LMFDB - Embedded newform 1008.4.bt.d.593.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,4,Mod(17,1008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 3, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1008.17");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1008.bt (of order $6$, degree $2$, 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:	$59.4739252858$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 504)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			593.16
Character	$\chi$	$=$	1008.593
Dual form			1008.4.bt.d.17.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(3.30930 - 5.73188i) q^{5} +(4.31556 + 18.0104i) q^{7} +O(q^{10})$	$q+(3.30930 - 5.73188i) q^{5} +(4.31556 + 18.0104i) q^{7} +(1.54712 - 0.893230i) q^{11} -4.72915i q^{13} +(-23.8462 - 41.3029i) q^{17} +(40.1443 + 23.1773i) q^{19} +(30.2011 + 17.4366i) q^{23} +(40.5970 + 70.3161i) q^{25} -48.3459i q^{29} +(107.814 - 62.2462i) q^{31} +(117.515 + 34.8657i) q^{35} +(-137.933 + 238.908i) q^{37} -37.3352 q^{41} +215.883 q^{43} +(-53.1037 + 91.9783i) q^{47} +(-305.752 + 155.450i) q^{49} +(233.771 - 134.968i) q^{53} -11.8239i q^{55} +(149.592 + 259.101i) q^{59} +(-292.459 - 168.851i) q^{61} +(-27.1069 - 15.6502i) q^{65} +(188.686 + 326.815i) q^{67} +816.814i q^{71} +(-596.355 + 344.306i) q^{73} +(22.7641 + 24.0095i) q^{77} +(307.064 - 531.851i) q^{79} +1323.90 q^{83} -315.658 q^{85} +(480.368 - 832.022i) q^{89} +(85.1741 - 20.4089i) q^{91} +(265.699 - 153.401i) q^{95} +449.508i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 24 q^{7}+O(q^{10}) $ 48 * q + 24 * q^7	$ 48 q + 24 q^{7} - 540 q^{19} - 924 q^{25} - 648 q^{31} - 132 q^{37} + 792 q^{43} + 672 q^{49} + 12 q^{67} + 2412 q^{73} - 1680 q^{79} + 480 q^{85} - 1404 q^{91}+O(q^{100}) $ 48 * q + 24 * q^7 - 540 * q^19 - 924 * q^25 - 648 * q^31 - 132 * q^37 + 792 * q^43 + 672 * q^49 + 12 * q^67 + 2412 * q^73 - 1680 * q^79 + 480 * q^85 - 1404 * q^91

Display 100 coefficients 10 coefficients

Character values

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

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$1$	$e\left(\frac{5}{6}\right)$	$1$	$-1$

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
$5$	3.30930	−	5.73188i	0.295993	−	0.512675i	−0.679222	−	0.733932i	$-0.737683\pi$
$5$	3.30930	−	5.73188i	0.295993	−	0.512675i	0.975215	+	0.221258i	$0.0710162\pi$
$6$		0			0
$7$	4.31556	+	18.0104i	0.233019	+	0.972472i
$7$	4.31556	+	18.0104i	0.233019	+	0.972472i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.54712	−	0.893230i	0.0424067	−	0.0244835i	−0.478647	−	0.878008i	$-0.658873\pi$
$11$	1.54712	−	0.893230i	0.0424067	−	0.0244835i	0.521053	+	0.853524i	$0.325539\pi$
$12$		0			0
$13$		−	4.72915i		−	0.100895i	−0.998727	−	0.0504473i	$-0.983935\pi$
$13$		−	4.72915i		−	0.100895i	0.998727	−	0.0504473i	$-0.0160647\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−23.8462	−	41.3029i	−0.340210	−	0.589260i	0.644262	−	0.764805i	$-0.277165\pi$
$17$	−23.8462	−	41.3029i	−0.340210	−	0.589260i	−0.984471	+	0.175545i	$0.943831\pi$
$18$		0			0
$19$	40.1443	+	23.1773i	0.484722	+	0.279855i	0.722382	−	0.691494i	$-0.243047\pi$
$19$	40.1443	+	23.1773i	0.484722	+	0.279855i	−0.237660	+	0.971348i	$0.576380\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	30.2011	+	17.4366i	0.273798	+	0.158078i	0.630613	−	0.776098i	$-0.282804\pi$
$23$	30.2011	+	17.4366i	0.273798	+	0.158078i	−0.356814	+	0.934175i	$0.616137\pi$
$24$		0			0
$25$	40.5970	+	70.3161i	0.324776	+	0.562529i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	48.3459i		−	0.309573i	−0.987948	−	0.154786i	$-0.950531\pi$
$29$		−	48.3459i		−	0.309573i	0.987948	−	0.154786i	$-0.0494689\pi$
$30$		0			0
$31$	107.814	−	62.2462i	0.624642	−	0.360637i	−0.154032	−	0.988066i	$-0.549226\pi$
$31$	107.814	−	62.2462i	0.624642	−	0.360637i	0.778674	+	0.627429i	$0.215893\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	117.515	+	34.8657i	0.567534	+	0.168382i
$36$		0			0
$37$	−137.933	+	238.908i	−0.612868	+	1.06152i	0.377887	+	0.925852i	$0.376651\pi$
$37$	−137.933	+	238.908i	−0.612868	+	1.06152i	−0.990755	+	0.135667i	$0.956682\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−37.3352			−0.142214			−0.0711071	−	0.997469i	$-0.522653\pi$
$41$	−37.3352			−0.142214			−0.0711071	+	0.997469i	$0.522653\pi$
$42$		0			0
$43$	215.883			0.765625			0.382812	−	0.923826i	$-0.374956\pi$
$43$	215.883			0.765625			0.382812	+	0.923826i	$0.374956\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−53.1037	+	91.9783i	−0.164808	+	0.285456i	−0.936587	−	0.350435i	$-0.886034\pi$
$47$	−53.1037	+	91.9783i	−0.164808	+	0.285456i	0.771779	+	0.635891i	$0.219367\pi$
$48$		0			0
$49$	−305.752	+	155.450i	−0.891405	+	0.453208i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	233.771	−	134.968i	0.605867	−	0.349797i	−0.165479	−	0.986213i	$-0.552917\pi$
$53$	233.771	−	134.968i	0.605867	−	0.349797i	0.771346	+	0.636416i	$0.219584\pi$
$54$		0			0
$55$		−	11.8239i		−	0.0289878i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	149.592	+	259.101i	0.330088	+	0.571729i	0.982529	−	0.186110i	$-0.0595882\pi$
$59$	149.592	+	259.101i	0.330088	+	0.571729i	−0.652441	+	0.757840i	$0.726255\pi$
$60$		0			0
$61$	−292.459	−	168.851i	−0.613861	−	0.354413i	0.160614	−	0.987017i	$-0.448653\pi$
$61$	−292.459	−	168.851i	−0.613861	−	0.354413i	−0.774475	+	0.632604i	$0.781986\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−27.1069	−	15.6502i	−0.0517261	−	0.0298641i
$66$		0			0
$67$	188.686	+	326.815i	0.344055	+	0.595922i	0.985182	−	0.171514i	$-0.0548658\pi$
$67$	188.686	+	326.815i	0.344055	+	0.595922i	−0.641126	+	0.767435i	$0.721533\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			816.814i			1.36532i	0.730734	+	0.682662i	$0.239178\pi$
$71$			816.814i			1.36532i	−0.730734	+	0.682662i	$0.760822\pi$
$72$		0			0
$73$	−596.355	+	344.306i	−0.956139	+	0.552027i	−0.894982	−	0.446101i	$-0.852812\pi$
