LMFDB - Embedded newform 624.2.d.e.287.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [624,2,Mod(287,624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(624, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("624.287");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	624.d (of order $2$, degree $1$, 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:	$4.98266508613$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			287.2
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	624.287
Dual form			624.2.d.e.287.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 + 1.41421i) q^{3} +1.41421i q^{5} +1.41421i q^{7} +(-1.00000 + 2.82843i) q^{9} +O(q^{10})$	$q+(1.00000 + 1.41421i) q^{3} +1.41421i q^{5} +1.41421i q^{7} +(-1.00000 + 2.82843i) q^{9} -4.00000 q^{11} +1.00000 q^{13} +(-2.00000 + 1.41421i) q^{15} +4.24264i q^{19} +(-2.00000 + 1.41421i) q^{21} -2.00000 q^{23} +3.00000 q^{25} +(-5.00000 + 1.41421i) q^{27} -8.48528i q^{29} +1.41421i q^{31} +(-4.00000 - 5.65685i) q^{33} -2.00000 q^{35} +2.00000 q^{37} +(1.00000 + 1.41421i) q^{39} +7.07107i q^{41} +8.48528i q^{43} +(-4.00000 - 1.41421i) q^{45} +8.00000 q^{47} +5.00000 q^{49} -5.65685i q^{53} -5.65685i q^{55} +(-6.00000 + 4.24264i) q^{57} -12.0000 q^{59} +(-4.00000 - 1.41421i) q^{63} +1.41421i q^{65} +4.24264i q^{67} +(-2.00000 - 2.82843i) q^{69} +8.00000 q^{71} +10.0000 q^{73} +(3.00000 + 4.24264i) q^{75} -5.65685i q^{77} -2.82843i q^{79} +(-7.00000 - 5.65685i) q^{81} +(12.0000 - 8.48528i) q^{87} +9.89949i q^{89} +1.41421i q^{91} +(-2.00000 + 1.41421i) q^{93} -6.00000 q^{95} +6.00000 q^{97} +(4.00000 - 11.3137i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^9	$ 2 q + 2 q^{3} - 2 q^{9} - 8 q^{11} + 2 q^{13} - 4 q^{15} - 4 q^{21} - 4 q^{23} + 6 q^{25} - 10 q^{27} - 8 q^{33} - 4 q^{35} + 4 q^{37} + 2 q^{39} - 8 q^{45} + 16 q^{47} + 10 q^{49} - 12 q^{57} - 24 q^{59} - 8 q^{63} - 4 q^{69} + 16 q^{71} + 20 q^{73} + 6 q^{75} - 14 q^{81} + 24 q^{87} - 4 q^{93} - 12 q^{95} + 12 q^{97} + 8 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^9 - 8 * q^11 + 2 * q^13 - 4 * q^15 - 4 * q^21 - 4 * q^23 + 6 * q^25 - 10 * q^27 - 8 * q^33 - 4 * q^35 + 4 * q^37 + 2 * q^39 - 8 * q^45 + 16 * q^47 + 10 * q^49 - 12 * q^57 - 24 * q^59 - 8 * q^63 - 4 * q^69 + 16 * q^71 + 20 * q^73 + 6 * q^75 - 14 * q^81 + 24 * q^87 - 4 * q^93 - 12 * q^95 + 12 * q^97 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/624\mathbb{Z}\right)^\times$.

$n$	$79$	$145$	$209$	$469$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$7$			1.41421i			0.534522i	0.963624	+	0.267261i	$0.0861187\pi$
$7$			1.41421i			0.534522i	−0.963624	+	0.267261i	$0.913881\pi$
$8$		0			0
$9$	−1.00000	+	2.82843i	−0.333333	+	0.942809i
$10$		0			0
$11$	−4.00000			−1.20605			−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000			−1.20605			−0.603023	+	0.797724i	$0.706037\pi$
$12$		0			0
$13$	1.00000			0.277350
$13$	1.00000			0.277350
$14$		0			0
$15$	−2.00000	+	1.41421i	−0.516398	+	0.365148i
$16$		0			0
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$			4.24264i			0.973329i	0.873589	+	0.486664i	$0.161786\pi$
$19$			4.24264i			0.973329i	−0.873589	+	0.486664i	$0.838214\pi$
$20$		0			0
$21$	−2.00000	+	1.41421i	−0.436436	+	0.308607i
$22$		0			0
$23$	−2.00000			−0.417029			−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000			−0.417029			−0.208514	+	0.978019i	$0.566863\pi$
$24$		0			0
$25$	3.00000			0.600000
$26$		0			0
$27$	−5.00000	+	1.41421i	−0.962250	+	0.272166i
$28$		0			0
$29$		−	8.48528i		−	1.57568i	−0.615882	−	0.787839i	$-0.711200\pi$
$29$		−	8.48528i		−	1.57568i	0.615882	−	0.787839i	$-0.288800\pi$
$30$		0			0
$31$			1.41421i			0.254000i	0.991903	+	0.127000i	$0.0405349\pi$
$31$			1.41421i			0.254000i	−0.991903	+	0.127000i	$0.959465\pi$
$32$		0			0
$33$	−4.00000	−	5.65685i	−0.696311	−	0.984732i
$34$		0			0
$35$	−2.00000			−0.338062
$36$		0			0
$37$	2.00000			0.328798			0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000			0.328798			0.164399	+	0.986394i	$0.447432\pi$
$38$		0			0
$39$	1.00000	+	1.41421i	0.160128	+	0.226455i
$40$		0			0
$41$			7.07107i			1.10432i	0.833740	+	0.552158i	$0.186195\pi$
$41$			7.07107i			1.10432i	−0.833740	+	0.552158i	$0.813805\pi$
$42$		0			0
$43$			8.48528i			1.29399i	0.762493	+	0.646997i	$0.223975\pi$
$43$			8.48528i			1.29399i	−0.762493	+	0.646997i	$0.776025\pi$
$44$		0			0
$45$	−4.00000	−	1.41421i	−0.596285	−	0.210819i
$46$		0			0
$47$	8.00000			1.16692			0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000			1.16692			0.583460	+	0.812142i	$0.301699\pi$
$48$		0			0
$49$	5.00000			0.714286
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	5.65685i		−	0.777029i	−0.921443	−	0.388514i	$-0.872988\pi$
$53$		−	5.65685i		−	0.777029i	0.921443	−	0.388514i	$-0.127012\pi$
$54$		0			0
$55$		−	5.65685i		−	0.762770i
$56$		0			0
$57$	−6.00000	+	4.24264i	−0.794719	+	0.561951i
$58$		0			0
$59$	−12.0000			−1.56227			−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000			−1.56227			−0.781133	+	0.624364i	$0.785358\pi$
$60$		0			0
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$		0			0
$63$	−4.00000	−	1.41421i	−0.503953	−	0.178174i
$64$		0			0
$65$			1.41421i			0.175412i
$66$		0			0
$67$			4.24264i			0.518321i	0.965834	+	0.259161i	$0.0834459\pi$
$67$			4.24264i			0.518321i	−0.965834	+	0.259161i	$0.916554\pi$
$68$		0			0
$69$	−2.00000	−	2.82843i	−0.240772	−	0.340503i
$70$		0			0
$71$	8.00000			0.949425			0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000			0.949425			0.474713	+	0.880141i	$0.342552\pi$
$72$		0			0
