LMFDB - Embedded newform 672.2.i.a.209.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [672,2,Mod(209,672)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(672, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("672.209");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 672 = 2^{5} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	672.i (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$5.36594701583$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			209.1
Root			$1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	672.209
Dual form			672.2.i.a.209.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 - 1.41421i) q^{3} +1.41421i q^{5} +(2.00000 - 1.73205i) q^{7} +(-1.00000 + 2.82843i) q^{9} +O(q^{10})$	\(q+(-1.00000 - 1.41421i) q^{3} +1.41421i q^{5} +(2.00000 - 1.73205i) q^{7} +(-1.00000 + 2.82843i) q^{9} +2.44949 q^{11} -2.00000 q^{13} +(2.00000 - 1.41421i) q^{15} +7.34847 q^{17} -4.00000 q^{19} +(-4.44949 - 1.09638i) q^{21} +1.41421i q^{23} +3.00000 q^{25} +(5.00000 - 1.41421i) q^{27} +4.89898 q^{29} -6.92820i q^{31} +(-2.44949 - 3.46410i) q^{33} +(2.44949 + 2.82843i) q^{35} -10.3923i q^{37} +(2.00000 + 2.82843i) q^{39} -2.44949 q^{41} +3.46410i q^{43} +(-4.00000 - 1.41421i) q^{45} +4.89898 q^{47} +(1.00000 - 6.92820i) q^{49} +(-7.34847 - 10.3923i) q^{51} +3.46410i q^{55} +(4.00000 + 5.65685i) q^{57} -5.65685i q^{59} +10.0000 q^{61} +(2.89898 + 7.38891i) q^{63} -2.82843i q^{65} -3.46410i q^{67} +(2.00000 - 1.41421i) q^{69} +1.41421i q^{71} +(-3.00000 - 4.24264i) q^{75} +(4.89898 - 4.24264i) q^{77} -8.00000 q^{79} +(-7.00000 - 5.65685i) q^{81} +11.3137i q^{83} +10.3923i q^{85} +(-4.89898 - 6.92820i) q^{87} -7.34847 q^{89} +(-4.00000 + 3.46410i) q^{91} +(-9.79796 + 6.92820i) q^{93} -5.65685i q^{95} +13.8564i q^{97} +(-2.44949 + 6.92820i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 8 q^{7} - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 + 8 * q^7 - 4 * q^9	$ 4 q - 4 q^{3} + 8 q^{7} - 4 q^{9} - 8 q^{13} + 8 q^{15} - 16 q^{19} - 8 q^{21} + 12 q^{25} + 20 q^{27} + 8 q^{39} - 16 q^{45} + 4 q^{49} + 16 q^{57} + 40 q^{61} - 8 q^{63} + 8 q^{69} - 12 q^{75} - 32 q^{79} - 28 q^{81} - 16 q^{91}+O(q^{100}) $ 4 * q - 4 * q^3 + 8 * q^7 - 4 * q^9 - 8 * q^13 + 8 * q^15 - 16 * q^19 - 8 * q^21 + 12 * q^25 + 20 * q^27 + 8 * q^39 - 16 * q^45 + 4 * q^49 + 16 * q^57 + 40 * q^61 - 8 * q^63 + 8 * q^69 - 12 * q^75 - 32 * q^79 - 28 * q^81 - 16 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/672\mathbb{Z}\right)^\times$.

$n$	$127$	$421$	$449$	$577$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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
$5$			1.41421i			0.632456i	0.948683	+	0.316228i	$0.102416\pi$
$5$			1.41421i			0.632456i	−0.948683	+	0.316228i	$0.897584\pi$
$6$		0			0
$7$	2.00000	−	1.73205i	0.755929	−	0.654654i
$7$	2.00000	−	1.73205i	0.755929	−	0.654654i
$8$		0			0
$9$	−1.00000	+	2.82843i	−0.333333	+	0.942809i
$10$		0			0
$11$	2.44949			0.738549			0.369274	−	0.929320i	$-0.379606\pi$
$11$	2.44949			0.738549			0.369274	+	0.929320i	$0.379606\pi$
$12$		0			0
$13$	−2.00000			−0.554700			−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000			−0.554700			−0.277350	+	0.960769i	$0.589456\pi$
$14$		0			0
$15$	2.00000	−	1.41421i	0.516398	−	0.365148i
$16$		0			0
$17$	7.34847			1.78227			0.891133	−	0.453743i	$-0.149911\pi$
$17$	7.34847			1.78227			0.891133	+	0.453743i	$0.149911\pi$
$18$		0			0
$19$	−4.00000			−0.917663			−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000			−0.917663			−0.458831	+	0.888523i	$0.651732\pi$
$20$		0			0
$21$	−4.44949	−	1.09638i	−0.970958	−	0.239249i
$22$		0			0
$23$			1.41421i			0.294884i	0.989071	+	0.147442i	$0.0471040\pi$
$23$			1.41421i			0.294884i	−0.989071	+	0.147442i	$0.952896\pi$
$24$		0			0
$25$	3.00000			0.600000
$26$		0			0
$27$	5.00000	−	1.41421i	0.962250	−	0.272166i
$28$		0			0
$29$	4.89898			0.909718			0.454859	−	0.890564i	$-0.349690\pi$
$29$	4.89898			0.909718			0.454859	+	0.890564i	$0.349690\pi$
$30$		0			0
$31$		−	6.92820i		−	1.24434i	−0.782881	−	0.622171i	$-0.786251\pi$
$31$		−	6.92820i		−	1.24434i	0.782881	−	0.622171i	$-0.213749\pi$
$32$		0			0
$33$	−2.44949	−	3.46410i	−0.426401	−	0.603023i
$34$		0			0
$35$	2.44949	+	2.82843i	0.414039	+	0.478091i
$36$		0			0
$37$		−	10.3923i		−	1.70848i	−0.519875	−	0.854242i	$-0.674022\pi$
$37$		−	10.3923i		−	1.70848i	0.519875	−	0.854242i	$-0.325978\pi$
$38$		0			0
$39$	2.00000	+	2.82843i	0.320256	+	0.452911i
$40$		0			0
$41$	−2.44949			−0.382546			−0.191273	−	0.981537i	$-0.561262\pi$
$41$	−2.44949			−0.382546			−0.191273	+	0.981537i	$0.561262\pi$
$42$		0			0
$43$			3.46410i			0.528271i	0.964486	+	0.264135i	$0.0850865\pi$
$43$			3.46410i			0.528271i	−0.964486	+	0.264135i	$0.914913\pi$
$44$		0			0
$45$	−4.00000	−	1.41421i	−0.596285	−	0.210819i
$46$		0			0
$47$	4.89898			0.714590			0.357295	−	0.933992i	$-0.383699\pi$
$47$	4.89898			0.714590			0.357295	+	0.933992i	$0.383699\pi$
$48$		0			0
$49$	1.00000	−	6.92820i	0.142857	−	0.989743i
$50$		0			0
$51$	−7.34847	−	10.3923i	−1.02899	−	1.45521i
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$			3.46410i			0.467099i
$56$		0			0
$57$	4.00000	+	5.65685i	0.529813	+	0.749269i
$58$		0			0
$59$		−	5.65685i		−	0.736460i	−0.929735	−	0.368230i	$-0.879964\pi$
$59$		−	5.65685i		−	0.736460i	0.929735	−	0.368230i	$-0.120036\pi$
$60$		0			0
$61$	10.0000			1.28037			0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000			1.28037			0.640184	+	0.768221i	$0.278858\pi$
$62$		0			0
$63$	2.89898	+	7.38891i	0.365237	+	0.930915i
$64$		0			0
$65$		−	2.82843i		−	0.350823i
$66$		0			0
$67$		−	3.46410i		−	0.423207i	−0.977356	−	0.211604i	$-0.932131\pi$
$67$		−	3.46410i		−	0.423207i	0.977356	−	0.211604i	$-0.0678686\pi$
$68$		0			0
$69$	2.00000	−	1.41421i	0.240772	−	0.170251i
