LMFDB - Embedded newform 2112.2.m.h.1121.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2112,2,Mod(1121,2112)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2112, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 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("2112.1121"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2112 = 2^{6} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2112.m (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0, 0,0,32,0,0,0,0,0,-48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(39)] == traces)

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

Self dual:	no
Analytic conductor:	$16.8644049069$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1121.4
Root			$0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	2112.1121
Dual form			2112.2.m.h.1121.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.41421 + 1.00000i) q^{3} +2.44949 q^{5} +4.24264i q^{7} +(1.00000 - 2.82843i) q^{9} +(-2.82843 - 1.73205i) q^{11} +4.24264 q^{13} +(-3.46410 + 2.44949i) q^{15} -6.92820 q^{17} -4.89898 q^{19} +(-4.24264 - 6.00000i) q^{21} -2.44949i q^{23} +1.00000 q^{25} +(1.41421 + 5.00000i) q^{27} +6.00000i q^{29} -10.3923 q^{31} +(5.73205 - 0.378937i) q^{33} +10.3923i q^{35} +(-6.00000 + 4.24264i) q^{39} -3.46410 q^{41} +9.79796 q^{43} +(2.44949 - 6.92820i) q^{45} -2.44949i q^{47} -11.0000 q^{49} +(9.79796 - 6.92820i) q^{51} -2.44949 q^{53} +(-6.92820 - 4.24264i) q^{55} +(6.92820 - 4.89898i) q^{57} +2.82843 q^{59} -4.24264 q^{61} +(12.0000 + 4.24264i) q^{63} +10.3923 q^{65} -4.00000i q^{67} +(2.44949 + 3.46410i) q^{69} -7.34847i q^{71} +4.89898i q^{73} +(-1.41421 + 1.00000i) q^{75} +(7.34847 - 12.0000i) q^{77} +12.7279i q^{79} +(-7.00000 - 5.65685i) q^{81} -16.9706 q^{85} +(-6.00000 - 8.48528i) q^{87} +2.82843i q^{89} +18.0000i q^{91} +(14.6969 - 10.3923i) q^{93} -12.0000 q^{95} -16.0000 q^{97} +(-7.72741 + 6.26795i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{9} + 8 q^{25} + 32 q^{33} - 48 q^{39} - 88 q^{49} + 96 q^{63} - 56 q^{81} - 48 q^{87} - 96 q^{95} - 128 q^{97}+O(q^{100}) $ 8 * q + 8 * q^9 + 8 * q^25 + 32 * q^33 - 48 * q^39 - 88 * q^49 + 96 * q^63 - 56 * q^81 - 48 * q^87 - 96 * q^95 - 128 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2112\mathbb{Z}\right)^\times$.

$n$	$133$	$1409$	$1729$	$2047$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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
$3$	−1.41421	+	1.00000i	−0.816497	+	0.577350i
$3$	−1.41421	+	1.00000i	−0.816497	+	0.577350i
$4$		0			0
$5$	2.44949			1.09545			0.547723	−	0.836660i	$-0.315495\pi$
$5$	2.44949			1.09545			0.547723	+	0.836660i	$0.315495\pi$
$6$		0			0
$7$			4.24264i			1.60357i	0.597614	+	0.801784i	$0.296115\pi$
$7$			4.24264i			1.60357i	−0.597614	+	0.801784i	$0.703885\pi$
$8$		0			0
$9$	1.00000	−	2.82843i	0.333333	−	0.942809i
$10$		0			0
$11$	−2.82843	−	1.73205i	−0.852803	−	0.522233i
$11$	−2.82843	−	1.73205i	−0.852803	−	0.522233i
$12$		0			0
$13$	4.24264			1.17670			0.588348	−	0.808608i	$-0.299778\pi$
$13$	4.24264			1.17670			0.588348	+	0.808608i	$0.299778\pi$
$14$		0			0
$15$	−3.46410	+	2.44949i	−0.894427	+	0.632456i
$16$		0			0
$17$	−6.92820			−1.68034			−0.840168	−	0.542326i	$-0.817544\pi$
$17$	−6.92820			−1.68034			−0.840168	+	0.542326i	$0.817544\pi$
$18$		0			0
$19$	−4.89898			−1.12390			−0.561951	−	0.827170i	$-0.689949\pi$
$19$	−4.89898			−1.12390			−0.561951	+	0.827170i	$0.689949\pi$
$20$		0			0
$21$	−4.24264	−	6.00000i	−0.925820	−	1.30931i
$22$		0			0
$23$		−	2.44949i		−	0.510754i	−0.966842	−	0.255377i	$-0.917800\pi$
$23$		−	2.44949i		−	0.510754i	0.966842	−	0.255377i	$-0.0821996\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	1.41421	+	5.00000i	0.272166	+	0.962250i
$28$		0			0
$29$			6.00000i			1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$			6.00000i			1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$		0			0
$31$	−10.3923			−1.86651			−0.933257	−	0.359211i	$-0.883046\pi$
$31$	−10.3923			−1.86651			−0.933257	+	0.359211i	$0.883046\pi$
$32$		0			0
$33$	5.73205	−	0.378937i	0.997822	−	0.0659645i
$34$		0			0
$35$			10.3923i			1.75662i
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$	−6.00000	+	4.24264i	−0.960769	+	0.679366i
$40$		0			0
$41$	−3.46410			−0.541002			−0.270501	−	0.962720i	$-0.587189\pi$
$41$	−3.46410			−0.541002			−0.270501	+	0.962720i	$0.587189\pi$
$42$		0			0
$43$	9.79796			1.49417			0.747087	−	0.664726i	$-0.231452\pi$
$43$	9.79796			1.49417			0.747087	+	0.664726i	$0.231452\pi$
$44$		0			0
$45$	2.44949	−	6.92820i	0.365148	−	1.03280i
$46$		0			0
$47$		−	2.44949i		−	0.357295i	−0.983913	−	0.178647i	$-0.942828\pi$
$47$		−	2.44949i		−	0.357295i	0.983913	−	0.178647i	$-0.0571721\pi$
$48$		0			0
$49$	−11.0000			−1.57143
$50$		0			0
$51$	9.79796	−	6.92820i	1.37199	−	0.970143i
$52$		0			0
$53$	−2.44949			−0.336463			−0.168232	−	0.985747i	$-0.553806\pi$
$53$	−2.44949			−0.336463			−0.168232	+	0.985747i	$0.553806\pi$
$54$		0			0
$55$	−6.92820	−	4.24264i	−0.934199	−	0.572078i
$56$		0			0
$57$	6.92820	−	4.89898i	0.917663	−	0.648886i
$58$		0			0
$59$	2.82843			0.368230			0.184115	−	0.982905i	$-0.441058\pi$
$59$	2.82843			0.368230			0.184115	+	0.982905i	$0.441058\pi$
$60$		0			0
$61$	−4.24264			−0.543214			−0.271607	−	0.962408i	$-0.587555\pi$
$61$	−4.24264			−0.543214			−0.271607	+	0.962408i	$0.587555\pi$
$62$		0			0
$63$	12.0000	+	4.24264i	1.51186	+	0.534522i
$64$		0			0
$65$	10.3923			1.28901
$66$		0			0
$67$		−	4.00000i		−	0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$		−	4.00000i		−	0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$		0			0
$69$	2.44949	+	3.46410i	0.294884	+	0.417029i
$70$		0			0
$71$		−	7.34847i		−	0.872103i	−0.899922	−	0.436051i	$-0.856377\pi$