$73$	10.0000			1.17041			0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000			1.17041			0.585206	+	0.810885i	$0.301014\pi$
$74$		0			0
$75$	3.00000	+	4.24264i	0.346410	+	0.489898i
$76$		0			0
$77$		−	5.65685i		−	0.644658i
$78$		0			0
$79$		−	2.82843i		−	0.318223i	−0.987261	−	0.159111i	$-0.949137\pi$
$79$		−	2.82843i		−	0.318223i	0.987261	−	0.159111i	$-0.0508629\pi$
$80$		0			0
$81$	−7.00000	−	5.65685i	−0.777778	−	0.628539i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	12.0000	−	8.48528i	1.28654	−	0.909718i
$88$		0			0
$89$			9.89949i			1.04934i	0.851304	+	0.524672i	$0.175812\pi$
$89$			9.89949i			1.04934i	−0.851304	+	0.524672i	$0.824188\pi$
$90$		0			0
$91$			1.41421i			0.148250i
$92$		0			0
$93$	−2.00000	+	1.41421i	−0.207390	+	0.146647i
$94$		0			0
$95$	−6.00000			−0.615587
$96$		0			0
$97$	6.00000			0.609208			0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000			0.609208			0.304604	+	0.952479i	$0.401476\pi$
$98$		0			0
$99$	4.00000	−	11.3137i	0.402015	−	1.13707i
$100$		0			0
$101$			19.7990i			1.97007i	0.172345	+	0.985037i	$0.444865\pi$
$101$			19.7990i			1.97007i	−0.172345	+	0.985037i	$0.555135\pi$
$102$		0			0
$103$		−	14.1421i		−	1.39347i	−0.717331	−	0.696733i	$-0.754636\pi$
$103$		−	14.1421i		−	1.39347i	0.717331	−	0.696733i	$-0.245364\pi$
$104$		0			0
$105$	−2.00000	−	2.82843i	−0.195180	−	0.276026i
$106$		0			0
$107$	12.0000			1.16008			0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000			1.16008			0.580042	+	0.814587i	$0.303036\pi$
$108$		0			0
$109$	10.0000			0.957826			0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000			0.957826			0.478913	+	0.877862i	$0.341031\pi$
$110$		0			0
$111$	2.00000	+	2.82843i	0.189832	+	0.268462i
$112$		0			0
$113$		−	2.82843i		−	0.266076i	−0.991111	−	0.133038i	$-0.957527\pi$
$113$		−	2.82843i		−	0.266076i	0.991111	−	0.133038i	$-0.0424732\pi$
$114$		0			0
$115$		−	2.82843i		−	0.263752i
$116$		0			0
$117$	−1.00000	+	2.82843i	−0.0924500	+	0.261488i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	5.00000			0.454545
$122$		0			0
$123$	−10.0000	+	7.07107i	−0.901670	+	0.637577i
$124$		0			0
$125$			11.3137i			1.01193i
$126$		0			0
$127$		−	11.3137i		−	1.00393i	−0.864888	−	0.501965i	$-0.832611\pi$
$127$		−	11.3137i		−	1.00393i	0.864888	−	0.501965i	$-0.167389\pi$
$128$		0			0
$129$	−12.0000	+	8.48528i	−1.05654	+	0.747087i
$130$		0			0
$131$	12.0000			1.04844			0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000			1.04844			0.524222	+	0.851581i	$0.324356\pi$
$132$		0			0
$133$	−6.00000			−0.520266
$134$		0			0
$135$	−2.00000	−	7.07107i	−0.172133	−	0.608581i
$136$		0			0
$137$		−	21.2132i		−	1.81237i	−0.422885	−	0.906183i	$-0.638983\pi$
$137$		−	21.2132i		−	1.81237i	0.422885	−	0.906183i	$-0.361017\pi$
$138$		0			0
$139$		−	11.3137i		−	0.959616i	−0.877373	−	0.479808i	$-0.840706\pi$
$139$		−	11.3137i		−	0.959616i	0.877373	−	0.479808i	$-0.159294\pi$
$140$		0			0
$141$	8.00000	+	11.3137i	0.673722	+	0.952786i
$142$		0			0
$143$	−4.00000			−0.334497
$144$		0			0
$145$	12.0000			0.996546
$146$		0			0
$147$	5.00000	+	7.07107i	0.412393	+	0.583212i
$148$		0			0
$149$			1.41421i			0.115857i	0.998321	+	0.0579284i	$0.0184495\pi$
$149$			1.41421i			0.115857i	−0.998321	+	0.0579284i	$0.981550\pi$
$150$		0			0
$151$		−	24.0416i		−	1.95648i	−0.207476	−	0.978240i	$-0.566525\pi$
$151$		−	24.0416i		−	1.95648i	0.207476	−	0.978240i	$-0.433475\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.00000			−0.160644
$156$		0			0
$157$	−12.0000			−0.957704			−0.478852	−	0.877896i	$-0.658947\pi$
$157$	−12.0000			−0.957704			−0.478852	+	0.877896i	$0.658947\pi$
$158$		0			0
$159$	8.00000	−	5.65685i	0.634441	−	0.448618i
$160$		0			0
$161$		−	2.82843i		−	0.222911i
$162$		0			0
$163$			15.5563i			1.21847i	0.792991	+	0.609234i	$0.208523\pi$
$163$			15.5563i			1.21847i	−0.792991	+	0.609234i	$0.791477\pi$
$164$		0			0
$165$	8.00000	−	5.65685i	0.622799	−	0.440386i
$166$		0			0
$167$	8.00000			0.619059			0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000			0.619059			0.309529	+	0.950890i	$0.399829\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$	−12.0000	−	4.24264i	−0.917663	−	0.324443i
$172$		0			0
$173$		−	14.1421i		−	1.07521i	−0.843198	−	0.537603i	$-0.819330\pi$
$173$		−	14.1421i		−	1.07521i	0.843198	−	0.537603i	$-0.180670\pi$
$174$		0			0
$175$			4.24264i			0.320713i
$176$		0			0
$177$	−12.0000	−	16.9706i	−0.901975	−	1.27559i
$178$		0			0
$179$	4.00000			0.298974			0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000			0.298974			0.149487	+	0.988764i	$0.452238\pi$
$180$		0			0
$181$	8.00000			0.594635			0.297318	−	0.954779i	$-0.403908\pi$
$181$	8.00000			0.594635			0.297318	+	0.954779i	$0.403908\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			2.82843i			0.207950i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−2.00000	−	7.07107i	−0.145479	−	0.514344i
$190$		0			0
$191$	26.0000			1.88129			0.940647	−	0.339387i	$-0.110219\pi$
$191$	26.0000			1.88129			0.940647	+	0.339387i	$0.110219\pi$
$192$		0			0
$193$	−10.0000			−0.719816			−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000			−0.719816			−0.359908	+	0.932988i	$0.617192\pi$
$194$		0			0
$195$	−2.00000	+	1.41421i	−0.143223	+	0.101274i
$196$		0			0
$197$		−	7.07107i		−	0.503793i	−0.967754	−	0.251896i	$-0.918946\pi$
$197$		−	7.07107i		−	0.503793i	0.967754	−	0.251896i	$-0.0810542\pi$
$198$		0			0
$199$			11.3137i			0.802008i	0.916076	+	0.401004i	$0.131339\pi$
$199$			11.3137i			0.802008i	−0.916076	+	0.401004i	$0.868661\pi$
$200$		0			0
$201$	−6.00000	+	4.24264i	−0.423207	+	0.299253i
$202$		0			0