$70$		0			0
$71$			1.41421i			0.167836i	0.996473	+	0.0839181i	$0.0267434\pi$
$71$			1.41421i			0.167836i	−0.996473	+	0.0839181i	$0.973257\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	−3.00000	−	4.24264i	−0.346410	−	0.489898i
$76$		0			0
$77$	4.89898	−	4.24264i	0.558291	−	0.483494i
$78$		0			0
$79$	−8.00000			−0.900070			−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000			−0.900070			−0.450035	+	0.893011i	$0.648589\pi$
$80$		0			0
$81$	−7.00000	−	5.65685i	−0.777778	−	0.628539i
$82$		0			0
$83$			11.3137i			1.24184i	0.783874	+	0.620920i	$0.213241\pi$
$83$			11.3137i			1.24184i	−0.783874	+	0.620920i	$0.786759\pi$
$84$		0			0
$85$			10.3923i			1.12720i
$86$		0			0
$87$	−4.89898	−	6.92820i	−0.525226	−	0.742781i
$88$		0			0
$89$	−7.34847			−0.778936			−0.389468	−	0.921040i	$-0.627341\pi$
$89$	−7.34847			−0.778936			−0.389468	+	0.921040i	$0.627341\pi$
$90$		0			0
$91$	−4.00000	+	3.46410i	−0.419314	+	0.363137i
$92$		0			0
$93$	−9.79796	+	6.92820i	−1.01600	+	0.718421i
$94$		0			0
$95$		−	5.65685i		−	0.580381i
$96$		0			0
$97$			13.8564i			1.40690i	0.710742	+	0.703452i	$0.248359\pi$
$97$			13.8564i			1.40690i	−0.710742	+	0.703452i	$0.751641\pi$
$98$		0			0
$99$	−2.44949	+	6.92820i	−0.246183	+	0.696311i
$100$		0			0
$101$			9.89949i			0.985037i	0.870302	+	0.492518i	$0.163924\pi$
$101$			9.89949i			0.985037i	−0.870302	+	0.492518i	$0.836076\pi$
$102$		0			0
$103$			13.8564i			1.36531i	0.730740	+	0.682656i	$0.239175\pi$
$103$			13.8564i			1.36531i	−0.730740	+	0.682656i	$0.760825\pi$
$104$		0			0
$105$	1.55051	−	6.29253i	0.151314	−	0.614088i
$106$		0			0
$107$	7.34847			0.710403			0.355202	−	0.934790i	$-0.384412\pi$
$107$	7.34847			0.710403			0.355202	+	0.934790i	$0.384412\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$	−14.6969	+	10.3923i	−1.39497	+	0.986394i
$112$		0			0
$113$		−	14.1421i		−	1.33038i	−0.746674	−	0.665190i	$-0.768350\pi$
$113$		−	14.1421i		−	1.33038i	0.746674	−	0.665190i	$-0.231650\pi$
$114$		0			0
$115$	−2.00000			−0.186501
$116$		0			0
$117$	2.00000	−	5.65685i	0.184900	−	0.522976i
$118$		0			0
$119$	14.6969	−	12.7279i	1.34727	−	1.16677i
$120$		0			0
$121$	−5.00000			−0.454545
$122$		0			0
$123$	2.44949	+	3.46410i	0.220863	+	0.312348i
$124$		0			0
$125$			11.3137i			1.01193i
$126$		0			0
$127$	4.00000			0.354943			0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000			0.354943			0.177471	+	0.984126i	$0.443208\pi$
$128$		0			0
$129$	4.89898	−	3.46410i	0.431331	−	0.304997i
$130$		0			0
$131$			11.3137i			0.988483i	0.869325	+	0.494242i	$0.164554\pi$
$131$			11.3137i			0.988483i	−0.869325	+	0.494242i	$0.835446\pi$
$132$		0			0
$133$	−8.00000	+	6.92820i	−0.693688	+	0.600751i
$134$		0			0
$135$	2.00000	+	7.07107i	0.172133	+	0.608581i
$136$		0			0
$137$			11.3137i			0.966595i	0.875456	+	0.483298i	$0.160561\pi$
$137$			11.3137i			0.966595i	−0.875456	+	0.483298i	$0.839439\pi$
$138$		0			0
$139$	14.0000			1.18746			0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000			1.18746			0.593732	+	0.804663i	$0.297654\pi$
$140$		0			0
$141$	−4.89898	−	6.92820i	−0.412568	−	0.583460i
$142$		0			0
$143$	−4.89898			−0.409673
$144$		0			0
$145$			6.92820i			0.575356i
$146$		0			0
$147$	−10.7980	+	5.51399i	−0.890601	+	0.454786i
$148$		0			0
$149$	9.79796			0.802680			0.401340	−	0.915929i	$-0.368545\pi$
$149$	9.79796			0.802680			0.401340	+	0.915929i	$0.368545\pi$
$150$		0			0
$151$	−8.00000			−0.651031			−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000			−0.651031			−0.325515	+	0.945537i	$0.605538\pi$
$152$		0			0
$153$	−7.34847	+	20.7846i	−0.594089	+	1.68034i
$154$		0			0
$155$	9.79796			0.786991
$156$		0			0
$157$	−14.0000			−1.11732			−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000			−1.11732			−0.558661	+	0.829396i	$0.688685\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	2.44949	+	2.82843i	0.193047	+	0.222911i
$162$		0			0
$163$			10.3923i			0.813988i	0.913431	+	0.406994i	$0.133423\pi$
$163$			10.3923i			0.813988i	−0.913431	+	0.406994i	$0.866577\pi$
$164$		0			0
$165$	4.89898	−	3.46410i	0.381385	−	0.269680i
$166$		0			0
$167$	−19.5959			−1.51638			−0.758189	−	0.652035i	$-0.773915\pi$
$167$	−19.5959			−1.51638			−0.758189	+	0.652035i	$0.773915\pi$
$168$		0			0
$169$	−9.00000			−0.692308
$170$		0			0
$171$	4.00000	−	11.3137i	0.305888	−	0.865181i
$172$		0			0
$173$			1.41421i			0.107521i	0.998554	+	0.0537603i	$0.0171207\pi$
$173$			1.41421i			0.107521i	−0.998554	+	0.0537603i	$0.982879\pi$
$174$		0			0
$175$	6.00000	−	5.19615i	0.453557	−	0.392792i
$176$		0			0
$177$	−8.00000	+	5.65685i	−0.601317	+	0.425195i
$178$		0			0
$179$	−22.0454			−1.64775			−0.823876	−	0.566771i	$-0.808193\pi$
$179$	−22.0454			−1.64775			−0.823876	+	0.566771i	$0.808193\pi$
$180$		0			0
$181$	22.0000			1.63525			0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000			1.63525			0.817624	+	0.575753i	$0.195291\pi$
$182$		0			0
$183$	−10.0000	−	14.1421i	−0.739221	−	1.04542i
$184$		0			0
$185$	14.6969			1.08054
$186$		0			0
$187$	18.0000			1.31629
$188$		0			0
$189$	7.55051	−	11.4887i	0.549219	−	0.835679i
$190$		0			0
$191$		−	15.5563i		−	1.12562i	−0.826587	−	0.562809i	$-0.809721\pi$
$191$		−	15.5563i		−	1.12562i	0.826587	−	0.562809i	$-0.190279\pi$
$192$		0			0
$193$	−4.00000			−0.287926			−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000			−0.287926			−0.143963	+	0.989583i	$0.545985\pi$
$194$		0			0
$195$	−4.00000	+	2.82843i	−0.286446	+	0.202548i
$196$		0			0
$197$	−14.6969			−1.04711			−0.523557	−	0.851991i	$-0.675395\pi$
$197$	−14.6969			−1.04711			−0.523557	+	0.851991i	$0.675395\pi$
$198$		0			0
$199$		−	10.3923i		−	0.736691i	−0.929689	−	0.368345i	$-0.879924\pi$
$199$		−	10.3923i		−	0.736691i	0.929689	−	0.368345i	$-0.120076\pi$
$200$		0			0
$201$	−4.89898	+	3.46410i	−0.345547	+	0.244339i
