LMFDB - Embedded newform 2688.2.w.b.2017.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2688,2,Mod(673,2688)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2688, base_ring=CyclotomicField(4))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2688.673");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2688 = 2^{7} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2688.w (of order $4$, 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:	$21.4637880633$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 4 x^{19} + 8 x^{18} - 16 x^{17} + 35 x^{16} - 56 x^{15} + 64 x^{14} - 84 x^{13} + 125 x^{12} + \cdots + 1024 $ x^20 - 4x^19 + 8x^18 - 16x^17 + 35x^16 - 56x^15 + 64x^14 - 84x^13 + 125x^12 - 120x^11 + 100x^10 - 240x^9 + 500x^8 - 672x^7 + 1024x^6 - 1792x^5 + 2240x^4 - 2048x^3 + 2048x^2 - 2048*x + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	no (minimal twist has level 336)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			2017.9
Root			$-0.603113 + 1.27916i$ of defining polynomial
Character	$\chi$	$=$	2688.2017
Dual form			2688.2.w.b.673.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 - 0.707107i) q^{3} +(0.894131 + 0.894131i) q^{5} -1.00000i q^{7} -1.00000i q^{9} +O(q^{10})$	\(q+(0.707107 - 0.707107i) q^{3} +(0.894131 + 0.894131i) q^{5} -1.00000i q^{7} -1.00000i q^{9} +(2.02397 + 2.02397i) q^{11} +(-3.23067 + 3.23067i) q^{13} +1.26449 q^{15} +0.119728 q^{17} +(4.85319 - 4.85319i) q^{19} +(-0.707107 - 0.707107i) q^{21} +9.33278i q^{23} -3.40106i q^{25} +(-0.707107 - 0.707107i) q^{27} +(5.18402 - 5.18402i) q^{29} -0.957732 q^{31} +2.86233 q^{33} +(0.894131 - 0.894131i) q^{35} +(-0.136949 - 0.136949i) q^{37} +4.56885i q^{39} +3.36970i q^{41} +(8.82429 + 8.82429i) q^{43} +(0.894131 - 0.894131i) q^{45} -3.50414 q^{47} -1.00000 q^{49} +(0.0846608 - 0.0846608i) q^{51} +(5.46741 + 5.46741i) q^{53} +3.61939i q^{55} -6.86345i q^{57} +(5.48985 + 5.48985i) q^{59} +(8.06921 - 8.06921i) q^{61} -1.00000 q^{63} -5.77728 q^{65} +(-8.09022 + 8.09022i) q^{67} +(6.59927 + 6.59927i) q^{69} -3.42115i q^{71} +2.48540i q^{73} +(-2.40491 - 2.40491i) q^{75} +(2.02397 - 2.02397i) q^{77} +6.89542 q^{79} -1.00000 q^{81} +(11.1660 - 11.1660i) q^{83} +(0.107053 + 0.107053i) q^{85} -7.33131i q^{87} +3.31067i q^{89} +(3.23067 + 3.23067i) q^{91} +(-0.677219 + 0.677219i) q^{93} +8.67878 q^{95} -5.52833 q^{97} +(2.02397 - 2.02397i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q+O(q^{10}) $ 20 * q	$ 20 q + 12 q^{11} - 8 q^{15} + 8 q^{19} - 12 q^{29} - 24 q^{33} - 12 q^{37} + 4 q^{43} - 20 q^{49} - 8 q^{51} + 36 q^{53} - 8 q^{61} - 20 q^{63} + 16 q^{65} - 12 q^{67} + 16 q^{69} - 16 q^{75} + 12 q^{77} - 24 q^{79} - 20 q^{81} + 40 q^{83} + 16 q^{85} - 72 q^{97} + 12 q^{99}+O(q^{100}) $ 20 * q + 12 * q^11 - 8 * q^15 + 8 * q^19 - 12 * q^29 - 24 * q^33 - 12 * q^37 + 4 * q^43 - 20 * q^49 - 8 * q^51 + 36 * q^53 - 8 * q^61 - 20 * q^63 + 16 * q^65 - 12 * q^67 + 16 * q^69 - 16 * q^75 + 12 * q^77 - 24 * q^79 - 20 * q^81 + 40 * q^83 + 16 * q^85 - 72 * q^97 + 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2688\mathbb{Z}\right)^\times$.

$n$	$127$	$1793$	$1921$	$2437$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{1}{4}\right)$

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
$3$	0.707107	−	0.707107i	0.408248	−	0.408248i
$3$	0.707107	−	0.707107i	0.408248	−	0.408248i
$4$		0			0
$5$	0.894131	+	0.894131i	0.399867	+	0.399867i	0.878186	−	0.478319i	$-0.158754\pi$
$5$	0.894131	+	0.894131i	0.399867	+	0.399867i	−0.478319	+	0.878186i	$0.658754\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$		−	1.00000i		−	0.333333i
$10$		0			0
$11$	2.02397	+	2.02397i	0.610250	+	0.610250i	0.943011	−	0.332761i	$-0.107980\pi$
$11$	2.02397	+	2.02397i	0.610250	+	0.610250i	−0.332761	+	0.943011i	$0.607980\pi$
$12$		0			0
$13$	−3.23067	+	3.23067i	−0.896026	+	0.896026i	−0.995082	−	0.0990557i	$-0.968418\pi$
$13$	−3.23067	+	3.23067i	−0.896026	+	0.896026i	0.0990557	+	0.995082i	$0.468418\pi$
$14$		0			0
$15$	1.26449			0.326490
$16$		0			0
$17$	0.119728			0.0290384			0.0145192	−	0.999895i	$-0.495378\pi$
$17$	0.119728			0.0290384			0.0145192	+	0.999895i	$0.495378\pi$
$18$		0			0
$19$	4.85319	−	4.85319i	1.11340	−	1.11340i	0.120711	−	0.992688i	$-0.461483\pi$
$19$	4.85319	−	4.85319i	1.11340	−	1.11340i	0.992688	−	0.120711i	$-0.0385174\pi$
$20$		0			0
$21$	−0.707107	−	0.707107i	−0.154303	−	0.154303i
$22$		0			0
$23$			9.33278i			1.94602i	0.230767	+	0.973009i	$0.425877\pi$
$23$			9.33278i			1.94602i	−0.230767	+	0.973009i	$0.574123\pi$
$24$		0			0
$25$		−	3.40106i		−	0.680212i
$26$		0			0
$27$	−0.707107	−	0.707107i	−0.136083	−	0.136083i
$28$		0			0
$29$	5.18402	−	5.18402i	0.962648	−	0.962648i	−0.0366787	−	0.999327i	$-0.511678\pi$
$29$	5.18402	−	5.18402i	0.962648	−	0.962648i	0.999327	+	0.0366787i	$0.0116778\pi$
$30$		0			0
$31$	−0.957732			−0.172014			−0.0860069	−	0.996295i	$-0.527411\pi$
$31$	−0.957732			−0.172014			−0.0860069	+	0.996295i	$0.527411\pi$
$32$		0			0
$33$	2.86233			0.498267
$34$		0			0
$35$	0.894131	−	0.894131i	0.151136	−	0.151136i
$36$		0			0
$37$	−0.136949	−	0.136949i	−0.0225143	−	0.0225143i	0.695760	−	0.718274i	$-0.255068\pi$
$37$	−0.136949	−	0.136949i	−0.0225143	−	0.0225143i	−0.718274	+	0.695760i	$0.755068\pi$
$38$		0			0
$39$			4.56885i			0.731602i
$40$		0			0
$41$			3.36970i			0.526258i	0.964761	+	0.263129i	$0.0847546\pi$
$41$			3.36970i			0.526258i	−0.964761	+	0.263129i	$0.915245\pi$
$42$		0			0
$43$	8.82429	+	8.82429i	1.34569	+	1.34569i	0.890282	+	0.455409i	$0.150507\pi$
$43$	8.82429	+	8.82429i	1.34569	+	1.34569i	0.455409	+	0.890282i	$0.349493\pi$
$44$		0			0
$45$	0.894131	−	0.894131i	0.133289	−	0.133289i
$46$		0			0
$47$	−3.50414			−0.511132			−0.255566	−	0.966792i	$-0.582262\pi$
$47$	−3.50414			−0.511132			−0.255566	+	0.966792i	$0.582262\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	0.0846608	−	0.0846608i	0.0118549	−	0.0118549i
$52$		0			0
$53$	5.46741	+	5.46741i	0.751006	+	0.751006i	0.974667	−	0.223661i	$-0.0718008\pi$
$53$	5.46741	+	5.46741i	0.751006	+	0.751006i	−0.223661	+	0.974667i	$0.571801\pi$
$54$		0			0
$55$			3.61939i			0.488038i
$56$		0			0
$57$		−	6.86345i		−	0.909086i
$58$		0			0
$59$	5.48985	+	5.48985i	0.714718	+	0.714718i	0.967518	−	0.252801i	$-0.0813517\pi$
