LMFDB - Embedded newform 1248.2.n.a.1247.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1248,2,Mod(1247,1248)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1248, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1248.1247"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1248 = 2^{5} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1248.n (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,-4,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$9.96533017226$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1247.1
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1248.1247
Dual form			1248.2.n.a.1247.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.00000 - 1.41421i) q^{3} -3.41421 q^{5} +3.41421 q^{7} +(-1.00000 + 2.82843i) q^{9} -0.828427i q^{11} +(3.00000 - 2.00000i) q^{13} +(3.41421 + 4.82843i) q^{15} -2.82843i q^{17} -2.24264 q^{19} +(-3.41421 - 4.82843i) q^{21} +3.65685 q^{23} +6.65685 q^{25} +(5.00000 - 1.41421i) q^{27} -6.82843i q^{29} -9.07107 q^{31} +(-1.17157 + 0.828427i) q^{33} -11.6569 q^{35} -1.65685i q^{37} +(-5.82843 - 2.24264i) q^{39} -3.41421 q^{41} +8.00000i q^{43} +(3.41421 - 9.65685i) q^{45} +3.17157i q^{47} +4.65685 q^{49} +(-4.00000 + 2.82843i) q^{51} -12.0000i q^{53} +2.82843i q^{55} +(2.24264 + 3.17157i) q^{57} +3.17157i q^{59} -13.6569 q^{61} +(-3.41421 + 9.65685i) q^{63} +(-10.2426 + 6.82843i) q^{65} -12.5858 q^{67} +(-3.65685 - 5.17157i) q^{69} -8.82843i q^{71} -4.00000i q^{73} +(-6.65685 - 9.41421i) q^{75} -2.82843i q^{77} -12.4853i q^{79} +(-7.00000 - 5.65685i) q^{81} +8.82843i q^{83} +9.65685i q^{85} +(-9.65685 + 6.82843i) q^{87} -11.4142 q^{89} +(10.2426 - 6.82843i) q^{91} +(9.07107 + 12.8284i) q^{93} +7.65685 q^{95} +(2.34315 + 0.828427i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 8 q^{5} + 8 q^{7} - 4 q^{9} + 12 q^{13} + 8 q^{15} + 8 q^{19} - 8 q^{21} - 8 q^{23} + 4 q^{25} + 20 q^{27} - 8 q^{31} - 16 q^{33} - 24 q^{35} - 12 q^{39} - 8 q^{41} + 8 q^{45} - 4 q^{49}+ \cdots + 32 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 - 8 * q^5 + 8 * q^7 - 4 * q^9 + 12 * q^13 + 8 * q^15 + 8 * q^19 - 8 * q^21 - 8 * q^23 + 4 * q^25 + 20 * q^27 - 8 * q^31 - 16 * q^33 - 24 * q^35 - 12 * q^39 - 8 * q^41 + 8 * q^45 - 4 * q^49 - 16 * q^51 - 8 * q^57 - 32 * q^61 - 8 * q^63 - 24 * q^65 - 56 * q^67 + 8 * q^69 - 4 * q^75 - 28 * q^81 - 16 * q^87 - 40 * q^89 + 24 * q^91 + 8 * q^93 + 8 * q^95 + 32 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1248\mathbb{Z}\right)^\times$.

$n$	$703$	$769$	$833$	$1093$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−1.00000	−	1.41421i	−0.577350	−	0.816497i
$3$	−1.00000	−	1.41421i	−0.577350	−	0.816497i
$4$		0			0
$5$	−3.41421			−1.52688			−0.763441	−	0.645877i	$-0.776492\pi$
$5$	−3.41421			−1.52688			−0.763441	+	0.645877i	$0.776492\pi$
$6$		0			0
$7$	3.41421			1.29045			0.645226	−	0.763992i	$-0.276763\pi$
$7$	3.41421			1.29045			0.645226	+	0.763992i	$0.276763\pi$
$8$		0			0
$9$	−1.00000	+	2.82843i	−0.333333	+	0.942809i
$10$		0			0
$11$		−	0.828427i		−	0.249780i	−0.992171	−	0.124890i	$-0.960142\pi$
$11$		−	0.828427i		−	0.249780i	0.992171	−	0.124890i	$-0.0398578\pi$
$12$		0			0
$13$	3.00000	−	2.00000i	0.832050	−	0.554700i
$13$	3.00000	−	2.00000i	0.832050	−	0.554700i
$14$		0			0
$15$	3.41421	+	4.82843i	0.881546	+	1.24669i
$16$		0			0
$17$		−	2.82843i		−	0.685994i	−0.939336	−	0.342997i	$-0.888558\pi$
$17$		−	2.82843i		−	0.685994i	0.939336	−	0.342997i	$-0.111442\pi$
$18$		0			0
$19$	−2.24264			−0.514497			−0.257249	−	0.966345i	$-0.582816\pi$
$19$	−2.24264			−0.514497			−0.257249	+	0.966345i	$0.582816\pi$
$20$		0			0
$21$	−3.41421	−	4.82843i	−0.745042	−	1.05365i
$22$		0			0
$23$	3.65685			0.762507			0.381253	−	0.924471i	$-0.375493\pi$
$23$	3.65685			0.762507			0.381253	+	0.924471i	$0.375493\pi$
$24$		0			0
$25$	6.65685			1.33137
$26$		0			0
$27$	5.00000	−	1.41421i	0.962250	−	0.272166i
$28$		0			0
$29$		−	6.82843i		−	1.26801i	−0.773330	−	0.634004i	$-0.781410\pi$
$29$		−	6.82843i		−	1.26801i	0.773330	−	0.634004i	$-0.218590\pi$
$30$		0			0
$31$	−9.07107			−1.62921			−0.814606	−	0.580015i	$-0.803047\pi$
$31$	−9.07107			−1.62921			−0.814606	+	0.580015i	$0.803047\pi$
$32$		0			0
$33$	−1.17157	+	0.828427i	−0.203945	+	0.144211i
$34$		0			0
$35$	−11.6569			−1.97037
$36$		0			0
$37$		−	1.65685i		−	0.272385i	−0.990682	−	0.136193i	$-0.956513\pi$
$37$		−	1.65685i		−	0.272385i	0.990682	−	0.136193i	$-0.0434866\pi$
$38$		0			0
$39$	−5.82843	−	2.24264i	−0.933295	−	0.359110i
$40$		0			0
$41$	−3.41421			−0.533211			−0.266605	−	0.963806i	$-0.585902\pi$
$41$	−3.41421			−0.533211			−0.266605	+	0.963806i	$0.585902\pi$
$42$		0			0
$43$			8.00000i			1.21999i	0.792406	+	0.609994i	$0.208828\pi$
$43$			8.00000i			1.21999i	−0.792406	+	0.609994i	$0.791172\pi$
$44$		0			0
$45$	3.41421	−	9.65685i	0.508961	−	1.43956i
$46$		0			0
$47$			3.17157i			0.462621i	0.972880	+	0.231311i	$0.0743014\pi$
$47$			3.17157i			0.462621i	−0.972880	+	0.231311i	$0.925699\pi$
$48$		0			0
$49$	4.65685			0.665265
$50$		0			0
$51$	−4.00000	+	2.82843i	−0.560112	+	0.396059i
$52$		0			0
$53$		−	12.0000i		−	1.64833i	−0.566352	−	0.824163i	$-0.691646\pi$
$53$		−	12.0000i		−	1.64833i	0.566352	−	0.824163i	$-0.308354\pi$
$54$		0			0
$55$			2.82843i			0.381385i
$56$		0			0
$57$	2.24264	+	3.17157i	0.297045	+	0.420085i
$58$		0			0
$59$			3.17157i			0.412904i	0.978457	+	0.206452i	$0.0661917\pi$
$59$			3.17157i			0.412904i	−0.978457	+	0.206452i	$0.933808\pi$
$60$		0			0
$61$	−13.6569			−1.74858			−0.874291	−	0.485403i	$-0.838673\pi$
$61$	−13.6569			−1.74858			−0.874291	+	0.485403i	$0.838673\pi$
$62$		0			0
$63$	−3.41421	+	9.65685i	−0.430150	+	1.21665i
$64$		0			0
$65$	−10.2426	+	6.82843i	−1.27044	+	0.846962i
$66$		0			0
$67$	−12.5858			−1.53760			−0.768799	−	0.639490i	$-0.779145\pi$