$203$	12.0000			0.842235
$204$		0			0
$205$	−10.0000			−0.698430
$206$		0			0
$207$	2.00000	−	5.65685i	0.139010	−	0.393179i
$208$		0			0
$209$		−	16.9706i		−	1.17388i
$210$		0			0
$211$		−	25.4558i		−	1.75245i	−0.481900	−	0.876226i	$-0.660053\pi$
$211$		−	25.4558i		−	1.75245i	0.481900	−	0.876226i	$-0.339947\pi$
$212$		0			0
$213$	8.00000	+	11.3137i	0.548151	+	0.775203i
$214$		0			0
$215$	−12.0000			−0.818393
$216$		0			0
$217$	−2.00000			−0.135769
$218$		0			0
$219$	10.0000	+	14.1421i	0.675737	+	0.955637i
$220$		0			0
$221$		0			0
$222$		0			0
$223$			26.8701i			1.79935i	0.436558	+	0.899676i	$0.356197\pi$
$223$			26.8701i			1.79935i	−0.436558	+	0.899676i	$0.643803\pi$
$224$		0			0
$225$	−3.00000	+	8.48528i	−0.200000	+	0.565685i
$226$		0			0
$227$	−24.0000			−1.59294			−0.796468	−	0.604681i	$-0.793301\pi$
$227$	−24.0000			−1.59294			−0.796468	+	0.604681i	$0.793301\pi$
$228$		0			0
$229$	−2.00000			−0.132164			−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000			−0.132164			−0.0660819	+	0.997814i	$0.521050\pi$
$230$		0			0
$231$	8.00000	−	5.65685i	0.526361	−	0.372194i
$232$		0			0
$233$			2.82843i			0.185296i	0.995699	+	0.0926482i	$0.0295332\pi$
$233$			2.82843i			0.185296i	−0.995699	+	0.0926482i	$0.970467\pi$
$234$		0			0
$235$			11.3137i			0.738025i
$236$		0			0
$237$	4.00000	−	2.82843i	0.259828	−	0.183726i
$238$		0			0
$239$	−8.00000			−0.517477			−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000			−0.517477			−0.258738	+	0.965947i	$0.583307\pi$
$240$		0			0
$241$	−6.00000			−0.386494			−0.193247	−	0.981150i	$-0.561902\pi$
$241$	−6.00000			−0.386494			−0.193247	+	0.981150i	$0.561902\pi$
$242$		0			0
$243$	1.00000	−	15.5563i	0.0641500	−	0.997940i
$244$		0			0
$245$			7.07107i			0.451754i
$246$		0			0
$247$			4.24264i			0.269953i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−22.0000			−1.38863			−0.694314	−	0.719672i	$-0.744292\pi$
$251$	−22.0000			−1.38863			−0.694314	+	0.719672i	$0.744292\pi$
$252$		0			0
$253$	8.00000			0.502956
$254$		0			0
$255$		0			0
$256$		0			0
$257$			2.82843i			0.176432i	0.996101	+	0.0882162i	$0.0281166\pi$
$257$			2.82843i			0.176432i	−0.996101	+	0.0882162i	$0.971883\pi$
$258$		0			0
$259$			2.82843i			0.175750i
$260$		0			0
$261$	24.0000	+	8.48528i	1.48556	+	0.525226i
$262$		0			0
$263$	−8.00000			−0.493301			−0.246651	−	0.969104i	$-0.579330\pi$
$263$	−8.00000			−0.493301			−0.246651	+	0.969104i	$0.579330\pi$
$264$		0			0
$265$	8.00000			0.491436
$266$		0			0
$267$	−14.0000	+	9.89949i	−0.856786	+	0.605839i
$268$		0			0
$269$		−	2.82843i		−	0.172452i	−0.996276	−	0.0862261i	$-0.972519\pi$
$269$		−	2.82843i		−	0.172452i	0.996276	−	0.0862261i	$-0.0274808\pi$
$270$		0			0
$271$			4.24264i			0.257722i	0.991663	+	0.128861i	$0.0411321\pi$
$271$			4.24264i			0.257722i	−0.991663	+	0.128861i	$0.958868\pi$
$272$		0			0
$273$	−2.00000	+	1.41421i	−0.121046	+	0.0855921i
$274$		0			0
$275$	−12.0000			−0.723627
$276$		0			0
$277$	−22.0000			−1.32185			−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000			−1.32185			−0.660926	+	0.750451i	$0.729836\pi$
$278$		0			0
$279$	−4.00000	−	1.41421i	−0.239474	−	0.0846668i
$280$		0			0
$281$			1.41421i			0.0843649i	0.999110	+	0.0421825i	$0.0134311\pi$
$281$			1.41421i			0.0843649i	−0.999110	+	0.0421825i	$0.986569\pi$
$282$		0			0
$283$			22.6274i			1.34506i	0.740070	+	0.672530i	$0.234792\pi$
$283$			22.6274i			1.34506i	−0.740070	+	0.672530i	$0.765208\pi$
$284$		0			0
$285$	−6.00000	−	8.48528i	−0.355409	−	0.502625i
$286$		0			0
$287$	−10.0000			−0.590281
$288$		0			0
$289$	17.0000			1.00000
$290$		0			0
$291$	6.00000	+	8.48528i	0.351726	+	0.497416i
$292$		0			0
$293$			24.0416i			1.40453i	0.711917	+	0.702264i	$0.247827\pi$
$293$			24.0416i			1.40453i	−0.711917	+	0.702264i	$0.752173\pi$
$294$		0			0
$295$		−	16.9706i		−	0.988064i
$296$		0			0
$297$	20.0000	−	5.65685i	1.16052	−	0.328244i
$298$		0			0
$299$	−2.00000			−0.115663
$300$		0			0
$301$	−12.0000			−0.691669
$302$		0			0
$303$	−28.0000	+	19.7990i	−1.60856	+	1.13742i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	32.5269i		−	1.85641i	−0.372071	−	0.928204i	$-0.621352\pi$
$307$		−	32.5269i		−	1.85641i	0.372071	−	0.928204i	$-0.378648\pi$
$308$		0			0
$309$	20.0000	−	14.1421i	1.13776	−	0.804518i
$310$		0			0
$311$	18.0000			1.02069			0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000			1.02069			0.510343	+	0.859971i	$0.329518\pi$
$312$		0			0
$313$	−26.0000			−1.46961			−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−26.0000			−1.46961			−0.734803	+	0.678280i	$0.762726\pi$
$314$		0			0
$315$	2.00000	−	5.65685i	0.112687	−	0.318728i
$316$		0			0
$317$			9.89949i			0.556011i	0.960579	+	0.278006i	$0.0896734\pi$
$317$			9.89949i			0.556011i	−0.960579	+	0.278006i	$0.910327\pi$
$318$		0			0
$319$			33.9411i			1.90034i
$320$		0			0
$321$	12.0000	+	16.9706i	0.669775	+	0.947204i
$322$		0			0
$323$		0			0
$324$		0			0
$325$	3.00000			0.166410
$326$		0			0
$327$	10.0000	+	14.1421i	0.553001	+	0.782062i
$328$		0			0
$329$			11.3137i			0.623745i
$330$		0			0
$331$			21.2132i			1.16598i	0.812478	+	0.582992i	$0.198118\pi$
$331$			21.2132i			1.16598i	−0.812478	+	0.582992i	$0.801882\pi$
$332$		0			0
$333$	−2.00000	+	5.65685i	−0.109599	+	0.309994i
$334$		0			0
$335$	−6.00000			−0.327815
$336$		0			0
$337$	10.0000			0.544735			0.272367	−	0.962193i	$-0.412193\pi$
$337$	10.0000			0.544735			0.272367	+	0.962193i	$0.412193\pi$
$338$		0			0
$339$	4.00000	−	2.82843i	0.217250	−	0.153619i
$340$		0			0
$341$		−	5.65685i		−	0.306336i
$342$		0			0
$343$			16.9706i			0.916324i