$59$	5.48985	+	5.48985i	0.714718	+	0.714718i	−0.252801	+	0.967518i	$0.581352\pi$
$60$		0			0
$61$	8.06921	−	8.06921i	1.03316	−	1.03316i	0.0337247	−	0.999431i	$-0.489263\pi$
$61$	8.06921	−	8.06921i	1.03316	−	1.03316i	0.999431	−	0.0337247i	$-0.0107369\pi$
$62$		0			0
$63$	−1.00000			−0.125988
$64$		0			0
$65$	−5.77728			−0.716583
$66$		0			0
$67$	−8.09022	+	8.09022i	−0.988378	+	0.988378i	−0.999933	−	0.0115550i	$-0.996322\pi$
$67$	−8.09022	+	8.09022i	−0.988378	+	0.988378i	0.0115550	+	0.999933i	$0.496322\pi$
$68$		0			0
$69$	6.59927	+	6.59927i	0.794459	+	0.794459i
$70$		0			0
$71$		−	3.42115i		−	0.406016i	−0.979177	−	0.203008i	$-0.934928\pi$
$71$		−	3.42115i		−	0.406016i	0.979177	−	0.203008i	$-0.0650717\pi$
$72$		0			0
$73$			2.48540i			0.290894i	0.989366	+	0.145447i	$0.0464620\pi$
$73$			2.48540i			0.290894i	−0.989366	+	0.145447i	$0.953538\pi$
$74$		0			0
$75$	−2.40491	−	2.40491i	−0.277695	−	0.277695i
$76$		0			0
$77$	2.02397	−	2.02397i	0.230653	−	0.230653i
$78$		0			0
$79$	6.89542			0.775795			0.387898	−	0.921703i	$-0.373201\pi$
$79$	6.89542			0.775795			0.387898	+	0.921703i	$0.373201\pi$
$80$		0			0
$81$	−1.00000			−0.111111
$82$		0			0
$83$	11.1660	−	11.1660i	1.22563	−	1.22563i	0.260032	−	0.965600i	$-0.416267\pi$
$83$	11.1660	−	11.1660i	1.22563	−	1.22563i	0.965600	−	0.260032i	$-0.0837333\pi$
$84$		0			0
$85$	0.107053	+	0.107053i	0.0116115	+	0.0116115i
$86$		0			0
$87$		−	7.33131i		−	0.785999i
$88$		0			0
$89$			3.31067i			0.350930i	0.984486	+	0.175465i	$0.0561429\pi$
$89$			3.31067i			0.350930i	−0.984486	+	0.175465i	$0.943857\pi$
$90$		0			0
$91$	3.23067	+	3.23067i	0.338666	+	0.338666i
$92$		0			0
$93$	−0.677219	+	0.677219i	−0.0702243	+	0.0702243i
$94$		0			0
$95$	8.67878			0.890424
$96$		0			0
$97$	−5.52833			−0.561317			−0.280659	−	0.959808i	$-0.590553\pi$
$97$	−5.52833			−0.561317			−0.280659	+	0.959808i	$0.590553\pi$
$98$		0			0
$99$	2.02397	−	2.02397i	0.203417	−	0.203417i
$100$		0			0
$101$	−3.93841	−	3.93841i	−0.391887	−	0.391887i	0.483473	−	0.875359i	$-0.339375\pi$
$101$	−3.93841	−	3.93841i	−0.391887	−	0.391887i	−0.875359	+	0.483473i	$0.839375\pi$
$102$		0			0
$103$			3.09565i			0.305024i	0.988302	+	0.152512i	$0.0487362\pi$
$103$			3.09565i			0.305024i	−0.988302	+	0.152512i	$0.951264\pi$
$104$		0			0
$105$		−	1.26449i		−	0.123402i
$106$		0			0
$107$	−0.776589	−	0.776589i	−0.0750757	−	0.0750757i	0.668572	−	0.743648i	$-0.266906\pi$
$107$	−0.776589	−	0.776589i	−0.0750757	−	0.0750757i	−0.743648	+	0.668572i	$0.766906\pi$
$108$		0			0
$109$	7.63298	−	7.63298i	0.731107	−	0.731107i	−0.239732	−	0.970839i	$-0.577060\pi$
$109$	7.63298	−	7.63298i	0.731107	−	0.731107i	0.970839	+	0.239732i	$0.0770596\pi$
$110$		0			0
$111$	−0.193675			−0.0183828
$112$		0			0
$113$	−1.64790			−0.155022			−0.0775109	−	0.996992i	$-0.524697\pi$
$113$	−1.64790			−0.155022			−0.0775109	+	0.996992i	$0.524697\pi$
$114$		0			0
$115$	−8.34472	+	8.34472i	−0.778149	+	0.778149i
$116$		0			0
$117$	3.23067	+	3.23067i	0.298675	+	0.298675i
$118$		0			0
$119$		−	0.119728i		−	0.0109755i
$120$		0			0
$121$		−	2.80708i		−	0.255189i
$122$		0			0
$123$	2.38274	+	2.38274i	0.214844	+	0.214844i
$124$		0			0
$125$	7.51165	−	7.51165i	0.671862	−	0.671862i
$126$		0			0
$127$	−11.9725			−1.06239			−0.531195	−	0.847250i	$-0.678257\pi$
$127$	−11.9725			−1.06239			−0.531195	+	0.847250i	$0.678257\pi$
$128$		0			0
$129$	12.4794			1.09875
$130$		0			0
$131$	−3.95526	+	3.95526i	−0.345573	+	0.345573i	−0.858458	−	0.512885i	$-0.828577\pi$
$131$	−3.95526	+	3.95526i	−0.345573	+	0.345573i	0.512885	+	0.858458i	$0.328577\pi$
$132$		0			0
$133$	−4.85319	−	4.85319i	−0.420825	−	0.420825i
$134$		0			0
$135$		−	1.26449i		−	0.108830i
$136$		0			0
$137$		−	6.13628i		−	0.524258i	−0.965033	−	0.262129i	$-0.915575\pi$
$137$		−	6.13628i		−	0.524258i	0.965033	−	0.262129i	$-0.0844245\pi$
$138$		0			0
$139$	9.98810	+	9.98810i	0.847179	+	0.847179i	0.989780	−	0.142601i	$-0.0455465\pi$
$139$	9.98810	+	9.98810i	0.847179	+	0.847179i	−0.142601	+	0.989780i	$0.545547\pi$
$140$		0			0
$141$	−2.47780	+	2.47780i	−0.208669	+	0.208669i
$142$		0			0
$143$	−13.0776			−1.09360
$144$		0			0
$145$	9.27038			0.769863
$146$		0			0
$147$	−0.707107	+	0.707107i	−0.0583212	+	0.0583212i
$148$		0			0
$149$	8.39975	+	8.39975i	0.688134	+	0.688134i	0.961819	−	0.273685i	$-0.0882426\pi$
$149$	8.39975	+	8.39975i	0.688134	+	0.688134i	−0.273685	+	0.961819i	$0.588243\pi$
$150$		0			0
$151$			0.310686i			0.0252833i	0.999920	+	0.0126416i	$0.00402406\pi$
$151$			0.310686i			0.0252833i	−0.999920	+	0.0126416i	$0.995976\pi$
$152$		0			0
$153$		−	0.119728i		−	0.00967947i
$154$		0			0
$155$	−0.856338	−	0.856338i	−0.0687827	−	0.0687827i
$156$		0			0
$157$	−5.62609	+	5.62609i	−0.449011	+	0.449011i	−0.895026	−	0.446015i	$-0.852843\pi$
$157$	−5.62609	+	5.62609i	−0.449011	+	0.449011i	0.446015	+	0.895026i	$0.352843\pi$
$158$		0			0
$159$	7.73208			0.613194
$160$		0			0
$161$	9.33278			0.735526
$162$		0			0
$163$	−10.6342	+	10.6342i	−0.832934	+	0.832934i	−0.987917	−	0.154983i	$-0.950468\pi$
$163$	−10.6342	+	10.6342i	−0.832934	+	0.832934i	0.154983	+	0.987917i	$0.450468\pi$
$164$		0			0
$165$	2.55929	+	2.55929i	0.199241	+	0.199241i
$166$		0			0
$167$			16.4161i			1.27031i	0.772383	+	0.635157i	$0.219065\pi$
$167$			16.4161i			1.27031i	−0.772383	+	0.635157i	$0.780935\pi$
$168$		0			0
$169$		−	7.87443i		−	0.605726i
$170$		0			0
$171$	−4.85319	−	4.85319i	−0.371133	−	0.371133i
$172$		0			0
$173$	14.0510	−	14.0510i	1.06828	−	1.06828i	0.0707873	−	0.997491i	$-0.477449\pi$
$173$	14.0510	−	14.0510i	1.06828	−	1.06828i	0.997491	−	0.0707873i	$-0.0225512\pi$
$174$		0			0
$175$	−3.40106			−0.257096
$176$		0			0
$177$	7.76382			0.583565
$178$		0			0
$179$	9.72673	−	9.72673i	0.727010	−	0.727010i	−0.243013	−	0.970023i	$-0.578136\pi$
$179$	9.72673	−	9.72673i	0.727010	−	0.727010i	0.970023	+	0.243013i	$0.0781358\pi$
$180$		0			0
$181$	−16.3503	−	16.3503i	−1.21531	−	1.21531i	−0.969258	−	0.246047i	$-0.920868\pi$
$181$	−16.3503	−	16.3503i	−1.21531	−	1.21531i	−0.246047	−	0.969258i	$-0.579132\pi$
$182$		0			0
$183$		−	11.4116i		−	0.843568i
$184$		0			0
$185$		−	0.244901i		−	0.0180055i
$186$		0			0
$187$	0.242327	+	0.242327i	0.0177207	+	0.0177207i
$188$		0			0
$189$	−0.707107	+	0.707107i	−0.0514344	+	0.0514344i
$190$		0			0
$191$	3.56107			0.257670			0.128835	−	0.991666i	$-0.458876\pi$