$67$	−12.5858			−1.53760			−0.768799	+	0.639490i	$0.779145\pi$
$68$		0			0
$69$	−3.65685	−	5.17157i	−0.440234	−	0.622584i
$70$		0			0
$71$		−	8.82843i		−	1.04774i	−0.851798	−	0.523871i	$-0.824487\pi$
$71$		−	8.82843i		−	1.04774i	0.851798	−	0.523871i	$-0.175513\pi$
$72$		0			0
$73$		−	4.00000i		−	0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$		−	4.00000i		−	0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$		0			0
$75$	−6.65685	−	9.41421i	−0.768667	−	1.08706i
$76$		0			0
$77$		−	2.82843i		−	0.322329i
$78$		0			0
$79$		−	12.4853i		−	1.40470i	−0.711830	−	0.702352i	$-0.752133\pi$
$79$		−	12.4853i		−	1.40470i	0.711830	−	0.702352i	$-0.247867\pi$
$80$		0			0
$81$	−7.00000	−	5.65685i	−0.777778	−	0.628539i
$82$		0			0
$83$			8.82843i			0.969046i	0.874779	+	0.484523i	$0.161007\pi$
$83$			8.82843i			0.969046i	−0.874779	+	0.484523i	$0.838993\pi$
$84$		0			0
$85$			9.65685i			1.04743i
$86$		0			0
$87$	−9.65685	+	6.82843i	−1.03532	+	0.732084i
$88$		0			0
$89$	−11.4142			−1.20990			−0.604952	−	0.796262i	$-0.706808\pi$
$89$	−11.4142			−1.20990			−0.604952	+	0.796262i	$0.706808\pi$
$90$		0			0
$91$	10.2426	−	6.82843i	1.07372	−	0.715814i
$92$		0			0
$93$	9.07107	+	12.8284i	0.940626	+	1.33025i
$94$		0			0
$95$	7.65685			0.785577
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$	2.34315	+	0.828427i	0.235495	+	0.0832601i
$100$		0			0
$101$			17.6569i			1.75692i	0.477813	+	0.878461i	$0.341429\pi$
$101$			17.6569i			1.75692i	−0.477813	+	0.878461i	$0.658571\pi$
$102$		0			0
$103$			1.65685i			0.163255i	0.996663	+	0.0816274i	$0.0260117\pi$
$103$			1.65685i			0.163255i	−0.996663	+	0.0816274i	$0.973988\pi$
$104$		0			0
$105$	11.6569	+	16.4853i	1.13759	+	1.60880i
$106$		0			0
$107$	1.65685			0.160174			0.0800871	−	0.996788i	$-0.474480\pi$
$107$	1.65685			0.160174			0.0800871	+	0.996788i	$0.474480\pi$
$108$		0			0
$109$		−	16.9706i		−	1.62549i	−0.582623	−	0.812743i	$-0.697974\pi$
$109$		−	16.9706i		−	1.62549i	0.582623	−	0.812743i	$-0.302026\pi$
$110$		0			0
$111$	−2.34315	+	1.65685i	−0.222402	+	0.157262i
$112$		0			0
$113$		−	8.48528i		−	0.798228i	−0.916901	−	0.399114i	$-0.869318\pi$
$113$		−	8.48528i		−	0.798228i	0.916901	−	0.399114i	$-0.130682\pi$
$114$		0			0
$115$	−12.4853			−1.16426
$116$		0			0
$117$	2.65685	+	10.4853i	0.245626	+	0.969365i
$118$		0			0
$119$		−	9.65685i		−	0.885242i
$120$		0			0
$121$	10.3137			0.937610
$122$		0			0
$123$	3.41421	+	4.82843i	0.307849	+	0.435365i
$124$		0			0
$125$	−5.65685			−0.505964
$126$		0			0
$127$		−	20.4853i		−	1.81777i	−0.417042	−	0.908887i	$-0.636933\pi$
$127$		−	20.4853i		−	1.81777i	0.417042	−	0.908887i	$-0.363067\pi$
$128$		0			0
$129$	11.3137	−	8.00000i	0.996116	−	0.704361i
$130$		0			0
$131$	1.65685			0.144760			0.0723800	−	0.997377i	$-0.476941\pi$
$131$	1.65685			0.144760			0.0723800	+	0.997377i	$0.476941\pi$
$132$		0			0
$133$	−7.65685			−0.663933
$134$		0			0
$135$	−17.0711	+	4.82843i	−1.46924	+	0.415565i
$136$		0			0
$137$	−13.5563			−1.15820			−0.579099	−	0.815258i	$-0.696595\pi$
$137$	−13.5563			−1.15820			−0.579099	+	0.815258i	$0.696595\pi$
$138$		0			0
$139$			2.34315i			0.198743i	0.995050	+	0.0993715i	$0.0316832\pi$
$139$			2.34315i			0.198743i	−0.995050	+	0.0993715i	$0.968317\pi$
$140$		0			0
$141$	4.48528	−	3.17157i	0.377729	−	0.267095i
$142$		0			0
$143$	−1.65685	−	2.48528i	−0.138553	−	0.207830i
$144$		0			0
$145$			23.3137i			1.93610i
$146$		0			0
$147$	−4.65685	−	6.58579i	−0.384091	−	0.543187i
$148$		0			0
$149$	−6.72792			−0.551173			−0.275586	−	0.961276i	$-0.588872\pi$
$149$	−6.72792			−0.551173			−0.275586	+	0.961276i	$0.588872\pi$
$150$		0			0
$151$	12.5858			1.02422			0.512108	−	0.858921i	$-0.328865\pi$
$151$	12.5858			1.02422			0.512108	+	0.858921i	$0.328865\pi$
$152$		0			0
$153$	8.00000	+	2.82843i	0.646762	+	0.228665i
$154$		0			0
$155$	30.9706			2.48762
$156$		0			0
$157$	−1.65685			−0.132231			−0.0661157	−	0.997812i	$-0.521061\pi$
$157$	−1.65685			−0.132231			−0.0661157	+	0.997812i	$0.521061\pi$
$158$		0			0
$159$	−16.9706	+	12.0000i	−1.34585	+	0.951662i
$160$		0			0
$161$	12.4853			0.983978
$162$		0			0
$163$	10.2426			0.802266			0.401133	−	0.916020i	$-0.368617\pi$
$163$	10.2426			0.802266			0.401133	+	0.916020i	$0.368617\pi$
$164$		0			0
$165$	4.00000	−	2.82843i	0.311400	−	0.220193i
$166$		0			0
$167$			8.82843i			0.683164i	0.939852	+	0.341582i	$0.110963\pi$
$167$			8.82843i			0.683164i	−0.939852	+	0.341582i	$0.889037\pi$
$168$		0			0
$169$	5.00000	−	12.0000i	0.384615	−	0.923077i
$170$		0			0
$171$	2.24264	−	6.34315i	0.171499	−	0.485072i
$172$		0			0
$173$		−	23.3137i		−	1.77251i	−0.463199	−	0.886254i	$-0.653299\pi$
$173$		−	23.3137i		−	1.77251i	0.463199	−	0.886254i	$-0.346701\pi$
$174$		0			0
$175$	22.7279			1.71807
$176$		0			0
$177$	4.48528	−	3.17157i	0.337134	−	0.238390i
$178$		0			0
$179$	9.65685			0.721787			0.360894	−	0.932607i	$-0.382472\pi$
$179$	9.65685			0.721787			0.360894	+	0.932607i	$0.382472\pi$
$180$		0			0
$181$	−12.9706			−0.964094			−0.482047	−	0.876145i	$-0.660107\pi$
$181$	−12.9706			−0.964094			−0.482047	+	0.876145i	$0.660107\pi$
$182$		0			0
$183$	13.6569	+	19.3137i	1.00954	+	1.42771i
$184$		0			0
$185$			5.65685i			0.415900i
$186$		0			0
$187$	−2.34315			−0.171348
$188$		0			0
$189$	17.0711	−	4.82843i	1.24174	−	0.351216i
$190$		0			0
$191$	3.65685			0.264601			0.132300	−	0.991210i	$-0.457764\pi$
$191$	3.65685			0.264601			0.132300	+	0.991210i	$0.457764\pi$
$192$		0			0
$193$			4.00000i			0.287926i	0.989583	+	0.143963i	$0.0459847\pi$
$193$			4.00000i			0.287926i	−0.989583	+	0.143963i	$0.954015\pi$
$194$		0			0
$195$	19.8995	+	7.65685i	1.42503	+	0.548319i
$196$		0			0
$197$	19.4142			1.38321			0.691603	−	0.722278i	$-0.256905\pi$
$197$	19.4142			1.38321			0.691603	+	0.722278i	$0.256905\pi$