$202$		0			0
$203$	9.79796	−	8.48528i	0.687682	−	0.595550i
$204$		0			0
$205$		−	3.46410i		−	0.241943i
$206$		0			0
$207$	−4.00000	−	1.41421i	−0.278019	−	0.0982946i
$208$		0			0
$209$	−9.79796			−0.677739
$210$		0			0
$211$		−	24.2487i		−	1.66935i	−0.550743	−	0.834675i	$-0.685655\pi$
$211$		−	24.2487i		−	1.66935i	0.550743	−	0.834675i	$-0.314345\pi$
$212$		0			0
$213$	2.00000	−	1.41421i	0.137038	−	0.0969003i
$214$		0			0
$215$	−4.89898			−0.334108
$216$		0			0
$217$	−12.0000	−	13.8564i	−0.814613	−	0.940634i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−14.6969			−0.988623
$222$		0			0
$223$			17.3205i			1.15987i	0.814664	+	0.579934i	$0.196921\pi$
$223$			17.3205i			1.15987i	−0.814664	+	0.579934i	$0.803079\pi$
$224$		0			0
$225$	−3.00000	+	8.48528i	−0.200000	+	0.565685i
$226$		0			0
$227$			2.82843i			0.187729i	0.995585	+	0.0938647i	$0.0299221\pi$
$227$			2.82843i			0.187729i	−0.995585	+	0.0938647i	$0.970078\pi$
$228$		0			0
$229$	−14.0000			−0.925146			−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000			−0.925146			−0.462573	+	0.886581i	$0.653074\pi$
$230$		0			0
$231$	−10.8990	−	2.68556i	−0.717100	−	0.176697i
$232$		0			0
$233$		−	14.1421i		−	0.926482i	−0.886232	−	0.463241i	$-0.846686\pi$
$233$		−	14.1421i		−	0.926482i	0.886232	−	0.463241i	$-0.153314\pi$
$234$		0			0
$235$			6.92820i			0.451946i
$236$		0			0
$237$	8.00000	+	11.3137i	0.519656	+	0.734904i
$238$		0			0
$239$			26.8701i			1.73808i	0.494742	+	0.869040i	$0.335262\pi$
$239$			26.8701i			1.73808i	−0.494742	+	0.869040i	$0.664738\pi$
$240$		0			0
$241$		−	6.92820i		−	0.446285i	−0.974786	−	0.223142i	$-0.928369\pi$
$241$		−	6.92820i		−	0.446285i	0.974786	−	0.223142i	$-0.0716315\pi$
$242$		0			0
$243$	−1.00000	+	15.5563i	−0.0641500	+	0.997940i
$244$		0			0
$245$	9.79796	+	1.41421i	0.625969	+	0.0903508i
$246$		0			0
$247$	8.00000			0.509028
$248$		0			0
$249$	16.0000	−	11.3137i	1.01396	−	0.716977i
$250$		0			0
$251$		−	14.1421i		−	0.892644i	−0.894873	−	0.446322i	$-0.852734\pi$
$251$		−	14.1421i		−	0.892644i	0.894873	−	0.446322i	$-0.147266\pi$
$252$		0			0
$253$			3.46410i			0.217786i
$254$		0			0
$255$	14.6969	−	10.3923i	0.920358	−	0.650791i
$256$		0			0
$257$	−2.44949			−0.152795			−0.0763975	−	0.997077i	$-0.524342\pi$
$257$	−2.44949			−0.152795			−0.0763975	+	0.997077i	$0.524342\pi$
$258$		0			0
$259$	−18.0000	−	20.7846i	−1.11847	−	1.29149i
$260$		0			0
$261$	−4.89898	+	13.8564i	−0.303239	+	0.857690i
$262$		0			0
$263$			9.89949i			0.610429i	0.952284	+	0.305215i	$0.0987282\pi$
$263$			9.89949i			0.610429i	−0.952284	+	0.305215i	$0.901272\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	7.34847	+	10.3923i	0.449719	+	0.635999i
$268$		0			0
$269$		−	24.0416i		−	1.46584i	−0.680313	−	0.732922i	$-0.738156\pi$
$269$		−	24.0416i		−	1.46584i	0.680313	−	0.732922i	$-0.261844\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$	8.89898	+	2.19275i	0.538591	+	0.132711i
$274$		0			0
$275$	7.34847			0.443129
$276$		0			0
$277$		−	3.46410i		−	0.208138i	−0.994570	−	0.104069i	$-0.966814\pi$
$277$		−	3.46410i		−	0.208138i	0.994570	−	0.104069i	$-0.0331862\pi$
$278$		0			0
$279$	19.5959	+	6.92820i	1.17318	+	0.414781i
$280$		0			0
$281$			2.82843i			0.168730i	0.996435	+	0.0843649i	$0.0268861\pi$
$281$			2.82843i			0.168730i	−0.996435	+	0.0843649i	$0.973114\pi$
$282$		0			0
$283$	−4.00000			−0.237775			−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000			−0.237775			−0.118888	+	0.992908i	$0.537933\pi$
$284$		0			0
$285$	−8.00000	+	5.65685i	−0.473879	+	0.335083i
$286$		0			0
$287$	−4.89898	+	4.24264i	−0.289178	+	0.250435i
$288$		0			0
$289$	37.0000			2.17647
$290$		0			0
$291$	19.5959	−	13.8564i	1.14873	−	0.812277i
$292$		0			0
$293$			1.41421i			0.0826192i	0.999146	+	0.0413096i	$0.0131530\pi$
$293$			1.41421i			0.0826192i	−0.999146	+	0.0413096i	$0.986847\pi$
$294$		0			0
$295$	8.00000			0.465778
$296$		0			0
$297$	12.2474	−	3.46410i	0.710669	−	0.201008i
$298$		0			0
$299$		−	2.82843i		−	0.163572i
$300$		0			0
$301$	6.00000	+	6.92820i	0.345834	+	0.399335i
$302$		0			0
$303$	14.0000	−	9.89949i	0.804279	−	0.568711i
$304$		0			0
$305$			14.1421i			0.809776i
$306$		0			0
$307$	−4.00000			−0.228292			−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000			−0.228292			−0.114146	+	0.993464i	$0.536413\pi$
$308$		0			0
$309$	19.5959	−	13.8564i	1.11477	−	0.788263i
$310$		0			0
$311$	24.4949			1.38898			0.694489	−	0.719503i	$-0.255630\pi$
$311$	24.4949			1.38898			0.694489	+	0.719503i	$0.255630\pi$
$312$		0			0
$313$		−	6.92820i		−	0.391605i	−0.980643	−	0.195803i	$-0.937269\pi$
$313$		−	6.92820i		−	0.391605i	0.980643	−	0.195803i	$-0.0627312\pi$
$314$		0			0
$315$	−10.4495	+	4.09978i	−0.588762	+	0.230996i
$316$		0			0
$317$	19.5959			1.10062			0.550308	−	0.834962i	$-0.314510\pi$
$317$	19.5959			1.10062			0.550308	+	0.834962i	$0.314510\pi$
$318$		0			0
$319$	12.0000			0.671871
$320$		0			0
$321$	−7.34847	−	10.3923i	−0.410152	−	0.580042i
$322$		0			0
$323$	−29.3939			−1.63552
$324$		0			0
$325$	−6.00000			−0.332820
$326$		0			0
$327$		0			0
$328$		0			0
$329$	9.79796	−	8.48528i	0.540179	−	0.467809i
$330$		0			0
$331$			24.2487i			1.33283i	0.745581	+	0.666415i	$0.232172\pi$
$331$			24.2487i			1.33283i	−0.745581	+	0.666415i	$0.767828\pi$
$332$		0			0
$333$	29.3939	+	10.3923i	1.61077	+	0.569495i
$334$		0			0
$335$	4.89898			0.267660
$336$		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
$339$	−20.0000	+	14.1421i	−1.08625	+	0.768095i
$340$		0			0
$341$		−	16.9706i		−	0.919007i
$342$		0			0
$343$	−10.0000	−	15.5885i	−0.539949	−	0.841698i
$344$		0			0
$345$	2.00000	+	2.82843i	0.107676	+	0.152277i
$346$		0			0
$347$	−2.44949			−0.131495			−0.0657477	−	0.997836i	$-0.520943\pi$
$347$	−2.44949			−0.131495			−0.0657477	+	0.997836i	$0.520943\pi$
$348$		0			0