$191$	3.56107			0.257670			0.128835	+	0.991666i	$0.458876\pi$
$192$		0			0
$193$	−3.87386			−0.278846			−0.139423	−	0.990233i	$-0.544525\pi$
$193$	−3.87386			−0.278846			−0.139423	+	0.990233i	$0.544525\pi$
$194$		0			0
$195$	−4.08515	+	4.08515i	−0.292544	+	0.292544i
$196$		0			0
$197$	2.00899	+	2.00899i	0.143135	+	0.143135i	0.775043	−	0.631908i	$-0.217728\pi$
$197$	2.00899	+	2.00899i	0.143135	+	0.143135i	−0.631908	+	0.775043i	$0.717728\pi$
$198$		0			0
$199$		−	3.24060i		−	0.229720i	−0.993382	−	0.114860i	$-0.963358\pi$
$199$		−	3.24060i		−	0.229720i	0.993382	−	0.114860i	$-0.0366419\pi$
$200$		0			0
$201$			11.4413i			0.807007i
$202$		0			0
$203$	−5.18402	−	5.18402i	−0.363847	−	0.363847i
$204$		0			0
$205$	−3.01295	+	3.01295i	−0.210434	+	0.210434i
$206$		0			0
$207$	9.33278			0.648673
$208$		0			0
$209$	19.6454			1.35890
$210$		0			0
$211$	−15.4495	+	15.4495i	−1.06358	+	1.06358i	−0.0657479	+	0.997836i	$0.520943\pi$
$211$	−15.4495	+	15.4495i	−1.06358	+	1.06358i	−0.997836	+	0.0657479i	$0.979057\pi$
$212$		0			0
$213$	−2.41912	−	2.41912i	−0.165755	−	0.165755i
$214$		0			0
$215$			15.7801i			1.07620i
$216$		0			0
$217$			0.957732i			0.0650151i
$218$		0			0
$219$	1.75744	+	1.75744i	0.118757	+	0.118757i
$220$		0			0
$221$	−0.386803	+	0.386803i	−0.0260192	+	0.0260192i
$222$		0			0
$223$	28.2506			1.89180			0.945901	−	0.324454i	$-0.105181\pi$
$223$	28.2506			1.89180			0.945901	+	0.324454i	$0.105181\pi$
$224$		0			0
$225$	−3.40106			−0.226737
$226$		0			0
$227$	2.00765	−	2.00765i	0.133252	−	0.133252i	−0.637335	−	0.770587i	$-0.719963\pi$
$227$	2.00765	−	2.00765i	0.133252	−	0.133252i	0.770587	+	0.637335i	$0.219963\pi$
$228$		0			0
$229$	−10.0493	−	10.0493i	−0.664075	−	0.664075i	0.292263	−	0.956338i	$-0.405592\pi$
$229$	−10.0493	−	10.0493i	−0.664075	−	0.664075i	−0.956338	+	0.292263i	$0.905592\pi$
$230$		0			0
$231$		−	2.86233i		−	0.188327i
$232$		0			0
$233$			8.71002i			0.570613i	0.958436	+	0.285306i	$0.0920953\pi$
$233$			8.71002i			0.570613i	−0.958436	+	0.285306i	$0.907905\pi$
$234$		0			0
$235$	−3.13316	−	3.13316i	−0.204385	−	0.204385i
$236$		0			0
$237$	4.87580	−	4.87580i	0.316717	−	0.316717i
$238$		0			0
$239$	14.0831			0.910962			0.455481	−	0.890246i	$-0.349467\pi$
$239$	14.0831			0.910962			0.455481	+	0.890246i	$0.349467\pi$
$240$		0			0
$241$	1.02384			0.0659515			0.0329757	−	0.999456i	$-0.489502\pi$
$241$	1.02384			0.0659515			0.0329757	+	0.999456i	$0.489502\pi$
$242$		0			0
$243$	−0.707107	+	0.707107i	−0.0453609	+	0.0453609i
$244$		0			0
$245$	−0.894131	−	0.894131i	−0.0571239	−	0.0571239i
$246$		0			0
$247$			31.3581i			1.99527i
$248$		0			0
$249$		−	15.7912i		−	1.00072i
$250$		0			0
$251$	16.8358	+	16.8358i	1.06266	+	1.06266i	0.997901	+	0.0647626i	$0.0206290\pi$
$251$	16.8358	+	16.8358i	1.06266	+	1.06266i	0.0647626	+	0.997901i	$0.479371\pi$
$252$		0			0
$253$	−18.8893	+	18.8893i	−1.18756	+	1.18756i
$254$		0			0
$255$	0.151396			0.00948076
$256$		0			0
$257$	−18.0887			−1.12834			−0.564171	−	0.825658i	$-0.690804\pi$
$257$	−18.0887			−1.12834			−0.564171	+	0.825658i	$0.690804\pi$
$258$		0			0
$259$	−0.136949	+	0.136949i	−0.00850960	+	0.00850960i
$260$		0			0
$261$	−5.18402	−	5.18402i	−0.320883	−	0.320883i
$262$		0			0
$263$		−	7.95967i		−	0.490814i	−0.969420	−	0.245407i	$-0.921078\pi$
$263$		−	7.95967i		−	0.490814i	0.969420	−	0.245407i	$-0.0789217\pi$
$264$		0			0
$265$			9.77716i			0.600606i
$266$		0			0
$267$	2.34100	+	2.34100i	0.143267	+	0.143267i
$268$		0			0
$269$	5.18208	−	5.18208i	0.315957	−	0.315957i	−0.531255	−	0.847212i	$-0.678279\pi$
$269$	5.18208	−	5.18208i	0.315957	−	0.315957i	0.847212	+	0.531255i	$0.178279\pi$
$270$		0			0
$271$	−12.3595			−0.750787			−0.375394	−	0.926865i	$-0.622493\pi$
$271$	−12.3595			−0.750787			−0.375394	+	0.926865i	$0.622493\pi$
$272$		0			0
$273$	4.56885			0.276520
$274$		0			0
$275$	6.88365	−	6.88365i	0.415100	−	0.415100i
$276$		0			0
$277$	9.03949	+	9.03949i	0.543130	+	0.543130i	0.924445	−	0.381315i	$-0.124529\pi$
$277$	9.03949	+	9.03949i	0.543130	+	0.543130i	−0.381315	+	0.924445i	$0.624529\pi$
$278$		0			0
$279$			0.957732i			0.0573379i
$280$		0			0
$281$		−	28.8938i		−	1.72366i	−0.507200	−	0.861828i	$-0.669319\pi$
$281$		−	28.8938i		−	1.72366i	0.507200	−	0.861828i	$-0.330681\pi$
$282$		0			0
$283$	−6.61846	−	6.61846i	−0.393427	−	0.393427i	0.482480	−	0.875907i	$-0.339736\pi$
$283$	−6.61846	−	6.61846i	−0.393427	−	0.393427i	−0.875907	+	0.482480i	$0.839736\pi$
$284$		0			0
$285$	6.13682	−	6.13682i	0.363514	−	0.363514i
$286$		0			0
$287$	3.36970			0.198907
$288$		0			0
$289$	−16.9857			−0.999157
$290$		0			0
$291$	−3.90912	+	3.90912i	−0.229157	+	0.229157i
$292$		0			0
$293$	−13.1240	−	13.1240i	−0.766712	−	0.766712i	0.210814	−	0.977526i	$-0.432388\pi$
$293$	−13.1240	−	13.1240i	−0.766712	−	0.766712i	−0.977526	+	0.210814i	$0.932388\pi$
$294$		0			0
$295$			9.81729i			0.571585i
$296$		0			0
$297$		−	2.86233i		−	0.166089i
$298$		0			0
$299$	−30.1511	−	30.1511i	−1.74368	−	1.74368i
$300$		0			0
$301$	8.82429	−	8.82429i	0.508623	−	0.508623i
$302$		0			0
$303$	−5.56976			−0.319974
$304$		0			0
$305$	14.4298			0.826251
$306$		0			0
$307$	7.54597	−	7.54597i	0.430671	−	0.430671i	−0.458185	−	0.888857i	$-0.651500\pi$
$307$	7.54597	−	7.54597i	0.430671	−	0.430671i	0.888857	+	0.458185i	$0.151500\pi$
$308$		0			0
$309$	2.18896	+	2.18896i	0.124525	+	0.124525i
$310$		0			0
$311$		−	32.5271i		−	1.84444i	−0.386665	−	0.922220i	$-0.626373\pi$
$311$		−	32.5271i		−	1.84444i	0.386665	−	0.922220i	$-0.373627\pi$
$312$		0			0
$313$		−	10.5957i		−	0.598907i	−0.954111	−	0.299453i	$-0.903196\pi$
$313$		−	10.5957i		−	0.598907i	0.954111	−	0.299453i	$-0.0968043\pi$
$314$		0			0
$315$	−0.894131	−	0.894131i	−0.0503786	−	0.0503786i
$316$		0			0
$317$	−11.5297	+	11.5297i	−0.647571	+	0.647571i	−0.952405	−	0.304834i	$-0.901399\pi$
$317$	−11.5297	+	11.5297i	−0.647571	+	0.647571i	0.304834	+	0.952405i	$0.401399\pi$
$318$		0			0
$319$	20.9846			1.17491
$320$		0			0
$321$	−1.09826			−0.0612991
$322$		0			0
$323$	0.581065	−	0.581065i	0.0323313	−	0.0323313i
$324$		0			0
$325$	10.9877	+	10.9877i	0.609488	+	0.609488i
$326$		0			0
$327$		−	10.7947i		−	0.596946i
$328$		0			0
$329$			3.50414i			0.193190i
$330$		0			0
$331$	−2.46209	−	2.46209i	−0.135328	−	0.135328i	0.636198	−	0.771526i	$-0.280506\pi$