$198$		0			0
$199$			12.4853i			0.885058i	0.896754	+	0.442529i	$0.145919\pi$
$199$			12.4853i			0.885058i	−0.896754	+	0.442529i	$0.854081\pi$
$200$		0			0
$201$	12.5858	+	17.7990i	0.887733	+	1.25544i
$202$		0			0
$203$		−	23.3137i		−	1.63630i
$204$		0			0
$205$	11.6569			0.814150
$206$		0			0
$207$	−3.65685	+	10.3431i	−0.254169	+	0.718898i
$208$		0			0
$209$			1.85786i			0.128511i
$210$		0			0
$211$			27.7990i			1.91376i	0.290481	+	0.956881i	$0.406185\pi$
$211$			27.7990i			1.91376i	−0.290481	+	0.956881i	$0.593815\pi$
$212$		0			0
$213$	−12.4853	+	8.82843i	−0.855477	+	0.604914i
$214$		0			0
$215$		−	27.3137i		−	1.86278i
$216$		0			0
$217$	−30.9706			−2.10242
$218$		0			0
$219$	−5.65685	+	4.00000i	−0.382255	+	0.270295i
$220$		0			0
$221$	−5.65685	−	8.48528i	−0.380521	−	0.570782i
$222$		0			0
$223$	6.72792			0.450535			0.225267	−	0.974297i	$-0.427674\pi$
$223$	6.72792			0.450535			0.225267	+	0.974297i	$0.427674\pi$
$224$		0			0
$225$	−6.65685	+	18.8284i	−0.443790	+	1.25523i
$226$		0			0
$227$		−	26.4853i		−	1.75789i	−0.476923	−	0.878945i	$-0.658248\pi$
$227$		−	26.4853i		−	1.75789i	0.476923	−	0.878945i	$-0.341752\pi$
$228$		0			0
$229$		−	25.6569i		−	1.69545i	−0.530434	−	0.847726i	$-0.677971\pi$
$229$		−	25.6569i		−	1.69545i	0.530434	−	0.847726i	$-0.322029\pi$
$230$		0			0
$231$	−4.00000	+	2.82843i	−0.263181	+	0.186097i
$232$		0			0
$233$			2.34315i			0.153505i	0.997050	+	0.0767523i	$0.0244551\pi$
$233$			2.34315i			0.153505i	−0.997050	+	0.0767523i	$0.975545\pi$
$234$		0			0
$235$		−	10.8284i		−	0.706369i
$236$		0			0
$237$	−17.6569	+	12.4853i	−1.14694	+	0.811006i
$238$		0			0
$239$		−	4.82843i		−	0.312325i	−0.987731	−	0.156162i	$-0.950088\pi$
$239$		−	4.82843i		−	0.312325i	0.987731	−	0.156162i	$-0.0499124\pi$
$240$		0			0
$241$		−	21.6569i		−	1.39504i	−0.716565	−	0.697520i	$-0.754287\pi$
$241$		−	21.6569i		−	1.39504i	0.716565	−	0.697520i	$-0.245713\pi$
$242$		0			0
$243$	−1.00000	+	15.5563i	−0.0641500	+	0.997940i
$244$		0			0
$245$	−15.8995			−1.01578
$246$		0			0
$247$	−6.72792	+	4.48528i	−0.428087	+	0.285392i
$248$		0			0
$249$	12.4853	−	8.82843i	0.791223	−	0.559479i
$250$		0			0
$251$	−20.6274			−1.30199			−0.650996	−	0.759082i	$-0.725648\pi$
$251$	−20.6274			−1.30199			−0.650996	+	0.759082i	$0.725648\pi$
$252$		0			0
$253$		−	3.02944i		−	0.190459i
$254$		0			0
$255$	13.6569	−	9.65685i	0.855225	−	0.604736i
$256$		0			0
$257$			3.31371i			0.206703i	0.994645	+	0.103352i	$0.0329567\pi$
$257$			3.31371i			0.206703i	−0.994645	+	0.103352i	$0.967043\pi$
$258$		0			0
$259$		−	5.65685i		−	0.351500i
$260$		0			0
$261$	19.3137	+	6.82843i	1.19549	+	0.422669i
$262$		0			0
$263$	−19.3137			−1.19093			−0.595467	−	0.803380i	$-0.703033\pi$
$263$	−19.3137			−1.19093			−0.595467	+	0.803380i	$0.703033\pi$
$264$		0			0
$265$			40.9706i			2.51680i
$266$		0			0
$267$	11.4142	+	16.1421i	0.698539	+	0.987883i
$268$		0			0
$269$			1.17157i			0.0714321i	0.999362	+	0.0357160i	$0.0113712\pi$
$269$			1.17157i			0.0714321i	−0.999362	+	0.0357160i	$0.988629\pi$
$270$		0			0
$271$	2.24264			0.136231			0.0681154	−	0.997677i	$-0.478301\pi$
$271$	2.24264			0.136231			0.0681154	+	0.997677i	$0.478301\pi$
$272$		0			0
$273$	−19.8995	−	7.65685i	−1.20437	−	0.463414i
$274$		0			0
$275$		−	5.51472i		−	0.332550i
$276$		0			0
$277$	−6.00000			−0.360505			−0.180253	−	0.983620i	$-0.557691\pi$
$277$	−6.00000			−0.360505			−0.180253	+	0.983620i	$0.557691\pi$
$278$		0			0
$279$	9.07107	−	25.6569i	0.543071	−	1.53604i
$280$		0			0
$281$	−5.55635			−0.331464			−0.165732	−	0.986171i	$-0.552999\pi$
$281$	−5.55635			−0.331464			−0.165732	+	0.986171i	$0.552999\pi$
$282$		0			0
$283$			15.5147i			0.922254i	0.887334	+	0.461127i	$0.152555\pi$
$283$			15.5147i			0.922254i	−0.887334	+	0.461127i	$0.847445\pi$
$284$		0			0
$285$	−7.65685	−	10.8284i	−0.453553	−	0.641421i
$286$		0			0
$287$	−11.6569			−0.688082
$288$		0			0
$289$	9.00000			0.529412
$290$		0			0
$291$		0			0
$292$		0			0
$293$	30.7279			1.79514			0.897572	−	0.440868i	$-0.145329\pi$
$293$	30.7279			1.79514			0.897572	+	0.440868i	$0.145329\pi$
$294$		0			0
$295$		−	10.8284i		−	0.630455i
$296$		0			0
$297$	−1.17157	−	4.14214i	−0.0679816	−	0.240351i
$298$		0			0
$299$	10.9706	−	7.31371i	0.634444	−	0.422963i
$300$		0			0
$301$			27.3137i			1.57434i
$302$		0			0
$303$	24.9706	−	17.6569i	1.43452	−	1.01436i
$304$		0			0
$305$	46.6274			2.66988
$306$		0			0
$307$	−10.2426			−0.584578			−0.292289	−	0.956330i	$-0.594417\pi$
$307$	−10.2426			−0.584578			−0.292289	+	0.956330i	$0.594417\pi$
$308$		0			0
$309$	2.34315	−	1.65685i	0.133297	−	0.0942551i
$310$		0			0
$311$	4.34315			0.246277			0.123139	−	0.992389i	$-0.460704\pi$
$311$	4.34315			0.246277			0.123139	+	0.992389i	$0.460704\pi$
$312$		0			0
$313$	−10.0000			−0.565233			−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000			−0.565233			−0.282617	+	0.959233i	$0.591202\pi$
$314$		0			0
$315$	11.6569	−	32.9706i	0.656789	−	1.85768i
$316$		0			0
$317$	−20.5858			−1.15621			−0.578106	−	0.815961i	$-0.696208\pi$
$317$	−20.5858			−1.15621			−0.578106	+	0.815961i	$0.696208\pi$
$318$		0			0
$319$	−5.65685			−0.316723
$320$		0			0
$321$	−1.65685	−	2.34315i	−0.0924766	−	0.130782i
$322$		0			0
$323$			6.34315i			0.352942i
$324$		0			0
$325$	19.9706	−	13.3137i	1.10777	−	0.738512i
$326$		0			0
$327$	−24.0000	+	16.9706i	−1.32720	+	0.938474i
$328$		0			0
$329$			10.8284i			0.596991i
$330$		0			0
$331$	31.6985			1.74231			0.871153	−	0.491011i	$-0.163373\pi$
$331$	31.6985			1.74231			0.871153	+	0.491011i	$0.163373\pi$
$332$		0			0
$333$	4.68629	+	1.65685i	0.256807	+	0.0907951i
$334$		0			0
$335$	42.9706			2.34773
$336$		0			0
$337$	2.00000			0.108947			0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000			0.108947			0.0544735	+	0.998515i	$0.482652\pi$
$338$		0			0
