LMFDB - Embedded newform 1008.4.bt.d.17.18

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			17.18
Character	$\chi$	$=$	1008.17
Dual form			1008.4.bt.d.593.18

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(5.00025 + 8.66069i) q^{5} +(-1.56940 + 18.4536i) q^{7} +O(q^{10})$	$q+(5.00025 + 8.66069i) q^{5} +(-1.56940 + 18.4536i) q^{7} +(8.94164 + 5.16246i) q^{11} -52.4866i q^{13} +(-0.584477 + 1.01234i) q^{17} +(86.7124 - 50.0634i) q^{19} +(90.1373 - 52.0408i) q^{23} +(12.4950 - 21.6419i) q^{25} -187.555i q^{29} +(107.524 + 62.0789i) q^{31} +(-167.669 + 78.6808i) q^{35} +(16.0459 + 27.7922i) q^{37} +415.597 q^{41} +193.264 q^{43} +(196.466 + 340.289i) q^{47} +(-338.074 - 57.9222i) q^{49} +(74.7758 + 43.1718i) q^{53} +103.254i q^{55} +(-102.286 + 177.165i) q^{59} +(-183.527 + 105.959i) q^{61} +(454.570 - 262.446i) q^{65} +(-364.857 + 631.951i) q^{67} +315.022i q^{71} +(-899.220 - 519.165i) q^{73} +(-109.299 + 156.904i) q^{77} +(607.787 + 1052.72i) q^{79} -333.797 q^{83} -11.6901 q^{85} +(168.355 + 291.599i) q^{89} +(968.568 + 82.3723i) q^{91} +(867.167 + 500.659i) q^{95} +893.179i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

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

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
$5$	5.00025	+	8.66069i	0.447236	+	0.774636i	0.998205	−	0.0598902i	$-0.0190751\pi$
$5$	5.00025	+	8.66069i	0.447236	+	0.774636i	−0.550969	+	0.834526i	$0.685742\pi$
$6$		0			0
$7$	−1.56940	+	18.4536i	−0.0847395	+	0.996403i
$7$	−1.56940	+	18.4536i	−0.0847395	+	0.996403i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	8.94164	+	5.16246i	0.245092	+	0.141504i	0.617515	−	0.786559i	$-0.288140\pi$
$11$	8.94164	+	5.16246i	0.245092	+	0.141504i	−0.372423	+	0.928063i	$0.621473\pi$
$12$		0			0
$13$		−	52.4866i		−	1.11978i	−0.828567	−	0.559891i	$-0.810843\pi$
$13$		−	52.4866i		−	1.11978i	0.828567	−	0.559891i	$-0.189157\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.584477	+	1.01234i	−0.00833862	+	0.0144429i	−0.870165	−	0.492761i	$-0.835988\pi$
$17$	−0.584477	+	1.01234i	−0.00833862	+	0.0144429i	0.861826	+	0.507204i	$0.169321\pi$
$18$		0			0
$19$	86.7124	−	50.0634i	1.04701	−	0.604491i	0.125199	−	0.992132i	$-0.460043\pi$
$19$	86.7124	−	50.0634i	1.04701	−	0.604491i	0.921811	+	0.387640i	$0.126710\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	90.1373	−	52.0408i	0.817171	−	0.471794i	−0.0322689	−	0.999479i	$-0.510273\pi$
$23$	90.1373	−	52.0408i	0.817171	−	0.471794i	0.849440	+	0.527685i	$0.176940\pi$
$24$		0			0
$25$	12.4950	−	21.6419i	0.0999598	−	0.173135i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	187.555i		−	1.20097i	−0.799636	−	0.600485i	$-0.794974\pi$
$29$		−	187.555i		−	1.20097i	0.799636	−	0.600485i	$-0.205026\pi$
$30$		0			0
$31$	107.524	+	62.0789i	0.622963	+	0.359668i	0.778022	−	0.628238i	$-0.216223\pi$
$31$	107.524	+	62.0789i	0.622963	+	0.359668i	−0.155059	+	0.987905i	$0.549557\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−167.669	+	78.6808i	−0.809748	+	0.379985i
$36$		0			0
$37$	16.0459	+	27.7922i	0.0712952	+	0.123487i	0.899469	−	0.436984i	$-0.143953\pi$
$37$	16.0459	+	27.7922i	0.0712952	+	0.123487i	−0.828174	+	0.560471i	$0.810620\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	415.597			1.58306			0.791528	−	0.611133i	$-0.209286\pi$
$41$	415.597			1.58306			0.791528	+	0.611133i	$0.209286\pi$
$42$		0			0
$43$	193.264			0.685407			0.342703	−	0.939444i	$-0.388657\pi$
$43$	193.264			0.685407			0.342703	+	0.939444i	$0.388657\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	196.466	+	340.289i	0.609735	+	1.05609i	0.991284	+	0.131743i	$0.0420573\pi$
$47$	196.466	+	340.289i	0.609735	+	1.05609i	−0.381549	+	0.924348i	$0.624609\pi$
$48$		0			0
$49$	−338.074	−	57.9222i	−0.985638	−	0.168869i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	74.7758	+	43.1718i	0.193797	+	0.111889i	0.593759	−	0.804643i	$-0.297643\pi$
$53$	74.7758	+	43.1718i	0.193797	+	0.111889i	−0.399962	+	0.916532i	$0.630977\pi$
$54$		0			0
$55$			103.254i			0.253142i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−102.286	+	177.165i	−0.225704	+	0.390931i	−0.956531	−	0.291632i	$-0.905802\pi$
$59$	−102.286	+	177.165i	−0.225704	+	0.390931i	0.730826	+	0.682564i	$0.239135\pi$
$60$		0			0
$61$	−183.527	+	105.959i	−0.385217	+	0.222405i	−0.680086	−	0.733133i	$-0.738057\pi$
$61$	−183.527	+	105.959i	−0.385217	+	0.222405i	0.294869	+	0.955538i	$0.404724\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	454.570	−	262.446i	0.867422	−	0.500806i
$66$		0			0
$67$	−364.857	+	631.951i	−0.665290	+	1.15232i	0.313917	+	0.949450i	$0.398359\pi$
$67$	−364.857	+	631.951i	−0.665290	+	1.15232i	−0.979207	+	0.202865i	$0.934975\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			315.022i			0.526566i	0.964719	+	0.263283i	$0.0848053\pi$
$71$			315.022i			0.526566i	−0.964719	+	0.263283i	$0.915195\pi$
$72$		0			0
$73$	−899.220	−	519.165i	−1.44172	−	0.832379i	−0.443758	−	0.896147i	$-0.646355\pi$
$73$	−899.220	−	519.165i	−1.44172	−	0.832379i	−0.997965	+	0.0637680i	$0.979688\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−109.299	+	156.904i	−0.161764	+	0.232219i
$78$		0			0
$79$	607.787	+	1052.72i	0.865587	+	1.49924i	0.866463	+	0.499241i	$0.166388\pi$
$79$	607.787	+	1052.72i	0.865587	+	1.49924i	−0.000876764		1.00000i	$0.500279\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−333.797			−0.441434			−0.220717	−	0.975338i	$-0.570840\pi$
$83$	−333.797			−0.441434			−0.220717	+	0.975338i	$0.570840\pi$
$84$		0			0
$85$	−11.6901			−0.0149173
$86$		0			0
$87$		0			0
$88$		0			0
$89$	168.355	+	291.599i	0.200512	+	0.347297i	0.948693	−	0.316197i	$-0.102406\pi$
$89$	168.355	+	291.599i	0.200512	+	0.347297i	−0.748182	+	0.663494i	$0.769073\pi$