$331$	−2.46209	−	2.46209i	−0.135328	−	0.135328i	−0.771526	+	0.636198i	$0.780506\pi$
$332$		0			0
$333$	−0.136949	+	0.136949i	−0.00750476	+	0.00750476i
$334$		0			0
$335$	−14.4674			−0.790440
$336$		0			0
$337$	16.9229			0.921850			0.460925	−	0.887439i	$-0.347518\pi$
$337$	16.9229			0.921850			0.460925	+	0.887439i	$0.347518\pi$
$338$		0			0
$339$	−1.16524	+	1.16524i	−0.0632874	+	0.0632874i
$340$		0			0
$341$	−1.93842	−	1.93842i	−0.104971	−	0.104971i
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$			11.8012i			0.635356i
$346$		0			0
$347$	−23.6687	−	23.6687i	−1.27060	−	1.27060i	−0.945772	−	0.324832i	$-0.894692\pi$
$347$	−23.6687	−	23.6687i	−1.27060	−	1.27060i	−0.324832	−	0.945772i	$-0.605308\pi$
$348$		0			0
$349$	−16.1446	+	16.1446i	−0.864201	+	0.864201i	−0.991823	−	0.127622i	$-0.959266\pi$
$349$	−16.1446	+	16.1446i	−0.864201	+	0.864201i	0.127622	+	0.991823i	$0.459266\pi$
$350$		0			0
$351$	4.56885			0.243867
$352$		0			0
$353$	33.5199			1.78409			0.892043	−	0.451950i	$-0.149271\pi$
$353$	33.5199			1.78409			0.892043	+	0.451950i	$0.149271\pi$
$354$		0			0
$355$	3.05895	−	3.05895i	0.162352	−	0.162352i
$356$		0			0
$357$	−0.0846608	−	0.0846608i	−0.00448072	−	0.00448072i
$358$		0			0
$359$		−	16.0137i		−	0.845170i	−0.906323	−	0.422585i	$-0.861123\pi$
$359$		−	16.0137i		−	0.845170i	0.906323	−	0.422585i	$-0.138877\pi$
$360$		0			0
$361$		−	28.1069i		−	1.47931i
$362$		0			0
$363$	−1.98491	−	1.98491i	−0.104181	−	0.104181i
$364$		0			0
$365$	−2.22227	+	2.22227i	−0.116319	+	0.116319i
$366$		0			0
$367$	−11.1537			−0.582220			−0.291110	−	0.956690i	$-0.594024\pi$
$367$	−11.1537			−0.582220			−0.291110	+	0.956690i	$0.594024\pi$
$368$		0			0
$369$	3.36970			0.175419
$370$		0			0
$371$	5.46741	−	5.46741i	0.283854	−	0.283854i
$372$		0			0
$373$	−17.3690	−	17.3690i	−0.899331	−	0.899331i	0.0960455	−	0.995377i	$-0.469381\pi$
$373$	−17.3690	−	17.3690i	−0.899331	−	0.899331i	−0.995377	+	0.0960455i	$0.969381\pi$
$374$		0			0
$375$		−	10.6231i		−	0.548573i
$376$		0			0
$377$			33.4957i			1.72512i
$378$		0			0
$379$	25.7851	+	25.7851i	1.32449	+	1.32449i	0.910098	+	0.414394i	$0.136006\pi$
$379$	25.7851	+	25.7851i	1.32449	+	1.32449i	0.414394	+	0.910098i	$0.363994\pi$
$380$		0			0
$381$	−8.46586	+	8.46586i	−0.433719	+	0.433719i
$382$		0			0
$383$	−4.93117			−0.251971			−0.125986	−	0.992032i	$-0.540209\pi$
$383$	−4.93117			−0.251971			−0.125986	+	0.992032i	$0.540209\pi$
$384$		0			0
$385$	3.61939			0.184461
$386$		0			0
$387$	8.82429	−	8.82429i	0.448564	−	0.448564i
$388$		0			0
$389$	−9.90961	−	9.90961i	−0.502437	−	0.502437i	0.409757	−	0.912195i	$-0.365613\pi$
$389$	−9.90961	−	9.90961i	−0.502437	−	0.502437i	−0.912195	+	0.409757i	$0.865613\pi$
$390$		0			0
$391$			1.11740i			0.0565093i
$392$		0			0
$393$			5.59359i			0.282159i
$394$		0			0
$395$	6.16540	+	6.16540i	0.310215	+	0.310215i
$396$		0			0
$397$	−23.1340	+	23.1340i	−1.16106	+	1.16106i	−0.176816	+	0.984244i	$0.556580\pi$
$397$	−23.1340	+	23.1340i	−1.16106	+	1.16106i	−0.984244	+	0.176816i	$0.943420\pi$
$398$		0			0
$399$	−6.86345			−0.343602
$400$		0			0
$401$	−25.3078			−1.26381			−0.631905	−	0.775046i	$-0.717727\pi$
$401$	−25.3078			−1.26381			−0.631905	+	0.775046i	$0.717727\pi$
$402$		0			0
$403$	3.09411	−	3.09411i	0.154129	−	0.154129i
$404$		0			0
$405$	−0.894131	−	0.894131i	−0.0444297	−	0.0444297i
$406$		0			0
$407$		−	0.554362i		−	0.0274787i
$408$		0			0
$409$		−	11.8729i		−	0.587077i	−0.955947	−	0.293538i	$-0.905167\pi$
$409$		−	11.8729i		−	0.587077i	0.955947	−	0.293538i	$-0.0948328\pi$
$410$		0			0
$411$	−4.33900	−	4.33900i	−0.214027	−	0.214027i
$412$		0			0
$413$	5.48985	−	5.48985i	0.270138	−	0.270138i
$414$		0			0
$415$	19.9678			0.980181
$416$		0			0
$417$	14.1253			0.691719
$418$		0			0
$419$	−20.7205	+	20.7205i	−1.01226	+	1.01226i	−0.0123379	+	0.999924i	$0.503927\pi$
$419$	−20.7205	+	20.7205i	−1.01226	+	1.01226i	−0.999924	+	0.0123379i	$0.996073\pi$
$420$		0			0
$421$	7.56576	+	7.56576i	0.368733	+	0.368733i	0.867015	−	0.498282i	$-0.166036\pi$
$421$	7.56576	+	7.56576i	0.368733	+	0.368733i	−0.498282	+	0.867015i	$0.666036\pi$
$422$		0			0
$423$			3.50414i			0.170377i
$424$		0			0
$425$		−	0.407204i		−	0.0197523i
$426$		0			0
$427$	−8.06921	−	8.06921i	−0.390496	−	0.390496i
$428$		0			0
$429$	−9.24723	+	9.24723i	−0.446460	+	0.446460i
$430$		0			0
$431$	−10.9428			−0.527098			−0.263549	−	0.964646i	$-0.584893\pi$
$431$	−10.9428			−0.527098			−0.263549	+	0.964646i	$0.584893\pi$
$432$		0			0
$433$	−30.4908			−1.46529			−0.732647	−	0.680609i	$-0.761715\pi$
$433$	−30.4908			−1.46529			−0.732647	+	0.680609i	$0.761715\pi$
$434$		0			0
$435$	6.55515	−	6.55515i	0.314295	−	0.314295i
$436$		0			0
$437$	45.2937	+	45.2937i	2.16669	+	2.16669i
$438$		0			0
$439$			29.1948i			1.39339i	0.717366	+	0.696696i	$0.245347\pi$
$439$			29.1948i			1.39339i	−0.717366	+	0.696696i	$0.754653\pi$
$440$		0			0
$441$			1.00000i			0.0476190i
$442$		0			0
$443$	11.6570	+	11.6570i	0.553843	+	0.553843i	0.927548	−	0.373705i	$-0.121913\pi$
$443$	11.6570	+	11.6570i	0.553843	+	0.553843i	−0.373705	+	0.927548i	$0.621913\pi$
$444$		0			0
$445$	−2.96017	+	2.96017i	−0.140325	+	0.140325i
$446$		0			0
$447$	11.8790			0.561859
$448$		0			0
$449$	−8.71500			−0.411286			−0.205643	−	0.978627i	$-0.565929\pi$
$449$	−8.71500			−0.411286			−0.205643	+	0.978627i	$0.565929\pi$
$450$		0			0
$451$	−6.82017	+	6.82017i	−0.321149	+	0.321149i
$452$		0			0
$453$	0.219688	+	0.219688i	0.0103219	+	0.0103219i
$454$		0			0
$455$			5.77728i			0.270843i
$456$		0			0
$457$			31.7466i			1.48504i	0.669822	+	0.742522i	$0.266370\pi$
$457$			31.7466i			1.48504i	−0.669822	+	0.742522i	$0.733630\pi$
$458$		0			0
$459$	−0.0846608	−	0.0846608i	−0.00395163	−	0.00395163i
$460$		0			0
$461$	−12.5034	+	12.5034i	−0.582340	+	0.582340i	−0.935546	−	0.353206i	$-0.885092\pi$
$461$	−12.5034	+	12.5034i	−0.582340	+	0.582340i	0.353206	+	0.935546i	$0.385092\pi$
$462$		0			0
$463$	−4.11446			−0.191215			−0.0956077	−	0.995419i	$-0.530479\pi$
$463$	−4.11446			−0.191215			−0.0956077	+	0.995419i	$0.530479\pi$
$464$		0			0
$465$	−1.21104			−0.0561608
$466$		0			0
$467$	10.9183	−	10.9183i	0.505238	−	0.505238i	−0.407823	−	0.913061i	$-0.633712\pi$
$467$	10.9183	−	10.9183i	0.505238	−	0.505238i	0.913061	+	0.407823i	$0.133712\pi$
$468$		0			0
$469$	8.09022	+	8.09022i	0.373572	+	0.373572i