$339$	−12.0000	+	8.48528i	−0.651751	+	0.460857i
$340$		0			0
$341$			7.51472i			0.406945i
$342$		0			0
$343$	−8.00000			−0.431959
$344$		0			0
$345$	12.4853	+	17.6569i	0.672185	+	0.950613i
$346$		0			0
$347$	6.68629			0.358939			0.179469	−	0.983764i	$-0.442562\pi$
$347$	6.68629			0.358939			0.179469	+	0.983764i	$0.442562\pi$
$348$		0			0
$349$			14.3431i			0.767771i	0.923381	+	0.383885i	$0.125414\pi$
$349$			14.3431i			0.767771i	−0.923381	+	0.383885i	$0.874586\pi$
$350$		0			0
$351$	12.1716	−	14.2426i	0.649670	−	0.760216i
$352$		0			0
$353$	2.44365			0.130062			0.0650312	−	0.997883i	$-0.479285\pi$
$353$	2.44365			0.130062			0.0650312	+	0.997883i	$0.479285\pi$
$354$		0			0
$355$			30.1421i			1.59978i
$356$		0			0
$357$	−13.6569	+	9.65685i	−0.722797	+	0.511095i
$358$		0			0
$359$			24.1421i			1.27417i	0.770792	+	0.637087i	$0.219861\pi$
$359$			24.1421i			1.27417i	−0.770792	+	0.637087i	$0.780139\pi$
$360$		0			0
$361$	−13.9706			−0.735293
$362$		0			0
$363$	−10.3137	−	14.5858i	−0.541329	−	0.765555i
$364$		0			0
$365$			13.6569i			0.714832i
$366$		0			0
$367$		−	4.00000i		−	0.208798i	−0.994535	−	0.104399i	$-0.966708\pi$
$367$		−	4.00000i		−	0.208798i	0.994535	−	0.104399i	$-0.0332919\pi$
$368$		0			0
$369$	3.41421	−	9.65685i	0.177737	−	0.502716i
$370$		0			0
$371$		−	40.9706i		−	2.12709i
$372$		0			0
$373$	24.6274			1.27516			0.637580	−	0.770384i	$-0.279936\pi$
$373$	24.6274			1.27516			0.637580	+	0.770384i	$0.279936\pi$
$374$		0			0
$375$	5.65685	+	8.00000i	0.292119	+	0.413118i
$376$		0			0
$377$	−13.6569	−	20.4853i	−0.703364	−	1.05505i
$378$		0			0
$379$	14.7279			0.756523			0.378261	−	0.925699i	$-0.376522\pi$
$379$	14.7279			0.756523			0.378261	+	0.925699i	$0.376522\pi$
$380$		0			0
$381$	−28.9706	+	20.4853i	−1.48421	+	1.04949i
$382$		0			0
$383$			13.7990i			0.705095i	0.935794	+	0.352548i	$0.114685\pi$
$383$			13.7990i			0.705095i	−0.935794	+	0.352548i	$0.885315\pi$
$384$		0			0
$385$			9.65685i			0.492159i
$386$		0			0
$387$	−22.6274	−	8.00000i	−1.15022	−	0.406663i
$388$		0			0
$389$			15.7990i			0.801041i	0.916288	+	0.400520i	$0.131171\pi$
$389$			15.7990i			0.801041i	−0.916288	+	0.400520i	$0.868829\pi$
$390$		0			0
$391$		−	10.3431i		−	0.523075i
$392$		0			0
$393$	−1.65685	−	2.34315i	−0.0835772	−	0.118196i
$394$		0			0
$395$			42.6274i			2.14482i
$396$		0			0
$397$			32.9706i			1.65475i	0.561654	+	0.827373i	$0.310165\pi$
$397$			32.9706i			1.65475i	−0.561654	+	0.827373i	$0.689835\pi$
$398$		0			0
$399$	7.65685	+	10.8284i	0.383322	+	0.542099i
$400$		0			0
$401$	1.07107			0.0534866			0.0267433	−	0.999642i	$-0.491486\pi$
$401$	1.07107			0.0534866			0.0267433	+	0.999642i	$0.491486\pi$
$402$		0			0
$403$	−27.2132	+	18.1421i	−1.35559	+	0.903724i
$404$		0			0
$405$	23.8995	+	19.3137i	1.18758	+	0.959706i
$406$		0			0
$407$	−1.37258			−0.0680364
$408$		0			0
$409$			12.9706i			0.641353i	0.947189	+	0.320677i	$0.103910\pi$
$409$			12.9706i			0.641353i	−0.947189	+	0.320677i	$0.896090\pi$
$410$		0			0
$411$	13.5563	+	19.1716i	0.668685	+	0.945664i
$412$		0			0
$413$			10.8284i			0.532832i
$414$		0			0
$415$		−	30.1421i		−	1.47962i
$416$		0			0
$417$	3.31371	−	2.34315i	0.162273	−	0.114744i
$418$		0			0
$419$	28.9706			1.41530			0.707652	−	0.706561i	$-0.249754\pi$
$419$	28.9706			1.41530			0.707652	+	0.706561i	$0.249754\pi$
$420$		0			0
$421$			16.9706i			0.827095i	0.910483	+	0.413547i	$0.135710\pi$
$421$			16.9706i			0.827095i	−0.910483	+	0.413547i	$0.864290\pi$
$422$		0			0
$423$	−8.97056	−	3.17157i	−0.436164	−	0.154207i
$424$		0			0
$425$		−	18.8284i		−	0.913313i
$426$		0			0
$427$	−46.6274			−2.25646
$428$		0			0
$429$	−1.85786	+	4.82843i	−0.0896985	+	0.233119i
$430$		0			0
$431$		−	38.4853i		−	1.85377i	−0.375344	−	0.926885i	$-0.622476\pi$
$431$		−	38.4853i		−	1.85377i	0.375344	−	0.926885i	$-0.377524\pi$
$432$		0			0
$433$	20.9706			1.00778			0.503890	−	0.863768i	$-0.331902\pi$
$433$	20.9706			1.00778			0.503890	+	0.863768i	$0.331902\pi$
$434$		0			0
$435$	32.9706	−	23.3137i	1.58082	−	1.11781i
$436$		0			0
$437$	−8.20101			−0.392308
$438$		0			0
$439$			12.9706i			0.619051i	0.950891	+	0.309526i	$0.100170\pi$
$439$			12.9706i			0.619051i	−0.950891	+	0.309526i	$0.899830\pi$
$440$		0			0
$441$	−4.65685	+	13.1716i	−0.221755	+	0.627218i
$442$		0			0
$443$	6.34315			0.301372			0.150686	−	0.988582i	$-0.451852\pi$
$443$	6.34315			0.301372			0.150686	+	0.988582i	$0.451852\pi$
$444$		0			0
$445$	38.9706			1.84738
$446$		0			0
$447$	6.72792	+	9.51472i	0.318220	+	0.450031i
$448$		0			0
$449$	−32.8701			−1.55123			−0.775617	−	0.631204i	$-0.782561\pi$
$449$	−32.8701			−1.55123			−0.775617	+	0.631204i	$0.782561\pi$
$450$		0			0
$451$			2.82843i			0.133185i
$452$		0			0
$453$	−12.5858	−	17.7990i	−0.591332	−	0.836269i
$454$		0			0
$455$	−34.9706	+	23.3137i	−1.63945	+	1.09296i
$456$		0			0
$457$		−	4.97056i		−	0.232513i	−0.993219	−	0.116257i	$-0.962911\pi$
$457$		−	4.97056i		−	0.232513i	0.993219	−	0.116257i	$-0.0370895\pi$
$458$		0			0
$459$	−4.00000	−	14.1421i	−0.186704	−	0.660098i
$460$		0			0
$461$	22.9289			1.06791			0.533953	−	0.845514i	$-0.320706\pi$
$461$	22.9289			1.06791			0.533953	+	0.845514i	$0.320706\pi$
$462$		0			0
$463$	30.7279			1.42805			0.714024	−	0.700121i	$-0.246871\pi$
$463$	30.7279			1.42805			0.714024	+	0.700121i	$0.246871\pi$
$464$		0			0
$465$	−30.9706	−	43.7990i	−1.43623	−	2.03113i
$466$		0			0
$467$	−30.0000			−1.38823			−0.694117	−	0.719862i	$-0.744205\pi$
$467$	−30.0000			−1.38823			−0.694117	+	0.719862i	$0.744205\pi$
$468$		0			0
$469$	−42.9706			−1.98420
$470$		0			0
$471$	1.65685	+	2.34315i	0.0763438	+	0.107966i
$472$		0			0
$473$	6.62742			0.304729
$474$		0			0
$475$	−14.9289			−0.684986
$476$		0			0
$477$	33.9411	+	12.0000i	1.55406	+	0.549442i
$478$		0			0