$73$	−596.355	+	344.306i	−0.956139	+	0.552027i	−0.0611561	+	0.998128i	$0.519479\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	22.7641	+	24.0095i	0.0336911	+	0.0355342i
$78$		0			0
$79$	307.064	−	531.851i	0.437309	−	0.757442i	−0.560172	−	0.828377i	$-0.689265\pi$
$79$	307.064	−	531.851i	0.437309	−	0.757442i	0.997481	+	0.0709347i	$0.0225982\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	1323.90			1.75081			0.875406	−	0.483389i	$-0.160594\pi$
$83$	1323.90			1.75081			0.875406	+	0.483389i	$0.160594\pi$
$84$		0			0
$85$	−315.658			−0.402798
$86$		0			0
$87$		0			0
$88$		0			0
$89$	480.368	−	832.022i	0.572123	−	0.990946i	−0.424225	−	0.905557i	$-0.639453\pi$
$89$	480.368	−	832.022i	0.572123	−	0.990946i	0.996348	−	0.0853887i	$-0.0272132\pi$
$90$		0			0
$91$	85.1741	−	20.4089i	0.0981172	−	0.0235103i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	265.699	−	153.401i	0.286949	−	0.165670i
$96$		0			0
$97$			449.508i			0.470522i	0.971932	+	0.235261i	$0.0755945\pi$
$97$			449.508i			0.470522i	−0.971932	+	0.235261i	$0.924406\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	178.518	+	309.203i	0.175874	+	0.304622i	0.940463	−	0.339895i	$-0.110392\pi$
$101$	178.518	+	309.203i	0.175874	+	0.304622i	−0.764590	+	0.644517i	$0.777058\pi$
$102$		0			0
$103$	506.565	+	292.466i	0.484596	+	0.279782i	0.722330	−	0.691549i	$-0.243071\pi$
$103$	506.565	+	292.466i	0.484596	+	0.279782i	−0.237734	+	0.971330i	$0.576405\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1105.24	+	638.108i	0.998571	+	0.576525i	0.907825	−	0.419349i	$-0.137742\pi$
$107$	1105.24	+	638.108i	0.998571	+	0.576525i	0.0907460	+	0.995874i	$0.471075\pi$
$108$		0			0
$109$	318.386	+	551.461i	0.279779	+	0.484591i	0.971330	−	0.237737i	$-0.0764056\pi$
$109$	318.386	+	551.461i	0.279779	+	0.484591i	−0.691551	+	0.722328i	$0.743072\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	961.802i		−	0.800696i	−0.916363	−	0.400348i	$-0.868889\pi$
$113$		−	961.802i		−	0.800696i	0.916363	−	0.400348i	$-0.131111\pi$
$114$		0			0
$115$	199.889	−	115.406i	0.162085	−	0.0935797i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	640.973	−	607.726i	0.493764	−	0.468153i
$120$		0			0
$121$	−663.904	+	1149.92i	−0.498801	+	0.863949i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1364.72			0.976512
$126$		0			0
$127$	−144.006			−0.100618			−0.0503089	−	0.998734i	$-0.516021\pi$
$127$	−144.006			−0.100618			−0.0503089	+	0.998734i	$0.516021\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	186.107	−	322.347i	0.124124	−	0.214989i	−0.797266	−	0.603628i	$-0.793721\pi$
$131$	186.107	−	322.347i	0.124124	−	0.214989i	0.921390	+	0.388639i	$0.127055\pi$
$132$		0			0
$133$	−244.188	+	823.039i	−0.159202	+	0.536590i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	478.822	−	276.448i	0.298602	−	0.172398i	−0.343213	−	0.939258i	$-0.611515\pi$
$137$	478.822	−	276.448i	0.298602	−	0.172398i	0.641815	+	0.766860i	$0.278182\pi$
$138$		0			0
$139$			2526.72i			1.54183i	0.636941	+	0.770913i	$0.280200\pi$
$139$			2526.72i			1.54183i	−0.636941	+	0.770913i	$0.719800\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−4.22422	−	7.31656i	−0.00247026	−	0.00427861i
$144$		0			0
$145$	−277.113	−	159.991i	−0.158710	−	0.0916314i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	2384.00	+	1376.40i	1.31077	+	0.756774i	0.982224	−	0.187714i	$-0.0601077\pi$
$149$	2384.00	+	1376.40i	1.31077	+	0.756774i	0.328547	+	0.944488i	$0.393441\pi$
$150$		0			0
$151$	395.046	+	684.240i	0.212903	+	0.368759i	0.952622	−	0.304157i	$-0.0983748\pi$
$151$	395.046	+	684.240i	0.212903	+	0.368759i	−0.739719	+	0.672916i	$0.765041\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	823.966i		−	0.426984i
$156$		0			0
$157$	871.100	−	502.930i	0.442811	−	0.255657i	−0.261978	−	0.965074i	$-0.584375\pi$
$157$	871.100	−	502.930i	0.442811	−	0.255657i	0.704789	+	0.709417i	$0.251042\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−183.706	+	619.184i	−0.0899260	+	0.303096i
$162$		0			0
$163$	8.76430	−	15.1802i	0.00421149	−	0.00729452i	−0.863912	−	0.503643i	$-0.831993\pi$
$163$	8.76430	−	15.1802i	0.00421149	−	0.00729452i	0.868123	+	0.496348i	$0.165326\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	195.663			0.0906639			0.0453320	−	0.998972i	$-0.485565\pi$
$167$	195.663			0.0906639			0.0453320	+	0.998972i	$0.485565\pi$
$168$		0			0
$169$	2174.64			0.989820
$170$		0			0
$171$		0			0
$172$		0			0
$173$	420.902	−	729.024i	0.184975	−	0.320385i	−0.758593	−	0.651564i	$-0.774113\pi$
$173$	420.902	−	729.024i	0.184975	−	0.320385i	0.943568	+	0.331179i	$0.107446\pi$
$174$		0			0
$175$	−1091.23	+	1034.62i	−0.471365	+	0.446916i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−766.552	+	442.569i	−0.320083	+	0.184800i	−0.651429	−	0.758709i	$-0.725830\pi$
$179$	−766.552	+	442.569i	−0.320083	+	0.184800i	0.331347	+	0.943509i	$0.392497\pi$
$180$		0			0
$181$			1011.03i			0.415190i	0.978215	+	0.207595i	$0.0665635\pi$
$181$			1011.03i			0.415190i	−0.978215	+	0.207595i	$0.933436\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	912.927	+	1581.24i	0.362809	+	0.628404i
$186$		0			0
$187$	−73.7859	−	42.6003i	−0.0288543	−	0.0166591i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2760.51	+	1593.78i	1.04578	+	0.603779i	0.921464	−	0.388464i	$-0.126994\pi$
$191$	2760.51	+	1593.78i	1.04578	+	0.603779i	0.124312	+	0.992243i	$0.460327\pi$
$192$		0			0
$193$	475.271	+	823.193i	0.177258	+	0.307019i	0.940940	−	0.338573i	$-0.109944\pi$
$193$	475.271	+	823.193i	0.177258	+	0.307019i	−0.763683	+	0.645592i	$0.776611\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			3589.52i			1.29819i	0.760709	+	0.649093i	$0.224851\pi$
$197$			3589.52i			1.29819i	−0.760709	+	0.649093i	$0.775149\pi$
$198$		0			0
$199$	−3985.72	+	2301.16i	−1.41980	+	0.819722i	−0.996281	−	0.0861652i	$-0.972539\pi$