$71$		−	7.34847i		−	0.872103i	0.899922	−	0.436051i	$-0.143623\pi$
$72$		0			0
$73$			4.89898i			0.573382i	0.958023	+	0.286691i	$0.0925553\pi$
$73$			4.89898i			0.573382i	−0.958023	+	0.286691i	$0.907445\pi$
$74$		0			0
$75$	−1.41421	+	1.00000i	−0.163299	+	0.115470i
$76$		0			0
$77$	7.34847	−	12.0000i	0.837436	−	1.36753i
$78$		0			0
$79$			12.7279i			1.43200i	0.698099	+	0.716002i	$0.254030\pi$
$79$			12.7279i			1.43200i	−0.698099	+	0.716002i	$0.745970\pi$
$80$		0			0
$81$	−7.00000	−	5.65685i	−0.777778	−	0.628539i
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$	−16.9706			−1.84072
$86$		0			0
$87$	−6.00000	−	8.48528i	−0.643268	−	0.909718i
$88$		0			0
$89$			2.82843i			0.299813i	0.988700	+	0.149906i	$0.0478972\pi$
$89$			2.82843i			0.299813i	−0.988700	+	0.149906i	$0.952103\pi$
$90$		0			0
$91$			18.0000i			1.88691i
$92$		0			0
$93$	14.6969	−	10.3923i	1.52400	−	1.07763i
$94$		0			0
$95$	−12.0000			−1.23117
$96$		0			0
$97$	−16.0000			−1.62455			−0.812277	−	0.583272i	$-0.801772\pi$
$97$	−16.0000			−1.62455			−0.812277	+	0.583272i	$0.801772\pi$
$98$		0			0
$99$	−7.72741	+	6.26795i	−0.776634	+	0.629953i
$100$		0			0
$101$		−	6.00000i		−	0.597022i	−0.954406	−	0.298511i	$-0.903510\pi$
$101$		−	6.00000i		−	0.597022i	0.954406	−	0.298511i	$-0.0964900\pi$
$102$		0			0
$103$	−3.46410			−0.341328			−0.170664	−	0.985329i	$-0.554591\pi$
$103$	−3.46410			−0.341328			−0.170664	+	0.985329i	$0.554591\pi$
$104$		0			0
$105$	−10.3923	−	14.6969i	−1.01419	−	1.43427i
$106$		0			0
$107$			17.3205i			1.67444i	0.546869	+	0.837218i	$0.315820\pi$
$107$			17.3205i			1.67444i	−0.546869	+	0.837218i	$0.684180\pi$
$108$		0			0
$109$	4.24264			0.406371			0.203186	−	0.979140i	$-0.434871\pi$
$109$	4.24264			0.406371			0.203186	+	0.979140i	$0.434871\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			14.1421i			1.33038i	0.746674	+	0.665190i	$0.231650\pi$
$113$			14.1421i			1.33038i	−0.746674	+	0.665190i	$0.768350\pi$
$114$		0			0
$115$		−	6.00000i		−	0.559503i
$116$		0			0
$117$	4.24264	−	12.0000i	0.392232	−	1.10940i
$118$		0			0
$119$		−	29.3939i		−	2.69453i
$120$		0			0
$121$	5.00000	+	9.79796i	0.454545	+	0.890724i
$122$		0			0
$123$	4.89898	−	3.46410i	0.441726	−	0.312348i
$124$		0			0
$125$	−9.79796			−0.876356
$126$		0			0
$127$		−	4.24264i		−	0.376473i	−0.982124	−	0.188237i	$-0.939723\pi$
$127$		−	4.24264i		−	0.376473i	0.982124	−	0.188237i	$-0.0602772\pi$
$128$		0			0
$129$	−13.8564	+	9.79796i	−1.21999	+	0.862662i
$130$		0			0
$131$		−	20.7846i		−	1.81596i	−0.419014	−	0.907980i	$-0.637624\pi$
$131$		−	20.7846i		−	1.81596i	0.419014	−	0.907980i	$-0.362376\pi$
$132$		0			0
$133$		−	20.7846i		−	1.80225i
$134$		0			0
$135$	3.46410	+	12.2474i	0.298142	+	1.05409i
$136$		0			0
$137$		−	19.7990i		−	1.69154i	−0.533546	−	0.845771i	$-0.679141\pi$
$137$		−	19.7990i		−	1.69154i	0.533546	−	0.845771i	$-0.320859\pi$
$138$		0			0
$139$	−9.79796			−0.831052			−0.415526	−	0.909581i	$-0.636402\pi$
$139$	−9.79796			−0.831052			−0.415526	+	0.909581i	$0.636402\pi$
$140$		0			0
$141$	2.44949	+	3.46410i	0.206284	+	0.291730i
$142$		0			0
$143$	−12.0000	−	7.34847i	−1.00349	−	0.614510i
$144$		0			0
$145$			14.6969i			1.22051i
$146$		0			0
$147$	15.5563	−	11.0000i	1.28307	−	0.907265i
$148$		0			0
$149$			18.0000i			1.47462i	0.675556	+	0.737309i	$0.263904\pi$
$149$			18.0000i			1.47462i	−0.675556	+	0.737309i	$0.736096\pi$
$150$		0			0
$151$			4.24264i			0.345261i	0.984987	+	0.172631i	$0.0552267\pi$
$151$			4.24264i			0.345261i	−0.984987	+	0.172631i	$0.944773\pi$
$152$		0			0
$153$	−6.92820	+	19.5959i	−0.560112	+	1.58424i
$154$		0			0
$155$	−25.4558			−2.04466
$156$		0			0
$157$		−	13.8564i		−	1.10586i	−0.833227	−	0.552931i	$-0.813509\pi$
$157$		−	13.8564i		−	1.10586i	0.833227	−	0.552931i	$-0.186491\pi$
$158$		0			0
$159$	3.46410	−	2.44949i	0.274721	−	0.194257i
$160$		0			0
$161$	10.3923			0.819028
$162$		0			0
$163$		−	10.0000i		−	0.783260i	−0.920123	−	0.391630i	$-0.871911\pi$
$163$		−	10.0000i		−	0.783260i	0.920123	−	0.391630i	$-0.128089\pi$
$164$		0			0
$165$	14.0406	−	0.928203i	1.09306	−	0.0722605i
$166$		0			0
$167$	−12.0000			−0.928588			−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000			−0.928588			−0.464294	+	0.885681i	$0.653692\pi$
$168$		0			0
$169$	5.00000			0.384615
$170$		0			0
$171$	−4.89898	+	13.8564i	−0.374634	+	1.05963i
$172$		0			0
$173$			6.00000i			0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$			6.00000i			0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$		0			0
$175$			4.24264i			0.320713i
$176$		0			0
$177$	−4.00000	+	2.82843i	−0.300658	+	0.212598i
$178$		0			0
$179$	−2.82843			−0.211407			−0.105703	−	0.994398i	$-0.533709\pi$
$179$	−2.82843			−0.211407			−0.105703	+	0.994398i	$0.533709\pi$
$180$		0			0
$181$		−	13.8564i		−	1.02994i	−0.857209	−	0.514969i	$-0.827803\pi$
$181$		−	13.8564i		−	1.02994i	0.857209	−	0.514969i	$-0.172197\pi$
$182$		0			0
$183$	6.00000	−	4.24264i	0.443533	−	0.313625i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	19.5959	+	12.0000i	1.43300	+	0.877527i
$188$		0			0
$189$	−21.2132	+	6.00000i	−1.54303	+	0.436436i
$190$		0			0
$191$			17.1464i			1.24067i	0.784336	+	0.620336i	$0.213004\pi$
$191$			17.1464i			1.24067i	−0.784336	+	0.620336i	$0.786996\pi$
$192$		0			0
$193$			4.89898i			0.352636i	0.984333	+	0.176318i	$0.0564187\pi$
$193$			4.89898i			0.352636i	−0.984333	+	0.176318i	$0.943581\pi$
$194$		0			0
$195$	−14.6969	+	10.3923i	−1.05247	+	0.744208i
$196$		0			0
$197$			6.00000i			0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$			6.00000i			0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$		0			0
$199$	24.2487			1.71895			0.859473	−	0.511182i	$-0.170792\pi$
$199$	24.2487			1.71895			0.859473	+	0.511182i	$0.170792\pi$