$479$			21.5147i			0.983033i	0.870868	+	0.491516i	$0.163557\pi$
$479$			21.5147i			0.983033i	−0.870868	+	0.491516i	$0.836443\pi$
$480$		0			0
$481$	−3.31371	−	4.97056i	−0.151092	−	0.226638i
$482$		0			0
$483$	−12.4853	−	17.6569i	−0.568100	−	0.803415i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−2.24264			−0.101624			−0.0508119	−	0.998708i	$-0.516181\pi$
$487$	−2.24264			−0.101624			−0.0508119	+	0.998708i	$0.516181\pi$
$488$		0			0
$489$	−10.2426	−	14.4853i	−0.463188	−	0.655047i
$490$		0			0
$491$	−24.6274			−1.11142			−0.555710	−	0.831376i	$-0.687553\pi$
$491$	−24.6274			−1.11142			−0.555710	+	0.831376i	$0.687553\pi$
$492$		0			0
$493$	−19.3137			−0.869846
$494$		0			0
$495$	−8.00000	−	2.82843i	−0.359573	−	0.127128i
$496$		0			0
$497$		−	30.1421i		−	1.35206i
$498$		0			0
$499$	−20.3848			−0.912548			−0.456274	−	0.889839i	$-0.650816\pi$
$499$	−20.3848			−0.912548			−0.456274	+	0.889839i	$0.650816\pi$
$500$		0			0
$501$	12.4853	−	8.82843i	0.557801	−	0.394425i
$502$		0			0
$503$	30.6274			1.36561			0.682805	−	0.730601i	$-0.260760\pi$
$503$	30.6274			1.36561			0.682805	+	0.730601i	$0.260760\pi$
$504$		0			0
$505$		−	60.2843i		−	2.68261i
$506$		0			0
$507$	−21.9706	+	4.92893i	−0.975747	+	0.218902i
$508$		0			0
$509$	9.07107			0.402068			0.201034	−	0.979584i	$-0.435570\pi$
$509$	9.07107			0.402068			0.201034	+	0.979584i	$0.435570\pi$
$510$		0			0
$511$		−	13.6569i		−	0.604144i
$512$		0			0
$513$	−11.2132	+	3.17157i	−0.495075	+	0.140028i
$514$		0			0
$515$		−	5.65685i		−	0.249271i
$516$		0			0
$517$	2.62742			0.115554
$518$		0			0
$519$	−32.9706	+	23.3137i	−1.44725	+	1.02336i
$520$		0			0
$521$		−	15.5147i		−	0.679712i	−0.940477	−	0.339856i	$-0.889622\pi$
$521$		−	15.5147i		−	0.679712i	0.940477	−	0.339856i	$-0.110378\pi$
$522$		0			0
$523$		−	16.9706i		−	0.742071i	−0.928619	−	0.371035i	$-0.879003\pi$
$523$		−	16.9706i		−	0.742071i	0.928619	−	0.371035i	$-0.120997\pi$
$524$		0			0
$525$	−22.7279	−	32.1421i	−0.991928	−	1.40280i
$526$		0			0
$527$			25.6569i			1.11763i
$528$		0			0
$529$	−9.62742			−0.418583
$530$		0			0
$531$	−8.97056	−	3.17157i	−0.389289	−	0.137635i
$532$		0			0
$533$	−10.2426	+	6.82843i	−0.443658	+	0.295772i
$534$		0			0
$535$	−5.65685			−0.244567
$536$		0			0
$537$	−9.65685	−	13.6569i	−0.416724	−	0.589337i
$538$		0			0
$539$		−	3.85786i		−	0.166170i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	12.9706	+	18.3431i	0.556620	+	0.787180i
$544$		0			0
$545$			57.9411i			2.48193i
$546$		0			0
$547$		−	16.9706i		−	0.725609i	−0.931865	−	0.362804i	$-0.881819\pi$
$547$		−	16.9706i		−	0.725609i	0.931865	−	0.362804i	$-0.118181\pi$
$548$		0			0
$549$	13.6569	−	38.6274i	0.582860	−	1.64858i
$550$		0			0
$551$			15.3137i			0.652386i
$552$		0			0
$553$		−	42.6274i		−	1.81270i
$554$		0			0
$555$	8.00000	−	5.65685i	0.339581	−	0.240120i
$556$		0			0
$557$	−16.1005			−0.682200			−0.341100	−	0.940027i	$-0.610800\pi$
$557$	−16.1005			−0.682200			−0.341100	+	0.940027i	$0.610800\pi$
$558$		0			0
$559$	16.0000	+	24.0000i	0.676728	+	1.01509i
$560$		0			0
$561$	2.34315	+	3.31371i	0.0989277	+	0.139905i
$562$		0			0
$563$	−28.9706			−1.22096			−0.610482	−	0.792030i	$-0.709024\pi$
$563$	−28.9706			−1.22096			−0.610482	+	0.792030i	$0.709024\pi$
$564$		0			0
$565$			28.9706i			1.21880i
$566$		0			0
$567$	−23.8995	−	19.3137i	−1.00368	−	0.811100i
$568$		0			0
$569$		−	19.3137i		−	0.809673i	−0.914389	−	0.404836i	$-0.867328\pi$
$569$		−	19.3137i		−	0.809673i	0.914389	−	0.404836i	$-0.132672\pi$
$570$		0			0
$571$			7.51472i			0.314481i	0.987560	+	0.157241i	$0.0502598\pi$
$571$			7.51472i			0.314481i	−0.987560	+	0.157241i	$0.949740\pi$
$572$		0			0
$573$	−3.65685	−	5.17157i	−0.152767	−	0.216046i
$574$		0			0
$575$	24.3431			1.01518
$576$		0			0
$577$		−	37.9411i		−	1.57951i	−0.613423	−	0.789755i	$-0.710208\pi$
$577$		−	37.9411i		−	1.57951i	0.613423	−	0.789755i	$-0.289792\pi$
$578$		0			0
$579$	5.65685	−	4.00000i	0.235091	−	0.166234i
$580$		0			0
$581$			30.1421i			1.25051i
$582$		0			0
$583$	−9.94113			−0.411719
$584$		0			0
$585$	−9.07107	−	35.7990i	−0.375042	−	1.48011i
$586$		0			0
$587$		−	9.79899i		−	0.404448i	−0.979339	−	0.202224i	$-0.935183\pi$
$587$		−	9.79899i		−	0.404448i	0.979339	−	0.202224i	$-0.0648168\pi$
$588$		0			0
$589$	20.3431			0.838225
$590$		0			0
$591$	−19.4142	−	27.4558i	−0.798594	−	1.12938i
$592$		0			0
$593$	5.55635			0.228172			0.114086	−	0.993471i	$-0.463606\pi$
$593$	5.55635			0.228172			0.114086	+	0.993471i	$0.463606\pi$
$594$		0			0
$595$			32.9706i			1.35166i
$596$		0			0
$597$	17.6569	−	12.4853i	0.722647	−	0.510989i
$598$		0			0
$599$	−22.6274			−0.924531			−0.462266	−	0.886742i	$-0.652963\pi$
$599$	−22.6274			−0.924531			−0.462266	+	0.886742i	$0.652963\pi$
$600$		0			0
$601$	−6.00000			−0.244745			−0.122373	−	0.992484i	$-0.539050\pi$
$601$	−6.00000			−0.244745			−0.122373	+	0.992484i	$0.539050\pi$
$602$		0			0
$603$	12.5858	−	35.5980i	0.512533	−	1.44966i
$604$		0			0
$605$	−35.2132			−1.43162
$606$		0			0
$607$		−	25.6569i		−	1.04138i	−0.853746	−	0.520690i	$-0.825675\pi$
$607$		−	25.6569i		−	1.04138i	0.853746	−	0.520690i	$-0.174325\pi$
$608$		0			0
$609$	−32.9706	+	23.3137i	−1.33603	+	0.944719i
$610$		0			0
$611$	6.34315	+	9.51472i	0.256616	+	0.384924i
$612$		0			0
$613$		−	14.3431i		−	0.579314i	−0.957130	−	0.289657i	$-0.906459\pi$
$613$		−	14.3431i		−	0.579314i	0.957130	−	0.289657i	$-0.0935413\pi$
$614$		0			0
$615$	−11.6569	−	16.4853i	−0.470050	−	0.664751i
$616$		0			0
$617$	29.5563			1.18989			0.594947	−	0.803765i	$-0.297173\pi$
$617$	29.5563			1.18989			0.594947	+	0.803765i	$0.297173\pi$
$618$		0			0
$619$	17.2721			0.694223			0.347112	−	0.937824i	$-0.387162\pi$
$619$	17.2721			0.694223			0.347112	+	0.937824i	$0.387162\pi$
$620$		0			0
$621$	18.2843	−	5.17157i	0.733723	−	0.207528i
$622$		0			0