$349$	−14.0000			−0.749403			−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000			−0.749403			−0.374701	+	0.927146i	$0.622255\pi$
$350$		0			0
$351$	−10.0000	+	2.82843i	−0.533761	+	0.150970i
$352$		0			0
$353$	−12.2474			−0.651866			−0.325933	−	0.945393i	$-0.605678\pi$
$353$	−12.2474			−0.651866			−0.325933	+	0.945393i	$0.605678\pi$
$354$		0			0
$355$	−2.00000			−0.106149
$356$		0			0
$357$	−32.6969	−	8.05669i	−1.73051	−	0.426405i
$358$		0			0
$359$			26.8701i			1.41815i	0.705134	+	0.709074i	$0.250887\pi$
$359$			26.8701i			1.41815i	−0.705134	+	0.709074i	$0.749113\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		0			0
$363$	5.00000	+	7.07107i	0.262432	+	0.371135i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	3.46410i		−	0.180825i	−0.995904	−	0.0904123i	$-0.971182\pi$
$367$		−	3.46410i		−	0.180825i	0.995904	−	0.0904123i	$-0.0288185\pi$
$368$		0			0
$369$	2.44949	−	6.92820i	0.127515	−	0.360668i
$370$		0			0
$371$		0			0
$372$		0			0
$373$			13.8564i			0.717458i	0.933442	+	0.358729i	$0.116790\pi$
$373$			13.8564i			0.717458i	−0.933442	+	0.358729i	$0.883210\pi$
$374$		0			0
$375$	16.0000	−	11.3137i	0.826236	−	0.584237i
$376$		0			0
$377$	−9.79796			−0.504621
$378$		0			0
$379$		−	10.3923i		−	0.533817i	−0.963722	−	0.266908i	$-0.913998\pi$
$379$		−	10.3923i		−	0.533817i	0.963722	−	0.266908i	$-0.0860021\pi$
$380$		0			0
$381$	−4.00000	−	5.65685i	−0.204926	−	0.289809i
$382$		0			0
$383$	−4.89898			−0.250326			−0.125163	−	0.992136i	$-0.539945\pi$
$383$	−4.89898			−0.250326			−0.125163	+	0.992136i	$0.539945\pi$
$384$		0			0
$385$	6.00000	+	6.92820i	0.305788	+	0.353094i
$386$		0			0
$387$	−9.79796	−	3.46410i	−0.498058	−	0.176090i
$388$		0			0
$389$	−24.4949			−1.24194			−0.620970	−	0.783834i	$-0.713261\pi$
$389$	−24.4949			−1.24194			−0.620970	+	0.783834i	$0.713261\pi$
$390$		0			0
$391$			10.3923i			0.525561i
$392$		0			0
$393$	16.0000	−	11.3137i	0.807093	−	0.570701i
$394$		0			0
$395$		−	11.3137i		−	0.569254i
$396$		0			0
$397$	−2.00000			−0.100377			−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000			−0.100377			−0.0501886	+	0.998740i	$0.515982\pi$
$398$		0			0
$399$	17.7980	+	4.38551i	0.891012	+	0.219550i
$400$		0			0
$401$			11.3137i			0.564980i	0.959270	+	0.282490i	$0.0911603\pi$
$401$			11.3137i			0.564980i	−0.959270	+	0.282490i	$0.908840\pi$
$402$		0			0
$403$			13.8564i			0.690237i
$404$		0			0
$405$	8.00000	−	9.89949i	0.397523	−	0.491910i
$406$		0			0
$407$		−	25.4558i		−	1.26180i
$408$		0			0
$409$		−	34.6410i		−	1.71289i	−0.516240	−	0.856444i	$-0.672669\pi$
$409$		−	34.6410i		−	1.71289i	0.516240	−	0.856444i	$-0.327331\pi$
$410$		0			0
$411$	16.0000	−	11.3137i	0.789222	−	0.558064i
$412$		0			0
$413$	−9.79796	−	11.3137i	−0.482126	−	0.556711i
$414$		0			0
$415$	−16.0000			−0.785409
$416$		0			0
$417$	−14.0000	−	19.7990i	−0.685583	−	0.969561i
$418$		0			0
$419$			11.3137i			0.552711i	0.961056	+	0.276355i	$0.0891267\pi$
$419$			11.3137i			0.552711i	−0.961056	+	0.276355i	$0.910873\pi$
$420$		0			0
$421$		−	24.2487i		−	1.18181i	−0.806741	−	0.590905i	$-0.798771\pi$
$421$		−	24.2487i		−	1.18181i	0.806741	−	0.590905i	$-0.201229\pi$
$422$		0			0
$423$	−4.89898	+	13.8564i	−0.238197	+	0.673722i
$424$		0			0
$425$	22.0454			1.06936
$426$		0			0
$427$	20.0000	−	17.3205i	0.967868	−	0.838198i
$428$		0			0
$429$	4.89898	+	6.92820i	0.236525	+	0.334497i
$430$		0			0
$431$			18.3848i			0.885564i	0.896629	+	0.442782i	$0.146008\pi$
$431$			18.3848i			0.885564i	−0.896629	+	0.442782i	$0.853992\pi$
$432$		0			0
$433$			20.7846i			0.998845i	0.866359	+	0.499422i	$0.166454\pi$
$433$			20.7846i			0.998845i	−0.866359	+	0.499422i	$0.833546\pi$
$434$		0			0
$435$	9.79796	−	6.92820i	0.469776	−	0.332182i
$436$		0			0
$437$		−	5.65685i		−	0.270604i
$438$		0			0
$439$		−	24.2487i		−	1.15733i	−0.815566	−	0.578664i	$-0.803574\pi$
$439$		−	24.2487i		−	1.15733i	0.815566	−	0.578664i	$-0.196426\pi$
$440$		0			0
$441$	18.5959	+	9.75663i	0.885520	+	0.464601i
$442$		0			0
$443$	17.1464			0.814651			0.407326	−	0.913283i	$-0.366461\pi$
$443$	17.1464			0.814651			0.407326	+	0.913283i	$0.366461\pi$
$444$		0			0
$445$		−	10.3923i		−	0.492642i
$446$		0			0
$447$	−9.79796	−	13.8564i	−0.463428	−	0.655386i
$448$		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
$451$	−6.00000			−0.282529
$452$		0			0
$453$	8.00000	+	11.3137i	0.375873	+	0.531564i
$454$		0			0
$455$	−4.89898	−	5.65685i	−0.229668	−	0.265197i
$456$		0			0
$457$	−10.0000			−0.467780			−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000			−0.467780			−0.233890	+	0.972263i	$0.575146\pi$
$458$		0			0
$459$	36.7423	−	10.3923i	1.71499	−	0.485071i
$460$		0			0
$461$			9.89949i			0.461065i	0.973065	+	0.230533i	$0.0740469\pi$
$461$			9.89949i			0.461065i	−0.973065	+	0.230533i	$0.925953\pi$
$462$		0			0
$463$	4.00000			0.185896			0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000			0.185896			0.0929479	+	0.995671i	$0.470371\pi$
$464$		0			0
$465$	−9.79796	−	13.8564i	−0.454369	−	0.642575i
$466$		0			0
$467$		−	31.1127i		−	1.43972i	−0.694117	−	0.719862i	$-0.744205\pi$
$467$		−	31.1127i		−	1.43972i	0.694117	−	0.719862i	$-0.255795\pi$
$468$		0			0
$469$	−6.00000	−	6.92820i	−0.277054	−	0.319915i
$470$		0			0
$471$	14.0000	+	19.7990i	0.645086	+	0.912289i
$472$		0			0
$473$			8.48528i			0.390154i
$474$		0			0
$475$	−12.0000			−0.550598
$476$		0			0
$477$		0			0
$478$		0			0
$479$	4.89898			0.223840			0.111920	−	0.993717i	$-0.464300\pi$
$479$	4.89898			0.223840			0.111920	+	0.993717i	$0.464300\pi$
$480$		0			0
$481$			20.7846i			0.947697i
$482$		0			0
$483$	1.55051	−	6.29253i	0.0705507	−	0.286320i
$484$		0			0
$485$	−19.5959			−0.889805
$486$		0			0
$487$	4.00000			0.181257			0.0906287	−	0.995885i	$-0.471112\pi$
$487$	4.00000			0.181257			0.0906287	+	0.995885i	$0.471112\pi$