$344$		0			0
$345$	4.00000	−	2.82843i	0.215353	−	0.152277i
$346$		0			0
$347$	−2.00000			−0.107366			−0.0536828	−	0.998558i	$-0.517096\pi$
$347$	−2.00000			−0.107366			−0.0536828	+	0.998558i	$0.517096\pi$
$348$		0			0
$349$	−26.0000			−1.39175			−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000			−1.39175			−0.695874	+	0.718164i	$0.744983\pi$
$350$		0			0
$351$	−5.00000	+	1.41421i	−0.266880	+	0.0754851i
$352$		0			0
$353$			26.8701i			1.43015i	0.699048	+	0.715074i	$0.253607\pi$
$353$			26.8701i			1.43015i	−0.699048	+	0.715074i	$0.746393\pi$
$354$		0			0
$355$			11.3137i			0.600469i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	16.0000			0.844448			0.422224	−	0.906492i	$-0.361250\pi$
$359$	16.0000			0.844448			0.422224	+	0.906492i	$0.361250\pi$
$360$		0			0
$361$	1.00000			0.0526316
$362$		0			0
$363$	5.00000	+	7.07107i	0.262432	+	0.371135i
$364$		0			0
$365$			14.1421i			0.740233i
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$	−20.0000	−	7.07107i	−1.04116	−	0.368105i
$370$		0			0
$371$	8.00000			0.415339
$372$		0			0
$373$	−38.0000			−1.96757			−0.983783	−	0.179364i	$-0.942596\pi$
$373$	−38.0000			−1.96757			−0.983783	+	0.179364i	$0.942596\pi$
$374$		0			0
$375$	−16.0000	+	11.3137i	−0.826236	+	0.584237i
$376$		0			0
$377$		−	8.48528i		−	0.437014i
$378$		0			0
$379$			4.24264i			0.217930i	0.994046	+	0.108965i	$0.0347536\pi$
$379$			4.24264i			0.217930i	−0.994046	+	0.108965i	$0.965246\pi$
$380$		0			0
$381$	16.0000	−	11.3137i	0.819705	−	0.579619i
$382$		0			0
$383$	−12.0000			−0.613171			−0.306586	−	0.951843i	$-0.599187\pi$
$383$	−12.0000			−0.613171			−0.306586	+	0.951843i	$0.599187\pi$
$384$		0			0
$385$	8.00000			0.407718
$386$		0			0
$387$	−24.0000	−	8.48528i	−1.21999	−	0.431331i
$388$		0			0
$389$		−	22.6274i		−	1.14726i	−0.819116	−	0.573628i	$-0.805536\pi$
$389$		−	22.6274i		−	1.14726i	0.819116	−	0.573628i	$-0.194464\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	12.0000	+	16.9706i	0.605320	+	0.856052i
$394$		0			0
$395$	4.00000			0.201262
$396$		0			0
$397$	10.0000			0.501886			0.250943	−	0.968002i	$-0.419259\pi$
$397$	10.0000			0.501886			0.250943	+	0.968002i	$0.419259\pi$
$398$		0			0
$399$	−6.00000	−	8.48528i	−0.300376	−	0.424795i
$400$		0			0
$401$		−	26.8701i		−	1.34183i	−0.741536	−	0.670913i	$-0.765902\pi$
$401$		−	26.8701i		−	1.34183i	0.741536	−	0.670913i	$-0.234098\pi$
$402$		0			0
$403$			1.41421i			0.0704470i
$404$		0			0
$405$	8.00000	−	9.89949i	0.397523	−	0.491910i
$406$		0			0
$407$	−8.00000			−0.396545
$408$		0			0
$409$	2.00000			0.0988936			0.0494468	−	0.998777i	$-0.484254\pi$
$409$	2.00000			0.0988936			0.0494468	+	0.998777i	$0.484254\pi$
$410$		0			0
$411$	30.0000	−	21.2132i	1.47979	−	1.04637i
$412$		0			0
$413$		−	16.9706i		−	0.835067i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	16.0000	−	11.3137i	0.783523	−	0.554035i
$418$		0			0
$419$	36.0000			1.75872			0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000			1.75872			0.879358	+	0.476162i	$0.157972\pi$
$420$		0			0
$421$	−18.0000			−0.877266			−0.438633	−	0.898666i	$-0.644537\pi$
$421$	−18.0000			−0.877266			−0.438633	+	0.898666i	$0.644537\pi$
$422$		0			0
$423$	−8.00000	+	22.6274i	−0.388973	+	1.10018i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−4.00000	−	5.65685i	−0.193122	−	0.273115i
$430$		0			0
$431$	−36.0000			−1.73406			−0.867029	−	0.498257i	$-0.833974\pi$
$431$	−36.0000			−1.73406			−0.867029	+	0.498257i	$0.833974\pi$
$432$		0			0
$433$	16.0000			0.768911			0.384455	−	0.923144i	$-0.374389\pi$
$433$	16.0000			0.768911			0.384455	+	0.923144i	$0.374389\pi$
$434$		0			0
$435$	12.0000	+	16.9706i	0.575356	+	0.813676i
$436$		0			0
$437$		−	8.48528i		−	0.405906i
$438$		0			0
$439$		−	25.4558i		−	1.21494i	−0.794342	−	0.607471i	$-0.792184\pi$
$439$		−	25.4558i		−	1.21494i	0.794342	−	0.607471i	$-0.207816\pi$
$440$		0			0
$441$	−5.00000	+	14.1421i	−0.238095	+	0.673435i
$442$		0			0
$443$	−28.0000			−1.33032			−0.665160	−	0.746701i	$-0.731637\pi$
$443$	−28.0000			−1.33032			−0.665160	+	0.746701i	$0.731637\pi$
$444$		0			0
$445$	−14.0000			−0.663664
$446$		0			0
$447$	−2.00000	+	1.41421i	−0.0945968	+	0.0668900i
$448$		0			0
$449$			29.6985i			1.40156i	0.713378	+	0.700779i	$0.247164\pi$
$449$			29.6985i			1.40156i	−0.713378	+	0.700779i	$0.752836\pi$
$450$		0			0
$451$		−	28.2843i		−	1.33185i
$452$		0			0
$453$	34.0000	−	24.0416i	1.59746	−	1.12957i
$454$		0			0
$455$	−2.00000			−0.0937614
$456$		0			0
$457$	−18.0000			−0.842004			−0.421002	−	0.907060i	$-0.638322\pi$
$457$	−18.0000			−0.842004			−0.421002	+	0.907060i	$0.638322\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			1.41421i			0.0658665i	0.999458	+	0.0329332i	$0.0104849\pi$
$461$			1.41421i			0.0658665i	−0.999458	+	0.0329332i	$0.989515\pi$
$462$		0			0
$463$			1.41421i			0.0657241i	0.999460	+	0.0328620i	$0.0104622\pi$
$463$			1.41421i			0.0657241i	−0.999460	+	0.0328620i	$0.989538\pi$
$464$		0			0
$465$	−2.00000	−	2.82843i	−0.0927478	−	0.131165i
$466$		0			0
$467$	26.0000			1.20314			0.601568	−	0.798821i	$-0.294543\pi$
$467$	26.0000			1.20314			0.601568	+	0.798821i	$0.294543\pi$
$468$		0			0
$469$	−6.00000			−0.277054
$470$		0			0
$471$	−12.0000	−	16.9706i	−0.552931	−	0.781962i
$472$		0			0
$473$		−	33.9411i		−	1.56061i
$474$		0			0
$475$			12.7279i			0.583997i
$476$		0			0
$477$	16.0000	+	5.65685i	0.732590	+	0.259010i
$478$		0			0
$479$	−28.0000			−1.27935			−0.639676	−	0.768644i	$-0.720932\pi$
$479$	−28.0000			−1.27935			−0.639676	+	0.768644i	$0.720932\pi$
$480$		0			0
$481$	2.00000			0.0911922
$482$		0			0