$90$		0			0
$91$	968.568	+	82.3723i	1.11575	+	0.0948897i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	867.167	+	500.659i	0.936521	+	0.540701i
$96$		0			0
$97$			893.179i			0.934934i	0.884011	+	0.467467i	$0.154833\pi$
$97$			893.179i			0.934934i	−0.884011	+	0.467467i	$0.845167\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	463.543	−	802.881i	0.456676	−	0.790986i	−0.542107	−	0.840310i	$-0.682373\pi$
$101$	463.543	−	802.881i	0.456676	−	0.790986i	0.998783	+	0.0493234i	$0.0157065\pi$
$102$		0			0
$103$	−487.361	+	281.378i	−0.466224	+	0.269175i	−0.714658	−	0.699474i	$-0.753418\pi$
$103$	−487.361	+	281.378i	−0.466224	+	0.269175i	0.248434	+	0.968649i	$0.420084\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1290.34	−	744.979i	1.16581	−	0.673082i	0.213123	−	0.977025i	$-0.431637\pi$
$107$	1290.34	−	744.979i	1.16581	−	0.673082i	0.952690	+	0.303943i	$0.0983033\pi$
$108$		0			0
$109$	−726.305	+	1258.00i	−0.638233	+	1.10545i	0.347587	+	0.937648i	$0.387001\pi$
$109$	−726.305	+	1258.00i	−0.638233	+	1.10545i	−0.985820	+	0.167804i	$0.946332\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	1761.48i		−	1.46642i	−0.680001	−	0.733211i	$-0.738021\pi$
$113$		−	1761.48i		−	1.46642i	0.680001	−	0.733211i	$-0.261979\pi$
$114$		0			0
$115$	901.419	+	520.434i	0.730937	+	0.422007i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−17.7642	−	12.3745i	−0.0136844	−	0.00953251i
$120$		0			0
$121$	−612.198	−	1060.36i	−0.459953	−	0.796663i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1499.97			1.07329
$126$		0			0
$127$	−68.3256			−0.0477395			−0.0238697	−	0.999715i	$-0.507599\pi$
$127$	−68.3256			−0.0477395			−0.0238697	+	0.999715i	$0.507599\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−242.249	−	419.588i	−0.161568	−	0.279844i	0.773863	−	0.633353i	$-0.218322\pi$
$131$	−242.249	−	419.588i	−0.161568	−	0.279844i	−0.935431	+	0.353509i	$0.884988\pi$
$132$		0			0
$133$	787.766	+	1678.73i	0.513594	+	1.09447i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1247.91	+	720.479i	0.778217	+	0.449304i	0.835798	−	0.549037i	$-0.185005\pi$
$137$	1247.91	+	720.479i	0.778217	+	0.449304i	−0.0575809	+	0.998341i	$0.518339\pi$
$138$		0			0
$139$			1445.47i			0.882035i	0.897499	+	0.441017i	$0.145382\pi$
$139$			1445.47i			0.882035i	−0.897499	+	0.441017i	$0.854618\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	270.960	−	469.316i	0.158453	−	0.274449i
$144$		0			0
$145$	1624.36	−	937.824i	0.930314	−	0.537117i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	846.151	−	488.526i	0.465231	−	0.268601i	−0.249010	−	0.968501i	$-0.580105\pi$
$149$	846.151	−	488.526i	0.465231	−	0.268601i	0.714241	+	0.699900i	$0.246772\pi$
$150$		0			0
$151$	−667.146	+	1155.53i	−0.359547	+	0.622753i	−0.987885	−	0.155187i	$-0.950402\pi$
$151$	−667.146	+	1155.53i	−0.359547	+	0.622753i	0.628338	+	0.777940i	$0.283735\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			1241.64i			0.643425i
$156$		0			0
$157$	1296.09	+	748.295i	0.658846	+	0.380385i	0.791837	−	0.610732i	$-0.209125\pi$
$157$	1296.09	+	748.295i	0.658846	+	0.380385i	−0.132991	+	0.991117i	$0.542458\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	818.881	+	1745.04i	0.400850	+	0.854211i
$162$		0			0
$163$	−1894.73	−	3281.76i	−0.910469	−	1.57698i	−0.813403	−	0.581700i	$-0.802388\pi$
$163$	−1894.73	−	3281.76i	−0.910469	−	1.57698i	−0.0970652	−	0.995278i	$-0.530946\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1202.66			0.557273			0.278637	−	0.960397i	$-0.410117\pi$
$167$	1202.66			0.557273			0.278637	+	0.960397i	$0.410117\pi$
$168$		0			0
$169$	−557.839			−0.253910
$170$		0			0
$171$		0			0
$172$		0			0
$173$	765.529	+	1325.94i	0.336428	+	0.582711i	0.983758	−	0.179499i	$-0.0574477\pi$
$173$	765.529	+	1325.94i	0.336428	+	0.582711i	−0.647330	+	0.762210i	$0.724114\pi$
$174$		0			0
$175$	379.763	+	264.543i	0.164042	+	0.114272i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	1595.31	+	921.053i	0.666141	+	0.384596i	0.794613	−	0.607117i	$-0.207674\pi$
$179$	1595.31	+	921.053i	0.666141	+	0.384596i	−0.128472	+	0.991713i	$0.541007\pi$
$180$		0			0
$181$			325.049i			0.133485i	0.997770	+	0.0667423i	$0.0212605\pi$
$181$			325.049i			0.133485i	−0.997770	+	0.0667423i	$0.978739\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−160.467	+	277.936i	−0.0637716	+	0.110456i
$186$		0			0
$187$	−10.4524	+	6.03468i	−0.00408745	+	0.00235989i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	3045.91	−	1758.56i	1.15390	−	0.666203i	0.204063	−	0.978958i	$-0.434585\pi$
$191$	3045.91	−	1758.56i	1.15390	−	0.666203i	0.949834	+	0.312755i	$0.101252\pi$
$192$		0			0
$193$	−502.491	+	870.339i	−0.187410	+	0.324603i	−0.944386	−	0.328839i	$-0.893343\pi$
$193$	−502.491	+	870.339i	−0.187410	+	0.324603i	0.756976	+	0.653442i	$0.226676\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			2808.91i			1.01587i	0.861395	+	0.507935i	$0.169591\pi$
$197$			2808.91i			1.01587i	−0.861395	+	0.507935i	$0.830409\pi$
$198$		0			0
$199$	210.249	+	121.387i	0.0748954	+	0.0432409i	0.536980	−	0.843595i	$-0.319565\pi$
$199$	210.249	+	121.387i	0.0748954	+	0.0432409i	−0.462085	+	0.886836i	$0.652898\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	3461.08	+	294.349i	1.19665	+	0.101770i
$204$		0			0
$205$	2078.09	+	3599.35i	0.707999	+	1.22629i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1033.80			0.342151
$210$		0			0
$211$	−2238.46			−0.730342			−0.365171	−	0.930940i	$-0.618989\pi$
$211$	−2238.46			−0.730342			−0.365171	+	0.930940i	$0.618989\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	966.369	+	1673.80i	0.306539	+	0.530941i
$216$		0			0
$217$	−1314.33	+	1886.78i	−0.411164	+	0.590244i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	53.1345	+	30.6772i	0.0161729	+	0.00933743i