$488$		0			0
$489$	14.6969	−	10.3923i	0.664619	−	0.469956i
$490$		0			0
$491$	−17.1464			−0.773807			−0.386904	−	0.922120i	$-0.626455\pi$
$491$	−17.1464			−0.773807			−0.386904	+	0.922120i	$0.626455\pi$
$492$		0			0
$493$	36.0000			1.62136
$494$		0			0
$495$	−9.79796	−	3.46410i	−0.440386	−	0.155700i
$496$		0			0
$497$	2.44949	+	2.82843i	0.109875	+	0.126872i
$498$		0			0
$499$		−	3.46410i		−	0.155074i	−0.996989	−	0.0775372i	$-0.975294\pi$
$499$		−	3.46410i		−	0.155074i	0.996989	−	0.0775372i	$-0.0247057\pi$
$500$		0			0
$501$	19.5959	+	27.7128i	0.875481	+	1.23812i
$502$		0			0
$503$	−14.6969			−0.655304			−0.327652	−	0.944798i	$-0.606257\pi$
$503$	−14.6969			−0.655304			−0.327652	+	0.944798i	$0.606257\pi$
$504$		0			0
$505$	−14.0000			−0.622992
$506$		0			0
$507$	9.00000	+	12.7279i	0.399704	+	0.565267i
$508$		0			0
$509$			18.3848i			0.814891i	0.913230	+	0.407445i	$0.133580\pi$
$509$			18.3848i			0.814891i	−0.913230	+	0.407445i	$0.866420\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−20.0000	+	5.65685i	−0.883022	+	0.249756i
$514$		0			0
$515$	−19.5959			−0.863499
$516$		0			0
$517$	12.0000			0.527759
$518$		0			0
$519$	2.00000	−	1.41421i	0.0877903	−	0.0620771i
$520$		0			0
$521$	−36.7423			−1.60971			−0.804856	−	0.593471i	$-0.797757\pi$
$521$	−36.7423			−1.60971			−0.804856	+	0.593471i	$0.797757\pi$
$522$		0			0
$523$	−22.0000			−0.961993			−0.480996	−	0.876723i	$-0.659725\pi$
$523$	−22.0000			−0.961993			−0.480996	+	0.876723i	$0.659725\pi$
$524$		0			0
$525$	−13.3485	−	3.28913i	−0.582575	−	0.143549i
$526$		0			0
$527$		−	50.9117i		−	2.21775i
$528$		0			0
$529$	21.0000			0.913043
$530$		0			0
$531$	16.0000	+	5.65685i	0.694341	+	0.245487i
$532$		0			0
$533$	4.89898			0.212198
$534$		0			0
$535$			10.3923i			0.449299i
$536$		0			0
$537$	22.0454	+	31.1769i	0.951330	+	1.34538i
$538$		0			0
$539$	2.44949	−	16.9706i	0.105507	−	0.730974i
$540$		0			0
$541$		−	31.1769i		−	1.34040i	−0.742180	−	0.670200i	$-0.766208\pi$
$541$		−	31.1769i		−	1.34040i	0.742180	−	0.670200i	$-0.233792\pi$
$542$		0			0
$543$	−22.0000	−	31.1127i	−0.944110	−	1.33517i
$544$		0			0
$545$		0			0
$546$		0			0
$547$			24.2487i			1.03680i	0.855138	+	0.518400i	$0.173472\pi$
$547$			24.2487i			1.03680i	−0.855138	+	0.518400i	$0.826528\pi$
$548$		0			0
$549$	−10.0000	+	28.2843i	−0.426790	+	1.20714i
$550$		0			0
$551$	−19.5959			−0.834814
$552$		0			0
$553$	−16.0000	+	13.8564i	−0.680389	+	0.589234i
$554$		0			0
$555$	−14.6969	−	20.7846i	−0.623850	−	0.882258i
$556$		0			0
$557$	−29.3939			−1.24546			−0.622729	−	0.782437i	$-0.713976\pi$
$557$	−29.3939			−1.24546			−0.622729	+	0.782437i	$0.713976\pi$
$558$		0			0
$559$		−	6.92820i		−	0.293032i
$560$		0			0
$561$	−18.0000	−	25.4558i	−0.759961	−	1.07475i
$562$		0			0
$563$			11.3137i			0.476816i	0.971165	+	0.238408i	$0.0766255\pi$
$563$			11.3137i			0.476816i	−0.971165	+	0.238408i	$0.923374\pi$
$564$		0			0
$565$	20.0000			0.841406
$566$		0			0
$567$	−23.7980	+	0.810647i	−0.999420	+	0.0340440i
$568$		0			0
$569$		−	14.1421i		−	0.592869i	−0.955053	−	0.296435i	$-0.904202\pi$
$569$		−	14.1421i		−	0.592869i	0.955053	−	0.296435i	$-0.0957977\pi$
$570$		0			0
$571$			17.3205i			0.724841i	0.932015	+	0.362420i	$0.118050\pi$
$571$			17.3205i			0.724841i	−0.932015	+	0.362420i	$0.881950\pi$
$572$		0			0
$573$	−22.0000	+	15.5563i	−0.919063	+	0.649876i
$574$		0			0
$575$			4.24264i			0.176930i
$576$		0			0
$577$			20.7846i			0.865275i	0.901568	+	0.432637i	$0.142417\pi$
$577$			20.7846i			0.865275i	−0.901568	+	0.432637i	$0.857583\pi$
$578$		0			0
$579$	4.00000	+	5.65685i	0.166234	+	0.235091i
$580$		0			0
$581$	19.5959	+	22.6274i	0.812976	+	0.938743i
$582$		0			0
$583$		0			0
$584$		0			0
$585$	8.00000	+	2.82843i	0.330759	+	0.116941i
$586$		0			0
$587$			2.82843i			0.116742i	0.998295	+	0.0583708i	$0.0185906\pi$
$587$			2.82843i			0.116742i	−0.998295	+	0.0583708i	$0.981409\pi$
$588$		0			0
$589$			27.7128i			1.14189i
$590$		0			0
$591$	14.6969	+	20.7846i	0.604551	+	0.854965i
$592$		0			0
$593$	−7.34847			−0.301765			−0.150883	−	0.988552i	$-0.548212\pi$
$593$	−7.34847			−0.301765			−0.150883	+	0.988552i	$0.548212\pi$
$594$		0			0
$595$	18.0000	+	20.7846i	0.737928	+	0.852086i
$596$		0			0
$597$	−14.6969	+	10.3923i	−0.601506	+	0.425329i
$598$		0			0
$599$			18.3848i			0.751182i	0.926786	+	0.375591i	$0.122560\pi$
$599$			18.3848i			0.751182i	−0.926786	+	0.375591i	$0.877440\pi$
$600$		0			0
$601$		−	6.92820i		−	0.282607i	−0.989966	−	0.141304i	$-0.954871\pi$
$601$		−	6.92820i		−	0.282607i	0.989966	−	0.141304i	$-0.0451294\pi$
$602$		0			0
$603$	9.79796	+	3.46410i	0.399004	+	0.141069i
$604$		0			0
$605$		−	7.07107i		−	0.287480i
$606$		0			0
$607$			3.46410i			0.140604i	0.997526	+	0.0703018i	$0.0223962\pi$
$607$			3.46410i			0.140604i	−0.997526	+	0.0703018i	$0.977604\pi$
$608$		0			0
$609$	−21.7980	−	5.37113i	−0.883298	−	0.217649i
$610$		0			0
$611$	−9.79796			−0.396383
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$	−4.89898	+	3.46410i	−0.197546	+	0.139686i
$616$		0			0
$617$		−	5.65685i		−	0.227736i	−0.993496	−	0.113868i	$-0.963676\pi$
$617$		−	5.65685i		−	0.227736i	0.993496	−	0.113868i	$-0.0363242\pi$
$618$		0			0
$619$	−10.0000			−0.401934			−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−10.0000			−0.401934			−0.200967	+	0.979598i	$0.564408\pi$
$620$		0			0
$621$	2.00000	+	7.07107i	0.0802572	+	0.283752i
$622$		0			0
$623$	−14.6969	+	12.7279i	−0.588820	+	0.509933i
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$	9.79796	+	13.8564i	0.391293	+	0.553372i
$628$		0			0
$629$		−	76.3675i		−	3.04497i
$630$		0			0
$631$	−32.0000			−1.27390			−0.636950	−	0.770905i	$-0.719804\pi$
$631$	−32.0000			−1.27390			−0.636950	+	0.770905i	$0.719804\pi$
$632$		0			0
$633$	−34.2929	+	24.2487i	−1.36302	+	0.963800i
$634$		0			0