$483$	4.00000	−	2.82843i	0.182006	−	0.128698i
$484$		0			0
$485$			8.48528i			0.385297i
$486$		0			0
$487$		−	12.7279i		−	0.576757i	−0.957516	−	0.288379i	$-0.906884\pi$
$487$		−	12.7279i		−	0.576757i	0.957516	−	0.288379i	$-0.0931162\pi$
$488$		0			0
$489$	−22.0000	+	15.5563i	−0.994874	+	0.703482i
$490$		0			0
$491$	14.0000			0.631811			0.315906	−	0.948791i	$-0.397692\pi$
$491$	14.0000			0.631811			0.315906	+	0.948791i	$0.397692\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	16.0000	+	5.65685i	0.719147	+	0.254257i
$496$		0			0
$497$			11.3137i			0.507489i
$498$		0			0
$499$		−	26.8701i		−	1.20287i	−0.798922	−	0.601434i	$-0.794596\pi$
$499$		−	26.8701i		−	1.20287i	0.798922	−	0.601434i	$-0.205404\pi$
$500$		0			0
$501$	8.00000	+	11.3137i	0.357414	+	0.505459i
$502$		0			0
$503$	16.0000			0.713405			0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000			0.713405			0.356702	+	0.934218i	$0.383901\pi$
$504$		0			0
$505$	−28.0000			−1.24598
$506$		0			0
$507$	1.00000	+	1.41421i	0.0444116	+	0.0628074i
$508$		0			0
$509$		−	35.3553i		−	1.56710i	−0.621330	−	0.783549i	$-0.713407\pi$
$509$		−	35.3553i		−	1.56710i	0.621330	−	0.783549i	$-0.286593\pi$
$510$		0			0
$511$			14.1421i			0.625611i
$512$		0			0
$513$	−6.00000	−	21.2132i	−0.264906	−	0.936586i
$514$		0			0
$515$	20.0000			0.881305
$516$		0			0
$517$	−32.0000			−1.40736
$518$		0			0
$519$	20.0000	−	14.1421i	0.877903	−	0.620771i
$520$		0			0
$521$			19.7990i			0.867409i	0.901055	+	0.433705i	$0.142794\pi$
$521$			19.7990i			0.867409i	−0.901055	+	0.433705i	$0.857206\pi$
$522$		0			0
$523$		−	28.2843i		−	1.23678i	−0.785869	−	0.618392i	$-0.787784\pi$
$523$		−	28.2843i		−	1.23678i	0.785869	−	0.618392i	$-0.212216\pi$
$524$		0			0
$525$	−6.00000	+	4.24264i	−0.261861	+	0.185164i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−19.0000			−0.826087
$530$		0			0
$531$	12.0000	−	33.9411i	0.520756	−	1.47292i
$532$		0			0
$533$			7.07107i			0.306282i
$534$		0			0
$535$			16.9706i			0.733701i
$536$		0			0
$537$	4.00000	+	5.65685i	0.172613	+	0.244111i
$538$		0			0
$539$	−20.0000			−0.861461
$540$		0			0
$541$	42.0000			1.80572			0.902861	−	0.429934i	$-0.141463\pi$
$541$	42.0000			1.80572			0.902861	+	0.429934i	$0.141463\pi$
$542$		0			0
$543$	8.00000	+	11.3137i	0.343313	+	0.485518i
$544$		0			0
$545$			14.1421i			0.605783i
$546$		0			0
$547$		−	16.9706i		−	0.725609i	−0.931865	−	0.362804i	$-0.881819\pi$
$547$		−	16.9706i		−	0.725609i	0.931865	−	0.362804i	$-0.118181\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	36.0000			1.53365
$552$		0			0
$553$	4.00000			0.170097
$554$		0			0
$555$	−4.00000	+	2.82843i	−0.169791	+	0.120060i
$556$		0			0
$557$		−	18.3848i		−	0.778988i	−0.921029	−	0.389494i	$-0.872650\pi$
$557$		−	18.3848i		−	0.778988i	0.921029	−	0.389494i	$-0.127350\pi$
$558$		0			0
$559$			8.48528i			0.358889i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−4.00000			−0.168580			−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000			−0.168580			−0.0842900	+	0.996441i	$0.526862\pi$
$564$		0			0
$565$	4.00000			0.168281
$566$		0			0
$567$	8.00000	−	9.89949i	0.335968	−	0.415740i
$568$		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
$571$		−	11.3137i		−	0.473464i	−0.971575	−	0.236732i	$-0.923924\pi$
$571$		−	11.3137i		−	0.473464i	0.971575	−	0.236732i	$-0.0760763\pi$
$572$		0			0
$573$	26.0000	+	36.7696i	1.08617	+	1.53607i
$574$		0			0
$575$	−6.00000			−0.250217
$576$		0			0
$577$	30.0000			1.24892			0.624458	−	0.781058i	$-0.285320\pi$
$577$	30.0000			1.24892			0.624458	+	0.781058i	$0.285320\pi$
$578$		0			0
$579$	−10.0000	−	14.1421i	−0.415586	−	0.587727i
$580$		0			0
$581$		0			0
$582$		0			0
$583$			22.6274i			0.937132i
$584$		0			0
$585$	−4.00000	−	1.41421i	−0.165380	−	0.0584705i
$586$		0			0
$587$	32.0000			1.32078			0.660391	−	0.750922i	$-0.270391\pi$
$587$	32.0000			1.32078			0.660391	+	0.750922i	$0.270391\pi$
$588$		0			0
$589$	−6.00000			−0.247226
$590$		0			0
$591$	10.0000	−	7.07107i	0.411345	−	0.290865i
$592$		0			0
$593$		−	12.7279i		−	0.522673i	−0.965248	−	0.261337i	$-0.915837\pi$
$593$		−	12.7279i		−	0.522673i	0.965248	−	0.261337i	$-0.0841632\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−16.0000	+	11.3137i	−0.654836	+	0.463039i
$598$		0			0
$599$	−16.0000			−0.653742			−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000			−0.653742			−0.326871	+	0.945069i	$0.605994\pi$
$600$		0			0
$601$	−6.00000			−0.244745			−0.122373	−	0.992484i	$-0.539050\pi$
$601$	−6.00000			−0.244745			−0.122373	+	0.992484i	$0.539050\pi$
$602$		0			0
$603$	−12.0000	−	4.24264i	−0.488678	−	0.172774i
$604$		0			0
$605$			7.07107i			0.287480i
$606$		0			0
$607$		−	33.9411i		−	1.37763i	−0.724938	−	0.688814i	$-0.758132\pi$
$607$		−	33.9411i		−	1.37763i	0.724938	−	0.688814i	$-0.241868\pi$
$608$		0			0
$609$	12.0000	+	16.9706i	0.486265	+	0.687682i
$610$		0			0
$611$	8.00000			0.323645
$612$		0			0
$613$	−2.00000			−0.0807792			−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000			−0.0807792			−0.0403896	+	0.999184i	$0.512860\pi$
$614$		0			0
$615$	−10.0000	−	14.1421i	−0.403239	−	0.570266i
$616$		0			0
$617$			7.07107i			0.284670i	0.989819	+	0.142335i	$0.0454611\pi$
$617$			7.07107i			0.284670i	−0.989819	+	0.142335i	$0.954539\pi$
$618$		0			0
$619$		−	29.6985i		−	1.19368i	−0.802359	−	0.596841i	$-0.796422\pi$
$619$		−	29.6985i		−	1.19368i	0.802359	−	0.596841i	$-0.203578\pi$
$620$		0			0
$621$	10.0000	−	2.82843i	0.401286	−	0.113501i
$622$		0			0
$623$	−14.0000			−0.560898
