LMFDB - Embedded newform 1008.3.o.b.1007.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1008,3,Mod(1007,1008)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 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("1008.1007"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1008.o (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$27.4660106475$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1007.6
Character	$\chi$	$=$	1008.1007
Dual form			1008.3.o.b.1007.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-4.34452 q^{5} +(0.646021 - 6.97013i) q^{7} +7.76492 q^{11} +13.4815i q^{13} +28.9344 q^{17} -32.7004 q^{19} +16.8486 q^{23} -6.12514 q^{25} -38.3891i q^{29} +50.1080 q^{31} +(-2.80665 + 30.2819i) q^{35} -39.1653 q^{37} -63.0188 q^{41} -7.01205i q^{43} -23.4545i q^{47} +(-48.1653 - 9.00569i) q^{49} -90.9488i q^{53} -33.7349 q^{55} -78.1723i q^{59} -49.9251i q^{61} -58.5705i q^{65} -15.8052i q^{67} +62.1280 q^{71} +59.7725i q^{73} +(5.01630 - 54.1225i) q^{77} -137.286i q^{79} +102.790i q^{83} -125.706 q^{85} +26.6516 q^{89} +(93.9675 + 8.70930i) q^{91} +142.068 q^{95} -105.697i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 264 q^{25} - 96 q^{37} - 312 q^{49} - 96 q^{85}+O(q^{100}) $ 24 * q + 264 * q^25 - 96 * q^37 - 312 * q^49 - 96 * q^85

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−4.34452			−0.868904			−0.434452	−	0.900695i	$-0.643058\pi$
$5$	−4.34452			−0.868904			−0.434452	+	0.900695i	$0.643058\pi$
$6$		0			0
$7$	0.646021	−	6.97013i	0.0922887	−	0.995732i
$7$	0.646021	−	6.97013i	0.0922887	−	0.995732i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	7.76492			0.705902			0.352951	−	0.935642i	$-0.385178\pi$
$11$	7.76492			0.705902			0.352951	+	0.935642i	$0.385178\pi$
$12$		0			0
$13$			13.4815i			1.03704i	0.855067	+	0.518518i	$0.173516\pi$
$13$			13.4815i			1.03704i	−0.855067	+	0.518518i	$0.826484\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	28.9344			1.70202			0.851011	−	0.525147i	$-0.175990\pi$
$17$	28.9344			1.70202			0.851011	+	0.525147i	$0.175990\pi$
$18$		0			0
$19$	−32.7004			−1.72107			−0.860537	−	0.509389i	$-0.829872\pi$
$19$	−32.7004			−1.72107			−0.860537	+	0.509389i	$0.829872\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	16.8486			0.732546			0.366273	−	0.930507i	$-0.380634\pi$
$23$	16.8486			0.732546			0.366273	+	0.930507i	$0.380634\pi$
$24$		0			0
$25$	−6.12514			−0.245005
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	38.3891i		−	1.32376i	−0.749608	−	0.661882i	$-0.769758\pi$
$29$		−	38.3891i		−	1.32376i	0.749608	−	0.661882i	$-0.230242\pi$
$30$		0			0
$31$	50.1080			1.61639			0.808193	−	0.588917i	$-0.200446\pi$
$31$	50.1080			1.61639			0.808193	+	0.588917i	$0.200446\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.80665	+	30.2819i	−0.0801900	+	0.865196i
$36$		0			0
$37$	−39.1653			−1.05852			−0.529261	−	0.848459i	$-0.677531\pi$
$37$	−39.1653			−1.05852			−0.529261	+	0.848459i	$0.677531\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−63.0188			−1.53704			−0.768522	−	0.639824i	$-0.779007\pi$
$41$	−63.0188			−1.53704			−0.768522	+	0.639824i	$0.779007\pi$
$42$		0			0
$43$		−	7.01205i		−	0.163071i	−0.996670	−	0.0815355i	$-0.974018\pi$
$43$		−	7.01205i		−	0.163071i	0.996670	−	0.0815355i	$-0.0259824\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	23.4545i		−	0.499032i	−0.968371	−	0.249516i	$-0.919728\pi$
$47$		−	23.4545i		−	0.499032i	0.968371	−	0.249516i	$-0.0802715\pi$
$48$		0			0
$49$	−48.1653	−	9.00569i	−0.982966	−	0.183790i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	90.9488i		−	1.71602i	−0.513637	−	0.858008i	$-0.671702\pi$
$53$		−	90.9488i		−	1.71602i	0.513637	−	0.858008i	$-0.328298\pi$
$54$		0			0
$55$	−33.7349			−0.613361
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	78.1723i		−	1.32495i	−0.749082	−	0.662477i	$-0.769505\pi$
$59$		−	78.1723i		−	1.32495i	0.749082	−	0.662477i	$-0.230495\pi$
$60$		0			0
$61$		−	49.9251i		−	0.818444i	−0.912435	−	0.409222i	$-0.865800\pi$
$61$		−	49.9251i		−	0.818444i	0.912435	−	0.409222i	$-0.134200\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	58.5705i		−	0.901084i
$66$		0			0
$67$		−	15.8052i		−	0.235898i	−0.993020	−	0.117949i	$-0.962368\pi$
$67$		−	15.8052i		−	0.235898i	0.993020	−	0.117949i	$-0.0376320\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	62.1280			0.875042			0.437521	−	0.899208i	$-0.355857\pi$
$71$	62.1280			0.875042			0.437521	+	0.899208i	$0.355857\pi$
$72$		0			0
$73$			59.7725i			0.818801i	0.912355	+	0.409401i	$0.134262\pi$
$73$			59.7725i			0.818801i	−0.912355	+	0.409401i	$0.865738\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	5.01630	−	54.1225i	0.0651468	−	0.702889i
$78$		0			0
$79$		−	137.286i		−	1.73780i	−0.494989	−	0.868899i	$-0.664828\pi$
$79$		−	137.286i		−	1.73780i	0.494989	−	0.868899i	$-0.335172\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			102.790i			1.23844i	0.785218	+	0.619219i	$0.212551\pi$
$83$			102.790i			1.23844i	−0.785218	+	0.619219i	$0.787449\pi$