$470$		0			0
$471$			7.95649i			0.366616i
$472$		0			0
$473$			35.7202i			1.64242i
$474$		0			0
$475$	−16.5060	−	16.5060i	−0.757347	−	0.757347i
$476$		0			0
$477$	5.46741	−	5.46741i	0.250335	−	0.250335i
$478$		0			0
$479$	−26.7048			−1.22018			−0.610088	−	0.792334i	$-0.708866\pi$
$479$	−26.7048			−1.22018			−0.610088	+	0.792334i	$0.708866\pi$
$480$		0			0
$481$	0.884874			0.0403468
$482$		0			0
$483$	6.59927	−	6.59927i	0.300277	−	0.300277i
$484$		0			0
$485$	−4.94305	−	4.94305i	−0.224452	−	0.224452i
$486$		0			0
$487$		−	17.7727i		−	0.805359i	−0.915341	−	0.402680i	$-0.868079\pi$
$487$		−	17.7727i		−	0.805359i	0.915341	−	0.402680i	$-0.131921\pi$
$488$		0			0
$489$			15.0390i			0.680088i
$490$		0			0
$491$	12.1323	+	12.1323i	0.547521	+	0.547521i	0.925723	−	0.378202i	$-0.123457\pi$
$491$	12.1323	+	12.1323i	0.547521	+	0.547521i	−0.378202	+	0.925723i	$0.623457\pi$
$492$		0			0
$493$	0.620674	−	0.620674i	0.0279538	−	0.0279538i
$494$		0			0
$495$	3.61939			0.162679
$496$		0			0
$497$	−3.42115			−0.153459
$498$		0			0
$499$	−13.2987	+	13.2987i	−0.595331	+	0.595331i	−0.939067	−	0.343736i	$-0.888308\pi$
$499$	−13.2987	+	13.2987i	−0.595331	+	0.595331i	0.343736	+	0.939067i	$0.388308\pi$
$500$		0			0
$501$	11.6079	+	11.6079i	0.518604	+	0.518604i
$502$		0			0
$503$			9.05699i			0.403831i	0.979403	+	0.201916i	$0.0647167\pi$
$503$			9.05699i			0.403831i	−0.979403	+	0.201916i	$0.935283\pi$
$504$		0			0
$505$		−	7.04291i		−	0.313405i
$506$		0			0
$507$	−5.56807	−	5.56807i	−0.247286	−	0.247286i
$508$		0			0
$509$	5.57709	−	5.57709i	0.247200	−	0.247200i	−0.572620	−	0.819821i	$-0.694073\pi$
$509$	5.57709	−	5.57709i	0.247200	−	0.247200i	0.819821	+	0.572620i	$0.194073\pi$
$510$		0			0
$511$	2.48540			0.109948
$512$		0			0
$513$	−6.86345			−0.303029
$514$		0			0
$515$	−2.76792	+	2.76792i	−0.121969	+	0.121969i
$516$		0			0
$517$	−7.09228	−	7.09228i	−0.311918	−	0.311918i
$518$		0			0
$519$		−	19.8711i		−	0.872246i
$520$		0			0
$521$		−	27.4233i		−	1.20144i	−0.799461	−	0.600718i	$-0.794882\pi$
$521$		−	27.4233i		−	1.20144i	0.799461	−	0.600718i	$-0.205118\pi$
$522$		0			0
$523$	−8.04026	−	8.04026i	−0.351576	−	0.351576i	0.509120	−	0.860696i	$-0.329971\pi$
$523$	−8.04026	−	8.04026i	−0.351576	−	0.351576i	−0.860696	+	0.509120i	$0.829971\pi$
$524$		0			0
$525$	−2.40491	+	2.40491i	−0.104959	+	0.104959i
$526$		0			0
$527$	−0.114668			−0.00499500
$528$		0			0
$529$	−64.1007			−2.78699
$530$		0			0
$531$	5.48985	−	5.48985i	0.238239	−	0.238239i
$532$		0			0
$533$	−10.8864	−	10.8864i	−0.471541	−	0.471541i
$534$		0			0
$535$		−	1.38874i		−	0.0600406i
$536$		0			0
$537$		−	13.7557i		−	0.593601i
$538$		0			0
$539$	−2.02397	−	2.02397i	−0.0871786	−	0.0871786i
$540$		0			0
$541$	−22.3119	+	22.3119i	−0.959265	+	0.959265i	−0.999202	−	0.0399372i	$-0.987284\pi$
$541$	−22.3119	+	22.3119i	−0.959265	+	0.959265i	0.0399372	+	0.999202i	$0.487284\pi$
$542$		0			0
$543$	−23.1228			−0.992293
$544$		0			0
$545$	13.6498			0.584692
$546$		0			0
$547$	−2.44272	+	2.44272i	−0.104443	+	0.104443i	−0.757397	−	0.652954i	$-0.773529\pi$
$547$	−2.44272	+	2.44272i	−0.104443	+	0.104443i	0.652954	+	0.757397i	$0.273529\pi$
$548$		0			0
$549$	−8.06921	−	8.06921i	−0.344385	−	0.344385i
$550$		0			0
$551$		−	50.3181i		−	2.14362i
$552$		0			0
$553$		−	6.89542i		−	0.293223i
$554$		0			0
$555$	−0.173171	−	0.173171i	−0.00735070	−	0.00735070i
$556$		0			0
$557$	−1.16381	+	1.16381i	−0.0493120	+	0.0493120i	−0.731333	−	0.682021i	$-0.761101\pi$
$557$	−1.16381	+	1.16381i	−0.0493120	+	0.0493120i	0.682021	+	0.731333i	$0.261101\pi$
$558$		0			0
$559$	−57.0167			−2.41155
$560$		0			0
$561$	0.342702			0.0144689
$562$		0			0
$563$	6.27044	−	6.27044i	0.264267	−	0.264267i	−0.562518	−	0.826785i	$-0.690167\pi$
$563$	6.27044	−	6.27044i	0.264267	−	0.264267i	0.826785	+	0.562518i	$0.190167\pi$
$564$		0			0
$565$	−1.47344	−	1.47344i	−0.0619882	−	0.0619882i
$566$		0			0
$567$			1.00000i			0.0419961i
$568$		0			0
$569$			33.1529i			1.38984i	0.719086	+	0.694921i	$0.244561\pi$
$569$			33.1529i			1.38984i	−0.719086	+	0.694921i	$0.755439\pi$
$570$		0			0
$571$	−1.59895	−	1.59895i	−0.0669141	−	0.0669141i	0.672858	−	0.739772i	$-0.265067\pi$
$571$	−1.59895	−	1.59895i	−0.0669141	−	0.0669141i	−0.739772	+	0.672858i	$0.765067\pi$
$572$		0			0
$573$	2.51806	−	2.51806i	0.105193	−	0.105193i
$574$		0			0
$575$	31.7413			1.32371
$576$		0			0
$577$	−35.4434			−1.47553			−0.737763	−	0.675059i	$-0.764118\pi$
$577$	−35.4434			−1.47553			−0.737763	+	0.675059i	$0.764118\pi$
$578$		0			0
$579$	−2.73923	+	2.73923i	−0.113839	+	0.113839i
$580$		0			0
$581$	−11.1660	−	11.1660i	−0.463245	−	0.463245i
$582$		0			0
$583$			22.1318i			0.916603i
$584$		0			0
$585$			5.77728i			0.238861i
$586$		0			0
$587$	−6.93808	−	6.93808i	−0.286365	−	0.286365i	0.549276	−	0.835641i	$-0.314904\pi$
$587$	−6.93808	−	6.93808i	−0.286365	−	0.286365i	−0.835641	+	0.549276i	$0.814904\pi$
$588$		0			0
$589$	−4.64806	+	4.64806i	−0.191520	+	0.191520i
$590$		0			0
$591$	2.84114			0.116869
$592$		0			0
$593$	1.28186			0.0526398			0.0263199	−	0.999654i	$-0.491621\pi$
$593$	1.28186			0.0526398			0.0263199	+	0.999654i	$0.491621\pi$
$594$		0			0
$595$	0.107053	−	0.107053i	0.00438874	−	0.00438874i
$596$		0			0
$597$	−2.29145	−	2.29145i	−0.0937827	−	0.0937827i
$598$		0			0
$599$		−	4.29013i		−	0.175290i	−0.996152	−	0.0876450i	$-0.972066\pi$
$599$		−	4.29013i		−	0.175290i	0.996152	−	0.0876450i	$-0.0279341\pi$
$600$		0			0
$601$			1.23806i			0.0505017i	0.999681	+	0.0252509i	$0.00803845\pi$
$601$			1.23806i			0.0505017i	−0.999681	+	0.0252509i	$0.991962\pi$
$602$		0			0
$603$	8.09022	+	8.09022i	0.329459	+	0.329459i
$604$		0			0
$605$	2.50990	−	2.50990i	0.102042	−	0.102042i
$606$		0			0
$607$	−27.4249			−1.11314			−0.556571	−	0.830800i	$-0.687883\pi$
$607$	−27.4249			−1.11314			−0.556571	+	0.830800i	$0.687883\pi$
$608$		0			0
$609$	−7.33131			−0.297080
$610$		0			0
$611$	11.3207	−	11.3207i	0.457987	−	0.457987i
$612$		0			0
$613$	−1.60492	−	1.60492i	−0.0648222	−	0.0648222i	0.673953	−	0.738775i	$-0.264595\pi$
$613$	−1.60492	−	1.60492i	−0.0648222	−	0.0648222i	−0.738775	+	0.673953i	$0.764595\pi$
$614$		0			0
$615$			4.26096i			0.171818i
$616$		0			0
$617$			26.8011i			1.07897i	0.841994	+	0.539486i	$0.181381\pi$
$617$			26.8011i			1.07897i	−0.841994	+	0.539486i	$0.818619\pi$