$199$	−3985.72	+	2301.16i	−1.41980	+	0.819722i	−0.423519	+	0.905887i	$0.639205\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	870.731	−	208.640i	0.301051	−	0.0721362i
$204$		0			0
$205$	−123.554	+	214.001i	−0.0420944	+	0.0729097i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	82.8106			0.0274073
$210$		0			0
$211$	−3074.01			−1.00296			−0.501478	−	0.865171i	$-0.667210\pi$
$211$	−3074.01			−1.00296			−0.501478	+	0.865171i	$0.667210\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	714.422	−	1237.42i	0.226620	−	0.392517i
$216$		0			0
$217$	1586.36	+	1673.14i	0.496263	+	0.523412i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−195.328	+	112.772i	−0.0594532	+	0.0343253i
$222$		0			0
$223$			2057.55i			0.617866i	0.951084	+	0.308933i	$0.0999718\pi$
$223$			2057.55i			0.617866i	−0.951084	+	0.308933i	$0.900028\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−892.967	−	1546.66i	−0.261094	−	0.452228i	0.705439	−	0.708771i	$-0.250750\pi$
$227$	−892.967	−	1546.66i	−0.261094	−	0.452228i	−0.966533	+	0.256543i	$0.917417\pi$
$228$		0			0
$229$	−1288.59	−	743.966i	−0.371844	−	0.214684i	0.302420	−	0.953175i	$-0.402206\pi$
$229$	−1288.59	−	743.966i	−0.371844	−	0.214684i	−0.674264	+	0.738491i	$0.735539\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	231.289	+	133.535i	0.0650312	+	0.0375458i	0.532163	−	0.846642i	$-0.321379\pi$
$233$	231.289	+	133.535i	0.0650312	+	0.0375458i	−0.467132	+	0.884188i	$0.654713\pi$
$234$		0			0
$235$	351.472	+	608.768i	0.0975639	+	0.168986i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	1832.70i		−	0.496013i	−0.968758	−	0.248007i	$-0.920225\pi$
$239$		−	1832.70i		−	0.496013i	0.968758	−	0.248007i	$-0.0797755\pi$
$240$		0			0
$241$	5639.44	−	3255.93i	1.50734	−	0.870261i	0.507373	−	0.861726i	$-0.330617\pi$
$241$	5639.44	−	3255.93i	1.50734	−	0.870261i	0.999964	−	0.00853503i	$-0.00271682\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−120.802	+	2266.96i	−0.0315011	+	0.591147i
$246$		0			0
$247$	109.609	−	189.848i	0.0282358	−	0.0489059i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−2590.15			−0.651350			−0.325675	−	0.945482i	$-0.605591\pi$
$251$	−2590.15			−0.651350			−0.325675	+	0.945482i	$0.605591\pi$
$252$		0			0
$253$	62.2996			0.0154812
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2367.41	−	4100.48i	0.574612	−	0.995256i	−0.421472	−	0.906841i	$-0.638486\pi$
$257$	2367.41	−	4100.48i	0.574612	−	0.995256i	0.996084	−	0.0884151i	$-0.0281802\pi$
$258$		0			0
$259$	−4898.09	−	1453.22i	−1.17511	−	0.348644i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	1176.48	−	679.239i	0.275835	−	0.159253i	−0.355701	−	0.934600i	$-0.615758\pi$
$263$	1176.48	−	679.239i	0.275835	−	0.159253i	0.631536	+	0.775346i	$0.282425\pi$
$264$		0			0
$265$		−	1786.60i		−	0.414150i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	2517.48	+	4360.41i	0.570608	+	0.988323i	0.996504	+	0.0835501i	$0.0266259\pi$
$269$	2517.48	+	4360.41i	0.570608	+	0.988323i	−0.425895	+	0.904772i	$0.640041\pi$
$270$		0			0
$271$	1678.51	+	969.091i	0.376245	+	0.217225i	0.676183	−	0.736733i	$-0.263633\pi$
$271$	1678.51	+	969.091i	0.376245	+	0.217225i	−0.299938	+	0.953959i	$0.596966\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	125.617	+	72.5250i	0.0275454	+	0.0159033i
$276$		0			0
$277$	−986.169	−	1708.09i	−0.213910	−	0.370503i	0.739025	−	0.673678i	$-0.235287\pi$
$277$	−986.169	−	1708.09i	−0.213910	−	0.370503i	−0.952935	+	0.303175i	$0.901953\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	255.262i		−	0.0541910i	−0.999633	−	0.0270955i	$-0.991374\pi$
$281$		−	255.262i		−	0.0541910i	0.999633	−	0.0270955i	$-0.00862582\pi$
$282$		0			0
$283$	−181.330	+	104.691i	−0.0380882	+	0.0219902i	−0.518923	−	0.854821i	$-0.673667\pi$
$283$	−181.330	+	104.691i	−0.0380882	+	0.0219902i	0.480835	+	0.876811i	$0.340334\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−161.123	−	672.424i	−0.0331386	−	0.138299i
$288$		0			0
$289$	1319.21	−	2284.95i	0.268515	−	0.465082i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	7355.30			1.46656			0.733279	−	0.679928i	$-0.237989\pi$
$293$	7355.30			1.46656			0.733279	+	0.679928i	$0.237989\pi$
$294$		0			0
$295$	1980.18			0.390815
$296$		0			0
$297$		0			0
$298$		0			0
$299$	82.4603	−	142.826i	0.0159492	−	0.0276248i
$300$		0			0
$301$	931.657	+	3888.15i	0.178405	+	0.744549i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1935.67	+	1117.56i	−0.363397	+	0.209807i
$306$		0			0
$307$			397.297i			0.0738598i	0.999318	+	0.0369299i	$0.0117578\pi$
$307$			397.297i			0.0738598i	−0.999318	+	0.0369299i	$0.988242\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−3349.51	−	5801.52i	−0.610717	−	1.05779i	−0.991120	−	0.132972i	$-0.957548\pi$
$311$	−3349.51	−	5801.52i	−0.610717	−	1.05779i	0.380402	−	0.924821i	$-0.375786\pi$
$312$		0			0
$313$	−3683.02	−	2126.39i	−0.665100	−	0.383996i	0.129117	−	0.991629i	$-0.458786\pi$
$313$	−3683.02	−	2126.39i	−0.665100	−	0.383996i	−0.794218	+	0.607633i	$0.792119\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−6472.98	−	3737.17i	−1.14687	−	0.662147i	−0.198749	−	0.980050i	$-0.563688\pi$
$317$	−6472.98	−	3737.17i	−1.14687	−	0.662147i	−0.948123	+	0.317903i	$0.897021\pi$
$318$		0			0
$319$	−43.1840	−	74.7969i	−0.00757944	−	0.0131280i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	2210.77i		−	0.380837i
$324$		0			0
$325$	332.536	−	191.989i	0.0567562	−	0.0327682i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1885.74	−	559.482i	−0.316001	−	0.0937546i
$330$		0			0
$331$	4036.45	−	6991.34i	0.670283	−	1.16096i	−0.307541	−	0.951535i	$-0.599506\pi$
$331$	4036.45	−	6991.34i	0.670283	−	1.16096i	0.977824	−	0.209429i	$-0.0671604\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	2497.68			0.407352
$336$		0			0
$337$	5669.65			0.916455			0.458228	−	0.888835i	$-0.348484\pi$
$337$	5669.65			0.916455			0.458228	+	0.888835i	$0.348484\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	111.200	−	192.605i	0.0176593	−	0.0305869i