$84$		0			0
$85$	−125.706			−1.47889
$86$		0			0
$87$		0			0
$88$		0			0
$89$	26.6516			0.299456			0.149728	−	0.988727i	$-0.452160\pi$
$89$	26.6516			0.299456			0.149728	+	0.988727i	$0.452160\pi$
$90$		0			0
$91$	93.9675	+	8.70930i	1.03261	+	0.0957066i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	142.068			1.49545
$96$		0			0
$97$		−	105.697i		−	1.08966i	−0.838547	−	0.544829i	$-0.816595\pi$
$97$		−	105.697i		−	1.08966i	0.838547	−	0.544829i	$-0.183405\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−95.6259			−0.946791			−0.473396	−	0.880850i	$-0.656972\pi$
$101$	−95.6259			−0.946791			−0.473396	+	0.880850i	$0.656972\pi$
$102$		0			0
$103$	−0.211707			−0.00205541			−0.00102770	−	0.999999i	$-0.500327\pi$
$103$	−0.211707			−0.00205541			−0.00102770	+	0.999999i	$0.500327\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	121.393			1.13451			0.567257	−	0.823541i	$-0.308005\pi$
$107$	121.393			1.13451			0.567257	+	0.823541i	$0.308005\pi$
$108$		0			0
$109$	−28.2904			−0.259545			−0.129773	−	0.991544i	$-0.541425\pi$
$109$	−28.2904			−0.259545			−0.129773	+	0.991544i	$0.541425\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	150.551i		−	1.33231i	−0.745813	−	0.666156i	$-0.767939\pi$
$113$		−	150.551i		−	1.33231i	0.745813	−	0.666156i	$-0.232061\pi$
$114$		0			0
$115$	−73.1989			−0.636512
$116$		0			0
$117$		0			0
$118$		0			0
$119$	18.6922	−	201.676i	0.157077	−	1.69476i
$120$		0			0
$121$	−60.7060			−0.501703
$122$		0			0
$123$		0			0
$124$		0			0
$125$	135.224			1.08179
$126$		0			0
$127$		−	16.1406i		−	0.127091i	−0.997979	−	0.0635456i	$-0.979759\pi$
$127$		−	16.1406i		−	0.127091i	0.997979	−	0.0635456i	$-0.0202408\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			71.5271i			0.546008i	0.962013	+	0.273004i	$0.0880172\pi$
$131$			71.5271i			0.546008i	−0.962013	+	0.273004i	$0.911983\pi$
$132$		0			0
$133$	−21.1251	+	227.926i	−0.158836	+	1.71373i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	71.2984i		−	0.520426i	−0.965551	−	0.260213i	$-0.916207\pi$
$137$		−	71.2984i		−	0.520426i	0.965551	−	0.260213i	$-0.0837928\pi$
$138$		0			0
$139$	−193.103			−1.38923			−0.694615	−	0.719381i	$-0.744425\pi$
$139$	−193.103			−1.38923			−0.694615	+	0.719381i	$0.744425\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			104.682i			0.732045i
$144$		0			0
$145$			166.782i			1.15022i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	7.33327i		−	0.0492166i	−0.999697	−	0.0246083i	$-0.992166\pi$
$149$		−	7.33327i		−	0.0492166i	0.999697	−	0.0246083i	$-0.00783385\pi$
$150$		0			0
$151$		−	121.565i		−	0.805064i	−0.915406	−	0.402532i	$-0.868130\pi$
$151$		−	121.565i		−	0.805064i	0.915406	−	0.402532i	$-0.131870\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−217.695			−1.40449
$156$		0			0
$157$		−	23.9122i		−	0.152307i	−0.997096	−	0.0761535i	$-0.975736\pi$
$157$		−	23.9122i		−	0.152307i	0.997096	−	0.0761535i	$-0.0242639\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	10.8845	−	117.437i	0.0676057	−	0.729420i
$162$		0			0
$163$			92.4900i			0.567423i	0.958910	+	0.283712i	$0.0915659\pi$
$163$			92.4900i			0.567423i	−0.958910	+	0.283712i	$0.908434\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			85.9811i			0.514857i	0.966297	+	0.257428i	$0.0828751\pi$
$167$			85.9811i			0.514857i	−0.966297	+	0.257428i	$0.917125\pi$
$168$		0			0
$169$	−12.7497			−0.0754422
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−110.721			−0.640007			−0.320004	−	0.947416i	$-0.603684\pi$
$173$	−110.721			−0.640007			−0.320004	+	0.947416i	$0.603684\pi$
$174$		0			0
$175$	−3.95697	+	42.6930i	−0.0226112	+	0.243960i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−134.658			−0.752281			−0.376141	−	0.926563i	$-0.622749\pi$
$179$	−134.658			−0.752281			−0.376141	+	0.926563i	$0.622749\pi$
$180$		0			0
$181$		−	177.051i		−	0.978181i	−0.872233	−	0.489090i	$-0.837329\pi$
$181$		−	177.051i		−	0.978181i	0.872233	−	0.489090i	$-0.162671\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	170.155			0.919754
$186$		0			0
$187$	224.673			1.20146
$188$		0			0
$189$		0			0
$190$		0			0
$191$	343.530			1.79859			0.899293	−	0.437346i	$-0.144081\pi$
$191$	343.530			1.79859			0.899293	+	0.437346i	$0.144081\pi$
$192$		0			0
$193$	−175.536			−0.909514			−0.454757	−	0.890616i	$-0.650274\pi$
$193$	−175.536			−0.909514			−0.454757	+	0.890616i	$0.650274\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	59.3117i		−	0.301075i	−0.988604	−	0.150537i	$-0.951900\pi$
$197$		−	59.3117i		−	0.301075i	0.988604	−	0.150537i	$-0.0481004\pi$
$198$		0			0
$199$	172.077			0.864708			0.432354	−	0.901704i	$-0.357683\pi$
$199$	172.077			0.864708			0.432354	+	0.901704i	$0.357683\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−267.577	−	24.8002i	−1.31811	−	0.122168i
$204$		0			0
$205$	273.786			1.33554
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−253.916			−1.21491
$210$		0			0
$211$			181.084i			0.858218i	0.903253	+	0.429109i	$0.141172\pi$