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$	24.0000	−	16.9706i	0.958468	−	0.677739i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		−	15.5563i		−	0.619288i	−0.950852	−	0.309644i	$-0.899790\pi$
$631$		−	15.5563i		−	0.619288i	0.950852	−	0.309644i	$-0.100210\pi$
$632$		0			0
$633$	36.0000	−	25.4558i	1.43087	−	1.01178i
$634$		0			0
$635$	16.0000			0.634941
$636$		0			0
$637$	5.00000			0.198107
$638$		0			0
$639$	−8.00000	+	22.6274i	−0.316475	+	0.895127i
$640$		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
$643$			24.0416i			0.948109i	0.880495	+	0.474055i	$0.157210\pi$
$643$			24.0416i			0.948109i	−0.880495	+	0.474055i	$0.842790\pi$
$644$		0			0
$645$	−12.0000	−	16.9706i	−0.472500	−	0.668215i
$646$		0			0
$647$	24.0000			0.943537			0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000			0.943537			0.471769	+	0.881722i	$0.343616\pi$
$648$		0			0
$649$	48.0000			1.88416
$650$		0			0
$651$	−2.00000	−	2.82843i	−0.0783862	−	0.110855i
$652$		0			0
$653$			42.4264i			1.66027i	0.557560	+	0.830137i	$0.311738\pi$
$653$			42.4264i			1.66027i	−0.557560	+	0.830137i	$0.688262\pi$
$654$		0			0
$655$			16.9706i			0.663095i
$656$		0			0
$657$	−10.0000	+	28.2843i	−0.390137	+	1.10347i
$658$		0			0
$659$	42.0000			1.63609			0.818044	−	0.575156i	$-0.195059\pi$
$659$	42.0000			1.63609			0.818044	+	0.575156i	$0.195059\pi$
$660$		0			0
$661$	−42.0000			−1.63361			−0.816805	−	0.576913i	$-0.804257\pi$
$661$	−42.0000			−1.63361			−0.816805	+	0.576913i	$0.804257\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	8.48528i		−	0.329045i
$666$		0			0
$667$			16.9706i			0.657103i
$668$		0			0
$669$	−38.0000	+	26.8701i	−1.46916	+	1.03886i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−28.0000			−1.07932			−0.539660	−	0.841883i	$-0.681447\pi$
$673$	−28.0000			−1.07932			−0.539660	+	0.841883i	$0.681447\pi$
$674$		0			0
$675$	−15.0000	+	4.24264i	−0.577350	+	0.163299i
$676$		0			0
$677$			22.6274i			0.869642i	0.900517	+	0.434821i	$0.143188\pi$
$677$			22.6274i			0.869642i	−0.900517	+	0.434821i	$0.856812\pi$
$678$		0			0
$679$			8.48528i			0.325635i
$680$		0			0
$681$	−24.0000	−	33.9411i	−0.919682	−	1.30063i
$682$		0			0
$683$	−12.0000			−0.459167			−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000			−0.459167			−0.229584	+	0.973289i	$0.573736\pi$
$684$		0			0
$685$	30.0000			1.14624
$686$		0			0
$687$	−2.00000	−	2.82843i	−0.0763048	−	0.107911i
$688$		0			0
$689$		−	5.65685i		−	0.215509i
$690$		0			0
$691$		−	43.8406i		−	1.66778i	−0.551934	−	0.833888i	$-0.686110\pi$
$691$		−	43.8406i		−	1.66778i	0.551934	−	0.833888i	$-0.313890\pi$
$692$		0			0
$693$	16.0000	+	5.65685i	0.607790	+	0.214886i
$694$		0			0
$695$	16.0000			0.606915
$696$		0			0
$697$		0			0
$698$		0			0
$699$	−4.00000	+	2.82843i	−0.151294	+	0.106981i
$700$		0			0
$701$			28.2843i			1.06828i	0.845395	+	0.534141i	$0.179365\pi$
$701$			28.2843i			1.06828i	−0.845395	+	0.534141i	$0.820635\pi$
$702$		0			0
$703$			8.48528i			0.320028i
$704$		0			0
$705$	−16.0000	+	11.3137i	−0.602595	+	0.426099i
$706$		0			0
$707$	−28.0000			−1.05305
$708$		0			0
$709$	14.0000			0.525781			0.262891	−	0.964826i	$-0.415324\pi$
$709$	14.0000			0.525781			0.262891	+	0.964826i	$0.415324\pi$
$710$		0			0
$711$	8.00000	+	2.82843i	0.300023	+	0.106074i
$712$		0			0
$713$		−	2.82843i		−	0.105925i
$714$		0			0
$715$		−	5.65685i		−	0.211554i
$716$		0			0
$717$	−8.00000	−	11.3137i	−0.298765	−	0.422518i
$718$		0			0
$719$	30.0000			1.11881			0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000			1.11881			0.559406	+	0.828894i	$0.311029\pi$
$720$		0			0
$721$	20.0000			0.744839
$722$		0			0
$723$	−6.00000	−	8.48528i	−0.223142	−	0.315571i
$724$		0			0
$725$		−	25.4558i		−	0.945406i
$726$		0			0
$727$		−	22.6274i		−	0.839204i	−0.907708	−	0.419602i	$-0.862170\pi$
$727$		−	22.6274i		−	0.839204i	0.907708	−	0.419602i	$-0.137830\pi$
$728$		0			0
$729$	23.0000	−	14.1421i	0.851852	−	0.523783i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	2.00000			0.0738717			0.0369358	−	0.999318i	$-0.488240\pi$
$733$	2.00000			0.0738717			0.0369358	+	0.999318i	$0.488240\pi$
$734$		0			0
$735$	−10.0000	+	7.07107i	−0.368856	+	0.260820i
$736$		0			0
$737$		−	16.9706i		−	0.625119i
$738$		0			0
$739$		−	4.24264i		−	0.156068i	−0.996951	−	0.0780340i	$-0.975136\pi$
$739$		−	4.24264i		−	0.156068i	0.996951	−	0.0780340i	$-0.0248643\pi$
$740$		0			0
$741$	−6.00000	+	4.24264i	−0.220416	+	0.155857i
$742$		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
$745$	−2.00000			−0.0732743
$746$		0			0
$747$		0			0
$748$		0			0
$749$			16.9706i			0.620091i
$750$		0			0
$751$		−	8.48528i		−	0.309632i	−0.987943	−	0.154816i	$-0.950521\pi$
$751$		−	8.48528i		−	0.309632i	0.987943	−	0.154816i	$-0.0494785\pi$
$752$		0			0
$753$	−22.0000	−	31.1127i	−0.801725	−	1.13381i
$754$		0			0
$755$	34.0000			1.23739
$756$		0			0
$757$	24.0000			0.872295			0.436147	−	0.899875i	$-0.356343\pi$
$757$	24.0000			0.872295			0.436147	+	0.899875i	$0.356343\pi$
$758$		0			0
$759$	8.00000	+	11.3137i	0.290382	+	0.410662i
$760$		0			0
$761$		−	24.0416i		−	0.871508i	−0.900066	−	0.435754i	$-0.856482\pi$
$761$		−	24.0416i		−	0.871508i	0.900066	−	0.435754i	$-0.143518\pi$
$762$		0			0
$763$			14.1421i			0.511980i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−12.0000			−0.433295
$768$		0			0
$769$	38.0000			1.37032			0.685158	−	0.728395i	$-0.259733\pi$
$769$	38.0000			1.37032			0.685158	+	0.728395i	$0.259733\pi$
$770$		0			0
$771$	−4.00000	+	2.82843i	−0.144056	+	0.101863i
$772$		0			0
$773$		−	35.3553i		−	1.27164i	−0.771836	−	0.635822i	$-0.780661\pi$
$773$		−	35.3553i		−	1.27164i	0.771836	−	0.635822i	$-0.219339\pi$