$342$		0			0
$343$	−4119.22	−	4835.87i	−0.648446	−	0.761261i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−5238.57	+	3024.49i	−0.810436	+	0.467905i	−0.847107	−	0.531422i	$-0.821658\pi$
$347$	−5238.57	+	3024.49i	−0.810436	+	0.467905i	0.0366712	+	0.999327i	$0.488325\pi$
$348$		0			0
$349$			492.028i			0.0754660i	0.999288	+	0.0377330i	$0.0120136\pi$
$349$			492.028i			0.0754660i	−0.999288	+	0.0377330i	$0.987986\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−6002.09	−	10395.9i	−0.904982	−	1.56748i	−0.820941	−	0.571013i	$-0.806550\pi$
$353$	−6002.09	−	10395.9i	−0.904982	−	1.56748i	−0.0840416	−	0.996462i	$-0.526783\pi$
$354$		0			0
$355$	4681.88	+	2703.08i	0.699967	+	0.404126i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	3458.42	+	1996.72i	0.508436	+	0.293545i	0.732190	−	0.681100i	$-0.238498\pi$
$359$	3458.42	+	1996.72i	0.508436	+	0.293545i	−0.223755	+	0.974645i	$0.571831\pi$
$360$		0			0
$361$	−2355.13	−	4079.20i	−0.343363	−	0.594722i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			4557.65i			0.653584i
$366$		0			0
$367$	640.131	−	369.580i	0.0910478	−	0.0525665i	−0.453785	−	0.891111i	$-0.649927\pi$
$367$	640.131	−	369.580i	0.0910478	−	0.0525665i	0.544833	+	0.838545i	$0.316593\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	3439.68	+	3627.86i	0.481346	+	0.507679i
$372$		0			0
$373$	−6623.43	+	11472.1i	−0.919432	+	1.59250i	−0.119152	+	0.992876i	$0.538018\pi$
$373$	−6623.43	+	11472.1i	−0.919432	+	1.59250i	−0.800280	+	0.599627i	$0.795316\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−228.635			−0.0312342
$378$		0			0
$379$	1420.35			0.192502			0.0962512	−	0.995357i	$-0.469315\pi$
$379$	1420.35			0.192502			0.0962512	+	0.995357i	$0.469315\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−5835.44	+	10107.3i	−0.778530	+	1.34845i	0.154258	+	0.988031i	$0.450701\pi$
$383$	−5835.44	+	10107.3i	−0.778530	+	1.34845i	−0.932789	+	0.360424i	$0.882632\pi$
$384$		0			0
$385$	212.953	−	51.0266i	0.0281898	−	0.00675470i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−8840.84	+	5104.26i	−1.15231	+	0.665286i	−0.949449	−	0.313921i	$-0.898357\pi$
$389$	−8840.84	+	5104.26i	−1.15231	+	0.665286i	−0.202861	+	0.979208i	$0.565024\pi$
$390$		0			0
$391$		−	1663.19i		−	0.215118i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−2032.34	−	3520.11i	−0.258881	−	0.448395i
$396$		0			0
$397$	−11903.6	−	6872.57i	−1.50485	−	0.868827i	−0.999984	−	0.00563049i	$-0.998208\pi$
$397$	−11903.6	−	6872.57i	−1.50485	−	0.868827i	−0.504868	−	0.863196i	$-0.668459\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−11514.5	−	6647.91i	−1.43394	−	0.827883i	−0.436517	−	0.899696i	$-0.643788\pi$
$401$	−11514.5	−	6647.91i	−1.43394	−	0.827883i	−0.997418	+	0.0718132i	$0.977121\pi$
$402$		0			0
$403$	−294.372	−	509.867i	−0.0363863	−	0.0630230i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			492.825i			0.0600207i
$408$		0			0
$409$	9944.36	−	5741.38i	1.20224	−	0.694115i	0.241189	−	0.970478i	$-0.422463\pi$
$409$	9944.36	−	5741.38i	1.20224	−	0.694115i	0.961053	+	0.276364i	$0.0891294\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−4020.94	+	3812.38i	−0.479074	+	0.454225i
$414$		0			0
$415$	4381.20	−	7588.46i	0.518228	−	0.897597i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−9092.21			−1.06010			−0.530052	−	0.847965i	$-0.677828\pi$
$419$	−9092.21			−1.06010			−0.530052	+	0.847965i	$0.677828\pi$
$420$		0			0
$421$	−7373.44			−0.853585			−0.426793	−	0.904349i	$-0.640357\pi$
$421$	−7373.44			−0.853585			−0.426793	+	0.904349i	$0.640357\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1936.17	−	3353.55i	0.220984	−	0.382756i
$426$		0			0
$427$	1778.96	−	5996.00i	0.201616	−	0.679548i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−14205.0	+	8201.24i	−1.58754	+	0.916566i	−0.593827	+	0.804592i	$0.702384\pi$
$431$	−14205.0	+	8201.24i	−1.58754	+	0.916566i	−0.993711	+	0.111973i	$0.964283\pi$
$432$		0			0
$433$		−	5691.74i		−	0.631703i	−0.948809	−	0.315852i	$-0.897710\pi$
$433$		−	5691.74i		−	0.631703i	0.948809	−	0.315852i	$-0.102290\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	808.267	+	1399.96i	0.0884775	+	0.153248i
$438$		0			0
$439$	8196.36	+	4732.17i	0.891095	+	0.514474i	0.874301	−	0.485385i	$-0.161321\pi$
$439$	8196.36	+	4732.17i	0.891095	+	0.514474i	0.0167945	+	0.999859i	$0.494654\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	9130.65	+	5271.58i	0.979255	+	0.565373i	0.902045	−	0.431642i	$-0.142065\pi$
$443$	9130.65	+	5271.58i	0.979255	+	0.565373i	0.0772101	+	0.997015i	$0.475399\pi$
$444$		0			0
$445$	−3179.37	−	5506.82i	−0.338689	−	0.586626i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	9305.47i		−	0.978067i	−0.872265	−	0.489034i	$-0.837350\pi$
$449$		−	9305.47i		−	0.978067i	0.872265	−	0.489034i	$-0.162650\pi$
$450$		0			0
$451$	−57.7621	+	33.3489i	−0.00603084	+	0.00348191i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	164.885	−	555.747i	0.0169889	−	0.0572611i
$456$		0			0
$457$	−4589.88	+	7949.91i	−0.469815	+	0.813744i	−0.999404	−	0.0345103i	$-0.989013\pi$
$457$	−4589.88	+	7949.91i	−0.469815	+	0.813744i	0.529589	+	0.848254i	$0.322346\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−6418.60			−0.648469			−0.324234	−	0.945977i	$-0.605107\pi$
$461$	−6418.60			−0.648469			−0.324234	+	0.945977i	$0.605107\pi$
$462$		0			0
$463$	9649.90			0.968615			0.484307	−	0.874898i	$-0.339072\pi$
$463$	9649.90			0.968615			0.484307	+	0.874898i	$0.339072\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−4082.03	+	7070.29i	−0.404484	+	0.700587i	−0.994261	−	0.106979i	$-0.965882\pi$
$467$	−4082.03	+	7070.29i	−0.404484	+	0.700587i	0.589777	+	0.807566i	$0.299216\pi$
$468$		0			0
$469$	−5071.78	+	4808.71i	−0.499346	+	0.473445i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	333.997	−	192.833i	0.0324676	−	0.0187452i