$211$			181.084i			0.858218i	−0.903253	+	0.429109i	$0.858828\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			30.4640i			0.141693i
$216$		0			0
$217$	32.3708	−	349.259i	0.149174	−	1.60949i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			390.078i			1.76506i
$222$		0			0
$223$	−142.430			−0.638699			−0.319350	−	0.947637i	$-0.603464\pi$
$223$	−142.430			−0.638699			−0.319350	+	0.947637i	$0.603464\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	130.464i		−	0.574731i	−0.957821	−	0.287366i	$-0.907220\pi$
$227$		−	130.464i		−	0.574731i	0.957821	−	0.287366i	$-0.0927795\pi$
$228$		0			0
$229$		−	130.393i		−	0.569402i	−0.958616	−	0.284701i	$-0.908106\pi$
$229$		−	130.393i		−	0.569402i	0.958616	−	0.284701i	$-0.0918943\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			333.516i			1.43140i	0.698409	+	0.715699i	$0.253892\pi$
$233$			333.516i			1.43140i	−0.698409	+	0.715699i	$0.746108\pi$
$234$		0			0
$235$			101.899i			0.433611i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	7.02757			0.0294040			0.0147020	−	0.999892i	$-0.495320\pi$
$239$	7.02757			0.0294040			0.0147020	+	0.999892i	$0.495320\pi$
$240$		0			0
$241$			63.5726i			0.263787i	0.991264	+	0.131893i	$0.0421057\pi$
$241$			63.5726i			0.263787i	−0.991264	+	0.131893i	$0.957894\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	209.255	+	39.1254i	0.854103	+	0.159696i
$246$		0			0
$247$		−	440.849i		−	1.78481i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	354.382i		−	1.41188i	−0.708272	−	0.705940i	$-0.750525\pi$
$251$		−	354.382i		−	1.41188i	0.708272	−	0.705940i	$-0.249475\pi$
$252$		0			0
$253$	130.828			0.517105
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−84.5204			−0.328873			−0.164437	−	0.986388i	$-0.552581\pi$
$257$	−84.5204			−0.328873			−0.164437	+	0.986388i	$0.552581\pi$
$258$		0			0
$259$	−25.3016	+	272.987i	−0.0976896	+	1.05400i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	68.3572			0.259913			0.129957	−	0.991520i	$-0.458516\pi$
$263$	68.3572			0.259913			0.129957	+	0.991520i	$0.458516\pi$
$264$		0			0
$265$			395.129i			1.49105i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−81.6067			−0.303371			−0.151685	−	0.988429i	$-0.548470\pi$
$269$	−81.6067			−0.303371			−0.151685	+	0.988429i	$0.548470\pi$
$270$		0			0
$271$	−394.777			−1.45674			−0.728371	−	0.685183i	$-0.759722\pi$
$271$	−394.777			−1.45674			−0.728371	+	0.685183i	$0.759722\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−47.5612			−0.172950
$276$		0			0
$277$	266.537			0.962228			0.481114	−	0.876658i	$-0.340232\pi$
$277$	266.537			0.962228			0.481114	+	0.876658i	$0.340232\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			347.356i			1.23614i	0.786123	+	0.618071i	$0.212085\pi$
$281$			347.356i			1.23614i	−0.786123	+	0.618071i	$0.787915\pi$
$282$		0			0
$283$	342.554			1.21044			0.605220	−	0.796059i	$-0.293085\pi$
$283$	342.554			1.21044			0.605220	+	0.796059i	$0.293085\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−40.7114	+	439.249i	−0.141852	+	1.53048i
$288$		0			0
$289$	548.198			1.89688
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−84.3867			−0.288009			−0.144005	−	0.989577i	$-0.545998\pi$
$293$	−84.3867			−0.288009			−0.144005	+	0.989577i	$0.545998\pi$
$294$		0			0
$295$			339.621i			1.15126i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			227.143i			0.759676i
$300$		0			0
$301$	−48.8749	−	4.52993i	−0.162375	−	0.0150496i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			216.901i			0.711150i
$306$		0			0
$307$	397.950			1.29626			0.648128	−	0.761532i	$-0.275552\pi$
$307$	397.950			1.29626			0.648128	+	0.761532i	$0.275552\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	188.336i		−	0.605583i	−0.953057	−	0.302792i	$-0.902081\pi$
$311$		−	188.336i		−	0.605583i	0.953057	−	0.302792i	$-0.0979186\pi$
$312$		0			0
$313$		−	346.942i		−	1.10844i	−0.832370	−	0.554220i	$-0.813017\pi$
$313$		−	346.942i		−	1.10844i	0.832370	−	0.554220i	$-0.186983\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			486.216i			1.53380i	0.641764	+	0.766902i	$0.278203\pi$
$317$			486.216i			1.53380i	−0.641764	+	0.766902i	$0.721797\pi$
$318$		0			0
$319$		−	298.089i		−	0.934447i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−946.166			−2.92931
$324$		0			0
$325$		−	82.5758i		−	0.254079i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−163.481	−	15.1521i	−0.496902	−	0.0460550i
$330$		0			0
$331$			47.1991i			0.142595i	0.997455	+	0.0712977i	$0.0227140\pi$
$331$			47.1991i			0.142595i	−0.997455	+	0.0712977i	$0.977286\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			68.6660i			0.204973i
$336$		0			0
$337$	−215.332			−0.638967			−0.319483	−	0.947592i	$-0.603509\pi$
$337$	−215.332			−0.638967			−0.319483	+	0.947592i	$0.603509\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	389.084			1.14101
$342$		0			0
$343$	−93.8866	+	329.900i	−0.273722	+	0.961809i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−190.548			−0.549131			−0.274566	−	0.961568i	$-0.588534\pi$