$200$		0			0
$201$	4.00000	+	5.65685i	0.282138	+	0.399004i
$202$		0			0
$203$	−25.4558			−1.78665
$204$		0			0
$205$	−8.48528			−0.592638
$206$		0			0
$207$	−6.92820	−	2.44949i	−0.481543	−	0.170251i
$208$		0			0
$209$	13.8564	+	8.48528i	0.958468	+	0.586939i
$210$		0			0
$211$	−14.6969			−1.01178			−0.505889	−	0.862598i	$-0.668836\pi$
$211$	−14.6969			−1.01178			−0.505889	+	0.862598i	$0.668836\pi$
$212$		0			0
$213$	7.34847	+	10.3923i	0.503509	+	0.712069i
$214$		0			0
$215$	24.0000			1.63679
$216$		0			0
$217$		−	44.0908i		−	2.99308i
$218$		0			0
$219$	−4.89898	−	6.92820i	−0.331042	−	0.468165i
$220$		0			0
$221$	−29.3939			−1.97725
$222$		0			0
$223$	3.46410			0.231973			0.115987	−	0.993251i	$-0.462997\pi$
$223$	3.46410			0.231973			0.115987	+	0.993251i	$0.462997\pi$
$224$		0			0
$225$	1.00000	−	2.82843i	0.0666667	−	0.188562i
$226$		0			0
$227$			3.46410i			0.229920i	0.993370	+	0.114960i	$0.0366741\pi$
$227$			3.46410i			0.229920i	−0.993370	+	0.114960i	$0.963326\pi$
$228$		0			0
$229$			6.92820i			0.457829i	0.973447	+	0.228914i	$0.0735176\pi$
$229$			6.92820i			0.457829i	−0.973447	+	0.228914i	$0.926482\pi$
$230$		0			0
$231$	1.60770	+	24.3190i	0.105779	+	1.60007i
$232$		0			0
$233$	−17.3205			−1.13470			−0.567352	−	0.823475i	$-0.692032\pi$
$233$	−17.3205			−1.13470			−0.567352	+	0.823475i	$0.692032\pi$
$234$		0			0
$235$		−	6.00000i		−	0.391397i
$236$		0			0
$237$	−12.7279	−	18.0000i	−0.826767	−	1.16923i
$238$		0			0
$239$	−12.0000			−0.776215			−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000			−0.776215			−0.388108	+	0.921614i	$0.626871\pi$
$240$		0			0
$241$		−	4.89898i		−	0.315571i	−0.987473	−	0.157786i	$-0.949565\pi$
$241$		−	4.89898i		−	0.315571i	0.987473	−	0.157786i	$-0.0504355\pi$
$242$		0			0
$243$	15.5563	+	1.00000i	0.997940	+	0.0641500i
$244$		0			0
$245$	−26.9444			−1.72141
$246$		0			0
$247$	−20.7846			−1.32249
$248$		0			0
$249$		0			0
$250$		0			0
$251$	5.65685			0.357057			0.178529	−	0.983935i	$-0.442866\pi$
$251$	5.65685			0.357057			0.178529	+	0.983935i	$0.442866\pi$
$252$		0			0
$253$	−4.24264	+	6.92820i	−0.266733	+	0.435572i
$254$		0			0
$255$	24.0000	−	16.9706i	1.50294	−	1.06274i
$256$		0			0
$257$		−	2.82843i		−	0.176432i	−0.996101	−	0.0882162i	$-0.971883\pi$
$257$		−	2.82843i		−	0.176432i	0.996101	−	0.0882162i	$-0.0281166\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	16.9706	+	6.00000i	1.05045	+	0.371391i
$262$		0			0
$263$	−24.0000			−1.47990			−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000			−1.47990			−0.739952	+	0.672660i	$0.765152\pi$
$264$		0			0
$265$	−6.00000			−0.368577
$266$		0			0
$267$	−2.82843	−	4.00000i	−0.173097	−	0.244796i
$268$		0			0
$269$	12.2474			0.746740			0.373370	−	0.927682i	$-0.378202\pi$
$269$	12.2474			0.746740			0.373370	+	0.927682i	$0.378202\pi$
$270$		0			0
$271$		−	12.7279i		−	0.773166i	−0.922255	−	0.386583i	$-0.873655\pi$
$271$		−	12.7279i		−	0.773166i	0.922255	−	0.386583i	$-0.126345\pi$
$272$		0			0
$273$	−18.0000	−	25.4558i	−1.08941	−	1.54066i
$274$		0			0
$275$	−2.82843	−	1.73205i	−0.170561	−	0.104447i
$276$		0			0
$277$	12.7279			0.764747			0.382373	−	0.924008i	$-0.375107\pi$
$277$	12.7279			0.764747			0.382373	+	0.924008i	$0.375107\pi$
$278$		0			0
$279$	−10.3923	+	29.3939i	−0.622171	+	1.75977i
$280$		0			0
$281$	20.7846			1.23991			0.619953	−	0.784639i	$-0.287152\pi$
$281$	20.7846			1.23991			0.619953	+	0.784639i	$0.287152\pi$
$282$		0			0
$283$	14.6969			0.873642			0.436821	−	0.899548i	$-0.356104\pi$
$283$	14.6969			0.873642			0.436821	+	0.899548i	$0.356104\pi$
$284$		0			0
$285$	16.9706	−	12.0000i	1.00525	−	0.710819i
$286$		0			0
$287$		−	14.6969i		−	0.867533i
$288$		0			0
$289$	31.0000			1.82353
$290$		0			0
$291$	22.6274	−	16.0000i	1.32644	−	0.937937i
$292$		0			0
$293$		−	18.0000i		−	1.05157i	−0.850617	−	0.525786i	$-0.823771\pi$
$293$		−	18.0000i		−	1.05157i	0.850617	−	0.525786i	$-0.176229\pi$
$294$		0			0
$295$	6.92820			0.403376
$296$		0			0
$297$	4.66025	−	16.5916i	0.270415	−	0.962744i
$298$		0			0
$299$		−	10.3923i		−	0.601003i
$300$		0			0
$301$			41.5692i			2.39601i
$302$		0			0
$303$	6.00000	+	8.48528i	0.344691	+	0.487467i
$304$		0			0
$305$	−10.3923			−0.595062
$306$		0			0
$307$	9.79796			0.559199			0.279600	−	0.960117i	$-0.409798\pi$
$307$	9.79796			0.559199			0.279600	+	0.960117i	$0.409798\pi$
$308$		0			0
$309$	4.89898	−	3.46410i	0.278693	−	0.197066i
$310$		0			0
$311$			12.2474i			0.694489i	0.937775	+	0.347245i	$0.112883\pi$
$311$			12.2474i			0.694489i	−0.937775	+	0.347245i	$0.887117\pi$
$312$		0			0
$313$	10.0000			0.565233			0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000			0.565233			0.282617	+	0.959233i	$0.408798\pi$
$314$		0			0
$315$	29.3939	+	10.3923i	1.65616	+	0.585540i
$316$		0			0
$317$	−17.1464			−0.963039			−0.481520	−	0.876435i	$-0.659915\pi$
$317$	−17.1464			−0.963039			−0.481520	+	0.876435i	$0.659915\pi$
$318$		0			0
$319$	10.3923	−	16.9706i	0.581857	−	0.950169i
$320$		0			0
$321$	−17.3205	−	24.4949i	−0.966736	−	1.36717i
$322$		0			0
$323$	33.9411			1.88853
$324$		0			0
$325$	4.24264			0.235339
$326$		0			0
$327$	−6.00000	+	4.24264i	−0.331801	+	0.234619i
$328$		0			0
$329$	10.3923			0.572946
$330$		0			0
$331$			14.0000i			0.769510i	0.923019	+	0.384755i	$0.125714\pi$
$331$			14.0000i			0.769510i	−0.923019	+	0.384755i	$0.874286\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	9.79796i		−	0.535320i
$336$		0			0
$337$			4.89898i			0.266864i	0.991058	+	0.133432i	$0.0425998\pi$
$337$			4.89898i			0.266864i	−0.991058	+	0.133432i	$0.957400\pi$
$338$		0			0
$339$	−14.1421	−	20.0000i	−0.768095	−	1.08625i
$340$		0			0
$341$	29.3939	+	18.0000i	1.59177	+	0.974755i
$342$		0			0