$618$		0			0
$619$	−33.4026	−	33.4026i	−1.34256	−	1.34256i	−0.893499	−	0.449066i	$-0.851757\pi$
$619$	−33.4026	−	33.4026i	−1.34256	−	1.34256i	−0.449066	−	0.893499i	$-0.648243\pi$
$620$		0			0
$621$	6.59927	−	6.59927i	0.264820	−	0.264820i
$622$		0			0
$623$	3.31067			0.132639
$624$		0			0
$625$	−3.57252			−0.142901
$626$		0			0
$627$	13.8914	−	13.8914i	0.554770	−	0.554770i
$628$		0			0
$629$	−0.0163967	−	0.0163967i	−0.000653779	−	0.000653779i
$630$		0			0
$631$		−	32.2658i		−	1.28448i	−0.766503	−	0.642241i	$-0.778005\pi$
$631$		−	32.2658i		−	1.28448i	0.766503	−	0.642241i	$-0.221995\pi$
$632$		0			0
$633$			21.8488i			0.868413i
$634$		0			0
$635$	−10.7050	−	10.7050i	−0.424815	−	0.424815i
$636$		0			0
$637$	3.23067	−	3.23067i	0.128004	−	0.128004i
$638$		0			0
$639$	−3.42115			−0.135339
$640$		0			0
$641$	16.5808			0.654904			0.327452	−	0.944868i	$-0.393810\pi$
$641$	16.5808			0.654904			0.327452	+	0.944868i	$0.393810\pi$
$642$		0			0
$643$	21.3939	−	21.3939i	0.843692	−	0.843692i	−0.145645	−	0.989337i	$-0.546526\pi$
$643$	21.3939	−	21.3939i	0.843692	−	0.843692i	0.989337	+	0.145645i	$0.0465258\pi$
$644$		0			0
$645$	11.1582	+	11.1582i	0.439355	+	0.439355i
$646$		0			0
$647$		−	42.6361i		−	1.67620i	−0.545517	−	0.838100i	$-0.683667\pi$
$647$		−	42.6361i		−	1.67620i	0.545517	−	0.838100i	$-0.316333\pi$
$648$		0			0
$649$			22.2226i			0.872313i
$650$		0			0
$651$	0.677219	+	0.677219i	0.0265423	+	0.0265423i
$652$		0			0
$653$	4.48193	−	4.48193i	0.175391	−	0.175391i	−0.613952	−	0.789343i	$-0.710421\pi$
$653$	4.48193	−	4.48193i	0.175391	−	0.175391i	0.789343	+	0.613952i	$0.210421\pi$
$654$		0			0
$655$	−7.07305			−0.276367
$656$		0			0
$657$	2.48540			0.0969646
$658$		0			0
$659$	−24.7529	+	24.7529i	−0.964238	+	0.964238i	−0.999382	−	0.0351443i	$-0.988811\pi$
$659$	−24.7529	+	24.7529i	−0.964238	+	0.964238i	0.0351443	+	0.999382i	$0.488811\pi$
$660$		0			0
$661$	−34.2858	−	34.2858i	−1.33356	−	1.33356i	−0.902159	−	0.431404i	$-0.858018\pi$
$661$	−34.2858	−	34.2858i	−1.33356	−	1.33356i	−0.431404	−	0.902159i	$-0.641982\pi$
$662$		0			0
$663$			0.547022i			0.0212446i
$664$		0			0
$665$		−	8.67878i		−	0.336548i
$666$		0			0
$667$	48.3813	+	48.3813i	1.87333	+	1.87333i
$668$		0			0
$669$	19.9762	−	19.9762i	0.772325	−	0.772325i
$670$		0			0
$671$	32.6637			1.26097
$672$		0			0
$673$	−18.8904			−0.728172			−0.364086	−	0.931365i	$-0.618619\pi$
$673$	−18.8904			−0.728172			−0.364086	+	0.931365i	$0.618619\pi$
$674$		0			0
$675$	−2.40491	+	2.40491i	−0.0925651	+	0.0925651i
$676$		0			0
$677$	15.3312	+	15.3312i	0.589227	+	0.589227i	0.937422	−	0.348195i	$-0.113205\pi$
$677$	15.3312	+	15.3312i	0.589227	+	0.589227i	−0.348195	+	0.937422i	$0.613205\pi$
$678$		0			0
$679$			5.52833i			0.212158i
$680$		0			0
$681$		−	2.83925i		−	0.108800i
$682$		0			0
$683$	−2.13282	−	2.13282i	−0.0816102	−	0.0816102i	0.665123	−	0.746734i	$-0.268379\pi$
$683$	−2.13282	−	2.13282i	−0.0816102	−	0.0816102i	−0.746734	+	0.665123i	$0.768379\pi$
$684$		0			0
$685$	5.48664	−	5.48664i	0.209634	−	0.209634i
$686$		0			0
$687$	−14.2118			−0.542215
$688$		0			0
$689$	−35.3268			−1.34584
$690$		0			0
$691$	17.0497	−	17.0497i	0.648602	−	0.648602i	−0.304053	−	0.952655i	$-0.598340\pi$
$691$	17.0497	−	17.0497i	0.648602	−	0.648602i	0.952655	+	0.304053i	$0.0983401\pi$
$692$		0			0
$693$	−2.02397	−	2.02397i	−0.0768843	−	0.0768843i
$694$		0			0
$695$			17.8613i			0.677519i
$696$		0			0
$697$			0.403449i			0.0152817i
$698$		0			0
$699$	6.15892	+	6.15892i	0.232952	+	0.232952i
$700$		0			0
$701$	−34.0605	+	34.0605i	−1.28645	+	1.28645i	−0.349518	+	0.936930i	$0.613655\pi$
$701$	−34.0605	+	34.0605i	−1.28645	+	1.28645i	−0.936930	+	0.349518i	$0.886345\pi$
$702$		0			0
$703$	−1.32928			−0.0501347
$704$		0			0
$705$	−4.43096			−0.166880
$706$		0			0
$707$	−3.93841	+	3.93841i	−0.148119	+	0.148119i
$708$		0			0
$709$	7.04022	+	7.04022i	0.264401	+	0.264401i	0.826839	−	0.562438i	$-0.190137\pi$
$709$	7.04022	+	7.04022i	0.264401	+	0.264401i	−0.562438	+	0.826839i	$0.690137\pi$
$710$		0			0
$711$		−	6.89542i		−	0.258598i
$712$		0			0
$713$		−	8.93830i		−	0.334742i
$714$		0			0
$715$	−11.6930	−	11.6930i	−0.437295	−	0.437295i
$716$		0			0
$717$	9.95827	−	9.95827i	0.371898	−	0.371898i
$718$		0			0
$719$	27.4221			1.02267			0.511335	−	0.859381i	$-0.329151\pi$
$719$	27.4221			1.02267			0.511335	+	0.859381i	$0.329151\pi$
$720$		0			0
$721$	3.09565			0.115288
$722$		0			0
$723$	0.723966	−	0.723966i	0.0269246	−	0.0269246i
$724$		0			0
$725$	−17.6312	−	17.6312i	−0.654805	−	0.654805i
$726$		0			0
$727$			1.52531i			0.0565706i	0.999600	+	0.0282853i	$0.00900469\pi$
$727$			1.52531i			0.0565706i	−0.999600	+	0.0282853i	$0.990995\pi$
$728$		0			0
$729$			1.00000i			0.0370370i
$730$		0			0
$731$	1.05652	+	1.05652i	0.0390767	+	0.0390767i
$732$		0			0
$733$	20.5365	−	20.5365i	0.758532	−	0.758532i	−0.217523	−	0.976055i	$-0.569798\pi$
$733$	20.5365	−	20.5365i	0.758532	−	0.758532i	0.976055	+	0.217523i	$0.0697979\pi$
$734$		0			0
$735$	−1.26449			−0.0466415
$736$		0			0
$737$	−32.7488			−1.20632
$738$		0			0
$739$	−10.9539	+	10.9539i	−0.402947	+	0.402947i	−0.879270	−	0.476324i	$-0.841969\pi$
$739$	−10.9539	+	10.9539i	−0.402947	+	0.402947i	0.476324	+	0.879270i	$0.341969\pi$
$740$		0			0
$741$	22.1735	+	22.1735i	0.814565	+	0.814565i
$742$		0			0
$743$		−	34.0256i		−	1.24828i	−0.781314	−	0.624138i	$-0.785450\pi$
$743$		−	34.0256i		−	1.24828i	0.781314	−	0.624138i	$-0.214550\pi$
$744$		0			0
$745$			15.0209i			0.550325i
$746$		0			0
$747$	−11.1660	−	11.1660i	−0.408544	−	0.408544i
$748$		0			0
$749$	−0.776589	+	0.776589i	−0.0283759	+	0.0283759i
$750$		0			0
$751$	3.00646			0.109707			0.0548537	−	0.998494i	$-0.482531\pi$
$751$	3.00646			0.109707			0.0548537	+	0.998494i	$0.482531\pi$
$752$		0			0
$753$	23.8094			0.867661
$754$		0			0
$755$	−0.277794	+	0.277794i	−0.0101100	+	0.0101100i
$756$		0			0
$757$	17.3543	+	17.3543i	0.630751	+	0.630751i	0.948257	−	0.317505i	$-0.102845\pi$
$757$	17.3543	+	17.3543i	0.630751	+	0.630751i	−0.317505	+	0.948257i	$0.602845\pi$
$758$		0			0
$759$			26.7135i			0.969637i
$760$		0			0
$761$		−	17.6334i		−	0.639212i	−0.947551	−	0.319606i	$-0.896449\pi$
$761$		−	17.6334i		−	0.639212i	0.947551	−	0.319606i	$-0.103551\pi$
$762$		0			0
$763$	−7.63298	−	7.63298i	−0.276332	−	0.276332i