$474$		0			0
$475$			3763.72i			0.363561i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−3688.53	−	6388.72i	−0.351844	−	0.609412i	0.634729	−	0.772735i	$-0.281112\pi$
$479$	−3688.53	−	6388.72i	−0.351844	−	0.609412i	−0.986573	+	0.163324i	$0.947779\pi$
$480$		0			0
$481$	1129.83	+	652.308i	0.107101	+	0.0618351i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	2576.53	+	1487.56i	0.241225	+	0.139271i
$486$		0			0
$487$	−5919.10	−	10252.2i	−0.550760	−	0.953944i	−0.998220	−	0.0596401i	$-0.981005\pi$
$487$	−5919.10	−	10252.2i	−0.550760	−	0.953944i	0.447460	−	0.894304i	$-0.352329\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	3932.63i		−	0.361460i	−0.983533	−	0.180730i	$-0.942154\pi$
$491$		−	3932.63i		−	0.361460i	0.983533	−	0.180730i	$-0.0578461\pi$
$492$		0			0
$493$	−1996.83	+	1152.87i	−0.182419	+	0.105320i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−14711.2	+	3525.01i	−1.32774	+	0.318146i
$498$		0			0
$499$	7852.94	−	13601.7i	0.704500	−	1.22023i	−0.262371	−	0.964967i	$-0.584505\pi$
$499$	7852.94	−	13601.7i	0.704500	−	1.22023i	0.966872	−	0.255263i	$-0.0821622\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−1461.52			−0.129555			−0.0647773	−	0.997900i	$-0.520634\pi$
$503$	−1461.52			−0.129555			−0.0647773	+	0.997900i	$0.520634\pi$
$504$		0			0
$505$	2363.08			0.208229
$506$		0			0
$507$		0			0
$508$		0			0
$509$	3690.38	−	6391.93i	0.321362	−	0.556616i	−0.659407	−	0.751786i	$-0.729193\pi$
$509$	3690.38	−	6391.93i	0.321362	−	0.556616i	0.980769	+	0.195170i	$0.0625260\pi$
$510$		0			0
$511$	−8774.71	−	9254.75i	−0.759629	−	0.801186i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	3352.76	−	1935.71i	0.286874	−	0.165627i
$516$		0			0
$517$			189.735i			0.0161403i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−5116.53	−	8862.09i	−0.430248	−	0.745211i	0.566646	−	0.823961i	$-0.308241\pi$
$521$	−5116.53	−	8862.09i	−0.430248	−	0.745211i	−0.996894	+	0.0787498i	$0.974907\pi$
$522$		0			0
$523$	16679.3	+	9629.78i	1.39452	+	0.805126i	0.993812	−	0.111080i	$-0.0354308\pi$
$523$	16679.3	+	9629.78i	1.39452	+	0.805126i	0.400708	+	0.916206i	$0.368764\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−5141.90	−	2968.68i	−0.425018	−	0.245384i
$528$		0			0
$529$	−5475.43	−	9483.72i	−0.450023	−	0.779463i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			176.564i			0.0143487i
$534$		0			0
$535$	7315.11	−	4223.38i	0.591140	−	0.341295i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−334.182	+	513.607i	−0.0267054	+	0.0410438i
$540$		0			0
$541$	−5087.07	+	8811.07i	−0.404270	+	0.700217i	−0.994236	−	0.107211i	$-0.965808\pi$
$541$	−5087.07	+	8811.07i	−0.404270	+	0.700217i	0.589966	+	0.807428i	$0.299141\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	4214.55			0.331250
$546$		0			0
$547$	−14637.4			−1.14415			−0.572076	−	0.820201i	$-0.693862\pi$
$547$	−14637.4			−1.14415			−0.572076	+	0.820201i	$0.693862\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1120.53	−	1940.81i	0.0866354	−	0.150057i
$552$		0			0
$553$	10904.0	+	3235.13i	0.838492	+	0.248773i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−2573.96	+	1486.07i	−0.195803	+	0.113047i	−0.594696	−	0.803951i	$-0.702728\pi$
$557$	−2573.96	+	1486.07i	−0.195803	+	0.113047i	0.398894	+	0.916997i	$0.369394\pi$
$558$		0			0
$559$		−	1020.94i		−	0.0772474i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−6713.27	−	11627.7i	−0.502541	−	0.870427i	−0.999996	−	0.00293687i	$-0.999065\pi$
$563$	−6713.27	−	11627.7i	−0.502541	−	0.870427i	0.497454	−	0.867490i	$-0.334268\pi$
$564$		0			0
$565$	−5512.93	−	3182.89i	−0.410497	−	0.237001i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−20925.0	−	12081.0i	−1.54169	−	0.890094i	−0.998732	−	0.0503329i	$-0.983972\pi$
$569$	−20925.0	−	12081.0i	−1.54169	−	0.890094i	−0.542956	−	0.839761i	$-0.682695\pi$
$570$		0			0
$571$	−2748.17	−	4759.96i	−0.201414	−	0.348859i	0.747571	−	0.664182i	$-0.231220\pi$
$571$	−2748.17	−	4759.96i	−0.201414	−	0.348859i	−0.948984	+	0.315324i	$0.897887\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			2831.50i			0.205359i
$576$		0			0
$577$	18614.4	−	10747.0i	1.34303	−	0.775398i	0.355778	−	0.934571i	$-0.384216\pi$
$577$	18614.4	−	10747.0i	1.34303	−	0.775398i	0.987251	+	0.159173i	$0.0508826\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	5713.39	+	23844.1i	0.407971	+	1.70262i
$582$		0			0
$583$	241.114	−	417.623i	0.0171285	−	0.0296675i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	15656.5			1.10087			0.550437	−	0.834877i	$-0.314461\pi$
$587$	15656.5			1.10087			0.550437	+	0.834877i	$0.314461\pi$
$588$		0			0
$589$	5770.80			0.403704
$590$		0			0
$591$		0			0
$592$		0			0
$593$	12134.6	−	21017.8i	0.840319	−	1.45547i	−0.0493069	−	0.998784i	$-0.515701\pi$
$593$	12134.6	−	21017.8i	0.840319	−	1.45547i	0.889625	−	0.456691i	$-0.150965\pi$
$594$		0			0
$595$	−1362.24	−	5685.13i	−0.0938595	−	0.391710i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−11862.8	+	6848.98i	−0.809182	+	0.467181i	−0.846672	−	0.532116i	$-0.821397\pi$
$599$	−11862.8	+	6848.98i	−0.809182	+	0.467181i	0.0374897	+	0.999297i	$0.488064\pi$
$600$		0			0
$601$		−	8670.98i		−	0.588514i	−0.955726	−	0.294257i	$-0.904928\pi$
$601$		−	8670.98i		−	0.588514i	0.955726	−	0.294257i	$-0.0950721\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	4394.12	+	7610.84i	0.295283	+	0.511446i
$606$		0			0
$607$	10404.7	+	6007.18i	0.695742	+	0.401687i	0.805760	−	0.592243i	$-0.201757\pi$
$607$	10404.7	+	6007.18i	0.695742	+	0.401687i	−0.110017	+	0.993930i	$0.535091\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	434.979	+	251.135i	0.0288009	+	0.0166282i
$612$		0			0
$613$	−8647.55	−	14978.0i	−0.569773	−	0.986877i	−0.996588	−	0.0825374i	$-0.973698\pi$
$613$	−8647.55	−	14978.0i	−0.569773	−	0.986877i	0.426815	−	0.904339i	$-0.359636\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	9886.44i		−	0.645078i	−0.946556	−	0.322539i	$-0.895464\pi$