$343$		−	16.9706i		−	0.916324i
$344$		0			0
$345$	6.00000	+	8.48528i	0.323029	+	0.456832i
$346$		0			0
$347$			13.8564i			0.743851i	0.928263	+	0.371925i	$0.121302\pi$
$347$			13.8564i			0.743851i	−0.928263	+	0.371925i	$0.878698\pi$
$348$		0			0
$349$	21.2132			1.13552			0.567758	−	0.823195i	$-0.307811\pi$
$349$	21.2132			1.13552			0.567758	+	0.823195i	$0.307811\pi$
$350$		0			0
$351$	6.00000	+	21.2132i	0.320256	+	1.13228i
$352$		0			0
$353$			28.2843i			1.50542i	0.658352	+	0.752710i	$0.271254\pi$
$353$			28.2843i			1.50542i	−0.658352	+	0.752710i	$0.728746\pi$
$354$		0			0
$355$		−	18.0000i		−	0.955341i
$356$		0			0
$357$	29.3939	+	41.5692i	1.55569	+	2.20008i
$358$		0			0
$359$	24.0000			1.26667			0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000			1.26667			0.633336	+	0.773877i	$0.281685\pi$
$360$		0			0
$361$	5.00000			0.263158
$362$		0			0
$363$	−16.8690	−	8.85641i	−0.885394	−	0.464841i
$364$		0			0
$365$			12.0000i			0.628109i
$366$		0			0
$367$	3.46410			0.180825			0.0904123	−	0.995904i	$-0.471182\pi$
$367$	3.46410			0.180825			0.0904123	+	0.995904i	$0.471182\pi$
$368$		0			0
$369$	−3.46410	+	9.79796i	−0.180334	+	0.510061i
$370$		0			0
$371$		−	10.3923i		−	0.539542i
$372$		0			0
$373$	4.24264			0.219676			0.109838	−	0.993950i	$-0.464967\pi$
$373$	4.24264			0.219676			0.109838	+	0.993950i	$0.464967\pi$
$374$		0			0
$375$	13.8564	−	9.79796i	0.715542	−	0.505964i
$376$		0			0
$377$			25.4558i			1.31104i
$378$		0			0
$379$			34.0000i			1.74646i	0.487306	+	0.873231i	$0.337980\pi$
$379$			34.0000i			1.74646i	−0.487306	+	0.873231i	$0.662020\pi$
$380$		0			0
$381$	4.24264	+	6.00000i	0.217357	+	0.307389i
$382$		0			0
$383$		−	26.9444i		−	1.37679i	−0.725334	−	0.688397i	$-0.758315\pi$
$383$		−	26.9444i		−	1.37679i	0.725334	−	0.688397i	$-0.241685\pi$
$384$		0			0
$385$	18.0000	−	29.3939i	0.917365	−	1.49805i
$386$		0			0
$387$	9.79796	−	27.7128i	0.498058	−	1.40872i
$388$		0			0
$389$	−22.0454			−1.11775			−0.558873	−	0.829253i	$-0.688766\pi$
$389$	−22.0454			−1.11775			−0.558873	+	0.829253i	$0.688766\pi$
$390$		0			0
$391$			16.9706i			0.858238i
$392$		0			0
$393$	20.7846	+	29.3939i	1.04844	+	1.48272i
$394$		0			0
$395$			31.1769i			1.56868i
$396$		0			0
$397$			34.6410i			1.73858i	0.494300	+	0.869291i	$0.335424\pi$
$397$			34.6410i			1.73858i	−0.494300	+	0.869291i	$0.664576\pi$
$398$		0			0
$399$	20.7846	+	29.3939i	1.04053	+	1.47153i
$400$		0			0
$401$		−	11.3137i		−	0.564980i	−0.959270	−	0.282490i	$-0.908840\pi$
$401$		−	11.3137i		−	0.564980i	0.959270	−	0.282490i	$-0.0911603\pi$
$402$		0			0
$403$	−44.0908			−2.19632
$404$		0			0
$405$	−17.1464	−	13.8564i	−0.852013	−	0.688530i
$406$		0			0
$407$		0			0
$408$		0			0
$409$			19.5959i			0.968956i	0.874804	+	0.484478i	$0.160990\pi$
$409$			19.5959i			0.968956i	−0.874804	+	0.484478i	$0.839010\pi$
$410$		0			0
$411$	19.7990	+	28.0000i	0.976612	+	1.38114i
$412$		0			0
$413$			12.0000i			0.590481i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	13.8564	−	9.79796i	0.678551	−	0.479808i
$418$		0			0
$419$	−5.65685			−0.276355			−0.138178	−	0.990407i	$-0.544125\pi$
$419$	−5.65685			−0.276355			−0.138178	+	0.990407i	$0.544125\pi$
$420$		0			0
$421$			13.8564i			0.675320i	0.941268	+	0.337660i	$0.109635\pi$
$421$			13.8564i			0.675320i	−0.941268	+	0.337660i	$0.890365\pi$
$422$		0			0
$423$	−6.92820	−	2.44949i	−0.336861	−	0.119098i
$424$		0			0
$425$	−6.92820			−0.336067
$426$		0			0
$427$		−	18.0000i		−	0.871081i
$428$		0			0
$429$	24.3190	−	1.60770i	1.17413	−	0.0776203i
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	−2.00000			−0.0961139			−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000			−0.0961139			−0.0480569	+	0.998845i	$0.515303\pi$
$434$		0			0
$435$	−14.6969	−	20.7846i	−0.704664	−	0.996546i
$436$		0			0
$437$			12.0000i			0.574038i
$438$		0			0
$439$		−	21.2132i		−	1.01245i	−0.862401	−	0.506225i	$-0.831040\pi$
$439$		−	21.2132i		−	1.01245i	0.862401	−	0.506225i	$-0.168960\pi$
$440$		0			0
$441$	−11.0000	+	31.1127i	−0.523810	+	1.48156i
$442$		0			0
$443$	5.65685			0.268765			0.134383	−	0.990930i	$-0.457095\pi$
$443$	5.65685			0.268765			0.134383	+	0.990930i	$0.457095\pi$
$444$		0			0
$445$			6.92820i			0.328428i
$446$		0			0
$447$	−18.0000	−	25.4558i	−0.851371	−	1.20402i
$448$		0			0
$449$			36.7696i			1.73526i	0.497208	+	0.867631i	$0.334358\pi$
$449$			36.7696i			1.73526i	−0.497208	+	0.867631i	$0.665642\pi$
$450$		0			0
$451$	9.79796	+	6.00000i	0.461368	+	0.282529i
$452$		0			0
$453$	−4.24264	−	6.00000i	−0.199337	−	0.281905i
$454$		0			0
$455$			44.0908i			2.06701i
$456$		0			0
$457$			19.5959i			0.916658i	0.888783	+	0.458329i	$0.151552\pi$
$457$			19.5959i			0.916658i	−0.888783	+	0.458329i	$0.848448\pi$
$458$		0			0
$459$	−9.79796	−	34.6410i	−0.457330	−	1.61690i
$460$		0			0
$461$			6.00000i			0.279448i	0.990190	+	0.139724i	$0.0446215\pi$
$461$			6.00000i			0.279448i	−0.990190	+	0.139724i	$0.955378\pi$
$462$		0			0
$463$	24.2487			1.12693			0.563467	−	0.826139i	$-0.309467\pi$
$463$	24.2487			1.12693			0.563467	+	0.826139i	$0.309467\pi$
$464$		0			0
$465$	36.0000	−	25.4558i	1.66946	−	1.18049i
$466$		0			0
$467$	5.65685			0.261768			0.130884	−	0.991398i	$-0.458218\pi$
$467$	5.65685			0.261768			0.130884	+	0.991398i	$0.458218\pi$
$468$		0			0
$469$	16.9706			0.783628
$470$		0			0
$471$	13.8564	+	19.5959i	0.638470	+	0.902932i
$472$		0			0
$473$	−27.7128	−	16.9706i	−1.27424	−	0.780307i
$474$		0			0
$475$	−4.89898			−0.224781
$476$		0			0
$477$	−2.44949	+	6.92820i	−0.112154	+	0.317221i
$478$		0			0
$479$	−36.0000			−1.64488			−0.822441	−	0.568850i	$-0.807388\pi$
$479$	−36.0000			−1.64488			−0.822441	+	0.568850i	$0.807388\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	−14.6969	+	10.3923i	−0.668734	+	0.472866i