$623$	−38.9706			−1.56132
$624$		0			0
$625$	−13.9706			−0.558823
$626$		0			0
$627$	2.62742	−	1.85786i	0.104929	−	0.0741960i
$628$		0			0
$629$	−4.68629			−0.186855
$630$		0			0
$631$	20.3848			0.811505			0.405753	−	0.913983i	$-0.367009\pi$
$631$	20.3848			0.811505			0.405753	+	0.913983i	$0.367009\pi$
$632$		0			0
$633$	39.3137	−	27.7990i	1.56258	−	1.10491i
$634$		0			0
$635$			69.9411i			2.77553i
$636$		0			0
$637$	13.9706	−	9.31371i	0.553534	−	0.369023i
$638$		0			0
$639$	24.9706	+	8.82843i	0.987820	+	0.349247i
$640$		0			0
$641$			10.8284i			0.427697i	0.976867	+	0.213849i	$0.0685999\pi$
$641$			10.8284i			0.427697i	−0.976867	+	0.213849i	$0.931400\pi$
$642$		0			0
$643$	5.55635			0.219121			0.109561	−	0.993980i	$-0.465056\pi$
$643$	5.55635			0.219121			0.109561	+	0.993980i	$0.465056\pi$
$644$		0			0
$645$	−38.6274	+	27.3137i	−1.52095	+	1.07548i
$646$		0			0
$647$	−20.6863			−0.813262			−0.406631	−	0.913592i	$-0.633297\pi$
$647$	−20.6863			−0.813262			−0.406631	+	0.913592i	$0.633297\pi$
$648$		0			0
$649$	2.62742			0.103135
$650$		0			0
$651$	30.9706	+	43.7990i	1.21383	+	1.71662i
$652$		0			0
$653$			40.7696i			1.59544i	0.603031	+	0.797718i	$0.293960\pi$
$653$			40.7696i			1.59544i	−0.603031	+	0.797718i	$0.706040\pi$
$654$		0			0
$655$	−5.65685			−0.221032
$656$		0			0
$657$	11.3137	+	4.00000i	0.441390	+	0.156055i
$658$		0			0
$659$	−14.6863			−0.572097			−0.286048	−	0.958215i	$-0.592342\pi$
$659$	−14.6863			−0.572097			−0.286048	+	0.958215i	$0.592342\pi$
$660$		0			0
$661$		−	24.0000i		−	0.933492i	−0.884391	−	0.466746i	$-0.845426\pi$
$661$		−	24.0000i		−	0.933492i	0.884391	−	0.466746i	$-0.154574\pi$
$662$		0			0
$663$	−6.34315	+	16.4853i	−0.246347	+	0.640235i
$664$		0			0
$665$	26.1421			1.01375
$666$		0			0
$667$		−	24.9706i		−	0.966864i
$668$		0			0
$669$	−6.72792	−	9.51472i	−0.260116	−	0.367860i
$670$		0			0
$671$			11.3137i			0.436761i
$672$		0			0
$673$	−8.97056			−0.345790			−0.172895	−	0.984940i	$-0.555312\pi$
$673$	−8.97056			−0.345790			−0.172895	+	0.984940i	$0.555312\pi$
$674$		0			0
$675$	33.2843	−	9.41421i	1.28111	−	0.362353i
$676$		0			0
$677$		−	35.5980i		−	1.36814i	−0.729416	−	0.684071i	$-0.760208\pi$
$677$		−	35.5980i		−	1.36814i	0.729416	−	0.684071i	$-0.239792\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−37.4558	+	26.4853i	−1.43531	+	1.01492i
$682$		0			0
$683$			12.1421i			0.464606i	0.972643	+	0.232303i	$0.0746261\pi$
$683$			12.1421i			0.464606i	−0.972643	+	0.232303i	$0.925374\pi$
$684$		0			0
$685$	46.2843			1.76843
$686$		0			0
$687$	−36.2843	+	25.6569i	−1.38433	+	0.978870i
$688$		0			0
$689$	−24.0000	−	36.0000i	−0.914327	−	1.37149i
$690$		0			0
$691$	12.5858			0.478786			0.239393	−	0.970923i	$-0.423052\pi$
$691$	12.5858			0.478786			0.239393	+	0.970923i	$0.423052\pi$
$692$		0			0
$693$	8.00000	+	2.82843i	0.303895	+	0.107443i
$694$		0			0
$695$		−	8.00000i		−	0.303457i
$696$		0			0
$697$			9.65685i			0.365779i
$698$		0			0
$699$	3.31371	−	2.34315i	0.125336	−	0.0886259i
$700$		0			0
$701$		−	42.1421i		−	1.59169i	−0.605503	−	0.795843i	$-0.707028\pi$
$701$		−	42.1421i		−	1.59169i	0.605503	−	0.795843i	$-0.292972\pi$
$702$		0			0
$703$			3.71573i			0.140141i
$704$		0			0
$705$	−15.3137	+	10.8284i	−0.576748	+	0.407822i
$706$		0			0
$707$			60.2843i			2.26722i
$708$		0			0
$709$		−	2.34315i		−	0.0879987i	−0.999032	−	0.0439993i	$-0.985990\pi$
$709$		−	2.34315i		−	0.0879987i	0.999032	−	0.0439993i	$-0.0140099\pi$
$710$		0			0
$711$	35.3137	+	12.4853i	1.32437	+	0.468235i
$712$		0			0
$713$	−33.1716			−1.24229
$714$		0			0
$715$	5.65685	+	8.48528i	0.211554	+	0.317332i
$716$		0			0
$717$	−6.82843	+	4.82843i	−0.255012	+	0.180321i
$718$		0			0
$719$	24.3431			0.907846			0.453923	−	0.891041i	$-0.350024\pi$
$719$	24.3431			0.907846			0.453923	+	0.891041i	$0.350024\pi$
$720$		0			0
$721$			5.65685i			0.210672i
$722$		0			0
$723$	−30.6274	+	21.6569i	−1.13905	+	0.805427i
$724$		0			0
$725$		−	45.4558i		−	1.68819i
$726$		0			0
$727$		−	1.17157i		−	0.0434512i	−0.999764	−	0.0217256i	$-0.993084\pi$
$727$		−	1.17157i		−	0.0434512i	0.999764	−	0.0217256i	$-0.00691602\pi$
$728$		0			0
$729$	23.0000	−	14.1421i	0.851852	−	0.523783i
$730$		0			0
$731$	22.6274			0.836905
$732$		0			0
$733$		−	29.9411i		−	1.10590i	−0.833214	−	0.552950i	$-0.813502\pi$
$733$		−	29.9411i		−	1.10590i	0.833214	−	0.552950i	$-0.186498\pi$
$734$		0			0
$735$	15.8995	+	22.4853i	0.586462	+	0.829382i
$736$		0			0
$737$			10.4264i			0.384062i
$738$		0			0
$739$	30.7279			1.13034			0.565172	−	0.824973i	$-0.308810\pi$
$739$	30.7279			1.13034			0.565172	+	0.824973i	$0.308810\pi$
$740$		0			0
$741$	13.0711	+	5.02944i	0.480178	+	0.184761i
$742$		0			0
$743$		−	27.1716i		−	0.996828i	−0.866939	−	0.498414i	$-0.833916\pi$
$743$		−	27.1716i		−	0.996828i	0.866939	−	0.498414i	$-0.166084\pi$
$744$		0			0
$745$	22.9706			0.841576
$746$		0			0
$747$	−24.9706	−	8.82843i	−0.913625	−	0.323015i
$748$		0			0
$749$	5.65685			0.206697
$750$		0			0
$751$		−	25.6569i		−	0.936232i	−0.883667	−	0.468116i	$-0.844933\pi$
$751$		−	25.6569i		−	0.936232i	0.883667	−	0.468116i	$-0.155067\pi$
$752$		0			0
$753$	20.6274	+	29.1716i	0.751705	+	1.06307i
$754$		0			0
$755$	−42.9706			−1.56386
$756$		0			0
$757$	12.2843			0.446479			0.223240	−	0.974764i	$-0.428337\pi$
$757$	12.2843			0.446479			0.223240	+	0.974764i	$0.428337\pi$
$758$		0			0
$759$	−4.28427	+	3.02944i	−0.155509	+	0.109962i
$760$		0			0
$761$	53.5563			1.94142			0.970708	−	0.240262i	$-0.0772334\pi$
$761$	53.5563			1.94142			0.970708	+	0.240262i	$0.0772334\pi$
$762$		0			0
$763$		−	57.9411i		−	2.09761i
$764$		0			0
$765$	−27.3137	−	9.65685i	−0.987529	−	0.349144i
$766$		0			0
$767$	6.34315	+	9.51472i	0.229038	+	0.343557i
$768$		0			0
$769$			21.9411i			0.791217i	0.918419	+	0.395609i	$0.129466\pi$