$222$		0			0
$223$		−	724.158i		−	0.217458i	−0.994071	−	0.108729i	$-0.965322\pi$
$223$		−	724.158i		−	0.217458i	0.994071	−	0.108729i	$-0.0346781\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1234.34	−	2137.94i	0.360908	−	0.625111i	−0.627202	−	0.778856i	$-0.715800\pi$
$227$	1234.34	−	2137.94i	0.360908	−	0.625111i	0.988111	+	0.153745i	$0.0491335\pi$
$228$		0			0
$229$	3595.98	−	2076.14i	1.03768	−	0.599106i	0.118507	−	0.992953i	$-0.462189\pi$
$229$	3595.98	−	2076.14i	1.03768	−	0.599106i	0.919176	+	0.393847i	$0.128856\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−5140.63	+	2967.94i	−1.44538	+	0.834492i	−0.998201	−	0.0599522i	$-0.980905\pi$
$233$	−5140.63	+	2967.94i	−1.44538	+	0.834492i	−0.447180	+	0.894444i	$0.647572\pi$
$234$		0			0
$235$	−1964.76	+	3403.06i	−0.545391	+	0.944644i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	4306.49i		−	1.16554i	−0.812638	−	0.582769i	$-0.801969\pi$
$239$		−	4306.49i		−	1.16554i	0.812638	−	0.582769i	$-0.198031\pi$
$240$		0			0
$241$	674.728	+	389.554i	0.180345	+	0.104122i	0.587455	−	0.809257i	$-0.300130\pi$
$241$	674.728	+	389.554i	0.180345	+	0.104122i	−0.407110	+	0.913379i	$0.633463\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1188.81	−	3217.58i	−0.310001	−	0.839035i
$246$		0			0
$247$	−2627.66	−	4551.23i	−0.676898	−	1.17242i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	804.927			0.202416			0.101208	−	0.994865i	$-0.467729\pi$
$251$	804.927			0.202416			0.101208	+	0.994865i	$0.467729\pi$
$252$		0			0
$253$	1074.63			0.267042
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2967.92	+	5140.59i	0.720365	+	1.24771i	0.960854	+	0.277057i	$0.0893589\pi$
$257$	2967.92	+	5140.59i	0.720365	+	1.24771i	−0.240489	+	0.970652i	$0.577308\pi$
$258$		0			0
$259$	−538.050	+	252.487i	−0.129084	+	0.0605745i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−6468.89	−	3734.82i	−1.51669	−	0.875660i	−0.999808	−	0.0196026i	$-0.993760\pi$
$263$	−6468.89	−	3734.82i	−1.51669	−	0.875660i	−0.516880	−	0.856058i	$-0.672907\pi$
$264$		0			0
$265$			863.480i			0.200163i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	3714.97	−	6434.52i	0.842029	−	1.45844i	−0.0461472	−	0.998935i	$-0.514694\pi$
$269$	3714.97	−	6434.52i	0.842029	−	1.45844i	0.888176	−	0.459503i	$-0.151972\pi$
$270$		0			0
$271$	−1076.21	+	621.349i	−0.241236	+	0.139278i	−0.615745	−	0.787946i	$-0.711145\pi$
$271$	−1076.21	+	621.349i	−0.241236	+	0.139278i	0.374509	+	0.927223i	$0.377811\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	223.451	−	129.010i	0.0489986	−	0.0282894i
$276$		0			0
$277$	−339.471	+	587.980i	−0.0736347	+	0.127539i	−0.900492	−	0.434873i	$-0.856793\pi$
$277$	−339.471	+	587.980i	−0.0736347	+	0.127539i	0.826857	+	0.562412i	$0.190127\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			2157.54i			0.458036i	0.973422	+	0.229018i	$0.0735515\pi$
$281$			2157.54i			0.458036i	−0.973422	+	0.229018i	$0.926449\pi$
$282$		0			0
$283$	843.962	+	487.262i	0.177273	+	0.102349i	0.586011	−	0.810303i	$-0.300698\pi$
$283$	843.962	+	487.262i	0.177273	+	0.102349i	−0.408738	+	0.912652i	$0.634031\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−652.236	+	7669.27i	−0.134147	+	1.57736i
$288$		0			0
$289$	2455.82	+	4253.60i	0.499861	+	0.865785i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−6457.16			−1.28748			−0.643740	−	0.765245i	$-0.722618\pi$
$293$	−6457.16			−1.28748			−0.643740	+	0.765245i	$0.722618\pi$
$294$		0			0
$295$	−2045.83			−0.403773
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2731.44	−	4731.00i	−0.528306	−	0.915053i
$300$		0			0
$301$	−303.308	+	3566.43i	−0.0580810	+	0.682942i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1835.36	−	1059.65i	−0.344566	−	0.198935i
$306$		0			0
$307$			4133.66i			0.768471i	0.923235	+	0.384235i	$0.125535\pi$
$307$			4133.66i			0.768471i	−0.923235	+	0.384235i	$0.874465\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	1222.12	−	2116.77i	0.222829	−	0.385952i	−0.732837	−	0.680405i	$-0.761804\pi$
$311$	1222.12	−	2116.77i	0.222829	−	0.385952i	0.955666	+	0.294453i	$0.0951374\pi$
$312$		0			0
$313$	3072.40	−	1773.85i	0.554831	−	0.320332i	−0.196237	−	0.980556i	$-0.562872\pi$
$313$	3072.40	−	1773.85i	0.554831	−	0.320332i	0.751068	+	0.660225i	$0.229539\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	6357.24	−	3670.36i	1.12637	−	0.650308i	0.183348	−	0.983048i	$-0.441306\pi$
$317$	6357.24	−	3670.36i	1.12637	−	0.650308i	0.943019	+	0.332740i	$0.107973\pi$
$318$		0			0
$319$	968.247	−	1677.05i	0.169942	−	0.294348i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			117.044i			0.0201625i
$324$		0			0
$325$	−1135.91	−	655.818i	−0.193874	−	0.111933i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−6587.91	+	3091.47i	−1.10396	+	0.518049i
$330$		0			0
$331$	−2815.66	−	4876.87i	−0.467561	−	0.809840i	0.531752	−	0.846900i	$-0.321534\pi$
$331$	−2815.66	−	4876.87i	−0.467561	−	0.809840i	−0.999313	+	0.0370602i	$0.988201\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−7297.51			−1.19017
$336$		0			0
$337$	−9281.97			−1.50036			−0.750180	−	0.661234i	$-0.770033\pi$
$337$	−9281.97			−1.50036			−0.750180	+	0.661234i	$0.770033\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	640.960	+	1110.17i	0.101789	+	0.176303i
$342$		0			0
$343$	1599.45	−	6147.79i	0.251784	−	0.967783i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	4441.01	+	2564.02i	0.687049	+	0.396668i	0.802506	−	0.596644i	$-0.203500\pi$
$347$	4441.01	+	2564.02i	0.687049	+	0.396668i	−0.115456	+	0.993313i	$0.536833\pi$
$348$		0			0
$349$		−	11096.5i		−	1.70196i	−0.525199	−	0.850980i	$-0.676009\pi$