$774$		0			0
$775$			4.24264i			0.152400i
$776$		0			0
$777$	−4.00000	+	2.82843i	−0.143499	+	0.101469i
$778$		0			0
$779$	−30.0000			−1.07486
$780$		0			0
$781$	−32.0000			−1.14505
$782$		0			0
$783$	12.0000	+	42.4264i	0.428845	+	1.51620i
$784$		0			0
$785$		−	16.9706i		−	0.605705i
$786$		0			0
$787$			1.41421i			0.0504113i	0.999682	+	0.0252056i	$0.00802405\pi$
$787$			1.41421i			0.0504113i	−0.999682	+	0.0252056i	$0.991976\pi$
$788$		0			0
$789$	−8.00000	−	11.3137i	−0.284808	−	0.402779i
$790$		0			0
$791$	4.00000			0.142224
$792$		0			0
$793$		0			0
$794$		0			0
$795$	8.00000	+	11.3137i	0.283731	+	0.401256i
$796$		0			0
$797$		−	11.3137i		−	0.400752i	−0.979719	−	0.200376i	$-0.935784\pi$
$797$		−	11.3137i		−	0.400752i	0.979719	−	0.200376i	$-0.0642164\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−28.0000	−	9.89949i	−0.989331	−	0.349781i
$802$		0			0
$803$	−40.0000			−1.41157
$804$		0			0
$805$	4.00000			0.140981
$806$		0			0
$807$	4.00000	−	2.82843i	0.140807	−	0.0995654i
$808$		0			0
$809$			8.48528i			0.298327i	0.988813	+	0.149163i	$0.0476580\pi$
$809$			8.48528i			0.298327i	−0.988813	+	0.149163i	$0.952342\pi$
$810$		0			0
$811$		−	15.5563i		−	0.546257i	−0.961978	−	0.273129i	$-0.911942\pi$
$811$		−	15.5563i		−	0.546257i	0.961978	−	0.273129i	$-0.0880585\pi$
$812$		0			0
$813$	−6.00000	+	4.24264i	−0.210429	+	0.148796i
$814$		0			0
$815$	−22.0000			−0.770626
$816$		0			0
$817$	−36.0000			−1.25948
$818$		0			0
$819$	−4.00000	−	1.41421i	−0.139771	−	0.0494166i
$820$		0			0
$821$		−	35.3553i		−	1.23391i	−0.786998	−	0.616955i	$-0.788366\pi$
$821$		−	35.3553i		−	1.23391i	0.786998	−	0.616955i	$-0.211634\pi$
$822$		0			0
$823$			31.1127i			1.08452i	0.840211	+	0.542260i	$0.182431\pi$
$823$			31.1127i			1.08452i	−0.840211	+	0.542260i	$0.817569\pi$
$824$		0			0
$825$	−12.0000	−	16.9706i	−0.417786	−	0.590839i
$826$		0			0
$827$	−8.00000			−0.278187			−0.139094	−	0.990279i	$-0.544419\pi$
$827$	−8.00000			−0.278187			−0.139094	+	0.990279i	$0.544419\pi$
$828$		0			0
$829$	28.0000			0.972480			0.486240	−	0.873825i	$-0.338368\pi$
$829$	28.0000			0.972480			0.486240	+	0.873825i	$0.338368\pi$
$830$		0			0
$831$	−22.0000	−	31.1127i	−0.763172	−	1.07929i
$832$		0			0
$833$		0			0
$834$		0			0
$835$			11.3137i			0.391527i
$836$		0			0
$837$	−2.00000	−	7.07107i	−0.0691301	−	0.244412i
$838$		0			0
$839$	−24.0000			−0.828572			−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000			−0.828572			−0.414286	+	0.910147i	$0.635969\pi$
$840$		0			0
$841$	−43.0000			−1.48276
$842$		0			0
$843$	−2.00000	+	1.41421i	−0.0688837	+	0.0487081i
$844$		0			0
$845$			1.41421i			0.0486504i
$846$		0			0
$847$			7.07107i			0.242965i
$848$		0			0
$849$	−32.0000	+	22.6274i	−1.09824	+	0.776571i
$850$		0			0
$851$	−4.00000			−0.137118
$852$		0			0
$853$	26.0000			0.890223			0.445112	−	0.895475i	$-0.353164\pi$
$853$	26.0000			0.890223			0.445112	+	0.895475i	$0.353164\pi$
$854$		0			0
$855$	6.00000	−	16.9706i	0.205196	−	0.580381i
$856$		0			0
$857$			19.7990i			0.676321i	0.941089	+	0.338160i	$0.109805\pi$
$857$			19.7990i			0.676321i	−0.941089	+	0.338160i	$0.890195\pi$
$858$		0			0
$859$			8.48528i			0.289514i	0.989467	+	0.144757i	$0.0462401\pi$
$859$			8.48528i			0.289514i	−0.989467	+	0.144757i	$0.953760\pi$
$860$		0			0
$861$	−10.0000	−	14.1421i	−0.340799	−	0.481963i
$862$		0			0
$863$	−48.0000			−1.63394			−0.816970	−	0.576681i	$-0.804348\pi$
$863$	−48.0000			−1.63394			−0.816970	+	0.576681i	$0.804348\pi$
$864$		0			0
$865$	20.0000			0.680020
$866$		0			0
$867$	17.0000	+	24.0416i	0.577350	+	0.816497i
$868$		0			0
$869$			11.3137i			0.383791i
$870$		0			0
$871$			4.24264i			0.143756i
$872$		0			0
$873$	−6.00000	+	16.9706i	−0.203069	+	0.574367i
$874$		0			0
$875$	−16.0000			−0.540899
$876$		0			0
$877$	42.0000			1.41824			0.709120	−	0.705088i	$-0.249093\pi$
$877$	42.0000			1.41824			0.709120	+	0.705088i	$0.249093\pi$
$878$		0			0
$879$	−34.0000	+	24.0416i	−1.14679	+	0.810904i
$880$		0			0
$881$			2.82843i			0.0952921i	0.998864	+	0.0476461i	$0.0151720\pi$
$881$			2.82843i			0.0952921i	−0.998864	+	0.0476461i	$0.984828\pi$
$882$		0			0
$883$			33.9411i			1.14221i	0.820877	+	0.571105i	$0.193485\pi$
$883$			33.9411i			1.14221i	−0.820877	+	0.571105i	$0.806515\pi$
$884$		0			0
$885$	24.0000	−	16.9706i	0.806751	−	0.570459i
$886$		0			0
$887$	6.00000			0.201460			0.100730	−	0.994914i	$-0.467882\pi$
$887$	6.00000			0.201460			0.100730	+	0.994914i	$0.467882\pi$
$888$		0			0
$889$	16.0000			0.536623
$890$		0			0
$891$	28.0000	+	22.6274i	0.938035	+	0.758047i
$892$		0			0
$893$			33.9411i			1.13580i
$894$		0			0
$895$			5.65685i			0.189088i
$896$		0			0
$897$	−2.00000	−	2.82843i	−0.0667781	−	0.0944384i
$898$		0			0
$899$	12.0000			0.400222
$900$		0			0
$901$		0			0
$902$		0			0
$903$	−12.0000	−	16.9706i	−0.399335	−	0.564745i
$904$		0			0
$905$			11.3137i			0.376080i
$906$		0			0
$907$			45.2548i			1.50266i	0.659925	+	0.751331i	$0.270588\pi$
$907$			45.2548i			1.50266i	−0.659925	+	0.751331i	$0.729412\pi$
$908$		0			0
$909$	−56.0000	−	19.7990i	−1.85740	−	0.656691i
$910$		0			0
$911$	42.0000			1.39152			0.695761	−	0.718273i	$-0.255067\pi$
$911$	42.0000			1.39152			0.695761	+	0.718273i	$0.255067\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$			16.9706i			0.560417i
$918$		0			0
$919$			31.1127i			1.02631i	0.858295	+	0.513157i	$0.171524\pi$
$919$			31.1127i			1.02631i	−0.858295	+	0.513157i	$0.828476\pi$
$920$		0			0
$921$	46.0000	−	32.5269i	1.51575	−	1.07180i
$922$		0			0
$923$	8.00000			0.263323
$924$		0			0
$925$	6.00000			0.197279
$926$		0			0