$347$	−190.548			−0.549131			−0.274566	+	0.961568i	$0.588534\pi$
$348$		0			0
$349$			123.762i			0.354620i	0.984155	+	0.177310i	$0.0567395\pi$
$349$			123.762i			0.354620i	−0.984155	+	0.177310i	$0.943260\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	399.030			1.13040			0.565198	−	0.824955i	$-0.308800\pi$
$353$	399.030			1.13040			0.565198	+	0.824955i	$0.308800\pi$
$354$		0			0
$355$	−269.916			−0.760328
$356$		0			0
$357$		0			0
$358$		0			0
$359$	493.736			1.37531			0.687654	−	0.726039i	$-0.258641\pi$
$359$	493.736			1.37531			0.687654	+	0.726039i	$0.258641\pi$
$360$		0			0
$361$	708.315			1.96209
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	259.683i		−	0.711460i
$366$		0			0
$367$	−592.907			−1.61555			−0.807775	−	0.589491i	$-0.799328\pi$
$367$	−592.907			−1.61555			−0.807775	+	0.589491i	$0.799328\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−633.925	−	58.7549i	−1.70869	−	0.158369i
$372$		0			0
$373$	−73.4924			−0.197031			−0.0985153	−	0.995136i	$-0.531409\pi$
$373$	−73.4924			−0.197031			−0.0985153	+	0.995136i	$0.531409\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	517.542			1.37279
$378$		0			0
$379$			559.383i			1.47594i	0.674831	+	0.737972i	$0.264217\pi$
$379$			559.383i			1.47594i	−0.674831	+	0.737972i	$0.735783\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	521.424i		−	1.36142i	−0.732552	−	0.680711i	$-0.761671\pi$
$383$		−	521.424i		−	1.36142i	0.732552	−	0.680711i	$-0.238329\pi$
$384$		0			0
$385$	−21.7934	+	235.136i	−0.0566063	+	0.610743i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			173.567i			0.446189i	0.974797	+	0.223094i	$0.0716158\pi$
$389$			173.567i			0.446189i	−0.974797	+	0.223094i	$0.928384\pi$
$390$		0			0
$391$	487.503			1.24681
$392$		0			0
$393$		0			0
$394$		0			0
$395$			596.442i			1.50998i
$396$		0			0
$397$			655.243i			1.65049i	0.564778	+	0.825243i	$0.308962\pi$
$397$			655.243i			1.65049i	−0.564778	+	0.825243i	$0.691038\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			760.249i			1.89588i	0.318444	+	0.947942i	$0.396840\pi$
$401$			760.249i			1.89588i	−0.318444	+	0.947942i	$0.603160\pi$
$402$		0			0
$403$			675.529i			1.67625i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−304.115			−0.747212
$408$		0			0
$409$			158.456i			0.387423i	0.981059	+	0.193711i	$0.0620526\pi$
$409$			158.456i			0.387423i	−0.981059	+	0.193711i	$0.937947\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−544.871	−	50.5009i	−1.31930	−	0.122278i
$414$		0			0
$415$		−	446.575i		−	1.07608i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			599.526i			1.43085i	0.698690	+	0.715425i	$0.253767\pi$
$419$			599.526i			1.43085i	−0.698690	+	0.715425i	$0.746233\pi$
$420$		0			0
$421$	56.9482			0.135269			0.0676344	−	0.997710i	$-0.478455\pi$
$421$	56.9482			0.135269			0.0676344	+	0.997710i	$0.478455\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−177.227			−0.417005
$426$		0			0
$427$	−347.984	−	32.2527i	−0.814951	−	0.0755331i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	584.112			1.35525			0.677625	−	0.735408i	$-0.263009\pi$
$431$	584.112			1.35525			0.677625	+	0.735408i	$0.263009\pi$
$432$		0			0
$433$		−	285.740i		−	0.659909i	−0.943997	−	0.329954i	$-0.892967\pi$
$433$		−	285.740i		−	0.659909i	0.943997	−	0.329954i	$-0.107033\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−550.954			−1.26076
$438$		0			0
$439$	813.164			1.85231			0.926155	−	0.377142i	$-0.123093\pi$
$439$	813.164			1.85231			0.926155	+	0.377142i	$0.123093\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	482.275			1.08866			0.544328	−	0.838872i	$-0.316785\pi$
$443$	482.275			1.08866			0.544328	+	0.838872i	$0.316785\pi$
$444$		0			0
$445$	−115.789			−0.260199
$446$		0			0
$447$		0			0
$448$		0			0
$449$			67.5218i			0.150383i	0.997169	+	0.0751913i	$0.0239568\pi$
$449$			67.5218i			0.150383i	−0.997169	+	0.0751913i	$0.976043\pi$
$450$		0			0
$451$	−489.336			−1.08500
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−408.244	−	37.8378i	−0.897239	−	0.0831599i
$456$		0			0
$457$	198.590			0.434552			0.217276	−	0.976110i	$-0.430283\pi$
$457$	198.590			0.434552			0.217276	+	0.976110i	$0.430283\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	263.672			0.571957			0.285979	−	0.958236i	$-0.407681\pi$
$461$	263.672			0.571957			0.285979	+	0.958236i	$0.407681\pi$
$462$		0			0
$463$		−	372.637i		−	0.804831i	−0.915457	−	0.402415i	$-0.868171\pi$
$463$		−	372.637i		−	0.804831i	0.915457	−	0.402415i	$-0.131829\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	801.489i		−	1.71625i	−0.513441	−	0.858125i	$-0.671629\pi$
$467$		−	801.489i		−	1.71625i	0.513441	−	0.858125i	$-0.328371\pi$
$468$		0			0
$469$	−110.164	−	10.2105i	−0.234892	−	0.0217708i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	54.4480i		−	0.115112i
$474$		0			0
$475$	200.294			0.421672
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	127.437i		−	0.266047i	−0.991113	−	0.133024i	$-0.957531\pi$