$349$		−	11096.5i		−	1.70196i	0.525199	−	0.850980i	$-0.323991\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	401.521	−	695.454i	0.0605405	−	0.104859i	−0.834167	−	0.551512i	$-0.814051\pi$
$353$	401.521	−	695.454i	0.0605405	−	0.104859i	0.894707	+	0.446653i	$0.147384\pi$
$354$		0			0
$355$	−2728.30	+	1575.19i	−0.407897	+	0.235499i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1596.45	−	921.712i	0.234701	−	0.135504i	−0.378038	−	0.925790i	$-0.623401\pi$
$359$	1596.45	−	921.712i	0.234701	−	0.135504i	0.612739	+	0.790286i	$0.290068\pi$
$360$		0			0
$361$	1583.19	−	2742.16i	0.230819	−	0.399791i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	10383.8i		−	1.48908i
$366$		0			0
$367$	10118.6	+	5841.98i	1.43920	+	0.830924i	0.997794	−	0.0663827i	$-0.0211458\pi$
$367$	10118.6	+	5841.98i	1.43920	+	0.830924i	0.441408	+	0.897307i	$0.354479\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−914.030	+	1312.13i	−0.127909	+	0.183619i
$372$		0			0
$373$	541.327	+	937.606i	0.0751444	+	0.130154i	0.901149	−	0.433509i	$-0.142725\pi$
$373$	541.327	+	937.606i	0.0751444	+	0.130154i	−0.826005	+	0.563663i	$0.809392\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−9844.13			−1.34482
$378$		0			0
$379$	13929.3			1.88786			0.943929	−	0.330148i	$-0.107099\pi$
$379$	13929.3			1.88786			0.943929	+	0.330148i	$0.107099\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1693.36	−	2932.98i	−0.225918	−	0.391301i	0.730677	−	0.682724i	$-0.239205\pi$
$383$	−1693.36	−	2932.98i	−0.225918	−	0.391301i	−0.956594	+	0.291423i	$0.905871\pi$
$384$		0			0
$385$	−1905.42	−	162.047i	−0.252232	−	0.0214511i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	6652.61	+	3840.89i	0.867097	+	0.500619i	0.866382	−	0.499381i	$-0.166439\pi$
$389$	6652.61	+	3840.89i	0.867097	+	0.500619i	0.000714618		1.00000i	$0.499773\pi$
$390$		0			0
$391$			121.667i			0.0157364i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−6078.17	+	10527.7i	−0.774243	+	1.34103i
$396$		0			0
$397$	8528.01	−	4923.65i	1.07811	−	0.622446i	0.147722	−	0.989029i	$-0.452806\pi$
$397$	8528.01	−	4923.65i	1.07811	−	0.622446i	0.930385	+	0.366583i	$0.119472\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−11009.5	+	6356.33i	−1.37104	+	0.791571i	−0.991059	−	0.133423i	$-0.957403\pi$
$401$	−11009.5	+	6356.33i	−1.37104	+	0.791571i	−0.379982	+	0.924994i	$0.624070\pi$
$402$		0			0
$403$	3258.31	−	5643.55i	0.402749	−	0.697582i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			331.344i			0.0403541i
$408$		0			0
$409$	−13544.0	−	7819.64i	−1.63743	−	0.945370i	−0.981714	−	0.190362i	$-0.939034\pi$
$409$	−13544.0	−	7819.64i	−1.63743	−	0.945370i	−0.655715	−	0.755009i	$-0.727633\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−3108.82	−	2165.60i	−0.370399	−	0.258020i
$414$		0			0
$415$	−1669.07	−	2890.91i	−0.197425	−	0.341950i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−10568.3			−1.23221			−0.616105	−	0.787664i	$-0.711290\pi$
$419$	−10568.3			−1.23221			−0.616105	+	0.787664i	$0.711290\pi$
$420$		0			0
$421$	−3890.36			−0.450367			−0.225184	−	0.974316i	$-0.572298\pi$
$421$	−3890.36			−0.450367			−0.225184	+	0.974316i	$0.572298\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	14.6061	+	25.2984i	0.00166705	+	0.00288742i
$426$		0			0
$427$	−1667.31	−	3553.04i	−0.188962	−	0.402678i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	2554.42	+	1474.79i	0.285480	+	0.164822i	0.635902	−	0.771770i	$-0.280628\pi$
$431$	2554.42	+	1474.79i	0.285480	+	0.164822i	−0.350422	+	0.936592i	$0.613962\pi$
$432$		0			0
$433$		−	14292.9i		−	1.58632i	−0.609017	−	0.793158i	$-0.708436\pi$
$433$		−	14292.9i		−	1.58632i	0.609017	−	0.793158i	$-0.291564\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	5210.68	−	9025.17i	0.570391	−	0.987946i
$438$		0			0
$439$	−10289.2	+	5940.50i	−1.11863	+	0.645842i	−0.941051	−	0.338265i	$-0.890160\pi$
$439$	−10289.2	+	5940.50i	−1.11863	+	0.645842i	−0.177580	+	0.984106i	$0.556827\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−8000.43	+	4619.05i	−0.858040	+	0.495390i	−0.863356	−	0.504596i	$-0.831641\pi$
$443$	−8000.43	+	4619.05i	−0.858040	+	0.495390i	0.00531517	+	0.999986i	$0.498308\pi$
$444$		0			0
$445$	−1683.63	+	2916.14i	−0.179352	+	0.310647i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			5478.22i			0.575798i	0.957661	+	0.287899i	$0.0929568\pi$
$449$			5478.22i			0.575798i	−0.957661	+	0.287899i	$0.907043\pi$
$450$		0			0
$451$	3716.12	+	2145.50i	0.387994	+	0.224008i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	4129.68	+	8800.35i	0.425500	+	0.906740i
$456$		0			0
$457$	−4423.49	−	7661.71i	−0.452784	−	0.784244i	0.545774	−	0.837932i	$-0.316236\pi$
$457$	−4423.49	−	7661.71i	−0.452784	−	0.784244i	−0.998558	+	0.0536880i	$0.982902\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	15927.8			1.60917			0.804587	−	0.593834i	$-0.202386\pi$
$461$	15927.8			1.60917			0.804587	+	0.593834i	$0.202386\pi$
$462$		0			0
$463$	10491.3			1.05307			0.526535	−	0.850153i	$-0.323491\pi$
$463$	10491.3			1.05307			0.526535	+	0.850153i	$0.323491\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−7862.95	−	13619.0i	−0.779130	−	1.34949i	−0.932444	−	0.361316i	$-0.882328\pi$
$467$	−7862.95	−	13619.0i	−0.779130	−	1.34949i	0.153313	−	0.988178i	$-0.451006\pi$
$468$		0			0
$469$	−11089.2	−	7724.73i	−1.09179	−	0.760543i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1728.10	+	997.718i	0.167987	+	0.0969876i
$474$		0			0
$475$		−	2502.16i		−	0.241699i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−8758.87	+	15170.8i	−0.835496	+	1.44712i	0.0581291	+	0.998309i	$0.481487\pi$
$479$	−8758.87	+	15170.8i	−0.835496	+	1.44712i	−0.893626	+	0.448813i	$0.851847\pi$
$480$		0			0