$764$		0			0
$765$	0.107053	−	0.107053i	0.00387050	−	0.00387050i
$766$		0			0
$767$	−35.4718			−1.28081
$768$		0			0
$769$	−49.0940			−1.77037			−0.885187	−	0.465235i	$-0.845970\pi$
$769$	−49.0940			−1.77037			−0.885187	+	0.465235i	$0.845970\pi$
$770$		0			0
$771$	−12.7906	+	12.7906i	−0.460644	+	0.460644i
$772$		0			0
$773$	7.85287	+	7.85287i	0.282448	+	0.282448i	0.834085	−	0.551637i	$-0.185996\pi$
$773$	7.85287	+	7.85287i	0.282448	+	0.282448i	−0.551637	+	0.834085i	$0.685996\pi$
$774$		0			0
$775$			3.25730i			0.117006i
$776$		0			0
$777$			0.193675i			0.00694806i
$778$		0			0
$779$	16.3538	+	16.3538i	0.585935	+	0.585935i
$780$		0			0
$781$	6.92430	−	6.92430i	0.247771	−	0.247771i
$782$		0			0
$783$	−7.33131			−0.262000
$784$		0			0
$785$	−10.0609			−0.359090
$786$		0			0
$787$	7.65872	−	7.65872i	0.273004	−	0.273004i	−0.557304	−	0.830308i	$-0.688164\pi$
$787$	7.65872	−	7.65872i	0.273004	−	0.273004i	0.830308	+	0.557304i	$0.188164\pi$
$788$		0			0
$789$	−5.62834	−	5.62834i	−0.200374	−	0.200374i
$790$		0			0
$791$			1.64790i			0.0585928i
$792$		0			0
$793$			52.1378i			1.85147i
$794$		0			0
$795$	6.91349	+	6.91349i	0.245196	+	0.245196i
$796$		0			0
$797$	14.1964	−	14.1964i	0.502862	−	0.502862i	−0.409464	−	0.912326i	$-0.634284\pi$
$797$	14.1964	−	14.1964i	0.502862	−	0.502862i	0.912326	+	0.409464i	$0.134284\pi$
$798$		0			0
$799$	−0.419545			−0.0148424
$800$		0			0
$801$	3.31067			0.116977
$802$		0			0
$803$	−5.03037	+	5.03037i	−0.177518	+	0.177518i
$804$		0			0
$805$	8.34472	+	8.34472i	0.294113	+	0.294113i
$806$		0			0
$807$		−	7.32857i		−	0.257978i
$808$		0			0
$809$		−	39.3970i		−	1.38513i	−0.721357	−	0.692563i	$-0.756481\pi$
$809$		−	39.3970i		−	1.38513i	0.721357	−	0.692563i	$-0.243519\pi$
$810$		0			0
$811$	−13.5619	−	13.5619i	−0.476222	−	0.476222i	0.427699	−	0.903921i	$-0.359324\pi$
$811$	−13.5619	−	13.5619i	−0.476222	−	0.476222i	−0.903921	+	0.427699i	$0.859324\pi$
$812$		0			0
$813$	−8.73950	+	8.73950i	−0.306508	+	0.306508i
$814$		0			0
$815$	−19.0167			−0.666126
$816$		0			0
$817$	85.6519			2.99658
$818$		0			0
$819$	3.23067	−	3.23067i	0.112889	−	0.112889i
$820$		0			0
$821$	−5.06432	−	5.06432i	−0.176746	−	0.176746i	0.613190	−	0.789936i	$-0.289886\pi$
$821$	−5.06432	−	5.06432i	−0.176746	−	0.176746i	−0.789936	+	0.613190i	$0.789886\pi$
$822$		0			0
$823$			17.1851i			0.599035i	0.954091	+	0.299518i	$0.0968257\pi$
$823$			17.1851i			0.599035i	−0.954091	+	0.299518i	$0.903174\pi$
$824$		0			0
$825$		−	9.73495i		−	0.338927i
$826$		0			0
$827$	−15.6454	−	15.6454i	−0.544044	−	0.544044i	0.380668	−	0.924712i	$-0.375694\pi$
$827$	−15.6454	−	15.6454i	−0.544044	−	0.544044i	−0.924712	+	0.380668i	$0.875694\pi$
$828$		0			0
$829$	34.8216	−	34.8216i	1.20941	−	1.20941i	0.238186	−	0.971219i	$-0.423447\pi$
$829$	34.8216	−	34.8216i	1.20941	−	1.20941i	0.971219	−	0.238186i	$-0.0765529\pi$
$830$		0			0
$831$	12.7838			0.443464
$832$		0			0
$833$	−0.119728			−0.00414834
$834$		0			0
$835$	−14.6781	+	14.6781i	−0.507957	+	0.507957i
$836$		0			0
$837$	0.677219	+	0.677219i	0.0234081	+	0.0234081i
$838$		0			0
$839$			35.3335i			1.21985i	0.792460	+	0.609923i	$0.208800\pi$
$839$			35.3335i			1.21985i	−0.792460	+	0.609923i	$0.791200\pi$
$840$		0			0
$841$		−	24.7481i		−	0.853384i
$842$		0			0
$843$	−20.4310	−	20.4310i	−0.703680	−	0.703680i
$844$		0			0
$845$	7.04077	−	7.04077i	0.242210	−	0.242210i
$846$		0			0
$847$	−2.80708			−0.0964525
$848$		0			0
$849$	−9.35991			−0.321231
$850$		0			0
$851$	1.27811	−	1.27811i	0.0438132	−	0.0438132i
$852$		0			0
$853$	−6.94647	−	6.94647i	−0.237843	−	0.237843i	0.578114	−	0.815956i	$-0.303789\pi$
$853$	−6.94647	−	6.94647i	−0.237843	−	0.237843i	−0.815956	+	0.578114i	$0.803789\pi$
$854$		0			0
$855$		−	8.67878i		−	0.296808i
$856$		0			0
$857$		−	54.3462i		−	1.85643i	−0.372041	−	0.928216i	$-0.621342\pi$
$857$		−	54.3462i		−	1.85643i	0.372041	−	0.928216i	$-0.378658\pi$
$858$		0			0
$859$	4.78678	+	4.78678i	0.163323	+	0.163323i	0.784037	−	0.620714i	$-0.213157\pi$
$859$	4.78678	+	4.78678i	0.163323	+	0.163323i	−0.620714	+	0.784037i	$0.713157\pi$
$860$		0			0
$861$	2.38274	−	2.38274i	0.0812034	−	0.0812034i
$862$		0			0
$863$	−6.00992			−0.204580			−0.102290	−	0.994755i	$-0.532617\pi$
$863$	−6.00992			−0.204580			−0.102290	+	0.994755i	$0.532617\pi$
$864$		0			0
$865$	25.1269			0.854340
$866$		0			0
$867$	−12.0107	+	12.0107i	−0.407904	+	0.407904i
$868$		0			0
$869$	13.9561	+	13.9561i	0.473429	+	0.473429i
$870$		0			0
$871$		−	52.2737i		−	1.77123i
$872$		0			0
$873$			5.52833i			0.187106i
$874$		0			0
$875$	−7.51165	−	7.51165i	−0.253940	−	0.253940i
$876$		0			0
$877$	8.29062	−	8.29062i	0.279954	−	0.279954i	−0.553136	−	0.833091i	$-0.686569\pi$
$877$	8.29062	−	8.29062i	0.279954	−	0.279954i	0.833091	+	0.553136i	$0.186569\pi$
$878$		0			0
$879$	−18.5601			−0.626018
$880$		0			0
$881$	−27.6859			−0.932762			−0.466381	−	0.884584i	$-0.654442\pi$
$881$	−27.6859			−0.932762			−0.466381	+	0.884584i	$0.654442\pi$
$882$		0			0
$883$	33.1342	−	33.1342i	1.11505	−	1.11505i	0.122599	−	0.992456i	$-0.460877\pi$
$883$	33.1342	−	33.1342i	1.11505	−	1.11505i	0.992456	−	0.122599i	$-0.0391228\pi$
$884$		0			0
$885$	6.94187	+	6.94187i	0.233348	+	0.233348i
$886$		0			0
$887$		−	37.8556i		−	1.27107i	−0.772073	−	0.635534i	$-0.780780\pi$
$887$		−	37.8556i		−	1.27107i	0.772073	−	0.635534i	$-0.219220\pi$
$888$		0			0
$889$			11.9725i			0.401546i
$890$		0			0
$891$	−2.02397	−	2.02397i	−0.0678056	−	0.0678056i
$892$		0			0
$893$	−17.0063	+	17.0063i	−0.569093	+	0.569093i
$894$		0			0
$895$	17.3939			0.581415
$896$		0			0
$897$	−42.6401			−1.42371
$898$		0			0
$899$	−4.96490	+	4.96490i	−0.165589	+	0.165589i
$900$		0			0
$901$	0.654604	+	0.654604i	0.0218080	+	0.0218080i
$902$		0			0
$903$		−	12.4794i		−	0.415289i
$904$		0			0
$905$		−	29.2385i		−	0.971922i
$906$		0			0
$907$	10.8409	+	10.8409i	0.359965	+	0.359965i	0.863800	−	0.503835i	$-0.168078\pi$
$907$	10.8409	+	10.8409i	0.359965	+	0.359965i	−0.503835	+	0.863800i	$0.668078\pi$
$908$		0			0
$909$	−3.93841	+	3.93841i	−0.130629	+	0.130629i
$910$		0			0
$911$	−33.9443			−1.12463			−0.562313	−	0.826925i	$-0.690088\pi$
$911$	−33.9443			−1.12463			−0.562313	+	0.826925i	$0.690088\pi$
$912$		0			0
$913$	45.1995			1.49588
$914$		0			0
$915$	10.2034	−	10.2034i	0.337315	−	0.337315i
$916$		0			0