$479$		−	127.437i		−	0.266047i	0.991113	−	0.133024i	$-0.0424686\pi$
$480$		0			0
$481$		−	528.006i		−	1.09772i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			459.202i			0.946809i
$486$		0			0
$487$		−	290.901i		−	0.597332i	−0.954358	−	0.298666i	$-0.903458\pi$
$487$		−	290.901i		−	0.597332i	0.954358	−	0.298666i	$-0.0965417\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−306.900			−0.625052			−0.312526	−	0.949909i	$-0.601175\pi$
$491$	−306.900			−0.625052			−0.312526	+	0.949909i	$0.601175\pi$
$492$		0			0
$493$		−	1110.77i		−	2.25308i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	40.1360	−	433.040i	0.0807565	−	0.871308i
$498$		0			0
$499$		−	770.288i		−	1.54366i	−0.635827	−	0.771832i	$-0.719341\pi$
$499$		−	770.288i		−	1.54366i	0.635827	−	0.771832i	$-0.280659\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	698.797i		−	1.38926i	−0.719368	−	0.694630i	$-0.755568\pi$
$503$		−	698.797i		−	1.38926i	0.719368	−	0.694630i	$-0.244432\pi$
$504$		0			0
$505$	415.449			0.822671
$506$		0			0
$507$		0			0
$508$		0			0
$509$	645.281			1.26774			0.633871	−	0.773439i	$-0.281465\pi$
$509$	645.281			1.26774			0.633871	+	0.773439i	$0.281465\pi$
$510$		0			0
$511$	416.622	+	38.6143i	0.815307	+	0.0755661i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	0.919766			0.00178595
$516$		0			0
$517$		−	182.122i		−	0.352268i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−862.570			−1.65560			−0.827802	−	0.561020i	$-0.810409\pi$
$521$	−862.570			−1.65560			−0.827802	+	0.561020i	$0.810409\pi$
$522$		0			0
$523$	147.952			0.282890			0.141445	−	0.989946i	$-0.454825\pi$
$523$	147.952			0.282890			0.141445	+	0.989946i	$0.454825\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1449.84			2.75113
$528$		0			0
$529$	−245.126			−0.463377
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	849.585i		−	1.59397i
$534$		0			0
$535$	−527.394			−0.985783
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−374.000	−	69.9285i	−0.693877	−	0.129737i
$540$		0			0
$541$	483.118			0.893009			0.446505	−	0.894781i	$-0.352669\pi$
$541$	483.118			0.893009			0.446505	+	0.894781i	$0.352669\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	122.908			0.225520
$546$		0			0
$547$			825.387i			1.50893i	0.656338	+	0.754467i	$0.272104\pi$
$547$			825.387i			1.50893i	−0.656338	+	0.754467i	$0.727896\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			1255.34i			2.27829i
$552$		0			0
$553$	−956.901	−	88.6897i	−1.73038	−	0.160379i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	233.711i		−	0.419589i	−0.977745	−	0.209795i	$-0.932720\pi$
$557$		−	233.711i		−	0.419589i	0.977745	−	0.209795i	$-0.0672795\pi$
$558$		0			0
$559$	94.5326			0.169110
$560$		0			0
$561$		0			0
$562$		0			0
$563$			510.026i			0.905908i	0.891534	+	0.452954i	$0.149630\pi$
$563$			510.026i			0.905908i	−0.891534	+	0.452954i	$0.850370\pi$
$564$		0			0
$565$			654.073i			1.15765i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	129.208i		−	0.227079i	−0.993533	−	0.113539i	$-0.963781\pi$
$569$		−	129.208i		−	0.227079i	0.993533	−	0.113539i	$-0.0362188\pi$
$570$		0			0
$571$		−	742.862i		−	1.30098i	−0.759513	−	0.650492i	$-0.774563\pi$
$571$		−	742.862i		−	1.30098i	0.759513	−	0.650492i	$-0.225437\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−103.200			−0.179478
$576$		0			0
$577$			549.384i			0.952138i	0.879408	+	0.476069i	$0.157939\pi$
$577$			549.384i			0.952138i	−0.879408	+	0.476069i	$0.842061\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	716.462	+	66.4047i	1.23315	+	0.114294i
$582$		0			0
$583$		−	706.210i		−	1.21134i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	697.804i		−	1.18876i	−0.804183	−	0.594381i	$-0.797397\pi$
$587$		−	697.804i		−	1.18876i	0.804183	−	0.594381i	$-0.202603\pi$
$588$		0			0
$589$	−1638.55			−2.78192
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−631.537			−1.06499			−0.532493	−	0.846434i	$-0.678745\pi$
$593$	−631.537			−1.06499			−0.532493	+	0.846434i	$0.678745\pi$
$594$		0			0
$595$	−81.2087	+	876.187i	−0.136485	+	1.47258i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	105.663			0.176400			0.0881998	−	0.996103i	$-0.471889\pi$
$599$	105.663			0.176400			0.0881998	+	0.996103i	$0.471889\pi$
$600$		0			0
$601$			988.391i			1.64458i	0.569071	+	0.822288i	$0.307303\pi$
$601$			988.391i			1.64458i	−0.569071	+	0.822288i	$0.692697\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	263.739			0.435932
$606$		0			0
$607$	164.700			0.271334			0.135667	−	0.990754i	$-0.456682\pi$
$607$	164.700			0.271334			0.135667	+	0.990754i	$0.456682\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	316.201			0.517514
$612$		0			0
$613$	413.251			0.674146			0.337073	−	0.941479i	$-0.390563\pi$
$613$	413.251			0.674146			0.337073	+	0.941479i	$0.390563\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	1093.33i		−	1.77201i	−0.463677	−	0.886005i	$-0.653470\pi$
$617$		−	1093.33i		−	1.77201i	0.463677	−	0.886005i	$-0.346530\pi$
$618$		0			0