$617$		−	9886.44i		−	0.645078i	0.946556	−	0.322539i	$-0.104536\pi$
$618$		0			0
$619$	−8751.96	+	5052.94i	−0.568289	+	0.328102i	−0.756466	−	0.654033i	$-0.773076\pi$
$619$	−8751.96	+	5052.94i	−0.568289	+	0.328102i	0.188177	+	0.982135i	$0.439742\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	17058.1	+	5061.00i	1.09698	+	0.325465i
$624$		0			0
$625$	−558.371	+	967.126i	−0.0357357	+	0.0618961i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	13156.8			0.834014
$630$		0			0
$631$	−22892.4			−1.44427			−0.722133	−	0.691754i	$-0.756838\pi$
$631$	−22892.4			−1.44427			−0.722133	+	0.691754i	$0.756838\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−476.559	+	825.424i	−0.0297821	+	0.0515842i
$636$		0			0
$637$	735.148	+	1445.95i	0.0457263	+	0.0899379i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−12723.7	+	7346.04i	−0.784019	+	0.452654i	−0.837853	−	0.545896i	$-0.816189\pi$
$641$	−12723.7	+	7346.04i	−0.784019	+	0.452654i	0.0538337	+	0.998550i	$0.482856\pi$
$642$		0			0
$643$			28388.0i			1.74108i	0.492099	+	0.870539i	$0.336230\pi$
$643$			28388.0i			1.74108i	−0.492099	+	0.870539i	$0.663770\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−15705.6	−	27202.9i	−0.954331	−	1.65295i	−0.735893	−	0.677098i	$-0.763237\pi$
$647$	−15705.6	−	27202.9i	−0.954331	−	1.65295i	−0.218438	−	0.975851i	$-0.570096\pi$
$648$		0			0
$649$	462.873	+	267.240i	0.0279959	+	0.0161634i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	20102.2	+	11606.0i	1.20468	+	0.695525i	0.961593	−	0.274479i	$-0.0885052\pi$
$653$	20102.2	+	11606.0i	1.20468	+	0.695525i	0.243091	+	0.970003i	$0.421839\pi$
$654$		0			0
$655$	−1231.77	−	2133.49i	−0.0734797	−	0.127271i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	4092.54i		−	0.241916i	−0.992658	−	0.120958i	$-0.961403\pi$
$659$		−	4092.54i		−	0.241916i	0.992658	−	0.120958i	$-0.0385966\pi$
$660$		0			0
$661$	17750.7	−	10248.4i	1.04451	−	0.603050i	0.123405	−	0.992356i	$-0.460619\pi$
$661$	17750.7	−	10248.4i	1.04451	−	0.603050i	0.921108	+	0.389306i	$0.127285\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	3909.47	+	4123.34i	0.227974	+	0.240446i
$666$		0			0
$667$	842.989	−	1460.10i	0.0489365	−	0.0847606i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−603.292			−0.0347091
$672$		0			0
$673$	11264.6			0.645198			0.322599	−	0.946536i	$-0.395443\pi$
$673$	11264.6			0.645198			0.322599	+	0.946536i	$0.395443\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−7916.23	+	13711.3i	−0.449403	+	0.778388i	−0.998347	−	0.0574706i	$-0.981696\pi$
$677$	−7916.23	+	13711.3i	−0.449403	+	0.778388i	0.548945	+	0.835859i	$0.315030\pi$
$678$		0			0
$679$	−8095.84	+	1939.88i	−0.457570	+	0.109640i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−10545.0	+	6088.16i	−0.590766	+	0.341079i	−0.765400	−	0.643554i	$-0.777459\pi$
$683$	−10545.0	+	6088.16i	−0.590766	+	0.341079i	0.174634	+	0.984633i	$0.444126\pi$
$684$		0			0
$685$		−	3659.40i		−	0.204114i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−638.283	−	1105.54i	−0.0352927	−	0.0611287i
$690$		0			0
$691$	−7657.82	−	4421.24i	−0.421588	−	0.243404i	0.274169	−	0.961682i	$-0.411597\pi$
$691$	−7657.82	−	4421.24i	−0.421588	−	0.243404i	−0.695756	+	0.718278i	$0.744931\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	14482.9	+	8361.68i	0.790455	+	0.456369i
$696$		0			0
$697$	890.305	+	1542.05i	0.0483826	+	0.0838012i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			22086.4i			1.19000i	0.803726	+	0.595000i	$0.202848\pi$
$701$			22086.4i			1.19000i	−0.803726	+	0.595000i	$0.797152\pi$
$702$		0			0
$703$	−11074.5	+	6393.85i	−0.594142	+	0.343028i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−4798.47	+	4549.57i	−0.255255	+	0.242015i
$708$		0			0
$709$	−14042.2	+	24321.9i	−0.743818	+	1.28833i	0.206927	+	0.978356i	$0.433654\pi$
$709$	−14042.2	+	24321.9i	−0.743818	+	1.28833i	−0.950745	+	0.309974i	$0.899680\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	4341.45			0.228035
$714$		0			0
$715$	−55.9168			−0.00292471
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−2227.62	+	3858.36i	−0.115544	+	0.200129i	−0.917997	−	0.396587i	$-0.870195\pi$
$719$	−2227.62	+	3858.36i	−0.115544	+	0.200129i	0.802453	+	0.596715i	$0.203528\pi$
$720$		0			0
$721$	−3081.32	+	10385.6i	−0.159160	+	0.536450i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	3399.50	−	1962.70i	0.174144	−	0.100542i
$726$		0			0
$727$			11915.8i			0.607884i	0.952690	+	0.303942i	$0.0983030\pi$
$727$			11915.8i			0.607884i	−0.952690	+	0.303942i	$0.901697\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−5148.00	−	8916.60i	−0.260473	−	0.451152i
$732$		0			0
$733$	6157.69	+	3555.14i	0.310286	+	0.179143i	0.647054	−	0.762444i	$-0.276001\pi$
$733$	6157.69	+	3555.14i	0.310286	+	0.179143i	−0.336769	+	0.941587i	$0.609334\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	583.841	+	337.081i	0.0291805	+	0.0168474i
$738$		0			0
$739$	−15356.0	−	26597.3i	−0.764382	−	1.32395i	−0.940573	−	0.339593i	$-0.889711\pi$
$739$	−15356.0	−	26597.3i	−0.764382	−	1.32395i	0.176190	−	0.984356i	$-0.443623\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	14280.3i		−	0.705104i	−0.935792	−	0.352552i	$-0.885314\pi$
$743$		−	14280.3i		−	0.705104i	0.935792	−	0.352552i	$-0.114686\pi$
$744$		0			0
$745$	15778.7	−	9109.87i	0.775958	−	0.448000i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−6722.89	+	22659.6i	−0.327969	+	1.10542i
$750$		0			0
$751$	−9233.47	+	15992.8i	−0.448647	+	0.777080i	−0.998298	−	0.0583146i	$-0.981427\pi$
$751$	−9233.47	+	15992.8i	−0.448647	+	0.777080i	0.549651	+	0.835394i	$0.314761\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	5229.31			0.252071
$756$		0			0
$757$	9297.14			0.446381			0.223190	−	0.974775i	$-0.428353\pi$
$757$	9297.14			0.446381			0.223190	+	0.974775i	$0.428353\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1751.42	−	3033.56i	0.0834285	−	0.144502i	−0.821292	−	0.570508i	$-0.806746\pi$