$635$			5.65685i			0.224485i
$636$		0			0
$637$	−2.00000	+	13.8564i	−0.0792429	+	0.549011i
$638$		0			0
$639$	−4.00000	−	1.41421i	−0.158238	−	0.0559454i
$640$		0			0
$641$		−	22.6274i		−	0.893729i	−0.894602	−	0.446865i	$-0.852541\pi$
$641$		−	22.6274i		−	0.893729i	0.894602	−	0.446865i	$-0.147459\pi$
$642$		0			0
$643$	20.0000			0.788723			0.394362	−	0.918955i	$-0.370966\pi$
$643$	20.0000			0.788723			0.394362	+	0.918955i	$0.370966\pi$
$644$		0			0
$645$	4.89898	+	6.92820i	0.192897	+	0.272798i
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$		−	13.8564i		−	0.543912i
$650$		0			0
$651$	−7.59592	+	30.8270i	−0.297707	+	1.20820i
$652$		0			0
$653$	9.79796			0.383424			0.191712	−	0.981451i	$-0.438596\pi$
$653$	9.79796			0.383424			0.191712	+	0.981451i	$0.438596\pi$
$654$		0			0
$655$	−16.0000			−0.625172
$656$		0			0
$657$		0			0
$658$		0			0
$659$	31.8434			1.24044			0.620221	−	0.784427i	$-0.287043\pi$
$659$	31.8434			1.24044			0.620221	+	0.784427i	$0.287043\pi$
$660$		0			0
$661$	−14.0000			−0.544537			−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000			−0.544537			−0.272268	+	0.962221i	$0.587774\pi$
$662$		0			0
$663$	14.6969	+	20.7846i	0.570782	+	0.807207i
$664$		0			0
$665$	−9.79796	−	11.3137i	−0.379949	−	0.438727i
$666$		0			0
$667$			6.92820i			0.268261i
$668$		0			0
$669$	24.4949	−	17.3205i	0.947027	−	0.669650i
$670$		0			0
$671$	24.4949			0.945615
$672$		0			0
$673$	8.00000			0.308377			0.154189	−	0.988041i	$-0.450724\pi$
$673$	8.00000			0.308377			0.154189	+	0.988041i	$0.450724\pi$
$674$		0			0
$675$	15.0000	−	4.24264i	0.577350	−	0.163299i
$676$		0			0
$677$			26.8701i			1.03270i	0.856378	+	0.516350i	$0.172710\pi$
$677$			26.8701i			1.03270i	−0.856378	+	0.516350i	$0.827290\pi$
$678$		0			0
$679$	24.0000	+	27.7128i	0.921035	+	1.06352i
$680$		0			0
$681$	4.00000	−	2.82843i	0.153280	−	0.108386i
$682$		0			0
$683$	−51.4393			−1.96827			−0.984135	−	0.177423i	$-0.943224\pi$
$683$	−51.4393			−1.96827			−0.984135	+	0.177423i	$0.943224\pi$
$684$		0			0
$685$	−16.0000			−0.611329
$686$		0			0
$687$	14.0000	+	19.7990i	0.534133	+	0.755379i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−10.0000			−0.380418			−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000			−0.380418			−0.190209	+	0.981744i	$0.560917\pi$
$692$		0			0
$693$	7.10102	+	18.0990i	0.269745	+	0.687526i
$694$		0			0
$695$			19.7990i			0.751018i
$696$		0			0
$697$	−18.0000			−0.681799
$698$		0			0
$699$	−20.0000	+	14.1421i	−0.756469	+	0.534905i
$700$		0			0
$701$	−29.3939			−1.11019			−0.555096	−	0.831786i	$-0.687318\pi$
$701$	−29.3939			−1.11019			−0.555096	+	0.831786i	$0.687318\pi$
$702$		0			0
$703$			41.5692i			1.56781i
$704$		0			0
$705$	9.79796	−	6.92820i	0.369012	−	0.260931i
$706$		0			0
$707$	17.1464	+	19.7990i	0.644858	+	0.744618i
$708$		0			0
$709$		−	13.8564i		−	0.520388i	−0.965556	−	0.260194i	$-0.916213\pi$
$709$		−	13.8564i		−	0.520388i	0.965556	−	0.260194i	$-0.0837866\pi$
$710$		0			0
$711$	8.00000	−	22.6274i	0.300023	−	0.848594i
$712$		0			0
$713$	9.79796			0.366936
$714$		0			0
$715$		−	6.92820i		−	0.259100i
$716$		0			0
$717$	38.0000	−	26.8701i	1.41914	−	1.00348i
$718$		0			0
$719$	−14.6969			−0.548103			−0.274052	−	0.961715i	$-0.588364\pi$
$719$	−14.6969			−0.548103			−0.274052	+	0.961715i	$0.588364\pi$
$720$		0			0
$721$	24.0000	+	27.7128i	0.893807	+	1.03208i
$722$		0			0
$723$	−9.79796	+	6.92820i	−0.364390	+	0.257663i
$724$		0			0
$725$	14.6969			0.545831
$726$		0			0
$727$		−	13.8564i		−	0.513906i	−0.966424	−	0.256953i	$-0.917281\pi$
$727$		−	13.8564i		−	0.513906i	0.966424	−	0.256953i	$-0.0827185\pi$
$728$		0			0
$729$	23.0000	−	14.1421i	0.851852	−	0.523783i
$730$		0			0
$731$			25.4558i			0.941518i
$732$		0			0
$733$	−14.0000			−0.517102			−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000			−0.517102			−0.258551	+	0.965998i	$0.583245\pi$
$734$		0			0
$735$	−7.79796	−	15.2706i	−0.287632	−	0.563265i
$736$		0			0
$737$		−	8.48528i		−	0.312559i
$738$		0			0
$739$		−	31.1769i		−	1.14686i	−0.819254	−	0.573431i	$-0.805612\pi$
$739$		−	31.1769i		−	1.14686i	0.819254	−	0.573431i	$-0.194388\pi$
$740$		0			0
$741$	−8.00000	−	11.3137i	−0.293887	−	0.415619i
$742$		0			0
$743$		−	7.07107i		−	0.259412i	−0.991552	−	0.129706i	$-0.958597\pi$
$743$		−	7.07107i		−	0.259412i	0.991552	−	0.129706i	$-0.0414034\pi$
$744$		0			0
$745$			13.8564i			0.507659i
$746$		0			0
$747$	−32.0000	−	11.3137i	−1.17082	−	0.413947i
$748$		0			0
$749$	14.6969	−	12.7279i	0.537014	−	0.465068i
$750$		0			0
$751$	−32.0000			−1.16770			−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000			−1.16770			−0.583848	+	0.811863i	$0.698454\pi$
$752$		0			0
$753$	−20.0000	+	14.1421i	−0.728841	+	0.515368i
$754$		0			0
$755$		−	11.3137i		−	0.411748i
$756$		0			0
$757$			41.5692i			1.51086i	0.655230	+	0.755429i	$0.272572\pi$
$757$			41.5692i			1.51086i	−0.655230	+	0.755429i	$0.727428\pi$
$758$		0			0
$759$	4.89898	−	3.46410i	0.177822	−	0.125739i
$760$		0			0
$761$	−17.1464			−0.621558			−0.310779	−	0.950482i	$-0.600590\pi$
$761$	−17.1464			−0.621558			−0.310779	+	0.950482i	$0.600590\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−29.3939	−	10.3923i	−1.06274	−	0.375735i
$766$		0			0
$767$			11.3137i			0.408514i
$768$		0			0
$769$		−	34.6410i		−	1.24919i	−0.780950	−	0.624593i	$-0.785265\pi$
$769$		−	34.6410i		−	1.24919i	0.780950	−	0.624593i	$-0.214735\pi$
$770$		0			0
$771$	2.44949	+	3.46410i	0.0882162	+	0.124757i
$772$		0			0
$773$			26.8701i			0.966449i	0.875497	+	0.483224i	$0.160534\pi$
$773$			26.8701i			0.966449i	−0.875497	+	0.483224i	$0.839466\pi$
$774$		0			0
$775$		−	20.7846i		−	0.746605i
$776$		0			0
$777$	−11.3939	+	46.2405i	−0.408753	+	1.65887i
$778$		0			0
$779$	9.79796			0.351048
$780$		0			0
$781$			3.46410i			0.123955i
$782$		0			0