$619$	75.5055			0.121980			0.0609899	−	0.998138i	$-0.480574\pi$
$619$	75.5055			0.121980			0.0609899	+	0.998138i	$0.480574\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	17.2175	−	185.765i	0.0276365	−	0.298179i
$624$		0			0
$625$	−434.354			−0.694967
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−1133.22			−1.80163
$630$		0			0
$631$			808.303i			1.28099i	0.767963	+	0.640494i	$0.221270\pi$
$631$			808.303i			1.28099i	−0.767963	+	0.640494i	$0.778730\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			70.1231i			0.110430i
$636$		0			0
$637$	121.410	−	649.339i	0.190596	−	1.01937i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	412.768i		−	0.643944i	−0.946749	−	0.321972i	$-0.895654\pi$
$641$		−	412.768i		−	0.643944i	0.946749	−	0.321972i	$-0.104346\pi$
$642$		0			0
$643$	−256.621			−0.399099			−0.199550	−	0.979888i	$-0.563948\pi$
$643$	−256.621			−0.399099			−0.199550	+	0.979888i	$0.563948\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	113.389i		−	0.175254i	−0.996153	−	0.0876271i	$-0.972072\pi$
$647$		−	113.389i		−	0.175254i	0.996153	−	0.0876271i	$-0.0279284\pi$
$648$		0			0
$649$		−	607.001i		−	0.935287i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	524.029i		−	0.802495i	−0.915970	−	0.401247i	$-0.868577\pi$
$653$		−	524.029i		−	0.802495i	0.915970	−	0.401247i	$-0.131423\pi$
$654$		0			0
$655$		−	310.751i		−	0.474429i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−569.919			−0.864823			−0.432412	−	0.901676i	$-0.642337\pi$
$659$	−569.919			−0.864823			−0.432412	+	0.901676i	$0.642337\pi$
$660$		0			0
$661$			478.373i			0.723712i	0.932234	+	0.361856i	$0.117857\pi$
$661$			478.373i			0.723712i	−0.932234	+	0.361856i	$0.882143\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	91.7786	−	990.229i	0.138013	−	1.48907i
$666$		0			0
$667$		−	646.802i		−	0.969718i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	387.664i		−	0.577741i
$672$		0			0
$673$	−137.786			−0.204735			−0.102367	−	0.994747i	$-0.532642\pi$
$673$	−137.786			−0.204735			−0.102367	+	0.994747i	$0.532642\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−612.441			−0.904639			−0.452319	−	0.891856i	$-0.649403\pi$
$677$	−612.441			−0.904639			−0.452319	+	0.891856i	$0.649403\pi$
$678$		0			0
$679$	−736.720	−	68.2824i	−1.08501	−	0.100563i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	823.775			1.20611			0.603056	−	0.797699i	$-0.293950\pi$
$683$	823.775			1.20611			0.603056	+	0.797699i	$0.293950\pi$
$684$		0			0
$685$			309.757i			0.452201i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1226.12			1.77957
$690$		0			0
$691$	−570.994			−0.826330			−0.413165	−	0.910656i	$-0.635577\pi$
$691$	−570.994			−0.826330			−0.413165	+	0.910656i	$0.635577\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	838.941			1.20711
$696$		0			0
$697$	−1823.41			−2.61608
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	516.520i		−	0.736833i	−0.929661	−	0.368417i	$-0.879900\pi$
$701$		−	516.520i		−	0.736833i	0.929661	−	0.368417i	$-0.120100\pi$
$702$		0			0
$703$	1280.72			1.82179
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−61.7763	+	666.525i	−0.0873781	+	0.942750i
$708$		0			0
$709$	529.200			0.746403			0.373202	−	0.927750i	$-0.378260\pi$
$709$	529.200			0.746403			0.373202	+	0.927750i	$0.378260\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	844.247			1.18408
$714$		0			0
$715$		−	454.795i		−	0.636077i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	29.1058i		−	0.0404810i	−0.999795	−	0.0202405i	$-0.993557\pi$
$719$		−	29.1058i		−	0.0404810i	0.999795	−	0.0202405i	$-0.00644319\pi$
$720$		0			0
$721$	−0.136767	+	1.47562i	−0.000189691	+	0.00204664i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			235.139i			0.324329i
$726$		0			0
$727$	−914.369			−1.25773			−0.628864	−	0.777515i	$-0.716480\pi$
$727$	−914.369			−1.25773			−0.628864	+	0.777515i	$0.716480\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	202.889i		−	0.277550i
$732$		0			0
$733$			995.630i			1.35830i	0.734002	+	0.679148i	$0.237650\pi$
$733$			995.630i			1.35830i	−0.734002	+	0.679148i	$0.762350\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	122.726i		−	0.166521i
$738$		0			0
$739$		−	407.011i		−	0.550760i	−0.961336	−	0.275380i	$-0.911196\pi$
$739$		−	407.011i		−	0.550760i	0.961336	−	0.275380i	$-0.0888036\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−645.381			−0.868615			−0.434308	−	0.900765i	$-0.643007\pi$
$743$	−645.381			−0.868615			−0.434308	+	0.900765i	$0.643007\pi$
$744$		0			0
$745$			31.8595i			0.0427645i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	78.4224	−	846.124i	0.104703	−	1.12967i
$750$		0			0
$751$		−	883.405i		−	1.17630i	−0.808750	−	0.588152i	$-0.799855\pi$
$751$		−	883.405i		−	1.17630i	0.808750	−	0.588152i	$-0.200145\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			528.140i			0.699524i
$756$		0			0
$757$	−651.461			−0.860583			−0.430292	−	0.902690i	$-0.641589\pi$