$761$	1751.42	−	3033.56i	0.0834285	−	0.144502i	0.904720	+	0.426006i	$0.140080\pi$
$762$		0			0
$763$	−8558.05	+	8114.15i	−0.406058	+	0.384996i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1225.33	−	707.442i	0.0576844	−	0.0333041i
$768$		0			0
$769$			16857.8i			0.790517i	0.918570	+	0.395258i	$0.129345\pi$
$769$			16857.8i			0.790517i	−0.918570	+	0.395258i	$0.870655\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	7410.77	+	12835.8i	0.344821	+	0.597248i	0.985321	−	0.170710i	$-0.0546062\pi$
$773$	7410.77	+	12835.8i	0.344821	+	0.597248i	−0.640500	+	0.767958i	$0.721273\pi$
$774$		0			0
$775$	8753.83	+	5054.02i	0.405738	+	0.234253i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1498.80	−	865.330i	−0.0689344	−	0.0397993i
$780$		0			0
$781$	729.602	+	1263.71i	0.0334279	+	0.0578989i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	6657.39i		−	0.302691i
$786$		0			0
$787$	−7423.05	+	4285.70i	−0.336218	+	0.194115i	−0.658598	−	0.752495i	$-0.728850\pi$
$787$	−7423.05	+	4285.70i	−0.336218	+	0.194115i	0.322381	+	0.946610i	$0.395517\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	17322.5	−	4150.72i	0.778655	−	0.186577i
$792$		0			0
$793$	−798.523	+	1383.08i	−0.0357584	+	0.0619353i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	44804.8			1.99130			0.995651	−	0.0931611i	$-0.0296972\pi$
$797$	44804.8			1.99130			0.995651	+	0.0931611i	$0.0296972\pi$
$798$		0			0
$799$	5065.29			0.224277
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−615.088	+	1065.36i	−0.0270311	+	0.0468193i
$804$		0			0
$805$	2941.15	+	3102.05i	0.128772	+	0.135817i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	482.120	−	278.352i	0.0209523	−	0.0120968i	−0.489487	−	0.872010i	$-0.662816\pi$
$809$	482.120	−	278.352i	0.0209523	−	0.0120968i	0.510440	+	0.859914i	$0.329483\pi$
$810$		0			0
$811$			16892.8i			0.731426i	0.930728	+	0.365713i	$0.119175\pi$
$811$			16892.8i			0.731426i	−0.930728	+	0.365713i	$0.880825\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−58.0074	−	100.472i	−0.00249314	−	0.00431825i
$816$		0			0
$817$	8666.47	+	5003.59i	0.371115	+	0.214264i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	4445.18	+	2566.43i	0.188962	+	0.109097i	0.591497	−	0.806308i	$-0.298537\pi$
$821$	4445.18	+	2566.43i	0.188962	+	0.109097i	−0.402535	+	0.915405i	$0.631871\pi$
$822$		0			0
$823$	20153.2	+	34906.4i	0.853581	+	1.47845i	0.877955	+	0.478743i	$0.158907\pi$
$823$	20153.2	+	34906.4i	0.853581	+	1.47845i	−0.0243743	+	0.999703i	$0.507759\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	24795.5i		−	1.04259i	−0.853375	−	0.521297i	$-0.825448\pi$
$827$		−	24795.5i		−	1.04259i	0.853375	−	0.521297i	$-0.174552\pi$
$828$		0			0
$829$	−413.706	+	238.854i	−0.0173325	+	0.0100069i	−0.508641	−	0.860979i	$-0.669852\pi$
$829$	−413.706	+	238.854i	−0.0173325	+	0.0100069i	0.491309	+	0.870985i	$0.336519\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	13711.6	+	8921.53i	0.570322	+	0.371084i
$834$		0			0
$835$	647.509	−	1121.52i	0.0268359	−	0.0464811i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	46068.4			1.89566			0.947829	−	0.318780i	$-0.103273\pi$
$839$	46068.4			1.89566			0.947829	+	0.318780i	$0.103273\pi$
$840$		0			0
$841$	22051.7			0.904165
$842$		0			0
$843$		0			0
$844$		0			0
$845$	7196.52	−	12464.7i	0.292980	−	0.507456i
$846$		0			0
$847$	−23575.6	−	6994.67i	−0.956396	−	0.283754i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−8331.48	+	4810.18i	−0.335605	+	0.193761i
$852$		0			0
$853$			5319.60i			0.213528i	0.994284	+	0.106764i	$0.0340490\pi$
$853$			5319.60i			0.213528i	−0.994284	+	0.106764i	$0.965951\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−18690.5	−	32372.9i	−0.744989	−	1.29036i	−0.950200	−	0.311641i	$-0.899121\pi$
$857$	−18690.5	−	32372.9i	−0.744989	−	1.29036i	0.205211	−	0.978718i	$-0.434212\pi$
$858$		0			0
$859$	32236.0	+	18611.5i	1.28042	+	0.739249i	0.976925	−	0.213585i	$-0.0685139\pi$
$859$	32236.0	+	18611.5i	1.28042	+	0.739249i	0.303493	+	0.952834i	$0.401847\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	8993.16	+	5192.20i	0.354728	+	0.204803i	0.666766	−	0.745267i	$-0.267678\pi$
$863$	8993.16	+	5192.20i	0.354728	+	0.204803i	−0.312037	+	0.950070i	$0.601011\pi$
$864$		0			0
$865$	−2785.78	−	4825.12i	−0.109502	−	0.189664i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	1097.12i		−	0.0428275i
$870$		0			0
$871$	1545.55	−	892.326i	0.0601253	−	0.0347133i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	5889.52	+	24579.2i	0.227545	+	0.949631i
$876$		0			0
$877$	−5046.94	+	8741.56i	−0.194325	+	0.336581i	−0.946679	−	0.322178i	$-0.895585\pi$
$877$	−5046.94	+	8741.56i	−0.194325	+	0.336581i	0.752354	+	0.658759i	$0.228918\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	27322.1			1.04484			0.522420	−	0.852688i	$-0.325029\pi$
$881$	27322.1			1.04484			0.522420	+	0.852688i	$0.325029\pi$
$882$		0			0
$883$	−638.558			−0.0243366			−0.0121683	−	0.999926i	$-0.503873\pi$
$883$	−638.558			−0.0243366			−0.0121683	+	0.999926i	$0.503873\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	4090.12	−	7084.29i	0.154828	−	0.268170i	−0.778168	−	0.628056i	$-0.783851\pi$
$887$	4090.12	−	7084.29i	0.154828	−	0.268170i	0.932996	+	0.359885i	$0.117184\pi$
$888$		0			0
$889$	−621.466	−	2593.61i	−0.0234458	−	0.0978480i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−4263.62	+	2461.60i	−0.159772	+	0.0922444i
$894$		0			0
$895$			5858.38i			0.218798i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−3009.35	−	5212.35i	−0.111643	−	0.193372i
$900$		0			0
$901$	−11149.1	−	6436.95i	−0.412243	−	0.238009i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	5795.11	+	3345.81i	0.212857	+	0.122893i
$906$		0			0
$907$	−4866.48	−	8429.00i	−0.178158	−	0.308578i	0.763092	−	0.646290i	$-0.223680\pi$