$769$			21.9411i			0.791217i	−0.918419	+	0.395609i	$0.870534\pi$
$770$		0			0
$771$	4.68629	−	3.31371i	0.168773	−	0.119340i
$772$		0			0
$773$	29.3553			1.05584			0.527919	−	0.849295i	$-0.322972\pi$
$773$	29.3553			1.05584			0.527919	+	0.849295i	$0.322972\pi$
$774$		0			0
$775$	−60.3848			−2.16909
$776$		0			0
$777$	−8.00000	+	5.65685i	−0.286998	+	0.202939i
$778$		0			0
$779$	7.65685			0.274335
$780$		0			0
$781$	−7.31371			−0.261705
$782$		0			0
$783$	−9.65685	−	34.1421i	−0.345108	−	1.22014i
$784$		0			0
$785$	5.65685			0.201902
$786$		0			0
$787$	11.4142			0.406873			0.203436	−	0.979088i	$-0.434789\pi$
$787$	11.4142			0.406873			0.203436	+	0.979088i	$0.434789\pi$
$788$		0			0
$789$	19.3137	+	27.3137i	0.687586	+	0.972394i
$790$		0			0
$791$		−	28.9706i		−	1.03007i
$792$		0			0
$793$	−40.9706	+	27.3137i	−1.45491	+	0.969938i
$794$		0			0
$795$	57.9411	−	40.9706i	2.05496	−	1.45308i
$796$		0			0
$797$			32.7696i			1.16076i	0.814347	+	0.580379i	$0.197095\pi$
$797$			32.7696i			1.16076i	−0.814347	+	0.580379i	$0.802905\pi$
$798$		0			0
$799$	8.97056			0.317356
$800$		0			0
$801$	11.4142	−	32.2843i	0.403301	−	1.14071i
$802$		0			0
$803$	−3.31371			−0.116938
$804$		0			0
$805$	−42.6274			−1.50242
$806$		0			0
$807$	1.65685	−	1.17157i	0.0583240	−	0.0412413i
$808$		0			0
$809$			49.4558i			1.73877i	0.494131	+	0.869387i	$0.335486\pi$
$809$			49.4558i			1.73877i	−0.494131	+	0.869387i	$0.664514\pi$
$810$		0			0
$811$	−34.0416			−1.19536			−0.597682	−	0.801734i	$-0.703911\pi$
$811$	−34.0416			−1.19536			−0.597682	+	0.801734i	$0.703911\pi$
$812$		0			0
$813$	−2.24264	−	3.17157i	−0.0786528	−	0.111232i
$814$		0			0
$815$	−34.9706			−1.22497
$816$		0			0
$817$		−	17.9411i		−	0.627681i
$818$		0			0
$819$	9.07107	+	35.7990i	0.316969	+	1.25092i
$820$		0			0
$821$	−8.87006			−0.309567			−0.154784	−	0.987948i	$-0.549468\pi$
$821$	−8.87006			−0.309567			−0.154784	+	0.987948i	$0.549468\pi$
$822$		0			0
$823$		−	20.0000i		−	0.697156i	−0.937280	−	0.348578i	$-0.886665\pi$
$823$		−	20.0000i		−	0.697156i	0.937280	−	0.348578i	$-0.113335\pi$
$824$		0			0
$825$	−7.79899	+	5.51472i	−0.271526	+	0.191998i
$826$		0			0
$827$			25.7990i			0.897119i	0.893753	+	0.448559i	$0.148063\pi$
$827$			25.7990i			0.897119i	−0.893753	+	0.448559i	$0.851937\pi$
$828$		0			0
$829$	13.6569			0.474322			0.237161	−	0.971470i	$-0.423783\pi$
$829$	13.6569			0.474322			0.237161	+	0.971470i	$0.423783\pi$
$830$		0			0
$831$	6.00000	+	8.48528i	0.208138	+	0.294351i
$832$		0			0
$833$		−	13.1716i		−	0.456368i
$834$		0			0
$835$		−	30.1421i		−	1.04311i
$836$		0			0
$837$	−45.3553	+	12.8284i	−1.56771	+	0.443415i
$838$		0			0
$839$		−	42.7696i		−	1.47657i	−0.674489	−	0.738284i	$-0.735636\pi$
$839$		−	42.7696i		−	1.47657i	0.674489	−	0.738284i	$-0.264364\pi$
$840$		0			0
$841$	−17.6274			−0.607842
$842$		0			0
$843$	5.55635	+	7.85786i	0.191371	+	0.270639i
$844$		0			0
$845$	−17.0711	+	40.9706i	−0.587263	+	1.40943i
$846$		0			0
$847$	35.2132			1.20994
$848$		0			0
$849$	21.9411	−	15.5147i	0.753017	−	0.532464i
$850$		0			0
$851$		−	6.05887i		−	0.207696i
$852$		0			0
$853$		−	6.62742i		−	0.226918i	−0.993543	−	0.113459i	$-0.963807\pi$
$853$		−	6.62742i		−	0.226918i	0.993543	−	0.113459i	$-0.0361931\pi$
$854$		0			0
$855$	−7.65685	+	21.6569i	−0.261859	+	0.740649i
$856$		0			0
$857$		−	31.5980i		−	1.07937i	−0.841868	−	0.539683i	$-0.818544\pi$
$857$		−	31.5980i		−	1.07937i	0.841868	−	0.539683i	$-0.181456\pi$
$858$		0			0
$859$		−	18.8284i		−	0.642418i	−0.947008	−	0.321209i	$-0.895911\pi$
$859$		−	18.8284i		−	0.642418i	0.947008	−	0.321209i	$-0.104089\pi$
$860$		0			0
$861$	11.6569	+	16.4853i	0.397265	+	0.561817i
$862$		0			0
$863$			29.7990i			1.01437i	0.861837	+	0.507185i	$0.169314\pi$
$863$			29.7990i			1.01437i	−0.861837	+	0.507185i	$0.830686\pi$
$864$		0			0
$865$			79.5980i			2.70641i
$866$		0			0
$867$	−9.00000	−	12.7279i	−0.305656	−	0.432263i
$868$		0			0
$869$	−10.3431			−0.350867
$870$		0			0
$871$	−37.7574	+	25.1716i	−1.27936	+	0.852906i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−19.3137			−0.652923
$876$		0			0
$877$			16.6863i			0.563456i	0.959494	+	0.281728i	$0.0909076\pi$
$877$			16.6863i			0.563456i	−0.959494	+	0.281728i	$0.909092\pi$
$878$		0			0
$879$	−30.7279	−	43.4558i	−1.03643	−	1.46573i
$880$		0			0
$881$			18.3431i			0.617996i	0.951063	+	0.308998i	$0.0999937\pi$
$881$			18.3431i			0.617996i	−0.951063	+	0.308998i	$0.900006\pi$
$882$		0			0
$883$			25.4558i			0.856657i	0.903623	+	0.428329i	$0.140897\pi$
$883$			25.4558i			0.856657i	−0.903623	+	0.428329i	$0.859103\pi$
$884$		0			0
$885$	−15.3137	+	10.8284i	−0.514765	+	0.363994i
$886$		0			0
$887$	−45.5980			−1.53103			−0.765515	−	0.643418i	$-0.777516\pi$
$887$	−45.5980			−1.53103			−0.765515	+	0.643418i	$0.777516\pi$
$888$		0			0
$889$		−	69.9411i		−	2.34575i
$890$		0			0
$891$	−4.68629	+	5.79899i	−0.156997	+	0.194273i
$892$		0			0
$893$		−	7.11270i		−	0.238017i
$894$		0			0
$895$	−32.9706			−1.10208
$896$		0			0
$897$	−21.3137	−	8.20101i	−0.711644	−	0.273824i
$898$		0			0
$899$			61.9411i			2.06585i
$900$		0			0
$901$	−33.9411			−1.13074
$902$		0			0
$903$	38.6274	−	27.3137i	1.28544	−	0.908943i
$904$		0			0
$905$	44.2843			1.47206
$906$		0			0
$907$			39.1127i			1.29872i	0.760483	+	0.649358i	$0.224962\pi$
$907$			39.1127i			1.29872i	−0.760483	+	0.649358i	$0.775038\pi$
$908$		0			0
$909$	−49.9411	−	17.6569i	−1.65644	−	0.585641i
$910$		0			0
$911$	22.9706			0.761049			0.380524	−	0.924771i	$-0.375744\pi$
$911$	22.9706			0.761049			0.380524	+	0.924771i	$0.375744\pi$
$912$		0			0
$913$	7.31371			0.242048
$914$		0			0
$915$	−46.6274	−	65.9411i	−1.54145	−	2.17995i
$916$		0			0
$917$	5.65685			0.186806
$918$		0			0
$919$		−	29.4558i		−	0.971659i	−0.874054	−	0.485829i	$-0.838518\pi$