$783$	24.4949	−	6.92820i	0.875376	−	0.247594i
$784$		0			0
$785$		−	19.7990i		−	0.706656i
$786$		0			0
$787$	2.00000			0.0712923			0.0356462	−	0.999364i	$-0.488651\pi$
$787$	2.00000			0.0712923			0.0356462	+	0.999364i	$0.488651\pi$
$788$		0			0
$789$	14.0000	−	9.89949i	0.498413	−	0.352431i
$790$		0			0
$791$	−24.4949	−	28.2843i	−0.870938	−	1.00567i
$792$		0			0
$793$	−20.0000			−0.710221
$794$		0			0
$795$		0			0
$796$		0			0
$797$			52.3259i			1.85348i	0.375705	+	0.926739i	$0.377401\pi$
$797$			52.3259i			1.85348i	−0.375705	+	0.926739i	$0.622599\pi$
$798$		0			0
$799$	36.0000			1.27359
$800$		0			0
$801$	7.34847	−	20.7846i	0.259645	−	0.734388i
$802$		0			0
$803$		0			0
$804$		0			0
$805$	−4.00000	+	3.46410i	−0.140981	+	0.122094i
$806$		0			0
$807$	−34.0000	+	24.0416i	−1.19686	+	0.846305i
$808$		0			0
$809$			19.7990i			0.696095i	0.937477	+	0.348048i	$0.113155\pi$
$809$			19.7990i			0.696095i	−0.937477	+	0.348048i	$0.886845\pi$
$810$		0			0
$811$	2.00000			0.0702295			0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000			0.0702295			0.0351147	+	0.999383i	$0.488820\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−14.6969			−0.514811
$816$		0			0
$817$		−	13.8564i		−	0.484774i
$818$		0			0
$819$	−5.79796	−	14.7778i	−0.202597	−	0.516378i
$820$		0			0
$821$	48.9898			1.70976			0.854878	−	0.518829i	$-0.173632\pi$
$821$	48.9898			1.70976			0.854878	+	0.518829i	$0.173632\pi$
$822$		0			0
$823$	−20.0000			−0.697156			−0.348578	−	0.937280i	$-0.613335\pi$
$823$	−20.0000			−0.697156			−0.348578	+	0.937280i	$0.613335\pi$
$824$		0			0
$825$	−7.34847	−	10.3923i	−0.255841	−	0.361814i
$826$		0			0
$827$	−7.34847			−0.255531			−0.127766	−	0.991804i	$-0.540781\pi$
$827$	−7.34847			−0.255531			−0.127766	+	0.991804i	$0.540781\pi$
$828$		0			0
$829$	34.0000			1.18087			0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000			1.18087			0.590434	+	0.807086i	$0.298956\pi$
$830$		0			0
$831$	−4.89898	+	3.46410i	−0.169944	+	0.120168i
$832$		0			0
$833$	7.34847	−	50.9117i	0.254609	−	1.76399i
$834$		0			0
$835$		−	27.7128i		−	0.959041i
$836$		0			0
$837$	−9.79796	−	34.6410i	−0.338667	−	1.19737i
$838$		0			0
$839$	48.9898			1.69132			0.845658	−	0.533726i	$-0.179208\pi$
$839$	48.9898			1.69132			0.845658	+	0.533726i	$0.179208\pi$
$840$		0			0
$841$	−5.00000			−0.172414
$842$		0			0
$843$	4.00000	−	2.82843i	0.137767	−	0.0974162i
$844$		0			0
$845$		−	12.7279i		−	0.437854i
$846$		0			0
$847$	−10.0000	+	8.66025i	−0.343604	+	0.297570i
$848$		0			0
$849$	4.00000	+	5.65685i	0.137280	+	0.194143i
$850$		0			0
$851$	14.6969			0.503805
$852$		0			0
$853$	10.0000			0.342393			0.171197	−	0.985237i	$-0.445237\pi$
$853$	10.0000			0.342393			0.171197	+	0.985237i	$0.445237\pi$
$854$		0			0
$855$	16.0000	+	5.65685i	0.547188	+	0.193460i
$856$		0			0
$857$	31.8434			1.08775			0.543874	−	0.839167i	$-0.316957\pi$
$857$	31.8434			1.08775			0.543874	+	0.839167i	$0.316957\pi$
$858$		0			0
$859$	20.0000			0.682391			0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000			0.682391			0.341196	+	0.939992i	$0.389168\pi$
$860$		0			0
$861$	10.8990	+	2.68556i	0.371436	+	0.0915237i
$862$		0			0
$863$			1.41421i			0.0481404i	0.999710	+	0.0240702i	$0.00766252\pi$
$863$			1.41421i			0.0481404i	−0.999710	+	0.0240702i	$0.992337\pi$
$864$		0			0
$865$	−2.00000			−0.0680020
$866$		0			0
$867$	−37.0000	−	52.3259i	−1.25659	−	1.77708i
$868$		0			0
$869$	−19.5959			−0.664746
$870$		0			0
$871$			6.92820i			0.234753i
$872$		0			0
$873$	−39.1918	−	13.8564i	−1.32644	−	0.468968i
$874$		0			0
$875$	19.5959	+	22.6274i	0.662463	+	0.764946i
$876$		0			0
$877$			13.8564i			0.467898i	0.972249	+	0.233949i	$0.0751648\pi$
$877$			13.8564i			0.467898i	−0.972249	+	0.233949i	$0.924835\pi$
$878$		0			0
$879$	2.00000	−	1.41421i	0.0674583	−	0.0477002i
$880$		0			0
$881$	22.0454			0.742729			0.371364	−	0.928487i	$-0.378890\pi$
$881$	22.0454			0.742729			0.371364	+	0.928487i	$0.378890\pi$
$882$		0			0
$883$			10.3923i			0.349729i	0.984593	+	0.174864i	$0.0559487\pi$
$883$			10.3923i			0.349729i	−0.984593	+	0.174864i	$0.944051\pi$
$884$		0			0
$885$	−8.00000	−	11.3137i	−0.268917	−	0.380306i
$886$		0			0
$887$	39.1918			1.31593			0.657967	−	0.753047i	$-0.271417\pi$
$887$	39.1918			1.31593			0.657967	+	0.753047i	$0.271417\pi$
$888$		0			0
$889$	8.00000	−	6.92820i	0.268311	−	0.232364i
$890$		0			0
$891$	−17.1464	−	13.8564i	−0.574427	−	0.464207i
$892$		0			0
$893$	−19.5959			−0.655752
$894$		0			0
$895$		−	31.1769i		−	1.04213i
$896$		0			0
$897$	−4.00000	+	2.82843i	−0.133556	+	0.0944384i
$898$		0			0
$899$		−	33.9411i		−	1.13200i
$900$		0			0
$901$		0			0
$902$		0			0
$903$	3.79796	−	15.4135i	0.126388	−	0.512929i
$904$		0			0
$905$			31.1127i			1.03422i
$906$		0			0
$907$		−	38.1051i		−	1.26526i	−0.774454	−	0.632630i	$-0.781975\pi$
$907$		−	38.1051i		−	1.26526i	0.774454	−	0.632630i	$-0.218025\pi$
$908$		0			0
$909$	−28.0000	−	9.89949i	−0.928701	−	0.328346i
$910$		0			0
$911$		−	15.5563i		−	0.515405i	−0.966224	−	0.257702i	$-0.917035\pi$
$911$		−	15.5563i		−	0.515405i	0.966224	−	0.257702i	$-0.0829654\pi$
$912$		0			0
$913$			27.7128i			0.917160i
$914$		0			0
$915$	20.0000	−	14.1421i	0.661180	−	0.467525i
$916$		0			0
$917$	19.5959	+	22.6274i	0.647114	+	0.747223i
$918$		0			0
$919$	40.0000			1.31948			0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000			1.31948			0.659739	+	0.751495i	$0.270667\pi$
$920$		0			0
$921$	4.00000	+	5.65685i	0.131804	+	0.186400i
$922$		0			0
$923$		−	2.82843i		−	0.0930988i
$924$		0			0
$925$		−	31.1769i		−	1.02509i
$926$		0			0
$927$	−39.1918	−	13.8564i	−1.28723	−	0.455104i
$928$		0			0
$929$	31.8434			1.04475			0.522373	−	0.852717i	$-0.325047\pi$
$929$	31.8434			1.04475			0.522373	+	0.852717i	$0.325047\pi$
$930$		0			0
$931$	−4.00000	+	27.7128i	−0.131095	+	0.908251i
$932$		0			0