$484$		0			0
$485$	−39.1918			−1.77961
$486$		0			0
$487$	10.3923			0.470920			0.235460	−	0.971884i	$-0.424340\pi$
$487$	10.3923			0.470920			0.235460	+	0.971884i	$0.424340\pi$
$488$		0			0
$489$	10.0000	+	14.1421i	0.452216	+	0.639529i
$490$		0			0
$491$		−	17.3205i		−	0.781664i	−0.920462	−	0.390832i	$-0.872187\pi$
$491$		−	17.3205i		−	0.781664i	0.920462	−	0.390832i	$-0.127813\pi$
$492$		0			0
$493$		−	41.5692i		−	1.87218i
$494$		0			0
$495$	−18.9282	+	15.3533i	−0.850759	+	0.690078i
$496$		0			0
$497$	31.1769			1.39848
$498$		0			0
$499$		−	14.0000i		−	0.626726i	−0.949633	−	0.313363i	$-0.898544\pi$
$499$		−	14.0000i		−	0.626726i	0.949633	−	0.313363i	$-0.101456\pi$
$500$		0			0
$501$	16.9706	−	12.0000i	0.758189	−	0.536120i
$502$		0			0
$503$	12.0000			0.535054			0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000			0.535054			0.267527	+	0.963550i	$0.413794\pi$
$504$		0			0
$505$		−	14.6969i		−	0.654005i
$506$		0			0
$507$	−7.07107	+	5.00000i	−0.314037	+	0.222058i
$508$		0			0
$509$	−2.44949			−0.108572			−0.0542859	−	0.998525i	$-0.517288\pi$
$509$	−2.44949			−0.108572			−0.0542859	+	0.998525i	$0.517288\pi$
$510$		0			0
$511$	−20.7846			−0.919457
$512$		0			0
$513$	−6.92820	−	24.4949i	−0.305888	−	1.08148i
$514$		0			0
$515$	−8.48528			−0.373906
$516$		0			0
$517$	−4.24264	+	6.92820i	−0.186591	+	0.304702i
$518$		0			0
$519$	−6.00000	−	8.48528i	−0.263371	−	0.372463i
$520$		0			0
$521$			11.3137i			0.495663i	0.968803	+	0.247831i	$0.0797179\pi$
$521$			11.3137i			0.495663i	−0.968803	+	0.247831i	$0.920282\pi$
$522$		0			0
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$		0			0
$525$	−4.24264	−	6.00000i	−0.185164	−	0.261861i
$526$		0			0
$527$	72.0000			3.13637
$528$		0			0
$529$	17.0000			0.739130
$530$		0			0
$531$	2.82843	−	8.00000i	0.122743	−	0.347170i
$532$		0			0
$533$	−14.6969			−0.636595
$534$		0			0
$535$			42.4264i			1.83425i
$536$		0			0
$537$	4.00000	−	2.82843i	0.172613	−	0.122056i
$538$		0			0
$539$	31.1127	+	19.0526i	1.34012	+	0.820652i
$540$		0			0
$541$	4.24264			0.182405			0.0912027	−	0.995832i	$-0.470929\pi$
$541$	4.24264			0.182405			0.0912027	+	0.995832i	$0.470929\pi$
$542$		0			0
$543$	13.8564	+	19.5959i	0.594635	+	0.840941i
$544$		0			0
$545$	10.3923			0.445157
$546$		0			0
$547$	24.4949			1.04733			0.523663	−	0.851925i	$-0.324565\pi$
$547$	24.4949			1.04733			0.523663	+	0.851925i	$0.324565\pi$
$548$		0			0
$549$	−4.24264	+	12.0000i	−0.181071	+	0.512148i
$550$		0			0
$551$		−	29.3939i		−	1.25222i
$552$		0			0
$553$	−54.0000			−2.29631
$554$		0			0
$555$		0			0
$556$		0			0
$557$			30.0000i			1.27114i	0.772043	+	0.635570i	$0.219235\pi$
$557$			30.0000i			1.27114i	−0.772043	+	0.635570i	$0.780765\pi$
$558$		0			0
$559$	41.5692			1.75819
$560$		0			0
$561$	−39.7128	+	2.62536i	−1.67668	+	0.110843i
$562$		0			0
$563$			3.46410i			0.145994i	0.997332	+	0.0729972i	$0.0232564\pi$
$563$			3.46410i			0.145994i	−0.997332	+	0.0729972i	$0.976744\pi$
$564$		0			0
$565$			34.6410i			1.45736i
$566$		0			0
$567$	24.0000	−	29.6985i	1.00791	−	1.24722i
$568$		0			0
$569$	6.92820			0.290445			0.145223	−	0.989399i	$-0.453610\pi$
$569$	6.92820			0.290445			0.145223	+	0.989399i	$0.453610\pi$
$570$		0			0
$571$	−19.5959			−0.820064			−0.410032	−	0.912071i	$-0.634482\pi$
$571$	−19.5959			−0.820064			−0.410032	+	0.912071i	$0.634482\pi$
$572$		0			0
$573$	−17.1464	−	24.2487i	−0.716302	−	1.01300i
$574$		0			0
$575$		−	2.44949i		−	0.102151i
$576$		0			0
$577$	20.0000			0.832611			0.416305	−	0.909225i	$-0.363325\pi$
$577$	20.0000			0.832611			0.416305	+	0.909225i	$0.363325\pi$
$578$		0			0
$579$	−4.89898	−	6.92820i	−0.203595	−	0.287926i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	6.92820	+	4.24264i	0.286937	+	0.175712i
$584$		0			0
$585$	10.3923	−	29.3939i	0.429669	−	1.21529i
$586$		0			0
$587$	2.82843			0.116742			0.0583708	−	0.998295i	$-0.481409\pi$
$587$	2.82843			0.116742			0.0583708	+	0.998295i	$0.481409\pi$
$588$		0			0
$589$	50.9117			2.09778
$590$		0			0
$591$	−6.00000	−	8.48528i	−0.246807	−	0.349038i
$592$		0			0
$593$	−20.7846			−0.853522			−0.426761	−	0.904365i	$-0.640345\pi$
$593$	−20.7846			−0.853522			−0.426761	+	0.904365i	$0.640345\pi$
$594$		0			0
$595$		−	72.0000i		−	2.95171i
$596$		0			0
$597$	−34.2929	+	24.2487i	−1.40351	+	0.992434i
$598$		0			0
$599$			22.0454i			0.900751i	0.892839	+	0.450375i	$0.148710\pi$
$599$			22.0454i			0.900751i	−0.892839	+	0.450375i	$0.851290\pi$
$600$		0			0
$601$			4.89898i			0.199834i	0.994996	+	0.0999168i	$0.0318577\pi$
$601$			4.89898i			0.199834i	−0.994996	+	0.0999168i	$0.968142\pi$
$602$		0			0
$603$	−11.3137	−	4.00000i	−0.460730	−	0.162893i
$604$		0			0
$605$	12.2474	+	24.0000i	0.497930	+	0.975739i
$606$		0			0
$607$			38.1838i			1.54983i	0.632065	+	0.774916i	$0.282208\pi$
$607$			38.1838i			1.54983i	−0.632065	+	0.774916i	$0.717792\pi$
$608$		0			0
$609$	36.0000	−	25.4558i	1.45879	−	1.03152i
$610$		0			0
$611$		−	10.3923i		−	0.420428i
$612$		0			0
$613$	−38.1838			−1.54223			−0.771114	−	0.636697i	$-0.780300\pi$
$613$	−38.1838			−1.54223			−0.771114	+	0.636697i	$0.780300\pi$
$614$		0			0
$615$	12.0000	−	8.48528i	0.483887	−	0.342160i
$616$		0			0
$617$		−	14.1421i		−	0.569341i	−0.958625	−	0.284670i	$-0.908116\pi$
$617$		−	14.1421i		−	0.569341i	0.958625	−	0.284670i	$-0.0918842\pi$
$618$		0			0
$619$		−	20.0000i		−	0.803868i	−0.915669	−	0.401934i	$-0.868338\pi$
$619$		−	20.0000i		−	0.803868i	0.915669	−	0.401934i	$-0.131662\pi$
$620$		0			0
$621$	12.2474	−	3.46410i	0.491473	−	0.139010i
$622$		0			0
$623$	−12.0000			−0.480770
$624$		0			0
$625$	−29.0000			−1.16000
$626$		0			0
$627$	−28.0812	+	1.85641i	−1.12146	+	0.0741377i