$481$	1458.72	−	842.192i	0.138278	−	0.0798350i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−7735.54	+	4466.12i	−0.724233	+	0.418136i
$486$		0			0
$487$	−7026.77	+	12170.7i	−0.653826	+	1.13246i	0.328360	+	0.944553i	$0.393504\pi$
$487$	−7026.77	+	12170.7i	−0.653826	+	1.13246i	−0.982187	+	0.187908i	$0.939829\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			9854.09i			0.905720i	0.891582	+	0.452860i	$0.149596\pi$
$491$			9854.09i			0.905720i	−0.891582	+	0.452860i	$0.850404\pi$
$492$		0			0
$493$	189.871	+	109.622i	0.0173455	+	0.0100144i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−5813.30	−	494.394i	−0.524672	−	0.0446209i
$498$		0			0
$499$	−6290.14	−	10894.8i	−0.564300	−	0.977396i	−0.997114	−	0.0759128i	$-0.975813\pi$
$499$	−6290.14	−	10894.8i	−0.564300	−	0.977396i	0.432815	−	0.901483i	$-0.357520\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−2545.29			−0.225624			−0.112812	−	0.993616i	$-0.535986\pi$
$503$	−2545.29			−0.225624			−0.112812	+	0.993616i	$0.535986\pi$
$504$		0			0
$505$	9271.33			0.816968
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−7388.51	−	12797.3i	−0.643399	−	1.11440i	−0.984669	−	0.174434i	$-0.944190\pi$
$509$	−7388.51	−	12797.3i	−0.643399	−	1.11440i	0.341270	−	0.939965i	$-0.389143\pi$
$510$		0			0
$511$	10991.7	−	15779.1i	0.951556	−	1.36600i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−4873.85	−	2813.92i	−0.417025	−	0.240769i
$516$		0			0
$517$			4056.99i			0.345119i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−25.4037	+	44.0005i	−0.00213619	+	0.00369999i	−0.867092	−	0.498149i	$-0.834013\pi$
$521$	−25.4037	+	44.0005i	−0.00213619	+	0.00369999i	0.864955	+	0.501849i	$0.167347\pi$
$522$		0			0
$523$	12714.2	−	7340.56i	1.06301	−	0.613729i	0.136746	−	0.990606i	$-0.456335\pi$
$523$	12714.2	−	7340.56i	1.06301	−	0.613729i	0.926263	+	0.376877i	$0.123002\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−125.690	+	72.5674i	−0.0103893	+	0.00599827i
$528$		0			0
$529$	−667.006	+	1155.29i	−0.0548209	+	0.0949526i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	21813.2i		−	1.77268i
$534$		0			0
$535$	12904.1	+	7450.16i	1.04279	+	0.602053i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−2723.92	−	2263.21i	−0.217676	−	0.180860i
$540$		0			0
$541$	8294.42	+	14366.4i	0.659159	+	1.14170i	0.980834	+	0.194846i	$0.0624208\pi$
$541$	8294.42	+	14366.4i	0.659159	+	1.14170i	−0.321675	+	0.946850i	$0.604246\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−14526.8			−1.14176
$546$		0			0
$547$	−2029.96			−0.158674			−0.0793371	−	0.996848i	$-0.525280\pi$
$547$	−2029.96			−0.158674			−0.0793371	+	0.996848i	$0.525280\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−9389.66	−	16263.4i	−0.725976	−	1.25743i
$552$		0			0
$553$	−20380.3	+	9563.75i	−1.56720	+	0.735428i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	2597.94	+	1499.92i	0.197627	+	0.114100i	0.595548	−	0.803320i	$-0.296935\pi$
$557$	2597.94	+	1499.92i	0.197627	+	0.114100i	−0.397921	+	0.917420i	$0.630268\pi$
$558$		0			0
$559$		−	10143.8i		−	0.767506i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	8333.02	−	14433.2i	0.623792	−	1.08044i	−0.364981	−	0.931015i	$-0.618925\pi$
$563$	8333.02	−	14433.2i	0.623792	−	1.08044i	0.988773	−	0.149424i	$-0.0477420\pi$
$564$		0			0
$565$	15255.6	−	8807.82i	1.13594	−	0.655837i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−16427.3	+	9484.30i	−1.21031	+	0.698774i	−0.962827	−	0.270118i	$-0.912937\pi$
$569$	−16427.3	+	9484.30i	−1.21031	+	0.698774i	−0.247485	+	0.968892i	$0.579604\pi$
$570$		0			0
$571$	−7667.11	+	13279.8i	−0.561924	+	0.973281i	0.435405	+	0.900235i	$0.356605\pi$
$571$	−7667.11	+	13279.8i	−0.561924	+	0.973281i	−0.997329	+	0.0730460i	$0.976728\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	2601.00i		−	0.188642i
$576$		0			0
$577$	−10576.2	−	6106.20i	−0.763076	−	0.440562i	0.0673231	−	0.997731i	$-0.478554\pi$
$577$	−10576.2	−	6106.20i	−0.763076	−	0.440562i	−0.830399	+	0.557169i	$0.811887\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	523.861	−	6159.78i	0.0374069	−	0.439846i
$582$		0			0
$583$	445.746	+	772.054i	0.0316653	+	0.0548460i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−23173.7			−1.62944			−0.814718	−	0.579857i	$-0.803108\pi$
$587$	−23173.7			−1.62944			−0.814718	+	0.579857i	$0.803108\pi$
$588$		0			0
$589$	12431.5			0.869664
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1894.64	−	3281.61i	−0.131203	−	0.227251i	0.792937	−	0.609303i	$-0.208551\pi$
$593$	−1894.64	−	3281.61i	−0.131203	−	0.227251i	−0.924141	+	0.382052i	$0.875217\pi$
$594$		0			0
$595$	18.3465	−	215.726i	0.00126409	−	0.0148637i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	13663.6	+	7888.67i	0.932018	+	0.538101i	0.887449	−	0.460906i	$-0.152475\pi$
$599$	13663.6	+	7888.67i	0.932018	+	0.538101i	0.0445687	+	0.999006i	$0.485809\pi$
$600$		0			0
$601$		−	23750.2i		−	1.61197i	−0.591939	−	0.805983i	$-0.701638\pi$
$601$		−	23750.2i		−	1.61197i	0.591939	−	0.805983i	$-0.298362\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	6122.29	−	10604.1i	0.411416	−	0.712593i
$606$		0			0
$607$	−21418.8	+	12366.1i	−1.43223	+	0.826897i	−0.997290	−	0.0735641i	$-0.976563\pi$
$607$	−21418.8	+	12366.1i	−1.43223	+	0.826897i	−0.434937	+	0.900461i	$0.643229\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	17860.6	−	10311.8i	1.18259	−	0.682769i
$612$		0			0
$613$	4659.04	−	8069.69i	0.306977	−	0.531700i	−0.670723	−	0.741708i	$-0.734016\pi$
$613$	4659.04	−	8069.69i	0.306977	−	0.531700i	0.977700	+	0.210009i	$0.0673492\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			16544.3i			1.07949i	0.841828	+	0.539746i	$0.181480\pi$
$617$			16544.3i			1.07949i	−0.841828	+	0.539746i	$0.818520\pi$
$618$		0			0