$927$	40.0000	+	14.1421i	1.31377	+	0.464489i
$928$		0			0
$929$			24.0416i			0.788780i	0.918943	+	0.394390i	$0.129044\pi$
$929$			24.0416i			0.788780i	−0.918943	+	0.394390i	$0.870956\pi$
$930$		0			0
$931$			21.2132i			0.695235i
$932$		0			0
$933$	18.0000	+	25.4558i	0.589294	+	0.833387i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−12.0000			−0.392023			−0.196011	−	0.980602i	$-0.562799\pi$
$937$	−12.0000			−0.392023			−0.196011	+	0.980602i	$0.562799\pi$
$938$		0			0
$939$	−26.0000	−	36.7696i	−0.848478	−	1.19993i
$940$		0			0
$941$			35.3553i			1.15255i	0.817255	+	0.576276i	$0.195494\pi$
$941$			35.3553i			1.15255i	−0.817255	+	0.576276i	$0.804506\pi$
$942$		0			0
$943$		−	14.1421i		−	0.460531i
$944$		0			0
$945$	10.0000	−	2.82843i	0.325300	−	0.0920087i
$946$		0			0
$947$		0			0			−	1.00000i	$-0.5\pi$
$947$		0			0				1.00000i	$0.5\pi$
$948$		0			0
$949$	10.0000			0.324614
$950$		0			0
$951$	−14.0000	+	9.89949i	−0.453981	+	0.321013i
$952$		0			0
$953$		−	48.0833i		−	1.55757i	−0.627291	−	0.778785i	$-0.715836\pi$
$953$		−	48.0833i		−	1.55757i	0.627291	−	0.778785i	$-0.284164\pi$
$954$		0			0
$955$			36.7696i			1.18983i
$956$		0			0
$957$	−48.0000	+	33.9411i	−1.55162	+	1.09716i
$958$		0			0
$959$	30.0000			0.968751
$960$		0			0
$961$	29.0000			0.935484
$962$		0			0
$963$	−12.0000	+	33.9411i	−0.386695	+	1.09374i
$964$		0			0
$965$		−	14.1421i		−	0.455251i
$966$		0			0
$967$		−	29.6985i		−	0.955039i	−0.878621	−	0.477519i	$-0.841536\pi$
$967$		−	29.6985i		−	0.955039i	0.878621	−	0.477519i	$-0.158464\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−22.0000			−0.706014			−0.353007	−	0.935621i	$-0.614841\pi$
$971$	−22.0000			−0.706014			−0.353007	+	0.935621i	$0.614841\pi$
$972$		0			0
$973$	16.0000			0.512936
$974$		0			0
$975$	3.00000	+	4.24264i	0.0960769	+	0.135873i
$976$		0			0
$977$			24.0416i			0.769160i	0.923092	+	0.384580i	$0.125654\pi$
$977$			24.0416i			0.769160i	−0.923092	+	0.384580i	$0.874346\pi$
$978$		0			0
$979$		−	39.5980i		−	1.26556i
$980$		0			0
$981$	−10.0000	+	28.2843i	−0.319275	+	0.903047i
$982$		0			0
$983$	−32.0000			−1.02064			−0.510321	−	0.859984i	$-0.670473\pi$
$983$	−32.0000			−1.02064			−0.510321	+	0.859984i	$0.670473\pi$
$984$		0			0
$985$	10.0000			0.318626
$986$		0			0
$987$	−16.0000	+	11.3137i	−0.509286	+	0.360119i
$988$		0			0
$989$		−	16.9706i		−	0.539633i
$990$		0			0
$991$			33.9411i			1.07818i	0.842250	+	0.539088i	$0.181231\pi$
$991$			33.9411i			1.07818i	−0.842250	+	0.539088i	$0.818769\pi$
$992$		0			0
$993$	−30.0000	+	21.2132i	−0.952021	+	0.673181i
$994$		0			0
$995$	−16.0000			−0.507234
$996$		0			0
$997$	−14.0000			−0.443384			−0.221692	−	0.975117i	$-0.571158\pi$
$997$	−14.0000			−0.443384			−0.221692	+	0.975117i	$0.571158\pi$
$998$		0			0
$999$	−10.0000	+	2.82843i	−0.316386	+	0.0894875i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.2.d.e.287.2	yes	2
3.2	odd	2		624.2.d.b.287.2	yes	2
4.3	odd	2		624.2.d.b.287.1	✓	2
8.3	odd	2		2496.2.d.f.1535.2		2
8.5	even	2		2496.2.d.c.1535.1		2
12.11	even	2	inner	624.2.d.e.287.1	yes	2
24.5	odd	2		2496.2.d.f.1535.1		2
24.11	even	2		2496.2.d.c.1535.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
624.2.d.b.287.1	✓	2	4.3	odd	2
624.2.d.b.287.2	yes	2	3.2	odd	2
624.2.d.e.287.1	yes	2	12.11	even	2	inner
624.2.d.e.287.2	yes	2	1.1	even	1	trivial
2496.2.d.c.1535.1		2	8.5	even	2
2496.2.d.c.1535.2		2	24.11	even	2
2496.2.d.f.1535.1		2	24.5	odd	2
2496.2.d.f.1535.2		2	8.3	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:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	624.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.41421i) q^{3} +1.41421i q^{5} +1.41421i q^{7} +(-1.00000 + 2.82843i) q^{9} +O(q^{10})\)	\(q+(1.00000 + 1.41421i) q^{3} +1.41421i q^{5} +1.41421i q^{7} +(-1.00000 + 2.82843i) q^{9} -4.00000 q^{11} +1.00000 q^{13} +(-2.00000 + 1.41421i) q^{15} +4.24264i q^{19} +(-2.00000 + 1.41421i) q^{21} -2.00000 q^{23} +3.00000 q^{25} +(-5.00000 + 1.41421i) q^{27} -8.48528i q^{29} +1.41421i q^{31} +(-4.00000 - 5.65685i) q^{33} -2.00000 q^{35} +2.00000 q^{37} +(1.00000 + 1.41421i) q^{39} +7.07107i q^{41} +8.48528i q^{43} +(-4.00000 - 1.41421i) q^{45} +8.00000 q^{47} +5.00000 q^{49} -5.65685i q^{53} -5.65685i q^{55} +(-6.00000 + 4.24264i) q^{57} -12.0000 q^{59} +(-4.00000 - 1.41421i) q^{63} +1.41421i q^{65} +4.24264i q^{67} +(-2.00000 - 2.82843i) q^{69} +8.00000 q^{71} +10.0000 q^{73} +(3.00000 + 4.24264i) q^{75} -5.65685i q^{77} -2.82843i q^{79} +(-7.00000 - 5.65685i) q^{81} +(12.0000 - 8.48528i) q^{87} +9.89949i q^{89} +1.41421i q^{91} +(-2.00000 + 1.41421i) q^{93} -6.00000 q^{95} +6.00000 q^{97} +(4.00000 - 11.3137i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{9} - 8 q^{11} + 2 q^{13} - 4 q^{15} - 4 q^{21} - 4 q^{23} + 6 q^{25} - 10 q^{27} - 8 q^{33} - 4 q^{35} + 4 q^{37} + 2 q^{39} - 8 q^{45} + 16 q^{47} + 10 q^{49} - 12 q^{57} - 24 q^{59} - 8 q^{63} - 4 q^{69} + 16 q^{71} + 20 q^{73} + 6 q^{75} - 14 q^{81} + 24 q^{87} - 4 q^{93} - 12 q^{95} + 12 q^{97} + 8 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^9 - 8 * q^11 + 2 * q^13 - 4 * q^15 - 4 * q^21 - 4 * q^23 + 6 * q^25 - 10 * q^27 - 8 * q^33 - 4 * q^35 + 4 * q^37 + 2 * q^39 - 8 * q^45 + 16 * q^47 + 10 * q^49 - 12 * q^57 - 24 * q^59 - 8 * q^63 - 4 * q^69 + 16 * q^71 + 20 * q^73 + 6 * q^75 - 14 * q^81 + 24 * q^87 - 4 * q^93 - 12 * q^95 + 12 * q^97 + 8 * q^99

Introduction

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Varieties

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Groups

Database

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