$757$	−651.461			−0.860583			−0.430292	+	0.902690i	$0.641589\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−137.234			−0.180333			−0.0901666	−	0.995927i	$-0.528740\pi$
$761$	−137.234			−0.180333			−0.0901666	+	0.995927i	$0.528740\pi$
$762$		0			0
$763$	−18.2762	+	197.188i	−0.0239531	+	0.258438i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1053.88			1.37402
$768$		0			0
$769$		−	705.493i		−	0.917416i	−0.888587	−	0.458708i	$-0.848312\pi$
$769$		−	705.493i		−	0.917416i	0.888587	−	0.458708i	$-0.151688\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	866.374			1.12079			0.560397	−	0.828224i	$-0.310648\pi$
$773$	866.374			1.12079			0.560397	+	0.828224i	$0.310648\pi$
$774$		0			0
$775$	−306.918			−0.396023
$776$		0			0
$777$		0			0
$778$		0			0
$779$	2060.74			2.64536
$780$		0			0
$781$	482.419			0.617694
$782$		0			0
$783$		0			0
$784$		0			0
$785$			103.887i			0.132340i
$786$		0			0
$787$	1027.15			1.30515			0.652575	−	0.757724i	$-0.273689\pi$
$787$	1027.15			1.30515			0.652575	+	0.757724i	$0.273689\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−1049.36	−	97.2592i	−1.32663	−	0.122957i
$792$		0			0
$793$	673.063			0.848755
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−611.097			−0.766747			−0.383373	−	0.923593i	$-0.625238\pi$
$797$	−611.097			−0.766747			−0.383373	+	0.923593i	$0.625238\pi$
$798$		0			0
$799$		−	678.641i		−	0.849364i
$800$		0			0
$801$		0			0
$802$		0			0
$803$			464.129i			0.577993i
$804$		0			0
$805$	−47.2880	+	510.206i	−0.0587429	+	0.633796i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	521.424i		−	0.644529i	−0.946650	−	0.322265i	$-0.895556\pi$
$809$		−	521.424i		−	0.644529i	0.946650	−	0.322265i	$-0.104444\pi$
$810$		0			0
$811$	−112.014			−0.138119			−0.0690594	−	0.997613i	$-0.522000\pi$
$811$	−112.014			−0.138119			−0.0690594	+	0.997613i	$0.522000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	401.825i		−	0.493037i
$816$		0			0
$817$			229.297i			0.280657i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			770.242i			0.938175i	0.883152	+	0.469088i	$0.155417\pi$
$821$			770.242i			0.938175i	−0.883152	+	0.469088i	$0.844583\pi$
$822$		0			0
$823$			135.515i			0.164660i	0.996605	+	0.0823298i	$0.0262361\pi$
$823$			135.515i			0.164660i	−0.996605	+	0.0823298i	$0.973764\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	428.008			0.517543			0.258771	−	0.965939i	$-0.416682\pi$
$827$	428.008			0.517543			0.258771	+	0.965939i	$0.416682\pi$
$828$		0			0
$829$		−	432.136i		−	0.521274i	−0.965437	−	0.260637i	$-0.916067\pi$
$829$		−	432.136i		−	0.521274i	0.965437	−	0.260637i	$-0.0839326\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1393.63	−	260.574i	−1.67303	−	0.312814i
$834$		0			0
$835$		−	373.547i		−	0.447361i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			383.809i			0.457460i	0.973490	+	0.228730i	$0.0734574\pi$
$839$			383.809i			0.457460i	−0.973490	+	0.228730i	$0.926543\pi$
$840$		0			0
$841$	−632.726			−0.752350
$842$		0			0
$843$		0			0
$844$		0			0
$845$	55.3915			0.0655520
$846$		0			0
$847$	−39.2174	+	423.129i	−0.0463015	+	0.499562i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−659.879			−0.775416
$852$		0			0
$853$		−	777.198i		−	0.911134i	−0.890201	−	0.455567i	$-0.849436\pi$
$853$		−	777.198i		−	0.911134i	0.890201	−	0.455567i	$-0.150564\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1261.11			−1.47154			−0.735769	−	0.677233i	$-0.763179\pi$
$857$	−1261.11			−1.47154			−0.735769	+	0.677233i	$0.763179\pi$
$858$		0			0
$859$	−102.095			−0.118854			−0.0594269	−	0.998233i	$-0.518927\pi$
$859$	−102.095			−0.118854			−0.0594269	+	0.998233i	$0.518927\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1073.55			1.24398			0.621989	−	0.783026i	$-0.286325\pi$
$863$	1073.55			1.24398			0.621989	+	0.783026i	$0.286325\pi$
$864$		0			0
$865$	481.031			0.556105
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	1066.01i		−	1.22671i
$870$		0			0
$871$	213.077			0.244635
$872$		0			0
$873$		0			0
$874$		0			0
$875$	87.3574	−	942.527i	0.0998370	−	1.07717i
$876$		0			0
$877$	−1408.87			−1.60647			−0.803234	−	0.595664i	$-0.796889\pi$
$877$	−1408.87			−1.60647			−0.803234	+	0.595664i	$0.796889\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−93.4281			−0.106048			−0.0530239	−	0.998593i	$-0.516886\pi$
$881$	−93.4281			−0.106048			−0.0530239	+	0.998593i	$0.516886\pi$
$882$		0			0
$883$			317.988i			0.360122i	0.983655	+	0.180061i	$0.0576295\pi$
$883$			317.988i			0.360122i	−0.983655	+	0.180061i	$0.942371\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	1129.36i		−	1.27324i	−0.771179	−	0.636618i	$-0.780333\pi$
$887$		−	1129.36i		−	1.27324i	0.771179	−	0.636618i	$-0.219667\pi$
$888$		0			0
$889$	−112.502	−	10.4272i	−0.126549	−	0.0117291i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			766.971i			0.858870i
$894$		0			0
$895$	585.026			0.653660
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	1923.60i		−	2.13971i
$900$		0			0
$901$		−	2631.55i		−	2.92070i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			769.201i			0.849945i