$619$	8835.49	+	5101.18i	0.573713	+	0.331234i	0.758631	−	0.651520i	$-0.225869\pi$
$619$	8835.49	+	5101.18i	0.573713	+	0.331234i	−0.184918	+	0.982754i	$0.559202\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−5645.28	+	2649.12i	−0.363039	+	0.170361i
$624$		0			0
$625$	5938.38	+	10285.6i	0.380056	+	0.658277i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−37.5137			−0.00237801
$630$		0			0
$631$	−14729.3			−0.929259			−0.464630	−	0.885505i	$-0.653813\pi$
$631$	−14729.3			−0.929259			−0.464630	+	0.885505i	$0.653813\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−341.645	−	591.746i	−0.0213508	−	0.0369807i
$636$		0			0
$637$	−3040.14	+	17744.3i	−0.189097	+	1.10370i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	25503.6	+	14724.5i	1.57150	+	0.907304i	0.995986	+	0.0895107i	$0.0285303\pi$
$641$	25503.6	+	14724.5i	1.57150	+	0.907304i	0.575511	+	0.817794i	$0.304803\pi$
$642$		0			0
$643$			4194.07i			0.257228i	0.991695	+	0.128614i	$0.0410529\pi$
$643$			4194.07i			0.257228i	−0.991695	+	0.128614i	$0.958947\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	10732.4	−	18589.0i	0.652137	−	1.12953i	−0.330466	−	0.943818i	$-0.607206\pi$
$647$	10732.4	−	18589.0i	0.652137	−	1.12953i	0.982603	−	0.185717i	$-0.0594608\pi$
$648$		0			0
$649$	−1829.22	+	1056.10i	−0.110636	+	0.0638760i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−28232.5	+	16300.1i	−1.69192	+	0.976832i	−0.738956	+	0.673753i	$0.764681\pi$
$653$	−28232.5	+	16300.1i	−1.69192	+	0.976832i	−0.952966	+	0.303078i	$0.901986\pi$
$654$		0			0
$655$	2422.61	−	4196.09i	0.144518	−	0.250313i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			5931.69i			0.350631i	0.984512	+	0.175315i	$0.0560945\pi$
$659$			5931.69i			0.350631i	−0.984512	+	0.175315i	$0.943905\pi$
$660$		0			0
$661$	16409.5	+	9474.01i	0.965589	+	0.557483i	0.897889	−	0.440223i	$-0.145100\pi$
$661$	16409.5	+	9474.01i	0.965589	+	0.557483i	0.0677004	+	0.997706i	$0.478434\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−10599.9	+	15216.7i	−0.618116	+	0.887334i
$666$		0			0
$667$	−9760.53	−	16905.7i	−0.566611	−	0.981398i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−2188.05			−0.125885
$672$		0			0
$673$	−19463.2			−1.11479			−0.557394	−	0.830248i	$-0.688199\pi$
$673$	−19463.2			−1.11479			−0.557394	+	0.830248i	$0.688199\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	7010.82	+	12143.1i	0.398003	+	0.689361i	0.993479	−	0.114012i	$-0.0363701\pi$
$677$	7010.82	+	12143.1i	0.398003	+	0.689361i	−0.595477	+	0.803373i	$0.703037\pi$
$678$		0			0
$679$	−16482.4	−	1401.75i	−0.931571	−	0.0792258i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	24229.0	+	13988.6i	1.35739	+	0.783688i	0.989271	−	0.146091i	$-0.0466693\pi$
$683$	24229.0	+	13988.6i	1.35739	+	0.783688i	0.368117	+	0.929780i	$0.380003\pi$
$684$		0			0
$685$			14410.3i			0.803780i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	2265.94	−	3924.72i	0.125291	−	0.217010i
$690$		0			0
$691$	−22352.5	+	12905.2i	−1.23058	+	0.710475i	−0.967151	−	0.254203i	$-0.918187\pi$
$691$	−22352.5	+	12905.2i	−1.23058	+	0.710475i	−0.263429	+	0.964679i	$0.584853\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−12518.7	+	7227.69i	−0.683255	+	0.394478i
$696$		0			0
$697$	−242.907	+	420.727i	−0.0132005	+	0.0228639i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	15653.8i		−	0.843420i	−0.906731	−	0.421710i	$-0.861430\pi$
$701$		−	15653.8i		−	0.843420i	0.906731	−	0.421710i	$-0.138570\pi$
$702$		0			0
$703$	2782.75	+	1606.62i	0.149293	+	0.0861946i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	14088.6	+	9814.10i	0.749443	+	0.522061i
$708$		0			0
$709$	16769.6	+	29045.8i	0.888286	+	1.53856i	0.841900	+	0.539633i	$0.181437\pi$
$709$	16769.6	+	29045.8i	0.888286	+	1.53856i	0.0463856	+	0.998924i	$0.485230\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	12922.5			0.678756
$714$		0			0
$715$	5419.47			0.283464
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−6620.60	−	11467.2i	−0.343403	−	0.594791i	0.641659	−	0.766990i	$-0.278246\pi$
$719$	−6620.60	−	11467.2i	−0.343403	−	0.594791i	−0.985062	+	0.172198i	$0.944913\pi$
$720$		0			0
$721$	−4427.58	−	9435.18i	−0.228699	−	0.487357i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−4059.06	−	2343.50i	−0.207931	−	0.120049i
$726$		0			0
$727$		−	33618.0i		−	1.71502i	−0.514464	−	0.857512i	$-0.672009\pi$
$727$		−	33618.0i		−	1.71502i	0.514464	−	0.857512i	$-0.327991\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−112.958	+	195.650i	−0.00571535	+	0.00989927i
$732$		0			0
$733$	3921.55	−	2264.11i	0.197607	−	0.114088i	−0.397932	−	0.917415i	$-0.630272\pi$
$733$	3921.55	−	2264.11i	0.197607	−	0.114088i	0.595539	+	0.803327i	$0.296939\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−6524.85	+	3767.12i	−0.326114	+	0.188282i
$738$		0			0
$739$	8723.03	−	15108.7i	0.434211	−	0.752075i	−0.563020	−	0.826443i	$-0.690361\pi$
$739$	8723.03	−	15108.7i	0.434211	−	0.752075i	0.997231	+	0.0743682i	$0.0236940\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			5516.23i			0.272370i	0.990683	+	0.136185i	$0.0434842\pi$
$743$			5516.23i			0.272370i	−0.990683	+	0.136185i	$0.956516\pi$
$744$		0			0
$745$	8461.94	+	4885.50i	0.416136	+	0.240256i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	11722.5	+	24980.7i	0.571871	+	1.21866i
$750$		0			0
$751$	1743.07	+	3019.09i	0.0846947	+	0.146695i	0.905261	−	0.424856i	$-0.139675\pi$
$751$	1743.07	+	3019.09i	0.0846947	+	0.146695i	−0.820566	+	0.571551i	$0.806342\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−13343.6			−0.643209
$756$		0			0
$757$	28650.9			1.37561			0.687805	−	0.725896i	$-0.258575\pi$
$757$	28650.9			1.37561			0.687805	+	0.725896i	$0.258575\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	9804.44	+	16981.8i	0.467031	+	0.808922i	0.999291	−	0.0376594i	$-0.0119902\pi$