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−24.2487			−0.965326			−0.482663	−	0.875806i	$-0.660330\pi$
$631$	−24.2487			−0.965326			−0.482663	+	0.875806i	$0.660330\pi$
$632$		0			0
$633$	20.7846	−	14.6969i	0.826114	−	0.584151i
$634$		0			0
$635$		−	10.3923i		−	0.412406i
$636$		0			0
$637$	−46.6690			−1.84909
$638$		0			0
$639$	−20.7846	−	7.34847i	−0.822226	−	0.290701i
$640$		0			0
$641$			22.6274i			0.893729i	0.894602	+	0.446865i	$0.147459\pi$
$641$			22.6274i			0.893729i	−0.894602	+	0.446865i	$0.852541\pi$
$642$		0			0
$643$			4.00000i			0.157745i	0.996885	+	0.0788723i	$0.0251319\pi$
$643$			4.00000i			0.157745i	−0.996885	+	0.0788723i	$0.974868\pi$
$644$		0			0
$645$	−33.9411	+	24.0000i	−1.33643	+	0.944999i
$646$		0			0
$647$		−	26.9444i		−	1.05929i	−0.848218	−	0.529647i	$-0.822325\pi$
$647$		−	26.9444i		−	1.05929i	0.848218	−	0.529647i	$-0.177675\pi$
$648$		0			0
$649$	−8.00000	−	4.89898i	−0.314027	−	0.192302i
$650$		0			0
$651$	44.0908	+	62.3538i	1.72806	+	2.44384i
$652$		0			0
$653$	22.0454			0.862703			0.431352	−	0.902184i	$-0.358037\pi$
$653$	22.0454			0.862703			0.431352	+	0.902184i	$0.358037\pi$
$654$		0			0
$655$		−	50.9117i		−	1.98928i
$656$		0			0
$657$	13.8564	+	4.89898i	0.540590	+	0.191127i
$658$		0			0
$659$			31.1769i			1.21448i	0.794518	+	0.607240i	$0.207723\pi$
$659$			31.1769i			1.21448i	−0.794518	+	0.607240i	$0.792277\pi$
$660$		0			0
$661$		−	20.7846i		−	0.808428i	−0.914665	−	0.404214i	$-0.867545\pi$
$661$		−	20.7846i		−	0.808428i	0.914665	−	0.404214i	$-0.132455\pi$
$662$		0			0
$663$	41.5692	−	29.3939i	1.61441	−	1.14156i
$664$		0			0
$665$		−	50.9117i		−	1.97427i
$666$		0			0
$667$	14.6969			0.569068
$668$		0			0
$669$	−4.89898	+	3.46410i	−0.189405	+	0.133930i
$670$		0			0
$671$	12.0000	+	7.34847i	0.463255	+	0.283685i
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	1.41421	+	5.00000i	0.0544331	+	0.192450i
$676$		0			0
$677$		−	42.0000i		−	1.61419i	−0.590421	−	0.807096i	$-0.701038\pi$
$677$		−	42.0000i		−	1.61419i	0.590421	−	0.807096i	$-0.298962\pi$
$678$		0			0
$679$		−	67.8823i		−	2.60508i
$680$		0			0
$681$	−3.46410	−	4.89898i	−0.132745	−	0.187729i
$682$		0			0
$683$	−45.2548			−1.73163			−0.865814	−	0.500366i	$-0.833199\pi$
$683$	−45.2548			−1.73163			−0.865814	+	0.500366i	$0.833199\pi$
$684$		0			0
$685$		−	48.4974i		−	1.85299i
$686$		0			0
$687$	−6.92820	−	9.79796i	−0.264327	−	0.373815i
$688$		0			0
$689$	−10.3923			−0.395915
$690$		0			0
$691$			4.00000i			0.152167i	0.997101	+	0.0760836i	$0.0242416\pi$
$691$			4.00000i			0.152167i	−0.997101	+	0.0760836i	$0.975758\pi$
$692$		0			0
$693$	−26.5927	−	32.7846i	−1.01017	−	1.24538i
$694$		0			0
$695$	−24.0000			−0.910372
$696$		0			0
$697$	24.0000			0.909065
$698$		0			0
$699$	24.4949	−	17.3205i	0.926482	−	0.655122i
$700$		0			0
$701$		−	42.0000i		−	1.58632i	−0.609015	−	0.793159i	$-0.708435\pi$
$701$		−	42.0000i		−	1.58632i	0.609015	−	0.793159i	$-0.291565\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	6.00000	+	8.48528i	0.225973	+	0.319574i
$706$		0			0
$707$	25.4558			0.957366
$708$		0			0
$709$			34.6410i			1.30097i	0.759519	+	0.650485i	$0.225434\pi$
$709$			34.6410i			1.30097i	−0.759519	+	0.650485i	$0.774566\pi$
$710$		0			0
$711$	36.0000	+	12.7279i	1.35011	+	0.477334i
$712$		0			0
$713$			25.4558i			0.953329i
$714$		0			0
$715$	−29.3939	−	18.0000i	−1.09927	−	0.673162i
$716$		0			0
$717$	16.9706	−	12.0000i	0.633777	−	0.448148i
$718$		0			0
$719$			41.6413i			1.55296i	0.630142	+	0.776480i	$0.282997\pi$
$719$			41.6413i			1.55296i	−0.630142	+	0.776480i	$0.717003\pi$
$720$		0			0
$721$		−	14.6969i		−	0.547343i
$722$		0			0
$723$	4.89898	+	6.92820i	0.182195	+	0.257663i
$724$		0			0
$725$			6.00000i			0.222834i
$726$		0			0
$727$	31.1769			1.15629			0.578144	−	0.815935i	$-0.303777\pi$
$727$	31.1769			1.15629			0.578144	+	0.815935i	$0.303777\pi$
$728$		0			0
$729$	−23.0000	+	14.1421i	−0.851852	+	0.523783i
$730$		0			0
$731$	−67.8823			−2.51072
$732$		0			0
$733$	21.2132			0.783528			0.391764	−	0.920066i	$-0.371865\pi$
$733$	21.2132			0.783528			0.391764	+	0.920066i	$0.371865\pi$
$734$		0			0
$735$	38.1051	−	26.9444i	1.40553	−	0.993859i
$736$		0			0
$737$	−6.92820	+	11.3137i	−0.255204	+	0.416746i
$738$		0			0
$739$	−34.2929			−1.26148			−0.630742	−	0.775993i	$-0.717249\pi$
$739$	−34.2929			−1.26148			−0.630742	+	0.775993i	$0.717249\pi$
$740$		0			0
$741$	29.3939	−	20.7846i	1.07981	−	0.763542i
$742$		0			0
$743$	−36.0000			−1.32071			−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000			−1.32071			−0.660356	+	0.750953i	$0.729595\pi$
$744$		0			0
$745$			44.0908i			1.61536i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−73.4847			−2.68507
$750$		0			0
$751$	−31.1769			−1.13766			−0.568831	−	0.822455i	$-0.692604\pi$
$751$	−31.1769			−1.13766			−0.568831	+	0.822455i	$0.692604\pi$
$752$		0			0
$753$	−8.00000	+	5.65685i	−0.291536	+	0.206147i
$754$		0			0
$755$			10.3923i			0.378215i
$756$		0			0
$757$		−	27.7128i		−	1.00724i	−0.863925	−	0.503620i	$-0.832001\pi$
$757$		−	27.7128i		−	1.00724i	0.863925	−	0.503620i	$-0.167999\pi$
$758$		0			0
$759$	−0.928203	−	14.0406i	−0.0336916	−	0.509641i
$760$		0			0
$761$	20.7846			0.753442			0.376721	−	0.926327i	$-0.377052\pi$
$761$	20.7846			0.753442			0.376721	+	0.926327i	$0.377052\pi$
$762$		0			0
$763$			18.0000i			0.651644i
$764$		0			0
$765$	−16.9706	+	48.0000i	−0.613572	+	1.73544i
$766$		0			0
$767$	12.0000			0.433295
$768$		0			0
$769$		−	14.6969i		−	0.529985i	−0.964250	−	0.264993i	$-0.914630\pi$
$769$		−	14.6969i		−	0.529985i	0.964250	−	0.264993i	$-0.0853695\pi$
$770$		0			0
$771$	2.82843	+	4.00000i	0.101863	+	0.144056i
$772$		0			0
$773$	−7.34847			−0.264306			−0.132153	−	0.991229i	$-0.542189\pi$
$773$	−7.34847			−0.264306			−0.132153	+	0.991229i	$0.542189\pi$
