31 - 152 * q^33 - 88 * q^37 + 8 * q^47 + 172 * q^49 - 152 * q^53 + 20 * q^55 + 16 * q^59 + 88 * q^67 + 8 * q^69 - 276 * q^71 - 40 * q^75 + 160 * q^77 - 320 * q^81 + 444 * q^89 + 40 * q^91 - 104 * q^93 + 312 * q^97 + 184 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more