$761$	9804.44	+	16981.8i	0.467031	+	0.808922i	−0.532259	+	0.846581i	$0.678657\pi$
$762$		0			0
$763$	−22074.8	−	15377.3i	−1.04739	−	0.729613i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	9298.80	+	5368.66i	0.437758	+	0.252739i
$768$		0			0
$769$		−	6180.49i		−	0.289823i	−0.989445	−	0.144912i	$-0.953710\pi$
$769$		−	6180.49i		−	0.289823i	0.989445	−	0.144912i	$-0.0462898\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	18600.6	−	32217.2i	0.865483	−	1.49906i	−0.00108453	−	0.999999i	$-0.500345\pi$
$773$	18600.6	−	32217.2i	0.865483	−	1.49906i	0.866567	−	0.499060i	$-0.166321\pi$
$774$		0			0
$775$	2687.01	−	1551.35i	0.124542	−	0.0719046i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	36037.4	−	20806.2i	1.65747	−	0.956943i
$780$		0			0
$781$	−1626.29	+	2816.81i	−0.0745110	+	0.129057i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			14966.7i			0.680488i
$786$		0			0
$787$	4876.40	+	2815.39i	0.220870	+	0.127519i	0.606353	−	0.795196i	$-0.292632\pi$
$787$	4876.40	+	2815.39i	0.220870	+	0.127519i	−0.385483	+	0.922715i	$0.625965\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	32505.6	+	2764.45i	1.46115	+	0.124264i
$792$		0			0
$793$	5561.45	+	9632.71i	0.249045	+	0.431359i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	482.766			0.0214560			0.0107280	−	0.999942i	$-0.496585\pi$
$797$	482.766			0.0214560			0.0107280	+	0.999942i	$0.496585\pi$
$798$		0			0
$799$	−459.320			−0.0203374
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−5360.34	−	9284.38i	−0.235569	−	0.408018i
$804$		0			0
$805$	−11018.6	+	15817.7i	−0.482428	+	0.692547i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	6666.35	+	3848.82i	0.289711	+	0.167265i	0.637812	−	0.770192i	$-0.279840\pi$
$809$	6666.35	+	3848.82i	0.289711	+	0.167265i	−0.348100	+	0.937457i	$0.613173\pi$
$810$		0			0
$811$			17872.8i			0.773858i	0.922110	+	0.386929i	$0.126464\pi$
$811$			17872.8i			0.773858i	−0.922110	+	0.386929i	$0.873536\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	18948.2	−	32819.3i	0.814389	−	1.41056i
$816$		0			0
$817$	16758.4	−	9675.46i	0.717628	−	0.414322i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	30510.4	−	17615.2i	1.29698	−	0.748811i	0.317097	−	0.948393i	$-0.397292\pi$
$821$	30510.4	−	17615.2i	1.29698	−	0.748811i	0.979881	+	0.199582i	$0.0639585\pi$
$822$		0			0
$823$	−10049.0	+	17405.3i	−0.425619	+	0.737194i	−0.996478	−	0.0838543i	$-0.973277\pi$
$823$	−10049.0	+	17405.3i	−0.425619	+	0.737194i	0.570859	+	0.821048i	$0.306610\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			4666.09i			0.196198i	0.995177	+	0.0980990i	$0.0312762\pi$
$827$			4666.09i			0.196198i	−0.995177	+	0.0980990i	$0.968724\pi$
$828$		0			0
$829$	25975.9	+	14997.2i	1.08828	+	0.628317i	0.933117	−	0.359573i	$-0.117078\pi$
$829$	25975.9	+	14997.2i	1.08828	+	0.628317i	0.155159	+	0.987889i	$0.450411\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	256.234	−	308.393i	0.0106578	−	0.0128274i
$834$		0			0
$835$	6013.60	+	10415.9i	0.249233	+	0.431684i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	13935.9			0.573446			0.286723	−	0.958014i	$-0.407434\pi$
$839$	13935.9			0.573446			0.286723	+	0.958014i	$0.407434\pi$
$840$		0			0
$841$	−10788.0			−0.442330
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−2789.34	−	4831.27i	−0.113558	−	0.196687i
$846$		0			0
$847$	20528.3	−	9633.16i	0.832773	−	0.390790i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2892.66	+	1670.08i	0.116521	+	0.0672733i
$852$		0			0
$853$		−	1372.58i		−	0.0550952i	−0.999620	−	0.0275476i	$-0.991230\pi$
$853$		−	1372.58i		−	0.0550952i	0.999620	−	0.0275476i	$-0.00876979\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	5619.68	−	9733.57i	0.223996	−	0.387973i	−0.732022	−	0.681281i	$-0.761423\pi$
$857$	5619.68	−	9733.57i	0.223996	−	0.387973i	0.956018	+	0.293309i	$0.0947564\pi$
$858$		0			0
$859$	−37877.1	+	21868.4i	−1.50448	+	0.868613i	−0.504496	+	0.863414i	$0.668322\pi$
$859$	−37877.1	+	21868.4i	−1.50448	+	0.868613i	−0.999986	+	0.00519912i	$0.998345\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−17076.7	+	9859.22i	−0.673577	+	0.388890i	−0.797430	−	0.603411i	$-0.793808\pi$
$863$	−17076.7	+	9859.22i	−0.673577	+	0.388890i	0.123854	+	0.992300i	$0.460475\pi$
$864$		0			0
$865$	−7655.68	+	13260.0i	−0.300926	+	0.521219i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			12550.7i			0.489935i
$870$		0			0
$871$	33168.9	+	19150.1i	1.29034	+	0.744979i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2354.06	+	27680.0i	−0.0909504	+	1.06943i
$876$		0			0
$877$	−11293.8	−	19561.4i	−0.434850	−	0.753183i	0.562433	−	0.826843i	$-0.309865\pi$
$877$	−11293.8	−	19561.4i	−0.434850	−	0.753183i	−0.997283	+	0.0736598i	$0.976532\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−39105.2			−1.49545			−0.747723	−	0.664011i	$-0.768853\pi$
$881$	−39105.2			−1.49545			−0.747723	+	0.664011i	$0.768853\pi$
$882$		0			0
$883$	37839.6			1.44213			0.721066	−	0.692866i	$-0.243652\pi$
$883$	37839.6			1.44213			0.721066	+	0.692866i	$0.243652\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	8185.69	+	14178.0i	0.309863	+	0.536698i	0.978332	−	0.207041i	$-0.0663835\pi$
$887$	8185.69	+	14178.0i	0.309863	+	0.536698i	−0.668469	+	0.743740i	$0.733050\pi$
$888$		0			0
$889$	107.230	−	1260.86i	0.00404542	−	0.0475678i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	34072.1	+	19671.5i	1.27680	+	0.737158i
$894$		0			0
$895$			18422.0i			0.688022i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	11643.2	−	20166.7i	0.431950	−	0.748160i
$900$		0			0
$901$	−87.4095	+	50.4659i	−0.00323200	+	0.00186600i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−2815.15	+	1625.33i	−0.103402	+	0.0596991i