$917$	3.95526	+	3.95526i	0.130614	+	0.130614i
$918$		0			0
$919$			40.1474i			1.32434i	0.749353	+	0.662171i	$0.230365\pi$
$919$			40.1474i			1.32434i	−0.749353	+	0.662171i	$0.769635\pi$
$920$		0			0
$921$		−	10.6716i		−	0.351642i
$922$		0			0
$923$	11.0526	+	11.0526i	0.363801	+	0.363801i
$924$		0			0
$925$	−0.465772	+	0.465772i	−0.0153145	+	0.0153145i
$926$		0			0
$927$	3.09565			0.101675
$928$		0			0
$929$	34.8692			1.14402			0.572011	−	0.820246i	$-0.306164\pi$
$929$	34.8692			1.14402			0.572011	+	0.820246i	$0.306164\pi$
$930$		0			0
$931$	−4.85319	+	4.85319i	−0.159057	+	0.159057i
$932$		0			0
$933$	−23.0001	−	23.0001i	−0.752990	−	0.752990i
$934$		0			0
$935$			0.433344i			0.0141719i
$936$		0			0
$937$		−	45.3449i		−	1.48135i	−0.671862	−	0.740676i	$-0.734505\pi$
$937$		−	45.3449i		−	1.48135i	0.671862	−	0.740676i	$-0.265495\pi$
$938$		0			0
$939$	−7.49232	−	7.49232i	−0.244503	−	0.244503i
$940$		0			0
$941$	−3.55463	+	3.55463i	−0.115878	+	0.115878i	−0.762668	−	0.646790i	$-0.776111\pi$
$941$	−3.55463	+	3.55463i	−0.115878	+	0.115878i	0.646790	+	0.762668i	$0.276111\pi$
$942$		0			0
$943$	−31.4486			−1.02411
$944$		0			0
$945$	−1.26449			−0.0411339
$946$		0			0
$947$	−0.315602	+	0.315602i	−0.0102557	+	0.0102557i	−0.712216	−	0.701960i	$-0.752308\pi$
$947$	−0.315602	+	0.315602i	−0.0102557	+	0.0102557i	0.701960	+	0.712216i	$0.252308\pi$
$948$		0			0
$949$	−8.02950	−	8.02950i	−0.260648	−	0.260648i
$950$		0			0
$951$			16.3054i			0.528739i
$952$		0			0
$953$			15.8425i			0.513189i	0.966519	+	0.256594i	$0.0826005\pi$
$953$			15.8425i			0.513189i	−0.966519	+	0.256594i	$0.917400\pi$
$954$		0			0
$955$	3.18407	+	3.18407i	0.103034	+	0.103034i
$956$		0			0
$957$	14.8384	−	14.8384i	0.479656	−	0.479656i
$958$		0			0
$959$	−6.13628			−0.198151
$960$		0			0
$961$	−30.0827			−0.970411
$962$		0			0
$963$	−0.776589	+	0.776589i	−0.0250252	+	0.0250252i
$964$		0			0
$965$	−3.46374	−	3.46374i	−0.111502	−	0.111502i
$966$		0			0
$967$			18.3606i			0.590436i	0.955430	+	0.295218i	$0.0953922\pi$
$967$			18.3606i			0.590436i	−0.955430	+	0.295218i	$0.904608\pi$
$968$		0			0
$969$		−	0.821750i		−	0.0263984i
$970$		0			0
$971$	16.5118	+	16.5118i	0.529888	+	0.529888i	0.920539	−	0.390651i	$-0.127750\pi$
$971$	16.5118	+	16.5118i	0.529888	+	0.529888i	−0.390651	+	0.920539i	$0.627750\pi$
$972$		0			0
$973$	9.98810	−	9.98810i	0.320204	−	0.320204i
$974$		0			0
$975$	15.5390			0.497645
$976$		0			0
$977$	39.3211			1.25799			0.628997	−	0.777408i	$-0.283466\pi$
$977$	39.3211			1.25799			0.628997	+	0.777408i	$0.283466\pi$
$978$		0			0
$979$	−6.70069	+	6.70069i	−0.214155	+	0.214155i
$980$		0			0
$981$	−7.63298	−	7.63298i	−0.243702	−	0.243702i
$982$		0			0
$983$			51.1394i			1.63109i	0.578692	+	0.815547i	$0.303564\pi$
$983$			51.1394i			1.63109i	−0.578692	+	0.815547i	$0.696436\pi$
$984$		0			0
$985$			3.59260i			0.114470i
$986$		0			0
$987$	2.47780	+	2.47780i	0.0788693	+	0.0788693i
$988$		0			0
$989$	−82.3551	+	82.3551i	−2.61874	+	2.61874i
$990$		0			0
$991$	−27.1183			−0.861442			−0.430721	−	0.902485i	$-0.641741\pi$
$991$	−27.1183			−0.861442			−0.430721	+	0.902485i	$0.641741\pi$
$992$		0			0
$993$	−3.48191			−0.110495
$994$		0			0
$995$	2.89752	−	2.89752i	0.0918574	−	0.0918574i
$996$		0			0
$997$	−8.87789	−	8.87789i	−0.281166	−	0.281166i	0.552408	−	0.833574i	$-0.313709\pi$
$997$	−8.87789	−	8.87789i	−0.281166	−	0.281166i	−0.833574	+	0.552408i	$0.813709\pi$
$998$		0			0
$999$			0.193675i			0.00612761i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2688.2.w.b.2017.9		20
4.3	odd	2		2688.2.w.a.2017.4		20
8.3	odd	2		336.2.w.a.85.3	✓	20
8.5	even	2		1344.2.w.a.1009.2		20
16.3	odd	4		2688.2.w.a.673.4		20
16.5	even	4		1344.2.w.a.337.2		20
16.11	odd	4		336.2.w.a.253.3	yes	20
16.13	even	4	inner	2688.2.w.b.673.9		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.w.a.85.3	✓	20	8.3	odd	2
336.2.w.a.253.3	yes	20	16.11	odd	4
1344.2.w.a.337.2		20	16.5	even	4
1344.2.w.a.1009.2		20	8.5	even	2
2688.2.w.a.673.4		20	16.3	odd	4
2688.2.w.a.2017.4		20	4.3	odd	2
2688.2.w.b.673.9		20	16.13	even	4	inner
2688.2.w.b.2017.9		20	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:	\( N \)	\(=\)	\( 2688 = 2^{7} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2688.w (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 - 0.707107i) q^{3} +(0.894131 + 0.894131i) q^{5} -1.00000i q^{7} -1.00000i q^{9} +O(q^{10})\)	\(q+(0.707107 - 0.707107i) q^{3} +(0.894131 + 0.894131i) q^{5} -1.00000i q^{7} -1.00000i q^{9} +(2.02397 + 2.02397i) q^{11} +(-3.23067 + 3.23067i) q^{13} +1.26449 q^{15} +0.119728 q^{17} +(4.85319 - 4.85319i) q^{19} +(-0.707107 - 0.707107i) q^{21} +9.33278i q^{23} -3.40106i q^{25} +(-0.707107 - 0.707107i) q^{27} +(5.18402 - 5.18402i) q^{29} -0.957732 q^{31} +2.86233 q^{33} +(0.894131 - 0.894131i) q^{35} +(-0.136949 - 0.136949i) q^{37} +4.56885i q^{39} +3.36970i q^{41} +(8.82429 + 8.82429i) q^{43} +(0.894131 - 0.894131i) q^{45} -3.50414 q^{47} -1.00000 q^{49} +(0.0846608 - 0.0846608i) q^{51} +(5.46741 + 5.46741i) q^{53} +3.61939i q^{55} -6.86345i q^{57} +(5.48985 + 5.48985i) q^{59} +(8.06921 - 8.06921i) q^{61} -1.00000 q^{63} -5.77728 q^{65} +(-8.09022 + 8.09022i) q^{67} +(6.59927 + 6.59927i) q^{69} -3.42115i q^{71} +2.48540i q^{73} +(-2.40491 - 2.40491i) q^{75} +(2.02397 - 2.02397i) q^{77} +6.89542 q^{79} -1.00000 q^{81} +(11.1660 - 11.1660i) q^{83} +(0.107053 + 0.107053i) q^{85} -7.33131i q^{87} +3.31067i q^{89} +(3.23067 + 3.23067i) q^{91} +(-0.677219 + 0.677219i) q^{93} +8.67878 q^{95} -5.52833 q^{97} +(2.02397 - 2.02397i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q+O(q^{10}) \) 20 * q	\( 20 q + 12 q^{11} - 8 q^{15} + 8 q^{19} - 12 q^{29} - 24 q^{33} - 12 q^{37} + 4 q^{43} - 20 q^{49} - 8 q^{51} + 36 q^{53} - 8 q^{61} - 20 q^{63} + 16 q^{65} - 12 q^{67} + 16 q^{69} - 16 q^{75} + 12 q^{77} - 24 q^{79} - 20 q^{81} + 40 q^{83} + 16 q^{85} - 72 q^{97} + 12 q^{99}+O(q^{100}) \) 20 * q + 12 * q^11 - 8 * q^15 + 8 * q^19 - 12 * q^29 - 24 * q^33 - 12 * q^37 + 4 * q^43 - 20 * q^49 - 8 * q^51 + 36 * q^53 - 8 * q^61 - 20 * q^63 + 16 * q^65 - 12 * q^67 + 16 * q^69 - 16 * q^75 + 12 * q^77 - 24 * q^79 - 20 * q^81 + 40 * q^83 + 16 * q^85 - 72 * q^97 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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