$774$		0			0
$775$	−10.3923			−0.373303
$776$		0			0
$777$		0			0
$778$		0			0
$779$	16.9706			0.608034
$780$		0			0
$781$	−12.7279	+	20.7846i	−0.455441	+	0.743732i
$782$		0			0
$783$	−30.0000	+	8.48528i	−1.07211	+	0.303239i
$784$		0			0
$785$		−	33.9411i		−	1.21141i
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$		0			0
$789$	33.9411	−	24.0000i	1.20834	−	0.854423i
$790$		0			0
$791$	−60.0000			−2.13335
$792$		0			0
$793$	−18.0000			−0.639199
$794$		0			0
$795$	8.48528	−	6.00000i	0.300942	−	0.212798i
$796$		0			0
$797$	36.7423			1.30148			0.650740	−	0.759300i	$-0.274459\pi$
$797$	36.7423			1.30148			0.650740	+	0.759300i	$0.274459\pi$
$798$		0			0
$799$			16.9706i			0.600375i
$800$		0			0
$801$	8.00000	+	2.82843i	0.282666	+	0.0999376i
$802$		0			0
$803$	8.48528	−	13.8564i	0.299439	−	0.488982i
$804$		0			0
$805$	25.4558			0.897201
$806$		0			0
$807$	−17.3205	+	12.2474i	−0.609711	+	0.431131i
$808$		0			0
$809$	−24.2487			−0.852539			−0.426270	−	0.904596i	$-0.640173\pi$
$809$	−24.2487			−0.852539			−0.426270	+	0.904596i	$0.640173\pi$
$810$		0			0
$811$	19.5959			0.688106			0.344053	−	0.938950i	$-0.388200\pi$
$811$	19.5959			0.688106			0.344053	+	0.938950i	$0.388200\pi$
$812$		0			0
$813$	12.7279	+	18.0000i	0.446388	+	0.631288i
$814$		0			0
$815$		−	24.4949i		−	0.858019i
$816$		0			0
$817$	−48.0000			−1.67931
$818$		0			0
$819$	50.9117	+	18.0000i	1.77900	+	0.628971i
$820$		0			0
$821$			30.0000i			1.04701i	0.852023	+	0.523504i	$0.175375\pi$
$821$			30.0000i			1.04701i	−0.852023	+	0.523504i	$0.824625\pi$
$822$		0			0
$823$	24.2487			0.845257			0.422628	−	0.906303i	$-0.361108\pi$
$823$	24.2487			0.845257			0.422628	+	0.906303i	$0.361108\pi$
$824$		0			0
$825$	5.73205	−	0.378937i	0.199564	−	0.0131929i
$826$		0			0
$827$		−	20.7846i		−	0.722752i	−0.932420	−	0.361376i	$-0.882307\pi$
$827$		−	20.7846i		−	0.722752i	0.932420	−	0.361376i	$-0.117693\pi$
$828$		0			0
$829$			13.8564i			0.481253i	0.970618	+	0.240626i	$0.0773529\pi$
$829$			13.8564i			0.481253i	−0.970618	+	0.240626i	$0.922647\pi$
$830$		0			0
$831$	−18.0000	+	12.7279i	−0.624413	+	0.441527i
$832$		0			0
$833$	76.2102			2.64053
$834$		0			0
$835$	−29.3939			−1.01722
$836$		0			0
$837$	−14.6969	−	51.9615i	−0.508001	−	1.79605i
$838$		0			0
$839$		−	12.2474i		−	0.422829i	−0.977397	−	0.211414i	$-0.932193\pi$
$839$		−	12.2474i		−	0.422829i	0.977397	−	0.211414i	$-0.0678070\pi$
$840$		0			0
$841$	−7.00000			−0.241379
$842$		0			0
$843$	−29.3939	+	20.7846i	−1.01238	+	0.715860i
$844$		0			0
$845$	12.2474			0.421325
$846$		0			0
$847$	−41.5692	+	21.2132i	−1.42834	+	0.728894i
$848$		0			0
$849$	−20.7846	+	14.6969i	−0.713326	+	0.504398i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−55.1543			−1.88845			−0.944224	−	0.329304i	$-0.893186\pi$
$853$	−55.1543			−1.88845			−0.944224	+	0.329304i	$0.893186\pi$
$854$		0			0
$855$	−12.0000	+	33.9411i	−0.410391	+	1.16076i
$856$		0			0
$857$	−51.9615			−1.77497			−0.887486	−	0.460835i	$-0.847550\pi$
$857$	−51.9615			−1.77497			−0.887486	+	0.460835i	$0.847550\pi$
$858$		0			0
$859$		−	14.0000i		−	0.477674i	−0.971060	−	0.238837i	$-0.923234\pi$
$859$		−	14.0000i		−	0.477674i	0.971060	−	0.238837i	$-0.0767661\pi$
$860$		0			0
$861$	14.6969	+	20.7846i	0.500870	+	0.708338i
$862$		0			0
$863$			36.7423i			1.25072i	0.780335	+	0.625362i	$0.215049\pi$
$863$			36.7423i			1.25072i	−0.780335	+	0.625362i	$0.784951\pi$
$864$		0			0
$865$			14.6969i			0.499711i
$866$		0			0
$867$	−43.8406	+	31.0000i	−1.48891	+	1.05282i
$868$		0			0
$869$	22.0454	−	36.0000i	0.747839	−	1.22122i
$870$		0			0
$871$		−	16.9706i		−	0.575026i
$872$		0			0
$873$	−16.0000	+	45.2548i	−0.541518	+	1.53164i
$874$		0			0
$875$		−	41.5692i		−	1.40530i
$876$		0			0
$877$	−4.24264			−0.143264			−0.0716319	−	0.997431i	$-0.522821\pi$
$877$	−4.24264			−0.143264			−0.0716319	+	0.997431i	$0.522821\pi$
$878$		0			0
$879$	18.0000	+	25.4558i	0.607125	+	0.858604i
$880$		0			0
$881$		−	22.6274i		−	0.762337i	−0.924506	−	0.381169i	$-0.875522\pi$
$881$		−	22.6274i		−	0.762337i	0.924506	−	0.381169i	$-0.124478\pi$
$882$		0			0
$883$			52.0000i			1.74994i	0.484178	+	0.874970i	$0.339119\pi$
$883$			52.0000i			1.74994i	−0.484178	+	0.874970i	$0.660881\pi$
$884$		0			0
$885$	−9.79796	+	6.92820i	−0.329355	+	0.232889i
$886$		0			0
$887$	−12.0000			−0.402921			−0.201460	−	0.979497i	$-0.564569\pi$
$887$	−12.0000			−0.402921			−0.201460	+	0.979497i	$0.564569\pi$
$888$		0			0
$889$	18.0000			0.603701
$890$		0			0
$891$	10.0010	+	28.1244i	0.335047	+	0.942201i
$892$		0			0
$893$			12.0000i			0.401565i
$894$		0			0
$895$	−6.92820			−0.231584
$896$		0			0
$897$	10.3923	+	14.6969i	0.346989	+	0.490716i
$898$		0			0
$899$		−	62.3538i		−	2.07962i
$900$		0			0
$901$	16.9706			0.565371
$902$		0			0
$903$	−41.5692	−	58.7878i	−1.38334	−	1.95633i
$904$		0			0
$905$		−	33.9411i		−	1.12824i
$906$		0			0
$907$			20.0000i			0.664089i	0.943264	+	0.332045i	$0.107738\pi$
$907$			20.0000i			0.664089i	−0.943264	+	0.332045i	$0.892262\pi$
$908$		0			0
$909$	−16.9706	−	6.00000i	−0.562878	−	0.199007i
$910$		0			0
$911$		−	12.2474i		−	0.405776i	−0.979202	−	0.202888i	$-0.934967\pi$
$911$		−	12.2474i		−	0.405776i	0.979202	−	0.202888i	$-0.0650327\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	14.6969	−	10.3923i	0.485866	−	0.343559i
$916$		0			0
$917$	88.1816			2.91201
$918$		0			0
$919$			55.1543i			1.81937i	0.415295	+	0.909687i	$0.363678\pi$
$919$			55.1543i			1.81937i	−0.415295	+	0.909687i	$0.636322\pi$
$920$		0			0
$921$	−13.8564	+	9.79796i	−0.456584	+	0.322854i
$922$		0			0
$923$		−	31.1769i		−	1.02620i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−3.46410	+	9.79796i	−0.113776	+	0.321807i
$928$		0			0