$906$		0			0
$907$	−13419.7	+	23243.6i	−0.491283	+	0.850927i	−0.999950	−	0.0100365i	$-0.996805\pi$
$907$	−13419.7	+	23243.6i	−0.491283	+	0.850927i	0.508667	+	0.860964i	$0.330139\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	45173.3i		−	1.64287i	−0.570300	−	0.821437i	$-0.693173\pi$
$911$		−	45173.3i		−	1.64287i	0.570300	−	0.821437i	$-0.306827\pi$
$912$		0			0
$913$	−2984.70	−	1723.22i	−0.108192	−	0.0624645i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	8123.11	−	3811.88i	0.292529	−	0.137273i
$918$		0			0
$919$	−10893.7	−	18868.4i	−0.391022	−	0.677270i	0.601563	−	0.798826i	$-0.294545\pi$
$919$	−10893.7	−	18868.4i	−0.391022	−	0.677270i	−0.992585	+	0.121556i	$0.961212\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	16534.4			0.589639
$924$		0			0
$925$	801.970			0.0285066
$926$		0			0
$927$		0			0
$928$		0			0
$929$	5860.89	+	10151.4i	0.206985	+	0.358509i	0.950763	−	0.309917i	$-0.100301\pi$
$929$	5860.89	+	10151.4i	0.206985	+	0.358509i	−0.743778	+	0.668427i	$0.766968\pi$
$930$		0			0
$931$	−32215.0	+	11902.6i	−1.13405	+	0.419002i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−104.529	−	60.3499i	−0.00365611	−	0.00211086i
$936$		0			0
$937$		−	9781.37i		−	0.341028i	−0.985355	−	0.170514i	$-0.945457\pi$
$937$		−	9781.37i		−	0.341028i	0.985355	−	0.170514i	$-0.0545429\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	14914.6	−	25832.9i	0.516687	−	0.894928i	−0.483125	−	0.875551i	$-0.660499\pi$
$941$	14914.6	−	25832.9i	0.516687	−	0.894928i	0.999812	−	0.0193766i	$-0.00616815\pi$
$942$		0			0
$943$	37460.8	−	21628.0i	1.29363	−	0.746876i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1073.64	+	619.868i	−0.0368413	+	0.0212703i	−0.518308	−	0.855194i	$-0.673438\pi$
$947$	−1073.64	+	619.868i	−0.0368413	+	0.0212703i	0.481466	+	0.876465i	$0.340104\pi$
$948$		0			0
$949$	−27249.2	+	47197.0i	−0.932082	+	1.61441i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	35161.2i		−	1.19516i	−0.801810	−	0.597578i	$-0.796130\pi$
$953$		−	35161.2i		−	1.19516i	0.801810	−	0.597578i	$-0.203870\pi$
$954$		0			0
$955$	30460.6	+	17586.5i	1.03213	+	0.595900i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−15253.9	+	21897.7i	−0.513634	+	0.737344i
$960$		0			0
$961$	−7187.92	−	12449.8i	−0.241278	−	0.417906i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−10050.3			−0.335265
$966$		0			0
$967$	−38907.9			−1.29389			−0.646946	−	0.762536i	$-0.723954\pi$
$967$	−38907.9			−1.29389			−0.646946	+	0.762536i	$0.723954\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	2540.26	+	4399.87i	0.0839557	+	0.145415i	0.904946	−	0.425527i	$-0.139911\pi$
$971$	2540.26	+	4399.87i	0.0839557	+	0.145415i	−0.820990	+	0.570942i	$0.806578\pi$
$972$		0			0
$973$	−26674.1	−	2268.51i	−0.878862	−	0.0747432i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	13364.5	+	7715.98i	0.437633	+	0.252667i	0.702593	−	0.711592i	$-0.252026\pi$
$977$	13364.5	+	7715.98i	0.437633	+	0.252667i	−0.264960	+	0.964259i	$0.585359\pi$
$978$		0			0
$979$			3476.50i			0.113493i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−16878.9	+	29235.2i	−0.547665	+	0.948584i	0.450769	+	0.892641i	$0.351150\pi$
$983$	−16878.9	+	29235.2i	−0.547665	+	0.948584i	−0.998434	+	0.0559429i	$0.982184\pi$
$984$		0			0
$985$	−24327.1	+	14045.2i	−0.786929	+	0.454334i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	17420.3	−	10057.6i	0.560095	−	0.323371i
$990$		0			0
$991$	−26259.7	+	45483.2i	−0.841743	+	1.45794i	0.0466765	+	0.998910i	$0.485137\pi$
$991$	−26259.7	+	45483.2i	−0.841743	+	1.45794i	−0.888420	+	0.459032i	$0.848196\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			2427.87i			0.0773555i
$996$		0			0
$997$	3156.68	+	1822.51i	0.100274	+	0.0578931i	0.549298	−	0.835626i	$-0.314895\pi$
$997$	3156.68	+	1822.51i	0.100274	+	0.0578931i	−0.449025	+	0.893519i	$0.648228\pi$
$998$		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.17.18		48
3.2	odd	2	inner	1008.4.bt.d.17.7		48
4.3	odd	2		504.4.bl.a.17.18	yes	48
7.5	odd	6	inner	1008.4.bt.d.593.7		48
12.11	even	2		504.4.bl.a.17.7	✓	48
21.5	even	6	inner	1008.4.bt.d.593.18		48
28.19	even	6		504.4.bl.a.89.7	yes	48
84.47	odd	6		504.4.bl.a.89.18	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.4.bl.a.17.7	✓	48	12.11	even	2
504.4.bl.a.17.18	yes	48	4.3	odd	2
504.4.bl.a.89.7	yes	48	28.19	even	6
504.4.bl.a.89.18	yes	48	84.47	odd	6
1008.4.bt.d.17.7		48	3.2	odd	2	inner
1008.4.bt.d.17.18		48	1.1	even	1	trivial
1008.4.bt.d.593.7		48	7.5	odd	6	inner
1008.4.bt.d.593.18		48	21.5	even	6	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(5.00025 + 8.66069i) q^{5} +(-1.56940 + 18.4536i) q^{7} +O(q^{10})\)	\(q+(5.00025 + 8.66069i) q^{5} +(-1.56940 + 18.4536i) q^{7} +(8.94164 + 5.16246i) q^{11} -52.4866i q^{13} +(-0.584477 + 1.01234i) q^{17} +(86.7124 - 50.0634i) q^{19} +(90.1373 - 52.0408i) q^{23} +(12.4950 - 21.6419i) q^{25} -187.555i q^{29} +(107.524 + 62.0789i) q^{31} +(-167.669 + 78.6808i) q^{35} +(16.0459 + 27.7922i) q^{37} +415.597 q^{41} +193.264 q^{43} +(196.466 + 340.289i) q^{47} +(-338.074 - 57.9222i) q^{49} +(74.7758 + 43.1718i) q^{53} +103.254i q^{55} +(-102.286 + 177.165i) q^{59} +(-183.527 + 105.959i) q^{61} +(454.570 - 262.446i) q^{65} +(-364.857 + 631.951i) q^{67} +315.022i q^{71} +(-899.220 - 519.165i) q^{73} +(-109.299 + 156.904i) q^{77} +(607.787 + 1052.72i) q^{79} -333.797 q^{83} -11.6901 q^{85} +(168.355 + 291.599i) q^{89} +(968.568 + 82.3723i) q^{91} +(867.167 + 500.659i) q^{95} +893.179i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 24 q^{7}+O(q^{10}) \) 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

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