$906$		0			0
$907$		−	643.806i		−	0.709819i	−0.934901	−	0.354910i	$-0.884512\pi$
$907$		−	643.806i		−	0.709819i	0.934901	−	0.354910i	$-0.115488\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1000.01			−1.09770			−0.548852	−	0.835920i	$-0.684935\pi$
$911$	−1000.01			−1.09770			−0.548852	+	0.835920i	$0.684935\pi$
$912$		0			0
$913$			798.159i			0.874215i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	498.553	+	46.2080i	0.543678	+	0.0503904i
$918$		0			0
$919$			1537.61i			1.67314i	0.547861	+	0.836569i	$0.315442\pi$
$919$			1537.61i			1.67314i	−0.547861	+	0.836569i	$0.684558\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			837.576i			0.907450i
$924$		0			0
$925$	239.893			0.259344
$926$		0			0
$927$		0			0
$928$		0			0
$929$	576.410			0.620463			0.310232	−	0.950661i	$-0.399593\pi$
$929$	576.410			0.620463			0.310232	+	0.950661i	$0.399593\pi$
$930$		0			0
$931$	1575.02	+	294.490i	1.69176	+	0.316315i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−976.097			−1.04395
$936$		0			0
$937$		−	398.967i		−	0.425792i	−0.977075	−	0.212896i	$-0.931710\pi$
$937$		−	398.967i		−	0.425792i	0.977075	−	0.212896i	$-0.0682896\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1021.05			−1.08507			−0.542533	−	0.840035i	$-0.682535\pi$
$941$	−1021.05			−1.08507			−0.542533	+	0.840035i	$0.682535\pi$
$942$		0			0
$943$	−1061.78			−1.12595
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−216.473			−0.228588			−0.114294	−	0.993447i	$-0.536461\pi$
$947$	−216.473			−0.228588			−0.114294	+	0.993447i	$0.536461\pi$
$948$		0			0
$949$	−805.820			−0.849126
$950$		0			0
$951$		0			0
$952$		0			0
$953$			989.112i			1.03789i	0.854807	+	0.518946i	$0.173676\pi$
$953$			989.112i			1.03789i	−0.854807	+	0.518946i	$0.826324\pi$
$954$		0			0
$955$	−1492.47			−1.56280
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−496.959	−	46.0603i	−0.518205	−	0.0480295i
$960$		0			0
$961$	1549.81			1.61271
$962$		0			0
$963$		0			0
$964$		0			0
$965$	762.620			0.790280
$966$		0			0
$967$			1338.63i			1.38431i	0.721748	+	0.692156i	$0.243339\pi$
$967$			1338.63i			1.38431i	−0.721748	+	0.692156i	$0.756661\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			571.990i			0.589074i	0.955640	+	0.294537i	$0.0951654\pi$
$971$			571.990i			0.589074i	−0.955640	+	0.294537i	$0.904835\pi$
$972$		0			0
$973$	−124.749	+	1345.95i	−0.128210	+	1.38330i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	637.338i		−	0.652342i	−0.945311	−	0.326171i	$-0.894242\pi$
$977$		−	637.338i		−	0.652342i	0.945311	−	0.326171i	$-0.105758\pi$
$978$		0			0
$979$	206.948			0.211387
$980$		0			0
$981$		0			0
$982$		0			0
$983$			1751.74i			1.78203i	0.453970	+	0.891017i	$0.350007\pi$
$983$			1751.74i			1.78203i	−0.453970	+	0.891017i	$0.649993\pi$
$984$		0			0
$985$			257.681i			0.261605i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	118.143i		−	0.119457i
$990$		0			0
$991$			622.310i			0.627962i	0.949429	+	0.313981i	$0.101663\pi$
$991$			622.310i			0.627962i	−0.949429	+	0.313981i	$0.898337\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−747.592			−0.751349
$996$		0			0
$997$			976.898i			0.979838i	0.871768	+	0.489919i	$0.162974\pi$
$997$			976.898i			0.979838i	−0.871768	+	0.489919i	$0.837026\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.3.o.b.1007.6	yes	24
3.2	odd	2	inner	1008.3.o.b.1007.18	yes	24
4.3	odd	2	inner	1008.3.o.b.1007.8	yes	24
7.6	odd	2	inner	1008.3.o.b.1007.17	yes	24
12.11	even	2	inner	1008.3.o.b.1007.20	yes	24
21.20	even	2	inner	1008.3.o.b.1007.5	✓	24
28.27	even	2	inner	1008.3.o.b.1007.19	yes	24
84.83	odd	2	inner	1008.3.o.b.1007.7	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.3.o.b.1007.5	✓	24	21.20	even	2	inner
1008.3.o.b.1007.6	yes	24	1.1	even	1	trivial
1008.3.o.b.1007.7	yes	24	84.83	odd	2	inner
1008.3.o.b.1007.8	yes	24	4.3	odd	2	inner
1008.3.o.b.1007.17	yes	24	7.6	odd	2	inner
1008.3.o.b.1007.18	yes	24	3.2	odd	2	inner
1008.3.o.b.1007.19	yes	24	28.27	even	2	inner
1008.3.o.b.1007.20	yes	24	12.11	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1008.o (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.34452 q^{5} +(0.646021 - 6.97013i) q^{7} +7.76492 q^{11} +13.4815i q^{13} +28.9344 q^{17} -32.7004 q^{19} +16.8486 q^{23} -6.12514 q^{25} -38.3891i q^{29} +50.1080 q^{31} +(-2.80665 + 30.2819i) q^{35} -39.1653 q^{37} -63.0188 q^{41} -7.01205i q^{43} -23.4545i q^{47} +(-48.1653 - 9.00569i) q^{49} -90.9488i q^{53} -33.7349 q^{55} -78.1723i q^{59} -49.9251i q^{61} -58.5705i q^{65} -15.8052i q^{67} +62.1280 q^{71} +59.7725i q^{73} +(5.01630 - 54.1225i) q^{77} -137.286i q^{79} +102.790i q^{83} -125.706 q^{85} +26.6516 q^{89} +(93.9675 + 8.70930i) q^{91} +142.068 q^{95} -105.697i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 264 q^{25} - 96 q^{37} - 312 q^{49} - 96 q^{85}+O(q^{100}) \) 24 * q + 264 * q^25 - 96 * q^37 - 312 * q^49 - 96 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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