$919$		−	29.4558i		−	0.971659i	0.874054	−	0.485829i	$-0.161482\pi$
$920$		0			0
$921$	10.2426	+	14.4853i	0.337506	+	0.477306i
$922$		0			0
$923$	−17.6569	−	26.4853i	−0.581182	−	0.871774i
$924$		0			0
$925$		−	11.0294i		−	0.362646i
$926$		0			0
$927$	−4.68629	−	1.65685i	−0.153918	−	0.0544182i
$928$		0			0
$929$	15.8995			0.521646			0.260823	−	0.965387i	$-0.416006\pi$
$929$	15.8995			0.521646			0.260823	+	0.965387i	$0.416006\pi$
$930$		0			0
$931$	−10.4437			−0.342277
$932$		0			0
$933$	−4.34315	−	6.14214i	−0.142188	−	0.201084i
$934$		0			0
$935$	8.00000			0.261628
$936$		0			0
$937$	4.97056			0.162381			0.0811906	−	0.996699i	$-0.474128\pi$
$937$	4.97056			0.162381			0.0811906	+	0.996699i	$0.474128\pi$
$938$		0			0
$939$	10.0000	+	14.1421i	0.326338	+	0.461511i
$940$		0			0
$941$	55.4975			1.80917			0.904583	−	0.426298i	$-0.140182\pi$
$941$	55.4975			1.80917			0.904583	+	0.426298i	$0.140182\pi$
$942$		0			0
$943$	−12.4853			−0.406577
$944$		0			0
$945$	−58.2843	+	16.4853i	−1.89599	+	0.536266i
$946$		0			0
$947$		−	9.79899i		−	0.318424i	−0.987244	−	0.159212i	$-0.949105\pi$
$947$		−	9.79899i		−	0.318424i	0.987244	−	0.159212i	$-0.0508954\pi$
$948$		0			0
$949$	−8.00000	−	12.0000i	−0.259691	−	0.389536i
$950$		0			0
$951$	20.5858	+	29.1127i	0.667540	+	0.944044i
$952$		0			0
$953$			24.0000i			0.777436i	0.921357	+	0.388718i	$0.127082\pi$
$953$			24.0000i			0.777436i	−0.921357	+	0.388718i	$0.872918\pi$
$954$		0			0
$955$	−12.4853			−0.404014
$956$		0			0
$957$	5.65685	+	8.00000i	0.182860	+	0.258603i
$958$		0			0
$959$	−46.2843			−1.49460
$960$		0			0
$961$	51.2843			1.65433
$962$		0			0
$963$	−1.65685	+	4.68629i	−0.0533914	+	0.151014i
$964$		0			0
$965$		−	13.6569i		−	0.439630i
$966$		0			0
$967$	37.7574			1.21419			0.607097	−	0.794627i	$-0.292334\pi$
$967$	37.7574			1.21419			0.607097	+	0.794627i	$0.292334\pi$
$968$		0			0
$969$	8.97056	−	6.34315i	0.288176	−	0.203771i
$970$		0			0
$971$	41.3137			1.32582			0.662910	−	0.748699i	$-0.269321\pi$
$971$	41.3137			1.32582			0.662910	+	0.748699i	$0.269321\pi$
$972$		0			0
$973$			8.00000i			0.256468i
$974$		0			0
$975$	−38.7990	−	14.9289i	−1.24256	−	0.478108i
$976$		0			0
$977$	−6.72792			−0.215245			−0.107623	−	0.994192i	$-0.534324\pi$
$977$	−6.72792			−0.215245			−0.107623	+	0.994192i	$0.534324\pi$
$978$		0			0
$979$			9.45584i			0.302210i
$980$		0			0
$981$	48.0000	+	16.9706i	1.53252	+	0.541828i
$982$		0			0
$983$			43.4558i			1.38603i	0.720925	+	0.693013i	$0.243717\pi$
$983$			43.4558i			1.38603i	−0.720925	+	0.693013i	$0.756283\pi$
$984$		0			0
$985$	−66.2843			−2.11199
$986$		0			0
$987$	15.3137	−	10.8284i	0.487441	−	0.344673i
$988$		0			0
$989$			29.2548i			0.930250i
$990$		0			0
$991$			45.9411i			1.45937i	0.683785	+	0.729684i	$0.260333\pi$
$991$			45.9411i			1.45937i	−0.683785	+	0.729684i	$0.739667\pi$
$992$		0			0
$993$	−31.6985	−	44.8284i	−1.00592	−	1.42259i
$994$		0			0
$995$		−	42.6274i		−	1.35138i
$996$		0			0
$997$	−2.68629			−0.0850757			−0.0425379	−	0.999095i	$-0.513544\pi$
$997$	−2.68629			−0.0850757			−0.0425379	+	0.999095i	$0.513544\pi$
$998$		0			0
$999$	−2.34315	−	8.28427i	−0.0741339	−	0.262103i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1248.2.n.a.1247.1	✓	4
3.2	odd	2		1248.2.n.d.1247.2	yes	4
4.3	odd	2		1248.2.n.c.1247.3	yes	4
12.11	even	2		1248.2.n.b.1247.4	yes	4
13.12	even	2		1248.2.n.b.1247.2	yes	4
39.38	odd	2		1248.2.n.c.1247.1	yes	4
52.51	odd	2		1248.2.n.d.1247.4	yes	4
156.155	even	2	inner	1248.2.n.a.1247.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1248.2.n.a.1247.1	✓	4	1.1	even	1	trivial
1248.2.n.a.1247.3	yes	4	156.155	even	2	inner
1248.2.n.b.1247.2	yes	4	13.12	even	2
1248.2.n.b.1247.4	yes	4	12.11	even	2
1248.2.n.c.1247.1	yes	4	39.38	odd	2
1248.2.n.c.1247.3	yes	4	4.3	odd	2
1248.2.n.d.1247.2	yes	4	3.2	odd	2
1248.2.n.d.1247.4	yes	4	52.51	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1248 = 2^{5} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1248.n (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.41421i) q^{3} -3.41421 q^{5} +3.41421 q^{7} +(-1.00000 + 2.82843i) q^{9} -0.828427i q^{11} +(3.00000 - 2.00000i) q^{13} +(3.41421 + 4.82843i) q^{15} -2.82843i q^{17} -2.24264 q^{19} +(-3.41421 - 4.82843i) q^{21} +3.65685 q^{23} +6.65685 q^{25} +(5.00000 - 1.41421i) q^{27} -6.82843i q^{29} -9.07107 q^{31} +(-1.17157 + 0.828427i) q^{33} -11.6569 q^{35} -1.65685i q^{37} +(-5.82843 - 2.24264i) q^{39} -3.41421 q^{41} +8.00000i q^{43} +(3.41421 - 9.65685i) q^{45} +3.17157i q^{47} +4.65685 q^{49} +(-4.00000 + 2.82843i) q^{51} -12.0000i q^{53} +2.82843i q^{55} +(2.24264 + 3.17157i) q^{57} +3.17157i q^{59} -13.6569 q^{61} +(-3.41421 + 9.65685i) q^{63} +(-10.2426 + 6.82843i) q^{65} -12.5858 q^{67} +(-3.65685 - 5.17157i) q^{69} -8.82843i q^{71} -4.00000i q^{73} +(-6.65685 - 9.41421i) q^{75} -2.82843i q^{77} -12.4853i q^{79} +(-7.00000 - 5.65685i) q^{81} +8.82843i q^{83} +9.65685i q^{85} +(-9.65685 + 6.82843i) q^{87} -11.4142 q^{89} +(10.2426 - 6.82843i) q^{91} +(9.07107 + 12.8284i) q^{93} +7.65685 q^{95} +(2.34315 + 0.828427i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 8 q^{5} + 8 q^{7} - 4 q^{9} + 12 q^{13} + 8 q^{15} + 8 q^{19} - 8 q^{21} - 8 q^{23} + 4 q^{25} + 20 q^{27} - 8 q^{31} - 16 q^{33} - 24 q^{35} - 12 q^{39} - 8 q^{41} + 8 q^{45} - 4 q^{49}+ \cdots + 32 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 - 8 * q^5 + 8 * q^7 - 4 * q^9 + 12 * q^13 + 8 * q^15 + 8 * q^19 - 8 * q^21 - 8 * q^23 + 4 * q^25 + 20 * q^27 - 8 * q^31 - 16 * q^33 - 24 * q^35 - 12 * q^39 - 8 * q^41 + 8 * q^45 - 4 * q^49 - 16 * q^51 - 8 * q^57 - 32 * q^61 - 8 * q^63 - 24 * q^65 - 56 * q^67 + 8 * q^69 - 4 * q^75 - 28 * q^81 - 16 * q^87 - 40 * q^89 + 24 * q^91 + 8 * q^93 + 8 * q^95 + 32 * q^99

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