$933$	−24.4949	−	34.6410i	−0.801927	−	1.13410i
$934$		0			0
$935$			25.4558i			0.832495i
$936$		0			0
$937$		−	41.5692i		−	1.35801i	−0.734135	−	0.679004i	$-0.762412\pi$
$937$		−	41.5692i		−	1.35801i	0.734135	−	0.679004i	$-0.237588\pi$
$938$		0			0
$939$	−9.79796	+	6.92820i	−0.319744	+	0.226093i
$940$		0			0
$941$			1.41421i			0.0461020i	0.999734	+	0.0230510i	$0.00733802\pi$
$941$			1.41421i			0.0461020i	−0.999734	+	0.0230510i	$0.992662\pi$
$942$		0			0
$943$		−	3.46410i		−	0.112807i
$944$		0			0
$945$	16.2474	+	10.6780i	0.528530	+	0.347356i
$946$		0			0
$947$	46.5403			1.51236			0.756178	−	0.654366i	$-0.227064\pi$
$947$	46.5403			1.51236			0.756178	+	0.654366i	$0.227064\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	−19.5959	−	27.7128i	−0.635441	−	0.898650i
$952$		0			0
$953$			45.2548i			1.46595i	0.680257	+	0.732974i	$0.261868\pi$
$953$			45.2548i			1.46595i	−0.680257	+	0.732974i	$0.738132\pi$
$954$		0			0
$955$	22.0000			0.711903
$956$		0			0
$957$	−12.0000	−	16.9706i	−0.387905	−	0.548580i
$958$		0			0
$959$	19.5959	+	22.6274i	0.632785	+	0.730677i
$960$		0			0
$961$	−17.0000			−0.548387
$962$		0			0
$963$	−7.34847	+	20.7846i	−0.236801	+	0.669775i
$964$		0			0
$965$		−	5.65685i		−	0.182101i
$966$		0			0
$967$	4.00000			0.128631			0.0643157	−	0.997930i	$-0.479514\pi$
$967$	4.00000			0.128631			0.0643157	+	0.997930i	$0.479514\pi$
$968$		0			0
$969$	29.3939	+	41.5692i	0.944267	+	1.33540i
$970$		0			0
$971$			36.7696i			1.17999i	0.807406	+	0.589996i	$0.200871\pi$
$971$			36.7696i			1.17999i	−0.807406	+	0.589996i	$0.799129\pi$
$972$		0			0
$973$	28.0000	−	24.2487i	0.897639	−	0.777378i
$974$		0			0
$975$	6.00000	+	8.48528i	0.192154	+	0.271746i
$976$		0			0
$977$			19.7990i			0.633426i	0.948521	+	0.316713i	$0.102579\pi$
$977$			19.7990i			0.633426i	−0.948521	+	0.316713i	$0.897421\pi$
$978$		0			0
$979$	−18.0000			−0.575282
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−39.1918			−1.25003			−0.625013	−	0.780615i	$-0.714906\pi$
$983$	−39.1918			−1.25003			−0.625013	+	0.780615i	$0.714906\pi$
$984$		0			0
$985$		−	20.7846i		−	0.662253i
$986$		0			0
$987$	−21.7980	−	5.37113i	−0.693837	−	0.170965i
$988$		0			0
$989$	−4.89898			−0.155778
$990$		0			0
$991$	52.0000			1.65183			0.825917	−	0.563791i	$-0.190658\pi$
$991$	52.0000			1.65183			0.825917	+	0.563791i	$0.190658\pi$
$992$		0			0
$993$	34.2929	−	24.2487i	1.08825	−	0.769510i
$994$		0			0
$995$	14.6969			0.465924
$996$		0			0
$997$	−62.0000			−1.96356			−0.981780	−	0.190022i	$-0.939144\pi$
$997$	−62.0000			−1.96356			−0.981780	+	0.190022i	$0.939144\pi$
$998$		0			0
$999$	−14.6969	−	51.9615i	−0.464991	−	1.64399i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.2.i.a.209.1		4
3.2	odd	2	inner	672.2.i.a.209.3		4
4.3	odd	2		168.2.i.c.125.1	yes	4
7.6	odd	2		672.2.i.c.209.4		4
8.3	odd	2		168.2.i.a.125.3	yes	4
8.5	even	2		672.2.i.c.209.3		4
12.11	even	2		168.2.i.c.125.4	yes	4
21.20	even	2		672.2.i.c.209.2		4
24.5	odd	2		672.2.i.c.209.1		4
24.11	even	2		168.2.i.a.125.2	yes	4
28.27	even	2		168.2.i.a.125.1	✓	4
56.13	odd	2	inner	672.2.i.a.209.2		4
56.27	even	2		168.2.i.c.125.3	yes	4
84.83	odd	2		168.2.i.a.125.4	yes	4
168.83	odd	2		168.2.i.c.125.2	yes	4
168.125	even	2	inner	672.2.i.a.209.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.i.a.125.1	✓	4	28.27	even	2
168.2.i.a.125.2	yes	4	24.11	even	2
168.2.i.a.125.3	yes	4	8.3	odd	2
168.2.i.a.125.4	yes	4	84.83	odd	2
168.2.i.c.125.1	yes	4	4.3	odd	2
168.2.i.c.125.2	yes	4	168.83	odd	2
168.2.i.c.125.3	yes	4	56.27	even	2
168.2.i.c.125.4	yes	4	12.11	even	2
672.2.i.a.209.1		4	1.1	even	1	trivial
672.2.i.a.209.2		4	56.13	odd	2	inner
672.2.i.a.209.3		4	3.2	odd	2	inner
672.2.i.a.209.4		4	168.125	even	2	inner
672.2.i.c.209.1		4	24.5	odd	2
672.2.i.c.209.2		4	21.20	even	2
672.2.i.c.209.3		4	8.5	even	2
672.2.i.c.209.4		4	7.6	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	672.i (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.41421i) q^{3} +1.41421i q^{5} +(2.00000 - 1.73205i) q^{7} +(-1.00000 + 2.82843i) q^{9} +O(q^{10})\)	\(q+(-1.00000 - 1.41421i) q^{3} +1.41421i q^{5} +(2.00000 - 1.73205i) q^{7} +(-1.00000 + 2.82843i) q^{9} +2.44949 q^{11} -2.00000 q^{13} +(2.00000 - 1.41421i) q^{15} +7.34847 q^{17} -4.00000 q^{19} +(-4.44949 - 1.09638i) q^{21} +1.41421i q^{23} +3.00000 q^{25} +(5.00000 - 1.41421i) q^{27} +4.89898 q^{29} -6.92820i q^{31} +(-2.44949 - 3.46410i) q^{33} +(2.44949 + 2.82843i) q^{35} -10.3923i q^{37} +(2.00000 + 2.82843i) q^{39} -2.44949 q^{41} +3.46410i q^{43} +(-4.00000 - 1.41421i) q^{45} +4.89898 q^{47} +(1.00000 - 6.92820i) q^{49} +(-7.34847 - 10.3923i) q^{51} +3.46410i q^{55} +(4.00000 + 5.65685i) q^{57} -5.65685i q^{59} +10.0000 q^{61} +(2.89898 + 7.38891i) q^{63} -2.82843i q^{65} -3.46410i q^{67} +(2.00000 - 1.41421i) q^{69} +1.41421i q^{71} +(-3.00000 - 4.24264i) q^{75} +(4.89898 - 4.24264i) q^{77} -8.00000 q^{79} +(-7.00000 - 5.65685i) q^{81} +11.3137i q^{83} +10.3923i q^{85} +(-4.89898 - 6.92820i) q^{87} -7.34847 q^{89} +(-4.00000 + 3.46410i) q^{91} +(-9.79796 + 6.92820i) q^{93} -5.65685i q^{95} +13.8564i q^{97} +(-2.44949 + 6.92820i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 8 q^{7} - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + 8 * q^7 - 4 * q^9	\( 4 q - 4 q^{3} + 8 q^{7} - 4 q^{9} - 8 q^{13} + 8 q^{15} - 16 q^{19} - 8 q^{21} + 12 q^{25} + 20 q^{27} + 8 q^{39} - 16 q^{45} + 4 q^{49} + 16 q^{57} + 40 q^{61} - 8 q^{63} + 8 q^{69} - 12 q^{75} - 32 q^{79} - 28 q^{81} - 16 q^{91}+O(q^{100}) \) 4 * q - 4 * q^3 + 8 * q^7 - 4 * q^9 - 8 * q^13 + 8 * q^15 - 16 * q^19 - 8 * q^21 + 12 * q^25 + 20 * q^27 + 8 * q^39 - 16 * q^45 + 4 * q^49 + 16 * q^57 + 40 * q^61 - 8 * q^63 + 8 * q^69 - 12 * q^75 - 32 * q^79 - 28 * q^81 - 16 * q^91

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