$907$	−4866.48	−	8429.00i	−0.178158	−	0.308578i	−0.941250	+	0.337712i	$0.890347\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			12619.6i			0.458953i	0.973314	+	0.229477i	$0.0737015\pi$
$911$			12619.6i			0.458953i	−0.973314	+	0.229477i	$0.926299\pi$
$912$		0			0
$913$	2048.24	−	1182.55i	0.0742462	−	0.0428660i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	6608.77	+	1960.76i	0.237994	+	0.0706108i
$918$		0			0
$919$	−12207.5	+	21144.1i	−0.438183	+	0.758954i	−0.997549	−	0.0699658i	$-0.977711\pi$
$919$	−12207.5	+	21144.1i	−0.438183	+	0.758954i	0.559367	+	0.828920i	$0.311044\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	3862.84			0.137754
$924$		0			0
$925$	−22398.8			−0.796180
$926$		0			0
$927$		0			0
$928$		0			0
$929$	9360.78	−	16213.4i	0.330589	−	0.572597i	−0.652038	−	0.758186i	$-0.726086\pi$
$929$	9360.78	−	16213.4i	0.330589	−	0.572597i	0.982628	+	0.185589i	$0.0594192\pi$
$930$		0			0
$931$	−15877.1	−	846.061i	−0.558916	−	0.0297836i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−488.360	+	281.955i	−0.0170814	+	0.00986193i
$936$		0			0
$937$			19693.0i			0.686598i	0.939226	+	0.343299i	$0.111544\pi$
$937$			19693.0i			0.686598i	−0.939226	+	0.343299i	$0.888456\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1511.66	−	2618.27i	−0.0523685	−	0.0907049i	0.838653	−	0.544666i	$-0.183344\pi$
$941$	−1511.66	−	2618.27i	−0.0523685	−	0.0907049i	−0.891021	+	0.453961i	$0.850010\pi$
$942$		0			0
$943$	−1127.57	−	651.000i	−0.0389380	−	0.0224809i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	34975.0	+	20192.8i	1.20014	+	0.692903i	0.960587	−	0.277979i	$-0.0896647\pi$
$947$	34975.0	+	20192.8i	1.20014	+	0.692903i	0.239556	+	0.970882i	$0.422998\pi$
$948$		0			0
$949$	1628.27	+	2820.25i	0.0556965	+	0.0964692i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	17126.4i		−	0.582137i	−0.956702	−	0.291069i	$-0.905989\pi$
$953$		−	17126.4i		−	0.582137i	0.956702	−	0.291069i	$-0.0940108\pi$
$954$		0			0
$955$	18270.7	−	10548.6i	0.619085	−	0.357429i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	7045.33	+	7430.76i	0.237232	+	0.250210i
$960$		0			0
$961$	−7146.32	+	12377.8i	−0.239882	+	0.415488i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	6291.26			0.209868
$966$		0			0
$967$	50418.3			1.67667			0.838336	−	0.545154i	$-0.183529\pi$
$967$	50418.3			1.67667			0.838336	+	0.545154i	$0.183529\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	25399.4	−	43993.0i	0.839448	−	1.45397i	−0.0509082	−	0.998703i	$-0.516212\pi$
$971$	25399.4	−	43993.0i	0.839448	−	1.45397i	0.890357	−	0.455264i	$-0.150455\pi$
$972$		0			0
$973$	−45507.4	+	10904.2i	−1.49938	+	0.359274i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−20845.4	+	12035.1i	−0.682605	+	0.394102i	−0.800836	−	0.598884i	$-0.795611\pi$
$977$	−20845.4	+	12035.1i	−0.682605	+	0.394102i	0.118231	+	0.992986i	$0.462278\pi$
$978$		0			0
$979$		−	1716.32i		−	0.0560303i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−2209.58	−	3827.10i	−0.0716933	−	0.124176i	0.827950	−	0.560802i	$-0.189507\pi$
$983$	−2209.58	−	3827.10i	−0.0716933	−	0.124176i	−0.899644	+	0.436625i	$0.856174\pi$
$984$		0			0
$985$	20574.7	+	11878.8i	0.665547	+	0.384254i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	6519.91	+	3764.27i	0.209627	+	0.121028i
$990$		0			0
$991$	−18648.8	−	32300.6i	−0.597777	−	1.03538i	−0.993148	−	0.116859i	$-0.962717\pi$
$991$	−18648.8	−	32300.6i	−0.597777	−	1.03538i	0.395371	−	0.918521i	$-0.370616\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			30460.9i			0.970528i
$996$		0			0
$997$	−9609.73	+	5548.18i	−0.305259	+	0.176241i	−0.644803	−	0.764349i	$-0.723061\pi$
$997$	−9609.73	+	5548.18i	−0.305259	+	0.176241i	0.339544	+	0.940590i	$0.389727\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.4.bt.d.593.16		48
3.2	odd	2	inner	1008.4.bt.d.593.9		48
4.3	odd	2		504.4.bl.a.89.16	yes	48
7.3	odd	6	inner	1008.4.bt.d.17.9		48
12.11	even	2		504.4.bl.a.89.9	yes	48
21.17	even	6	inner	1008.4.bt.d.17.16		48
28.3	even	6		504.4.bl.a.17.9	✓	48
84.59	odd	6		504.4.bl.a.17.16	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.4.bl.a.17.9	✓	48	28.3	even	6
504.4.bl.a.17.16	yes	48	84.59	odd	6
504.4.bl.a.89.9	yes	48	12.11	even	2
504.4.bl.a.89.16	yes	48	4.3	odd	2
1008.4.bt.d.17.9		48	7.3	odd	6	inner
1008.4.bt.d.17.16		48	21.17	even	6	inner
1008.4.bt.d.593.9		48	3.2	odd	2	inner
1008.4.bt.d.593.16		48	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1008.bt (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(3.30930 - 5.73188i) q^{5} +(4.31556 + 18.0104i) q^{7} +O(q^{10})\)	\(q+(3.30930 - 5.73188i) q^{5} +(4.31556 + 18.0104i) q^{7} +(1.54712 - 0.893230i) q^{11} -4.72915i q^{13} +(-23.8462 - 41.3029i) q^{17} +(40.1443 + 23.1773i) q^{19} +(30.2011 + 17.4366i) q^{23} +(40.5970 + 70.3161i) q^{25} -48.3459i q^{29} +(107.814 - 62.2462i) q^{31} +(117.515 + 34.8657i) q^{35} +(-137.933 + 238.908i) q^{37} -37.3352 q^{41} +215.883 q^{43} +(-53.1037 + 91.9783i) q^{47} +(-305.752 + 155.450i) q^{49} +(233.771 - 134.968i) q^{53} -11.8239i q^{55} +(149.592 + 259.101i) q^{59} +(-292.459 - 168.851i) q^{61} +(-27.1069 - 15.6502i) q^{65} +(188.686 + 326.815i) q^{67} +816.814i q^{71} +(-596.355 + 344.306i) q^{73} +(22.7641 + 24.0095i) q^{77} +(307.064 - 531.851i) q^{79} +1323.90 q^{83} -315.658 q^{85} +(480.368 - 832.022i) q^{89} +(85.1741 - 20.4089i) q^{91} +(265.699 - 153.401i) q^{95} +449.508i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 24 q^{7}+O(q^{10}) \) 48 * q + 24 * q^7	\( 48 q + 24 q^{7} - 540 q^{19} - 924 q^{25} - 648 q^{31} - 132 q^{37} + 792 q^{43} + 672 q^{49} + 12 q^{67} + 2412 q^{73} - 1680 q^{79} + 480 q^{85} - 1404 q^{91}+O(q^{100}) \) 48 * q + 24 * q^7 - 540 * q^19 - 924 * q^25 - 648 * q^31 - 132 * q^37 + 792 * q^43 + 672 * q^49 + 12 * q^67 + 2412 * q^73 - 1680 * q^79 + 480 * q^85 - 1404 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more