$929$			36.7696i			1.20637i	0.797601	+	0.603185i	$0.206102\pi$
$929$			36.7696i			1.20637i	−0.797601	+	0.603185i	$0.793898\pi$
$930$		0			0
$931$	53.8888			1.76613
$932$		0			0
$933$	−12.2474	−	17.3205i	−0.400963	−	0.567048i
$934$		0			0
$935$	48.0000	+	29.3939i	1.56977	+	0.961283i
$936$		0			0
$937$			29.3939i			0.960256i	0.877198	+	0.480128i	$0.159410\pi$
$937$			29.3939i			0.960256i	−0.877198	+	0.480128i	$0.840590\pi$
$938$		0			0
$939$	−14.1421	+	10.0000i	−0.461511	+	0.326338i
$940$		0			0
$941$		−	42.0000i		−	1.36916i	−0.728937	−	0.684580i	$-0.759985\pi$
$941$		−	42.0000i		−	1.36916i	0.728937	−	0.684580i	$-0.240015\pi$
$942$		0			0
$943$			8.48528i			0.276319i
$944$		0			0
$945$	−51.9615	+	14.6969i	−1.69031	+	0.478091i
$946$		0			0
$947$	2.82843			0.0919115			0.0459558	−	0.998943i	$-0.485367\pi$
$947$	2.82843			0.0919115			0.0459558	+	0.998943i	$0.485367\pi$
$948$		0			0
$949$			20.7846i			0.674697i
$950$		0			0
$951$	24.2487	−	17.1464i	0.786318	−	0.556011i
$952$		0			0
$953$	3.46410			0.112213			0.0561066	−	0.998425i	$-0.482131\pi$
$953$	3.46410			0.112213			0.0561066	+	0.998425i	$0.482131\pi$
$954$		0			0
$955$			42.0000i			1.35909i
$956$		0			0
$957$	2.27362	+	34.3923i	0.0734958	+	1.11175i
$958$		0			0
$959$	84.0000			2.71250
$960$		0			0
$961$	77.0000			2.48387
$962$		0			0
$963$	48.9898	+	17.3205i	1.57867	+	0.558146i
$964$		0			0
$965$			12.0000i			0.386294i
$966$		0			0
$967$			12.7279i			0.409302i	0.978835	+	0.204651i	$0.0656060\pi$
$967$			12.7279i			0.409302i	−0.978835	+	0.204651i	$0.934394\pi$
$968$		0			0
$969$	−48.0000	+	33.9411i	−1.54198	+	1.09035i
$970$		0			0
$971$	22.6274			0.726148			0.363074	−	0.931760i	$-0.381727\pi$
$971$	22.6274			0.726148			0.363074	+	0.931760i	$0.381727\pi$
$972$		0			0
$973$		−	41.5692i		−	1.33265i
$974$		0			0
$975$	−6.00000	+	4.24264i	−0.192154	+	0.135873i
$976$		0			0
$977$			39.5980i			1.26685i	0.773803	+	0.633426i	$0.218352\pi$
$977$			39.5980i			1.26685i	−0.773803	+	0.633426i	$0.781648\pi$
$978$		0			0
$979$	4.89898	−	8.00000i	0.156572	−	0.255681i
$980$		0			0
$981$	4.24264	−	12.0000i	0.135457	−	0.383131i
$982$		0			0
$983$		−	7.34847i		−	0.234380i	−0.993110	−	0.117190i	$-0.962611\pi$
$983$		−	7.34847i		−	0.234380i	0.993110	−	0.117190i	$-0.0373886\pi$
$984$		0			0
$985$			14.6969i			0.468283i
$986$		0			0
$987$	−14.6969	+	10.3923i	−0.467809	+	0.330791i
$988$		0			0
$989$		−	24.0000i		−	0.763156i
$990$		0			0
$991$	31.1769			0.990367			0.495184	−	0.868788i	$-0.335101\pi$
$991$	31.1769			0.990367			0.495184	+	0.868788i	$0.335101\pi$
$992$		0			0
$993$	−14.0000	−	19.7990i	−0.444277	−	0.628302i
$994$		0			0
$995$	59.3970			1.88301
$996$		0			0
$997$	29.6985			0.940560			0.470280	−	0.882517i	$-0.344153\pi$
$997$	29.6985			0.940560			0.470280	+	0.882517i	$0.344153\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2112.2.m.h.1121.4	yes	8
3.2	odd	2		2112.2.m.i.1121.7	yes	8
4.3	odd	2		2112.2.m.i.1121.6	yes	8
8.3	odd	2		2112.2.m.i.1121.3	yes	8
8.5	even	2	inner	2112.2.m.h.1121.5	yes	8
11.10	odd	2		2112.2.m.i.1121.4	yes	8
12.11	even	2	inner	2112.2.m.h.1121.1	✓	8
24.5	odd	2		2112.2.m.i.1121.2	yes	8
24.11	even	2	inner	2112.2.m.h.1121.8	yes	8
33.32	even	2	inner	2112.2.m.h.1121.7	yes	8
44.43	even	2	inner	2112.2.m.h.1121.6	yes	8
88.21	odd	2		2112.2.m.i.1121.5	yes	8
88.43	even	2	inner	2112.2.m.h.1121.3	yes	8
132.131	odd	2		2112.2.m.i.1121.1	yes	8
264.131	odd	2		2112.2.m.i.1121.8	yes	8
264.197	even	2	inner	2112.2.m.h.1121.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2112.2.m.h.1121.1	✓	8	12.11	even	2	inner
2112.2.m.h.1121.2	yes	8	264.197	even	2	inner
2112.2.m.h.1121.3	yes	8	88.43	even	2	inner
2112.2.m.h.1121.4	yes	8	1.1	even	1	trivial
2112.2.m.h.1121.5	yes	8	8.5	even	2	inner
2112.2.m.h.1121.6	yes	8	44.43	even	2	inner
2112.2.m.h.1121.7	yes	8	33.32	even	2	inner
2112.2.m.h.1121.8	yes	8	24.11	even	2	inner
2112.2.m.i.1121.1	yes	8	132.131	odd	2
2112.2.m.i.1121.2	yes	8	24.5	odd	2
2112.2.m.i.1121.3	yes	8	8.3	odd	2
2112.2.m.i.1121.4	yes	8	11.10	odd	2
2112.2.m.i.1121.5	yes	8	88.21	odd	2
2112.2.m.i.1121.6	yes	8	4.3	odd	2
2112.2.m.i.1121.7	yes	8	3.2	odd	2
2112.2.m.i.1121.8	yes	8	264.131	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:	\( N \)	\(=\)	\( 2112 = 2^{6} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2112.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.41421 + 1.00000i) q^{3} +2.44949 q^{5} +4.24264i q^{7} +(1.00000 - 2.82843i) q^{9} +(-2.82843 - 1.73205i) q^{11} +4.24264 q^{13} +(-3.46410 + 2.44949i) q^{15} -6.92820 q^{17} -4.89898 q^{19} +(-4.24264 - 6.00000i) q^{21} -2.44949i q^{23} +1.00000 q^{25} +(1.41421 + 5.00000i) q^{27} +6.00000i q^{29} -10.3923 q^{31} +(5.73205 - 0.378937i) q^{33} +10.3923i q^{35} +(-6.00000 + 4.24264i) q^{39} -3.46410 q^{41} +9.79796 q^{43} +(2.44949 - 6.92820i) q^{45} -2.44949i q^{47} -11.0000 q^{49} +(9.79796 - 6.92820i) q^{51} -2.44949 q^{53} +(-6.92820 - 4.24264i) q^{55} +(6.92820 - 4.89898i) q^{57} +2.82843 q^{59} -4.24264 q^{61} +(12.0000 + 4.24264i) q^{63} +10.3923 q^{65} -4.00000i q^{67} +(2.44949 + 3.46410i) q^{69} -7.34847i q^{71} +4.89898i q^{73} +(-1.41421 + 1.00000i) q^{75} +(7.34847 - 12.0000i) q^{77} +12.7279i q^{79} +(-7.00000 - 5.65685i) q^{81} -16.9706 q^{85} +(-6.00000 - 8.48528i) q^{87} +2.82843i q^{89} +18.0000i q^{91} +(14.6969 - 10.3923i) q^{93} -12.0000 q^{95} -16.0000 q^{97} +(-7.72741 + 6.26795i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{9} + 8 q^{25} + 32 q^{33} - 48 q^{39} - 88 q^{49} + 96 q^{63} - 56 q^{81} - 48 q^{87} - 96 q^{95} - 128 q^{97}+O(q^{100}) \) 8 * q + 8 * q^9 + 8 * q^25 + 32 * q^33 - 48 * q^39 - 88 * q^49 + 96 * q^63 - 56 * q^81 - 48 * q^87 - 96 * q^95 - 128 * q^97

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